Homogenization of inelastic composites with misaligned inclusions by using the optimal pseudo-grain discretization

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Homogenization of inelastic composites with misaligned inclusions by using the optimal pseudo-grain discretization Witold Ogierman∗, Grzegorz Kokot Silesian University of Technology, ul. Konarskiego 18A, 44-100 Gliwice, Poland

a r t i c l e

i n f o

Article history: Received 8 July 2016 Revised 17 January 2017 Available online xxx Keywords: Orientation averaging Two-step homogenization Distributed orientations Misaligned composites

a b s t r a c t The paper deals with micromechanics of composite materials reinforced with misaligned, non-spherical inclusions. The main goal of this study is focused on numerical treatment of misaligned orientations by using the concept of two-step homogenization. The paper is concerned with the novel method of pseudograin discretization based on the optimal selection of pseudo-grains discrete orientations and corresponding weights. In order to solve the optimization problem evolutionary algorithms are used. The proposed approach leads to reduction of the pseudo-grains amount which, in turn, especially in the case of inelastic materials, results in homogenization that is more computationally efficient. The accuracy of the proposed method is presented by analysis of exemplary microstructures. Both the orientation data reconstruction accuracy and stiffness prediction accuracy are discussed. In addition, cases of elastic-plastic material behaviour are analysed and two-step homogenization results are compared with results of direct finite element (FE) based homogenization of representative volume element (RVE) with complex geometry. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Misaligned orientation of inclusions is common in the case of composite materials reinforced with short fibres or non-spherical particles. The type of materials that are most frequently discussed in the literature are polymer matrix composites reinforced with short fibres manufactured by injection moulding (Advani and Sozer, 2003; Bourmaud et al., 2013; Thi et al., 2015). Other studies of composite with misaligned inclusions are devoted to the analysis of anisotropy of metal matrix composites reinforced with ceramic particles (Jeong et al., 1991; Ganesh and Chawla, 2005). Distributed orientation of the inclusions is also observed in steel fibre reinforced concrete (Wuest et al., 2009; Suuronen et al., 2013). In such cases simplification of orientation distribution and treating it as unidirectional or random can lead to unacceptable errors in predictions of the material behaviour. Therefore, numerical methods that allow to take into account complex spatial orientation of inclusions are introduced and discussed in the literature. One of the most popular methods of estimation of the composite effective properties is direct finite element (FE) analysis of representative volume element (RVE) (Segurado and Llorca, 2002; Pierard et al., 2007; Rassol and Böhm, 2012). Finite element approach ∗

Corresponding author. E-mail address: [email protected] (W. Ogierman).

can deal with any reinforcement shape and orientation but, on the other hand, it requires high computational cost. Alternatively, boundary element method (BEM) can be applied into homogenization with the main advantages of BEM being high accuracy for materials in complex stress state, easy modification of geometry and reduction of the number of discretizing elements as compared ´ to FEM (Fedelinski et al. 2011; Ptaszny, 2015). The computational effort is connected not only with the solution of boundary value problem, but also with the creation of the inclusions geometry. The generation of spherical or unidirectional oriented inclusions is rather trivial but in the case of misaligned distribution the situation is becoming more complicated. Moreover, finding the geometry that represents accurately prescribed orientation distribution can be cumbersome. Another approach of determining the material effective properties that accounts for misaligned orientations is orientation averaging procedure. In this case effective properties of the material are taken as the weighted average of unidirectional material properties with respect to orientation distribution of inclusions. Advani and Tucker (1987) presented explicit expressions that allow to determine elastic stiffness tensor of misaligned composite in terms of orientation tensors and stiffness tensor of unidirectional composite. There are a lot of works that present the effectiveness of this approach for linear material properties (Pierard et al., 2004; Laspalas et al., 2008; Ogierman and Kokot, 2015, 2016). The analysis of nonlinear constitutive behavior is more complex

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Please cite this article as: W. Ogierman, G. Kokot, Homogenization of inelastic composites with misaligned inclusions by using the optimal pseudo-grain discretization, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.03.008

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and requires orientation distribution function reconstruction and decomposition of the RVE into the so-called pseudo-grains and performing the homogenization in two steps (Lielens et al., 1998). Doghri and Tinel (20 05, 20 06) presented detailed methodology for two-step homogenization with the use of pseudo-grain discretization procedure. This approach was also applied in multi-scale simulations where FE program is linked at macro-scale to the homogenization procedure at micro-scale (Doghri and Tinel, 2006). The authors of works devoted to two-step homogenization, that were mentioned above, performed the homogenization generally with the use of mean field approaches based on an Eshelby solution (Eshelby, 1957). Kammoun et al. (2011) have extended the two-step homogenization procedure by accounting for a damage phenomenon. Moreover, the procedures proposed by Doghri and Tinel (20 05, 20 06) is implemented in commercial software Digimat (Adam et al., 2009). Quite similar, but simplified approach based on discrete orientations has been also presented in work of NottaCuvier et al. (2013). This paper introduces a novel method of optimal pseudo-grain discretization. In the proposed method, an optimal selection of pseudo-grains orientations and weights can reduce the required amount of pseudo-grains with no loss of precision in the case of orientation reconstruction. From the point of view of computational costs and multi-sale simulations the reduction of pseudo-grain amount is an important issue. During this study description of orientation distribution is expressed by orientation tensors (Advani and Tucker, 1987) and therefore discrete orientation of the pseudo-grains are identified by fitting the fourth order orientation tensor produced by pseudo-grains orientations to fourth order orientation tensor that describes given inclusions orientation distribution. The paper has the following outline. Section 2 presents general expressions connected with orientation averaging, two-step homogenization and pseudo-grain discretization. In addition, a scheme of RVE decomposition into pseudo-grains as part of the optimization involving evolutionary algorithm is discussed. Section 3 analyses the influence of pseudo-grain discretization on the reconstruction accuracy of orientation tensors. Four different orientations are taken into consideration: exemplary two misaligned orientations, random orientation and orientation analysed by Doghri and Tinel (2006) (in order to compare the results). Section 4 is devoted to the simulation of elastic-plastic material behaviour. The results of two-step homogenization are compared with results of direct FE based homogenization of a representative volume element RVE with complex geometry. Moreover, the behaviour of material with random orientation of inclusions is analysed and the obtained results are compared with the results presented in work of Doghri and Tinel (2006).

