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Stochastic homogenization analysis of a porous material with the perturbation method considering a microscopic geometrical random variation
International Journal of Mechanical Sciences 77 (2013) 145–154
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Stochastic homogenization analysis of a porous material with the perturbation method considering a microscopic geometrical random variation S. Sakata a,n, F. Ashida b, K. Ohsumimoto b a b
Kinki University, Department of Mechanical Engineering, 3-4-1 Kowakae, Higashi-Osaka City, Osaka 577-8502, Japan Graduate School of Shimane University, 1060, Nishikawatsu-cho, Matue, Shimane 690-8504, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 24 May 2013 Received in revised form 27 September 2013 Accepted 2 October 2013 Available online 12 October 2013
This paper discusses stochastic homogenization analysis of a periodic porous material fabricated using a rapid prototyping technique. A rapid prototyping system will be helpful to fabricate an order-made structure stably consisting of a porous material having a desired void distribution than a general porous material, but the influence of a geometrical random variation of pores should be still investigated, because some geometrical parameters are difficult to be perfectly controlled. In this paper, the stochastic homogenization analysis is performed for evaluation of the probabilistic characteristics of the homogenized elastic properties for a geometrical random variation in microstructure. The perturbation-based approach with the finite difference scheme is proposed for stochastic homogenization analysis of the porous material considering a parametric geometrical random variation. Influence of the random variations of microscopic geometry parameters on the homogenized elastic property is investigated, and accuracy of the finite difference based perturbation approach is discussed. In addition, a numerical result is compared to the experimental result, and applicability of the stochastic homogenization analysis to a practical problem is investigated. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Stochastic homogenization Porous material Perturbation Geometrical random variation Rapid prototyping
1. Introduction Porous materials are used for several purposes in a structure, for example, as the core of a lightweight structure or an implant biomaterial. These materials can be fabricated using techniques such as sintering, gas impregnation, and chemical reaction. Although these techniques enable to manufacture a porous material, it is difficult to control the microscopic geometry or density distribution of voids in a complex-shaped structure. Random geometrical variations at the microscopic level bring about unpredictable changes in a homogenized material property, its macroscopic response, and microscopic stress field; therefore, a microscopic random variation should be reduced. To this end, a rapid prototyping system would be helpful. A rapid prototyping system can be used for manufacturing complex shapes made of a porous material, and the pore distribution can be controlled as desired. As a micro-porous material, for example, Ohtsubo et al. reported the fabrication of a three-dimensional (3D) microstructure using microstereolithography [1]. In this report, a micro-porous material having a honeycomb shape with around 40
n
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0020-7403/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2013.10.001
voids in a 0:1 mm 0:1 mm square is introduced. This highlights the potential of the rapid prototyping technique for manufacturing a porous material having quasi-periodic voids. This porous material will have controlled voids. However, the microscopic random variation cannot be eliminated even if it can be reduced in comparison with typical porous materials, because the rapid prototyping process involves a photochemical reaction or other processes such as thermal ones and includes some mechanical errors. From this background, we investigate the influence of microscopic random variations observed in a porous material on the homogenized mechanical property or strength of the material. This type of analysis is called as “stochastic homogenization” or “stochastic multiscale stress analysis”. The stochastic homogenization analysis of composite materials has been discussed in recent years. For example, composite materials have complex microstructures, and there is uncertainty surrounding their homogenized material properties caused by random variations in their microstructure. The microscopic random variations decrease the reliability of a composite structure, and therefore the stochastic homogenization problem of a composite material should be analyzed. Kaminski et al. [2,3] and Sakata et al. [4–7] reported results of the stochastic homogenization analysis obtained from a Monte-Carlo simulation and/or a perturbation-based stochastic homogenization analysis of
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a composite material. Xu et al. [8] or Tootkaboni et al. [9] proposed different methods for the stochastic homogenization analysis of composite materials. Multiscale stochastic stress analysis of a particle reinforced composite material [10], stochastic stress analysis of composites using the Green function-based method [11], approximate stochastic homogenization analyses [12–14], microscopic failure probability analysis of a unidirectional fiber reinforced composite material through the multiscale stochastic stress analysis[15] and the inverse stochastic homogenization [16] analysis have been reported. However, such stochastic homogenization analysis of a porous material manufactured using rapid prototyping is yet to be discussed. In particular, for a porous material, the influence of microscopic geometrical variations such as those in size, shape, or location of voids on the probabilistic properties of homogenized elastic constants should be investigated in addition to random variations in the material properties of its component materials. Therefore, in this study, stochastic homogenization analysis of a porous material manufactured with a rapid prototyping system is carried out to evaluate the influence of microscopic geometrical randomness on the homogenized elastic property of a material. There are two aims of studies on multiscale problems considering the probabilistic aspect; one is considering non-uniform distributions of one microscopic quantity in a given structure, and the other is estimating the degree of a random variation, for example, coefficient of variance (CV) computed from a set of target structures. In the latter case, it can be considered that the observed values of a random variable are uniform over a structure, and the statistical properties computed from the observed values indicate a possible difference in each structure, for example, manufactured objects from different production lots. In this case, the homogenization theory can be applied to multiscale elastic analysis because the microstructure is still aligned uniformly and periodically. Also, the probabilistic characteristics can be estimated from a set of deterministic inputs which can be regarded as realized values of a random phenomenon. In this work, the second case is considered as the target problem. The analysis can be performed using a Monte-Carlo simulation or perturbation-based stochastic homogenization method. These methods have been applied to the stochastic homogenization analysis of composite materials, but a comprehensive perturbation-based stochastic homogenization analysis of a quasiperiodic porous material considering microscopic geometrical random variations is yet to be reported. In particular, the finite difference approach is employed for perturbation-based stochastic homogenization analysis considering parametric random variations in the microscopic microscopic geometry. In addition, a comparative result with an experimental data set is provided in this paper. In general, the accuracy and applicability of the perturbation-based stochastic homogenization analysis of composite materials have been investigated in comparison with the results of the Monte-Carlo simulation in general. Validation of the numerical results against an experimental result would be difficult in the case of heterogeneous materials, especially in the case of a composite material with a material property variation as reported previously, because it is difficult to measure microscopic random variations of material properties of component materials. Therefore, a comparison between numerical and experimental results has not yet been reported. To this end, as well as for the other purpose of investigating the influence of the microscopic geometrical random variation in a porous material on its homogenized elastic property, The numerical result is compared to an experimental result in order to investigate the applicability of the stochastic homogenization method to a practical problem. A rapid prototyping system enables
the fabrication of porous materials having quasi-periodic voids, and microscopic geometrical random variation can be measured experimentally, these measurements can then be compared with the results of the stochastic homogenization analysis. This experimental evaluation of the probabilistic characteristics of the equivalent elastic property for microscopic geometrical random variation of a porous material by a rapid prototyping technique is compatible with the stochastic homogenization analysis, and this is one of the key ideas of this study. In the following sections, an outline of stochastic homogenization analysis using the Monte-Carlo simulation and perturbation-based stochastic homogenization is described. In particular, a simple finite difference-based approach for parametric random variations in microscopic geometry is introduced. Next, the problem setting and some numerical results are provided. A detailed analysis of the influence of the microscopic geometrical random variation is performed, and the accuracy of the proposed perturbation approach for each geometrical random variation is investigated. Finally, from the numerical and experimental results, the influence of microscopic geometrical random variation and the applicability of the presented numerical method to a practical problem are discussed.
2. Porous material manufactured using a rapid prototyping system In general, various geometrical random variations are observed in a typical porous material. For instance, voids in a porous material are distributed randomly, and the shape or size of these voids is not uniform. Fig. 1 shows an enlarged schematic figure of a typical porous material. As shown in this figure, void shapes are distorted randomly, and their size and location vary randomly. These random variations bring about uncertainty in homogenized material property or strength of the material; this fact highlights the necessity of a statistical evaluation of the porous material considering these random variations in its microstructure. A rapid prototyping (RP) system allows for the fabrication of a resin-based structure having a complex, 3D shape. Only CAD data for a structure and a few additional processes would be required for the rapid prototyping. Owing to this feature, rapid prototyping is suitable for use in conjunction with finite element (FE) analysis because numerical models used in FE analysis can be easily converted to model data for rapid prototyping. As an example, a photograph of a manufactured 3D porous material and its finite element model are shown in Fig. 2. Fig. 2 (a) shows a sample of the 3D porous material having voids of a cube,1 mm each side. A few examples of smaller micro-porous structures fabricated using rapid prototyping are introduced in literature [1,17].
randomness location
void shape and size Fig. 1. Random variations in the porous material. (a) porous material and (b) randomness in voids.
S. Sakata et al. / International Journal of Mechanical Sciences 77 (2013) 145–154
147
Fig. 2. RP and FE model of a porous material. (a) 3D porous material by RP. (b) FE model of 3D porous material.
A rapid prototype system can be used to manufacture a porous material that has greater stability than a typical porous material having randomly distributed voids, but a microscopic geometric parameter in the material as the shape, size or location variation of the voids will still be difficult to control to the desired extent. Therefore, the influence of microscopic random variation of voids in a porous material on its homogenized material property should be investigated.
