Unit cell simulations and porous plasticity modelling for recrystallization textures in aluminium alloys

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21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy 21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy

Unit cell simulations and porous plasticity modelling for Unit cell simulations and porous plasticity XV Portuguese Conference on Fracture, PCF 2016, Februarymodelling 2016, Paço defor Arcos, Portugal recrystallization textures in10-12 aluminium alloys recrystallization textures in aluminium alloysa b a L.E.B Dæhlia,∗modeling , J. Faleskogof , T.aBørvik , O.S. Hopperstad Thermo-mechanical high pressure turbine blade of an a,∗ b a a L.E.B(SIMLab), DæhliCentre , J.forFaleskog , T. Børvik O.S. Hopperstad Structural Impact Laboratory Research-based Innovation (CRI),, Department of Structural Engineering, Norwegian gas(NTNU), turbine University of airplane Science and Technology NO-7491engine Trondheim, Norway Structural Impact Laboratory (SIMLab), Centre for Research-based Innovation (CRI), Department of Structural Engineering, Norwegian a a

b Department

of Solid Mechanics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden University of Science and Technology (NTNU), NO-7491 Trondheim, Norway a b c b Department of Solid Mechanics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden

P. Brandão , V. Infante , A.M. Deus *

a

Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Abstract The well-known Gurson model has been heuristically extendedPortugal to incorporate effects of matrix anisotropy on the macroscopic c yielding of porous ductile solids. Typical components of Superior recrystallization textures forde aluminium alloys to calibrate Department of Mechanical Engineering, Instituto Universidade Lisboa, Roviscowere Pais, 1, 1049-001 Lisboa, The CeFEMA, well-known Gurson model has been heuristically extended toTécnico, incorporate effects of matrixAv. anisotropy on used the macroscopic the Barlat Yld2004-18p yield criterion using a full-constraint Taylor homogenization method. The resulting yield surfaces were Portugal yielding of porous ductile solids. Typical components of recrystallization textures for aluminium alloys were used to calibrate

Abstract b

further employed in unityield cell simulations using the finite element method. Unit cell calculations invoked to investigate the the Barlat Yld2004-18p criterion using a full-constraint Taylor homogenization method. Theare resulting yield surfaces were evolution of the approximated microstructure under pre-defined loading conditions and to calibrate the proposed porous plasticity further employed in unit cell simulations using the finite element method. Unit cell calculations are invoked to investigate the Abstract model. Numerical results obtained from the unit cell analyses demonstrate that anisotropic plasticthe yielding has porous great impact on evolution of the approximated microstructure under pre-defined loading conditions and to calibrate proposed plasticity the mechanical response of the approximated microstructure. Despite the simplifying assumptions that underlie the proposed model. Numerical results obtained from the unit cell analyses demonstrate that anisotropic plastic yielding has great impact on During their operation, modern aircraft engine components are subjected increasingly operating conditions, constitutive model, it seems capture the overall macroscopic response unittocell. However, todemanding further enhance the the mechanical response oftothe approximated microstructure. Despite of thethesimplifying assumptions that underlie the numerical proposed especiallythe themodel high pressure turbine (HPT) blades. Such conditions cause these that partsaccounts to undergo different types of time-dependent predictions, should be supplemented with a void evolution expression for directional dependency, and a constitutive model, itofseems toiscapture the overallusing macroscopic response the unit cell. However, to further enhance theable numerical degradation, one which creep. A model the finite element of method (FEM) was developed, in order to be to predict void coalescence criterion in order to capture the last stages of deformation. predictions, the model should be supplemented with void evolution that aircraft, accounts provided for directional a creep behaviour of HPT Flight dataarecords (FDR) expression for a specific by a dependency, commercial and aviation c the  2016 The Authors. Published byblades. Elsevier B.V. void coalescence criterion in order to capture the last stages of deformation. Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license company,under were responsibility used to obtainofthermal and mechanical data for three different flight cycles. In order to create the 3D model Peer-review Scientific of ECF21. c needed  2016 The Published byathe Elsevier B.V. Committee (http://creativecommons.org/licenses/by-nc-nd/4.0/). forAuthors. the FEM analysis, HPT blade scrap was scanned, and its chemical composition and material properties were Peer-review under responsibility ofanisotropy; the Scientific Committee of model ECF21. Peer-review ofgathered the Scientific Committee of FEM ECF21. obtained.under The responsibility data that was was Unit fed cell into the and differentGurson simulations Porous plasticity; Plastic simulations; Barlat Yld2004-18p; model were run, first with a simplified 3D Keywords: rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The Keywords: Porous plasticity; Plastic anisotropy; Unit cell simulations; Barlat Yld2004-18p; Gurson model overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be useful in the goal of predicting turbine blade life, given a set of FDR data.

