micromechanics and fatigue life simulation of random

SESSION TITLE – WILL BE COMPLETED BY NAFEMS

MICROMECHANICS AND FATIGUE LIFE SIMULATION OF RANDOM FIBER REINFORCED COMPOSITES Atul Jain (LMS International N.V.; KU Leuven, Belgium); Stepan V. Lomov (KU Leuven, Belgium); Yasmine Abdin (KU Leuven, Belgium); Ignaas Verpoest (KU Leuven, Belgium); Wim Van Paepegem (Ghent University) Michael Hack (LMS International N.V.) Atul Jain; RTD engineer THEME Composites SUMMARY Good specific properties and relative ease for large scale manufacturing make random fiber reinforced composites (RFRC) very attractive for a wide range of industries including the automotive industry. There are a large number of processes by which RFRC composites could be manufactured. Each process leads to a different statistical distribution of fibers at different points in the composite component, which in turn leads to different material properties, mechanical and fatigue behavior at different points, for different loading directions and types. Micromechanical simulation methods of RFRC taking into account fiber orientation and length distribution have been developed keeping in mind the requirements of fatigue simulation of complex industrial sized composite components. Fatigue behavior of composites is very different from metals due to the presence of various damage events and modes before failure leading to continuous degradation of properties whereas the global stiffness of metal components remains more or less constant. Mean field homogenization is usually used to predict the effective stresses of composite materials. For correct damage and fatigue analysis both the predictions of effective properties and stresses inside of individual inclusions should be correctly estimated. A study of the different mean field methods and benchmarking with full “quasi-correct” FE calculations was conducted to ascertain these requirements and the Mori-Tanaka formulation was found to be most suitable. Different damage mechanisms and subsequent redistribution of stresses were studied; finite element simulation is used extensively for validation of the proposed models. Damage modes considered are fiber matrix debonding including complete pull-out and fiber breakage in RFRC composites. Accurate simulation of progressive damage in RFRC has led to better understanding of characteristics and nature of the SN curve for RFRC. A

MICROMECHANICS AND FATIGUE LIFE SIMULATION OF RANDOM FIBER REINFORCED COMPOSITES

multi-scale approach where the damage events occurring at the “micro” level are linked to properties of the component on a macro-scale is proposed for the fatigue analysis of RFRC. Prediction of SN curves for different configurations based on a micromechanics damage operator and a single input of a reference SN curve is proposed by a “hybrid multi-scale” concept of “scaling of SN curves”. Such a method is expected to be dependent on input of lesser number of SN curves as compared to purely test based interpolation methods. Finally, a roadmap for fatigue simulation is presented highlighting the various challenges in simulation of fatigue of composite materials and possible solutions for the same. KEYWORDS Fatigue, fiber reinforced composites, multi-scale damage, SN curves, mean field homogenization, Mori-Tanaka formulation 1: Introduction This article will describe all the aspects of micromechanics of short fiber composites: homogenization, including capabilities of the different methods of homogenization, initiation of different damage modes, damage progression and finally the relationship between the material fatigue properties and the micromechanics damage events. The need for fuel reduction and CO2 regulations has increased the relevance of composite materials for the automotive industry. The effective properties of a composite material depend upon a wide variety of factors like properties of the fiber and matrix, the architecture of the composite, the manufacturing process employed etc. Based on the architecture of fibers, we can divide composites into three major groups: i) Short fiber reinforced composites, ii) Continuous unidirectional composite and iii) Textile composites. Short fiber composites, which are the subject of the present paper, usually have glass or carbon fibers with length in the range from 0.1mm to 25mm, randomly dispersed in the matrix. Automotive industry places ease of manufacturing and cost of component as important criteria for choosing the type of composite material to be used as a component. Injection molded short fiber composites are cheap to make and also relatively easy to manufacture even for complex component shapes, making them an ideal candidate for industrial application. The process of injection molding results in different orientation of fiber at different locations.