3. Stochastic homogenization with the Monte-Carlo simulation In many reports, an equivalent material property of a porous or another heterogeneous material is evaluated with the homogenization methods. An expected homogenized property will be computed from a deterministic model of a microstructure, but an additional analysis will be needed for evaluation of the probabilistic properties of the homogenized properties of such a random media. In this paper, it is assumed that the expected value of the porous material can be estimated with the homogenization theory [18]. In this case, the homogenized elastic tensor of a composite material can be computed as Z 1 ∂χ CH ¼ dY ð1Þ C I jYj Y ∂y where the superscript H indicates a homogenized quantity, C is an elastic tensor of a microstructure, jYj is the volume of a unit cell and I is a unit tensor. χ is a characteristic displacement, which satisfies with the following equation, ∂ T ∂χ ∂T C ¼ C ∂y ∂y ∂y
a random variable as X n ¼ X 0 ð1 þ αÞ
ð4Þ
where X 0 is an expected value of the quantity, and α is a normalized random variable. Note that this expression assumes that the microscopic quantity varies uniformly within one material in a unit cell of a microstructure. In this case, the observed value of the homogenized elastic tensor becomes a function of the normalized random variable α, and the expected value Exp½ and variance Var½ of the homogenized elastic tensor can be approximately computed as, for example [19], Exp½C Hn
1 ∑C Hn ðαÞ n
ð5Þ
Var½C Hn
1 ∑ðC Hn ðαÞ Exp½C Hn Þ2 n 1
ð6Þ
From these quantities, the coefficient of variance CV½ can be also computed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ CV½C Hn ¼ V ar½C Hn =Exp½C Hn In case of assuming the Gaussian random variable, the random number is generated with the following formula [20]. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α ¼ 2s2 log U 1 sin 2π U 2 ð8Þ where 0 o U 1 r 1 and 0 r U 2 r 1 are observed values of a uniform random variable.
ð2Þ
4. Perturbation-based stochastic homogenization analysis for a parametric geometrical random variation
In the case of considering a microscopic uncertainty of a heterogeneous material, the homogenized elastic tensor also has a random variation. For example, if an elastic property and geometry of microstructure of the component materials have a random variation, the observed homogenized elastic tensor can be computed as Z 1 ∂ χn CH n ¼ dY ð3Þ Cn I jY j Y ∂y
In order to estimate the probabilistic characteristics of the homogenized elastic property of a porous material for a microscopic small random variation, the perturbation-based stochastic homogenization technique can be used. As described with Eq. (4), it is assumed that an observed value of an elastic property, for example Young's modulus E of a component material, can be expressed as the following equation.
In order to analyze the probabilistic characteristics of a homogenized material property caused by a microscopic random variation, as a first choice of an analysis method, the Monte-Carlo simulation can be usable. For example, it is assumed that an observed value of a microscopic quantity X n can be expressed as a linear function of
En ¼ E0 ð1 þ αÞ
ð9Þ
where the superscripts “n” and “0” indicate the observed and expected value. If a random variation in an elastic property of a component material is taken into account, Eq. (3) can be approximated with applying the asymptotic expansion form of the random responses
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equations.
as
Z
∂ ðχ0 þ χ1 α þ χ2 α2 þ ⋯Þ dY ∂y
CH1 ¼
dCHn CH ð1 þ ΔαÞ CH ð1 ΔαÞ dα 2Δα
ð17Þ
where superscript “i” indicates the ith order perturbation term. In this paper, all response functions such as Cn or χn are differentiable with respect to a random variable and the perturbation term can be approximated with the finite different scheme. In order to obtain the perturbation terms of the characteristic displacement, the following simultaneous equation is solved. 9 KY0 χ0 ¼ FY0 > = ð11Þ ðKY1 χ0 þ KY0 χ1 Þ ¼ FY1 > ; ⋮
CH2 ¼
d CH n CH ð1 þ ΔαÞ 2CH0 þ CH ð1 ΔαÞ dα2 Δα2
ð18Þ
where KY and FY are the stiffness matrix and load vector for the characteristic equation. For example, the equation for computing the first order perturbation term of the characteristic displacement for an elastic property variation is expressed as follows. Z Z Z BT C0 ðBχ1 ÞdY þ BT C1 ðI Bχ0 ÞdY ¼ BT C1 dY ð12Þ
Exp½CHn ¼ CH0 þ ∑ ∑ ½CH2 ij cov½αi ; αj þ ⋯
CH n
1 jYj
Y
ðC0 þC1 α þ C2 α2 þ ⋯Þ I
ð10Þ
2
Y
Y
Y
where B is the following the displacement-strain matrix. ∂ B¼ ∂y
Var½C ¼ ∑ ∑ ½C i
i
j
H1
i ½CH1 j cov½αi ; αj
j
þ 2∑ ∑ ∑ ½CH1 i ½CH2 jk M3 ½αi ; αj ; αk i
i
ð13Þ
ð14Þ Also, the first order perturbation term of the microscopic stiffness matrix with respect to the microscopic geometrical random variation can be written as Z Z ∂ KY1 ¼ BnT CBn dY ¼ ðBT Þ1 CB0 þ ðBT Þ0 CB1 dY ∂α Y Y α¼0 α¼0 ð15Þ Here, the first order perturbation term of Bn for a geometrical random variation will be difficult to be expressed by an analytical form, especially in the case that a geometrical variation can be given as a parametric variation of a random variable such as shape, size, volume fraction or location of voids when the finite element method is employed for the numerical analysis. From this reason, the first order perturbation term for a microscopic geometrical random variation is approximately computed with the finite difference scheme as dBn Bð1 þ ΔαÞ Bð1 ΔαÞ dα 2Δα
Hn
j
k
þ ∑ ∑ ∑ ∑f½CH2 ij ½CH2 kl ðM4 ½αi ; αj ; αk ; αl
It is recognized from this equation that the perturbation term of the characteristic displacement is not independent of that of the elastic tensor. In case of considering only a geometrical random variation, the first order perturbation term of the homogenized elastic tensor can be computed as Z Z ∂ 1 ∂ χn 1 CH1 ¼ dY C I ¼ CðB1 χ0 þB0 χ1 ÞdY ∂α jYj Y jYj Y ∂y α¼0
B1 ¼
It should be noted that pre-confirmation about influence of Δα on accuracy of the sensitivity approximation is needed, because the accuracy of the approximations depend on the choice of Δα. With this numerical procedure, the expectation or variance of the homogenized elastic properties of the porous material with considering a geometrical random variation of a microstructure, for example shape or size variation of voids, can be approximately computed with the nth order second moment method as follows.
ð16Þ
where Δα is the small non-zero positive finite difference of the random variable α. Also, the perturbation terms of the characteristic displacement or the homogenized elastic tensor can be computed with the same manner. For example, when the uniform microscopic random variation is considered, the first two perturbation terms of the homogenized elastic tensor can be expressed by the following
j
k
l
cov½αi ; αj cov½αk ; αl Þg þ ⋯
ð19Þ
where M 3 and M 4 are the third and fourth order probabilistic moment, respectively. In this paper, it is assumed that the normalized random variable Δα distributes according to the Gaussian distribution, and in this case, a higher order probabilistic moment can be generally computed as follows: ( 0 : n ¼ odd Mn ¼ ð20Þ 1 3 ⋯ ðn 1Þsn n ¼ even For a detailed explanation on a higher order perturbationbased approach, a textbook [21] can be referred.
5. Numerical examples 5.1. Problem setting The influence of random variation in size, shape, and location of voids on the stochastic properties of a porous material is investigated in detail. Given that the porous material made using rapid prototyping can have periodic voids, as explained in the previous section, a small unit cell is used for the analysis. To simplify the problem, we assume that the material has circular holes, as shown in Fig. 3. As illustrated in this figure, the variation in the diameters of the circular holes expresses a random variation in their volume fraction, whereas the shape variation refers to the change in the ratio of the lateral and vertical axes of the elliptical holes. Because the location variation of the hole cannot be considered with a unit cell including having one hole, a unit cell with 3 3 holes is used for the stochastic homogenization analysis with considering the location variation of the hole, as shown in Fig. 3(c). With the presented numerical analysis methods, the influences of those variations on the probabilistic characteristics of the homogenized elastic properties of the porous material are investigated. The elastic properties of the material used in the analysis correspond to those of epoxy resin, and the expected value of the volume fraction of holes is 0.2.
S. Sakata et al. / International Journal of Mechanical Sciences 77 (2013) 145–154
149
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
Vf
0
Vf
CV of the equivalent elastic coefficients
0.07 *
y
x
expected size
varied size
0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.00
0.02
0.04
0.06
0.08
0.10
CV of the shape of holes
2rx
0
2rx
Fig. 5. CV of the equivalent elastic coefficients for the random variation of shape of the holes.
*
y
expected shape
CV of the equivalent elastic coefficients
x
varied shape
0.0004 0.0003 0.0002 0.0001 0.0000 0.00
0.02
0.04
0.06
0.08
0.10
CV of the shape of locations
y
Fig. 6. CV of the equivalent elastic coefficients for the random variation of the location of holes.
d 0
x
1 expected location
0
1 varied location
Fig. 3. Schematic view of an analysis model for a periodic porous material. (a) size (volume fraction) variation. (b) shape variation. (c) location variation.
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
0.07
CV of the equivalent elastic coefficients
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
0.0005
0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.00
0.02
0.04
0.06
0.08
0.10
CV of the volume fraction of holes Fig. 4. CV of the equivalent elastic coefficients for the random variation of the volume fraction of the holes.