1. Introduction

© 2016 The Authors. Published by Elsevier B.V.

1. Peer-review Introduction underalloys responsibility the Scientificapplications Committee ofoften PCF exhibit 2016. anisotropic behaviour under plastic deformaWrought metal used forofengineering tions. Rolling and alloys subsequent annealing of aluminium alloys induces plastic anisotropy in which material axes Wrought metal used Blade; for engineering often exhibitSimulation. anisotropic behaviour underthe plastic deformaKeywords: High Pressure Turbine Creep; Finiteapplications Element Method; 3D Model; align with the rolling, normal, and transverse directions. alloys The resulting mainly composed of cube and goss tions. Rolling and subsequent annealing of aluminium inducestextures plastic are anisotropy in which the material axes generic textures (Barlat and Richmond, 1987)directions. in addition to resulting some degree of random texture. Such aluminium align with the rolling, normal, and transverse The textures are mainly composed of cube andalloys goss are frequently employed in structural applications, for instance in automotive marine industries. Hence, predictive generic textures (Barlat and Richmond, 1987) in addition to some degree of and random texture. Such aluminium alloys material models capable in of structural accounting for the microstructural is of and great importance for structural integrity are frequently employed applications, for instance evolution in automotive marine industries. Hence, predictive assessment and design. material models capable of accounting for the microstructural evolution is of great importance for structural integrity assessment and design. ∗

Corresponding author. Tel.: +47 73594677

address: [email protected] Corresponding author.Tel.: Tel.:+47 +351 218419991. ∗ *E-mail Corresponding author. 73594677

E-mail address: [email protected] E-mail address: [email protected]

2452-3216 © 2016 The Authors. Published by Elsevier B.V. c 2016 2452-3216  The Authors. Published by Elsevier B.V.Committee of PCF 2016. Peer-review under responsibility of the Scientific Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the Scientific Committee of ECF21. c 2016 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). 2452-3216  Peer reviewunder under responsibility of Scientific the Scientific Committee of ECF21. Peer-review responsibility of the Committee of ECF21. 10.1016/j.prostr.2016.06.317