Fig. 1. Orientation vector p in terms of spherical angles θ and ϕ .

ψ (p)dp. Orientation distribution function has the following properties:

ψ ( p)dp = ψ (−p)dp, 

ψ ( p)dp = 1.

(4)

While the description of orientation distribution function is cumbersome, the orientation tensor approach of Advani and Tucker (1987) represents the distribution function of fibres in a concise form. Orientation tensors are defined from the dyadic products of the unit vector p and the distribution function ψ (p) over the unit sphere as:



pi p j ψ ( p)dp,

ai j =  ai jkl =  ai j... =

(5)

pi p j pk pl ψ ( p)dp,

(6)

pi p j ...ψ ( p)dp.

(7)

There is an infinite number of these tensors in all the even orders but this work is limited to the usage of the second and fourth order tensors that are sufficient for most uses (Advani and Tucker, 1987). Elastic stiffness tensor Cijkl of misaligned composite estimation involving the second and fourth order orientation tensors can be determined by using the following relation:

(8)

2.1. Orientation averaging In orientation averaging procedure the volume average of any field μ in composite ω is taken as an average of μ(p) determined for unidirectional composite over all directions weighted by orientation distribution function ψ (p):



(3)

Ci jkl = B1 ai jkl + B2 (ai j δkl + akl δi j ) + B3 (aik δ jl + ail δ jk + a jl δik + a jk δil ) +B4 (δi j δkl ) + B5 (δik δ jl + δil δ jk )

2. Homogenization of misaligned composites

μω =

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μ( p)ψ ( p)dp.

(1)

Unidirectional composite orientation can be defined by vector p which can be described by two spherical angles θ and ϕ (Fig. 1):

p = [sin θ cos ϕ , sin θ sin ϕ , cos θ ] . T

(2)

Orientation distribution function is defined as the probability of finding an inclusion whose orientation is between p and (p +dp) is

where B1 –B5 are scalar constants related to the components of stiffness tensor of unidirectional composite that are presented in detail in work of Advani and Tucker (1987). 2.2. Two-step homogenization Homogenizing inelastic composite requires the knowledge of orientation distribution function. The orientation distribution function can be recovered from the orientation tensors aij and aijkl (Advani and Tucker, 1987; Onat and Leckie, 1988; Doghri and Tinel, 2006). The method that can deal with distributed orientation and nonlinear constitutive behaviour of the inclusions is a two-step homogenization. Misaligned inclusions are divided into groups of unidirectional oriented inclusions characterized by different orientation vectors p. In other words, the RVE is decomposed into a set

Please cite this article as: W. Ogierman, G. Kokot, Homogenization of inelastic composites with misaligned inclusions by using the optimal pseudo-grain discretization, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.03.008

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with respect to accounted variables:



min F (θ , ϕ , w ) =

ai jkl GIV EN − ai jkl F OUND (θ , ϕ , w ) ai jkl GIV EN

3

2 ,

where i, j, k, l = 1, 2, 3.

(9)

GIVEN

where aijkl is reference fourth order orientation tensor defined by Eq. (6) that is built knowing the exact distribution of inclusion orientations, aijkl FOUND is fourth order orientation tensor depending on variables that are spherical angles θ and ϕ and weights w of the pseudo-grains. The termination condition that indicates the end of evolutionary optimization is fulfilled when absolute relative percentage error between aijkl GIVEN and aijkl FOUND (for all tensor components) is at the prescribed level. While unit vector p is defined by two spherical angles Eq. (1) can be rewritten in the following form:

μω =

π 2π

μ(θ , ϕ )ψ (θ , ϕ ) sin θ dθ dϕ .

(10)

θ =0 ϕ =0

Fig. 2. Scheme of two-step homogenization.

of the so called pseudo-grains. Each pseudo-grain, is a two-phase composite containing the same volume fraction of matrix material as in RVE- v0 and those inclusions of volume fraction 1- v0 with orientation pi . After the discretization into pseudo-grains, the first step is to homogenize each pseudo grain individually. Then, in the second step, pseudo-grain average fields weighted by the orientation distribution function ψ (p) are computed. A scheme of the two-step homogenization is presented in Fig. 2. Pierard et al. (2007) concluded that the usage of Mori-Tanaka (Mori and Tanaka, 1973; Benveniste 1987) or double inclusion (Lielens et al., 1998) method for the first step leads to accurate results but the second step requires the use of other homogenization models, ones that are suitable for aggregates or polycrystals. In this work, the first step consists of determining the effective stiffness tensors, presented in Section 3, with the use of the MoriTanaka method. In the case of estimation of elastic-plastic material response FE based homogenization is performed (Section 4) to make a reliable comparison to the results obtained by analysis of complex RVE. It has to be underlined that the usage of the Mori-Tanaka method for elastic-plastic material is more complicated than in the case of elastic material due to Eshelby’s problem which has no analytical solution when the material displays nonlinear behaviour. These difficulties have been discussed and overcome in works of Gavazzi and Lagoudas (1990), Doghri and Ouaar (2003) and Petterman et al. (2010). In the second step, in all analyses, Voigt model is considered by assuming uniform strain for each pseudo-grain (Pierard et al., 2007).