5.2. Numerical results The estimated results under the presented problem setting are shown in Figs. 4–6. These figures illustrate the CVs of each homogenized elastic property for the case that the CV of the
random variables is between 0.01 and 0.1. These results are obtained from the Monte-Carlo simulation. Fig. 4 illustrates the CVs of the homogenized elastic properties for hole-size variation. From this figure, it is recognized that the size variation has a considerable influence on the equivalent Young's modulus of the porous material in the transverse direction H H EH x and E y , and the equivalent sharing modulus Gxy . For example, H approximately CVs of 0.04 are observed for Ex and EH y in terms of random variations in the V f of voids, CV[V f ] ¼0.06. The relationship between the CVs of the size variation and the equivalent elastic constants is almost linear. Fig. 5 illustrates the CVs of the homogenized elastic properties with respect to hole-shape variations. In this case, to clarify the influence of shape variation itself, it is assumed that the volume fraction of holes is constant. From Fig. 5, it can be seen that the random variation of the hole shape considerably influences the stochastic properties of the equivalent elastic constants of the porous material even under a constant volume fraction; therefore, as is the case with volume fraction variation, it should be considered in the reliability evaluation. Furthermore, the relationship between the CVs of the random variable and the equivalent elastic constants is almost linear, but minor non-linearity is H observed, except in the case of GH xy variation. The CVs of E z and H vzx are very small and can be ignored in case of a random variation in the hole shape. The relationship between the CVs of the equivalent elastic constants and the microscopic random variation of hole location is non-linear, as shown in Fig. 6. This result implies that lower-order perturbation may not accurately estimate the CVs of the
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0.006
×10−5
0.005
CV
0.004 0.003 0.002 0.001
2Δα = 0.05
0.000 0.00
0.02
0.04
0.06
0.08
0.10
2Δα
0.6
×10−5
0.5
dCV
0.4 0.3
2Δα = 0.05
0.2 0.1 0.0 0.00
0.02
0.04 2Δα
0.06
0.08
Fig. 7. Relationship between the estimated CV or ΔCV and Δα (a) relationship between CV and Δα. (b) relationship between ΔCV and Δα.
equivalent elastic constants for hole location variation, especially in the case of a large random variation. However, it can be confirmed that the absolute values of the CVs are very small compared with the other cases. In light of this finding, the microscopic variation of hole locations can be ignored for multiple random variations of the geometrical parameters in the stochastic homogenization analysis of a periodic porous material. 5.3. Accuracy of perturbation-based analysis for a geometrical random variation of holes in a porous material To investigate applicability of the perturbation-based approach to the stochastic homogenization analysis of the porous material for the microscopic geometrical random variations, we investigated the relative estimation error between the results estimated with the perturbation-based technique and the Monte-Carlo simulation. As examples, first-, second-, and third-order perturbation methods are employed. The computational conditions are the same as those in Section 5.2, and the same finite element model of the unit cell is used for this analysis. Since accuracy of the approximated perturbation terms with the finite difference scheme (Eqs.(17) and (18)) depends on the choice of the finite difference Δα, convergency of the perturbation terms of the equivalent elastic constants is checked. As an example, V f variation is assumed, and relationship between the estimated CV and Δα is investigated. Fig. 7(a) shows the relationship between the estimated CV for the volume fraction variation of holes and Δα, and Fig. 7(b) shows the relationship between the
ΔCV of the equivalent elastic properties for different Δα. ΔCV means the difference of CVs, for example, ΔCVΔα ¼ 0:01 ¼ CVΔα ¼ 0:02 CVΔα ¼ 0:01 . From Fig. 7(a), the estimated CVs are almost constant. On the other hand, form Fig. 12 (b), the ΔCV is fluctuated in case of smaller Δα. It is recognized from this figure that ΔCV is conversing as Δα increasing, and we considered 2Δα Z 0:05 will be better for stable computation. From these results, 2Δα ¼ 0:05 is adopted in this paper. The estimation errors of the perturbation-based technique for the case of V f variation of the holes are shown in Fig. 8. From these results, it can be inferred that the estimation error is very small even when using the first-order perturbation-based technique. The maximum relative estimation error is about 0.2% for 0.1 CV of the volume fraction. This result demonstrates the applicability of the perturbation-based stochastic homogenization method for the volume fraction variation. Next, the relative estimation error in the case of shape variation is investigated. Fig. 9 shows the estimation errors for each CV of the equivalent elastic constants in the case of shape variation. Fig. 9(a) shows the result of employing first-order perturbation, and Fig. 9(b) and (c) show the results of second- and third-order perturbations, respectively. In this result, it should be noted that first-order perturbation cannot estimate the CV of the GH xy . H H Furthermore, the estimation errors in the CVs of νH zx , Gzx , νxy , or EH x are large when using first-order perturbation. With the second-order perturbation, the accuracy of the CV estimation of GH xy improves drastically. From this result, it can be considered that at least second-order perturbation will be required for determining the shape variation of holes. Additionally, when using third-order perturbation, accuracy can be improved effectively, except in the case of GH xy variation. Third-order perturbation will be usable for determining the microscopic shape variation of the circular holes, but for estimating the CV of GH xy with greater accuracy, another stochastic homogenization methodology may need to be developed. In order to discuss a reason underlying the failure of lowerorder perturbations in accurately estimating the CV of GH xy , the relationship between GH xy and α is shown in Fig. 10. This figure shows the GH xy obtained from the deterministic homogenization analysis and the perturbation-based approximation. The legend “Exact” shows the result of the deterministic computation, “1st,” “2nd,” and “3rd” show the estimated GH xy with the first-, second-, and third-order perturbation–based approximations. From this figure, it is recognized that the response of GH xy is almost symmetric with respect toα, and that first-order perturbation cannot express the relationship between GH xy and α. Therefore, first-order perturbation cannot estimate the CV of GH xy . In contrast, second- or third- order perturbations yield better approximations and, thus, more accurate CVs of GH xy . For location variation, it is considered that first-order perturbation approach cannot estimate the CVs of the equivalent elastic constants appropriately using the central finite difference, as shown in Eq. (17), because the random variation is symmetric. Therefore, the estimation errors for both central and forward finite differences are investigated. Fig. 11 illustrates the relative estimation error for the location variation using the central and forward difference methods. From Fig. 11, it is recognized that the first-order perturbation approach with central finite difference cannot estimate the CVs for location variation. The forward finite difference can slightly improve the estimation accuracy; nonetheless, the estimation errors are very large. However, second-order perturbation can improve the accuracy, and it should be noted that the large error in the CV estimation of νH xy is observed with the forward finite difference. From this result, we can infer that the first-order perturbation approach with both the central and forward
Relative error in the estimated CV (%)
Relative error in the estimated CV (%)
S. Sakata et al. / International Journal of Mechanical Sciences 77 (2013) 145–154
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
0.8 0.4 0.0 -0.4 -0.8 0.00
0.02
0.04
0.06
0.08
0.10
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
0.8 0.4 0.0 -0.4 -0.8 0.00
0.02
CV of the volume fraction of holes
Relative error in the estimated CV (%)
151
0.04
0.06
0.08
0.10
CV of the volume fraction of holes
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
0.8 0.4 0.0 -0.4 -0.8 0.00
0.02
0.04
0.06
0.08
0.10
CV of the volume fraction of holes
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
100 98 96 94 92 90 8 6 4 2 0 0.00
0.02
0.04
0.06
0.08
0.10
Relative error in the estimated CV (%)
Relative error in the estimated CV (%)
Fig. 8. Relative error in CV estimation for V f variation (a) first order perturbation. (b) second order perturbation. (c) third order perturbation.
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
10 8 6 4 2 0 0.00
Relative error in the estimated CV (%)
CV of the volume fraction of holes
0.02
0.04
0.06
0.08
0.10
CV of the volume fraction of holes
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
10 8 6 4 2 0 0.00
0.02
0.04
0.06
0.08
0.10
CV of the volume fraction of holes Fig. 9. Relative error in CV estimation for shape variation. (a) first order perturbation. (b) second order perturbation. (c) third order perturbation.
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GHxy [GPa]
0.90 0.88 Exact 1st 2nd 3rd
0.86 0.84 0.82 -0.3
-0.2
-0.1
0.0 α
0.1
0.2
0.3
Relative error in the estimated CV (%)
H
0.92
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
35 30 25 20 15 10 5 0 0.00
0.02
0.04
0.06
0.08
0.10
CV of the volume fraction of holes
Fig. 10. Relationship between GH xy and α for the shape variation.
1st central 1st forward 2nd central 2nd forward
100 80 60 40 20 0
Ex
Ey
Gyz Gzx Gxy νyz
νzx
Fig. 11. Comparison between the estimation errors by the central and forward finite difference in the case of the location variation.
difference does not yield an acceptable estimation. From the result of the second-order perturbation, it is concluded that the central finite difference is preferable for determining location variation when higher-order perturbation approaches are employed. Next, for detailed investigations, the relative estimation errors when using second- and third-order perturbations are shown in Fig. 12. Fig. 12(a) shows the result of second-order perturbation, and Fig. 12(b) shows the result of third-order perturbation. It is confirmed that the estimation error is large, especially in the case of CV[GH xy ] estimation. This estimation error is not improved by applying third-order perturbation. Because the CVs of the equivalent elastic constants are very small for location variation, it is not a serious problem. However, this result demonstrates the application limit of lowerorder perturbation for a random variation of microscopic geometry.