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A commonly observed ductile fracture mechanism is by nucleation, growth and final coalescence of microscopic voids (Hancock and Mackenzie, 1976). This mechanism is largely dependent upon the degree of triaxiality of the imposed stress states (Rice and Tracey, 1969). Triaxial loading cases that commonly arise in practical application facilitates enlargement of pre-existing and nucleated voids, which becomes important due to the loss of load-carrying capacity and material softening. In addition, previous investigations (Zhang, 2001; Gao et al., 2010) have revealed a pronounced effect of the Lode angle on ductile fracture even for isotropic matrix materials that are independent of the third deviatoric stress invariant J3 . This effect is closely related to the void evolution, which differs substantially between states of generalized tension, shear and compression. Ductile fracture of anisotropic metal alloys has received considerable interest in the literature during the past decade. Numerical unit cell studies that incorporate plastically anisotropic matrix behaviour (Keralavarma and Benzerga, 2010; Steglich et al., 2010; Keralavarma et al., 2011) reveal that the mechanical response is markedly dependent upon the induced anisotropy during plastic deformations. This dependence is rather obvious in terms of stress-strain response, but anisotropy also affects the evolution of the voids; both growth rate and shape evolution is related to the degree of anisotropy and orientation of the material axes. In effect, this has profound influence on the aggregate material behaviour. Local approaches to ductile fracture include porous plasticity models that account for the evolution of damage during plastic deformation. One such constitutive relation is the Gurson model (Gurson, 1977) which has been widely employed in numerical studies in the literature and subjected to many modifications. Among these include extensions to incorporate plastic anisotropy effects of the matrix material. Benzerga and Besson (2001) used upper-bound limit analysis to derive a yield function for matrix materials governed by Hill’s yield criterion for anisotropic materials. Efforts have also been made by Benzerga et al. (2004), and in this study the model was also supplemented with a coalescence criterion. Monchiet et al. (2008) and Keralavarma and Benzerga (2010) provide a constitutive model for plastically anisotropic porous solids in the case of non-spherical voids, however restricted to remain spheroidal. We should also note that some studies have been devoted to extend the Gurson modelling framework for matrix materials that are governed by a crystal plasticity formulation (see e.g. Han et al. (2013); Paux et al. (2015)) which is inherently anisotropic. The present study is largely inspired by the work undertaken by Steglich et al. (2010), where combined use of unit cell simulations and a homogenized material model gave promising results for predictions of the direction-dependent deformation and crack propagation of an Al2198 sheet metal alloy. They used the Gurson model in its original form, however incorporating an equivalent stress measure that accounts for the plastic anisotropy of the matrix material. The matrix material was described by the anisotropy model introduced in Bron and Besson (2004). We propose a similar phenomenological extension of the Gurson model in order to incorporate plastic anisotropy. The linear-transformation based yield criterion by Barlat et al. (2005) is used to describe the plastic anisotropy of the matrix material. Only an isotropic distribution of spherical voids that undergo spherical void growth will be considered. Thus, any anisotropy effects of initial void and particle morphology are precluded. 2. Matrix description A hypoelastic-plastic framework is assumed for the material behaviour. The elastic deformations are approximated by Hooke’s law while the plastic response is governed by orthotropic plasticity using the Barlat Yld2004-18p constitutive model (Barlat et al., 2005) with the associated flow rule and isotropic work-hardening. The two generic textures employed in this study are the main components for a recrystallization texture in aluminium alloys (Barlat and Richmond, 1987). A full-constraint Taylor homogenization procedure was used to calibrate the yield surfaces for the two textures shown in Fig. 1. We note that this method assumes that all grains are subjected to the same deformation, thus neglecting possible effects of stress and strain gradients within the grains. Details regarding the yield surface calibration may be found in Saai et al. (2013). Table 1: Generic material parameters for the matrix material. E [MPa]

ν

σ0 [MPa]

Q [MPa]

C

70000

0.3

100

100

10



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150◦

120◦ Σ2

90◦

180◦

150◦

60◦

210◦ Σ3

60◦

30◦ Σ3

Σ1

300◦

90◦

210◦

0◦

270◦

Σ2

180◦

30◦

240◦

120◦

2537 3

Σ1

240◦

330◦

0◦

270◦

(a)

300◦

330◦

(b)

Fig. 1: Plot of the yield surface in the deviatoric plane for (a) cube and (b) goss textures. The yield surfaces are generated for the case when the material axes coincide with the principal stress directions. The deviatoric angles θ = 0◦ , 30◦ , . . . , 330◦ correspond to those investigated in this study.

The Barlat Yld2004-18p model consists of two linear transformations of the stress deviator s˜ = C : s

∧

s˜ = C : s

(1)

where s˜ and s˜ are the transformed deviatoric stress tensors, C and C are the fourth-order transformation tensors accounting for plastic anisotropy, and s is the stress deviator. The equivalent stress is then defined as  m1  3 3   1   σeq ≡ ϕ (σ) =  |S˜ i − S˜ j |m  4 i=1 j=1

(2)

where S˜ i and S˜ j are the principal values of the transformed deviatoric stress. The reader is referred to Barlat et al. (2005) for further details about the anisotropic yield function. In the present work, the anisotropic yield criterion is calibrated for plane stress states, reducing the number of independent coefficients in C and C and the computational cost of the calibration. Plastic yielding of the matrix material is governed by the yield function Φ(σ, σM ) = ϕ(σ) − σM ≤ 0 where the matrix flow stress is assumed to be described by a one-term Voce rule   σM = σ0 + Q 1 − exp (−C p)

with the material parameters found in Table 1. The accumulated plastic strain p is calculated from  t σ : dp σ : dp ⇒ p= dt p˙ = σeq σeq 0

