Eigenstrain reconstruction of residual stresses

Eigenstrain reconstruction of residual stresses induced by selective laser melting M.P. Fransen

P&E report number: 2788

Eigenstrain reconstruction of residual stresses induced by selective laser melting by

M.P. Fransen to obtain the degree of Master of Science at the Delft University of Technology, to be defended publicly on Thursday October 27, 2016 at 10:00 AM.

Student number: Project duration: Thesis committee:

4018761 May 1, 2015 – October 27, 2016 Prof. dr. ir. F. van Keulen, TU Delft Dr. C. Ayas, TU Delft supervisor Dr. ir. R.A.J. van Ostayen, TU Delft Dr. Y. Yang, TU Delft

An electronic version of this thesis is available at http://repository.tudelft.nl/.

Abstract Selective laser melting (SLM) is an additive manufacturing (AM) technique that is used for metallic products. These products are built in a layer by layer manner by means of selectively melting and hence fusing a layer of metal powder with a laser scanning along a predefined path. The SLM process is able to create products with complex geometries, high accuracy, and within a single manufacturing step. Although SLM presents a number of advantages, the thermal loading causing residual stresses and distortions is an important issue in SLM products. Residual stresses may lead to a limited load resistance and cracks leading to impaired toughness. Distortions may cause the SLM product to be manufactured outside of design tolerances. Therefore, for an SLM product it is essential to accurately predict residual stresses and distortion in the design phase. In literature a great variety of modelling techniques are proposed to capture the physics of the process. Two most widely used approaches to model the process are the melting pool scale and the layer by layer modelling approach. The latter is a computationally efficient way of modelling but does not contain complex process history whereas the former is computationally expensive but can capture the effects of scanning strategy and other process parameters. In this thesis a novel modelling approach is presented which combines the advantages of modelling at the layer scale and the melting pool scale. The general idea is that, the source of strain, giving rise to residual stresses during the SLM process for a given scanning strategy is reconstructed from a detailed simulation and mapped onto a layer. In order to do this the eigenstrain reconstruction method (ERM) is utilised. In this method an equivalent strain source is reconstructed for a residual stress field. The reconstructed equivalent strain source or eigenstrain can be used as a building block to describe the strain source for an entire layer or product. By defining the strain source for an entire product, the residual stresses in the product can be predicted directly. The novelty of the work presented in this thesis lies in the way the residual stress field is predicted. In literature the residual stress field caused by the SLM process is directly mapped to a layer or product. In this approach it is not the stress field that is mapped to the layer or product but the strain source. The advantage of this idea is that the resulting residual stress fields from the strain source are continuous in the SLM product. Although the ERM is widely used to model 1D and 2D residual stress fields for welding and peening processes, so far no fully 3D ERM has been implemented. In real SLM products the residual stress fields are always multi-axial. Therefore, it is necessary to extend ERM to 3D which is capable of dealing with the multidimensionality of the residual stress fields. One of the requirements of ERM is the input of a reference residual stress field. Therefore an expensive numerical reference model is developed which includes the effect of scanning strategy on the residual stress and warpage of a SLM product. The ERM model developed in this thesis is validated by reconstructing the residual stress fields for three geometries; a line, layer and ten layer geometry. The residual stress distributions calculated by the eigenstrain are compared with the results of the expensive numerical reference model. In this model the scanning strategy is incorporated by a point by point modelling approach. The performance of ERM is evaluated by calculating the corresponding residual stress field error between reference and ERM results, which lies within 0.5-4%. It is concluded that the ERM is a suitable method to reconstruct equivalent strain sources or eigenstrains for predicting 3D residual stress fields caused by the SLM process. The contribution to the field consists of an expansion of ERM to three dimensions and a hybrid model for the SLM process which combines the computational efficiency of the global approach with the detail of the local approach.

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Acknowledgements This thesis is the final part to obtain my master’s degree in Mechanical Engineering. It has been quite a ride with some ups and downs but in the end it all came together. Therefore, I would like to take this opportunity to thank the people that have contributed to this thesis. First of all, I would like to thank my daily supervisors Can Ayas and Yabin Yang for the time and energy that they have invested in this project. As a part of my final masters project I did an internship at NLR. This internship was a nice way to get acquainted with the practical side of selective laser melting and it lead to great insights for this thesis work. Therefore I would like to thank Rob Huls, my supervisor at NLR. This thesis would not have been possible without the occasional coffee break or discussions on random topics to clear my mind. Therefore I would like to thank my friends in Delft and fellow students at the office. Finally, I would like to thank some special people in my life. My mother, father, and sister for always supporting me and a special thank you to my lovely girlfriend who stood by my side during the entire process. Marc Fransen Delft, October 2016

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Nomenclature Symbols Symbol α C i j kl δ E ² ²eff ²all ²e ²th ²∗ ²tot Γ H Λ ν Ω r σ σy σVM Σ Si j S iJ j ti T Tini Tref ui U U0 xc ζiJ j

Units 1/K Pa (N /m 2 ) Pa (N /m 2 ) m Pa (N /m 2 ) Pa (N /m 2 ) Pa (N /m 2 ) Pa (N /m 2 ) K K K m J J m -

Description Thermal expansion coefficient Compliance tensor Error description Young’s modulus Strain Effective strain Allowable strain Elastic strain Thermal strain Eigenstrain Total strain Boundary of domain Functional Base plate Poisson ratio Domain or internal domain radius Stress Yield limit Von Mises stress Function space Total elastic strain solution Elastic strain solution J Traction vector Temperature Initial temperature Reference temperature Displacement vector Strain energy Strain energy density Location of center coordinate Eigenstrain component

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Acronyms AM BG ERM IG MBVP MSE NLR NoBF NRMSE RBF RMSE SLM SSE TU

Additive manufacturing Boundary grid Eigenstrain reconstruction method Internal grid Mechanical boundary value problem Mean squared error Netherlands aerospace center Number of basis functions Normalised root mean square error Radial basis function Root mean squared error Selective lase melting Sum squared error Technical university

Contents 1 Introduction

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1.1 Selective laser melting process . . . . . . . . . . . . . . . . . . . . . . 1.2 Literature review on SLM process . . . . . . . . . . . . . . . . . . . . . 1.2.1 Physical phenomena in SLM process . . . . . . . . . . . . . . . 1.2.2 Measurement of residual stress and deformation of SLM products. 1.2.3 Modelling approaches . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Thermomechanical SLM models. . . . . . . . . . . . . . . . . . 1.2.5 Eigenstrain theory . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Research objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Outline thesis work . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.1 Field quantities, thermoelasticity, and thermoelastic mechanical boundary value problem. 3.1.1 Stress, strain, and elastic strain energy . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Thermoelastic mechanical boundary value problem. . . . . . . . . . . . . . . . . 3.2 Eigenstrain theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The eigenstrain problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Eigenstrain reconstruction method - theory . . . . . . . . . . . . . . . . . . . . . 3.2.3 Function space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Error norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Disadvantages of ERM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Experimental and numerical work at NLR

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2.1 Internship goals. . . . . . . . . . . . . . . . . . 2.2 NLR model . . . . . . . . . . . . . . . . . . . . 2.2.1 Physical phenomena included in the model 2.2.2 Layer by layer approach . . . . . . . . . . 2.2.3 Mechanical boundary value problem . . . 2.3 Experimental work . . . . . . . . . . . . . . . . 2.4 Results . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Numerical results . . . . . . . . . . . . . 2.4.2 Experimental results . . . . . . . . . . . . 2.4.3 Comparison of results . . . . . . . . . . . 2.5 Conclusions. . . . . . . . . . . . . . . . . . . . 2.6 Implications for thesis work. . . . . . . . . . . .

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3 Eigenstrain reconstruction method

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4 SLM modelling

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4.1 SLM reference model . . . . . . . . . . . . . . . 4.1.1 Modelling strategy . . . . . . . . . . . . . 4.1.2 Working principle thermal strain approach 4.1.3 Line by line and point by point model . . . 4.1.4 Material model. . . . . . . . . . . . . . . 4.1.5 Geometries . . . . . . . . . . . . . . . . 4.1.6 Reference models in FE package . . . . . . 4.2 Eigenstrain reconstruction method . . . . . . . . 4.2.1 Thermal expansion mechanism in ERM . . 4.2.2 ERM - line on baseplate . . . . . . . . . . 4.2.3 ERM - layer on base plate . . . . . . . . . 4.2.4 ERM - Multilayer on baseplate . . . . . . . ix

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Contents 4.3 Error evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Results & Discussions

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5.1 Line geometry. . . . . . . . . . . . . 5.1.1 Reference results line geometry 5.1.2 Results ERM line geometry. . . 5.1.3 Discussion . . . . . . . . . . . 5.2 Layer geometry . . . . . . . . . . . . 5.2.1 Reference results layer model . 5.2.2 ERM results . . . . . . . . . . 5.2.3 Discussion . . . . . . . . . . . 5.3 Multilayer geometry . . . . . . . . . 5.3.1 Multilayer reference results . . 5.3.2 ERM results . . . . . . . . . . 5.3.3 Discussion . . . . . . . . . . .

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6 Conclusions and recommendations

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6.1 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.2 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A Additional figures A.1 A.2 A.3 A.4

Line geometry. . . . . . . . . . Layer geometry . . . . . . . . . Ten layer geometry . . . . . . . ERM results: ten layer geometry

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1 Introduction Chapter 1 starts with an introduction to the selective laser melting process in section 1.1. After the introduction, an overview is given on the literature regarding the SLM process in section 1.2. The literature review revealed the problems that are faced in research on the SLM process. These insights lead to the research objective and research questions described in section 1.3. At the end of this chapter an outline of the contents of this thesis is given in section 1.4.

1.1. Selective laser melting process Selective laser melting (SLM) is an additive manufacturing (AM) technique which is used to produce metal products. The characteristic feature of the SLM process is the layer by layer production of a product. AM is the generic name for processes which use the layer by layer approach to produce products. In Fig. 1.1 the SLM process environment is shown schematically. The SLM process takes place in a build chamber with an inert atmosphere. The SLM product is built on top of a base plate, the base plate is made of the same material as the SLM product. From a powder reservoir powder material is deposited onto the base plate. To get a uniform thickness the deposited powder is evenly distributed over the base plate by a scraper. Processing of the powder bed is done by scanning a laser over the surface of the powder bed, the path that is followed by the laser is called the scanning strategy. Positioning of the laser beam on the powder bed is done by mirrors and refocusing by a lens. By following the scanning strategy the powder material is melted along the path by a laser, when the laser moves forward the molten material that is left behind cools down and solidifies. After completing a layer the base plate is lowered in x 3 direction with the thickness of the built layer. Then the whole process is repeated until the entire product is built. When the product is finished the unprocessed powder material is removed. Then the product and base plate are removed from the SLM machine. If necessary the product and base plate are post-processed by post-processing techniques such as heat treatment, hot isostatic pressing (HIP), etc. After post-processing the product is removed from the base plate by wire electronic discharge milling (wire EDM) or a different cutting method. Fig. 1.2 zooms in on the actual melting process to show what occurs at the point where material is melted. The figure shows the laser beam which is pointed at the powder bed. At the point where the laser hits the powder bed the thermal energy of the laser causes the powder bed to melt. The molten material forms a melting pool which has the shape of a sideways droplet with a round front and a sharp tail. This characteristic shape is caused by the movement of the laser. When the laser moves along the scanning path molten material is left behind, this material gradually cools down and solidifies. In the figure it is shown that the melting pool also melts part of the previously processed material. This is essential in the SLM process to get well connected layers and eventually a high quality SLM product. The SLM process is an efficient way of building products because only the material needed to build the product is used. Compared to conventional manufacturing methods such as computer numerical controlled (CNC) machines the SLM machine has only one processing tool while a CNC machine has dozens of tools. Another advantage of the SLM process is that it is able to build customised products, small batches,and large quantities of small products. Whereas the CNC machines are only able to build one product at a time. The SLM machine can build as many products as desired as long as there is room on the base plate. The major advantage of the SLM process is the freedom to produce complex geometries in one single step. 1

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1. Introduction

Although the SLM process is a promising manufacturing technique it is not jet widely used. The two barriers that prevent the broad implementation of the SLM process are the residual stresses and large deformations that arise in SLM products due to processing, Kruth et al. and Papadakis et al. [14, 19]. The residual stresses may lead to a limited load resistance and thermal cracks. The large deformations in SLM products are called warpage or distortion, and may cause the product to exceed design tolerances. In the manufacturing industry a high product quality is required but due to these barriers the high quality of SLM products cannot be guaranteed. In practice, there are several ways to overcome the barriers of residual stress and warpage. The general approach in relaxation of residual stresses is heat treatment of the SLM product, Shiomil et al. [22]. To prevent warpage of a product one can adjust the geometry or the orientation in which the product is build. Finding the right orientation is a skill and requires experience in designing SLM products. The downside of postprocessing is that it is an extra step in the production process. These extra steps are time consuming and expensive which make them undesirable. By acquiring knowledge about the SLM process and its properties one can try to reduce undesired effects such as residual stress and warpage. This can be done by empirically optimizing the process parameters of the SLM process. Another approach is using models of the process that predict the amount of residual stress and warpage of the product. These models can be used to alter designs or adjust process parameters.

Figure 1.1: Global overview of the SLM process. The SLM product is build on top of a base plate. Powder material is deposited on the base plate from the powder reservoir. The scraper evenly distributes the powder material. The laser beam coming from the laser source is moved over the powder bed following a scanning strategy. Positioning of the beam is done by mirrors and refocusing by a lens. By following the scanning strategy the powder material melts and solidifies when the laser beam moves forward.

Figure 1.2: Schematic overview of the SLM process at melting pool level. The laser beam (orange area) is scanned over the powder bed (light grey area) and creates a melting pool (red area). When the laser beam moves further over the powder bed the molten material that is left behind starts to cool down and forms solid material. This is visualised as the processed material (dark grey).

1.2. Literature review on SLM process

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1.2. Literature review on SLM process To get acquainted with the topic selective laser melting, a literature review has been carried out. The purpose of this literature review was to understand the SLM process and to grasp the current state of SLM modelling. This section contains a short overview of the literature study that is carried out, for more detail the reader is referred to the full literature report [5].

1.2.1. Physical phenomena in SLM process To be able to develop a model for the SLM process it is important to understand the physical phenomena that occur during the process. The physical phenomena are divided into two groups; thermal and mechanical. The thermal phenomena describe the transfer of heat through the product during the SLM process. The mechanical phenomena describe the development of field quantities such as stress, strain, and deformations during the SLM process. Thermal phenomena In the SLM process powder material is melted by the incident energy of the laser beam. After the laser has moved further along the scanning path the molten material starts to solidify and cool down. During this process heat present in the melting pool is distributed within the body and base plate by conduction. Besides conduction other heat transfer mechanism such as convection are present as depicted in Fig. 1.3. The laser beam is positioned at the top of the figure, the laser moves from right to left. The laser beam Q r ad transfers heat through radiation. The radiative heat melts the powder bed and results in the melting pool. Because the melting pool is at such a high temperature it also emits heat to the build chamber through radiation. In the melting pool there is convective heat transfer Q conv . This type of convective heat transfer is called Marangoni convection and causes the liquid metal to rotate in the melting pool. Convective heat transfer is also present at the interface of the melting pool and the gas environment. This heat transfer is from the melting pool to the gas environment. The last heat transfer mechanism is conduction which is present in the powder bed and in the already processed material. The heat transfer in the SLM process is time dependent which makes it a transient. In the remainder of this thesis the thermal behaviour in the SLM process will be referred to as transient thermal behaviour.

Figure 1.3: Schematic overview of the melting pool in the SLM process and the heat transfer mechanisms that are present. At the top, the laser is shown and its direction of movement. The laser beam transfers thermal energy to the powder bed through radiation. Convective heat transfer is present at two locations; inside the melting pool and between the outer surface of the melting pool and the gas environment. Conductive heat transfer is present in the powder bed (light grey area) and in the processed material (dark grey area).

Mechanical phenomena The cooling process of the material is the cause of mechanical deformation of the SLM product. The deformations of the material caused by the SLM process are elasto-plastic when the yield limit σy of the material is reached. Due to the high difference between the processing temperature and the build chamber temperature, 1200-1800K, pronounced thermal contraction is anticipated. This causes high stresses in the material which reach the yield limit, therefore the material deforms both elastic and plastic. The metals used in the SLM process such as Ti6Al4V and Inconel 718 have properties that are temperature dependent. These properties are the Young’s modulus E , Poisson ratio ν, yield limit σy , and the thermal

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1. Introduction

expansion coefficient α. Small temperature fluctuations have no significant effect on the property values but the large differences in SLM processing makes the change in properties significant. The yield limit σy determines the transition from elastic to plastic behaviour and is therefore an important property. When the temperature of a material rises the yield limit goes down. This means that when the temperature of the material rises the amount of residual stress that can be contained in the material decreases. The elevated temperature during processing can trigger phenomena such as stress relaxation and creep in the SLM product. Creep is the tendency of a material to deform under constant stress. Relaxation is the decrease of stress in a material when it is under constant strain.

1.2.2. Measurement of residual stress and deformation of SLM products To gain insight on the effect the SLM process has on the mechanical behaviour of the product it is necessary to measure the residual stress and warpage. Measurement of residual stress can be carried out with two types of methods; destructive and non-destructive. The difference between destructive and non-destructive methods is that in destructive methods part of the product is damaged or destroyed, whereas non-destructive methods keep the product intact. A well known and widely used destructive method is hole drilling. This is method where strain gauges are placed on a product and a hole is drilled near the gauges. The stress relief measured by the strain gauges represents the residual stress in the product. A non-destructive method is x-ray diffraction which measures the lattice spacing. With this lattice spacing distribution stresses can be found. Destructive and non-destructive methods are able to obtain the residual stresses in a SLM product but there are disadvantages to both methods. Destructive measurement methods are not desired for measuring residual stresses in SLM products because the whole or part of the product is destroyed. This is inconvenient because one of the main advantages of the SLM process is that most products build are one-off or customized products. Therefor non-destructive methods are more convenient for measuring residual stresses. Warpage or deformation of SLM products is measured by using cameras or coordinate measuring machines. With cameras images are made during the process, from these images the deformation of the product can be obtained. A coordinate measuring machine uses a probe to measure the spatial location of the surface of the product. Application of measurement methods for residual stress and warpage is possible when the SLM product is finished. During the SLM process it is difficult to measure residual stresses using these methods because of the operating conditions. In the build chamber of the SLM machine there is little space to fit measurement devices such as a hole drilling setup, x-ray machines, and coordinate measuring machines. Besides the little space, measurement equipment must be able to resist the impact of high temperature powder material and other contamination in the build chamber such as smoke particles. During the SLM process it is difficult to track the development of residual stresses and warpage. Therefore the development of numerical models for the mechanical behaviour during the SLM process is attractive. Such a model can be validated with experimentally obtained residual stresses and warpage from a finished product.

1.2.3. Modelling approaches In literature on the SLM process different modelling approaches are used to model the process. These approaches are the global and local approach. The global approach is suitable for models which describe the global behaviour of the SLM product. The local approach is more suitable for modelling the complex physics in the vicinity of the melting pool. In the global approach the SLM process is modelled as a layer addition process. In these models an entire layer is built in one step, after building the first layer the second layer is built in the second step. This layer addition is continued until the entire product is built. The layer addition process is called the layer by layer approach. Examples of models using the layer by layer approach are the models by Zaeh et al. [26, 27] and a model by Mercelis & Kruth [17]. In the model by Zaeh et al. [26, 27] the layer by layer approach is combined with thermomechanical model which includes transient thermal behaviour and elasto-plastic material behaviour. The model by Mercelis & Kruth [17] uses a simple thermomechanical model based on the temperature difference between the melting and build chamber temperature. The layer by layer approach is a suitable method for modelling the global behaviour of the SLM product. The major advantage of the layer by layer approach is that it is computationally efficient. Another advantage is that the process history of a layer can be included if it is known a priori. The disadvantage of this approach

1.2. Literature review on SLM process

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Table 1.1: An overview of the thermomechanical SLM models found in literature. For each model it is defined if it is local or global or a combination. Furthermore it is stated what type of model is used for the thermal phenomena and which is used for the mechanical phenomena.

Matsumoto et al. [16] Zaeh et al. [27] Li et al. [15] Mercelis & Kruth [17] Papadakis et al. [19]

Global x x x x

Local x x -

Thermal behaviour transient including all heat transfer mechanisms transient including all heat transfer mechanisms transient including all heat transfer mechanisms thermal problem reduced to temperature difference in cooling transient including all heat transfer mechanisms

Mechanical behaviour thermoelastic thermoelasto-plastic thermoelasto-plastic thermoelasto-plastic thermoelastic

is that the process in the layer cannot be modelled on a local level. The local approach is used to model the SLM process in the vicinity of the melting pool. Models at this level are able to look at the transient thermal behaviour in the melting pool domain. This transient thermal behaviour is modelled by Gusarov et al. [7]. The models by Cervera et al. and Li et al. [2, 15] combine the melting pool approach with a thermomechanical model. The local approach is suitable for models that describe the thermal and mechanical behaviour in the vicinity of the melting pool. A disadvantage of the local approach is that the size of the domain is small (100200µm) while the product is far larger (5mm-1m). This makes use of models at the local level to describe the global behaviour require large computational resources. In general transient thermomechanical models are computationally expensive because of the time dependency and complex mechanisms that are included in these models.