Relative error in the estimated CV (%)
Relative estimation error (%)
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
35 30 25 20 15 10 5 0 0.00
0.02
0.04
0.06
0.08
0.10
CV of the volume fraction of holes Fig. 12. Relative error in CV estimation for location variation. (a) second order perturbation. (b) third order perturbation.
5.0mm
t=2.5mm
y x
20.0mm 70.0mm
y
5.4. Comparison with experimental results
x To validate the proposed computational procedure, we applied the proposed method to a stochastic homogenization problem measured from the experimental result. In this case, a porous material with two-dimensional distribution of void, fabricated using the rapid prototyping system, is adopted as the target material. A simple uniaxial tensile test is performed for estimating the equivalent Young's modulus of the periodic porous material. For the experiment, a specimen with many holes is fabricated to simulate a porous material having a 2D distribution of voids. Fig. 13 shows the input model for the rapid prototyping system and a photograph of the specimen. Square holes are arranged periodically in the material. The size and location of the holes are
Fig. 13. A porous material having 2-dimensionally distributed voids. (a) an input data for a rapid prototyping. (b) a test peace of a porous material manufactured with the rapid prototyping.
varied randomly in addition to varying the elastic property of the material. Furthermore, stochastic homogenization analysis is performed for estimating the probabilistic characteristics of the porous material's equivalent Young's modulus in the loading direction. The number of holes is not too large, but a macroscopic deformation of 80 16 or fewer cells (20 4, 40 8) can be accurately evaluated using the homogenization theory even if a bending moment is applied to a cantilever [22]. It is considered
S. Sakata et al. / International Journal of Mechanical Sciences 77 (2013) 145–154
153
0.6
Relative frequency
Relative frequency
0.7 0.5 0.4 0.3 0.2 0.1 0.0 -1.0
-0.5
0.0
0.5
0.3 0.2 0.1 0.0
1.0
-1.0
-0.5
Relative variation
0.0
0.5
Relative variation
Relative frequency
Relative frequency
0.7 0.3 0.2 0.1 0.0
-1.0
-0.5
0.0
0.6 0.5 0.4 0.3 0.2 0.1 0.0 -1.0
0.5
-0.5
0.0
0.5
1.0
Relative variation
Relative variation
Fig. 14. Histogram of the relative variations from the experimental results. (a) Young's modulus, without holes. (b) size of holes, x-direction. (c) size of holes, y-direction. (d) equivalent Young's modulus, with holes.
Table 1 Expectation and CVs of the size and Young's modulus of the material. Size
Table 2 Comparison between the numerical and experimental results. E
x
y
0.54 0.21
0.58 0.21
CV Exp CV
Experiment
1st order perturbation 2nd order perturbation
Error[%]
0.1105
0.1059 0.1069
4.2 3.3
228[MPa] 0.0481
that 10 3 cells can be acceptable as a target specimen for evaluating the equivalent elastic property with the homogenization theory in such a simple uniaxial tensile load case. For confirmation, the equivalent elastic moduli along x-direction in Fig. 13 obtained from a single-scale computational experiment and the homogenization analysis are compared. As a result, the difference between them is about 0.3%, and it is considered the presented homogenization-based approach can be applicable to this analysis. From the previous discussion, the influence of the location variation can be ignored in this case. The size variation is considered as the microscopic geometrical variation, and the Young's modulus variation of the component material is considered as the material property variation. The CV of the size variation is determined from a set of measured data of the holes of 32 specimens. The CV of Young's modulus variation of the component material is estimated from the results of a uniaxial tensile test of a hole-less specimen with the same shape as that shown in Fig. 13. The experimental results are shown in Fig. 14. The figure shows the histograms of the relative frequency of Young's modulus of resin, x- and y-sizes of the holes, and equivalent Young's modulus of the porous material. From these figures, it can be recognized that each quantity varies randomly, and it can be assumed that these variations approximately follow the Gaussian distribution. The expected values and the coefficient of variances of the Young's modulus of the resin and the hole sizes are listed in
Table 1. The aim of the stochastic homogenization analysis is to estimate the coefficient of variance of the equivalent Young's modulus of the porous material from the coefficient of variances of Young's modulus of resin and the x- and y-sizes of the holes. Table 2 shows a comparison of the CVs of the equivalent Young's modulus values obtained from the experiment and those obtained from numerical analysis using the perturbation-based technique. The estimation error is 4.2% when using first-order perturbation and 3.3% when using second-order perturbation. A higher-order perturbation gives a more accurate estimation, and from this result, it can be confirmed that the perturbation-based stochastic homogenization method has a potential to be used for the stochastic homogenization analysis of a practical periodic porous material.
6. Conclusion In this paper, the stochastic homogenization analysis is performed for a two-dimensional porous material. Influences of the microscopic geometrical randomness as shape, size or location of voids on the homogenized elastic properties are investigated using the Monte-Carlo simulation and the perturbation-based stochastic homogenization method. The influence of the microscopic geometrical random variation of a periodic porous material on the homogenized elastic property is investigated in detail. From the numerical results, a size
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variation in a periodic porous material may have a large influence of the probabilistic property of some homogenized elastic constants than the location randomness, so the stochastic homogenization analysis with respect to a microscopic random variation such as size or shape variation of holes will be important for a quasi-periodic porous material. Applicability of the perturbation-based technique is also investigated. Each perturbation term is computed with the finite difference scheme, and accuracy of the first or higher order perturbation techniques are compared with the results of the Monte-Carlo simulation. From the numerical results, the lower order perturbation based approach will be acceptable for the presented stochastic homogenization problem considering a microscopic geometrical random variation, but a higher order perturbation may be preferable for estimating the CV of the GH xy for the shape variation. In addition, applicability of the presented method to a practical stochastic homogenization analysis of a periodic porous material is investigated. From the comparison between the numerical and experimental results, it is confirmed the perturbation method has a potential for the CV of the equivalent Young's modulus for considering both the microscopic size variation of holes and Young's modulus variation of the component material. In a next step of this study, a multiscale stochastic stress analysis of a porous material will be performed. In particular, the stochastic homogenization and multiscale stochastic stress analysis considering a non-uniform random variation [23] will be important for a practical problem. Also, the stochastic homogenization and multiscale stochastic stress analysis should be performed for a 3D porous structure. For this purpose, in order to investigate validity of the presented numerical method, 3D fabricating system, which enables to prepare many accurate 3D porous specimens stably at a lower cost, will be needed. Acknowledgment The first author is pleased to acknowledge support in part by Grants-in-Aid for Young Scientists (B) (No.23760079) from the Ministry of Education, Culture, Sports Science and Technology, and MEXT-supported program for the Strategic Research Foundation at Private Universities, 2012–2014. References [1] Ohtsubo Y, Yamamoto M, Tadokoro S, Takamori T. Fabrication of 3D Microstructures using micro stereolithography (1st report). Seimitsu Kougaku Kaishi 1998;64(8):1181–5. [2] Kami´nski M, Kleiber M. Perturbation based stochastic finite element method for homogenization of two-phase elastic composites. Comput Struct 2000;78:811–26.
[3] Kami´nski M. Generalized perturbation-based stochastic finite element method in elastostatics. Comput Struct 2007;85:586–94. [4] Sakata S, Ashida F, Kojima T, Zako M. Influence of uncertainty in microscopic material property on homogenized elastic property of unidirectional fiber reinforced composites. Theor Appl Mech 2008;56:67–76. [5] Sakata S, Ashida F, Kojima T, Zako M. Three-dimensional stochastic analysis using a perturbation-based homogenization method for homogenized elastic property of inhomogeneous material considering microscopic uncertainty. Int J Solids Struct 2008;45(3/4):894–907. [6] Sakata S, Ashida F, Kojima T. On perturbation-based stochastic homogenization analysis for a homogenized equivalent elastic constant of frp using the equivalent inclution method. J Solid Mech Mater Eng 2009;3(3):552–62. [7] Sakata S, Ashida F, Kojima T. Stochastic homogenization analysis for thermal expansion coefficient of fiber reinforced composites using the equivalent inclusion method with perturbation-based approach. Comput. Struct. 2010;88(7–8):458–66. [8] Xu XF, Brady LG. Computational stochastic homogenization of random media elliptic problems using Fourier Galerkin method. Finite Elem Anal Des 2006;42:613–22. [9] Tootkaboni M, Brady LG. A multi-scale spectral stochastic method for homogenization of multi-phase periodic composites with random material properties. Int. J. Num. Meth. Eng. 2010;83:59–90. [10] Sakata S, Ashida F, En'ya K. Perturbation-based stochastic stress analysis of a particle reinforced composite material via the stochastic homogenization analysis considering uncertainty in material properties. Int J Multiscale Comput Eng 2011;9(4):395–408. [11] Xu FX, Chen X, Shen L. A green-function-based multiscale method for uncertainty quantification of finite body random heterogeneous materials. Comput. Struct. 2009;87:1416–26. [12] Sakata S, Ashida F, Zako M. Kriging-based approximate stochastic homogenization analysis for composite material. Comput Meth Appl Mech. Eng. 2008;197(21–24):1953–64. [13] Kaminski M. Sensitivity and randomness in homogenization of periodic fiberreinforced composites via the response function method. Int. J. Solids Struct. 2009;46(3–4):923–37. [14] Sakata S, Ashida F, Iwahashi D. Stochastic homogenization analysis of a particle reinforced composite material using and approximate monte-carlo simulation with the weighted least square method. J Comput Sci Technol. 2013;7(1):1–11. [15] Sakata S, Ashida F, Enya K. A microscopic failure probability analysis of a unidirectional fiber reinforced composite material via a multiscale stochastic stress analysis for a microscopic random variation of an elastic property. Comput. Mater. Sci. 2012;62:35–46. [16] Sakata S, Ashida F, Shimizu Y. Inverse stochastic homogenization analysis for a particle reinforced composite material with the Monte-Carlo simulation. Int J Multiscale Comput Eng 2011;9(4):409–23. [17] Kalita SJ, Bose S, Hosic HL, Bandyopadhayay A. Development of controlled porosity polymer-ceramic composite scaffolds via fused deposition modeling. Mater Sci Eng. C 2003;23 611-520. [18] Guedes M, Kikuchi N. Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comput. Methods Appl. Mech. Eng. 1990;83:143–98. [19] Kaminski M. Computational Mechanics of Composite Materials. London: Springer; 2003; 16. [20] Press WH, Teukolski SA Vetterling, WT Flannery, BP. Numerical Recipes in C. Gijutsu-Hyoron Sha; 1993. (Japanese Edition). [21] Kaminski M. The stochastic perturbation method for computational mechanics. Chichester: Wiley; 2013. [22] Terada K, Kikuchi N. Kinshitsukaho Nyumon. Maruzen 2003:125–9 (in Japanese). [23] Sakata S, Ashida F. Hierarchical stochastic homogenization analysis of a particle reinforced composite material considering non-uniform distribution of microscopic random quantities. Comput Mech 2011;48:529–40.