The numerical solution of Eq. (10) is generally performed by dividing the domain into finite number of discrete orientations defined by equal spherical angles increments. Then, for each discrete orientation the orientation distribution function value is computed. Doghri and Tinel (2006) applied isosize facets method where equal-sized increments of θ are considered, and the corresponding values of ϕ increments are computed so that the surface of the unit sphere is subdivided into facets of almost equal areas. The objective of this work is a decomposition of the problem domain into a relatively small amount of the pseudo grains. One idea is to not follow the fixed angles increments but find the optimal discrete orientations and corresponding weights w. Eq. (10) can be expressed in discrete form as follows:

μω =



During pseudo-grain discretization due to condition expressed by Eq. (3) only a half of sphere is considered. In the method proposed in this work, an optimal selection of pseudo-grains orientations and weights can reduce the required amount of pseudograins with no loss of precision in the case of orientation reconstruction. In order to identify optimal pseudo-grains parameters the optimization problem was solved. In particular, it was the minimization of the relative difference between given components of fourth order orientation tensor and components of fourth order orientation tensor that was determined during the computations

(11)

The domain is divided on arbitrary number of the pseudograins. Preliminary computations showed that nine pseudo-grains (that results in computation of 27 pseudo-grain parameters) are sufficient to find the optimum. In this case variables accounted during the optimization are nine θ and ϕ angles and nine weights w. Optimum is achieved when identified pseudo-grain parameters allow to reconstruct fourth order orientation tensor with prescribed accuracy (with respect to given fourth order orientation tensor). During this work error levels of 1.5%, 1%,0.5% and 0.1% were prescribed. Moreover simulation accounting larger number of 18 pseudo-grains has been conducted in order to investigate the influence of the amount of the pseudo-grains parameters on computational efficiency of evolutionary optimization. Constraints are applied on the accounted variables as follows:

 π π  π π θ = − , , ϕ = − , , w = 0, 1. 2

2.3. Pseudo-grain discretization

μ(θi , ϕi )w(θi , ϕi ).

i

2

2

2

(12)

The optimization problem was solved by using evolutionary algorithm which is a common approach in engineering practice ´ (Maletta and Pagnotta, 2004; Długosz and Burczynski, 2013; Beluch ´ and Burczynski, 2014; Makowski and Kus´ ; 2016). A scheme of proposed pseudo-grain decomposition is presented in Fig. 3. The evolutionary algorithm parameters that were used are introduced in Table 1. The initial population is created randomly. 3. Accuracy of orientation tensors reconstruction and elastic stiffness tensor prediction In this section an influence of pseudo-grain discretization on the accuracy of reconstructed orientation tensors is analysed. The

Please cite this article as: W. Ogierman, G. Kokot, Homogenization of inelastic composites with misaligned inclusions by using the optimal pseudo-grain discretization, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.03.008

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Fig. 3. Scheme of RVE decomposition.

Table 1 Evolutionary algorithm parameters. Population size Crossover fraction Selection procedure Elite count Crossover Mutation

150 0.9 Rank selection 5% of population size Heuristic Adaptive

solution of optimization problem lead to obtaining θ and ϕ angles and weights w. As a result of that the orientation tensors reconstructed by the identified θ , ϕ and w are different than the original one. Although the error is controlled by applying the termination condition of evolutionary algorithm, the reconstruction accuracy analysis is presented in detail by comparing the given orientation data with the reconstructed one. Several analyses considering two different pseudo-grain amounts and four different prescribed error levels were performed. Moreover, the stiffness tensors were determined by using two-step homogenization scheme. Considering linear-elastic material behaviour effective stiffness tensor C is computed in terms of stiffness tensor of unidirectional material CUNI , transformation tensor T dependent on spherical angles of pseudo-grains and weight of the pseudo-grains w in the following way:

C=

n 

T (θi , ϕi )C T

UNI

T (θi , ϕi )wi

Numerical simulations were carried out for the selected orientation distributions of inclusions defined by the fourth order orientation tensors (corresponding second order tensors are also presented). Due to nondeterministic nature of evolutionary optimization for each case of error level and amount of pseudo-grains ten independent optimization problems were solved. The same elastic properties of composite constituents were prescribed for all performed analyses: Young modulus and Poison ratio of matrix are 72.5 GPa and 0.34, Young modulus and Poison ratio of inclusions are 455 GPa and 0.15 respectively. 3.1. The first example The first simulations that were conducted are devoted to RVE (Representative Volume Element) with distributed inclusions presented in Fig 4. Volume fraction of the inclusions is 0.1 and aspect ratio is 10. Orientation tensors that are built knowing the exact distribution of inclusion orientations in accordance with the microstructure presented in Fig. 5 have the following form:



ai j

GIV EN

=

0.54416 0.04725 −0.01677

0.04725 0.40110 0.01219



−0.01677 0.01219 0.05474

ai jkl GIV EN =

(13)

i

where n indicates the number of pseudo-grains. The obtained stiffness tensors are compared with analytical solution connected with the use of Eq. (8). Stiffness tensor CUNI can be computed by using a wide range of homogenization methods but the results presented in this section are obtained by using the well-known Mori-Tanaka method with the Eshelby solution for prolate ellipsoid (Mura, 1987).