1.2.4. Thermomechanical SLM models In literature there are several SLM models which are used to predict residual stresses and warpage in SLM products. All the models described in literature are thermomechanical models [15–17, 19, 27]. In the models by Matsumoto et al., Zaeh et al., Papadakis et al., and Li et al. [15, 16, 19, 27] the thermal side of the model includes all three heat transfer mechanisms. Whereas the model by Mercelis & Kruth [17] reduces the thermal model to a temperature difference which corresponds to cooling of the metal. These models can also be divided into global or local models and also a combination of both. The global models are by Zaeh et al., Papadakis et al., and Mercelis & Kruth [17, 19, 27]. In these models the layer by layer approach is adopted in modelling the building process. The model by Matsumoto et al. [16] uses a local approach in modelling the SLM process. The model developed by Li et al. [15] is a multiscale approach and combines the local and global approach. The mechanical behaviour of the SLM process is modelled either thermoelastic or thermoelasto-plastic. In all models the information from the thermal models is transferred to the mechanical model through a thermal expansion mechanism. In the models by Matsumoto et al. [16] and Papadakis et al. [19] the mechanical behaviour is described by a thermoelastic model. In the models by Zaeh et al., Li et al., and Mercelis & Kruth [15, 17, 27] the plastic behaviour is added which means that the models are thermoelasto-plastic. In Table 1.1 an overview of the thermomechanical SLM models is given. From this overview it is clear to see that the models are either global or local except for the model by Li et al. [15]. This model has modelled the SLM behaviour at the local level in full complexity and uses these results to predict the global behaviour. During the literature review the models made for thermal processes similar to SLM are investigated. Processes such as welding and peening are widely described in literature. Examples are the work on welding by Korsunsky et al. [11, 13] and by Fukuda & Ueda [23, 24], and peening by Korsunsky et al. [10]. In these articles the use of eigenstrain theory to model the welding and peening process is discussed. Because the SLM process is reminiscent of welding there is a potential use for eigenstrain theory in modelling the SLM process.

1.2.5. Eigenstrain theory Eigenstrain theory is used to find equivalent strain fields for residual stresses caused by physical processes. In literature eigenstrain theory has been described extensively by Mura [18]. In eigenstrain theory the main problem is the eigenstrain problem which describes the relation between the inelastic eigenstrain and the residual elastic strains caused by the eigenstrain, Korsunsky et al. [13]. In this problem the eigenstrain is an unknown distribution for which a solution is found that causes a known resid-

6

1. Introduction

ual stress field. The eigenstrain problem can be solve by two approaches ;the direct approach and the inverse approach. The direct approaches solves the analytical eigenstrain problem and finds the exact solution for the eigenstrain problem. In the inverse approach the finite element method (FEM) is used to approximate the solution of the eigenstrain problem. To do this a set of linear residual strain solutions is calculated for eigenstrain components defined on the domain of the problem. These residual strain solutions are multiplied with undetermined coefficients and added together. Then a least squares minimization problem is defined which describes the error between the known strain field and the sum of residual strain solutions. Solving this problem leads to the coefficients for which the error between the known elastic strains and sum of residual strain solutions is at a minimum. Then the eigenstrain is reconstructed by multiplying the found coefficients with their corresponding eigenstrain components. The reconstructed eigenstrain is then a source for the known residual stress field. For an extensive description of the inverse approach the reader is referred to Korsunky et al. and Jun et al. [8, 13]. Solving the eigenstrain problem with the direct approach is only applicable to simple geometries for which an analytical solution can be found. The inverse approach is used when the geometries get more complex and the residual stress fields that need to be reconstructed have a complex shape. In literature the inverse approach is also called the eigenstrain reconstruction method (ERM) which is the name that is adopted in this thesis. The use of the eigenstrain reconstruction method in literature is limited to one and two dimensional cases. The one dimensional cases described by Korsunsky et al. [10–13] and by Fukuda & Ueda [23, 24] show high levels of accuracy. The two dimensional ERM method has been described by Song & Korsunsky [25] but the results are not convincing. The application of the ERM in three dimensions is not analysed to the best of authors knowledge.

1.3. Research objective The residual stresses and warpage in SLM products are the main problems in manufacturing products using the SLM process. Measurement of these quantities during the SLM process is difficult. Therefore detailed modelling of the process is the general approach taken in literature. The warpage and residual stresses and strains are mainly caused by the process at the melting pool level. The detailed information at this level is needed to predict warpage and residual stresses in the SLM product. Solving thermal and mechanical models at the melting pool level is a computational expensive task. Therefore it is unrealistic to solve these models for the entire SLM product. This leads to the main objective of this thesis work. Develop a simple model which describes the SLM process and predicts the residual stress and warpage in SLM products. The thermomechanical SLM models described in literature model the process at a local and global level. The models at the global level are computationally efficient but do not contain the process history of the layer. The local level models require large amounts of computational resources but describe the SLM process accurately. The challenge in this thesis is to develop a model that incorporates the scanning strategy of the SLM process in prediction of residual stresses and warpage without high computational costs. In literature the ERM is a frequently used method to reconstruct the eigenstrain source of residual stresses fields caused by a thermal process such as welding or peening. The SLM process is a thermal process which is similar to welding. Therefore the ERM is a potential method to reconstruct the eigenstrain source for a residual stress field caused by the SLM process. The objective for the ERM is to be able to describe a three dimensional residual stress field caused by the SLM process. For this objective it is necessary to expand the method to three dimensions. To the best of authors knowledge, the use of ERM as a modelling tool for the SLM process has not been described in literature. To conclude, the objective of this thesis is to develop a simple model which incorporates the scanning strategy and predicts the residual stress and warpage in SLM products.

1.4. Outline thesis work To reach the research goal described in section 1.3 a research strategy is defined. The research strategy is schematically shown in Fig. 1.4 and consists of three phases; orientation, development, and optimization. The orientation phase consists of the literature review and the internship at NLR. The purpose of the literature review is to get acquainted with the topic selective laser melting. The literature review is already described in section 1.2. The internship is used to gather insight in the practical use of the SLM process and numerical models of the process. The contents of the internship are described in chapter 2.

1.4. Outline thesis work

7

Figure 1.4: Overview of the phases in this thesis work; orientation, development, and optimization and validation. The orientation phase consists of the literature review and NLR internship. The development phase contains the theoretical background on the ERM, the detailed SLM model, and the ERM model. In the optimization and validation phase the ERM parameters such as function space and error norm are evaluated for several test cases.

The second phase is the development phase which contains the theoretical background on the ERM, the SLM model, and the ERM model. Chapter 3 contains the theoretical background on the ERM and consists of the eigenstrain concept and the mathematical formulation of the eigenstrain reconstruction method. The development of the SLM and ERM model are described in chapter 4. In this chapter the implementation of these models is discussed. The final phase in this thesis is optimization and validation of the developed ERM model. This consists of evaluating the different options for the ERM model. These options are the content of the function space and the error norms used in the ERM model. Finally the ERM model is validated for different geometries. The optimization and validation phase is entirely described in chapter 5. The results from the optimization and validation phase lead to the conclusions of this thesis and the recommendations for future work. These are described in chapter 6

(a)

(b)

(c)

Figure 1.5: Visualisation of the three geometries that are used to validate the ERM model. (a) A single line of material on a base plate. (b) An entire layer of material on a base plate. (c) A geometry that consists out of ten layers on a base plate.

2 Experimental and numerical work at NLR At the start of this thesis work it seemed convenient for the project to do an internship which is related to the topic of the thesis work. At SOM there are connections with NLR, the dutch areospace laboratory, due to a joined project on additive manufacturing. At NLR additive manufacturing is a topic of interest and especially the SLM process. At NLR research is performed on modelling of the SLM process and manufacturing using the SLM process. During the internship the in-house deformation model for SLM products was further investigated. The in-house model has been validated experimentally by building SLM products with different strategies. In section 2.1 the internship goals are discussed, this contains the research questions and hypothesis. The second section, 2.2, contains a short description of the used model. Section 2.3 describes the experimental setup and the manufactured SLM products. In section 2.4 the results from the numerical model and experiment are elaborated. From the results, conclusions can be drawn which are described in section 2.5. Finally, the implications of the results from the internship on the thesis work at TU Delft are elaborated. This entire chapter is an overview of the work performed at NLR, for a full report on the internship the reader can request the report at the author [6].

2.1. Internship goals During the NLR internship two parts regarding the SLM process were important. The first part is numerical modelling of the SLM process and the second part validation of these models by an experiment. In the modelling part the physics of the process and a general framework for the layer addition process were investigated. The desired outcome of this part are models that predict the residual stresses and deformations of an arbitrary geometry which is produced by using the SLM process. The second part consists of building a test geometry using the SLM process. The results from this work are compared to the numerical results for the test geometry in the model. The experimental results are also observed for physical phenomena and mechanical behavior. For the purpose of the research at NLR four research questions were defined. The first two apply to the modelling part of the SLM process and the other two are for the experimental work. The first question is if the layer based mechanical model is a suitable modelling approach for the SLM process. The layer based mechanical model is a suitable modelling approach because it includes the layer addition that is present in the SLM process. The shrinking process of each layer resembles the shrinking process that occurs during the SLM process. 1.0em The second question is, which physical mechanism and phenomena are dominant in the SLM process. In the SLM process thermal phenomena such as conduction, convection, and radiation are dominant. Mechanical phenomena such as elasto-plastic behaviour, thermal expansion, creep, and relaxation are dominant. Answers to this question are based on evaluation of the physics behind the SLM process. Building SLM products with the same scanning strategy results in products which qualitative and quantitatively the same deformation behaviour. The SLM process is a complex process for which the stability of the process conditions is of great importance. The current SLM machines are able to deal with most effects but it 9

10

2. Experimental and numerical work at NLR

is difficult to keep the conditions constant during the entire process. The scanning strategy changes the deformation behaviour of a SLM product. The SLM process gradually builds a product. Each time the laser moves further over the powder bed a new part of the product is created. The deformation process is likely to depend on the scanning strategy that is used.

2.2. NLR model At NLR a the numerical model used to predict deformation behaviour and residual strains and stresses is based on a layer by layer approach. This model is a one-way coupled thermomechanical model, a model which uses output from a thermal model in a mechanical model but does not couple the output from the mechanical model to the thermal model.

2.2.1. Physical phenomena included in the model In the SLM process a number of physical phenomena is present. As mentioned in chapter 1 the thermal phenomena that are present are convection, conduction, and radiation. The mechanical phenomena that are present in the SLM process are elastic and plastic behaviour, creep, relaxation, nonlinear material behaviour, and thermal expansion. This last one is a thermally coupled mechanical phenomena. The NLR model contains a thermal expansion model and elastic and plastic behaviour. For the thermal expansion model the thermal expansion coefficients are temperature dependent and the temperature difference is defined by the melting temperature and the build chamber temperature. The elastic material model includes a temperature dependent Young’s modulus and poisson’s ratio. The transition point from elastic to plastic behaviour is the yield limit which is also temperature dependent.

Material properties At NLR two powder materials are used, Inconel 718 which is a nickle based alloy and Ti6Al4V which is a titanium based alloy. The NLR model contains material models for Inconel 718 and Ti6Al4V. The material properties that are used are the Young’s modulus E , poisson ratio ν, thermal expansion coefficient α, and the yield strength σy of the material. The material properties are assumed to be temperature dependent based on data provided on these materials. In Fig. 2.1 graphs are shown which contain the material properties dependent on temperature. In the figures for the Young’s modulus and the yield strength, a strong dependency on temperature can be seen. The Young’s modulus of Inconel 718 shows a drop of 35% and over 60% for Ti6Al4V. The data for the yield strength show that increasing the temperature ultimately leads to no yield strength at all. The poisson ratio does not show large fluctuations but because it corresponds to the Young’s modulus it is also defined as temperature dependent. The thermal expansion coefficient is only known in a small range but is included in the NLR model. It is assumed that the data can be extrapolated linearly.

2.2.2. Layer by layer approach The NLR model uses a layer by layer approach for modelling the SLM process. In the layer by layer approach the geometry of the SLM product is divided into layers of equal thickness. These layers are subjected to a thermal strain which is calculated using Eq. (3.5). The layer approach has been used and described by Zaeh et al. [27] and Mercelis & Kruth [17]. To get the thermal strain the thermal expansion coefficient is multiplied with the difference between melting temperature of the material and the build chamber temperature of the SLM machine. In Fig. 2.2 the layer by layer approach is presented schematically. The left side of the figure shows how the product is build using the layer by layer approach. At the top left of the figure a block of material is shown which is constrained at the bottom. On this block a thermal strain ²th is imposed. At that point the block is no longer in equilibrium. By solving the mechanical boundary value problem (MBVP) the deformed equilibrium state of the block is obtained. The next step is adding a new genuine block of material. Applying the new block to the previously build material is difficult because the corner points of the deformed material and new material are not aligned. This discontinuity is solved by lowering the new block and deforming it such that the corners meet. This process does not induce any additional stresses. After the block is connected the thermal strain can be applied and building the product is continued. In the layer approach it is assumed that the processing history of the layer has no effect on the deformation behaviour.

2.3. Experimental work

11

Figure 2.1: Figures containing the temperature dependent material properties incorporated in the NLR model. Top left shows the Young’s modulus E , top right shows the poisson ratio ν, bottom left shows the yield strength of the material σy , and the bottom right figure shows the thermal expansion coefficient α.

2.2.3. Mechanical boundary value problem In the NLR model only the product geometry is modelled. The bottom surface of this geometry is assumed to be fixed in rotation and translation in all three directions. This assumption implies that the base plate is considered to be a rigid body. Material that is build on top of support structures is also assumed to be fixed in rotation and translation in all three directions. In the the model this means that the support structures are not included, instead the surface to which the support structures are connected is fixed in rotation and translation in three directions. After the product has been build it is removed from the base plate. To simulate this the NLR model has a final step in which the boundary conditions are changed. All previous boundary conditions are removed, then a corner point of the bottom surface is fixed in rotation and translation. This allows the body to relax its residual stress and deform.

2.3. Experimental work At NLR experimental work has been performed to validate the outcome of the numerical models used at NLR. In the numerical model a cantilever test geometry is constructed. The resulting deformations of the test geometry with respect to the undeformed geometry are obtained from the model and are available for comparison. The cantilever test geometry is shown in Fig. 2.3 with the dimensions in mm. At NLR two sets of cantilevers were build using Inconel 718 powder material. In a previous project the same test geometry has been fabricated using Ti6Al4V powder material. The resulting deformations for this geometry are available. The first set consists of four cantilevers which are made using a checkerboard pattern scanning strategy.

12


Figure 2.2: Schematic overview of the layer by layer approach. The layer by layer approach starts with building a layer on which a thermal strain is imposed. By solving the MBVP the deformed equilibrium state of the block is calculated. Applying a new layer is difficult because of discontinuities at the interface. Therefor, the layer is lowered and reconnected to the surface. This reconnecting process does not induce any additional stresses. After the block is connected the thermal strain is applied and the MBVP is solved. Then the building process is continued. Table 2.1: Names of the test geometries built with the SLM machine at NLR. A-I are the Inconel 718 cantilevers from the second set. T1-T5 are the Ti6Alv4 cantilevers. Inc1-Inc4 are the Inconel 718 cantilevers from the first set. For each cantilever the used scanning strategy is depicted.

Name A B C D E F G H I

Scanning strategy 0◦ (parallel) 0◦ (20mm) 0◦ (10mm) 0◦ (5mm) 90◦ (perpendicular) checkerboard no rotation checkerboard rotation 45◦ 45◦ alternating −45◦

Material Inconel 718 Inconel 718 Inconel 718 Inconel 718 Inconel 718 Inconel 718 Inconel 718 Inconel 718 Inconel 718

Name T1 T2 T3 T4 T5 Inc1 Inc2 Inc3 Inc4

Scanning strategy 0◦ (parallel) 0◦ (20mm) 0◦ (10mm) 0◦ (5mm) 90◦ (perpendicular) checkerboard rotation checkerboard rotation checkerboard rotation checkerboard rotation

Material Ti6Al4V Ti6Al4V Ti6Al4V Ti6Al4V Ti6Al4V Inconel 718 Inconel 718 Inconel 718 Inconel 718

This scanning strategy is shown in Fig. 2.4. The left part of the figure shows that the layer is divided into blocks which have a 90 degree rotated scanning direction w.r.t. there adjacent block. The dimension of the blocks is 10x10 mm, the dimension of these blocks can be changed depending on the products dimensions. The right part of the figure shows a rotation of the scanning strategy of 63 degrees. This rotation is applied to each added layer. The rotation is applied because it is assumed that this strategy yields the smallest product deformations and lowest amount of residual stresses in the product. In literature on the SLM process proof for this statement is not found during my thesis work. The second set consists of nine cantilevers which are made using a variety of scanning strategies. These scanning strategies are shown in Fig. 2.5. From top left to top right; parallel scanning strategy (angle zero degrees), perpendicular scanning strategy (90 degree angle w.r.t. x 1 axis), diagonal scanning strategy (45 degrees w.r.t. x 1 axis), mirrored diagonal scanning strategy. Bottom left to bottom right; checkerboard scanning strategy, parallel with scanning length of 20mm, parallel with scanning length of 10mm, and parallel with scanning length of 5mm. All eight scanning strategies are applied on the cantilevers and another one which is the checkerboard without rotation. The set of Ti6Al4V cantilevers which were already build at NLR are made using the parallel, perpendicular, 20mm, 10mm, and 5mm scanning strategy. The test geometries are build on top of a baseplate made of the same material as the products. The baseplate and products are shown in Fig. 2.6a. In the figure blue marked support structures can be seen. These support structures are made of the same material as the product and are build simultaneously with the layers. The support structures are point supports attached to the structure which are needed to prevent

2.4. Results

13

Figure 2.3: Schematic view of the cantilever test geometry with dimensions in mm

Figure 2.4: Schematic view of the checkerboard scanning strategy, Left figure: top view of layer filled with block with a 90 degree shift of scanning direction, blocks have dimension of 10x10 mm. Right figure: top view of layer, for each added layer the scanning strategy is rotated over an angle of 63 degrees

overhang failure during production. After production these supports can be snapped off due to the weak connection to the product. After the SLM process has finished all cantilevers, the cantilevers are still attached to the baseplate. After the cantilevers are finished no postprocessing steps are applied such that the residual stresses and deformations of the cantilevers are retained. To remove the cantilevers from the baseplate they are cut using a wire EDM process. The cut is performed at 2.5mm from the top surface of the cantilevers. The wire EDM process cuts right through the left and right base of the cantilevers as through the support structures. The cutting plane and support structures are visualised in Fig. 2.6b. The produced cantilevers are measured by a Brown & Sharp Scirocco model 20.10.09, 3D co-ordinate measuring machine. This machine uses a probe to feel the surface of the cantilevers and remembers the coordinates of a surface.

2.4. Results The results of the work carried out at NLR are presented in this section. The results consist of two parts which are the numerical results and experimental results. The numerical results are obtained from the layer model of the cantilever beam. The calculations are performed for the materials Inconel 718 and Ti6Al4V. The experimental results are deformations of the top surface of the build cantilevers in Inconel 718 and Ti6Al4V. The cantilevers are build using the scanning strategies described in 2.3. To evaluate the results from the NLR model it is convenient to have one value which represent the deformation of the cantilever. All results show a curvature of the slender part of the cantilever as shown in Fig. 2.7. The radius of curvature of the slender part is a good measure of deformation which can be compared with both numerical and experimental results. PN φ=

i =1 (r g

N

− r i )2

(2.1)

14


Figure 2.5: Scanning patterns used in the experimental work at NLR, from top left to top right; Parallel (0 deg), perpendicular (90 deg), diagonal (45 deg), diagonal (−45 deg). Bottom left to bottom right; Checkerboard, parallel scanning length 20mm, parallel scanning length 10mm, and parallel scanning length 5mm.

(a)

(b)

Figure 2.6: (a) Schematic view of the test geometries on the base plate. The blue striped part underneath the cantilevers represents the support structures that are needed to build overhanging parts such as the cantilever. (b) Schematic view of the side of the cantilever. Top figure shows the support structures beneath the bridge part of the cantilever. The bottom figure shows the cutting plane for the wire EDM process at 2.5 mm under the top of the cantilever surface.

ri =

q (x ai − x ac )2 + (y ai − y ac )2

(2.2)

The radius of curvature is obtained by processing the location data of the top surface of the cantilever. To determine the radius a least squares optimization is used to find the center location and radius of a circle that fits the curvature of the top surface. The objective function Eq. (2.1) describes the squared error between the initial guess of the radius R g and the radius to each data point. The radius is calculated through Eq. (2.2) and determines the distance from the center location x ac , y ac to the location of the data point. To find the radius of curvature the objective function needs to be minimized. By minimizing the objective function a radius and center location are found for which the distance to every data point is minimized.