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Stochastic homogenization analysis of a porous material with the perturbation method considering a microscopic geometrical random variation S. Sakata a,n, F. Ashida b, K. Ohsumimoto b a b
Kinki University, Department of Mechanical Engineering, 3-4-1 Kowakae, Higashi-Osaka City, Osaka 577-8502, Japan Graduate School of Shimane University, 1060, Nishikawatsu-cho, Matue, Shimane 690-8504, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 24 May 2013 Received in revised form 27 September 2013 Accepted 2 October 2013 Available online 12 October 2013
This paper discusses stochastic homogenization analysis of a periodic porous material fabricated using a rapid prototyping technique. A rapid prototyping system will be helpful to fabricate an order-made structure stably consisting of a porous material having a desired void distribution than a general porous material, but the influence of a geometrical random variation of pores should be still investigated, because some geometrical parameters are difficult to be perfectly controlled. In this paper, the stochastic homogenization analysis is performed for evaluation of the probabilistic characteristics of the homogenized elastic properties for a geometrical random variation in microstructure. The perturbation-based approach with the finite difference scheme is proposed for stochastic homogenization analysis of the porous material considering a parametric geometrical random variation. Influence of the random variations of microscopic geometry parameters on the homogenized elastic property is investigated, and accuracy of the finite difference based perturbation approach is discussed. In addition, a numerical result is compared to the experimental result, and applicability of the stochastic homogenization analysis to a practical problem is investigated. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Stochastic homogenization Porous material Perturbation Geometrical random variation Rapid prototyping
1. Introduction Porous materials are used for several purposes in a structure, for example, as the core of a lightweight structure or an implant biomaterial. These materials can be fabricated using techniques such as sintering, gas impregnation, and chemical reaction. Although these techniques enable to manufacture a porous material, it is difficult to control the microscopic geometry or density distribution of voids in a complex-shaped structure. Random geometrical variations at the microscopic level bring about unpredictable changes in a homogenized material property, its macroscopic response, and microscopic stress field; therefore, a microscopic random variation should be reduced. To this end, a rapid prototyping system would be helpful. A rapid prototyping system can be used for manufacturing complex shapes made of a porous material, and the pore distribution can be controlled as desired. As a micro-porous material, for example, Ohtsubo et al. reported the fabrication of a three-dimensional (3D) microstructure using microstereolithography [1]. In this report, a micro-porous material having a honeycomb shape with around 40
n
Corresponding author. Tel.: þ 81 667212332. E-mail address: [email protected] (S. Sakata).
0020-7403/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2013.10.001
voids in a 0:1 mm 0:1 mm square is introduced. This highlights the potential of the rapid prototyping technique for manufacturing a porous material having quasi-periodic voids. This porous material will have controlled voids. However, the microscopic random variation cannot be eliminated even if it can be reduced in comparison with typical porous materials, because the rapid prototyping process involves a photochemical reaction or other processes such as thermal ones and includes some mechanical errors. From this background, we investigate the influence of microscopic random variations observed in a porous material on the homogenized mechanical property or strength of the material. This type of analysis is called as “stochastic homogenization” or “stochastic multiscale stress analysis”. The stochastic homogenization analysis of composite materials has been discussed in recent years. For example, composite materials have complex microstructures, and there is uncertainty surrounding their homogenized material properties caused by random variations in their microstructure. The microscopic random variations decrease the reliability of a composite structure, and therefore the stochastic homogenization problem of a composite material should be analyzed. Kaminski et al. [2,3] and Sakata et al. [4–7] reported results of the stochastic homogenization analysis obtained from a Monte-Carlo simulation and/or a perturbation-based stochastic homogenization analysis of
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a composite material. Xu et al. [8] or Tootkaboni et al. [9] proposed different methods for the stochastic homogenization analysis of composite materials. Multiscale stochastic stress analysis of a particle reinforced composite material [10], stochastic stress analysis of composites using the Green function-based method [11], approximate stochastic homogenization analyses [12–14], microscopic failure probability analysis of a unidirectional fiber reinforced composite material through the multiscale stochastic stress analysis[15] and the inverse stochastic homogenization [16] analysis have been reported. However, such stochastic homogenization analysis of a porous material manufactured using rapid prototyping is yet to be discussed. In particular, for a porous material, the influence of microscopic geometrical variations such as those in size, shape, or location of voids on the probabilistic properties of homogenized elastic constants should be investigated in addition to random variations in the material properties of its component materials. Therefore, in this study, stochastic homogenization analysis of a porous material manufactured with a rapid prototyping system is carried out to evaluate the influence of microscopic geometrical randomness on the homogenized elastic property of a material. There are two aims of studies on multiscale problems considering the probabilistic aspect; one is considering non-uniform distributions of one microscopic quantity in a given structure, and the other is estimating the degree of a random variation, for example, coefficient of variance (CV) computed from a set of target structures. In the latter case, it can be considered that the observed values of a random variable are uniform over a structure, and the statistical properties computed from the observed values indicate a possible difference in each structure, for example, manufactured objects from different production lots. In this case, the homogenization theory can be applied to multiscale elastic analysis because the microstructure is still aligned uniformly and periodically. Also, the probabilistic characteristics can be estimated from a set of deterministic inputs which can be regarded as realized values of a random phenomenon. In this work, the second case is considered as the target problem. The analysis can be performed using a Monte-Carlo simulation or perturbation-based stochastic homogenization method. These methods have been applied to the stochastic homogenization analysis of composite materials, but a comprehensive perturbation-based stochastic homogenization analysis of a quasiperiodic porous material considering microscopic geometrical random variations is yet to be reported. In particular, the finite difference approach is employed for perturbation-based stochastic homogenization analysis considering parametric random variations in the microscopic microscopic geometry. In addition, a comparative result with an experimental data set is provided in this paper. In general, the accuracy and applicability of the perturbation-based stochastic homogenization analysis of composite materials have been investigated in comparison with the results of the Monte-Carlo simulation in general. Validation of the numerical results against an experimental result would be difficult in the case of heterogeneous materials, especially in the case of a composite material with a material property variation as reported previously, because it is difficult to measure microscopic random variations of material properties of component materials. Therefore, a comparison between numerical and experimental results has not yet been reported. To this end, as well as for the other purpose of investigating the influence of the microscopic geometrical random variation in a porous material on its homogenized elastic property, The numerical result is compared to an experimental result in order to investigate the applicability of the stochastic homogenization method to a practical problem. A rapid prototyping system enables
the fabrication of porous materials having quasi-periodic voids, and microscopic geometrical random variation can be measured experimentally, these measurements can then be compared with the results of the stochastic homogenization analysis. This experimental evaluation of the probabilistic characteristics of the equivalent elastic property for microscopic geometrical random variation of a porous material by a rapid prototyping technique is compatible with the stochastic homogenization analysis, and this is one of the key ideas of this study. In the following sections, an outline of stochastic homogenization analysis using the Monte-Carlo simulation and perturbation-based stochastic homogenization is described. In particular, a simple finite difference-based approach for parametric random variations in microscopic geometry is introduced. Next, the problem setting and some numerical results are provided. A detailed analysis of the influence of the microscopic geometrical random variation is performed, and the accuracy of the proposed perturbation approach for each geometrical random variation is investigated. Finally, from the numerical and experimental results, the influence of microscopic geometrical random variation and the applicability of the presented numerical method to a practical problem are discussed.