⎡

0.40969 ⎢0.11007 ⎢0.02439 ⎢ ⎢0.0 0 065 ⎣ −0.00306 0.03483

0.11007 0.27361 0.01742 0.00506 −0.00625 0.01075

0.02439 0.01742 0.01293 0.00647 −0.00746 0.00167

0.0 0 065 0.00506 0.00647 0.01742 0.00167 −0.00625

−0.00306 −0.00625 −0.00746 0.00167 0.02439 0.0 0 065

⎤

0.03483 0.01075 ⎥ 0.00167 ⎥ ⎥ −0.00625⎥ ⎦ 0.0 0 065 0.11007

Table 2 shows average generations required to find the optimum with respect to the amount of pseudo-grains and prescribed error, ten tests for each error level were performed. The amount of

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Table 2 Average generations required to find the optimum with respect to amount of pseudo-grains and prescribed error. Prescribed error 9 pseudo-grains 18 pseudo-grains

1.5% 1137 1385

1% 1516 1673

0.5% 1884 2451

0.1% 7724 4477

ence between given and reconstructed components of the orientation tensors. Results presented in Tables 5 and 6 show that identified pseudo-grain parameters allow to reconstruct the given orientation tensor with desired accuracy. Table 7 collects components of elastic stiffness tensors computed by using two-step homogenization method and absolute relative percentage differences related to the analytical solution CAT (obtained by using Eq. (8) and assuming ellipsoidal shape of the inclusions) that has the following form:

⎡

C AT Fig. 4. The first analysed RVE containing distributed orientation of inclusions.

130.771 61.613 60.552 −0.011 128.133 60.697 0.097 ⎢ ⎢ 123.167 0.126 =⎢ ⎢ 31.546 ⎣ sym.

−0.054 −0.094 −0.144 0.069 31.793

⎤

0.687 0.195 ⎥ −0.060⎥ ⎥. −0.140 ⎥ ⎦ 0.022 33.800

Obtained stiffness tensors shows that proposed pseudo-grain discretization method allows not only to reconstruct the given orientation tensors but also provide accurate in predictions of material behaviour. 3.2. The second example The second example is connected with analysis of inclusions distribution presented in Fig. 5. Volume fraction of the inclusions is 0.1 and aspect ratio is 4. Orientation tensors that are built knowing the exact distribution of inclusion orientations in accordance with the microstructure presented in Fig. 5 have the following form:



ai j GIV EN =

0.75632 0.06749 0.07862

0.06749 0.19227 0.05468



0.07862 0.05468 0.05141

ai jkl GIV EN =

⎡

Fig. 5. The second analysed RVE containing distributed orientation of inclusions.

required generations increases with decreasing prescribed error. In particular tremendous increase is observed when error is on very low level of 0.1%, considering larger number of pseudo-grains leads to achieve significantly faster convergence in this case. Spherical angles and weights identified during two independent evolutionary algorithm runs with consideration of different prescribed error levels are collected in Tables 3 and 4 respectively. Nine discrete orientations are sufficient for an appropriate reconstruction of orientation tensors therefore only parameters obtained for nine pseudo-grains are presented. Second and fourth order orientation tensors recomputed by accounting identified variables are collected in Tables 5 and 6. Moreover mentioned Tables present absolute relative percentage differ-

0.64960 ⎢0.07720 ⎢0.02952 ⎢ ⎢0.02446 ⎣ 0.06448 0.04394

0.07720 0.10100 0.01407 0.02400 0.00825 0.01646

0.02952 0.01407 0.00782 0.00622 0.00588 0.00709

0.02446 0.02400 0.00622 0.01407 0.00709 0.00825

0.06448 0.00825 0.00588 0.00709 0.02952 0.02446

⎤

0.04394 0.01646 ⎥ 0.00709 ⎥ ⎥ 0.00825⎥ ⎦ 0.02446 0.07720

The results are presented in the same manner as in the previous case: Table 8 presents average generations required to find the optimum with respect to amount of pseudo-grains and prescribed error, Tables 9 and 10 present obtained pseudo-grains parameters, Tables 11 and 12 reconstructed orientation tensors components. Table 13 presents components of elastic stiffness tensors computed by using two-step homogenization method and additionally absolute relative percentage differences related to the analytical solution (obtained by using Eq. (8)) that has the following form:

⎡

C AT

130.484 60.983 60.644 0.191 124.305 61.039 0.274 ⎢ ⎢ 123.198 0.096 ⎢ =⎢ 31.295 ⎣ sym.

0.693 0.006 0.108 0.125 31.900

⎤

0.480 0.207 ⎥ 0.005 ⎥ ⎥ 0.145 ⎥ ⎦ 0.288 32.489

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W. Ogierman, G. Kokot / International Journal of Solids and Structures 000 (2017) 1–11 Table 3 Obtained spherical angles and weights defining pseudo-grains (obtained during the first optimization with prescribed error of 1%).

θ1

θ2

θ3

θ4

θ5

θ6

θ7

θ8

θ9

–59.256°

–77.639°

61.345°

–53.526°

88.670°

85.928°

–47.056°

–79.824°

80.026°

48.560° w1 0.030526

11.802° w2 0.193282

7.164° w3 0.018348

–67.514° w4 0.043628

–45.751° w5 0.232343

69.582° w6 0.175889

–37.903° w7 0.018025

73.844° w8 0.074267

15.172° w9 0.209670

ϕ1

ϕ2

ϕ3

ϕ4

ϕ5

ϕ6

ϕ7

ϕ8

ϕ9

Table 4 Obtained spherical angles and weights defining pseudo-grains (obtained during the second optimization with prescribed error of 0.1%).