Figure 1 Short fiber composite, showing the different orientations and length of inclusions at different points

Manufacturing simulation software can predict reasonably accurately the orientations and length distribution of fibers in different points of the composite based on the rheology characteristics of resin and a host of other factors[1]. The stiffness and strength of a composite show a strong dependence on the orientation and length of fibers. There are a number of methods by which the effective properties of a composite can be estimated based on the output provided by manufacturing software. A brief overview of the different approaches, their relative merits and limitations is given in section 2. A comparison of prediction of effective properties by different homogenization methods is abundant in literature for example Klusemann [2]. However, a comparison of predictive capabilities of stresses in individual inclusions and matrix is not readily available. A brief description of the different homogenization schemes is presented in section 2. A comparison of different mean field methods for the same was undertaken and is presented in section 3. The stresses inside the individual inclusions are used to model micromechanics damage events like fiber-matrix debonding and fiber failure. Modelling of initiation and progression of different modes of damage is presented in the section 4. Micromechanics damage has led to greater insight to the fatigue properties of a composite material; this is described in section 5. A hybrid multi-scale method of scaling SN curves of a composite representative volume element (RVE) based on a reference input SN curve and performed by relating the damage on the constituent level to the fatigue behaviour of the composite is presented in section 6. Finally some outlook for future research is presented in the section 7 before listing the conclusions in section 8. 2: Homogenization Schemes A composite material consists of two phases, namely the fiber and the matrix. Fiber is also commonly named inclusions in the context of homogenization. The inclusions in a certain representative volume element (RVE) of the material are characterized by the volume fraction of the inclusions, their orientation and length distributions. The effective properties of a RVE


(homogenisation) are calculated by relating the applied volume average strain to the resultant volume average stress:  ij   Cijkl  kl 

1.

Where, < > bracket is the volume average of a quantity in an RVE. 1 2.  A   AdV V There are several methods of estimating the volume average stresses in a composite material. One could perform full finite element calculation on a RVE and calculate the stresses at each point and consequently calculating the volume average of stresses over the entire RVE. Finite element calculations are accurate and give complete description of stresses at every point. However, the computational cost required for a full FE calculation of a RVE is enormous and therefore not really feasible for a component level analysis[3]. An alternate and slightly faster method of computation is performing a voxel based calculation. Both methods are accurate but also quite expensive and therefore not really feasible for component level analysis. There are a series of methods which are used to calculate the mean stresses in a phase without calculation of the stresses in every point, such methods are collectively known as mean field homogenization schemes. All mean field homogenizations schemes are based on the work of Eshelby [4]. Eshelby relates strain inside an inclusion to the applied far-field strain. Closed form solutions exist for the Eshelby tensor if the shape of inclusions is ellipsoid. It is due to this reason that all mean field homogenization schemes require the fibers to be modelled as ellipsoidal inclusion. A few mean field homogenization schemes are Mori-Tanaka formulation[5], self-consistent schemes [6], Double Inclusion scheme [7]. Among the many mean field homogenization schemes, Mori-Tanaka formulation is the most commonly used. By using the Mori-Tanaka formulation one can calculate the effective stiffness of a composite by estimating first the strain concentration factor in the inclusions and then relating the effective stiffness of a composite to the strain concentration factor by the following relation M

C eff  C m   c (C   C m ) A

3.

 1

where, Ceff is the effective stiffness of the composite, Cm, Cα are the stiffness matrix of the matrix and inclusion respectively, cα is the volume fraction of individual inclusion, M is the total number of inclusions and Aα is the strain concentration factor which relates the strain in the inclusion to the applied