2. Porous material manufactured using a rapid prototyping system In general, various geometrical random variations are observed in a typical porous material. For instance, voids in a porous material are distributed randomly, and the shape or size of these voids is not uniform. Fig. 1 shows an enlarged schematic figure of a typical porous material. As shown in this figure, void shapes are distorted randomly, and their size and location vary randomly. These random variations bring about uncertainty in homogenized material property or strength of the material; this fact highlights the necessity of a statistical evaluation of the porous material considering these random variations in its microstructure. A rapid prototyping (RP) system allows for the fabrication of a resin-based structure having a complex, 3D shape. Only CAD data for a structure and a few additional processes would be required for the rapid prototyping. Owing to this feature, rapid prototyping is suitable for use in conjunction with finite element (FE) analysis because numerical models used in FE analysis can be easily converted to model data for rapid prototyping. As an example, a photograph of a manufactured 3D porous material and its finite element model are shown in Fig. 2. Fig. 2 (a) shows a sample of the 3D porous material having voids of a cube,1 mm each side. A few examples of smaller micro-porous structures fabricated using rapid prototyping are introduced in literature [1,17].
randomness location
void shape and size Fig. 1. Random variations in the porous material. (a) porous material and (b) randomness in voids.
S. Sakata et al. / International Journal of Mechanical Sciences 77 (2013) 145–154
147
Fig. 2. RP and FE model of a porous material. (a) 3D porous material by RP. (b) FE model of 3D porous material.
A rapid prototype system can be used to manufacture a porous material that has greater stability than a typical porous material having randomly distributed voids, but a microscopic geometric parameter in the material as the shape, size or location variation of the voids will still be difficult to control to the desired extent. Therefore, the influence of microscopic random variation of voids in a porous material on its homogenized material property should be investigated.
3. Stochastic homogenization with the Monte-Carlo simulation In many reports, an equivalent material property of a porous or another heterogeneous material is evaluated with the homogenization methods. An expected homogenized property will be computed from a deterministic model of a microstructure, but an additional analysis will be needed for evaluation of the probabilistic properties of the homogenized properties of such a random media. In this paper, it is assumed that the expected value of the porous material can be estimated with the homogenization theory [18]. In this case, the homogenized elastic tensor of a composite material can be computed as Z 1 ∂χ CH ¼ dY ð1Þ C I jYj Y ∂y where the superscript H indicates a homogenized quantity, C is an elastic tensor of a microstructure, jYj is the volume of a unit cell and I is a unit tensor. χ is a characteristic displacement, which satisfies with the following equation, ∂ T ∂χ ∂T C ¼ C ∂y ∂y ∂y
a random variable as X n ¼ X 0 ð1 þ αÞ
ð4Þ
where X 0 is an expected value of the quantity, and α is a normalized random variable. Note that this expression assumes that the microscopic quantity varies uniformly within one material in a unit cell of a microstructure. In this case, the observed value of the homogenized elastic tensor becomes a function of the normalized random variable α, and the expected value Exp½ and variance Var½ of the homogenized elastic tensor can be approximately computed as, for example [19], Exp½C Hn
1 ∑C Hn ðαÞ n
ð5Þ
Var½C Hn
1 ∑ðC Hn ðαÞ Exp½C Hn Þ2 n 1
ð6Þ
From these quantities, the coefficient of variance CV½ can be also computed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ CV½C Hn ¼ V ar½C Hn =Exp½C Hn In case of assuming the Gaussian random variable, the random number is generated with the following formula [20]. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α ¼ 2s2 log U 1 sin 2π U 2 ð8Þ where 0 o U 1 r 1 and 0 r U 2 r 1 are observed values of a uniform random variable.
ð2Þ
4. Perturbation-based stochastic homogenization analysis for a parametric geometrical random variation
In the case of considering a microscopic uncertainty of a heterogeneous material, the homogenized elastic tensor also has a random variation. For example, if an elastic property and geometry of microstructure of the component materials have a random variation, the observed homogenized elastic tensor can be computed as Z 1 ∂ χn CH n ¼ dY ð3Þ Cn I jY j Y ∂y
In order to estimate the probabilistic characteristics of the homogenized elastic property of a porous material for a microscopic small random variation, the perturbation-based stochastic homogenization technique can be used. As described with Eq. (4), it is assumed that an observed value of an elastic property, for example Young's modulus E of a component material, can be expressed as the following equation.
In order to analyze the probabilistic characteristics of a homogenized material property caused by a microscopic random variation, as a first choice of an analysis method, the Monte-Carlo simulation can be usable. For example, it is assumed that an observed value of a microscopic quantity X n can be expressed as a linear function of
En ¼ E0 ð1 þ αÞ
ð9Þ
where the superscripts “n” and “0” indicate the observed and expected value. If a random variation in an elastic property of a component material is taken into account, Eq. (3) can be approximated with applying the asymptotic expansion form of the random responses
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equations.
as
Z
∂ ðχ0 þ χ1 α þ χ2 α2 þ ⋯Þ dY ∂y
CH1 ¼
dCHn CH ð1 þ ΔαÞ CH ð1 ΔαÞ dα 2Δα
ð17Þ
where superscript “i” indicates the ith order perturbation term. In this paper, all response functions such as Cn or χn are differentiable with respect to a random variable and the perturbation term can be approximated with the finite different scheme. In order to obtain the perturbation terms of the characteristic displacement, the following simultaneous equation is solved. 9 KY0 χ0 ¼ FY0 > = ð11Þ ðKY1 χ0 þ KY0 χ1 Þ ¼ FY1 > ; ⋮
CH2 ¼
d CH n CH ð1 þ ΔαÞ 2CH0 þ CH ð1 ΔαÞ dα2 Δα2
ð18Þ
where KY and FY are the stiffness matrix and load vector for the characteristic equation. For example, the equation for computing the first order perturbation term of the characteristic displacement for an elastic property variation is expressed as follows. Z Z Z BT C0 ðBχ1 ÞdY þ BT C1 ðI Bχ0 ÞdY ¼ BT C1 dY ð12Þ
Exp½CHn ¼ CH0 þ ∑ ∑ ½CH2 ij cov½αi ; αj þ ⋯
CH n
1 jYj
Y
ðC0 þC1 α þ C2 α2 þ ⋯Þ I
ð10Þ
2
Y
Y
Y
where B is the following the displacement-strain matrix. ∂ B¼ ∂y
Var½C ¼ ∑ ∑ ½C i
i
j
H1
i ½CH1 j cov½αi ; αj
j
þ 2∑ ∑ ∑ ½CH1 i ½CH2 jk M3 ½αi ; αj ; αk i
i
ð13Þ
ð14Þ Also, the first order perturbation term of the microscopic stiffness matrix with respect to the microscopic geometrical random variation can be written as Z Z ∂ KY1 ¼ BnT CBn dY ¼ ðBT Þ1 CB0 þ ðBT Þ0 CB1 dY ∂α Y Y α¼0 α¼0 ð15Þ Here, the first order perturbation term of Bn for a geometrical random variation will be difficult to be expressed by an analytical form, especially in the case that a geometrical variation can be given as a parametric variation of a random variable such as shape, size, volume fraction or location of voids when the finite element method is employed for the numerical analysis. From this reason, the first order perturbation term for a microscopic geometrical random variation is approximately computed with the finite difference scheme as dBn Bð1 þ ΔαÞ Bð1 ΔαÞ dα 2Δα
Hn
j
k
þ ∑ ∑ ∑ ∑f½CH2 ij ½CH2 kl ðM4 ½αi ; αj ; αk ; αl
It is recognized from this equation that the perturbation term of the characteristic displacement is not independent of that of the elastic tensor. In case of considering only a geometrical random variation, the first order perturbation term of the homogenized elastic tensor can be computed as Z Z ∂ 1 ∂ χn 1 CH1 ¼ dY C I ¼ CðB1 χ0 þB0 χ1 ÞdY ∂α jYj Y jYj Y ∂y α¼0
B1 ¼
It should be noted that pre-confirmation about influence of Δα on accuracy of the sensitivity approximation is needed, because the accuracy of the approximations depend on the choice of Δα. With this numerical procedure, the expectation or variance of the homogenized elastic properties of the porous material with considering a geometrical random variation of a microstructure, for example shape or size variation of voids, can be approximately computed with the nth order second moment method as follows.
ð16Þ
where Δα is the small non-zero positive finite difference of the random variable α. Also, the perturbation terms of the characteristic displacement or the homogenized elastic tensor can be computed with the same manner. For example, when the uniform microscopic random variation is considered, the first two perturbation terms of the homogenized elastic tensor can be expressed by the following
j
k
l
cov½αi ; αj cov½αk ; αl Þg þ ⋯
ð19Þ
where M 3 and M 4 are the third and fourth order probabilistic moment, respectively. In this paper, it is assumed that the normalized random variable Δα distributes according to the Gaussian distribution, and in this case, a higher order probabilistic moment can be generally computed as follows: ( 0 : n ¼ odd Mn ¼ ð20Þ 1 3 ⋯ ðn 1Þsn n ¼ even For a detailed explanation on a higher order perturbationbased approach, a textbook [21] can be referred.
