Crack growth simulation of bulk and ultrafine grained ...

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Int. J. Materials and Product Technology, Vol. 44, Nos. 3/4, 2012

Crack growth simulation of bulk and ultrafine grained 7075 Al alloy by XFEM Prosenjit Das* Foundry Group, Central Mechanical Engineering Research Institute (CSIR), Durgapur 713209, India E-mail: [email protected] *Corresponding author

Indra Vir Singh Department of Mechanical and Industrial Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, India E-mail: [email protected]

R. Jayaganthan Department of Metallurgical and Materials Engineering and Centre of Nanotechnology Indian Institute of Technology Roorkee, Roorkee 247667, India E-mail: [email protected] Abstract: In the present work, the effect of cryorolling on tensile strength, impact toughness and fracture energy of 7075 Al alloy has been studied experimentally, and quasi-static crack growth simulation has been performed by extended finite element method (XFEM) for both UFG and bulk Al Alloys. The 7075 Al alloy is rolled for 40% and 70% thickness reduction at cryogenic temperature. The microstructural characterisation of the alloy was carried out by using field emission scanning electron microscopy (FESEM). The cryorolled Al alloy after 70% thickness reduction exhibits fully formed ultrafine grain structure (grain size 600 nanometres) throughout the cross section as observed from FESEM micrographs. The mechanical properties of both alloys are obtained by tensile and Charpy impact testing. In XFEM simulations, this impact energy is used as a crack growth criterion for elastic-plastic ductile fracture. In XFEM, a discontinuous function is used to model the region behind the crack tip, whereas a crack tip is modelled by near-tip asymptotic functions. Keywords: ultra-fine grained Al alloy; cryorolling; fracture energy; elastic-plastic fracture; extended finite element simulation. Reference to this paper should be made as follows: Das, P., Singh, I.V. and Jayaganthan, R. (2012) ‘Crack growth simulation of bulk and ultrafine grained 7075 Al alloy by XFEM’, Int. J. Materials and Product Technology, Vol. 44, Nos. 3/4, pp.252–276.

Copyright © 2012 Inderscience Enterprises Ltd.

Crack growth simulation of bulk and ultrafine grained 7075 Al alloy

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Biographical notes: Prosenjit Das is presently working as a Scientist at the Foundry Group at CSIR-Central Mechanical Engineering Research Institute, India. His areas of research are advanced materials and manufacturing processes, mechanics of materials, computational material science, casting and solidification etc. He has written 34 scientific papers till date in different international journals and international/national conferences. He graduated from Kalyani Govt. Engg. College, India and post graduation from Indian Institute of Technology Roorkee, India. Indra Vir Singh is an Assistant Professor at the Department of Mechanical and Industrial Engg. of Indian Institute Of Technology Roorkee. He mainly works in the area of computational fracture mechanics. He performs the modelling and simulations of fracture and fatigue behaviour of materials. He is also developing constitutive models for Al-alloys. He has written over 100 scientific papers till date. R. Jayaganthan obtained his PhD in Materials Engg from IISC Bangalore, India and currently is an Associate Professor at the Department of Metallurgical and Materials Engg & Centre of Naotechnology Indian Institute of Technology Roorkee. His area of research is nano materials and has written over 100 scientific papers.

1

Introduction

The 7XXX series aluminium alloys are widely used as structural materials due to their excellent mechanical properties such as low density, high strength, ductility, toughness, and resistance to fatigue. The 7075 aluminium alloy is one of the most important engineering alloys and has been utilised extensively in aircraft structures because of its high strength-to-density ratio (Li and Stranik, 2001; Valiev et al., 2000; Meyers et al., 2006). Ultrafine-grained (UFG) form of Cu, Ni, Al alloys shows improved tensile, hardness properties as reported in the earlier literature (Panigrahi and Jayaganthan, 2008; Jayaganthan and Panigrahi, 2008; Lee et al., 2004; Shanmugasundaram et al., 2006). The literature on tensile and impact properties of UFG Al 7000 processed by cryorolling are very limited (Panigrahi and Jayaganthan, 2008; Jayaganthan and Panigrahi, 2008). Therefore, tensile and impact properties of the bulk and UFG form of 7075 Al alloy are evaluated experimentally in the present work, and used subsequently for the crack growth simulation using XFEM. UFG 7075 Al alloy is prepared by doing conventional rolling at cryogenic (liquid nitrogen) temperature. Rolling at cryogenic temperature suppresses dynamic recovery, and density of accumulated dislocations reaches to higher level as compared to room temperature rolling. With the multiple cryorolling passes, these high density dislocations are converted into grain fragments or ultrafine grain structures with high angle grain boundaries (GBs). Experimental results show a significant improvement in the tensile and impact toughness as observed in the present work. There have been no reported studies on crack growth simulation of UFG 7075 Al alloy by XFEM. Therefore, in the present work, crack growth in UFG and bulk 7075 Al alloy has been investigated using XFEM. The fracture energy absorbed by Charpy impact specimens of both bulk and UFG alloys are used as the crack growth criterion for XFEM

