Stochastic analysis of laminated composite plate considering stochastic homogenization problem

Abstract

This paper discusses a multiscale stochastic analysis of a laminated composite plate consisting of unidirectional fiber reinforced composite laminae. In particular, influence of a microscopic random variation of the elastic properties of component materials on mechanical properties of the laminated plate is investigated. Laminated composites are widely used in civil engineering, and therefore multiscale stochastic analysis of laminated composites should be performed for reliability evaluation of a composite civil structure. This study deals with the stochastic response of a laminated composite plate against the microscopic random variation in addition to a random variation of fiber orientation in each lamina, and stochastic properties of the mechanical responses of the laminated plate is investigated. Halpin-Tsai formula and the homogenization theory-based finite element analysis are employed for estimation of effective elastic properties of lamina, and the classical laminate theory is employed for analysis of a laminated plate. The Monte-Carlo simulation and the first-order second moment method with sensitivity analysis are employed for the stochastic analysis. From the numerical results, importance of the multiscale stochastic analysis for reliability evaluation of a laminated composite structure and applicability of the sensitivity-based approach are discussed.

Keywords

stochastic homogenization / multiscale stochastic analysis / microscopic random variation / laminated composite plate

Introduction

Advanced heterogeneous materials such as composite materials have been widely used as a structural material, because of its preferable characteristics such as higher relative rigidity or strength. On the other hand, owing to complexity of microstructure, evaluation of a mechanical property or reliability of a composite structure will be more difficult compared to a structure made of a homogeneous material such as steel. From this reason, several researches on evaluation of equivalent material properties of composites and multiscale stress analysis methodology have been carried out. The classical theory of mixture or Halpin-Tsai formula [ 1] will be selectable as a first approach for estimating an effective elastic property of a unidirectional fiber reinforced composite material. Also for more complex microstructures, the equivalent inclusion method [ 2] with the Eshelby tensor or the homogenization theory [ 3] becomes a popular method for analysis of composite material. In particular, the homogenization theory enables to investigate a microscopic stress field via a multiscale stress analysis framework, and it will be helpful for reliability evaluation of a composite structure.

Since this type of averaging theory will estimate a material property of a material having simply arranged microstructures, a further effort for analysis of a real structure will be generally needed. For example, a unidirectional fiber reinforced composite material is usually used as the laminated composite material, and the laminate theory [ 4] will be used for analysis of a laminated structure. For a detailed analysis, a three-dimensional solid model may be usable, but from the fact that a general laminated composite consists of many thin layers of lamina, it is sometimes modeled as a thin-plate structure.

In this case, the classical laminate theory is one of first choices of an analysis model. With using the above averaging theory and the laminate theory, a structural analysis of a laminated composite structure can be carried out.

This analysis framework has been widely used for design of a structure made of a laminated composite material, but several conditions such as a material property of composite materials or volume fraction is assumed to be deterministic. Since a microscopic / macroscopic condition in a composite material is generally randomly varied, influence of the random variation through the different scales should be investigated from the viewpoint of reliability assessment of a composite structure.

From this background, studies on a stochastic homogenization / multiscale stress analysis have been noticed in recent. For example, stochastic homogenization analysis with the Monte-Carlo simulation [ 5], the perturbation-based approach for an elastic problem [ 6, 7] and thermal / thermoelastic problem [ 8– 10] have been reported. Also, Xu [ 11, 12], Tootkaboni [ 13] or Ma [ 14] reported some approaches for the stochastic homogenization analysis.

For reliability evaluation of a composite structure, the stochastic homogenization problem will be important especially in the case that a priori random variation or a posteriori effect such as aging or mechanical and chemical influences in a manufacturing process should be taken into account. Further, this type of uncertainty propagation through the different scales should be considered for the validation and verification of the numerical simulation from a viewpoint for assessment on a computational mechanics based analysis.

In those previous reports, influence of the microscopic randomness on the equivalent stiffness of a composite material is mainly discussed. In addition, a few reports on multiscale stochastic stress analysis for investigation of the influence on microscopic and macroscopic stress fields [ 15, 16] or inverse stochastic homogenization analysis have been reported.

However, from the author’s knowledge, multiscale stochastic analysis for a more complicated structure such as a laminated composite structure has not been well discussed yet. In particular, multiscale stochastic analysis of a laminated plate considering a microscopic random variation will be important for a large system like a civil structure or airplane.

From these backgrounds, this paper aims to discuss the multiscale stochastic stress analysis of a laminated composite plate considering a microscopic randomness. In particular, the influence of the random variation in elastic properties of component materials is compared with that of the fiber orientation, and necessity of the multiscale analysis considering the microscopic random variation is discussed with the numerical results.