5. Numerical examples 5.1. Problem setting The influence of random variation in size, shape, and location of voids on the stochastic properties of a porous material is investigated in detail. Given that the porous material made using rapid prototyping can have periodic voids, as explained in the previous section, a small unit cell is used for the analysis. To simplify the problem, we assume that the material has circular holes, as shown in Fig. 3. As illustrated in this figure, the variation in the diameters of the circular holes expresses a random variation in their volume fraction, whereas the shape variation refers to the change in the ratio of the lateral and vertical axes of the elliptical holes. Because the location variation of the hole cannot be considered with a unit cell including having one hole, a unit cell with 3 3 holes is used for the stochastic homogenization analysis with considering the location variation of the hole, as shown in Fig. 3(c). With the presented numerical analysis methods, the influences of those variations on the probabilistic characteristics of the homogenized elastic properties of the porous material are investigated. The elastic properties of the material used in the analysis correspond to those of epoxy resin, and the expected value of the volume fraction of holes is 0.2.
S. Sakata et al. / International Journal of Mechanical Sciences 77 (2013) 145–154
149
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
Vf
0
Vf
CV of the equivalent elastic coefficients
0.07 *
y
x
expected size
varied size
0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.00
0.02
0.04
0.06
0.08
0.10
CV of the shape of holes
2rx
0
2rx
Fig. 5. CV of the equivalent elastic coefficients for the random variation of shape of the holes.
*
y
expected shape
CV of the equivalent elastic coefficients
x
varied shape
0.0004 0.0003 0.0002 0.0001 0.0000 0.00
0.02
0.04
0.06
0.08
0.10
CV of the shape of locations
y
Fig. 6. CV of the equivalent elastic coefficients for the random variation of the location of holes.
d 0
x
1 expected location
0
1 varied location
Fig. 3. Schematic view of an analysis model for a periodic porous material. (a) size (volume fraction) variation. (b) shape variation. (c) location variation.
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
0.07
CV of the equivalent elastic coefficients
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
0.0005
0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.00
0.02
0.04
0.06
0.08
0.10
CV of the volume fraction of holes Fig. 4. CV of the equivalent elastic coefficients for the random variation of the volume fraction of the holes.
5.2. Numerical results The estimated results under the presented problem setting are shown in Figs. 4–6. These figures illustrate the CVs of each homogenized elastic property for the case that the CV of the
random variables is between 0.01 and 0.1. These results are obtained from the Monte-Carlo simulation. Fig. 4 illustrates the CVs of the homogenized elastic properties for hole-size variation. From this figure, it is recognized that the size variation has a considerable influence on the equivalent Young's modulus of the porous material in the transverse direction H H EH x and E y , and the equivalent sharing modulus Gxy . For example, H approximately CVs of 0.04 are observed for Ex and EH y in terms of random variations in the V f of voids, CV[V f ] ¼0.06. The relationship between the CVs of the size variation and the equivalent elastic constants is almost linear. Fig. 5 illustrates the CVs of the homogenized elastic properties with respect to hole-shape variations. In this case, to clarify the influence of shape variation itself, it is assumed that the volume fraction of holes is constant. From Fig. 5, it can be seen that the random variation of the hole shape considerably influences the stochastic properties of the equivalent elastic constants of the porous material even under a constant volume fraction; therefore, as is the case with volume fraction variation, it should be considered in the reliability evaluation. Furthermore, the relationship between the CVs of the random variable and the equivalent elastic constants is almost linear, but minor non-linearity is H observed, except in the case of GH xy variation. The CVs of E z and H vzx are very small and can be ignored in case of a random variation in the hole shape. The relationship between the CVs of the equivalent elastic constants and the microscopic random variation of hole location is non-linear, as shown in Fig. 6. This result implies that lower-order perturbation may not accurately estimate the CVs of the
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0.006
×10−5
0.005
CV
0.004 0.003 0.002 0.001
2Δα = 0.05
0.000 0.00
0.02
0.04
0.06
0.08
0.10
2Δα
0.6
×10−5
0.5
dCV
0.4 0.3
2Δα = 0.05
0.2 0.1 0.0 0.00
0.02
0.04 2Δα
0.06
0.08
Fig. 7. Relationship between the estimated CV or ΔCV and Δα (a) relationship between CV and Δα. (b) relationship between ΔCV and Δα.
equivalent elastic constants for hole location variation, especially in the case of a large random variation. However, it can be confirmed that the absolute values of the CVs are very small compared with the other cases. In light of this finding, the microscopic variation of hole locations can be ignored for multiple random variations of the geometrical parameters in the stochastic homogenization analysis of a periodic porous material. 5.3. Accuracy of perturbation-based analysis for a geometrical random variation of holes in a porous material To investigate applicability of the perturbation-based approach to the stochastic homogenization analysis of the porous material for the microscopic geometrical random variations, we investigated the relative estimation error between the results estimated with the perturbation-based technique and the Monte-Carlo simulation. As examples, first-, second-, and third-order perturbation methods are employed. The computational conditions are the same as those in Section 5.2, and the same finite element model of the unit cell is used for this analysis. Since accuracy of the approximated perturbation terms with the finite difference scheme (Eqs.(17) and (18)) depends on the choice of the finite difference Δα, convergency of the perturbation terms of the equivalent elastic constants is checked. As an example, V f variation is assumed, and relationship between the estimated CV and Δα is investigated. Fig. 7(a) shows the relationship between the estimated CV for the volume fraction variation of holes and Δα, and Fig. 7(b) shows the relationship between the
ΔCV of the equivalent elastic properties for different Δα. ΔCV means the difference of CVs, for example, ΔCVΔα ¼ 0:01 ¼ CVΔα ¼ 0:02 CVΔα ¼ 0:01 . From Fig. 7(a), the estimated CVs are almost constant. On the other hand, form Fig. 12 (b), the ΔCV is fluctuated in case of smaller Δα. It is recognized from this figure that ΔCV is conversing as Δα increasing, and we considered 2Δα Z 0:05 will be better for stable computation. From these results, 2Δα ¼ 0:05 is adopted in this paper. The estimation errors of the perturbation-based technique for the case of V f variation of the holes are shown in Fig. 8. From these results, it can be inferred that the estimation error is very small even when using the first-order perturbation-based technique. The maximum relative estimation error is about 0.2% for 0.1 CV of the volume fraction. This result demonstrates the applicability of the perturbation-based stochastic homogenization method for the volume fraction variation. Next, the relative estimation error in the case of shape variation is investigated. Fig. 9 shows the estimation errors for each CV of the equivalent elastic constants in the case of shape variation. Fig. 9(a) shows the result of employing first-order perturbation, and Fig. 9(b) and (c) show the results of second- and third-order perturbations, respectively. In this result, it should be noted that first-order perturbation cannot estimate the CV of the GH xy . H H Furthermore, the estimation errors in the CVs of νH zx , Gzx , νxy , or EH x are large when using first-order perturbation. With the second-order perturbation, the accuracy of the CV estimation of GH xy improves drastically. From this result, it can be considered that at least second-order perturbation will be required for determining the shape variation of holes. Additionally, when using third-order perturbation, accuracy can be improved effectively, except in the case of GH xy variation. Third-order perturbation will be usable for determining the microscopic shape variation of the circular holes, but for estimating the CV of GH xy with greater accuracy, another stochastic homogenization methodology may need to be developed. In order to discuss a reason underlying the failure of lowerorder perturbations in accurately estimating the CV of GH xy , the relationship between GH xy and α is shown in Fig. 10. This figure shows the GH xy obtained from the deterministic homogenization analysis and the perturbation-based approximation. The legend “Exact” shows the result of the deterministic computation, “1st,” “2nd,” and “3rd” show the estimated GH xy with the first-, second-, and third-order perturbation–based approximations. From this figure, it is recognized that the response of GH xy is almost symmetric with respect toα, and that first-order perturbation cannot express the relationship between GH xy and α. Therefore, first-order perturbation cannot estimate the CV of GH xy . In contrast, second- or third- order perturbations yield better approximations and, thus, more accurate CVs of GH xy . For location variation, it is considered that first-order perturbation approach cannot estimate the CVs of the equivalent elastic constants appropriately using the central finite difference, as shown in Eq. (17), because the random variation is symmetric. Therefore, the estimation errors for both central and forward finite differences are investigated. Fig. 11 illustrates the relative estimation error for the location variation using the central and forward difference methods. From Fig. 11, it is recognized that the first-order perturbation approach with central finite difference cannot estimate the CVs for location variation. The forward finite difference can slightly improve the estimation accuracy; nonetheless, the estimation errors are very large. However, second-order perturbation can improve the accuracy, and it should be noted that the large error in the CV estimation of νH xy is observed with the forward finite difference. From this result, we can infer that the first-order perturbation approach with both the central and forward
Relative error in the estimated CV (%)
Relative error in the estimated CV (%)
S. Sakata et al. / International Journal of Mechanical Sciences 77 (2013) 145–154
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
0.8 0.4 0.0 -0.4 -0.8 0.00
0.02
0.04
0.06
0.08
0.10
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
0.8 0.4 0.0 -0.4 -0.8 0.00
0.02
CV of the volume fraction of holes
Relative error in the estimated CV (%)
151
0.04
0.06
0.08
0.10
CV of the volume fraction of holes
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
0.8 0.4 0.0 -0.4 -0.8 0.00
0.02
0.04
0.06
0.08
0.10
CV of the volume fraction of holes
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
100 98 96 94 92 90 8 6 4 2 0 0.00
0.02
0.04
0.06
0.08
0.10
Relative error in the estimated CV (%)
Relative error in the estimated CV (%)
Fig. 8. Relative error in CV estimation for V f variation (a) first order perturbation. (b) second order perturbation. (c) third order perturbation.