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simulations. Several numerical examples are presented to show the accuracy and efficiency of XFEM. An explicit crack propagation criterion for elastic-plastic ductile fracture has been derived based on the Griffith energy concept. Apart from boundary changes due to the crack growth, two material non-linearities arise in the ductile fracture, i.e., the plastic deformation in the bulk material and crack surface separation in the fracture process zone, which are characterised by the nucleation, growth and coalescence of micro-voids (Yang, 2005; Fan et al., 2007; Li and Siegmund, 2002). Mesh independent crack propagation simulations has been carried out under quasi-static Mode-I loading using the deformation plasticity theory (Elguedj et al., 2006; Rempler et al., 2006; Wang and Nakamura, 2004). A brief description of XFEM and energy criterion for the crack growth is given in Section 2 and Section 3, respectively.

2

Extended finite element method

In recent decades, the finite element method (FEM) has been widely used for modelling of fracture mechanics problems. In comparison to the classical FEM, the XFEM provides significant benefits in modelling crack growth problems. In traditional FEM formulation, the presence of a crack is ensured by aligning the element edges with the crack, whereas in XFEM (Sukumar and Prevost, 2003; Huang et al., 2003), physically there is no crack and its presence is ensured by partition of unity (PU) enrichment. In XFEM, a crack is not aligned with the element edges, which provides flexibility and versatility in crack growth modelling (Moes and Belytschko, 2002; Zi and Belytschko, 2003). The method is based on the enrichment of the standard finite element approximation with known functions to ensure the presence of a discontinuity. These enrichment functions require additional degrees of freedom (DOFs) tied with the nodes of the elements intersected by the crack. In this way, a discontinuity/crack is included in the FE model without modifying the original mesh. In general, XFEM has following advantages over FEM: •

In FEM for crack growth modelling, a mesh needs to conform the crack geometry so remeshing is required when the crack growth takes place whereas in case of XFEM; a crack is modelled by enrichment functions only so an original mesh is used to model the crack growth.

•

In FEM, singular elements are used at crack tip to represent the asymptotic crack tip displacement fields whereas in case of XFEM, asymptotic crack tip functions obtained from the theoretical background of the problem are used as enrichment.

•

In FEM, tracking time history of points, which require remeshing, creates many challenges whereas in XFEM, there is no issue of remeshing as crack location is independent of mesh.

•

In FEM, as crack grows, one must recreate a new mesh around the crack tip, which is quite expensive whereas in XFEM, there is no need of remeshing around the crack tip.


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The touchstone of the XFEM is its ability to incorporate a priori knowledge of a model’s response (say discontinuities) into the analysis through enrichment. The main objective of using various types of enrichment functions in XFEM is to reproduce a singular field around a crack tip and maintaining the continuity of primary variables among adjacent finite elements. In XFEM, the following approximation is utilised to calculate the displacement at any point within the domain (Belytschko and Black, 1999). ⎡ ⎤ 4 ⎢ ⎥ Φ α (x) bIα ⎥ u h (x) = N I ( x ) ⎢u I + H ( x ) a I +