θ1 –79.928

θ2 °

θ3

87.394

ϕ1

°

–62.595

ϕ2

–45.242° w1 0.164072

θ4 °

ϕ3

12.622° w2 0.186652

16.260° w3 0.101851

θ5

89.994

°

ϕ4

54.235° w4 0.162968

48.921

θ6 °

θ7

89.992

ϕ5

°

78.152

ϕ6

88.220° w5 0.042017

θ8 °

78.050

ϕ7

74.141° w6 0.085246

θ9 °

ϕ8

1.812° w7 0.131462

81.437°

ϕ9

–81.880° w8 0.062284

–50.522° w9 0.063794

Table 5 Reconstructed components of the second order orientation tensor and sum of weights with corresponding errors.  wi Prescribed error a11 a22 a33 a12 a23 a13 i

1%

0.54002 0.766% 0.54447 0.056%

0.1%

0.40150 0.100% 0.40113 0.007%

0.05446 0.520% 0.05475 0.021%

0.04696 0.633% 0.04724 0.030%

0.01220 0.144% 0.01218 0.030%

–0.01682 0.292% –0.01676 0.057%

0.99598 0.402% 1.0 0 035 0.035%

Table 6 Reconstructed components of the fourth order orientation tensor with corresponding errors. Prescribed error

a1111

a2222

a3333

a1122

a1133

a2233

a2323

a1313

a1212

1%

0.40689 0.684% 0.41004 0.084%

0.27528 0.611% 0.27364 0.013%

0.01306 0.997% 0.01294 0.013%

0.10898 0.998% 0.11005 0.022%

0.02415 0.978% 0.02437 0.062%

0.01725 0.996% 0.01744 0.096%

0.01725 0.996% 0.01744 0.096%

0.02415 0.978% 0.02437 0.062%

0.10898 0.998% 0.11005 0.022%

0.1%

Table 7 Components of effective elastic stiffness tensors with corresponding errors, GPa. Prescribed error

C1111

C2222

C3333

C1122

C1133

C2233

C2323

C1313

C1212

1%

130.183 0.450% 130.712 0.045%

127.762 0.289% 128.162 0.022%

122.782 0.313% 123.229 0.051%

61.378 0.382% 61.702 0.144%

60.321 0.383% 60.607 0.091%

60.331 0.603% 60.664 0.053%

31.456 0.285% 31.756 0.666%

31.642 0.476% 31.802 0.028%

33.551 0.737% 33.661 0.412%

0.1%

Table 8 Average generations required to find the optimum with respect to amount of pseudo-grains and prescribed error.

3.3. The third example - random 3d orientation The next simulations that were conducted are devoted to random three-dimensional orientation of inclusions defined by the following orientation tensors:

Prescribed error 9 pseudo-grains 18 pseudo-grains

1.5% 845 960

1% 844 1056

0.5% 2607 1743

0.1% 6275 5448

Results presented for the second example lead to similar conclusions as in the previous case.

1 ai j GIV EN =

3

0 0

0 1 3

0

0 0



1 3

Table 9 Obtained spherical angles and weights defining pseudo-grains (obtained during the first optimization with prescribed error of 1%).

θ1

θ2

θ3

θ4

θ5

θ6

θ7

θ8

θ9

–70.620

–76.995

–87.229

29.993

–29.993

29.083

88.212

84.037

71.328

–54.388 w1 0.088693

13.623 w2 0.091357

37.687 w3 0.035957

–30.002 w4 0.005592

–29.914 w5 0.0 0 0541

84.917 w6 0.0 0 0607

–89.679 w7 0.043088

–4.871 w8 0.481909

34.677 w9 0.259167

ϕ1

ϕ2

ϕ3

ϕ4

ϕ5

ϕ6

ϕ7

ϕ8

ϕ9

Please cite this article as: W. Ogierman, G. Kokot, Homogenization of inelastic composites with misaligned inclusions by using the optimal pseudo-grain discretization, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.03.008

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Table 10 Obtained spherical angles and weights defining pseudo-grains (obtained during the second optimization with prescribed error of 0.1%).

θ1

θ2

θ3

θ4

θ5

θ6

θ7

θ8

θ9

–69.287

86.448

–88.131

–66.593

76.795

81.316

69.273

89.378

51.199

–35.287 w1 0.065846

11.851 w2 0.291175

83.385 w3 0.029917

30.319 w4 0.030210

11.023 w5 0.264629

–72.647 w6 0.006420

62.104 w7 0.114310

–25.146 w8 0.174658

10.017 w9 0.022769

ϕ1

ϕ2

ϕ3

ϕ4

ϕ5

ϕ6

ϕ7

ϕ8

ϕ9

Table 11 Reconstructed components of the second order orientation tensor and sum of weights with corresponding errors.  wi Prescribed error a11 a22 a33 a12 a23 a13 i