2.4.1. Numerical results The single layer model has been solved for different layer thicknesses and initial temperatures. The thicknesses are 50, 100, 200, 500, and 1000µm. The initial temperatures for Inconel 718 are 1573, 1373, and 1200K and for Ti6Al4V are 1878, 1573, and 1178K . The resulting radii are shown in Table 2.2. In Table 2.2a the resulting radii for the Ti6Al4V calculations are shown. First of all it is clear to see that the initial temperature has almost no effect on the radius of curvature. This is caused by the temperature dependent yield strength. The yield strength is approximately zero at a temperature of 1200K . This means that every initial temperature higher than this 1200K does not result in build up of residual stresses. The cantilever undergoes plastic deformations but these do not contain residual stresses. The radii of curvature show a clear dependency on the layer thickness that is used. Increasing the layer thickness decreases the radius of curvature so the cantilever bends more if the layer is thicker. This is logical

2.4. Results

15

Figure 2.7: Schematic view of the radius of curvature of the cantilever.

because each time the layer increases the volume that shrinks increases. Shrinking of a larger volume exerts a larger force which results in higher stresses. The build up residual stresses relaxes when the model is removed from the base plate. Due to the high amount of stored elastic energy for thicker layers the cantilever deforms more. The results for the Inconel 718 calculations with the NLR model are shown in Table 2.2b. The results clearly show that the initial temperature has no effect on the radius of curvature of the cantilever. The effect of layer thickness on the resulting radii of curvature for the Inconel 718 calculations is small. The radii lie in the range from 1246 to 1304 mm. One would expect that the behaviour is the same as for the Ti6Al4V cantilevers. The behaviour can be explained by the allowable elastic strain in the cantilevers for the two maσy terials. The allowable strain is equal to ²all = E and depends on the yield strength and Young’s modulus. At room temperature the yield strength for Ti6Al4V is higher than for Inconel 718. The Young’s modulus is almost twice as high for Inconel 718. This makes the allowable strain for Ti6Al4V much higher than for Inconel 718. A way to explain that the radii for Inconel 718 stays more or less constant is that the allowable strain is reached fast and that most deformations are in the plastic regime. The high allowable strain for Ti6Al4V also results in much larger deformations for Ti6Al4V. This can be seen because the radii for Ti6Al4V are much smaller than for Inconel 718. Table 2.2: (a) Resulting radii from the Ti6Al4V cantilever simulation for varying thickness and initial temperature. (b) Resulting radii from the Inconel 718 cantilever simulation for varying thickness and initial temperature. (a)

Ti6Al4V 50µm 100µm 200µm 500µm 1000µm

Tini =1178K 1266 1152 976 904

(b)

Tini =1573K 1273 1178 937 870

Tini =1878K 1398 1281 1176 949 876

Inconel 718 50µm 100µm 200µm 500µm 1000µm

Tini =1200K 1257 1246 1239 1304

Tini =1373K 1247 1256 1255 1304

Tini =1573K 1302 1272 1264 1258 1302

2.4.2. Experimental results To evaluate the experimental results the radius of curvature is used. The cantilevers A and T1, and E and T5 are build using the parallel and perpendicular scanning strategy. For the Inconel 718 cantilevers A and E the radius of curvature 2.5 times higher for the perpendicular scanned cantilever than for the parallel scanned cantilever. For the Ti6Al4V cantilevers this is equal to 2.8 times. This clearly shows that the chosen scanning strategy influences the deformation behaviour of the cantilever. A small radius of curvature means large bending of the cantilever. The parallel scanned cantilevers show much more bending than the perpendicular scanned cantilevers. This effect is explained by the way the material shrinks after the laser moved on. The melting pools are longer than they are wide and therefore have more shrinkage in the longitudinal direction than in the latitudinal direction. In the experimental Inconel 718 sets five cantilevers are build using the rotating checkerboard scanning pattern. The radii of curvature for these cantilevers lie in a wide range from 565 to 796 mm. This wide range indicates that the quality of the products that are produced is not consistent. It indicates that more research

16


Table 2.3: Tabulated radii of curvature for the cantilevers which are build using the SLM machine at NLR.

A B C D E F G H I

r (mm) 495 307 371 1231 545 629 658 689

T1 T2 T3 T4 T5 Inc1 Inc2 Inc3 Inc4

r (mm) 885 775 693 696 2554 796 719 602 565

should be focussed on process control. The parallel scanning strategies with scanning line length of 20, 10, and 5 mm were used for the Inconel 718 cantilevers B, C, and D. The same strategies were used for T2, T3, and T4. Unfortunately the resulting cantilevers are not usable to identify the effect of these scanning strategies. The cantilevers showed defects at the start stop point where the scanning lines ended. The 45 degree, H, and alternating 45 degree, I, scanning pattern are useful. The radii of curvature are 658 and 689 mm respectively and thus show the same amount of bending. The only difference is that for cantilever H the tip of the cantilever shows a large rotation where cantilever I is straight. This nicely shows that the scanning strategy certainly effects the deformation behaviour.

2.4.3. Comparison of results A comparison of the numerical and experimental results provides feedback on the model developed at NLR. This feedback can be used to update the or change the NLR model. However, the experimental results for Ti6Al4V cannot be compared to the calculations performed with the NLR model. This is not possible because the Ti6Al4V cantilevers are not built using the checkerboard pattern which resembles uniform deformation behaviour. For the experimental results with Inconel 718 cantilevers the five rotating checkerboard patterns can be compared with the calculations. The radii for the experiment were in the range from 565 to 796 mm where the calculation for the 50µm layer thickness the radius is 1302 mm. This shows that the NLR model overestimates the bending of the cantilevers.

2.5. Conclusions From the experimental results it can be concluded that the scanning strategy influences the deformation behaviour. The effect of different scanning patterns is clearly visible in the experimental results. For both the Ti6Al4V and Inconel 718 sets the radius of curvature is lower for the full-line scanning strategy than for the perpendicular scanning strategy. The rotating checkerboard pattern shows a radius of curvature in the range from 565 to 796 mm for five geometries. This observation indicates that the use of the same scanning strategy for several products in one or more batches does not result in the same amount of deformation. The general model that is developed at NLR uses the layer by layer approach. The NLR model is able to describe the same bending behaviour as is seen in the experimental results. This indicates that the general framework proposed in this research is able to qualitatively predict deformation during the SLM process. The effect of layer thickness and initial temperatures for the models is evaluated for the NLR model. For Ti6Al4V the simulation results show that the effect of layer thickness is large. The radius of curvature of the cantilever decreases fast when the layer thickness is increased. For the Inconel 718 simulations the effect was small and no clear trend was visible as a function of the layer thickness. The reason for this difference is currently unknown. The effect of different initial temperatures on the models is small. This supports the suggestion that the residual stresses will mainly build up in the last part of cooling. This is probably caused by the low yield strength at higher temperatures which prohibit the development of stress.

2.6. Implications for thesis work

17

In the NLR model the layer approach is used in which the process history of the layer is neglected. A comparison of the modelling and experimental results clearly shows that the layer model used at NLR is not able to describe the effect that the scanning strategy has on the deformation of the product. Neglecting the scanning strategy in the model is therefore inaccurate. The NLR model included the temperature dependency of material properties and plastic material behaviour. Including these properties made the model complex and increase the computation time. The added complexity of the model make evaluating the results more difficult.

2.6. Implications for thesis work The NLR intership has given valuable insight on modelling the SLM process and the behaviour of SLM products. The conclusions drawn from the internship have implications on the thesis work at TU Delft. These implications are discussed below. The experimental work at NLR shows that the scanning strategy affects the deformation behaviour of the SLM product. In the NLR model the scanning strategy was not included. Therefore in the thesis work at TU Delft, a way to include the scanning strategy is developed. In the NLR model the temperature dependent properties and plastic material behaviour cause the model to be more complex. The computational effort increases by 6 to 8 times using a plasticity model. The temperature dependency of material properties increases the computational effort for determining the thermal strain and solving the MBVP. Besides the computational effect the complex model makes it difficult to identify the cause of the behaviour. Therefore in the thesis work a model is developed which assumes constant material properties and no plasticity. The stress distributions in the NLR model and produced cantilevers are not investigated during the internship. In the thesis work at TU Delft, the focus is therefore shifted towards evaluation of the development of residual stresses in the developed model. The assumptions made with regard to the boundary conditions of the base plate and the support structures do restrict the deformation behaviour of the material. Observations during the experiments showed that both the base plate and the support structures deform when the cantilevers are build. Hence in the thesis work at TU Delft,the base plate is included in the model. The support structures are not considered because geometries are used which do not include support structures.

3 Eigenstrain reconstruction method In chapter 1 the use of the eigenstrain reconstruction method to model the residual stresses caused by the SLM process is proposed. Chapter 3 contains the necessary theoretical background to understand the working principle of the developed models. In section 3.1 the field quantities, thermal expansion mechanism, and the formulation of the mechanical boundary value problem (MBVP) are discussed. The eigenstrain concept and the mathematical formulation of the eigenstrain reconstruction method are presented in section 3.2.

3.1. Field quantities, thermoelasticity, and thermoelastic mechanical boundary value problem In subsection 3.1.1 the field quantities stress, strain, and strain energy that are used in this thesis are described. During the SLM process material is heated which causes thermal vibration in the atoms of the material. These thermal vibrations contribute to the total strain in the system in addition to the stress induced elastic strains. The effect of thermal vibrations is captured by assuming thermoelasticity. Therefore, a description of the thermal expansion mechanism is presented in subsection 3.1.2. In the final subsection of this section a description of the MBVP is given which includes thermal expansion. The thermal expansion

3.1.1. Stress, strain, and elastic strain energy In the field of structural mechanics the most important field quantities are stress and strain. Stress is a tensor that describes the internal forces in a continuum. The internal forces are defined as the forces that material points in a continuum exert on each other. Strain is also a tensor that describes the degree of deformation of a body. The stress and strain tensors σ and ² are both symmetric second order tensors and hence fully described by six components in three dimensions. The six stress and strain components are shown in Fig. 3.1a. Here an infinitesimal volume element is shown with three normal stress components; σ11 , σ22 , and σ33 . In the same infinitesimal volume element the shear stress components are shown; σ12 , σ23 , and σ13 . The strain components are shown in Fig. 3.1b. The six strain components consist of three normal strains; ²11 , ²22 , and ²33 . The shear strain components consist out of three shear strains. The strain state in a body is described by the total strain, ²tot = ²e + ²th , (3.1) where ²e represents the elastic strain and ²th the thermal strain. Depending on the type of model the total strain consists of several components. In a linear thermo-elastic model which undergoes thermal expansion the components are elastic ²e and thermal strain ²th . This means that Eq. (3.1) is only valid when thermoelasticity in the model is assumed. This description of the total strain is used in the layer by layer approach described by Mercelis & Kruth [17] and discussed in section 2.2. In chapter 5 the results for the ERM are compared with the results for the SLM model. The field quantities that are used to evaluate the error between the models are von Mises stress σVM , effective strain ²eff , an elastic strain energy U . The von Mises stress is presented in the following equation, 1 q σVM = p (σ11 − σ22 )2 + (σ33 − σ11 )2 + (σ22 − σ33 )2 + 6(τ212 + τ223 + τ213 ). (3.2) 2 19

20

3. Eigenstrain reconstruction method

(a)

(b)

Figure 3.1: (a) Schematic view of the normal stress components σ11 , σ22 , σ33 and the shear stress components σ12 , σ23 , σ13 on an infinitesimal volume. (b) Schematic view of the normal strain components ²11 , ²22 , ²33 and the shear strain components ²12 , ²23 , ²13 on an infinitesimal volume.

In this equation one sees all the stress components. The von Mises stress is a scalar that represents the equivalent stress in a volume. This description is used for error evaluation because it is more convenient than evaluating individual stress components. The following component is the effective strain ²eff , r 2 2 1 (²11 + ²222 + ²233 ) + (γ212 + γ223 + γ213 ). ²eff = (3.3) 3 3 This quantity is a scalar that represents the equivalent strain in a volume, which is calculated from all six strain components. The use of the effective strain is preferred over the individual strain components because it is more convenient to evaluate a scalar. The final value that is used to evaluate the results for the developed models is the strain energy, Z 1 U= (σ11 ²11 + σ22 ²22 + σ33 ²33 + τ12 γ12 + τ23 γ23 + τ13 γ13 )dV. (3.4) V 2 The strain energy is an interesting value because it represents the energy in the volume. It is calculated by using both the stress and strain in a volume. The von Mises stress, effective strain, and strain energy are shown in Eqs. (3.2), (3.3), and (3.27).

3.1.2. Thermoelasticity In the SLM process material is subjected to significant heating and cooling cycles where the resulting temperature change causes strains associated with thermal expansion given as, ²th i j = α(T − Tref )δi j .

(3.5)

is the thermal strain equation which consists of two terms. The first term is the thermal expansion coefficient α. The other is the imposed temperature difference ∆T = T − Tref . The thermal expansion mechanism described here is linear. For most material the thermal expansion coefficient is dependent on temperature. Including the temperature dependency makes the thermal expansion model nonlinear. Besides the linear and nonlinear variant three directional dependent thermal expansion models exist. These are the isotropic, orthotropic, and anisotropic thermal expansion model. The isotropic model is used when thermal expansion is the same in all directions. Therefore, for an isotropic model the thermal expansion coefficient α is a constant. When isotropic thermoelasticity is assumed there is no shear strain induced by the thermal expansion. Orthotropic thermal expansion is used for materials which expand differently in the normal directions. For an orthotropic model three thermal expansion coefficients are defined; α11 , α22 , and α33 . By assuming orthotropic thermoelasticity one again assumes that there are no induced shear strains due to thermal expansion. In an anisotropic thermal expansion model the expansion behaviour is described for six directions. Three normal components and three shear components. This results in six thermal expansion coefficients, α11 , α22 , and α33 for the normal components and α12 , α23 , and α13 for the shear components. In Fig. 3.2 the thermal expansion in all six directions are shown. On the left of the neutral block the expansion in normal directions are shown. The expansion in the shear directions are shown on the right. By assuming anisotropic thermoelasticity it is assumed that thermal expansion induces shear strains.

3.1. Field quantities, thermoelasticity, and thermoelastic mechanical boundary value problem

21

Figure 3.2: Schematic overview of thermal expansion in all directions. On the top row the expansion of the infinitesimal volume element in the normal directions is shown. On the bottom row the expansion of the volume in shear directions is shown.

Orthotropic and anisotropic thermal expansion are difficult to understand because intuitively one considers expansion as a uniform process. In general this is true but when one considers materials which are orthotropic or anisotropic non-uniform thermal expansion can occur. Examples are composit materials and materials that have a crystal structures less than cubic. These materials generally have different expansion coefficients in different directions.

3.1.3. Thermoelastic mechanical boundary value problem To describe a differential equation one can use two formulations, the strong formulation and weak formulation. The difference between the strong and weak form is that the strong form describes conditions at every material point while the weak form does this in an average or integral sense [3] page 136. In the finite element (FE) method the weak formulation is used because the integral expression can describe the problem at the level of a finite element. In Fig. 3.3 a schematic overview is given of the mechanical boundary value problem (MBVP) and solution

Figure 3.3: Schematic overview of the mechanical boundary value problem MBVP. On the left the domain Ω is shown which is fixed in translation on boundary Γ2 ∪ Γ3 . The boundaries Γ1 ∪ Γ4 are traction free, and the thermal strain ²th is imposed on the domain Ω. After solving the MBVP the result is a deformed configuration Ω0 which is in equilibrium

of the MBVP. In the left figure the domain Ω and its boundary Γ are described. Part Γ1 and Γ4 of the boundary contains a traction free boundary condition i.e. a Neumann boundary condition. Γ2 and Γ3 is the part of the boundary that is fixed in displacement u = 0. A thermal strain ²th as described in subsection 3.1.2 is imposed on the domain Ω. This defines the mechanical boundary value problem as follows, ∇·σ = 0

on Ω

(3.6)

u=0

on Γ2 ∪ Γ3

(3.7)

on Γ1 ∪ Γ4

(3.8)

f = f(² )

on Ω

(3.9)

σ = C²

on Ω

(3.10)

t=0 th

22


where the first equation describes the equilibrium on the domain Ω. The second equation the no displacement (Dirichlet) boundary condition on Γ2 ∪ Γ3 . The third line shows the traction free boundary condition on Γ1 ∪ Γ4 . The fourth line represents the equivalent force term for the thermal strain which is defined on Ω. The fifth and last line shows the tensor notation of Hookes law which describes the relation between stress and strain. The tensor C is the fourth order compliance tensor which contains materials constants such as Young’s modulus and Poisson ratio. The system that has been described is not in equilibrium due to the imposed thermal strain. To get to the equilibrium condition the MBVP needs to be solved. In this thesis the defined problems are solved using the FE method. This means that the discretised versions of the problem are solved. For a static problem the problem is described by a system of equations as described below, K u − f = 0.

(3.11)

The system of equations described in Eq. (3.11) contains the stiffness matrix K , the degrees of freedom u, and a force vector f which contains the contribution of the thermal strain as follows, f = f (²th ).

(3.12)

This force vector is defined by the thermal strain. In the FE method the general approach of inducing a thermal strain is by calculating an equivalent force vector for this thermal strain, [3] page 52. The force vector is determined by calculating the thermal stresses corresponding to the thermal strain and integrating these thermal stresses over the volume of the element. By solving the system of equations the displacements u are found for which the system is in equilibrium. This results in the deformed configuration Ω0 which is described on the right of Fig. 3.3. The deformed state Ω0 delivers the information on the residual stress state σres , the total strain ²tot , and the deformation u of the geometry.

3.2. Eigenstrain theory Eigenstrain theory has been developed to describe the relation between the strain source of mechanical behaviour and the resulting residual stress. In the early days of the introduction of eigenstrain theory it has been used to find analytical solutions for the strain source by Reissner [21] and Eshelby [4]. In the 70s another use of eigenstrain theory was developed which used the FE method to find eigenstrains corresponding to a stress state. For cases where the source of deformation is difficult to model or it is computationally expensive such as welding, peening, and the SLM process. For these kind of problem the use of eigenstrain theory is convenient.

3.2.1. The eigenstrain problem The eigenstrain problem is described as, ²ekl (x i ) = −²∗kl (x i ) −

Z

∞ −∞

C pqmn ²∗mn G kp,ql (x i − x i0 )d x i0 .

(3.13)

In this equation ²ekl (x i ) represents the residual elastic strain at location xi . ²∗kl describes the eigenstrain at location x i . The integral describes the effect of boundary conditions on the elastic strain. The integral contains the fourth order compliance tensor C pqmn which contains the material constants of the used material. The next component is the eigenstrain term, ²∗kl . G kp,ql is the Green’s function. This Green’s function is a mapping function which calculates the effect of the boundary conditions on the solution of the mechanical BVP. The eigenstrain problem can be solved in two ways; the direct approach and the inverse approach. With the direct approach the eigenstrain problem is solved analytically which results in the exact solution of the problem. The inverse approach uses the FE method to approximate the solution of the eigenstrain problem. The analytical problem can only be solved for simple geometries. This means that the direct approach is not suitable for complex geometries. In the case of the SLM process the products generally have complex geometries. Therefore the use of the inverse approach to solve the eigenstrain problem is needed. In literature the inverse approach is generally called the eigenstrain reconstruction method (ERM) and is the adopted term in this thesis.