strain. A detailed mathematical description of the Mori-Tanaka formulation can be found, for example, in [7]. However, the Mori-Tanaka formulation is often criticized for giving physically inadmissible solutions. Mori-Tanaka and self-consistent schemes were shown by Benveniste et al., [8] to yield a symmetric effective stiffness tensor, only if the composite had reinforcements of similar shape and alignment. The stiffness tensor must be symmetric and its inverse must lead to the compliance matrix. A number of other authors have reported mathematical and physical inadmissibility problems with the Mori-Tanaka formulation [9],[10]. Doghri [11-13] proposed a method to circumvent the mathematical problems of the Mori-Tanaka formulation. He discretized the representative volume element (RVE) to a number of “pseudo-grains”. A pseudo-grain is defined as a bi-phase composite consisting of inclusions having the same orientation and aspect ratio. Doghri applied Mori-Tanaka formulation individually on the “pseudo-grains” and then volume averaged the stiffness of the grains to get the effective properties of the short-fiber composite. The basic idea behind breaking the homogenization scheme into two steps is the following: if each step individually satisfies all the conditions of the homogenization scheme, then the procedure in itself will satisfy all the conditions required of for meanfield homogenization schemes. This approach eliminated the mathematical problems of the Mori-Tanaka formulation but introduced additional approximations with regards to the interactions between the inclusions. This method of homogenization was also applied for a number of material nonlinearity models, both time independent (plasticity) [13] and time dependent (viscosity) [12]. We will call this “pseudo-grain” formulation of the MoriTanaka method in short the “PGMT” formulation. All the mean field theorems are extremely cheap in terms of calculation cost compared to full finite element calculations. The average time for a mean-field computation is a fraction of a second, while the full finite element calculation of an RVE usually takes about an hour in a normal PC, if material non-linearity is also taken into account. Thus we decided to use mean field homogenization method as the principal method for homogenization and full FE is used to benchmark different homogenization schemes and also to validate the damage initiation and propagation laws. This is described in succeeding sections of this article. 3: Predictive capabilities of stresses inside of individual inclusions by Mori-Tanaka formulation and PGMT The predictions of various damage modes in short fiber composites components like fiber matrix debonding and fiber breakage requires the knowledge of both the effective properties of the composite and of the stresses


and strains in the individual inclusions. In this section we check the predictive capabilities for stresses inside individual inclusions of different mean field theorems namely the Mori-Tanaka formulation and PGMT in comparison to the exact solution obtained by Finite element calculations. To compare the inclusion average stresses predictions by the Mori-Tanaka formulation and PGMT a series of RVE were created. The volume fraction of the inclusions varied from 1% to 25%. For all the calculations, a second order orientation tensor “a”, was fed as an input to describe the orientation distribution of the inclusions [1]. Complete information about the orientations of inclusions is based on a fourth order orientation tensor. If the orientation is fixed or uniformly random the closure from second order orientation tensor a to the fourth order orientation tensor is exact. However for orientations which are neither uniformly random nor fixed, the estimation of the fourth order orientation tensor from the second order orientation tensor is not exact and some approximation is needed. In such cases, orthotropic closure method as described by Cintra [14] was used for both Mori-Tanaka and PGMT formulation. The orientation of a particular inclusion is defined by two angles, theta, θ and phi, φ. θ is the angle in degrees between axis 3 and the orientation vector p, while φ is the angle in degrees between axis 1 and the projection of the orientation vector p onto the (1,2)-plane. Figure 4 illustrates p, θ and φ. All the models built had at most 2-dimensional orientation distribution of inclusions, meaning that the value of φ for an inclusion could vary from 0-180 deg and that the value of θ is 90 degree for all inclusions.

Figure 2 Schematic representation of orientation of an inclusion, p is the orientation tensor and θ and φ are the spherical angles

Pseudo-grain discretization of such an RVE, consisting of inclusions with different orientation but with same aspect ratio was done with the number of angle increments equal to 30. PGMT calculations were performed using the software DIGIMAT[15]. For the Mori-Tanaka formulation a realization of 1000 inclusions was used in all the calculations. For building the finite element model, meshing ellipsoids in the finite element solver ABAQUS[16] proved to be challenging for inclusions with aspect ratio