(3)

(4)

(5)

where the stress and plastic rate-of-deformation tensors are denoted σ and d p , respectively. 3. Porous plasticity model

To approximate the unit cell response in a single material element subjected to homogeneous deformation, a phenomenological extension of the Gurson model is proposed     Σeq 2 3Σh + 2 f q1 cosh q2 (6) − 1 − (q1 f )2 ≤ 0 Φ(Σ, σM , f ) = σM 2σM

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Here, Σeq is the macroscopic equivalent stress governed by Equation (2), Σh is the macroscopic hydrostatic stress, σM is the matrix flow stress, f is the void volume fraction and qi are the material parameters introduced by Tvergaard (1981). The model parameters qi will be calibrated from unit cell analyses, to optimize the correspondence between the homogenized material model and the mechanical response of the approximated microstructure for a variety of loading cases. These results are presented in Section 5. In this work, we adopt the void evolution law f˙ = (1 − f ) trD p

(7)

where trD p denotes the volumetric plastic strain rate. Note that this relation does not account for non-spherical void evolution during plastic deformation. For the highly anisotropic matrix materials considered in this work, this assumption is evidently violated due to the directional dependency of the matrix flow stress. The matrix flow stress σM is described by a generic work-hardening law calculated from Equation (4) with the matrix parameters listed in Table 1. For the voided solid, the matrix accumulated plastic strain is calculated from  t Σ : Dp Σ : Dp ⇒ p= dt (8) p˙ = (1 − f ) σM 0 (1 − f ) σM where the macroscopic stress and plastic rate-of-deformation tensors are denoted Σ and D p , respectively.

4. Unit cell modelling The unit cell simulations were carried out under the assumption that the principal stress directions are collinear to the orthotropy axes which enable the use of symmetry conditions to reduce the discretized model to a one-eight model, as illustrated in Fig. 2. In the numerical procedure, the material axes of the unit cell (mi in Fig. 2a) remain fixed. The principal stress directions are then varied in different analyses by changing the deviatoric angle (θ), which is defined as the angle between Σ1 and the current stress state Σ in the deviatoric plane (see Fig. 1). Hence, effects of changing the main loading directions relative to the anisotropy axes may be elucidated. Additionally, the stress triaxiality (T ) is varied to demonstrate its effect on the aggregate mechanical response.

Σj

Σi

m2 Σk m1 m3

(a)

(b)

Fig. 2: Illustration of (a) the representative volume element used in this study, where the material axes are aligned with principal stress directions and (b) the corresponding FE model where symmetry conditions in all three directions have been utilized.

In order to control the macroscopic stress state imposed to the unit cell, we prescribe values of the stress triaxiality (T ) and the deviatoric angle (θ). To this end, the principal stress vector is written on the form            θ  θ    cos  cos Σ1    1 1      2    2 vm cos θ − 2π       vm  cos θ − 2π   1 = Σ + Σ + T (9) Σ2  = Σeq        1 h 3  3 eq               3 3  cos θ + 2π  cos θ + 2π   1 Σ3 1 3 3



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where Σvm eq is the macroscopic equivalent von Mises stress, Σh is the macroscopic hydrostatic stress and θ is the angle in the deviatoric plane, taking values on the range 0 ≤ θ ≤ 2π. The stress triaxiality T is defined on the usual form T=

Σh Σvm eq

(10)

The unit cells are subjected to a wide range of stress states, corresponding to θ = 0◦ , 30◦ , . . . , 330◦ and T = 0.667, 1.0, 1.667, 3.0, which results in a total of 48 simulations for each texture. A multi-point constraint (MPC) user subroutine was implemented in the implicit FE solver Abaqus/Standard to maintain the imposed stress state throughout the numerical simulations. Details regarding the MPC subroutine and its application to unit cell analyses may be found elsewhere, see for instance Faleskog et al. (1998), Barsoum and Faleskog (2007), and Dæhli et al. (2016). 15.10

11.57

θ θ θ θ

= 0◦ = 90◦ = 120◦ = 240◦

T = 1.0

f f0 8.05

(a)

(b) 4.53

1.00 0.0

0.2

0.4

0.6

0.8

vm Eeq

(e) (c)

(d)

Fig. 3: Deformed configurations for the Goss texture at maximum equivalent stress for loading situations corresponding to a stress triaxiality of T = 1.0 for the deviatoric angles (a) 0◦ , (b) 90◦ , (c) 120◦ , and (d) 240◦ . Fringes of accumulated plastic strain are shown on the deformed configurations. The corresponding void growth plots are presented in (e).