3.2.2. Eigenstrain reconstruction method - theory The ERM is a tool in eigenstrain theory which is used to find strain sources for residual stress fields caused by physical phenomena. The ERM can be considered as a black box shown in Fig. 3.4, the ERM box has two

3.2. Eigenstrain theory

(a)

23

(b)

Figure 3.4: (a) Schematic overview of the eigenstrain reconstruction method. The ERM is seen as a black box which has two inputs and one output. The inputs of the ERM box are a reference stress field σref on a domain Ω and a function space Σ which is valid on kl

the domain Ω. The output of the ERM box is an eigenstrain which is an approximation of the source of the reference stress field σref . kl (b) Schematic overview of the content of the ERM black box. The box contains the five steps to get to the eigenstrain. The first step is extraction of residual elastic strain ²ei j . The second step is calculating the elastic strain solution S iJ j . The third step is combining the the two solutions in a functional H which describes the sum of squared errors. The fourth step is minimization of this functional which results in the coefficients c J . In the fifth and final step the found coefficients c J and eigenstrain components ζiJ j are combined in the eigenstrain distribution ²∗ . ij

inputs and one output. The input to the ERM box is a reference stress field σref which is defined on a domain kl Ω and a function space Σ which is valid on a domain Ω. The residual stress field that is used as a reference is the most important input for the model because it is the stress field that is reconstructed. The output of the ERM box is the eigenstrain which is the approximate strain source for the reference stress field σref . kl The procedure that is followed to obtain the eigenstrain ²∗kl is schematically shown in Fig. 3.4b. The procedure taken in the ERM is divided into five steps. Each step is explained to get a good understanding of the ERM. The procedure described in this section is based on the description of the ERM by Korsunsky et al. [11, 13] and Jun & Korsunsky [8]. Step 1 - Extracting residual strain data from σref ij The first step is to obtain the elastic strains ²ekl from the reference stress field σref or have them available kl directly. The reference stress field that is used can be obtained from a numerical model or through measurement. For the ERM the origin of the data is not important. Step 2 - Calculating the residual strain solutions for the basis functions J The second step is to use the function space Σ which comprise of N basis functions ζkl where j = 1, .., N . The basis functions from the function space are the components that are used to construct the eigenstrain. Therefore the basis functions are called the eigenstrain components in the remainder of this thesis. Each eigenstrain component is valid on the domain Ω. Each of these eigenstrain components is imposed on the domain as a strain source. By solving mechanical J J boundary value problems one gets elastic strain solutions S kl for each eigenstrain component ζkl . By doing this for all basis functions one gets a residual strain solution for each eigenstrain component. J Every obtained elastic strain solution S kl is a linear solution. This is the case because a linear elastic model with small strain assumption is used to calculate the residual strain solutions of the eigenstrain components.

24


Step 3 - Combining the reference and elastic strain solutions The third step in the ERM is reconstruction of the residual elastic strains ²ekl by means of superposition of

J J elastic strain solutions S kl . The elastic strain solutions S kl are multiplied with an undetermined coefficient J c and summed in the total elastic strain solution S kl which is defined as follows,

S kl (x i ) =

N X J =1

J c J S kl (x i ).

(3.14)

In the third step the reference elastic strains and total elastic strain solution are combined in a functional H as described below, M X H [x i ] = [²ekl (x iI ) − S kl (x iI )]2 . (3.15) I =1

This functional H describes the sum of squared errors between the elastic strains ²ekl (x i ) and total elastic strain solution S kl (x i ). The summation for I = 1, .., M represents the summation for all data points I at location x iI . Step 4 - Finding the undetermined coefficients c J The fourth step in the ERM is finding the values of the coefficients c J for which the functional H is minimized. Minimization of the functional means that a set of coefficients c J is found which makes the error between the reference and total elastic strain solution go to zero. This can be done by taking the derivative of the functional H w.r.t. the undetermined coefficients c J and equating to zero, " # M N X X £ K I ¤ ∂H e I J J I =2 ²kl (x i ) − c S kl (x i ) −S kl (x i ) = 0 (3.16) ∂c J I =1 J =1 The quadratic form of the functional H , Eq. (3.15), is positive definite and therefore the system of linear equations described by Eq. (3.16) always has a unique solution corresponding to a minimum of H , Korsunsky et al. [13]. The derivative of the functional H w.r.t. the undetermined coefficients c J can be rewritten to the following form, ´ X M ³ M ¡ X ¢ J K K c J S kl (x iI )S kl (x iI ) = ²ekl (x iI )S kl (x iI ) , (3.17) I =1

I =1

where one has a left hand side with a known and unknown part and a right hand side which is known. The J K known part on the left hand side is the multiplication of the elastic strain solutions S kl S kl . The unknown part J is the vector of undetermined coefficients c . The right hand side shows the multiplication of the residual elastic strains ²iJ j and the residual strain solutions S iKj . The multiplication of the elastic strain solutions in left hand side results in a matrix with size (NxP) because J = 1, .., N and K = 1, .., P . The contents of the matrix are as follows, ¡ 1 I P I ¢ ¡ 1 I 1 I ¢ S kl (x i )S kl (x i ) . . . S kl (x i )S kl (x i ) M X  .. .. . .. A= (3.18)  . . . ¢ ¡ N I P I ¢ I =1 ¡ N I 1 I S kl (x i )S kl (x i ) . . . S kl (x i )S kl (x i ) For the vector c J containing the undetermined coefficients the content is described as,  c1  .  c =  ..  . cP 

(3.19)

In this vector the first entry is the coefficient of the first eigenstrain component and last entry contains the coeffcient of the last eigenstrain component P . The right hand side vector b is described as given below, ¡ e I 1 I ¢ ²kl (x i )S kl (x i ) M X   .. b= (3.20)  , . ¡ ¢ I =1 e I P I ²kl (x i )S kl (x i )


25

where the top entry of the vector contains a multiplication of the elastic strain ²ekl and the elastic strain solution from the first eigenstrain component. The last entry in the vector described the multiplication with the elastic strain solution for the last eigenstrain component. Together, the matrix A, vector c, and vector b describe the system of equations as follows, Ac = b. (3.21) Solving the system of equations Ac = b results in the values of the undetermined coefficientc c J of the residual J strain solutions S kl . The content of the system of equations Ac = b is described similarly by Jun & Korsunsky [8]. Step 5 - Reconstructing the eigenstrain The final step is the assembly of the eigenstrain which is described by the following equation, ²∗kl (x iI ) =

N X J =1

J c J ζkl .

(3.22)

For the residual strain solutions it is assumed that the principle of superposition can be used. The same analogy is used to state that the sum of eigenstrain components yields the same solution as the sum of the residual strain solution for each individual eigenstrain component. By using this principle the eigenstrain distribution is defined as given in Eq. 3.22. The assumptions made in the ERM are validated by comparing the reference stress field with the stress field that is caused by the eigenstrain distribution. To find the stress field caused by the eigenstrain the MBVP is solved.

3.2.3. Function space In the description of the ERM the term function space is mentioned several times. The function space is the basis for the inverse approach of solving the eigenstrain problem. The function space is filled with basis functions that are described in an n-dimensional space. In real processes such as the SLM process this comes down to four dimensions; three spatial dimensions and the dimension time. In this thesis the dimension time is not used so the function space is defined in three dimensions. The function space or basis is a set which contains a variety of functions. The functions contained in the function space can be polynomials as is the case for Chebyshev and Legendre polynomials, center based functions such as radial basis functions, or goniometric series such as the Fourier series. Examples of fields in which function spaces are used are numerical analysis and approximation theory. The finite element method is a tool from numerical analysis and uses the functions space for interpolation purposes. In approximation theory the function space is used to find a blending or representative function which describes the trend in an experimental or numerical dataset. In the ERM the function space is used in the same sense as in approximation theory. The ERM is used to reconstruct an eigenstrain which is the approximation of the reference residual stress field. The function space determines the solution space from which the eigenstrain can be constructed. Thus the function space is the key component that determines how well the ERM will perform. In literature regarding the ERM the choices made on the content of the function space are not the same. Korsunsky et. al. [10–13] use the SIMTRI method which is based on center based functions such as the radial basis function (RBF) and in other studies Chebyshev polynomials. Kartal et al. [9] use Legendre polynomials and Cao et al. [1] and Qian et al. [20] use a series of smooth basis functions. The problems discussed in the mentioned literature range from welding to shot peening and it is stated that for different problems different function spaces should be used. In designing the function space two aspects are important. The shape of the residual stress field that needs to be reconstructed and the shape of the residual strain solutions for the basis functions. If the reference field contains steep gradients residual strain solutions are needed that can describe those gradients. A smooth surface in the reference field requires more smooth basis functions that span a large area. The shape of the residual strain solution depends on the shape of the basis function and geometry that are used. The design of the function space used to model the SLM process mainly depends on the geometry and the stress fields in the geometry. Because the SLM process is a layer based process it is convenient to use basis functions that are defined in each layer. Because the SLM process is generally used to manufacture complex products it is assumed that in designing the function space an irregular domain should be considered. On an irregular domain it is not convenient to use Chebyshev, Legendre, or any kind of polynomials. The

26


polynomials are capable of capturing regular shaped surfaces but it is difficult to describe a complex geometry. Another reason is that the idea behind eigenstrains is that they can be used as a modelling tool. A tool to predict the residual stresses on a domain which undergoes the SLM process. The use of Fourier series leads to the same problems in implementation as the polynomials. Location based functions such as the radial basis function are very convenient because they can be positioned everywhere in the domain. Another advantage is that the properties of each basis function can be adjusted. In literature the research by Korsunsky et al. [12] and Jun & Korsunsky [8] uses a variety of radial basis functions which show good results from the ERM. Therefore the function space used in this thesis is based on radial basis functions.

φ(||x i ||) =

( ||x || 4||x || (1 − ri )4 ( r i + 1)

||x i || ≤ 0,

(3.23)

||x i || > r,

0

The radial basis function that is used in this thesis is Wendland’s C 2 compactly supported RBF which is shown in Eq. (3.23). The reason that this specific radial basis function is chosen is that it can easily be defined for one and two dimensional spaces and it is a smooth function. During the courses of the masters program this function has been used so it could be used directly. The choice of basis functions is recommended future work because the use of different types of RBFs might be beneficial. In Figs. 3.5a and 3.5b the one and two dimensional variant of the Wendland’s RBF are shown. 1

0.8

1

1

0.8

0.8

φ(||x i||)

0.6 φ(||xi||)

0.4

0.4

A (-)

A (-)

A (-)

0.6

r=||x i||

0 -1

-0.5

0.2

0

0

x1 (mm)

(a)

0 -1

xc

1 0

0.5

φ(||xi||)

0.4

0.2 0.2

0.6

1

x1 (mm)

0 1

-1

x2 (mm)

(b)

-1

1 0

x1 (mm)

0 1

-1

x2 (mm)

(c)

Figure 3.5: (a) Radial basis function as a function of x 1 with radius r = 1 and center location x c = 0. (b) Radial basis function as a function of x 1 and x 2 with radius r = 1 and center location x c = (0, 0). (c) Radial basis function which is defined along the edge x 2 . Radius r = 1 and the center location in x c = −1.25 in x 1 .

3.2.4. Error norms The functional H described in the Eq. (3.15) is based on an error between a reference value and the approximated value of the field quantity strain. This is not the only field quantity that can be used to describe the error. Other choices of field quantities that are stress and strain energy. δer r or = ²i j − ²0i j

(3.24)

Strain norm Strain is a dimensionless value which describes the elongation per unit length of a part. In the ERM the use of this norm chosen because it enables observing the error in strain separately in all six directions, three normal components and three shear components. The strain norm is shown in Eq. (3.24) where ²i j is the reference value and ²0i j the approximated value. δer r or = σi j − σ0i j

(3.25)

Stress norm The second norm that is discussed is the stress norm. This norm is based on the error between the reference and modelling stress values. The stress norm describes the error between the six stress components, three


27

normal compenents σ11 , σ22 , and σ33 and three shear components σ12 , σ13 , and σ23 . In Eq. (3.25) the error is described by δer r or and is based on the reference values σi j and the values from the FE solution σ0i j . 1 U0 = (σ11 ²11 + σ22 ²22 + σ33 ²33 + σ12 ²12 + σ23 ²23 + σ13 ²13 ) 2 Z V U= U0 dV

(3.27)

δer r or = U −U 0

(3.28)

(3.26)

0

Strain energy norm The strain energy norm is based on the difference in strain energy for the reference values and the approximated values. The strain energy that is present in a body or element is calculated by integrating the strain energy density, equation (3.26), over the volume of the body or element as shown in Eq. (3.27). It describes the difference in strain energy contained in a volume rather than a distance between two values such as strain, or stress.

3.2.5. Disadvantages of ERM The ERM is a method which is very useful in reconstructing residual stress and strain fields. Although the method is useful there are quite some disadvantages and pitfalls to the method. An important aspect to consider when the ERM is applied on the SLM process is the dimensionality of the ERM. In literature the examples of ERM are numerous but are only applied to 1D and in some cases to 2D. It is observed that the extension to higher dimensions introduces errors. Research by Song et al. [25] showed that the results were qualitatively good but that the magnitude of the solution did not match the reference results. Therefore in the development of the ERM a study on the performance of the ERM is carried out. In the ERM a reference data set is used and a set of residual strain solutions. Generally these data sets contain data points that are not aligned. This means that interpolation techniques need to be used. The use of interpolation or extrapolation techniques is a source of error. To see whether the ERM works properly the use of interpolation or extrapolation techniques should be avoided. The goal of this project is to develop a model that predicts residual stress and warpage of SLM products. Reconstruction of residual stress fields has proven to be accurate in one dimensional cases. Thus, it is not certain that the ERM is able to reconstruct the displacement field of the SLM product. The contents of the function space have already been discussed but are a pitfall of the ERM. Although the ERM is implemented correctly the choice of function space can give inaccurate results. This is a problem that needs to be considered in developing the ERM for three dimensional problems.

4 SLM modelling In this chapter the developed models are discussed. The first model is the detailed model of the SLM process which is used as a reference model. The second model is the ERM model for which the results from the reference model are needed as input.

4.1. SLM reference model The literature review summarised in chapter 1 and internship at NLR as discussed in chapter 2revealed that the scanning strategy of the SLM process has a large influence on the residual stress and warpage of SLM products. In the layer by layer SLM models described in literature the scanning strategy is not included. Therefore, in this thesis a SLM model is developed which can capture the effect of scanning strategy. In this section the working principle of the detailed SLM model is described.

4.1.1. Modelling strategy The development of the SLM model is based on the modelling techniques described in literature and the conclusions of the internship at NLR. These conclusions suggest that the scanning strategy should be included in the SLM model without increasing the computational costs. In literature two types of modelling approaches are used to model the SLM process; the global and local approach. The global layer by layer approach describes the SLM process as a layer addition process and neglects the process history and how the layer is manufactured. The local melting pool approach is a way of manufacturing a layer which is suitable for the behaviour in the vicinity of the melting pool of the SLM process. The results of the internship at NLR showed that the layer approach was not able to capture the effect the scanning strategy has on the deformations of a SLM machined product. This conclusion leads to the objective to develop a SLM model which includes the effect of the scanning strategy. While being kept simple to avoid large computational costs. The SLM models in literature that use the melting pool approach are transient thermomechanical models. From a physical point of view many details are included, such as the laser power distribution, scanning speed, etc.. However modelling the scanning strategy using the melting pool approach is not convenient for several reasons. Computationally, the use of these models is very demanding because incremental iterative solution techniques are needed to solve non-linear problems. The other reason is also a computational issue which is concerned with the size of the domain of the problem. SLM products generally have a size which ranges from millimeters to several centimeters. The melting process in SLM occurs in a domain which has a size from 50 to 200µm. The large difference in those scales requires to use fine spatial discretisation for numerical solutions of the governing equations in the melting pool approach for the entire SLM product. The other question that needs to be answered is if it is necessary to have a detailed solution at the melting pool level. The objective here is to obtain a global description of the residual stress depending on the scanning strategy. Therefore having a detailed transient solution at the melting pool level is not required. In literature the SLM process is modelled on a global level by using a layer by layer approach. These models are thermomachanical and have two approaches for solving the thermal problem. In the SLM model by Mercelis & Kruth [17] the thermal problem is reduced to a thermal expansion model. This approach is the 29

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4. SLM modelling

same as the one used for the NLR model which has been described in chapter 2. In the models by Zaeh et al. [26, 27] and Papadakis et al. [19] a transient thermomechanical model is used. In the transient thermomechanical models a uniform heat source is imposed on the layer for which the thermomechanical BVP is solved. The model by Mercelis & Kruth [17] assumes a thermal strain which is uniform over the layer for which the thermomechanical BVP is solved. The transient and time independent thermomechanical models that use the layer by layer approach neglect the process history of the layer. Computationally the uniform thermal strain approach used by Mercelis & Kruth [17] is less expensive than the transient thermomechanical approach used by Zaeh et al. [27]. This is because the thermal strain approach only solves the MBVP whereas the thermomechanical approach solves the coupled transient thermomechanical problem. The layer by layer model containing the thermal strain approach by Mercelis & Kruth [17] is a method which contains a simple description of the thermal model and focuses on the mechanical effect of the thermal load. Computationally this type of model is interesting because only the mechanical boundary value problem needs to be solved. Therefore this modelling approach is used to develop the reference models of the ERM.

4.1.2. Working principle thermal strain approach The thermal strain approach is used as the deformation source in the SLM model. Fig. 4.1 schematically illustrates how the thermal strain approach works. The process starts at the left side of the figure with an initial continuum on which a thermal strain is imposed. The first step is solving the MBVP to obtain the equilibrium state of the block under the imposed strain. The second step is adding another block of material. The third step is changing the shape of the pristine block of material such that it is perfectly bonded to the already deformed continuum without inducing any stress. The blocks are reconnected and a thermal strain is imposed on the added block of material. The fourth step is solving the MBVP to obtain the deformations at equilibrium. As the process continues the third and fourth step are repeated until the line scan is finished and a line of material is build. The thermal strain approach described here is a general description which can be utilised for modelling the building process of a product. In the following section the use of the thermal strain approach in a line by line and point by point model are discussed.

Figure 4.1: Schematic overview of the thermal strain approach: The building process starts an initial continuum of material on which a thermal strain is imposed. The first step is solving the MBVP which results in a deformed continuum which is in equilibrium. The second step is creating a new block, this block is not connected to the deformed continuum. Therefore the block is reconnected without inducing stresses. The third step is imposing a thermal strain on this reconnected block. The fourth step is solving the MBVP to get to the deformed equilibrium of the continuum.

4.1.3. Line by line and point by point model In the models that use the layer approach entire layers are added each step. A comparison of the numerical and experimental results during the internship showed that this approach is currently unable to describe the effect of scanning strategy. Therefore two approaches are developed which build layers in a more realistic way. The first approach is termed line by line approach. In this approach a layer is built by adding lines of material reminiscent of laser scanning vectors. The line by line approach is schematically shown in Fig. 4.2b and is a refinement compared to the layer by layer approach shown in Fig. 4.2a. The line by line approach is based on the thermal strain approach which is described in 4.1.2. The layer is divided into scanning lines. In each addition step the geometry of the entire scanning line is added. On this line a uniform thermal strain is imposed which represents cooling and shrinking of the line. By assuming a uniform thermal strain for the line the transient deformation behaviour in the scanning line is neglected.

4.1. SLM reference model

31

The second variant is the point by point approach in which a layer is built by adding small volumes of material. In Fig. 4.2c the point by point approach is shown. The point by point approach is based on the thermal strain approach described in 4.1.2. With the point approach the layer is divided into small volume elements. In each addition step a volume element of material is added as described in section 4.1.2. This allows building of the layer according to an arbitrary scanning strategy. On each volume element a thermal strain is imposed which corresponds to the cooling and shrinking of the material upon cooling. With this approach it is possible to closely mimic the building process of SLM.

(a)

(b)

(c)

Figure 4.2: (a) Schematic description of the layer by layer approach of building a layer. (b) Schematic description of the line by line approach of building a layer. (c) Schematic description of the point by point approach of building a layer.

4.1.4. Material model The constitutive relations utilised to describe the field quantities and material behaviour is described in this section. The material that is used is Inconel 718. This material is frequently used in aviation industry and especially in SLM. In the NLR internship cantilevers have been built using Inconel 718. Therefore Inconel 718 is used as the material for the modelled geometries. In chapter 3 the mechanical behaviour of the material has been described as elasto-plastic with temperature dependent material properties and isotropic thermal expansion behaviour. The properties which were temperature dependent are the Young’s modulus, Poisson ratio, yield strength, and thermal expansion coefficient. Ideally this model should also be used in the SLM model. Unfortunately this material model leads to expensive computations as observed during the internship. The material nonlinearities require an iterative solver which in general requires 7 to 8 iterations for each solution. In the application of the SLM model, this procedure is repeated for 2500 or 6250 times for the detailed reference models. Including the plasticity would require much computational effort. The goal of this thesis work is a model which approximates the residual stress, strains, and deformations of an SLM product. Ideally a thermoelasto-plastic model should be incorporated but from a practical point of view it is decided to use a thermoelastic model. Leaving out the nonlinearities and plasticity in the model is a practical choice which reduces the computational effort. If in further research it is possible to expand the model nonlinearities and plasticity can be added without much problems, it will only affect the computation time of the model. From the above discussion it is concluded that the linear elastic material including thermal expansion is the most pragmatic choice. In a linear elastic material the Young’s modulus E and the Poisson’s ratio ν are constant. The material is assumed to be isotropic such that mechanical behaviour is the same in all directions. The SLM model also includes thermal expansion. For thermal expansion the linear expansion coefficient α of the material needs to be defined. In the case of an isotropic material the thermal expansion coefficient is a constant and the same in all directions. For the SLM model it is assumed that the thermal expansion model is isotropic. The material properties of Inconel 718 that are used are tabulated in Table 4.1. The material properties that are used are the ones at room temperature from the data used during the NLR internship, Fig. 2.1.