⎢ ⎥ I =1 α =1 I ∈N r

⎥ ⎢ I ∈ N a ⎣ ⎦ N

∑

∑

(1)

where, uI is the unknown nodal displacement vector associated with the continuous part of the finite element solution, aI is the nodal enriched additional degree of freedom vector associated with discontinuous Heaviside function, and bIα is the nodal enriched additional degree of freedom vector associated with the asymptotic crack tip functions. The additional degrees of freedom are evaluated only for those elements, which are affected by the crack, and are obtained as a part of the solution similar to the unknown displacements. In the above equation, N is the set of all nodes in the mesh, Nr is the set of nodes whose shape functions are completely cut by the crack face, and Na is the set of nodes whose shape functions are cut by the crack tip. For node Nr, the support of the nodal shape function is completely cut by the crack. The Heaviside jump function H(x), is a discontinuous function across the crack surface and is constant on each side of the crack, i.e., +1 on one side and –1 on other. The crack tip enrichment is done by adding additional functions in the displacement approximation, which are given as 1 1 θ 1 θ 1 θ θ Φ1− 4 (x) = r n +1 cos , r n +1 sin , r n +1 sin sin θ , r n +1 cos sin θ 2 2 2 2

where r and θ are the local crack tip parameters of an evaluation point having x is the spatial coordinates and n is the Ramberg-Osgood parameter for the material. Figure 1

Enrichment of nodes in XFEM: circles nodes with 2 additional DOFs and squares nodes with 8 additional DOFs (see online version for colours)

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For crack modelling in XFEM, two types of enrichment functions are used, i.e., Heaviside function, H(x) and crack-tip functions. If no enrichment, then above equation N

(1) reduces to the classical finite element approximation: u h (x) =

∑ N (x)u . I

I

Hence,

I =1

XFEM retains most of the advantages of the FEM. Figure 1 shows the enriched nodes used in XFEM for crack modelling.

3

The energy criterion for crack growth

The fracture energy obtained by Charpy impact test is used as the crack growth criterion for quasi-static loading. The equations given here are limited to quasi-static crack growth only. The functional relationship of Charpy impact energy is given as ET = EI + EP

(2)

where, ET = total fracture energy; EI = fracture initiation energy; and EP is fracture propagation energy. The fundamental principle of energy conservation with respect to unit area crack extension (Tajally et al., 2010) is based on Griffith fracture mechanics criterion which can be mathematically stated as (Yang, 2005): ∂ (WF − U e − U P − Es ) with ∂A ⎧Tr > ET Crack propagates ⎫ ⎪ ⎪ ⎨Tr = ET Critical condition ⎬ ⎪T < E Crack does not prpagate ⎪ T ⎩ r ⎭

Tr =

(3)

where Tr is the total energy required for crack propagation, Ue is the elastic strain energy and Up is the plastic strain energy of the system respectively. WF is the work done by the externally applied loadings, Es is the surface dissipated energy and A is the total crack surface area (Michopoulos, 1988; Priest, 1998; Tzadka and Schulgasser, 2009). The result of first two terms in equation (3) is the strain energy release rate (G) which is considered as a driving force for crack growth. G=

∂ (WF − U e ) ∂A

(4)

The last two terms of equation (3) represent the plastic energy dissipation rate and the energy dissipation rate due to crack surface separation. The summation of these two terms represents the crack growth resistance, which is defined as ‘energy dissipation rate’ R. R=

∂ (U p + ES ) ∂A

(5)


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Equation (3) says that whenever the crack driving force exceeds the crack resistance, the crack will propagate. In context of FEM, the external work P and the total strain energy of the system U can be expressed respectively as, P = uT F

U = Ue + U p =

(6)

∫

V

WdV

(7)

where, V is the system volume, u is the nodal displacement vector, F is the nodal equivalent force vector respectively and W is the total strain energy density. The volume integration in equation (7) can be carried out in an element by element way. The strain energy of each element is calculated first for all the elements.

4

Results and discussion

4.1 Experimental results The optical micrograph of the bulk alloy (starting material) and SEM micrographs of the cryorolled (CR) Al 7075 alloy after 40% and 70% thickness reduction are shown in Figures 2(a) to 2(c). The microstructure of the bulk alloy exhibits lamellar grains lying parallel to the ingot axis. The average grain size is around 40 μm. The grain size is reduced to around 950 nm and 600 nm for the CR samples subjected to 40% and 70% thickness reduction, respectively as observed from the Figures 2(b) and 2(c). Since the dynamic recovery is effectively suppressed by rolling at liquid nitrogen temperature (–190°C), the CR Al 7075 alloy shows high fraction of high angle GBs and high amount of dislocation density as reported in our earlier work (Panigrahi et al., 2009; Das et al., 2010, 2011). Figure 3 shows the tensile properties of 7075 Al alloy cryorolled at different percentage of thickness reduction. It is observed that the tensile strength (UTS) has increased from 500 MPa to 530 MPa (nearly 6% increase) and yield strength (YS) has increased from 260 MPa to 430 MPa (nearly 66% increase) for the 40% reduction in the cryorolled samples. However, for 70% reduction (e = 1.8) in the CR samples, UTS has increased from 500 MPa to 550 MPa (nearly 10% increase) whereas YS increased from 260 MPa to 540 MPa (nearly 108% increase). The CR samples subject to 40% thickness reduction are also tested and compared with that of 70% reduction. Figure 4 shows the impact toughness properties of 7075 Al alloy cryorolled to different thickness reduction. It is evident that impact toughness of CR samples has increased due to breakage of the large aluminium dendrites, modified grain or grain fragment with high angle boundaries, ultrafine grain or grain fragments. Impact energy of starting bulk Al alloy was 17 J, which is increased to 21 J and 27 J after 40% and 70% thickness reduction, respectively.