Fig. 3 shows fringes of accumulated plastic strain on deformed configurations of the unit cell with the Goss texture for various deviatoric angles θ and a stress triaxiality of T = 1.0. The deformed configurations correspond to the first frame after the macroscopic equivalent stress reaches its maximum Σ˙ vm eq = 0, and consequently the macroscopic effective deformation is in general different between the shown configurations. Fig. 3e shows that the void growth for the Goss texture is linked to the magnitude of the imposed stress, from which it is evident that the void evolution is affected by the loading. Another interesting observation is that even though the loading states θ = 120◦ and θ = 240◦ are almost indistinguishable in terms of Σvm eq , seen from the radius of the yield surface in Fig. 1b, the void growth rate is very different. Also, their void shapes evolve quite differently. This indicates that the plastic flow direction, or correspondingly the normal to the yield surface, is important for the void evolution. However, the void shape is also affected by the local field quantities in the proximity of the void, and it is thus difficult to separate the various effects. In general terms, the interplay between the plastic flow and the equivalent stress seems decisive for the macroscopic behaviour of the unit cell. 5. Calibration of porous plasticity model The constitutive relation given by Equation (6) was calibrated to the numerical unit cell calculations by means of a non-linear least-square error routine. Model parameters qi were varied in the optimization procedure. The stress and void volume fraction residuals to be minimized were defined as  Eeqmax  Eeqmax UC |ΣGT | f GT − f UC |dEeq eq − Σeq |dE eq 0 eσ =  E max  , e f =  E0max  (11)   eq eq 1 1 GT UC GT + f UC dE Σ dE f + Σ eq eq eq eq 2 2 0 0

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2540 6

4.70

2.10 T = 1.0

1.58

3.78

Σvm eq σ0 1.05

f f0 2.85 θ = 0◦ θ = 60◦ θ = 120◦ θ = 180◦ θ = 240◦ θ = 300◦ Unit cell Gurson

0.52

0.00 0.000

0.125

0.250

0.375

θ = 0◦ θ = 60◦ θ = 120◦ θ = 180◦ θ = 240◦ θ = 300◦ Unit cell Gurson

1.92 T = 1.0 1.00 0.000

0.500

0.125

(a) 4.50 T = 1.0

1.58

3.62

Σvm eq σ0 1.05

f f0 2.75 θ = 30◦ θ = 90◦ θ = 150◦ θ = 210◦ θ = 270◦ θ = 330◦ Unit cell Gurson

0.52

0.150

0.300

0.375

0.500

(b)

2.10

0.00 0.000

0.250 vm Eeq

vm Eeq

0.450

θ = 30◦ θ = 90◦ θ = 150◦ θ = 210◦ θ = 270◦ θ = 330◦ Unit cell Gurson

1.88 T = 1.0 1.00 0.000

0.600

0.150

0.300

0.450

0.600

vm Eeq

vm Eeq

(c)

(d)

Fig. 4: Response of unit cell and the homogenized material model in terms of (a) and (c) equivalent von Mises stress, and (b) and (d) void volume fraction against the equivalent strain. All curves shown are for the Cube texture.

where the superscripts GT and UC denote Gurson-Tvergaard and unit cell, respectively, and |◦| denotes the magnitude. Note that the limit of the definite integral is the equivalent strain at maximum stress from the respective unit cell  max UC ˙ vm analyses Eeq = Eeq Σeq = 0 with the equivalent strain being defined by  2   Eeq = E E (12) 3 ij ij where Ei j are the macroscopic strain deviator components. A weighted residual on the form e = wσ eσ + w f e f