1%

0.76288 0.867% 0.75618 0.019%

0.1%

0.19268 0.214% 0.19237 0.050%

0.05135 0.118% 0.05139 0.036%

0.06745 0.059% 0.06747 0.037%

0.05510 0.769% 0.05466 0.043%

0.07917 0.699% 0.07864 0.025%

1.00691 0.691% 0.99993 0.007%

Table 12 Reconstructed components of the fourth order orientation tensor with corresponding errors. Prescribed error

a1111

a2222

a3333

a1122

a1133

a2233

a2323

a1313

a1212

1%

0.65606 0.994% 0.64942 0.027%

0.10089 0.108% 0.10106 0.056%

0.00789 0.991% 0.00781 0.013%

0.07758 0.493% 0.07725 0.060%

0.02924 0.941% 0.02951 0.042%

0.01421 0.991% 0.01407 0.038%

0.01421 0.991% 0.01407 0.038%

0.02924 0.941% 0.02951 0.042%

0.07758 0.493% 0.07725 0.060%

0.1%

Table 13 Components of effective elastic stiffness tensors with corresponding errors, GPa. Prescribed error

C1111

C2222

C3333

C1122

C1133

C2233

C2323

C1313

C1212

1%

131.351 0.664% 130.275 0.160%

125.206 0.725% 124.383 0.063%

124.126 0.753% 123.396 0.161%

61.443 0.755% 61.139 0.256%

61.074 0.709% 60.680 0.060%

61.369 0.541% 60.788 0.411%

31.905 1.947% 31.584 0.924%

32.095 0.609% 31.901 0.004%

32.306 0.566% 32.147 1.053%

0.1%

Table 14 Average generations required to find the optimum with respect to amount of pseudo-grains and prescribed error.

form:

Prescribed error 1.5% 160 331

9 pseudo-grains 18 pseudo-grains

⎡1 ai jkl

GIV EN

5 1 15 1 15

⎢ ⎢ =⎢ ⎢0 ⎣ 0 0

1 15 1 5 1 15

0 0 0

1 15 1 15 1 5

0 0 0

0 0 0 1 15

0 0

1% 176 412

0 0 0 0 1 15

0

0.5% 219 459

0.1% 380 596

⎤

0 0⎥ 0⎥ ⎥ 0⎥ ⎦ 0 1 15

The results are presented in the same manner as in the previous cases: Table 14 presents average generations required to find the optimum with respect to amount of pseudo-grains and prescribed error, Tables 15 and 16 present obtained pseudo-grains parameters, Tables 17 and 18 reconstructed orientation tensors components. Effective elastic stiffness tensor has been determined by considering ellipsoidal inclusions of aspect ratio 10 and volume fraction 0.1. Table 19 presents the components of elastic stiffness tensors computed by using two-step homogenization method and additionally absolute relative percentage differences related to the analytical solution (obtained by using Eq. (8)) that has the following

C AT

⎡

126.698 61.283 61.283 0.0 0 0 126.698 61.283 0.0 0 0 ⎢ ⎢ 126 . 698 0.0 0 0 =⎢ ⎢ 32.708 ⎣ sym.

0.0 0 0 0.0 0 0 0.0 0 0 0.0 0 0 32.708

⎤

0.0 0 0 0.0 0 0 ⎥ 0.0 0 0 ⎥ ⎥ 0.0 0 0 ⎥ ⎦ 0.0 0 0 32.708

Results presented for the example of random orientation lead to similar conclusions as in the previous cases however Table 14 shows significant decrease of generations required to find the optimum. 3.4. Orientation close to random 3d The main reason for analysing orientation defined by the following second order orientation tensor



ai j GIV EN =

0.338333 0.005 0.005

0.005 0.333333 0.005



0.005 0.005 0.328333

is to compare them with the results of Doghri and Tinel (2006). The work of Doghri and Tinel (2006) is devoted mainly to problems caused by the fact that only the second orientation tensor is known. In this case the fourth order orientation tensor is approximated by using closure approximation. To make the comparison reliable the fourth order orientation tensor is approximated by the same hybrid closure that was applied in work Doghri and Tinel (2006). Details connected with hybrid closure approximation can be found in work of Advani and Tucker (1987) and approximated fourth order tensor that is an input data to optimization is

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W. Ogierman, G. Kokot / International Journal of Solids and Structures 000 (2017) 1–11 Table 15 Obtained spherical angles and weights defining pseudo-grains (obtained during the first optimization with prescribed error of 1%).

θ1

θ2

θ3

θ4

θ5

θ6

θ7

θ8

θ9

–79.384°

–54.485°

–48.880°

28.403°

–16.944°

27.581°

57.076°

65.847°

85.192°

–57.631° w1 0.157082

4.529° w2 0.155749

81.801° w3 0.096924

–61.720° w4 0.071958

14.232° w5 0.021354

78.723° w6 0.151031

–49.542° w7 0.040877

0.110° w8 0.151246

58.759° w9 0.152524

ϕ1

ϕ2

ϕ3

ϕ4

ϕ5

ϕ6

ϕ7

ϕ8

ϕ9

Table 16 Obtained spherical angles and weights defining pseudo-grains (obtained during the second optimization with prescribed error of 0.1%).

θ1 66.865

θ2 °

θ3

–61.450

ϕ1

°

–64.379

ϕ2

71.629° w1 0.165623

θ4 °

64.783

ϕ3

–37.482° w2 0.160097

θ5 °

ϕ4

30.599° w3 0.143931

–63.999° w4 0.130063

70.169

θ6 °

3.393

ϕ5

θ7 °

72.137

ϕ6

–3.077° w5 0.074457

θ8 °

ϕ7

60.061° w6 0.161667

θ9

58.948

°

ϕ8

–75.994° w7 0.015593

4.824° w8 0.091891

–48.474°

ϕ9

73.255° w9 0.062629

Table 17 Reconstructed components of the second order orientation tensor and sum of weights with corresponding errors.  wi Prescribed error a11 a22 a33 i

1%

0.33255 0.236% 0.33320 0.039%

0.1%

0.33405 0.072% 0.33316 0.052%

0.33215 0.118% 0.33329 0.012%

0.9987 0.125% 0.9996 0.035%

in the following form:

Fig. 6. Finite element mesh discretizing inclusions corresponding to a) complex RVE, b) exemplary pseudo-grain.