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Figure 2

(a) Optical micrograph of starting material and FESEM image of (b) 40 CR and (c) 70 CR

Figure 3

Tensile properties of Al 7075 alloy after different percentage of thickness reduction (see online version for colours)

Crack growth simulation of bulk and ultrafine grained 7075 Al alloy Figure 4

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Effect of cryorolling on the impact energy of 7075 Al alloy (see online version for colours)

4.2 XFEM simulation The materials behaviour of UFG as well as bulk 7075 Al-alloy is assumed as elasto-plastic. Therefore, elasto-plastic XFEM simulation has been performed in the present work using ABAQUS software under plain-strain condition. The computation of the strain energy release rate, plastic dissipation, stress distribution ahead of the crack tip (Yang, 2005), and crack propagation simulations shows the improved fracture properties of UFG 7075 Al alloy as compared to the bulk alloy. The cracks growth direction is obtained by maximum hoop stress criterion. Table 1 shows the elastic-plastic material properties of the 7075 Al alloy in its UFG form and bulk form. The properties in UFG form are obtained after processing the bulk alloy by cryorolling technique. Table 1

Elastic–plastic material properties of the 7075 Al alloy

Material

E (GPa)

υ

σy (MPa)

N

Al 7075 alloy (Bulk)

72

0.33

260

10

Al 7075 alloy (UFG)

72

0.33

540

12

Mesh independent crack propagation simulations has been carried out under quasi-static Mode-I loading using the deformation plasticity theory based on Ramberg-Osgood relationship (Fan et al., 2007; Abaqus Analysis User’s Manual, 2009; Abaqus Theory Manual, 2009). In case of deformation plasticity, the material behaviour is modelled by a polynomial popularly known as Ramberg-Osgood relation. It models the material behaviour (elastic-plastic) by one function only. In this model, stress and strain involving plastic deformation has the following relationship, ⎛ σ ⎞ E ε = σ +α ⎜ ⎟ ⎝ σo ⎠

n −1

σ

(8)

where σ is the stress, ε is the strain, E is the Young’s modulus, α is the ‘yield’ offset (0.2%), and n (> 1) is the hardening exponent for the plastic deformation. The model is

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often used to obtain fully plastic solutions for small strain fracture mechanics problems where strain energy release is important. For evaluating strain energy release, required strain energy density is computed in this model as,

∫

W = σ dε

(9)

During analysis at each integration point, the final estimate of the kinematic solution is provided to the constitutive routines, which must provide the corresponding stress tensor calculated for the material model being used. Since this material model is non-linear, the method followed for stress solution is described below. Corresponding to a strain value of ε, σ may be found from equation (8). Since equation (8) is non-linear, σ is found by Newton-Raphson (Desikan and Sethuraman, 2000; Gopalakrishna and Greimann, 1988) method. In this method, a correction cσ to σ (Abaqus Theory Manual, 2009) is given as n −1 n −1 ⎡ ⎛ σ ⎞ ⎤ ⎛ σ ⎞ ⎢1 + nα ⎜ ⎟ ⎥ cσ = Eε − σ − α ⎜ ⎟ σ ⎝ σ o ⎠ ⎥⎦ ⎝ σo ⎠ ⎢⎣

(10)

σ = σ + cσ

(11)

Here, σ = Eε if Eε ≤ σo and σ = ( Eεσ o n −1 / α )

1/ n

if Eε > σo.