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Table 4.1: Material properties Inconel 718 at room temperature

E (GPa) 210

ν (-) 0.3056

α (K −1 ) 1.32e-5

4.1.5. Geometries To validate the ERM model that is developed in this thesis several geometries are used. The geometries are a single line scan, an entire layer, and a rectangular block consisting of ten layers. All these geometries are built on a deformable base plate which made up of Inconel 718, i.e. the identical material being printed.

Model basis In the SLM process the product is built by adding a layer of powder and processing it with a laser to build a single layer. Solidified layers have a thickness of 50 µm. Consequently, the layers in the numerical model are set to 50µm in undeformed configuration, i.e. before the thermal strain is imposed. In Fig. 4.3 the layers of the SLM product and the baseplate are shown. The layers are denoted by Ωn where the number n denotes the layer number. The layers are built on top of the baseplate, the baseplate is denoted by Λ. The shape of the layer Ωn is a rectangular prism and corresponds to the undeformed state of the layer. The volume of the layer in this state corresponds to the volume of the material when it is at melting temperature. The base plate on which the product is built during the SLM process is included in the SLM models. The NLR

Figure 4.3: Schematic overview of the contents of a SLM model. At the bottom the base plate Λ is positioned, this block of material is made of the same material as the product that is build. On top of the base plate a block of material is build which consists out of multiple layers Ωn .

model assumed the base plate to be an undeformable body. This assumption is not accurate enough because experimental work shows that the base plate deforms when a product is built. Therefore in the models used for this thesis work the base plate is included. The base plate is not included entirely because the base plate is far larger than the geometries that are modelled. Modelling the entire base plate would increase the number of elements which increases computation time. Moreover, Saint-Venant’s principle implies that deformation alternates away from the base plate and part interface. Recall that Saint Venant’s principle states that the effect of a load source decreases when the distance between the location of the source and the point of interest increases. The size of the base plate is chosen such that the effect on the edges of the base plate can be neglected. This size differs for each geometry that is used. The base plates are brick shaped. This means that the base plate has six surfaces: A top, bottom, and four side surfaces. The SLM product is built on the top surface. Besides the connection to the geometry, the top surface is free. In the SLM machine the base plate is fixed to the machine by bolts. To mimic the fixed state of the base plate, boundary conditions are applied to the bottom and four side surfaces. These five surfaces are constrained in translation and rotation for all directions. This set of boundary conditions is used for the base plate of each geometry. These boundary conditions can be imposed even though the size of the base plate is decreased. The size of the base plate is decreased such that no stresses will arise near the edges of the base plate.


33

Line on base plate geometry The geometry of the line on base plate model is shown in Fig. 4.5a. In this figure the dimensions of the line and base plate are shown. The line has a height of 50 µm, width of 50 µm, and a length of 2500 µm. The base plate has a height of 250 µm, width of 600 µm, and a length of 3200 µm. The line on base plate geometry is built using the point by point approach in which cubic blocks of 50µm are added each time. The total number of blocks that is built is 50. The domain of the line is (-1.25,1.25) which is along the x 1 axis. Building of the line starts at x 1 =-1.25 mm. After the geometry is build the residual stresses, total and elastic strains, strain energy, and displacements are extracted from the FE model. The extracted data is used as a reference for the ERM.

Layer on baseplate geometry The layer on base plate model is schematically shown in Fig. 4.5b. The geometry consists of a layer and a base plate. The layer has a height of 50 µm and both width and length are 2500 µm. The base plate in this geometry is shown in the lower right corner of the figure. The base plate has a height of 250 µm and a width and length which are both 3400 µm. In 4.1.3 the three approaches used to build a layer are described. The layer geometry is used to investigate the effect of these approaches on the stress, strain, and strain energy distributions in the layer after processing. With the layer by layer approach the entire layer is added and the MBVP is solved. The domain of the layer is defined as (-1.25,1.25; -1.25,1.25)mm for (x 1 , x 2 ). In Fig. 4.4a the domain is shown on which the layer is build. For the line by line approach the layer is divided into 50 lines with a length of 2500µm and a height and width of 50µm. Building of the lines starts at x 2 = -1.25 mm and the first scanning line is defined along the entire domain in x 1 direction. In Fig. 4.4b this is visualised by an initial line and the building direction denoted by red arrows. In the point by point approach the layer is build by adding cubic blocks of 50µm. The scanning strategy that is followed in the point by point approach is visualised in Fig. 4.4c. Building with the point by point approach starts at x 1 =-1.25 mm, and x 2 =-1.25 mm and proceeds building along the x 1 axis. When an entire line is finished the building continues again at x 1 =1.25 mm and x 2 =-1.2 mm. This proceeds until the entire layer is finished. After the layers are build using the different approaches the residual stresses, total and elastic strain, strain energy, and displacements of the layers are extracted. The results for the point approach are used as reference results for the ERM model of the layer geometry.

(a)

(b)

(c)

Figure 4.4: (a) Domain on which the entire layer is build in one step. (b) Visualisation of the scanning strategy to build the layer geometry by using the line by line approach. (c) Visualisation of the scanning strategy for which the layer is build using the point by point approach.

10 layers on baseplate geometry The 10 layers on baseplate model consists of a baseplate on which ten layers of material are built. In Fig. 4.5c an overview of the dimensions of the geometry is given. The layers in this geometry all have the same dimensions as shown in the figure. The layers have thickness of 50 µm and a width and length which is equal to 1250 µm. The total height of the ten layers is 500 µm. The bottom right figure shows the dimensions of the baseplate. The baseplate has a height of 250 µm and a width and length which is equal to 1250 µm.

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4. SLM modelling

The ten layer geometry is built using the point by point approach. The domain of each layer is (-0.625,0.625; -0.625,0.625) mm which is build according to the same strategy as in Fig. 4.4c only the starting point is now x 1 =-0.625 mm and x 2 =-0.625 mm. This scanning strategy is used for each layer in the ten layer geometry. In each layer 625 blocks of material are build with a size of 50µm. In total the amount of blocks that are build is 6250. After the ten layer geometry is built using the point by point approach, data is extracted from the FE model. The extracted data are the residual stresses, total and elastic strains, strain energy and displacements of the geometry.

4.1.6. Reference models in FE package The developed SLM model is implemented in the Abaqus FE software package. The Abaqus FE software package is used because it is the package that has been worked with during the NLR internship. In this subsection the implementation of the SLM model is explained. The implementation of the SLM model in the Abaqus FE software consists out of seven steps. The steps are listed below. 1. Create geometry 2. Partitioning and assigning sets 3. Initial and boundary conditions 4. Mesh generation 5. Deactivate material sections 6. Sequence of point by point addition 7. Data extraction

Step 1: Create FE model The first step is creating a geometry that is going to be built. This can be done by drawing the geometry in the Abaqus CAE environment. Another way is scripting the entire drawing process in python and let Abaqus run the script to create the geometry. When the geometry is described the material properties of the material of which the geometry consists are defined.

Step 2: Partitioning and assigning sets The second step is preparing the modelled geometry for the SLM procedure. Therefore the modelled geometry needs to be partitioned into sections which represent layers, lines, or points. In Abaqus a partitioning tool is available. After partitioning each created section is assigned to set with a specific name. This partitioning and set assignment is scripted in python such that Abaqus does it automatically when the script runs in Abaqus.

Step 3: Initial and boundary conditions The third step is applying the initial and boundary conditions to the geometry. The initial conditions are the temperature of the geometry that needs to be build and the base plate. The boundary conditions have been described in section 4.1.5 for each geometry and are imposed as such.

Step 4: Mesh generation Step four is mesh generation. The geometry that is described in Abaqus with given material properties representative of Inconel 718 and prescribed boundary conditions is partitioned. Subsequently, the domain is discretised (meshed) by dividing the geometry into small pieces, i.e. finite elements (FE). To mesh the geometry the type of finite elements needs to be chosen and the approximate size of the elements. In the case of the SLM model the elements that are used are linear brick elements (C3D8R) which are approximately 50x50x50µm. Each brick element consists out of eight nodes and due to reduced integration have only one integration point. The choice for linear elements is based on the assumption that there are only small deformations. This is realistic because the thermal strain imposed is between 0.002-0.005 (-). Also FE models with higher order elements such as quadratic elements take considerable more time than linear models. Another reason to use linear elements is that the behaviour of the material is linear elastic and the thermal expansion is also linear.


35

(a)

(b)

(c) Figure 4.5: Geometries with the corresponding dimensions; (a) Line on baseplate geometry. (b) Layer on baseplate geometry (c) 10 Layers on baseplate.

Step 5: Deactivate material sections The FE model is ready for use after the mesh generation. The starting point of the SLM procedure is the deactivation of all material sections of the geometry that needs to be built. In Abaqus this is performed by the Model Change tool.

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4. SLM modelling

Step 6: Sequence of section addition In step six the building sequence is defined. With the model change tool the first material section is activated. This section is at melting temperature and is cooled down to build chamber temperature. This cooling process is imposed on the material section by a thermal strain. The equilibrium state is found by solving the MBVP which results in the deformed and residual stress state of the material section. A new material section is activated and the whole process is repeated until the entire geometry is built.

Step 7: Data extraction After the entire structure is built, the data is extracted from the solution file (ODB file). The results are extracted by writing a python script which opens the ODB file, extracts the desired data, and writes this data to csv files. Csv files contain comma separated values which can be easily imported into Matlab which is used for post-processing.

4.2. Eigenstrain reconstruction method In section 3.2 the formulation of the eigenstrain reconstruction method is explained. This section describes how the ERM is implemented in a Matlab and Abaqus environment. The implementation of ERM consists of four steps. The first step is creating an FE model of the geometry and processing the reference data. The second step is calculating all the residual strain solutions for each eigenstrain component. The third step is using the reference and residual strain solutions to calculate the coefficients of the eigenstrain components. The fourth step is reconstruction of the eigenstrain with the found coefficients. The fifth step is solving the MBVP for the found eigenstrain to validate the results. 1. FE model of geometry and processing reference data 2. Solving MBVP for eigenstrain components to obtain residual strain solutions 3. Calculation of eigenstrain coefficients 4. Reconstruction of eigenstrain distribution 5. Validation of eigenstrain After the eigenstrain is reconstructed the relevant data such as residual stress, total and elastic strain, strain energy, and displacements are extracted from the solution. This data is used to validate the ERM and evaluate the error between the eigenstrain approximation and the reference data.

Step 1: FE model of geometry and reference data The first step in implementing the ERM is creating an initial geometry for the ERM. The geometries used in the ERM are the same as the geometries used in the SLM model. In the ERM the geometries described in section 4.1.5 are used. On these geometries the eigenstrain components are imposed in step 2. It is assumed that the calculated eigenstrain source mimics the deformation behaviour of the SLM model. Therefore it is needed to start in with the same initial geometry. An additional advantage of using the same geometry is that the same FE mesh can be used. The advantage of using the same FE mesh is that the data points of the reference data and the data produced with the ERM are aligned. Because the data points are aligned no interpolation or extrapolation techniques are needed to align the data. The material behaviour considered is linear elastic which is representative of Inconel 718. For the Young’s modulus and the poison ratio the same values are used as in the SLM model, see Table 4.1. In the ERM model a thermal expansion model is used to impose the eigenstrain components on the geometry. The used thermal expansion model needs some further explanation which is given in subsection 4.2.1. The geometries that are used consist of the base plate and the product that is built on top of the base plate. The boundary conditions imposed on the base plate are identical for the geometries in the SLM model. There is no translation or rotation on the bottom and side surfaces of the base plate. It is necessary to use the same boundary conditions for the ERM because the MBVP for the eigenstrain components must be solved under identical conditions as in the geometry build with the SLM model. The reference data set contains stress or strain distributions in the geometry. The reference data for the geometry used in the ERM model is provided by the SLM model. The geometry is build by the SLM model and after the build is finished the results are extracted for the finished product. This data is put in the ERM

4.2. Eigenstrain reconstruction method

37

model and is used as the reference. The performance of the ERM depends on the resolution of the reference data. More data points in the reference data results in the better the approximation of the eigenstrain is.

Step 2: Calculation of residual strain solutions for eigenstrain components Step 2 in the implementation of the ERM is the calculation of the residual strain solutions for the eigenstrain components. In the previous step, a FE mesh of the geometry is created. The eigenstrain components ζiJ j are imposed onto this mesh through a thermal expansion field. How the thermal expansion field works is explained in subsection 4.2.1. After the thermal expansion field is applied, the geometry on top of the base plate is cooled down from 1K to 0K . In this way the thermal strain is defined on the geometry. Then the MBVP is solved which results in the equilibrium state of the geometry. This procedure is repeated for each eigenstrain components ζiJ j that is present in the function space Σ. For each solution the stress, strain, and strain energy data is extracted. This data is extracted because it is the data needed for the stress, strain, and strain energy norm. The function spaces that are used for the three geometries is different. For the line geometry the function space consists of one dimensional functions which depend on x 1 . The function space for the layer geometry is two dimensional and is defined in x 1 and x 2 . The function space for the ten layer geometry is defined in three dimensions and depends on x 1 , x 2 , and x 3 . For each geometry the contents of the function space is described in a dedicated section; line in 4.2.2, layer in 4.2.3, and ten layer in 4.2.4.

Step 3: Calculation of eigenstrain coefficients When all residual strain solutions S iJ j are calculated by means of FE calculations, the data on stresses, strains, and strain energy for these solutions is extracted at the integration points of the elements. The integration point values are used because the value at this point represents the state of the entire element. Nodal point values in the Abaqus FE software are calculated by means of extrapolation and interpolation of the data at the integration points. The data that is extracted is used to determine the coefficients by solving the system of equations described in chapter 3. The system of equations is stated as follows, Ac = b,

(4.1)

where the collected data is used to construct the system of equations described in Eq. (4.1). The residual strain solutions are used to define the matrix A and both the residual strain solutions and reference solution are used to construct vector b. The contents of matrix A and b are given in Eqs. (3.18) and (3.20). For each of the described geometries there is a special structure in the matrix A and vector b. The shape of these structures is discussed for each geometry; line in 4.2.2, layer in 4.2.3, and ten layer in 4.2.4. The matrix A is a square (NxN) matrix where N is equal to the number of basis functions that are in the function space. The vectors b and c always have a length which is equal to the number of basis functions. When the system of equations is fully defined the coefficients c J from vector c are obtained by solving the system. Step three is entirely defined in Matlab. The data from the FE solutions and the reference solution is imported into Matlab. The data is then used to build the matrix A and the vector b. To solve the system of equations the matrix A is inverted and multiplied with vector b. This results in the coefficients c J .

Step 4: Reconstruction of eigenstrain distribution The reconstruction of the eigenstrain distribution is done in the following equation, ²∗i j = c 1 ζ1i j + c 2 ζ2i j + ... + c J −1 ζiJ −1 + c J ζiJ j , j

(4.2)

where ²i j is the eigenstrain, c J the coefficients, and ζiJ j . The coefficients c J that are calculated in step three are used to construct the final eigenstrain distribution. In Eq. (4.2) the the coefficients c J which are multiplied with their respective eigenstrain components ζi j . With this step the eigenstrain reconstruction is completed.

Step 5: Validation of eigenstrain

The final step in the ERM is the validation step. In the validation step the found eigenstrain ²∗i j is imposed on the geometry and the MBVP is solved. The solution of the MBVP is the predicted residual stress field by the reconstructed eigenstrain.

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4.2.1. Thermal expansion mechanism in ERM Each basis function in the function space is an eigenstrain component ζiJ j . Each of those eigenstrain components need to be imposed on the domain and subsequently the MBVP is solved to obtain the elastic strain solution S iJ j . This procedure is straightforward but unfortunately the implementation is not that simple. In Abaqus it is not possible to directly prescribe the eigenstrain component ζiJ j in a specific direction i j as an expansion field unless the user subroutine UEXPAN is used. This user subroutine UEXPAN is able to directly prescribe the eigenstrain component on the domain. However, the use of this user subroutine is not possible at TU Delft. Therefore a different way of defining a spatially dependent expansion field on the domain is needed. This problem has been encountered by chopping the geometry into small sections and assigning material models to each section. For each material section Ωm,n the anisotropic thermal expansion coefficients αi j can be defined. By mapping the eigenstrain component on the thermal expansion coefficients in each section the expansion field can be imposed on the geometry. This has been visualized in Fig. 4.6. To explain the procedure the line on base plate geometry is used which is shown on the left. To the right of this geometry a side view in the x 1 , x 3 plane of the line geometry is shown. This line is divided into square sections where the blue dots represent the section centers x c . Each of those square sections contains a material model for which the thermal expansion coefficients are defined by mapping the corresponding value of the eigenstrain component onto the thermal expansion coefficients. This makes imposing the eigenstrain component ζiJ j on the geometry possible. One important aspect of defining the expansion field is mapping of the values from the eigenstrain component to the thermal expansion coefficient. To map the value of an eigenstrain component to the thermal expansion coefficient the center locations of the sections are used. The thermal expansion coefficient αi j of section Ωm,n is defined by the function value of the eigenstrain component at the center location x c of the material section Ωn,m . Hence, the function value at the center location determines the thermal expansion coefficient for the whole section. This means that the thermal expansion coefficient is assumed to be uniform over the section. The choice to define the thermal expansion coefficient by this procedure is justified by the chosen size of the material sections. For the line, layer, and multilayer geometry there are 50, 2500, and 6250 material sections with a size of 50x50x50µm respectively. These amounts of material sections are suitable to give a high resolution description of the expansion field. Increasing the material sections in size decrease the resolution of the mapped expansion field which will result in less accurate results. In such a setting it is recommended to look at different ways of adopting the thermal expansion coefficient from the basis function. Other methods are integration of the expansion coefficient over the volume of the material section, using a Riemann sum to find the value, or an interpolation technique. After the thermal expansion coefficients are defined by mapping the function values of the eigenstrain components the geometry undergoes a uniform cooling step from 1K to 0K . To conclude, for all sections in the geometry a material model is defined. In this material model the Young’s modulus and Poisson ratio are the same for every section and have the values in Table 4.1. Each section has its own thermal expansion coefficients which are defined by mapping the function values of the eigenstrain components onto the thermal expansion coefficients.

4.2.2. ERM - line on baseplate The ERM model of the line on base plate is used to evaluate ERM parameters for the ERM. The evaluated parameters are the error norm, anisotropic or orthotropic ERM, number of basis functions (NoBF), and radius of the basis function. The anisotropic ERM reconstructs the eigenstrain distribution for both the normal and shear directions, hence one gets the eigenstrains ²∗11 , ²∗22 , ²∗33 , ²∗12 , ²∗23 , and ²∗13 . In the orthotropic ERM the eigenstrain distribution is only reconstructed for the normal directions, so one gets the eigenstrains ²∗11 , ²∗22 , and ²∗33 . The difference between these two methods is that the anisotropic ERM has three more eigenstrain distributions that can be used to reconstruct the reference stress field. By using the anisotropic ERM one uses twice the amount of eigenstrain components as for the orthotropic ERM because six and three eigenstrain distributions are used respectively. In literature on the ERM [1, 8, 11],the used error norm in the functional is the strain norm as described in Eq. (3.24). Other options are the stress and strain energy norm as described by Eqs. (3.25) and (3.28). The


39

Figure 4.6: This figure shows how the strain fields are translated from the basis functions to the FE model. On the left the line on base plate geometry is shown. To the right of the geometry a front view of the line is shown. This line geometry is divided into square sections for which the center coordinate is defined. The magnitude of the eigenstrain component ζi (x 1J ) at the center of the material section determines the thermal expansion coefficient α11 of the material. This is visualised by the arrows from the eigenstrain component to the centers of the sections.

choice of error norm which is used for the layer and ten layer geometry is based on the performance of the options on the line geometry. In the function space the one dimensional variant of the RBF described in chapter 3 is used. With the ERM on the line geometry the influence of radius and the number of basis functions on the solution of the ERM is evaluated. The results for the ERM line model that are extracted from the FE model are stress, total and elastic strain, strain energy, and displacement. The error in between the reference and ERM results indicate how well the ERM method performs.

Function space The function space for the line geometry is one dimensional because the geometry is a line for which the integration points are positioned on the x 1 axis. The basis functions are defined for each direction; 11, 22, 33, 12, 23, and 13. For each direction the number of basis functions, radius r , and center location x c is the same. So if there are five basis functions in 11 direction the total number of basis functions is thirty. The center locations of the RBFs are evenly distributed over the domain (-1.225,1.225) mm. In Fig. 4.7b the distribution of basis functions in 11 direction is shown. In this function space the same basis functions are used for each direction. This choice is made because the goal of this geometry is to gain insight in the behaviour of the ERM under changing properties. Directly starting to design function spaces for specific directions is not possible without insight on the behaviour.

Structure of system of equations h J O line = [ζ111 . . . ζ11 ]

...

[ζ1i j . . . ζiJ j ]

...