higher than 5. Keeping this in mind, the aspect ratio chosen for the majority of the calculations was 3. The placement of the inclusion centres was random. There was however some limit to the minimum distance between the two ellipsoids to ensure an acceptable mesh. The minimum distance between two inclusions was 0.0035 times the diameter of the inclusion [17]. Periodic boundary conditions were applied to all the three axis of the RVE cube. Under periodic boundary conditions, the number of inclusions required to completely characterize an RVE depends on the volume fraction of the inclusion phase. When the volume fraction is very low the behaviour of the composite can be characterized with an RVE made of a single inclusion. For higher volume fractions, to estimate the number of inclusions needed for a representative RVE we decided to look at the phase average stresses (not individual inclusions). We considered the number of inclusions as high enough if two conditions are fulfilled simultaneously: On the one hand, the variation of average stresses in the inclusion phase across different realizations of the RVE should be less than 5% different from each other; on the other hand, also the mean values of average stress in the complete inclusion phase across different realizations should not change even if the realizations consisted of a higher number of inclusions. For the case of volume fraction of inclusions equal to 25%, we found that a minimum of 20 inclusions are required to fulfil these conditions. Therefore the FE simulations were based on cubic RVEs with a random isotropic distribution of at least 20 inclusions. In most cases however we decided to use 30 inclusions. The ABAQUS solver was used to solve the finite element problem with C3D1ØM elements – a 10 node tetrahedron element (figure 3).

Figure 3 Finite element model of RVE containing 30 inclusions having uniform random distribution of inclusions and a volume fraction of 0.25. Note that the structure is periodic and inclusions intersecting a face of cube also appear on the opposite face.


The phase average stresses were calculated as a volume weighted average of the stresses in the elements forming the inclusion phase. For all the calculations, the inclusion phase was isotropic glass fiber and matrix was polyamide with a Young’s modulus of 72 GPa and 3 GPa respectively, while the Poisson’s ratio was 0.22 and 0.37 respectively. Results of comparisons For the case of uniform in-plane random orientation, the orientation distribution tensor is a11=a22=0.5. A RVE with 30 inclusions was created with a uniform random orientation of inclusions. The inclusion average stresses were calculated in the global co-ordinate system in which the orientation tensor is defined. The inclusion average of stresses in the global direction, S11 was found by all the three methods to follow a quasi-sinusoidal trend with the inclusions with orientations closer to 0° with respect to the loading axis were stressed higher than the ones with higher orientation angles. However the peak in the curve of PGMT was much lower than in the predictions by both MoriTanaka and full finite element calculations. The predictions by the PGMT were off significantly in all the cases. The predictions by both Mori-Tanaka formulation and FE were however in excellent agreement with each other for both the axial and transverse stresses. Similar trend was observed for a range of volume fractions between 0.01 and 0.25 for average stresses in axial direction (figures 3a, 3b, 3c) as well as transverse direction (figures 4a, 4b, 4c). The scatter in the FE results was found to be increasing with increase of the volume fraction. We were not able to perform calculations for higher volume fraction owing to difficulties in creating a FE model and determining the size of RVE with minimum scatter.


Figure 4 Inclusion average stresses in the global loading direction, S11 for random orientation of inclusions , applied load is 1% strain: (a) vf = 0.1, with orientation tensor (a11=0.51, a22=0.49); (b) vf = 0.01, with orientation tensor (a11=0.54, a22=0.46) ; (c) vf = 0.25, with orientation tensor (a11=0.52, a22=0.48)


Figure 5 Inclusion average stresses in the transverse to global loading direction, S11 for random orientation of inclusions , applied load is 1% strain: (a) vf = 0.1, with orientation tensor (a11=0.51, a22=0.49); (b) vf = 0.01, with orientation tensor (a11=0.54, a22=0.46) ; (c) vf = 0.25, with orientation tensor (a11=0.52, a22=0.48)

The average stiffness of the composite is a direct function of the phase average stress (and strain), both methods give similar values of the phase average. Thus both methods can be expected to give similar predictions of effective stiffness. The phase average stress in the inclusion phase was quite close for both the mean field methods and the finite element calculations as can be seen in table 1. Table 1: Comparison of prediction of average stress [MPa] in inclusion in the global coordinate system for different volume fractions subjected to uniaxial strain loading of 1%. Aspect ratio of inclusions in all the cases is 3. Sl. No.

Description

1

Stress component

MT

PGMT

FE

S11

78.9

78.9

78.2

S22

-6.6

-6.7

-6.7

vf =0.01; a11= 0.54, a22=0.46 2


3

S11

83.7

83.8

87.1

S22

-7.3

-6.9

-8.8

S11

101.0

100.6

104.2

S22

-5.4

-5.6.