In this case, the current value of material stiffness matrix is given by dσ E = dε 1 + nα ( σ / σ o )n −1

(12)

A comparison between deformation plasticity and other standard elasto plasticity models are shown below. Isotropic elasto-plasticity model is commonly used for metal plasticity calculations, either as a rate-dependent or rate-independent model. The strain rate is decomposed in two components as ε = ε el + ε pl

(13)

The elastic behaviour is assumed as linear and isotropic and, therefore can be written in terms of two temperature dependent material parameters. For the purpose of this development, it is appropriate to choose these parameters as bulk modulus, K, and shear modulus, G. These are computed readily from the user’s input of Young’s modulus, E, and Poisson’s ratio, ν as: K=

E E and G = 3(1 − 2ν ) 2(1 + ν)

(14)

Thus, both the rate-independent model and the integrated rate-dependent model give the general uniaxial form as q = σ(εpl). where εpl is the equivalent plastic strain (Abaqus Theory Manual, 2009). An increment in plastic flow is determined by evaluating q based on the elastic response of the material. The Johnson-Cook plasticity model is particularly suited to model high-strain-rate deformation of metals. This model is a particular type of Mises plasticity that includes


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analytical forms of the hardening law and rate dependence. A Mises yield surface with associated flow is used in the Johnson-Cook plasticity model. Johnson-Cook hardening is a particular type of isotropic hardening where the static yield stress, σy, is assumed to be of the form,

σ y = ⎡⎣ A + B ( ε pl )n ⎤⎦

(15)

where εpl is the equivalent plastic strain and A, B and n are material parameters measured experimentally. The present work is based on grain refinement into the ultrafine-grain regime. So, significant improvement is observed in the yield strength of the material and due to cryorolling, work hardening took place in the material. Hence, a model which best describes the improved damage properties of the UFG alloy is the deformation plasticity which takes into account the yield strength as well as hardening exponent of the material and models the material behaviour from elastic to plastic range using one polynomial function only. The other material models such as ductile damage based on void growth and nucleation assumes equivalent plastic strain at the onset of damage as a function of stress triaxiality and strain rate, which will not be useful to consider the effects of ultrafine-grain formation. Figure 5

Edge crack under uniaxial tension

4.2.1 Case 1: Single edge crack In first example, an edge-cracked plate is modelled under pure mode-I loading. Figure 5 shows a geometry of the edge cracked plate with height (h = 80 mm), width (b = 40 mm), thickness (t = 10 mm) and crack length (a = 5 mm). The applied value of uniaxial tensile stress (σ) is 200 MPa. Edge crack has been modelled by unstructured mesh using 41 elements bilinear quadrilateral along the edges. Mesh sensitivity analysis has been performed by refining the mesh (considering 81 elements along the edges), which reveals

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that the mesh independent crack growth capabilities of XFEM. Figure 6 shows XFEM capabilities of crack propagation without a need of mesh refinement. The assessment of the stress-strain near the crack tip is one of the most important tasks to analyse the fracture behaviour of structural components containing flaws. The present work concentrate on the elastic-plastic ductile fracture behaviour of bulk 7075 Al alloy and its UFG form. Crack tends to follow straight path here due to transition from tensile to shear dominated crack growth. Figure 6

Crack propagation simulations in a edge-cracked plate shows mesh independence, (a) initial crack at time scale factor: 0 (b) crack at time scale factor: 0.2 (c) crack at time scale factor: 0.6 (d) crack at time scale factor: 0.8 (e) crack at time scale factor: 1.0 (f) crack at time scale factor: 1.0 (refined mesh) (see online version for colours)

(a)

(b)

(c)

(d)

(e)

(f)

Crack growth simulation of bulk and ultrafine grained 7075 Al alloy Figure 7

Von-Mises stress distribution over the plate, (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (see online version for colours)

(a) Figure 8

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(b)

Maximum principal stress distribution over the plate, (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (see online version for colours)

(a)

(b)

A von-Mises stress distribution over the plate is shown in Figure 7 for both UFG and bulk alloy, whereas Figure 8 presents the maximum principal stress distribution over the plate for both UFG and bulk alloy. The stress plots show that the stress values along the crack path are higher in the vicinity of crack tip for both alloys. From the results presented in figures, it can be seen that the crack growth occurs in case of bulk alloy but not in case of UFG alloy. Enhanced plastic zone size ahead of the crack tip also supports the improved fracture resistance and crack arrest capabilities of the alloy in its UFG form which is shown in Figures 7 to 8. A wake of residual plastic deformation behind the crack