J [ζ113 . . . ζ13 ]

i

(4.3)

For each ERM of a geometry the basis functions are ordered in a specific way. The order of these basis functions determines the structure of the system matrix A and vector b. The order of the basis functions for the line geometry is shown in Eq. (4.3). It starts with the basis functions in 11 direction and proceeds to the basis functions in 13 direction.  1 1 1 1 S 11 S 11 + ... + S 13 S 13 M X  ..  .

i =1

N 1 N 1 S 11 + ... + S 13 S 13 S 11

··· .. . ···

 1 1  1 P 1 P  1 c (e 11 S 11 + ... + e 12 S 13 ) S 11 S 11 + ... + S 13 S 13 M X   ..    .. ..  .  =   . . i =1 P P P N P N P (e 11 S 11 + ... + e 12 S 13 ) S 11 S 11 + ... + S 12 S 12 c

(4.4)

In total there are P basis functions in the function space, so the matrix A has a size P xP and the vector b a size of P x1. In shape and contents of the matrix A and vector b are shown in Eq. (4.4). As can be seen the

40

4. SLM modelling

1

1

0.8

0.8

← ζ2ij (x1)

← ζ3ij (x1)

← ζ4ij (x1)

← ζ5ij (x1)

A (-)

0.6

A (-)

0.6

← ζ1ij (x1)

φ(||x i||) 0.4

0.4

0.2

0.2

r=||x i|| xc

0 -1

-0.5

0

0 0.5

1

-1

-0.5

x1 (mm)

0

0.5

1

x1 (mm)

(a)

(b)

Figure 4.7: (a) Figure containing a description of the one dimensional RBF with a description of the radius r and the center location x c . (b) Figure containing an example of the distribution of RBFs with evenly distributed center coordinates in the domain (-1.225,1.225) mm and a radius of r =1 mm.

matrix A and vector b have values in each entry.  A 1111 A1122 A 1133 A  2211 A 2222 A 2233   A 3311 A 3322 A 3333 A=  A 1211 A 1222 A 1233   A 2311 A 2322 A 2333 A 1311 A 1322 A 1333

A 1112 A 2212 A 3312 A 1212 A 2312 A 1312

A 1123 A 2223 A 3323 A 1223 A 2323 A 1323

 A 1113 A 2213    A 3313   A 1213   A 2313  A 1313

  B 11 B   22    B 33  B =  B 12    B 23  B 13

(4.5)

The order of the basis functions determines the internal structure of the matrix A and vector b. In the case of the line model the matrix consists of submatrices A i j kl and subvectors b mn as shown in Eq. (4.5). In submatrices A i j kl the indices i j represent the basis function in direction i j and indices kl represent the basis functions kl . On the diagonal of matrix A this results in matrices for which the two solutions are those from the basis functions in the same direction. The off diagonal terms represent the coupling between the basis functions in different directions. In subvector b mn the indices mn represents the basis function in mn direction. The coupling term is interesting to evaluate. The coupling terms arise when the MBVP is solved for an imposed strain in any direction. Taking the 11 as the imposed strain will result in residual strain solutions for J J all directions; S 11 ,...,S 13 . This is logical because the mechanical behaviour is coupled in each direction by the Poisson ratio.

4.2.3. ERM - layer on base plate The ERM model for the layer geometry is an expansion of the line model to a layer geometry. The results on anisotropic and orthotropic, number of basis functions, radius, and error norm from the line geometry are incorporated in this model. Due to the expansion to the layer geometry the function space becomes two dimensional. This requires that the two dimensional variant of the RBF needs to be used. The purpose of the ERM model for the layer on base plate geometry is to evaluate the performance of a function space which contains an internal and a boundary grid. On the internal grid the number of basis functions is varied and for the boundary grid the radius of the basis function on the boundary is varied. The function space which yields the best approximation is used as the function space for the ten layer geometry.

Function space In the ERM model for the layer geometry the function space contains two specific functions sets. These two function sets are called the internal grid and the boundary grid. The purpose of the internal grid is to capture the bulk stresses in the geomerty. The boundary grid is used to capture the large gradients that appear at the edges. The RBFs in the internal grid are two dimensional radial basis functions which are defined in the x 1 , x 2 plane and have a radius which is equal to 2 mm (2000 µm). The purpose of this internal grid is to capture the bulk value of the stresses that are present in the reference set. The center locations of these radial basis


41

functions are shown in Fig. 4.8a as blue dots and accompanied by a basis function ζii j . The RBFs in the boundary grid are one dimensional basis functions which are defined along the edge on

1

A (-)

0.8 0.6 ζJij (xi)

0.4 0.2 0 -1

1 0

x1 (mm)

0 1

(a)

-1

x2 (mm)

(b)

Figure 4.8: (a) Example of the internal grid with 16 RBFS with a radius r . (b) Visualisation of the two dimensional variant of the RBF with center location x c =(0,0) and radius r =1 mm.

which they are positioned. This function set is used to capture the sharp peak values of stress and strain which appear near the edges. The locations of these RBFs are shown in Fig. 4.9a. The functions are represented by the blue lines and the functions ζii j .

1

A (-)

0.8 0.6 ζJij (xi)

0.4 0.2 0 -1

1 0

x1 (mm) (a)

0 1

-1

x2 (mm)

(b)

Figure 4.9: (a) Position of the basis functions in the boundary grid. (b) Visualisation of a RBF at the boundary. This is the one dimensional RBF but continued allong the edge.

42

4. SLM modelling

Structure of system of equations In the ERM model for the layer geometry the function space is expanded from one to two dimensions. The internal structure of the matrix A and vector b for the two dimensional function space is the same as for the line geometry shown in Eq. (4.5). The only difference with the line geometry function space is that the functions are spatially dependent in two directions. The numbering of the basis functions is the same as shown in Figs. 4.8a and 4.9a. The first basis function is located at x c =(-1.225,-1.225) from which the other basis functions are numbered until the end of the row x 1 =(1.225,-1.225) mm is reached. This is followed by a shift in x 2 direction and than the numbering of the next row starts. This continues until all functions are numbered. The boundary functions are numbered after the layer is numbered. After the numbering for 11 direction is done the basis functions for the 22 direction are numbered. This numbering scheme leads to the characteristic structure in each submatrix as described below, " A i j kl =

A iI Ij kl

A Bi jIkl

A iI Bj kl

A Bi jBkl

#

bI b mn = mn B b mn ·

¸ (4.6)

where the matrix A i j kl consists out of four submatrices as shown in Eq. (4.6). Matrix A iI Ij kl contains the multiplication of the residual strain solutions for the eigenstrain components in the internal grid. In matrix ABB the multiplication of the residual strain solutions for the eigenstrain components in the boundary grid are ijkl present. The other two matrices A iI Bj kl and A Bi jIkl contain the coupling between the solutions for the boundary grid and the internal grid.

4.2.4. ERM - Multilayer on baseplate The ERM model of the multilayer geometry expands the function space from two to three dimensional because the height becomes a factor. The purpose of this model is to evaluate the performance of the ERM in a three dimensional setting. T The ERM model of the multilayer geometry makes use of the same function space as the layer ERM but has that function space defined for all layers in the model. This adds up to ten times the amount of basis functions which were originally in the function space. The dimensions of the 10 layer model are smaller than for the layer model because the reference calculation for the 10 layer model would take too much time.

Function space The geometry of the multilayer model consists of ten layers, the shape of the layers in this model is the same as for the layer geometry. The only difference is the size, the layers in the multilayer model are 1250x1250x50 µm, one fourth of the size of the layer in the layer geometry. The insights gathered from the ERM model of the layer geometry lead to the following function space. The internal grid consists out of sixteen RBFs with a radius r =2 mm. These RBFs are evenly distributed over the domain of the layer which is (-0.6,0.6; -0.6,0.6) mm. The boundary grid consists out of four RBFs which have a radius r =0.1 mm. This function space consists out of 20 eigenstrain components for each direction. In total this results in 120 eigenstrain components for each layer. It is assumed that the function space defined for the first layer is also able to describe the behaviour in remaining nine layers. This assumption is made because all ten layers are build with the same scanning strategy. This will result in similar residual stress fields in the layers. In total the number of eigenstrain components in the function space is 1200.

Structure of system of equations The expansion of the layer model to an accumulation of layers changes the internal structure of the matrix A and vector b. The order of the basis functions in the function space is divided into sections for each layer. Each layer is numbered according to the scheme and structure that is used in the ERM model for the layer geometry.  1   1,1 b A A 1,2 . . . A 1,9 A 1,10 2,2 2,9 2,10   b2   A 2,1 A . . . A A      .   . .. .. ..  ..  .   . (4.7) b = A= .  .   . . . .   9   9,1 9,2 9,9 9,10 b   A A ... A A b 10 A 10,1 A 10,2 . . . A 10,9 A 10,10

4.3. Error evaluation

43

(a)

(b)

Figure 4.10: (a) Internal grid for the multilayer model. (b) Boundary grid for the multilayer model.

The internal structure of the matrix A and vector b becomes a set of submatrices which are denoted by A i , j . The indices i and j represent the layer number. For the diagonal the submatrices A i , j contain the multiplications of the residual strain solutions for the the i t h layer. The off diagonal terms show the coupling between the residual strain solutions for two different layers. The internal structure of all submatrices is the same as described for the layer in Eq. (4.6). The subvectors b i contain the multiplication for the i t h layer with the reference results ²e .

4.3. Error evaluation Interpretation of results is an important aspect of evaluating the performance of a model. The results from the reference and ERM models are evaluated by means of the characteristics of the results and the error that is made between the reference and ERM models. Two types of error descriptions are applicable for every error set, the absolute and relative error. The absolute error is described as the absolute difference between the reference value and the approximation. The absolute error gives the magnitude of the error between reference and approximation results. The relative error is the absolute error divided by the reference value. This type of error gives an indication of the ratio of the error and the reference value at a data point. The relative error also used to describe the error percentage at a certain data point by multiplying the relative error with 100%. For small error sets the absolute and relative error provide all the information that is needed for a good evaluation. When the error sets get larger it becomes difficult to evaluate the errors using the absolute and relative error. Methods to evaluate large error sets resort to statistical methods using probability functions or residual error analysis. To use the statistical methods the error set must meet the requirements of the statistical methods. The example mentioned at the beginning of this section requires that the error set is normally distributed. In the case of the ERM method it is desired to have an error analysis tool which is valid for all geometries and its mechanical boundary value problems. The geometry of the problem influences the size and shape of the error set. This means that while for one geometry the error set is normally distributed it does not mean that the other set is also normally distributed. This uncertainty makes the use of statistical methods for error evaluation of the ERM method not robust and therefore not desirable. M SE =

N X i =1

(²ei j − S i j )2 r

R M SE = N R M SE =

M SE N

R M SE max(²ei j ) − mi n(²ei j )

(4.8)

(4.9) (4.10)

44

4. SLM modelling

When statistical methods are not suitable one can use residual error analysis. In residual error analysis the total error made by a model is evaluated. Eq. (4.8) describes the mean squared error (MSE) which is the sum of squares of all errors in the error set. This error gives information on the total error made between the reference results and the approximation. The root mean squared error is the root of the MSE and is shown in Eq. (4.9). This error gives an indication of the average error in the data points. This can be interpreted as an average absolute error of the error set. The last error that is described is the normalized variant of the the RMSE. In Eq. (4.10) this error is described, the error is the RMSE divided by the difference between the maximum and minimum value of the reference set. The result is a relative error which can be used to describe the error as a percentage. For the error evaluation between the reference and ERM models the use of residual error analysis is used in combination with the evaluation of local maximum and minimum errors by absolute and relative errors. The residual error analysis is suitable because the working principle is independent of the size and shape of the error set. The absolute and relative error are used because they can be used to locate regions of the domain where large errors occur.

5 Results & Discussions Chapter 5 contains the results of the selective laser melting (SLM) model described in section 4.1 and the eigenstrain reconstruction method (ERM) model elaborated in section 4.2. The SLM and ERM model are applied on the three geometries described in chapter 4. The results for the line geometry are discussed in section 5.1. For the layer geometry the results are presented in section 5.2. Finally, the results for the ten layer geometry are given in section 5.3. In the sections 5.1, 5.2, and 5.3 the results for the reference SLM model are discussed first. This is followed by the results for the ERM model. At the end of each of these sections the results are discussed.

5.1. Line geometry This section contains the results for the line geometry. First the results for the reference calculation on the line geometry are discussed. This is followed by the results for the ERM line geometry investigation in section 5.1.2. In this section the effect of ERM parameters on the ability of ERM to reconstruct the predictions by the SLM reference model is discussed.

5.1.1. Reference results line geometry The reference results for the line geometry are obtained by the point by point approach. The point by point approach is used to build a geometry point by point as described in section 4.1.2. In Fig. 5.1 the von Mises stress σVM , effective strain ²eff , and the elastic strain energy U are shown for the line on base plate geometry. All three results have similar shapes, a nearly constant stress, strain, and strain energy level in the center section. In the vicinity of the edges the stress, strain, and strain energy goes down in the direction of zero since at the outer edges of the line the surfaces are traction free. The shape of the solutions for stress and strain are similar because the linear elastic material depicts a linear relation between stress and strain. The elastic strain energy has a similar shape to the stress and strain curves because it is calculated with the strain and stress components as input. These similarities are expected for the other ERM results and therefore only the solutions for the von Mises stress will be visualized. The solutions for the effective strain and strain energy can be found in appendix A.

5.1.2. Results ERM line geometry The ERM line geometry is used to look at the effect of ERM parameters on the ability to reconstruct the reference stress fields. Based on the description of the ERM in section 3.2 several options for the ERM are investigated. As mentioned in section 4.2 the eigenstrain components are imposed on the domain through a thermal expansion model. In the ERM one can choose to impose eigenstrain components through orthotropic (only the normal directions) or anisotropic (normal and shear directions) thermal expansion by defining eigenstrain components for the available directions. By evaluating the results for both thermal expansion models one can identify which one predicts the residual stress field better. The second option consists of investigation the effect of the parameters of the eigenstrain components on the ability of the ERM to reconstruct the reference stress field. The eigenstrain components that are used are the radial basis functions (RBF) described in chapter 3 and 4. The parameters that are investigated are 45

46

5. Results & Discussions

700

4

600

3.5

150

3

400 300

100

2.5

U (J)

ǫ eff (-)

500

σ VM (MPa)

×10-3

2 1.5

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100

0.5

0

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0

x1 (mm)

x1 (mm)

(a)

(b)

0.5

1

-1

-0.5

0

0.5

1

x1 (mm)

(c)

Figure 5.1: (a) Von Mises stress σVM stress distribution in the line geometry. (b) Effective strain ²eff in the line geometry. (c) Elastic strain energy U in the line geometry. S, Mises (MPa) (Avg: 75%) 700 642 583 525 467 408 350 292 233 175 117 58 0 X3 X2

X1

(a) S, Mises (MPa) (Avg: 75%) 700 642 583 525 467 408 350 292 233 175 117 58 0 X3 X2

X1

(b) Figure 5.2: (a) Figure of the deformed configuration after the line is build using the SLM model. (b) Deformed configuration of the geometry after the eigenstrain is imposed on the geometry.

the radius r of the RBFs and the number of RBFs (NoBF). The third option that is investigated is the error norm that is used in the functional H which is described in Eq. (3.15). The error norms that are evaluated are the strain, stress, and strain energy norm described in section 3.2.4. After elaboration of the results on the options for ERM the deformations caused by the reconstructed eigenstrain and the deformations of the reference model are compared. The results for the ERM line geometry end with an evaluation of the behaviour of the eigenstrains themselves. Anisotropic vs orthotropic ERM results The anisotropic ERM reconstructs the eigenstrain distribution for both the normal and shear directions, hence one gets the eigenstrains ²∗11 , ²∗22 , ²∗33 , ²∗12 , ²∗23 , and ²∗13 . In the orthotropic ERM the eigenstrain distribution is only reconstructed for the normal directions, so one gets the eigenstrains ²∗11 , ²∗22 , and ²∗33 . The

5.1. Line geometry

47

difference between these two methods is that the anisotropic ERM has three more eigenstrain distributions that can be used to reconstruct the reference stress field. By using the anisotropic ERM one uses twice the amount of eigenstrain components as for the orthotropic ERM because six and three eigenstrain distributions are used respectively. The results for these two approaches will show if including the eigenstrain distributions in shear directions is necessary to obtain accurate results. The results of the approaches are evaluated by visual inspection of the solutions and comparing the solutions with the reference solution. For the ERM solutions the normalized root mean square error (NRMSE) percentage and the maximum error percentage is calculated w.r.t. the reference solution. The errors are described in section 4.3. This is performed for the quantities; von Mises stress σVM , effective strain ²eff , and the strain energy U which have been described in section 3.1. The ERM calculations are performed with five radial basis functions equally distributed over the domain (−1.225, 1.225) mm. The radius r of the radial basis functions is varied between 0.25, 0.5, 1, 2, and 5 mm. The identical range of radii is used to show the differences between anisotropic and orthotropic ERM predictions. In Figs. 5.3a and 5.3d the resulting von Mises stress for the reference and ERM solutions are shown for anisotropic ERM and orthotropic ERM, respectively. Comparison of the figures directly shows that the results for the smaller radii 0.25 and 0.5 mm give stress fields with large oscillations. The difference between the anisotropic and orthotropic approach is that the oscillations for the anisotropic ERM are lower in amplitude than for the orthotropic ERM. An explanation for the difference in oscillation amplitude is that the anisotropic ERM has three additional eigenstrain distributions which enable the anisotropic ERM to find a more accurate solution. Upon increasing the radii to 1, 2 and 5 mm oscillations disappear but mismatches between the reference and ERM stress fields are still visible. In both the anisotropic and orthotropic ERM they appear at the right side of the domain where the von Mises stress decreases. The mismatch between the ERM and reference stress field is much larger for the orthotropic ERM than for the anisotropic ERM. In Figs. 5.3b and 5.3e the NRMSE percentage for the von Mises stress σVM , effective strain ²eff , and the elastic strain energy U are shown. The quantities are shown as a function of radius r . In both figures it is clear to see that the error reaches a plateau for all three quantities. In Fig. 5.3b the NRMSE percentage error for the anisotropic ERM model is below 2%. For the orthotropic ERM model Fig. 5.3e the NRMSE percentage error is below 4%. This is a distinct difference which is expected if one recalls the observations on the shape of the solutions. The percentage of the maximum error is shown in Figs. 5.3c and 5.3f. For the anisotropic ERM model it can be seen that the maximum percentage error is below 7% where it is 14% for the orthotropic ERM model in the steady state regime. The location of the maximum errors is near the edges where the gradient of the field is highest. Evaluation of the results for the anisotropic and orthotropic ERM models leads to the following conclusion. The overall performance of the anisotropic ERM model is better than for the orthotropic model. Therefore the anisotropic model is adopted in the remainder of the line, layer, and 10 layer ERM geometries. Effect of increasing number of basis functions The function space specifies a number of basis functions (eigenstrain components). The number of basis functions (NoBF) that is used affects the accuracy of the solution of the ERM. The ERM line geometry is used to calculate the solutions for a number of basis functions which ranges from 2 to 20 for each strain direction. In these calculations the radius is kept constant at r = 1 and the anisotropic ERM is used. For the solutions of ERM with the varying number of basis functions the NRMSE percentage for the von Mises To evaluate the results, the solutions for the von Mises stress and the NRMSE percentage as a function of the number of basis functions are used. The solutions and the NRMSE are shown in Fig. 5.4. Fig. 5.4a shows the von Mises stress solutions for 2, 6, 10, 16, and 20 basis functions, where all have the identical radius of r = 1mm. Fig. 5.4b shows the NRMSE percentage for the von Mises stress, effective strain, and strain energy. The von Mises stress fields provided by the eigenstrains do not show a clear trend upon increasing the number of basis functions, see also Fig. 5.4b. When the number of basis functions is 2, 3, or 4 the results from the ERM method have a large error. This can be explained by the lack of overlap between the basis functions which results in oscillations in the solution. Looking at the solutions for 8, 12, and 20 basis functions the solutions are also inaccurate while the remaining solutions are quite accurate. The behaviour of the error is not consistent when the number of basis functions is increased. A possible cause for the inconsistent behaviour is the center location of the basis functions. This location affects which value of the basis function is mapped on the FE mesh as the thermal expansion coefficient. In Fig. 5.4c the percentage of the maximum error in the solution is shown. In this figure the same inconsistent behaviour is seen as for the NRMSE percentage.

48


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(c)

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1

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3

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0

1

2

r (mm)

(d)

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r (mm)

(e)

(f)

Figure 5.3: (a) Visualisation of the total strain solutions of the eigenstrain using basis functions with a radius which varies from 0.25 to 5 mm for the anisotropic ERM (b) Development of the normalized root mean square error percentage as a function of radius for the anisotropic ERM. (c) Development of the normalized root mean square error percentage as a function of radius for the anisotropic ERM. (d) Visualisation of the total strain solutions of the eigenstrain using basis functions with a radius which varies from 0.25 to 5 mm for the orthotropic ERM. (e) Development of the normalized root mean square error percentage as a function of radius for the orthotropic ERM. (f) Development of the normalized root mean square error percentage as a function of radius for the orthotropic ERM.