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
10 8 6 4 2 0 0.00
Relative error in the estimated CV (%)
CV of the volume fraction of holes
0.02
0.04
0.06
0.08
0.10
CV of the volume fraction of holes
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
10 8 6 4 2 0 0.00
0.02
0.04
0.06
0.08
0.10
CV of the volume fraction of holes Fig. 9. Relative error in CV estimation for shape variation. (a) first order perturbation. (b) second order perturbation. (c) third order perturbation.
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GHxy [GPa]
0.90 0.88 Exact 1st 2nd 3rd
0.86 0.84 0.82 -0.3
-0.2
-0.1
0.0 α
0.1
0.2
0.3
Relative error in the estimated CV (%)
H
0.92
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
35 30 25 20 15 10 5 0 0.00
0.02
0.04
0.06
0.08
0.10
CV of the volume fraction of holes
Fig. 10. Relationship between GH xy and α for the shape variation.
1st central 1st forward 2nd central 2nd forward
100 80 60 40 20 0
Ex
Ey
Gyz Gzx Gxy νyz
νzx
Fig. 11. Comparison between the estimation errors by the central and forward finite difference in the case of the location variation.
difference does not yield an acceptable estimation. From the result of the second-order perturbation, it is concluded that the central finite difference is preferable for determining location variation when higher-order perturbation approaches are employed. Next, for detailed investigations, the relative estimation errors when using second- and third-order perturbations are shown in Fig. 12. Fig. 12(a) shows the result of second-order perturbation, and Fig. 12(b) shows the result of third-order perturbation. It is confirmed that the estimation error is large, especially in the case of CV[GH xy ] estimation. This estimation error is not improved by applying third-order perturbation. Because the CVs of the equivalent elastic constants are very small for location variation, it is not a serious problem. However, this result demonstrates the application limit of lowerorder perturbation for a random variation of microscopic geometry.
Relative error in the estimated CV (%)
Relative estimation error (%)
H
E x H E y H E z H G yz H G zx H G xy H ν yz H ν zx H ν xy
35 30 25 20 15 10 5 0 0.00
0.02
0.04
0.06
0.08
0.10
CV of the volume fraction of holes Fig. 12. Relative error in CV estimation for location variation. (a) second order perturbation. (b) third order perturbation.
5.0mm
t=2.5mm
y x
20.0mm 70.0mm
y
5.4. Comparison with experimental results
x To validate the proposed computational procedure, we applied the proposed method to a stochastic homogenization problem measured from the experimental result. In this case, a porous material with two-dimensional distribution of void, fabricated using the rapid prototyping system, is adopted as the target material. A simple uniaxial tensile test is performed for estimating the equivalent Young's modulus of the periodic porous material. For the experiment, a specimen with many holes is fabricated to simulate a porous material having a 2D distribution of voids. Fig. 13 shows the input model for the rapid prototyping system and a photograph of the specimen. Square holes are arranged periodically in the material. The size and location of the holes are
Fig. 13. A porous material having 2-dimensionally distributed voids. (a) an input data for a rapid prototyping. (b) a test peace of a porous material manufactured with the rapid prototyping.
varied randomly in addition to varying the elastic property of the material. Furthermore, stochastic homogenization analysis is performed for estimating the probabilistic characteristics of the porous material's equivalent Young's modulus in the loading direction. The number of holes is not too large, but a macroscopic deformation of 80 16 or fewer cells (20 4, 40 8) can be accurately evaluated using the homogenization theory even if a bending moment is applied to a cantilever [22]. It is considered
S. Sakata et al. / International Journal of Mechanical Sciences 77 (2013) 145–154
153
0.6
Relative frequency
Relative frequency
0.7 0.5 0.4 0.3 0.2 0.1 0.0 -1.0
-0.5
0.0
0.5
0.3 0.2 0.1 0.0
1.0
-1.0
-0.5
Relative variation
0.0
0.5
Relative variation
Relative frequency
Relative frequency
0.7 0.3 0.2 0.1 0.0
-1.0
-0.5
0.0
0.6 0.5 0.4 0.3 0.2 0.1 0.0 -1.0
0.5
-0.5
0.0
0.5
1.0
Relative variation
Relative variation
Fig. 14. Histogram of the relative variations from the experimental results. (a) Young's modulus, without holes. (b) size of holes, x-direction. (c) size of holes, y-direction. (d) equivalent Young's modulus, with holes.
Table 1 Expectation and CVs of the size and Young's modulus of the material. Size
Table 2 Comparison between the numerical and experimental results. E
x
y
0.54 0.21
0.58 0.21
CV Exp CV
Experiment
1st order perturbation 2nd order perturbation
Error[%]
0.1105
0.1059 0.1069
4.2 3.3
228[MPa] 0.0481
that 10 3 cells can be acceptable as a target specimen for evaluating the equivalent elastic property with the homogenization theory in such a simple uniaxial tensile load case. For confirmation, the equivalent elastic moduli along x-direction in Fig. 13 obtained from a single-scale computational experiment and the homogenization analysis are compared. As a result, the difference between them is about 0.3%, and it is considered the presented homogenization-based approach can be applicable to this analysis. From the previous discussion, the influence of the location variation can be ignored in this case. The size variation is considered as the microscopic geometrical variation, and the Young's modulus variation of the component material is considered as the material property variation. The CV of the size variation is determined from a set of measured data of the holes of 32 specimens. The CV of Young's modulus variation of the component material is estimated from the results of a uniaxial tensile test of a hole-less specimen with the same shape as that shown in Fig. 13. The experimental results are shown in Fig. 14. The figure shows the histograms of the relative frequency of Young's modulus of resin, x- and y-sizes of the holes, and equivalent Young's modulus of the porous material. From these figures, it can be recognized that each quantity varies randomly, and it can be assumed that these variations approximately follow the Gaussian distribution. The expected values and the coefficient of variances of the Young's modulus of the resin and the hole sizes are listed in
Table 1. The aim of the stochastic homogenization analysis is to estimate the coefficient of variance of the equivalent Young's modulus of the porous material from the coefficient of variances of Young's modulus of resin and the x- and y-sizes of the holes. Table 2 shows a comparison of the CVs of the equivalent Young's modulus values obtained from the experiment and those obtained from numerical analysis using the perturbation-based technique. The estimation error is 4.2% when using first-order perturbation and 3.3% when using second-order perturbation. A higher-order perturbation gives a more accurate estimation, and from this result, it can be confirmed that the perturbation-based stochastic homogenization method has a potential to be used for the stochastic homogenization analysis of a practical periodic porous material.
6. Conclusion In this paper, the stochastic homogenization analysis is performed for a two-dimensional porous material. Influences of the microscopic geometrical randomness as shape, size or location of voids on the homogenized elastic properties are investigated using the Monte-Carlo simulation and the perturbation-based stochastic homogenization method. The influence of the microscopic geometrical random variation of a periodic porous material on the homogenized elastic property is investigated in detail. From the numerical results, a size
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variation in a periodic porous material may have a large influence of the probabilistic property of some homogenized elastic constants than the location randomness, so the stochastic homogenization analysis with respect to a microscopic random variation such as size or shape variation of holes will be important for a quasi-periodic porous material. Applicability of the perturbation-based technique is also investigated. Each perturbation term is computed with the finite difference scheme, and accuracy of the first or higher order perturbation techniques are compared with the results of the Monte-Carlo simulation. From the numerical results, the lower order perturbation based approach will be acceptable for the presented stochastic homogenization problem considering a microscopic geometrical random variation, but a higher order perturbation may be preferable for estimating the CV of the GH xy for the shape variation. In addition, applicability of the presented method to a practical stochastic homogenization analysis of a periodic porous material is investigated. From the comparison between the numerical and experimental results, it is confirmed the perturbation method has a potential for the CV of the equivalent Young's modulus for considering both the microscopic size variation of holes and Young's modulus variation of the component material. In a next step of this study, a multiscale stochastic stress analysis of a porous material will be performed. In particular, the stochastic homogenization and multiscale stochastic stress analysis considering a non-uniform random variation [23] will be important for a practical problem. Also, the stochastic homogenization and multiscale stochastic stress analysis should be performed for a 3D porous structure. For this purpose, in order to investigate validity of the presented numerical method, 3D fabricating system, which enables to prepare many accurate 3D porous specimens stably at a lower cost, will be needed. Acknowledgment The first author is pleased to acknowledge support in part by Grants-in-Aid for Young Scientists (B) (No.23760079) from the Ministry of Education, Culture, Sports Science and Technology, and MEXT-supported program for the Strategic Research Foundation at Private Universities, 2012–2014. References [1] Ohtsubo Y, Yamamoto M, Tadokoro S, Takamori T. Fabrication of 3D Microstructures using micro stereolithography (1st report). Seimitsu Kougaku Kaishi 1998;64(8):1181–5. [2] Kami´nski M, Kleiber M. Perturbation based stochastic finite element method for homogenization of two-phase elastic composites. Comput Struct 2000;78:811–26.
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