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tip is observed due to elastic unloading, and an irreversibility effect, which tends to stabilise the quasi-static crack growth. Experimental findings of crack arrest capabilities of the UFG alloy (grain size 600 nm) signifies that the fracture mechanism of the alloy is dominated by the interaction between a propagating crack and the GBs structure. The voids created by grain-boundary sliding which leaves wedges at the points of triple junction and also enhanced the plastic zone size ahead of the crack tip. The coarse grained (CG) bulk alloy with grain size ranging from 30 to 50 µm, the failure is dominated by the resistance of the grains with no influence of the GBs. Moreover, in comparison with the CG structure, a small grain size can potentially result in more homogeneous deformation, which can retard crack the nucleation reducing stress concentrations. In most planar slip materials, GBs provide a ‘topological obstacles to the slip’ (Vasudevan et al., 1997). In case of UFG alloy, the precipitations are smaller in size and well dispersed so a crack develops steps on the crack plane while bypassing the precipitates due to crack-precipitate interaction at the GBs. Figure 9(a) shows the microstructurally-informed model of crack deflection due to the diffused branching mechanism of crack, and a GB interaction is seen in case of UFG alloy. Enhanced amount of grain boundaries due to ultrafine-grain formation is responsible for crack branching mechanism. Figure 9(b) shows the experimental observation of crack deflection in a compact tension sample of UFG 7075 Al alloy and tendency of re-straightening due to shear dominated crack growth (Sutton et al., 1997; Pirondi and Dalle, 2001). Figure 9

(a) Crack paths for bulk (transgranular) and UFG (intergranular) 7075 Al alloy and (b) Experimental observation of crack deflection in a CT sample due to diffuse branching mechanism in case of UFG alloy and tendency of re-straightening due to shear dominated crack growth (see online version for colours)

Figure 10 shows the strain energy release for an edge crack under same loading and geometric conditions for two different mesh sizes. Strain energy release (Abaqus Analysis User’s Manual, 2009) was computed from stress intensity factor using the following equation, G=

KI , where, E ′ = E / (1 − ν 2 ) E′

(16)


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Figure 10 Plots of strain energy release for bulk and UFG 7075 Al alloys, (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (see online version for colours)

(a)

(b)

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Figure 11 Plots of plastic dissipation of whole model for both bulk and UFG7075 Al alloys, (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (see online version for colours)

(a)

(b)

In Abaqus, the stress intensity factors are extracted at the crack tip using interaction integral approach. Figure 11 shows a plot of plastic dissipation for both bulk and UFG 7075 Al alloy, whereas the stress along the crack path for both alloys are plotted in Figure 12. From the results presented in Figures 10 to 12, it can be seen that the external work is higher in case of UFG 7075 Al alloy in comparison to its bulk form. In UFG form of alloy, the plastic strain work is higher than its bulk form in the fracture process zone ahead of the crack tip. The plastic dissipation for the whole model is lesser in case of UFG form of the Al-alloy in comparison to bulk form of Al-alloy due to work hardening effect. Moreover, the UFG alloy shows a higher value of hardening exponent.


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Figure 12 Plots of stress along crack path of both bulk and UFG 7075 Al alloys, (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (see online version for colours)

(a)

(b)

4.2.2 Case 2: Two edge cracks Figure 13 shows a model with two edge cracks under pure mode-I loading. The data used for the model is given as: height, h = 80 mm, width, b = 40 mm, thickness, t = 10 mm, and crack length on both the edges a = 5 mm. The value of uniaxial tensile stress (σ) is 200 MPa. Von-Mises stress distribution for double edge cracked geometry is shown in Figure 14 for both UFG and bulk alloy, whereas Figure 15 presents the maximum principal stress distribution over the plate for both UFG and bulk alloy. Von-Mises and maximum principal stress distribution ahead of the crack tip is more in case of UFG alloy. The difference between stress distributions for both form of the alloy occurs due to their difference in elasto-plastic material properties.