100

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400 300 ref. 2BF 6BF 10BF 16BF 20BF

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(b)

(c)

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Figure 5.4: (a) reference solution and ERM solutions for the von Mises stress for 2, 6, 8, 12, and 20 basis functions in each direction in the function space with an identical radius r = 1 mm. (b) NRMSE percentage of the ERM solutions for 2 to 20 basis functions (NoBF) in each direction w.r.t. the reference solutions. The values for which the error is calculated are the von Mises stress, effective strain, and strain energy. (c) Percentage of the maximum error of the ERM solutions for 2 to 20 basis functions (NoBF) in each direction w.r.t. the reference solutions. The NRMSE error is calculated for the von Mises stress, effective strain, and strain energy.

5.1. Line geometry

49

Effect of increasing radius of basis functions The radius r of the basis functions plays an important role in the ability of the ERM to calculate an accurate eigenstrain. Basis functions with a small radius are able to describe steep gradients in the stress field whereas a large radius is able to describe large surfaces. In Fig. 5.3 the von Mises stress, NRMSE percentage, and the maximum error percentage are shown for the anisotropic and orthotropic ERM. The function space for these solutions consist of five basis functions for each strain direction. The basis functions are equally distributed over the domain (-1.225, 1.225). The range of radius r of the basis functions is 0.25, 0.5, 1, 2, and 5 mm. In Figs. 5.3a and 5.3d it is clear to see that the discrepancy between the reference solution and ERM predictions are decreasing when the radius is increased. The main reason for this improvement is the increase of overlap between the basis functions. The increase in overlap increases the ability of the function space to describe a smooth eigenstrain without large fluctuations over the domain. The results clearly show that the choices made in designing the function space play a major role in the performance of the ERM model. In this case the function space is constructed with radial basis functions. The important parameters for such a function space are the radius r, position xci , and number of basis functions that are used. Error norms The functional H given in Eq. (3.15), describes the sum of squared errors between the reference solution ²ei j and the total elastic strain solution S i j . The ERM described in literature uses the error in the strain components to describe the error. However, it is suggested that also the stress components or strain energy can be used to describe the sum of squared errors. Subsequently, the coefficients can be found by minimizing the functional H for either the strain, stress, or strain energy norm. The three error norms are described in section 3.2. An important side note to the use of different error norms is that the reference solution and total strain solution in the functional H are stress or strain energy instead of elastic strain solutions. These three error norms are evaluated for the ERM line geometry. The number of basis functions that are used is five and are distributed evenly over the domain (−1.225, 1.225) described in mm. The basis functions are radial basis functions with a radius r = 2 mm. The distribution of the basis functions has been shown in chapter 4 in Fig. 4.7b. The results from the ERM line geometry for the error norms are compared to the reference solution by visual inspection and the NRMSE percentage between the ERM and reference solutions. In Figs. 5.5a, 5.5b, and 5.5c the von Mises stress for the reference solution and the ERM solutions for the strain (a), stress (b), and strain energy (c) norm. The figure clearly shows that both the ERM with the strain and stress norm give an accurate prediction of the von Mises stress field. For the strain energy norm the ERM is not able to find the correct prediction of the von Mises stress field. To find out why this error norm does not perform as well as the strain and stress norm a closer look is given at the values that are used to determine the eigenstrain coefficients. The ERM is a method which is developed to reconstruct stress fields by finding the corresponding strain 600

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1

Figure 5.5: (a) Von Mises stress distribution for the reference solution and the ERM solution using the strain norm. (b) Von Mises stress distribution for the reference solution and the ERM solution using the stress norm. (c) Von Mises stress distribution for the reference solution and the ERM solution using the strain energy norm.

50


source distributions for normal and shear directions. This means that the ERM needs information on the strain in the normal and shear directions. The used values in the error norms are stress, strain, and strain energy. The stress and strain information obtained from the Abaqus FE program consists of the values for the stress and strain components in all six directions in an element. The strain energy U is a single value which represents the stored energy in an element. The important thing to notice is that with the stress or strain information and the volume description of the element the strain energy can be calculated. In the other direction this is not possible because from the scalar value U one cannot define the stress or strain state in the element. Hence there is no unique solution for the stress and strain state in the element. For the ERM to work it is essential that information on the stress and strain state is known. Therefore it is not possible to define the coefficients of the eigenstrain components using the strain energy norm. The previous statement is explained by the following example. Consider a solid continuum having a stress and strain state σ11 , ²11 , σ22 , ²22 , σ12 , and ²12 . For this continuum two cases are defined: • Case 1: σ11 =1, σ22 =2, σ12 = 0 (MPa) and ²11 =1, ²22 =2, ²12 = 0 (-) • Case 2: σ11 =2, σ22 =1, σ12 = 0 (MPa) and ²11 =2, ²22 =1, ²12 = 0 (-) For both case the strain energy is calculated, U = 12 (σ11 ²11 +σ22 ²22 +σ12 ²12 )V . In both cases the strain energy that is calculated is equal to 5 J, so U1 = U2 . However the stress and strain state in the continuum is different in both cases. This shows if one only knows the strain energy U one cannot describe the stress and strain state of a continuum. With regards to the ERM this leads to a problem. The ERM is used to define the eigenstrain distributions in all directions. The strain energy of the reference geometry does not hold any information on the strain or stress distributions in the geometry. For the ERM the way the stresses and strains are distributed is the key part of information that is needed to determine the coefficients of the eigenstrain components. Without the information of the strain distributions in all directions it is not possible to determine the coefficients of the eigenstrain components. Finally the NRMSE percentage for ERM with the stress, strain, and strain energy norms are evaluated. These error percentages are shown in table 5.1. The results clearly show that the error percentage for the strain and stress norm are similar. For the strain energy norm the percentage error is around 6000 percent which is extremely high. In Fig. 5.5 this is also visible for the solutions for the different error norms. The rationalisation of the strain energy norm and the results from the ERM calculations with the error norms show that the strain energy norm cannot be used in ERM. The stress and strain norm both perform well and have a NRMSE percentage of below 1% for the von Mises stress, effective strain, and strain energy. In the rest of the calculation the strain norm will be used because that performs slightly better than the stress norm. Table 5.1: Table containing the NRMSE % for the von Mises stress, effective strain, and strain energy for the three error norms; strain, stress, and strain energy.

error norm strain stress strain energy

NRMSE % σVM 0.85 0.88 6.1e3

NRMSE % ²eff 0.77 0.80 5.93e3

NRMSE % U 0.94 0.98 5.66e3

Warpage as a result of the eigenstrain The reference model build the line on base plate using the point by point approach. This results in the deformed state of the line as shown in Fig. 5.2a. The ERM line model uses the same initial geometry as the reference model. Applying the eigenstrain on this geometry results in a deformed state of the line. For the ERM of this line geometry with five basis functions and a radius r = 2 mm the deformed configuration is given in Fig. 5.2b. Comparison of the two figures clearly shows that the deformations of the line are not the same. In the ERM formulation it was assumed that the deformations are also approximated. The error percentages in Figs. 5.6a and 5.6b show clearly that the average error can go up to 7000%. The maximum error even goes to errors of 15000%. The results clearly show that the displacement field cannot be predicted. An explanation for the fact that this is not possible is discussed in section 5.1.3.

5.1. Line geometry

51

1.5

8000 7000

U3

×104

1 U1

Max error (%)

NRMSE (%)

6000 5000 4000 U1

3000

0.5 U2

0 -0.5

2000

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-1.5 0

1

2

3

4

5

0

1

2

3

r (mm)

r (mm)

(a)

(b)

4

5

Figure 5.6: (a) Development of the NRMSE % of the displacement in the ERM solution w.r.t. the reference solution.U1 , U2 , and U3 are the displacements in x 1 , x 2 , and x 3 direction. (b) The maximum error present in the deformation of the geometry as a percentage of the reference solution.

Eigenstrains The resulting von Mises stress, effective strain, and strain energy distributions are the result of the found eigenstrains. The eigenstrains found using the anisotropic ERM are shown in Fig. 5.8. In the solutions for the anisotropic ERM model, Fig. 5.3a, large oscillations were observed in the solutions with the basis functions and radius r=0.25 and r=0.5 mm. The eigenstrains corresponding to these oscillating solutions also show this oscillating behaviour. The ERM calculates the coefficients of the eigenstrain distributions in all directions. In Figs. 5.7 and 5.8 the residual stress, strain, and eigenstrain distributions are visualised. The eigenstrain distributions are constructed on a basis of five equally distributed basis functions over the domain (-1.225; 1.225) mm and for the radii 0.25, 0.5, 1, 2, and 5 mm. The reference residual strain field in 11 direction is reconstructed by the residual strain solutions for 11 direction of all basis functions. This reconstruction result in the coefficients for the basis functions in the eigenstrain in 11 direction. These eigenstrains are shown in Fig. 5.8a for all radii. The eigenstrain for r=0.25 and r=0.5 show oscillations in the eigenstrain and it is clearly visible that the basis functions show no overlap. Increasing the radius shows that these oscillations disappear and that the eigenstrains become more smooth functions. This is caused by the increased overlap between the basis functions in the basis set. For the eigenstrains in directions 22 and 33 these oscillations also disappear. The shear components in 12, 23, and 13 direction do not show rapid oscillations in all solutions. An explanation for the absence of oscillations in the shear components is that the reference data for the shear components is zero except at the edges. Therefore the three basis functions in the interior are not needed. It is observed that the interior basis functions are not needed, because the corresponding coefficients are close to zero. The resulting eigenstrains shown in Fig. 5.8 do not show a specific trend in shape. It appears that each time the function space is adjusted the eigenstrain distribution gets a different shape. However the solutions of these eigenstrains in Fig. 5.3a remain within 2% accuracy when compared to reference values. It is observed that the solution is highly dependent on the properties and content of the function space.

5.1.3. Discussion The ERM results for the line geometry have given insight on the behaviour of the method. The results of the anisotropic and orthotropic ERM show that the anisotropic ERM model is better in describing the eigenstrain than the orthotropic ERM model. This conclusion is based on the results for a function space for which the radius is varied and the number of basis functions kept constant for each direction. This is a point of discussion because the anisotropic model has twice the number of basis functions as the orthotropic model. The fact that three directions are added to the eigenstrain means that the anisotropic ERM has more possibilities to describe the eigenstrain. The properties and locations of the basis functions in the domain are crucial for the accuracy of ERM. If the function space is badly designed, the ERM will not yield eigenstrains that accurately describe the reference stress field. The most important property is the overlap between the basis functions. Sufficient overlap

52


×10-3

4

600 500

3

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σ (MPa)

ǫe (-)

2 1

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0 -1 ǫe11

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ǫe13

σ11

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σ33

σ12

σ23

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0

0.5

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-1

x1 (mm)

-0.5

0

0.5

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x1 (mm)

(a)

(b)

Figure 5.7: (a) Reference solutions for the strain components in all directions. (b) Reference solutions for the stress components in all directions.

ǫ*11 r=0.25

0.06

ǫ*11 r=0.5

ǫ*11 (-)

0.04 0.02

r=1 r=2 r=5

ǫ*22 (-)

ǫ*11 ǫ*11 ǫ*11

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-2 -1 -4 -6

ǫ*13 (-)

ǫ*23 r=0.25

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0

0.5

1

-1

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0

x1 (mm)

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(e)

(f)

0.5

1

Figure 5.8: (a) Eigenstrain distributions in 11 direction for r is 0.25, 0.5, 1, 2, 5 mm. (b) Eigenstrain distributions in 22 direction for r is 0.25, 0.5, 1, 2, 5 mm. (c) Eigenstrain distributions in 33 direction for r is 0.25, 0.5, 1, 2, 5 mm. (d) Eigenstrain distributions in 12 direction for r is 0.25, 0.5, 1, 2, 5 mm. (e) Eigenstrain distributions in 23 direction for r is 0.25, 0.5, 1, 2, 5 mm. (f) Eigenstrain distributions in 13 direction for r is 0.25, 0.5, 1, 2, 5 mm.

results in the ability of the function space to be smooth. To ensure that the overlap is sufficient the center location of a basis function must lie within the basis of another basis function. The relative size of the basis functions to the finite element size appears to have an effect on the accuracy of the ERM. In research by Jun & Korsunsky [8] and Song & Korsunsky [25] the basis functions are narrow and

5.1. Line geometry

53

only cover four to five elements. This lead to accurate results in the one dimensional case as demonstrated by Jun & Korsunsky [8] but inaccurate results for Song & Korsunsky [25] in a two dimensional case. The problem with basis functions with a small basis and a small amount of data points (four to five, one for each element) is that the amount of data points is insufficient to capture the shape of the basis function. In the ERM line model each basis function has ten or twenty data points. This results in accurate solutions for the reference stress distributions. The relation between the amount of data points to represent the function and the accuracy of ERM is recommended for further research. When ERM is utilised in the line geometry the choice of error norm has three options; strain, stress, and strain energy. The strain and stress norm almost showed the same result, the difference is within 0.1%. This is expected because the relation between strain and stress is linear. The strain energy norm implemented in the ERM showed that it is not able to determine the correct coefficients. The reason is that the strain energy does not hold information on the stress and strain state of a continuum. Therefore it is not possible to define the eigenstrain distributions in different directions. The entire explanation is given in section 5.1.2 In literature the eigenstrain reconstruction method is used to reconstruct residual stress fields [1, 8, 11– 13]. The research goal was to predict both residual stress and warpage of SLM products. The literature on the ERM does not address the deformation behaviour of the material in which the residual stress field is present. The results for the line model showed that ERM is not able to describe the deformations of the product. To explain why the ERM does not provide the correct deformations a schematic illustration of the process is shown in Fig. 5.9. On the left side of the figure the reference process is visualised. The initial geometry Ω is built according to a certain process. This process leads to deformations u iref and residual stresses σref in the ij product. The residual stresses σref are used as input for the ERM. The ERM calculates the eigenstrain and ij eig

eig

subsequently a MBVP is solved. This results in deformations u i and residual stresses σi j . The ERM uses the reference stresses in the deformed geometry to calculate an eigenstrain. The ERM does not use the deformations in the deformed geometry in the eigenstrain calculation. This is logical because the ERM is developed to find a source for the residual stresses. Initially it was assumed that when the residual stresses are correctly predicted by the eigenstrain the deformation will also be correct. The ERM is intro-

Figure 5.9: Schematic overview of the process of finding the eigenstrain. On the left side the reference process is visualised. The initial geometry Ω is build according to a certain process. This process leads to deformations u iref and residual stresses σref in the product. j ij The residual stresses σref are used as input for the ERM. The ERM produces the eigenstrain for which the MBVP is solved. This results in ij eig

deformations u i

eig

and residual stresses σi j .

duced as a method to find equivalent strain sources for residual stress fields. To the best of our knowledge

54


the use of ERM to predict warpage of products has not been described. Based on the way the ERM works and the insights gathered in this thesis work, it is not likely that the ERM is a suitable method for the prediction of warpage of products.

5.2. Layer geometry

55

5.2. Layer geometry The layer model is an expansion of the line model. In this chapter the results for the reference model using the layer by layer, line by line, and point by point approach are discussed. The resulting residual stresses from the layer build with the point approach are used in the ERM layer model. The results for the ERM calculations are discussed in 5.2.2.

5.2.1. Reference results layer model The reference results for the layer model are obtained by building the layer according to the layer by layer, line by line, and point by point approach. The layer approach has been described in section 2.2.2 and the line and point approach in section 4.1.3. The von Mises stresses for the different approaches are shown in Figs. 5.10a, 5.10b, and 5.10c. In Fig. 5.11 the same stress fields are shown but onto the layer geometry. The resulting figures show two major effects of the different building approaches. The first one is the magnitude of the von Mises stress in the layer after building the layer. The magnitude of the von Mises stress σVM is around σVM = 3000 MPa for the layer by layer approach, σVM = 1300 MPa for the line by line approach, and σVM = 800 MPa for the point by point approach. The decrease in magnitude of the stresses can be explained by looking at the approaches. In the layer by layer approach the entire layer shrinks which exerts large forces on the base plate. With the point by point approach small amounts of material are added, shrinking these small blocks exerts a much smaller and local force on the layer and base plate. Due to this smaller force the build up of stresses is much smaller. The second effect that can be observed is that the plateau stress field for the layer by layer approach is in the center of the layer. For the line by line approach it is clearly visible that the stress plateau is centered at the upper side of the layer. For the point by point approach the stress plateau is shifted to the lower left corner. The shape of the stress fields is expected because in the layer approach the entire layer undergoes a uniform shrinking process. In the line approach lines are added which should result in axi-symmetric behaviour. For the point approach it is obvious that the results are shifted towards a corner because building starts in a the lower left corner. In literature the mostly used approach is the layer by layer approach, [17, 19, 27]. The results for the different approaches clearly show that the layer by layer approach results in high von Mises stress whereas the von Mises stress is much lower for the point by point approach. The point by point approach is close to the actual building process which makes the resulting stresses more realistic. This indicates that the layer by layer approach overestimates the build up of stresses in SLM products.

(a)

(b)

(c)

Figure 5.10: Von Mises stress σVM for the layer, line, and point approach. (a) σVM in the layer when layer approach is used. (b) σVM in the layer when line approach is used. (c) σVM in the layer when point approach is used.

5.2.2. ERM results The ERM layer model is used to reconstruct the eigenstrain that represents the residual stresses and strains for the point by point built layer. The function space for the layer geometry is expanded to two dimensions and

56


S, Mises (MPa) (Avg: 75%) 3500 3208 2917 2625 2333 2042 1750 1458 1167 875 583 292 0 X3 X2

X1

(a)

S, Mises (MPa) (Avg: 75%) 3500 3208 2917 2625 2333 2042 1750 1458 1167 875 583 292 0 X3 X2

X1

(b)

S, Mises (MPa) (Avg: 75%) 3500 3208 2917 2625 2333 2042 1750 1458 1167 875 583 292 0 X3 X2

X1

(c) Figure 5.11: (a) σVM distribution in the layer when layer by layer approach is used. (b) σVM distribution in the layer when the line by line approach is used. (c) σVM distribution in the layer when the point by point approach is used.

is tested on the effect of increasing the number of basis functions. Besides the expansion to two dimensions the addition of a boundary grid is tested which takes care of the steep gradients near the boundaries. This boundary grid is a set of basis functions which are positioned on the edges. For the ERM layer geometry the anisotropic version of the method is adopted and the strain norm is used. These choices are motivated by the

5.2. Layer geometry

57

analysis of the results of the ERM scheme in the line geometry. Internal grid The first numerical results for the layer model show the effect of increasing the amount of basis functions on the domain. The range that is used goes from a 2x2, 4x4, 6x6, 8x8, and 10x10 grid of radial basis functions with a radius equal r = 2 mm. The centers of the radial basis functions are equally distributed over the domain which is (−1.225, 1.225; −1.225, 1.225) mm. In Fig. 5.12a the NRMSE percentage for the strain components is shown as a function of the number of basis functions. The results show that the smallest grid (2x2) show an error which ranges from 5 - 27 %. Increasing the number of basis functions in the grid reduces the error significantly. When the largest number of basis functions (10x10) is used the NRMSE percentage error is below 5% for all strain components. The layer model results show a consistent effect of increasing the number of basis functions while that effect was not clearly visible for the line model results. The effect of increasing the number of basis functions has on the maximum error percentage between the reference and ERM results for the von Mises stress, effective strain, and strain energy are shown in Fig. 5.12b. For the von Mises stress, effective strain, and strain energy the maximum error percentage decreases when the number of basis functions is increased.

40

250 NRMSE % σVM

35

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max % ǫeff

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max % U

max error (%)

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150

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10 50 5 0

0 0

20

40

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NoBF (a)

80

100

0

20

40

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80

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NoBF (b)

Figure 5.12: (a) The effect of increasing the number of basis functions in the internal grid on the NRMSE percentage of the von Mises stress, effective strain, and strain energy for the reference and ERM solutions. (b) The maximum error percentage of the von Mises stress, effective strain, and strain energy for the reference and ERM solutions of the internal grid as a function of increasing number of basis functions.