(13)

was adopted in the optimization process. In the present work, the residuals were given equal weight wσ = w f because the calibrated qi -values were only slightly affected by the residual weights. Table 2: Parameters retrieved from the optimization procedure. Texture

q1

q2

Cube Goss

1.912 1.282

0.791 0.842

The resulting parameters of the optimization procedure can be found in Table 2. For comparative reasons, we note that the calibrated parameters in the study by Steglich et al. (2010) were q1 = 1.22 and q2 = 1.16. An obvious reason



L.E.B Dæhli et al. / Procedia Structural Integrity 2 (2016) 2535–2542 L.E.B. Dæhli et al. / Structural Integrity Procedia 00 (2016) 000–000 4.60

2.20 T = 1.0

1.65

3.70

Σvm eq σ0 1.10

f f0 2.80 θ = 0◦ θ = 60◦ θ = 120◦ θ = 180◦ θ = 240◦ θ = 300◦ Unit cell Gurson

0.55

0.00 0.000

0.200

0.400

0.600

0.800

θ = 0◦ θ = 60◦ θ = 120◦ θ = 180◦ θ = 240◦ θ = 300◦ Unit cell Gurson

1.90 T = 1.0 1.00 0.000

0.200

0.400

0.600

(a)

(b) 4.10

2.70

θ = 30◦ θ = 90◦ θ = 150◦ θ = 210◦ θ = 270◦ θ = 330◦ Unit cell Gurson

T = 1.0

2.03

3.32

Σvm eq σ0 1.35

f f0 2.55 θ = 30◦ θ = 90◦ θ = 150◦ θ = 210◦ θ = 270◦ θ = 330◦ Unit cell Gurson

0.68

0.225

0.450 vm Eeq

(c)

0.800

vm Eeq

vm Eeq

0.00 0.000

2541 7

0.675

0.900

1.78 T = 1.0 1.00 0.000

0.225

0.450

0.675

0.900

vm Eeq

(d)

Fig. 5: Response of unit cell and the homogenized material model in terms of (a) and (c) equivalent von Mises stress, and (b) and (d) void volume fraction against the equivalent strain. All curves shown are for the Goss texture.

for the difference between the values found in the present study and these referred values is that we consider widely different materials, both in terms of texture and work-hardening properties. Also, the current work covers a wider range of stress states which may influence the optimized solution parameters. Figs. 4 and 5 compare the calibrated model response against the unit cell calculations for a stress triaxiality of T = 1 and for all deviatoric angles in terms of equivalent stress and void growth. From these curves we may observe that the model captures the general trends of the unit cell simulations in terms of equivalent stress-strain response. With reference to Figs. 4a, 4c, 5a, and 5c, the discrepancy between the respective curves is more pronounced for the generalized axisymmetric states than for the generalized shear states. Also, the cube texture seems somewhat better replicated by the proposed Gurson model. This is most likely due to the less extreme anisotropy of this texture as compared to the Goss texture (see Fig. 1), leaving it more compatible with the framework of the original Gurson model. We may readily see from Figs. 4b, 4d, 5b, and 5d that the void evolution is not accurately predicted by this model. This is presumably due to the assumption of spherical void growth which is employed in Equation 7. In the case of anisotropy, the void shape evolution will depend upon the orientation of the material axes (see Section 4). 6. Concluding remarks A heuristic extension of the Gurson model to account for plastic anisotropy of the matrix material is proposed. Unit cell analyses were employed to investigate effects of plastic anisotropy on the mechanical response and to calibrate the porous plasticity model. Unit cell calculations revealed the great influence of matrix anisotropy on the stressstrain response and the microstructural evolution, which in the present work is approximated by a single parameter accounting for the void volume fraction. The void growth rate is greatly affected by the equivalent stress magnitude.