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Figure 13 Double edge crack under uniaxial tension

Figure 14 Von-Mises stress distribution over the plate, (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (see online version for colours)

(a)

(b)

Figures 16 and 17 shows the strain energy release and plastic dissipation respectively for bulk and UFG alloys. A similar type of behaviour like edge crack has been observed under same loading and geometric conditions. The strain energy release and external work is higher in case of ultra-fine grained 7075 Al alloy. In this case, crack advancement in the right edge of the bulk alloy can be seen, whereas it gets arrested in case of UFG alloy. Stress values are plotted in Figure 18 along the crack path. The values of stresses are higher near the crack tip for both materials.


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Figure 15 Maximum principal stress distribution over the plate (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (see online version for colours)

(a)

(b)

Figure 16 Plots of strain energy release for both bulk and UFG 7075 Al alloys, (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (see online version for colours)

(a)

(b)

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Figure 17 Plots of plastic dissipation of whole model for both bulk and UFG7075 Al alloys, (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (see online version for colours)

(a)

(b)

4.2.3 Case 3: Centre crack in tension A plate with centre-crack is shown in Figure 19 under pure mode-I loading. The model geometry is having height, h and width, b along with a centre crack of length a. The dimensions of the cracked plate are taken as height, h = 80 mm, width, b = 40 mm, thickness, t = 10 mm and crack length, a = 6 mm. The value of uniaxial tensile stress (σ) is 500 MPa. Von-Mises and principal stresses distribution ahead of the crack tip for a centre crack problem are plotted in Figure 20 and Figure 21, respectively. The difference in the values of stresses is more in case of UFG alloy. The strain energy release and plastic dissipation for the centre cracked plate of Bulk and UFG alloy are plotted in Figure 22 and Figure 23, respectively. Stress values are plotted along the crack path in Figure 24 for both materials. Stress values at the crack tip are much higher for UFG alloy due to its high yield strength.


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Figure 18 Plots of stress along crack path of both bulk and UFG 7075 Al alloys, (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (see online version for colours)

(a)

(b) Figure 19 Centre crack under uniaxial tension

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Figure 20 Von-Mises stress distribution over the plate, (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (see online version for colours)

(a)

(b)

Figure 21 Maximum principal stress distribution over the plate, (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (see online version for colours)

(a)

(b)

The results obtained for bulk 7075 Al alloy and its UFG form are compared against each other to show the improved fracture behaviour of the UFG alloy. All these case studies shows that for the same loading, boundary and geometric conditions, the crack growth took place in case of bulk 7075 alloy, whereas it get arrested in case of UFG form of the alloy. Moreover, the size of the plastic zone ahead of the crack tip is much bigger in case of UFG form of the alloy, which indicates the improved fracture toughness due to grain refinement.


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Figure 22 Plots of strain energy release for bulk and UFG 7075 Al alloys, (a) bulk 7075 Al Alloy (b) UFG 7075 Al alloy (see online version for colours)

(a)

(b) Figure 23 Plots of plastic dissipation of whole model for both bulk and UFG7075 Al alloys, (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (see online version for colours)

(a)

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Figure 23 Plots of plastic dissipation of whole model for both bulk and UFG7075 Al alloys, (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (continued) (see online version for colours)

(b) Figure 24 Maximum principal stress distribution along the crack path, (a) bulk 7075 Al alloy (b) UFG 7075 Al alloy (see online version for colours)

(a)

(b)


5

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Conclusions

In the present study, tensile and impact toughness behaviour of bulk and UFG form of 7075 Al alloy has been evaluated. XFEM simulations were performed to study the crack growth behaviour. The impact energy obtained by Charpy test has been utilised as a crack growth criterion under quasi-static loading. Elastic-plastic material properties such as yield strength, hardening exponent have been evaluated by tensile test to carry out XFEM simulations. Three case studies were selected to demonstrate the use of energy based crack growth simulations. The results obtained from the case studies shows that the criterion implemented here for simulating the crack growth is quite efficient and versatile, and can be extended complex fracture mechanics problems. The computation of the strain energy release, plastic dissipation, stress distribution ahead of the crack tip and crack propagation simulations demonstrate the improved fracture properties of the UFG 7075 Al alloy in comparison to the bulk 7075 Al alloy. Numerical examples also shows that for the same loading, boundary and geometric conditions, the crack growth took place in case of bulk 7075 alloy whereas it got arrested in case of UFG form of the alloy. Moreover, the size of the plastic zone ahead of the crack tip has been found much bigger in case of UFG alloy. This study shows a significant improvement in the fracture toughness of UFG 7075 alloy, which is due to the effective grain refinement to the ultrafine grain regime.

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