EN aGIV i jkl

⎡

0.20420 ⎢0.06742 ⎢0.06671 ⎢ ⎢0.0 0 072 ⎣ 0.00214 0.00214

0.06742 0.19992 0.06599 0.00214 0.0 0 072 0.00214

0.06671 0.06599 0.19564 0.00214 0.00214 0.0 0 072

0.0 0 072 0.00214 0.00214 0.06589 0.0 0 071 0.0 0 071

0.00214 0.0 0 072 0.00214 0.0 0 071 0.06661 0.0 0 071

⎤

0.00214 0.00214 ⎥ 0.0 0 072⎥ ⎥ 0.0 0 071⎥ ⎦ 0.0 0 071 0.06732

The obtained pseudo-grains parameters are presented in Tables 20 and 21. Comparison of results of current study with results of method presented by Doghri and Tinel (2006) is presented in Table 22 that collects reconstructed components of the second order orientation tensor. Table 22 contains results obtained for 9 pseudo-grains defined by the parameters presented in the Tables 20 and 21 and the results obtained by Doghri and Tinel (2006) for 100, 846, 5196 pseudo-grains by using the isosize facets algorithm. Additionally, absolute relative percentage differences between components of reconstructed and given second order orientation tensor are presented. Presented results shows that nine pseudo-grains obtained by the method of optimal pseudo-grain discretization allow to reconstruct the second-order orientation tensor more accurately even than 5196 pseudo-grain determined by using the isosize facets algorithm.

4. Simulation of elastic-plastic behaviour To verify the accuracy of the two-step homogenization for inelastic material behaviour the obtained results were compared with results of direct FE based homogenization of complex RVE containing 100 inclusions. Accounted orientation distribution of the inclusions is this same as the one discussed in Section 3.2 and considered pseudo-grains parameters are included in Table 9. The analysed composite is reinforced with ellipsoidal particles of volume fraction 0.1 and aspect ratio 4. Inclusions are elastic with Young modulus 455 GPa and Poisson ratio 0.15. Matrix is elasticplastic with Young modulus 72.5 GPa and Poisson ratio 0.34. Von Mises yield criterion is considered by taking into account yield stress 250 MPa and power law isotropic hardening defined by constant k = 173 MPa and hardening exponent m = 0.46. To make the comparison reliable each pseudo-grain is also homogenized by using FEM at the first homogenization step. Every pseudo-grain contains 15 inclusions. Geometry is discretized by tetrahedral finite elements with quadratic shape functions. Finite element meshes for inclusions corresponding to the complex RVE and exemplary pseudo-grain are presented in Fig. 6. Periodic boundary conditions (Segurado and Llorca, 2002) were prescribed to enforce strain both for complex RVE and for pseudo-grains. FE computations were car-

Table 18 Reconstructed components of the fourth order orientation tensor with corresponding errors. Prescribed error

a1111

a2222

a3333

a1122

a1133

a2233

a2323

a1313

a1212

1%

0.19923 0.389% 0.19983 0.083%

0.20037 0.185% 0.19988 0.062%

0.19804 0.990% 0.20 0 01 0.003%

0.06645 0.333% 0.06668 0.022%

0.06688 0.314% 0.06669 0.029%

0.06723 0.844% 0.06660 0.098%

0.06723 0.844% 0.06660 0.098%

0.06688 0.314% 0.06669 0.029%

0.06645 0.333% 0.06668 0.022%

0.1%

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9

Table 19 Components of effective elastic stiffness tensors with corresponding errors, GPa. Prescribed error

C1111

C2222

C3333

C1122

C1133

C2233

C2323

C1313

C1212

1%

126.543 0.123% 126.665 0.026%

126.607 0.072% 126.674 0.019%

126.347 0.277% 126.702 0.003%

61.087 0.319% 61.269 0.022%

61.315 0.052% 61.243 0.065%

61.276 0.011% 61.234 0.080%

32.719 0.036% 32.689 0.056%

32.684 0.072% 32.620 0.270%

32.657 0.156% 32.741 0.103%

0.1%

Table 20 Obtained spherical angles and weights defining pseudo-grains (obtained during the second optimization with prescribed error of 1%).

θ1

θ2

θ3

θ4

θ5

θ6

θ7

θ8

θ9

–48.113°

–76.938°

–40.478°

–18.393°

–3.632°

–2.068°

73.076°

43.266°

81.079°

–55.479° w1 0.139166

7.741° w2 0.166967

67.174° w3 0.133768

–54.518° w4 0.019631

–5.862° w5 0.043630

81.900° w6 0.019065

–49.795° w7 0.159847

15.067° w8 0.154695

68.540° w9 0.164175

ϕ1

ϕ2

ϕ3

ϕ4

ϕ5

ϕ6

ϕ7

ϕ8

ϕ9

Table 21 Obtained spherical angles and weights defining pseudo-grains (obtained during the second optimization with prescribed error of 0.1%).