Boundary grid The second set of numerical results is for a 4x4 internal grid of radial basis functions and a boundary grid which contains for basis functions. The radius of the basis functions of the internal grid is equal r = 2 mm. The radius of the basis functions in the boundary grid is in a range of 0.1, 0.2, 0.5, 1, and 2 mm. In Fig. 5.13a the NRMSE for all strain components is shown as a function of the radius of the basis functions in the boundary grid. Decreasing the radius of the basis function shows that the error goes down to below 2%. The results also show that the error decreases when the radius gets higher than 1 mm. The maximum error percentage is shown in Fig. 5.13b shows the maximum . The two values are the percentage error between the reference and ERM results for the von Mises stress, effective strain, and strain energy. The results clearly show that the error decreases when the radius decreases. It also shows that it starts to decrease when the radius becomes larger then 1. To make sure the use of the boundary grid improves the results visual inspection is needed. For this inspection the reference solution , internal grid with 4x4 BF and r = 2 mm, internal grid with 4x4 BF and r = 2 mm combined with a boundary grid with r = 0.1, and an internal grid with 4x4 BF and radius r = 2 mm combined with a boundary grid with r = 2 mm are evaluated for the stress component in 11 direction, σ11 . These

58


70

10

60 8

max error (%)

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50 6

4 NRMSE % σVM

2

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40 30 20

max % σVM max % ǫeff

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0

0 0

0.5

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r (mm) (a)

1.5

2

0

0.5

1

1.5

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Figure 5.13: (a) The effect of increasing the radius r of the basis functions in the boundary grid on the NRMSE percentage for the von Mises stress, effective strain, and the elastic strain energy for the reference and ERM solutions (b) This figure shows the effect on the von Mises stress and strain energy in the layer by decreasing the radius of the basis functions in the boundary grid from 2 to 0.1mm.

results are shown in Fig. 5.14. The reference results σ11 for the point by point build layer are shown in the top left of Fig. 5.14. As discussed in section 5.2.2 the reference result show a steep strain gradient along the x 1 axis at x 2 = −1.225 mm. The top right part of the figure shows the solution for σ11 with the 4x4 IG which is not able to capture the steep gradient near the edge. Therefore the use of a boundary grid was proposed. The bottom two figures show the result for σ11 , the bottom left for a BG with radius r 2 = 0.1 and the bottom right for a BG with radius r 2 = 2. The solution for BG with radius r 2 = 0.1 captures the steep gradient near the edge. The solution for the BG with radius r 2 = 2 does not. The results presented in this section show that the use of a boundary grid is necessary to capture the steep gradients present in the reference solution.

5.2.3. Discussion The effect of increasing the number of basis functions on the grid is clear. Increasing the number of basis functions leads to a more accurate eigenstrain. In the study on the layer geometry the radius of the basis functions has not been investigated. In the results for the basis functions it is clearly visible that this leads to wiggles in the surface. Therefore it is suggested that the radius of the basis functions is also investigated. The results in section 5.2 show that building a layer using one of the approaches results in stress fields with large gradients near the boundary. The addition of a boundary grid showed that the ability of the ERM to find an accurate eigenstrain increased. The boundary grid that has been introduced consists of four basis functions which are defined on each edge. This is the most simple boundary grid one can think of. It is suggested that other designs of the boundary grid need further research.

5.3. Multilayer geometry

59

σ11

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Figure 5.14: (a) Reference results σ11 for the point by point layer model. (b) Layer model ERM result σ11 with a function space containing internal grid 4x4 with r = 2 mm. (c) Layer model ERM result σ11 with a function space containing internal grid 4x4 with r 1 = 2 mm and boundary grid with r 2 = 0.1 mm. (d) Layer model ERM result σ11 with a function space containing internal grid 4x4 with r 1 = 2 mm and BG with r 2 = 2 mm.

5.3. Multilayer geometry The results for the multilayer model contains the results for the point by point approach build multilayer geometry in section 5.2.2. These reference results are used in the ERM of the multilayer geometry. The results of the ERM model are discussed in section 5.3.2.

5.3.1. Multilayer reference results The multilayer model is built using the point approach. From this geometry the residual stresses and strains, and the elastic strain energy are extracted. From these results the von Mises stress σVM and effective strain ²eff are calculated. Together with the elastic strain energy these values are evaluated. This is done for the cross

60


(a)

(b)

Figure 5.15: (a) Cross section of the geometry along the x 1 axis at x 2 = 0. (b) Cross section of the geometry along the x 2 axis at x 1 = 0.

section of the geometry and the first layer. In Fig. 5.16 the results for the reference calculation on the ten layer geometry is shown for the von Mises stress. S, Mises (MPa) (Avg: 75%) 2000 1833 1667 1500 1333 1167 1000 833 667 500 333 167 0 X3 X2

X1

Figure 5.16: Visualisation of the multilayer geometry with the von Mises stress distribution.

Effect of layer addition on the stress and strain state in the first layer In research by Li et al. [15] the effect of layer addition on the previous build structure is mentioned as a field of interest. In their research it is stated that there is certainly an effect of added layers to previous layers. However they think that the effect is to complex to model. In most models this effect is neglected or just not evaluated. The multilayer reference model is used to find out if layer addition has an effect on the stresses and strains in the build geometry. The first hypotheses is that the addition and cooling of material decreases the magnitude of stresses and strains in the previously build layers of the structure. The motivation for this hypothesis is shown schematically in Fig. 5.17. In this figure the mechanism of stress generation is explained. The first layer Ω1 is applied on the base plate and a thermal strain is imposed, the subscript denotes the layer number. Solving the MBVP results in a deformed configuration Ω01 and a residual stress state in the first layer, σ11 . The accent denotes the deformed configuration. The next step is addition of layer Ω2 and impose a thermal strain ²th . Solving the layer results in the deformed configuration Ω02 and the tensile residual stress field σ02 in the layer. Due to shrinking of layer Ω2 a compressive force is exerted on layer Ω01 , this results in compressive stress and deforms the layer to the second deformed configuration Ω1 ". The new residual stress state σ1 " is the sum of the previous stress σ01 and the compressive stress ∆σcomp . The stresses σ1 " in layer Ω1 " are tensile stresses. Compressive stress has a negative sign and will decrease the tensile stress in the layer. This is valid because it is assumed that it remains in the linear elastic regime. The second hypotheses is that the effect of thermal expansion of the added layer on the previously build layers decreases when the distance from the added layer to the previous layers increases. This hypothesis is a inspired by Saint-Venant’s principle which states that the effect of a load on a region of the structure decreases when the distance from the load to the region increases. From the ten layer geometry build with the point approach the stresses are extracted for all build layers


61

Figure 5.17: In this figure the mechanism of stress degeneration is explained. The first layer Ω1 is applied on the base plate and a thermal strain is imposed. Solving the MBVP results in a deformed configuration Ω01 and a residual stress state in the first layer, σres 1 . The next

step is addition of layer Ω2 and impose a thermal strain ²th . Solving the layer results in the deformed configuration Ω02 and the tensile residual stress field σres 2 in the layer. Due to shrinking of layer Ω2 a compressive force is exerted on layer Ω1 , this results in compressive 0

res comp . stress in layer Ω1 . The new residual stress state σres 1 is the sum of the previous stress σ1 and the compressive stress ∆σ

each time a new layer was added in the reference model. From these results the stresses are obtained for the first layer. Fig. 5.21a shows the stresses in 11 direction for the first layer. The stress results shown are for No. 2, 4, 6, 8, and 9 added layers on top of the first layer. In this figure it is clearly visible that when layers are added on top of the first layer the residual stresses start to decrease. This shows that the first hypotheses is correct. The development of stress for other directions in the reference calculation can be found in appendix A. In the 22 and 33 direction the same behaviour as for the 11 direction is observed. The stress in 12 direction stays constant in magnitude and is not affected by the layer addition. This observation is anticipated because the ratio between the stresses in 11 and 22 direction does not change. In 23 direction the overall magnitude of the stress increases. This term represents the coupling between x 2 and x 3 directions. The stress in 33 decreases from a 0 stress level to -100 MPa which results in an increase of the shear stress in 23 direction. This same effect is observed for 13 direction because this couples the deformation in direction x 1 and x 2 . The second hypothesis was that the decrease of stress due to layer addition reduces when the distance between the first and added layer increases. This is also clearly visible in the results. The highest jump in stress is observed for between the stress state for the first layer and with two layers added. The change of the stress state eventually reaches a steady state near the point where nine layers are added. To quantify the decrease in stress the percentage of magnitude loss for the entire stress fields in 11 and 22 direction are calculated. For the 11 and 22 direction this results in -8.9% and -14.1% respectively. For the 33 direction it is difficult to calculate a decrease in percentage because the initial distribution has a value of 0 MPa. For the shear direction 12 the decrease in magnitude is -1.8% which shows that there is a slight change in the ratio between the stress in 11 and 22 direction. For the shear directions 23 and 13 the percentage is not intuitive because the values in the distribution both increase in negative and positive direction. Stress distributions in cross section The cross sections in Figs. 5.19a and 5.20a give a good view of the development of stress when the ten layer geometry is finished. In the both cross sections a large stress concentration is visible at the bottom. The stress concentrations are highest near the base plate because the added material is constrained in shrinking. The material is constrained by the perfect connection to the base plate. Going up the layers in the cross section shows that the stress reaches a constant level which is far lower than at the bottom.

5.3.2. ERM results The results for the ERM model of the multilayer geometry are given in this section. The ERM model of the multilayer model uses the same function space as the ERM layer model. This function space is described by the internal and boundary grid. The internal grid consists of 16 basis functions for each strain component. The basis functions are equally distributed on the domain (-0.6;0.6, -0.6;0.6) mm to a 4x4 grid. The boundary grid consists of 4 basis functions for each strain component. The basis functions are defined on each edge. The total number of basis function in each layer is equal to 120 basis functions. The multilayer geometry consists of ten layers, therefore the multilayer ERM model contains 1200 basis functions. The first ERM results that are shown are the stresses and strains in 11 and 22 direction for the first layer. These are compared with the resulting stress and strain distribution from the reference calculation. Also the development of the eigenstrain distribution in the first layer are evaluated. These are evaluated to see if the effect of layer addition is visible in the eigenstrain distribution.

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Shape of von Mises stress field in layer 1 In Fig. 5.18 the von Mises stress in the first layer of the multilayer geometry is shown. The left figure shows the von Mises stress in the first layer for the SLM reference model. The right figure shows the von Mises stress in the first layer in the ERM model. The general shape and magnitude of the von Mises stress in the first layer shows great resemblance. A close look at the figures shows that near the edges the magnitude of the von Mises stress is higher for the reference than for the ERM results. In the interior part of the eigenstrain result a dent is visible where this is absent in the reference results. These errors in magnitude and shape occur because the function space is not able to describe the residual stress more accurate.

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Figure 5.19: (a) Von Mises stress in the cross section CS(x1 ) from the SLM model reference calculation. (b) Von Mises stress in the cross section CS(x1 ) from the ERM model.


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Figure 5.20: (a) Von Mises stress in the cross section CS2(x 2 ) from the SLM model reference calculation. (b) Von Mises stress in the cross section CS2(x 2 ) from the ERM model.

Development of stress in first layer due to layer addition The reference model showed that stress degeneration is present in a layer on which new layers are applied. The degeneration is caused by a compressive force exerted on the previous material by the shrinking material. The degeneration stops after enough layers are added which is explained by Saint Venant’s principle. With the ERM the residual stress and strain field in the multilayer geometry are reconstructed. In Figs. 5.21b and 5.21e the reconstructed stress fields for σ11 are shown in x 1 and x 2 direction respectively. On the left side of these figures the reference results for both figures are shown. The results along x 1 direction show great resemblance with the reference results. The minor differences are the that the eigenstrain results shifted a little bit to the right and a wiggle in the results is visible for all lines. These wiggles are caused by the inability of the function space to produce a better fit to the reference results. The used internal grid is 4x4, increasing the number of basis function would result in a smoother fit to the reference results. In Fig. 5.21c the eigenstrains that correspond to the solutions in Fig. 5.21b are shown. In the solutions the degeneration of stress is clearly visible. This same trend is expected in the eigenstrain distributions. The lines in Fig. 5.21c do not show a clear trend. In the center of the eigenstrains the distribution is nearly constant closely above zero. Near the edges large peaks in the eigenstrains are observed. These peaks are small when only one layer is applied but do increase each time a layer is added. The direction that these peaks have is constant on the left but shifts from up to down on the right. For this effect no explanation has been found. In x 2 direction the results do not have a parabolic shape. In Fig. 5.21e the residual stress solutions for the eigenstrains are plotted. The general shape of the solutions resembles the reference results. Only the distribution in the first layer shows large errors in the two valleys on the left and right of the solution. Besides the reconstruction for the first layer the trend in stress degeneration is clearly visible. Fig. 5.21f shows the values of the eigenstrains along the line x 2 . Opposed to the eigenstrains in x 1 directions one directly sees a trend in the eigenstrain behaviour if one neglects the first eigenstrain. It can be seen that the eigenstrain starts increasing in magnitude when layers are added. Adding more layers finally results in a sort of steady state eigenstrain as can be observed for line L1-9AL. Comparing the magnitudes of the eigenstrains in x 1 and x 2 shows that they are similar in the center. However, at the edges the magnitude of the eigenstrains in x 1 direction are ten times as high than the overall value in x 2 direction. Von Mises stress in cross section The cross sections CS1 and CS2, give insight in the stress distribution in the multilayer block after it has been build. In Figs. 5.19 and 5.20 the von Mises stress for the reference calculation (a) and for the eigenstrain calculation (b) are shown. The red dotted line represents the boundary between two layers. The cross-sections

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Figure 5.21: a) Decrease of stress in 11 direction for the first layer in cross-section CS1. (b) Decrease of stress in 11 direction for the first layer in cross-section CS1 calculated by eigenstrain. (c) Eigenstrains in 11 direction in first layer for CS!. (d) Decrease of stress in 11 direction for the first layer in cross-section CS1. (e) Decrease of stress in 11 direction for the first layer in cross-section CS1 calculated by eigenstrain. (f) Eigenstrains in 11 direction in first layer for CS2.

for the reference and eigenstrain calculation are almost exactly the same. A visual inspection does only reveal one difference between the results and that is that the contour lines are shifted a tiny bit. These results show that the expansion of ERM to a three dimensions works as expected. Error between multilayer reference and ERM For the ten layer geometry a fixed number of basis functions with predefined properties is used. Comparing the reference calculation with the eigenstrain solution results in two errors. The first error is the NRMSE%, for which the eigenstrain solution overestimates the reference solution by 0.49%. The other error is the relative percentage of the maximum error, that error is equal to 16.77%. This maximum error is located at the edge of the first layer. These results are satisfying and tell that the ERM is a suitable method to reconstruct three dimensional stress fields.

5.3.3. Discussion The ERM model for the multilayer geometry is a single test case for the expansion of the ERM to three dimensions. From the insights gathered in the line and layer ERM calculations a single function space was designed to do the three dimensional ERM calculation. It is desired to do more calculations with different function spaces but due to limitations in time this was not possible. Therefore further investigation on the design of the function space for three dimensional ERM is necessary. This implementation of the ERM is a time consuming process because the MBVP for each eigenstrain component is calculated in series. To speed up the method it is possible to do all the calculations parallel because for each eigenstrain component an individual MBVP is defined. Parallelisation of the implemented ERM is recommended when the method is used in the future. The results of the three dimensional ERM are satisfactory. The overall error is low and the location of the


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maximum error is known. For the three dimensional ERM model only one function space is used. It might have been better to do iterations with the three dimensional model. This would lead to a function space which approximates the reference stress field more accurate and bring the maximum error down. The results of the ERM calculation show that the basis function in the boundary grid have difficulties with the stress distribution near the edges. By adjusting the radius and/or position of the radial basis function on the boundary the ERM can describe the stress field at the boundary better.

6 Conclusions and recommendations In this chapter the conclusions of this thesis work are presented in section 6.1. The recommendation for future work are discussed in section 6.2.

6.1. Conclusions The main goal of this thesis work is to develop a simple mechanical model which describes the residual stress and warpage of a SLM product. For this purpose two types of models are developed and applied on three types of geometry; a line, a layer, and a multilayer geometry which are all build on a base plate. The detailed SLM model in combination with the point by point building approach is able to include the effect of scanning strategy on the residual stress distribution of a SLM product. Comparison of the residual stress, strain, and strain energy fields showed that the layer models described in literature [17, 26, 27] overestimate the residual stress levels due to the SLM process. However, computational costs associated with the SLM model is significantly higher than for the layer models. The results of the detailed SLM model for the ten layer geometry showed that adding layers of material decreases the stresses in previously built layers. This decrease in tensile stress is caused by induced compressive stresses by the added layer on the previously built layers. The decrease in tensile stress becomes smaller when the distance between the added layer and the region of interest increases. In order to exploit the more realistic physics incorporated in the detailed SLM model, a three dimensional ERM employed. The ERM produces an eigenstrain that can be imposed onto a completely built layer which almost perfectly reproduces the residual stress field predicted by the detailed SLM model. The combination of ERM and the detailed SLM model inherent the computational tractability of the layer by layer approach and includes the SLM process history. The results of the ERM model for the different geometries show that the performance of the ERM model depends entirely on the characteristics of the function space. Therefore, when designing the function space the shape of the reference results must be taken into consideration. From the results in the ERM model three guidelines are determined; i) Smooth constant levels of stress in the reference data require basis functions with a large basis, ii) Steep gradients in the surface require basis functions with a small basis, and iii) The basis functions in the function space must overlap each other, which enables the function space to describe smooth surfaces. For the ERM the use of orthotropic and anisotropic expansion fields have been investigated. The anisotropic ERM model showed an error of 2% where the error for the orthotropic ERM model was 10%. Thus it is concluded that the anisotropic ERM model calculates the most accurate eigenstrains. The functional H described in ERM, Eq. (3.15) uses a strain, stress, or strain energy norm. The results showed that the ERM using the strain and stress norm shows similar results. The results for the strain energy norm is not accurate because the strain energy is unable to describe the stress or strain state in a geometry. Therefor it is concluded that the strain and stress norm are both suitable to reconstruct accurate eigenstrains while the strain energy norm cannot be used for the lack essential information. Evaluating the deformations extracted from the SLM and ERM model for the line geometry showed a significant discrepancy. Consequently, it is concluded that the ERM is not suitable to describe the deformation behaviour of a SLM product. 67

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6. Conclusions and recommendations

In literature [11, 25] the ERM has only been used for one and two dimensional problems. The results in this thesis show that the ERM is applicable in a three dimensional setting and gives accurate results. It is concluded that the ERM is a suitable method to find equivalent descriptions or eigenstrains for three dimensional residual stress fields. The accuracy of the ERM depends on the design of the function space. The systematic design of the function spaces in this thesis resulted in errors between the reference and ERM results between 0.5-4%.

6.2. Recommendations for future work The developed ERM model is an important step to get to a model that predicts the residual stress in SLM products. The application of the results of this model in a global model of a SLM product should be investigated. It is recommended to investigate the application of the eigenstrains in a multiscale approach such as described by Li et al. [15]. In this approach the residual stress fields are mapped to layers in the global model to predict residual stress in the product. In this study the reference stress fields are obtained from numerical models. It is suggested that the same type of research is repeated with the use of experimental data from the SLM process. This would be both a way to validate the developed SLM model as well for the ERM model. For the SLM and ERM model only one specific scanning strategy is used. To see the effect of changing the scanning strategy different scanning strategies should be used in the detailed SLM model to build the layer. With changing the scanning strategy one can think of start stops in scanning lines and the checkerboard patterns described in chapter 2. In the ERM and SLM model the same mesh density is used but that is not necessary. The mesh density of the geometry used in the ERM model can be coarsened because it describes a larger domain. It is suggested to vary the size of the finite elements in the ERM mesh and see how accurate the calculated eigenstrains are.

A Additional figures A.1. Line geometry This section contains the strain and stress components in the line geometry. The strain and stress components are shown in Figs. A.1a and A.1b respectively.

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A.2. Layer geometry In this section the contour plots for the von Mises stress, effective strain, and elastic strain energy are shown, see Figs. A.2, A.3, and A.4.

A.3. Ten layer geometry In this section the development of the stress levels in all principal directions is shown for the first layer due to layer addition. Fig. A.5 shows the development of stress in layer one for the reference calculation with the SLM model. Fig. A.7 shows the development of the stresses of the eigenstrain solution. Both figures show the results along the cross-section in x 1 , x 3 plane at x 2 = 0. 69

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Figure A.2: Von Mises stress σVM for the layer, line, and point approach. (a) σVM in the layer when layer approach is used. (b) σVM in the layer when line approach is used. (c) σVM in the layer when point approach is used.

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Figure A.3: Effective strain ²eff for the layer, line, and point approach for the reference calculation with the detailed SLM model. (a) ²eff in the layer when layer approach is used. (b) ²eff in the layer when line approach is used. (c) ²eff in the layer when point approach is used.

Figs. A.6 and A.8 show the results for the reference and eigenstrain solution along the cross-section of the first layer in x 2 , x 3 plane at x 1 = 0.

A.4. ERM results: ten layer geometry


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Figure A.4: Elastic strain energy U for the layer, line, and point approach for the reference detailed SLM model. (a) U in the layer when layer approach is used. (b) U in the layer when line approach is used. (c) U in the layer when point approach is used.

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