-7.3

vf = 0.1; a11=0.51, a22=0.49 4 5 vf = 0.25; a11=0.52, a22=0.48 6

Based on these findings it was decided to use Mori-Tanaka formulation as the principle mean-field homogenization scheme for all further calculations. For the modelling of damage and fatigue, the values of the stresses inside individual inclusions are important. 4: Damage modelling of short fiber composites Most of the short fiber composite polymers show a non-linear behaviour under mechanical loading. This non-linearity is due to a number of reasons; some of them are fiber-matrix debonding and fiber breakage; the present section describes models for these phenomena. 4a.) Fiber matrix-debonding: Fiber matrix debonding is the damage of the contact surface between the inclusion and matrix. Piggott [18] reviewed the importance of the fiber matrix interphase. He noted that much attention has been given to the fiber and polymer matrix and relatively less work has been done on the interphase. The interphase is at the heart of the composite since it transfers the stresses from the matrix to the fiber. There are severe changes in the distribution of stresses within the composite if the interface is damaged. Onset of fiber matrix debonding is estimated by a modified Coulomb criterion[19]. Mathematically, modified Coulomb criteria is written as

 N     C

4.

Where, σN is the normal component of interfacial stress, τ is the tangential component of the stress vector and β is the shear contribution co-efficient and is experimentally determined to be 0.5 for most polymers. σN and τ are dependent on the interfacial stress and are calculated by applying traction continuity conditions (figure 6.), a complete mathematical description of the equations can be found in [19].


Figure 6. Calculation of stresses in the interface by applying traction continuity conditions [19]

A critical question after calculating the onset of debonding is how to account for the redistribution of stresses due to onset of debonding. Mori-Tanaka formulation and indeed all the other mean field homogenization schemes assume perfect adhesion between the inclusion and the matrix. After onset of debonding; this condition is no longer valid. This problem was solved by replacing a partially debonded inclusion with a perfectly bonded inclusion with altered properties such that the two systems of inclusions are mechanically equivalent to each other. Below in figure 7 is a schematic representation of the idea

Figure 7 Schematic representation of treatment of debonded inclusion. A debonded inclusion A is replaced by a perfectly bonded equivalent inclusion B

Consider a composite material with inclusion phase consisting of inclusion A , which suffered debonding at certain applied load. For the next increment of load, Mori-Tanaka formulation cannot be used since the basic assumption of perfect adhesion is no longer valid. Thus for the next increment of load, the debonded inclusion A is replaced by an inclusion B, which is perfectly bonded. The mechanical properties of inclusion B should be estimated in such a way that the two composite systems are equivalent to each other. In this context,


equivalence means that the effective properties of both the composite systems are same and also that the stresses inside the inclusions (inclusion A and inclusion B) are the same. To achieve both conditions of equivalence, properties of inclusion B are estimated as a function of properties of inclusion A, location of debonding and region of debonding. Based on the location of onset of debonding, debonding propagation can be grouped into two categories: Case 1 where there is debonding from tip of inclusion towards centre and case 2 where the debonding starts from the centre of inclusion and propagates towards the tip.

Figure 8. Schematic representation of debonded inclusion: Case1. The onset of debonding is from the tip- dotted red surface denotes debonded surface; debonding is fully characterized by values Δy and Δx

For this case the properties of inclusion B are calculated using the following expressions x E x  Eincl (1  y )

y E y  Ez  Eincl (1  x / AR )

9.

xy Gxy  Gxz  Gincl (1  x / AR ) yz Gyz  Gincl (1  y )

Where, Ex, Ey, Ez are the Young’s moduli of the inclusion B and Gxy, Gxz are the shear moduli.