θ1

θ2

θ3

θ4

θ5

θ6

θ7

θ8

θ9

89.984°

–56.990°

58.613°

85.672°

–28.515°

32.449°

–90.0 0 0°

90.0 0 0°

–33.176°

60.482° w1 0.030666

–56.458° w2 0.159744

–56.601° w3 0.160716

80.856° w4 0.045339

25.572° w5 0.041554

32.263° w6 0.169677

0.459° w7 0.167366

59.416° w8 0.105053

37.027° w9 0.119778

ϕ1

ϕ2

ϕ3

ϕ4

ϕ5

ϕ6

ϕ7

ϕ8

ϕ9

Table 22 Reconstructed components of the second order orientation tensor, results of current study and comparison with results of method presented by Doghri and Tinel (2006).

9 discrete orientations, prescribed error 1% 9 discrete orientations, prescribed error 0.1% 100 discrete orientations, Doghri and Tinel (2006) 846 discrete orientations, Doghri and Tinel (2006) 5196 discrete orientations, Doghri and Tinel (2006)

a11

a22

a33

a12

a23

a13

0.339819 0.44% 0.338413 0.02% 0.324131 4.20% 0.332331 1.77% 0.335852 0.73%

0.333472 0.04% 0.333389 0.02% 0.319274 4.22% 0.327419 1.77% 0.330875 0.74%

0.327652 0.21% 0.32809 0.07% 0.312544 4.81% 0.3214422 2.10% 0.323458 1.48%

0.005019 0.37% 0.004997 0.06% 0.004827 3.47% 0.004911 1.78% 0.004966 0.68%

0.005026 0.53% 0.004996 0.07% 0.002698 46.03% 0.004676 6.48% 0.004912 1.76%

0.004989 0.21% 0.004999 0.03% 0.001230 75.39% 0.004505 9.90% 0.004884 2.32%

ried out by using Ansys software. At post-processing stage of FE analysis stress in the RVE is averaged in the following way:

σ RV E =

1

RV E



σ d

(14)

RV E

where RVE is volume of the RVE. The results of performed comparative analyses are presented in Fig. 7–9 containing the results for different loading directions. In addition, to make the results more clear, the results for different loading directions are presented together with results for unidirectional composite (transversally isotropic behaviour) in Fig. 10. Another example is devoted to material with random orientation of inclusions. The assumed properties of composite constituents are the same as in the previous case. The only difference is that inclusion length to diameter ratio is 10. Parameters of the pseudo-grains that were taken into account have been already presented in Table 4. The results of the performed simulation as well as the results obtained by Doghri and Tinel (2006) are presented in Fig. 11. Fig. 7. Elastic-plastic response of the material: loading direction 1.

5. Conclusions The paper is concerned with the novel method of pseudo-grain discretization based on optimal selection of pseudo-grains discrete orientations and corresponding weights. The proposed approach

leads to reduction of the pseudo-grains amount that in turn leads to computationally efficient homogenization. The results of performed computations show that the amount of nine pseudo-grains

Please cite this article as: W. Ogierman, G. Kokot, Homogenization of inelastic composites with misaligned inclusions by using the optimal pseudo-grain discretization, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.03.008

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Fig. 8. Elastic-plastic response of the material: loading direction 2.

Fig. 10. Elastic-plastic response of the material: summary of the results.

Fig. 9. Elastic-plastic response of the material: loading direction 3.

Fig. 11. Elastic-plastic response of the material with randomly distributed inclusions, comparison with the results of Doghri and Tinel (2006).

is sufficient for an appropriate representation of orientation distribution. The results presented in Section 3 are in agreement with the given fourth order orientation tensor and the reconstructed form of discrete pseudo-grains. Stiffness tensor predictions based on the two-step homogenization are also in good agreement with the analytical result. In comparison with the results of orientation tensor reconstruction performed by isosize facets algorithm (Section 3.4), the method proposed in this paper is more accurate and requires much less discrete orientations. On the other hand, it has to be underlined that, in general, it requires more computational effort than the method of orientation distribution reconstruction applied by Advani and Tucker (1987) and Doghri and Tinel (2006). In order to verify the accuracy of the proposed method in simulation of nonlinear material behaviour tensile response of elastic-plastic matrix reinforced with elastic particles was studied. The results of two-step homogenization in the form of stress-strain curves were compared with the results of FE based homogenization of complex RVE containing 100 particles. The results agree with each other and the maximum relative error is 1.3% (for the case presented in Fig. 8). In addition, homogenization of elastic-plastic material with random orientation of inclusions was analysed and the obtained results were compared with data pre-

sented in work of Doghri and Tinel (2006). In conclusion, optimal pseudo-grain discretization requires high computational effort in order to identify pseudo-grains’ discrete orientations and corresponding weights due to time consuming evolutionary algorithm. However, it provides more accurate orientation distribution function reconstruction, as compared to existing studies, and only nine discrete orientations have to be accounted for, making the homogenization more computationally efficient. Acknowledgment The research for this paper was financially supported by the Silesian University of Technology grants 10/040/BKM_15/2016 and 10/990/BK_16/0040. References Adam, L., Depouhon, A., Assaker, R., 2009. Multi-scale modeling of crash & failure of reinforced plastics parts with digimat to ls-dyna interface. 7th European LS– DYNA Conference. Advani, S.G., Tucker III, C.L., 1987. The use of tensors to describe and predict fibre orientation in short fibre composites. J. Rheol. 31, 751–784. Advani, S.G, Sozer, E.M, 2003. Process Modeling in Composites Manufacturing. Marcel Dekker Inc, New York, Basel.

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Please cite this article as: W. Ogierman, G. Kokot, Homogenization of inelastic composites with misaligned inclusions by using the optimal pseudo-grain discretization, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.03.008