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Numerical simulations also demonstrate the directional dependency of the void evolution, which is linked to the plastic flow direction. The calibrated porous plasticity model shows predictive capabilities. However, the equivalent stressstrain response is captured to a greater extent than the void growth. This is most likely related to the void evolution expression which does not account for the void shape evolution and directional dependency due to plastic anisotropy. It thus seems probable that the predictions would be enhanced if the void evolution is more correctly accounted for by the model. Also, it is noted that void coalescence is not accounted for in the present work. More involved unit cell analyses are needed to study this appropriately, which is considered for future work. Acknowledgement The financial support of this work from the Centres for Research-based Innovation (CRI) SIMLab and CASA at the Norwegian University of Science and Technology (NTNU) is gratefully acknowledged. References Barlat, F., Aretz, H., Yoon, J.W., Karabin, M.E., Brem, J.C., Dick, R.E., 2005. Linear transfomation-based anisotropic yield functions. International Journal of Plasticity 21, 1009–1039. Barlat, F., Richmond, O., 1987. Prediction of tricomponent plane stress yield surfaces and associated flow and failure behavior of strongly textured F.C.C. polycrystalline sheets. Materials Science and Engineering 95, 15–29. Barsoum, I., Faleskog, J., 2007. Rupture mechanisms in combined tension and shear-Micromechanics. International Journal of Solids and Structures 44, 5481–5498. Benzerga, A.A., Besson, J., 2001. Plastic potentials for anisotropic porous solids. European Journal of Mechanics, A/Solids 20, 397–434. Benzerga, A.A., Besson, J., Pineau, A., 2004. Anisotropic ductile fracture: Part II: Theory. Acta Materialia 52, 4639–4650. Bron, F., Besson, J., 2004. A yield function for anisotropic materials Application to aluminum alloys. International Journal of Plasticity 20, 937–963. Dæhli, L.E.B., Børvik, T., Hopperstad, O.S., 2016. Influence of loading path on ductile fracture of tensile specimens made from aluminium alloys. International Journal of Solids and Structures doi:10.1016/j.ijsolstr.2016.03.028. Faleskog, J., Gao, X., Shih, C.F., 1998. Cell model for nonlinear fracture analysis – I. Micromechanics calibration. International Journal of Fracture 89, 355–373. Gao, X., Zhang, G., Roe, C., 2010. A Study on the Effect of the Stress State on Ductile Fracture. International Journal of Damage Mechanics 19, 75–94. Gurson, A., 1977. Continuum Theory of Ductile Rupture by Void Nucelation and Growth: Part I – Yield Criteria and Flow Rules for Porous Ductile Media. Journal of Engineering Materials and Technology 99, 2—-15. Han, X., Besson, J., Forest, S., Tanguy, B., Bugat, S., 2013. A yield function for single crystals containing voids. International Journal of Solids and Structures 50, 2115–2131. Hancock, J., Mackenzie, A., 1976. On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states. Journal of the Mechanics and Physics of Solids 24, 147–160. Keralavarma, S.M., Benzerga, A.A., 2010. A constitutive model for plastically anisotropic solids with non-spherical voids. Journal of the Mechanics and Physics of Solids 58, 874–901. Keralavarma, S.M., Hoelscher, S., Benzerga, A.A., 2011. Void growth and coalescence in anisotropic plastic solids. International Journal of Solids and Structures 48, 1696–1710. Monchiet, V., Cazacu, O., Charkaluk, E., Kondo, D., 2008. Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids. International Journal of Plasticity 24, 1158–1189. Paux, J., Morin, L., Brenner, R., Kondo, D., 2015. An approximate yield criterion for porous single crystals. European Journal of Mechanics A/Solids 51, 1–10. Rice, J., Tracey, D., 1969. On the ductile enlargement of voids in triaxial stress fields. Journal of the Mechanics and Physics of Solids 17, 201–217. Saai, A., Dumoulin, S., Hopperstad, O.S., Lademo, O.G., 2013. Simulation of yield surfaces for aluminium sheets with rolling and recrystallization textures. Computational Materials Science 67, 424–433. Steglich, D., Wafai, H., Besson, J., 2010. Interaction between anisotropic plastic deformation and damage evolution in Al 2198 sheet metal. Engineering Fracture Mechanics 77, 3501–3518. Tvergaard, V., 1981. Influence of voids on shear band instabilities under plane strain conditions. International Journal of Fracture 17, 389–407. Zhang, K., 2001. Numerical analysis of the influence of the Lode parameter on void growth. International Journal of Solids and Structures 38, 5847–5856.