Figure 9. Schematic representation of debonded inclusion: Case2. The onset of debonding is from the centre- dotted red surface denotes debonded surface; debonding is fully characterized by values Δy and Δx

For case 2, properties of inclusion B are calculated by the following expressions x E x  Eincl (1  y )

y E y  E z  Eincl *

volbonded volincl

xy Gxy  Gxz  Gincl *

volbonded volincl

10.

yz Gyz  Gincl (1  y )

The elastic modulus of an orthotropic material can be fully described by 9 elastic constants (Three Young’s moduli, Three shear moduli and Three Poisson’s ratios) However, we cannot independently alter the values of each of the 9 elastic moduli, since the stiffness matrix of the inclusion must be always positive definite. Six elastic constants are changed as described above, while three Poisson’s ratios were altered to meet the positive definite conditions if required. Finite element models were used to validate the models proposed for physical equivalence. Inclusion A with debonded surface is modelled in finite elements, while the corresponding Inclusion B which is perfectly bonded is modelled by the Mori-Tanaka formulation. A finite element model with a single inclusion was created. The inclusion in finite element is modelled as a cylinder as opposed to an ellipsoid for Mori-Tanaka formulation. Cylinder is chosen as the shape for finite element formulation, since it is easier to model debonded partitions in cylindrical surface. The outer surface of the inclusion is segmented into a number of regions. A certain region can be modelled as perfectly bonded if the surface of that region is tied to the corresponding surface of matrix. Debonding condition is simulated if a contact surface allows the inclusion and


matrix to separate but not penetrate into each other. Periodic boundary conditions are applied to closely approximate the condition of infinite RVE.

Figure 10 Sketch of inclusion divided into various segments. Note that the shape of inclusion in finite element model is cylinder.

Same materials viz. glass fiber inclusion and polyamide matrix, which were used for comparison of different mean field methods in section 3, were used for the validations. Results of finite element calculations showed good match for the average stresses in the composite as well as for prediction of the stresses inside the inclusion for all types of uniaxial loads viz. axial, transverse and shear. Figure 11 shows comparison of average stresses in composite and inclusions by FE and equivalent system Mori-Tanaka formulation for different extent of debonding under an applied load of 1% in the axial direction.


Figure 11 Comparison of average stresses in composite and inclusions for the case of debonded inclusion in FE and perfectly bonded equivalent inclusion in MoriTanaka formulation. The inclusion was made of glass fiber in a resin of polyamide with a volume fraction of 0.1%

4b.) Fiber breakage: To account for fiber breakage, the fiber is assumed to be brittle in nature. The fiber is said to be broken if the applied principle stress in the inclusion is higher than the ultimate strength of the inclusion. Treatment of broken fiber is comparatively easier as compared to fiber matrix debonding. After a fiber is calculated to be broken, it is replaced by two fibers of half the length each. The physical properties of the broken fiber are assumed to be the same. The change in geometry of fiber leads to new Eshelby tensor and then subsequently new equivalent properties of the composite material. 5: Micromechanics and Fatigue of short fiber composite Micromechanics events at the constituent level are thought to have a big effect on the fatigue properties of the composite material. Most metals have a stress level at which it can sustain infinite number of load cycles, this stress is commonly known as the endurance strength of metal. However in the case of a composite, there is no clear cut stress level at which the composite is able to sustain infinite number of cycles. For composite materials endurance strength is defined here as the stress level at which the component is able to sustain one million cycles. The fatigue behaviour of composite materials can be linked to the onset of damage. In the case of metals the onset and propagation of a single crack leads to complete failure of a component. As a starting point it was hypothesized that the stress level at the onset of damage should correspond to the endurance strength of a composite. A number of composite configurations were considered and it was seen that the onset of fiber matrix debonding is the first damage event in all the composite configurations. However the stress levels at which debonding starts was found to be much lower than the measured “endurance strength” of composite. It can


be inferred from these calculations that for a composite material initiation of damage is not enough to cause failure in a composite. There are multiple damage events in a composite material before it propagates to cause final failure. Unlike a metal, damage in a composite is not localized. Figure 12 below depicts the onset of debonding for a variety of composite configurations with varying aspect ratio and orientation of inclusions with volume fraction 0.161 (weight fraction of 35%). The horizontal dotted lines are representative of typical ranges of “endurance strength” reported in experimental data presented in literature [20] for short fiber composites made of glass fiber and similar matrix system.

Figure 12 The onset of debonding for a variety of composite configurations with varying aspect ratio and orientation of inclusions with volume fraction 0.161 (weight fraction of 35%).

This observation leads to the conclusion that onset of a single damage is not enough to propagate and cause final failure even if the load is repeated for millions of cycles. For a composite component to fail there must be several damage events occurring at the RVE. Thus the damage in a composite is not local like metal, but is more spread and global in nature. 6: Scaling of SN curve So far in this article we have discussed the micromechanics of a composite RVE. We now consider a component level analysis of an injection moulded composite material. Injection moulded components have a different orientation and length distribution of inclusion at every material point, which is characterised by its own RVE. This leads to different fatigue properties at every point. Fatigue analysis could be done at three major scales namely micro,


meso, and macro. Micro-scale analysis consists of studying and analyzing events occurring at the constituent level i.e. fiber, matrix and the interface between them, this method is usually quite accurate and can also predict the mode of damage, but requires computational resources which are usually not available to researchers and engineers. Macro-scale analysis takes into account damage and failure at the component level, with no or little regard to events occurring at the micro-level; this method though computationally cheap does not capture events at the micro-level and sometimes cannot predict the mode of failure. Knowledge of mode and extent of damage at micro-level is important to understand the damage and fatigue characteristics of composite component. An intermediate level of analysis is the meso analysis. Meso-level is the intermediate scale of analysis between the micro and macro scales. The scale of analysis is usually a matter of choice and requirement, with different scale suitable for different type of composite materials. A combination of two scales viz. micro-macro scale is usually implemented for the case of static analysis of short fiber composite materials. Each point in the FE model is the center of a RVE, the effective properties of the composite are estimated using the Mori-Tanaka formulation. The stress levels in the FE model are calculated with the RVE properties. A similar strategy cannot be applied for the fatigue of a composite material due to the absence of an analytical method to compute the fatigue properties of a short fiber composite. Unlike Mori-Tanaka formulation which can be used to compute the effective static properties of a composite, there is no such formulation for the fatigue properties of the composite. One possible method of estimating the SN curves is to use the concept of scaling of SN-curves, as opposite to a purely test based method which is dependent on a number of tests. The concept is to proceed from micro to macro level using a scaling of a Master SN curve. Unlike purely test based methods the scaling will not just be deducted from interpolating extreme orientation situations, but start from a master SN-curve achieved from tests with coupons of random fiber orientation. In other words, for a given coupon with a given orientation, volume fraction and length distribution of fibers, a SN- curve is measured from tests. From that curve it is then proposed to scale SN curves for different orientations, volume fraction and length distribution of fibers. This proposed method is hybrid in nature involving both micro-mechanics and tests. Such a method could be more accurate and also depend on fewer tests. This method could also have the advantage of not depending on tests on coupons with specific orientations which are usually time consuming and difficult to perform. Instead, the input required for this method would be a SN curve derived from tests performed on coupons with random orientations.


Figure 13 Schematic representation of concept of scaling of SN-curves

This idea of scaling of SN-curve is based on the scaling of a damage operator that represents the loss of stiffness due to different damage events. The concept of scaling was tested against experimental data [20]. De Monte conducted a series of fatigue tests for glass fiber and polyamide short fiber composites with different orientations. The volume fraction of glass fiber was 19.7% (35% weight fraction), while the average aspect ratio of the inclusion was 26. For each of the cases one of the experimental SN curve was assumed to be the input SN curve for scaling and the predictions of the SN curves for two different orientations are indicated by dotted lines.

Figure 14 Scaling of SN curve with input of 90-deg coupon (indicated by red line). The coupons are injection moulded and made of polyamide and glass fiber with a weight fraction of 35%. Points in grey are estimated using the scaling based on damage parameter and SN-curves indicated by solid dotted lines are predicted by the scaling approach.


Figure 15 Scaling of SN curve with input of 30-deg coupon (indicated by blue line). The coupons are injection moulded and made of polyamide and glass fiber with a weight fraction of 35%. Points in grey are estimated using the scaling based on damage parameter and SN-curves indicated by solid dotted lines are predicted by the scaling approach.

Figure 16 Scaling of SN curve with input of 0-deg coupon (indicated by black line). The coupons are injection moulded and made of polyamide and glass fiber with a weight fraction of 35%. Points in grey are estimated using the scaling based on damage parameter and SN-curves indicated by solid dotted lines are predicted by the scaling approach.

It was seen that by scaling we were able to predict with good accuracy the SN curves for different orientations. Scaling of SN curve was validated for


different orientations of the input SN curve. Good accuracy is achieved irrespective of the orientation of the input SN curve. The input for such an analysis is only one input SN-curve and elastic moduli of the matrix and fiber. This concept of scaling is being implemented as a part of fatigue solver LMS Virtual.Lab Durability [27]. 7: Way forward, some perspectives As discussed in the above sections, a thorough study of micromechanics damage was undertaken and method to estimate the SN curves based on damage operator based scaling was developed. A next step in this direction should be correct representation of material non linearity of matrix. Scaling is based on quantifying and adding up the irreversible damage events. Differentiating between reversible material non-linearity and irreversible material non-linearity is a key challenge going forward. A key challenge in the durability analysis of composite material is the dependence on a lot of tests, experimental tests are usually quite time consuming and expensive, esp. for test specimen with fixed controlled fiber directions. There is need to for both reduction of time and cost of a fatigue test in composite materials. Some work has already been done in this direction [21, 22]; yet better experimental techniques need to be developed for proper testing of the models proposed. Damage causes degradation of material property and continuous stress redistribution, feedback of the changes at every cycle is not possible, efficient feedback algorithms which take into account events happening at the microstructure must be developed for short fiber composites. Feedback algorithms must be implemented and developed in fatigue solver LMS Virtual.Lab Durability [27]. An example of the feedback algorithm is the cycle jump algorithm [23] which was originally developed for constant amplitude load cycles. For real life load situations, which are both multi axial and variable amplitude locally not only the stiffness but also the fatigue resistance may change. The use of hysteresis operator based approaches [24, 25, and 26]; instead of traditional rainflow based approaches to fatigue accumulation is a promising approach. For variable amplitude loading it is very typical that the largest load cycles – that contribute most to the damage - take a very long time to complete, due to the many nested cycles inside. In this case the approach to only consider cycles when they are completed - like traditional rainflow based approaches - cannot account for occurrence of the cycle jump and changes in the fatigue behaviour during such a large cycle. Therefore the application of such hysteresis operator based approaches will be explored with respect to cycle jump based approaches. Hysteresis operators based analysis has already been implemented


in LMS Virtual.Lab [27] to analyze thermal fatigue in metals. The extension of such methods to composite materials will allow to also taking into account general load situations with good efficiency and accuracy at reduced computational costs. 8: Conclusions A thorough review of micromechanics of short fiber composites was presented. A comparison of the predictive abilities for stresses in individual inclusions of different mean field theorems with full finite element calculations as benchmark was undertaken and it was concluded that the Mori-Tanaka formulation should be the first choice mean field method. Initiation and propagation of different damage modes like fiber breakage and fiber matrix debonding was studied and new models for the same proposed. Models developed for the progressive fiber matrix debonding based on replacing the debonded inclusion with a perfectly bonded equivalent inclusion was presented. This model is compared with full finite element calculations and was found to be in excellent agreement with finite element calculations. New perspectives on fatigue of short fiber composites were gained based on the study of micromechanics damage. This perspective has led to better understanding of fatigue in short fiber composites. An improved method of predicting the SN curves of a short fiber composite was developed, based on the scaling of S-N curves with an appropriate damage indicator. This was validated against available experimental data. The proposed method was found to predict the SN curves of different configurations very accurately. The input for such an analysis is only one input SN-curve and mechanical properties of the matrix and fiber. CONTACT INFORMATION Atul Jain Researchpark Z1 Interleuvenlaan 68 3001 Leuven Belgium [email protected] Acknowledgments We gratefully acknowledge the support of IWT, Belgium for funding this research as a part of Baekeland project 100689 “Fatigue life prediction of


random fiber composites using hybrid multi-scale modeling methods (COMPFAT)”. The authors want to thank e-Xstream engineering for making modifications to software DIGIMAT, which was needed to making the comparison of Mori-Tanaka formulation and Pseudo-grain Mori-Tanaka formulation. The authors also want to place on record thanks to Prof. Issam Doghri for the discussions and explanations of the pseudo-grain concept. References: 1.

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