As the numerical methodology, Halpin-Tsai formula and the homogenization theory are employed as the averaging theory. To evaluate mechanical properties of a laminated plate, the classical laminate theory is employed as the first step of this study. The stochastic property is analyzed with the Monte-Carlo simulation, and applicability of the first order second moment method is also discussed.

Analysis of a laminated composite plate

Averaging theory of a unidirectional fiber reinforced composite lamina

In this case, a unidirectional fiber reinforced composite material which has square arranged fibers is considered as a target material of lamina. A schematic view of a laminated plate consisting of a set of unidirectional fiber reinforced lamina and its microstructure is illustrated in Fig. 1. For the purpose of efficient mechanical analysis of a laminated composite structure, some averaging methods for estimation of effective elastic properties of lamina and laminated plate were proposed.

Halpin-Tsai approach

The Halpin-Tsai formula is an improved version of the classical rule of mixture, and it is recognized that this approach gives good agreement with the experimental results of a unidirectional fiber reinforced composite material. The effective Young’s modulus of the longitudinal direction E_x and the transverse direction E_z are computed as follows.

(1) E_x=\left(\mathrm{1}-V_f\right)E_m+V_f{E}_f,E_z={\displaystyle \frac{E_m\left(\mathrm{1}+\xi {\eta }_E{V}_f\right)}{\left(\mathrm{1}-{\eta }_E{V}_f\right)}},

(2) {\eta }_E={\displaystyle \frac{\left(E_f/E_m\right)-\mathrm{1}}{\left(E_f/E_m\right)+\xi }},

where E_m and E_f are Young’s modulus of resin and fiber, \xi is an optional parameter. In this paper, it is assumed that \xi = 0.9. Also, the other elastic coefficients for each direction are

(3) G_{zx}=G_{xy}={\displaystyle \frac{G_m\left(\mathrm{1}+\xi {\eta }_G{V}_f\right)}{\left(\mathrm{1}-{\eta }_G{V}_f\right)}},\hspace{1em}{G}_{yz}={\displaystyle \frac{E_z}{\mathrm{2}\left(\mathrm{1}+{\nu }_{yz}\right)}},

(4) {\eta }_G={\displaystyle \frac{\left(G_f/G_m\right)-\mathrm{1}}{\left(G_f/G_m\right)+\xi }},

{\nu }_{xy}=\left(\mathrm{1}-V_f\right){\nu }_m+V_f{\nu }_f,{\nu }_{yz}=\mathrm{1}-{\nu }_{zx}-{\displaystyle \frac{E_y}{\mathrm{3}K}},\hspace{1em}

(5) {\nu }_{zx}=\left(V_f{\nu }_f+\left(\mathrm{1}-V_f\right){\nu }_m\right)\left({\displaystyle \frac{E_y}{E_x}}\right).

The homogenization theory

An effective equivalent elastic property of unidirectional fiber reinforced lamina can be also estimated with the homogenization theory. With the homogenization theory [ 3] the homogenized elastic tensor can be computed as follows. For a square arranged fiber, a finite element model illustrated in Fig. 1 is used as a unit cell.

(6) E^H={\displaystyle \frac{\mathrm{1}}{\left|Y\right|}}{\displaystyle {\int }_{Y}E\left(I-{\displaystyle \frac{\partial \chi }{\partial y}}\right)}\mathrm{d}Y,

where E is an elastic tensor of the microstructure, \left|Y\right| is the volume of a unit cell, and I is a unit tensor. \chi is a characteristic displacement, which is the solution of the following equation.

(7) {\displaystyle \frac{\partial }{\partial y}}E{\displaystyle \frac{\partial \chi }{\partial y}}={\displaystyle \frac{\partial }{\partial y}}E.

In the case of the orthotropic material, the equivalent elastic constants of the composite material can be computed from the homogenized elastic tensor computed from Eq. (6) as follows.

(8) \begin{array}{l}{E}_x^H={\displaystyle \frac{\mathrm{1}}{S_{\mathrm{11}}^H}},\hspace{1em}{E}_y^H={\displaystyle \frac{\mathrm{1}}{S_{\mathrm{22}}^H}},\hspace{1em}{E}_z^H={\displaystyle \frac{\mathrm{1}}{S_{\mathrm{33}}^H}}\hfill \\ {\nu }_{yz}^H=-E_y^H{S}_{\mathrm{23}}^H,\hspace{1em}{\nu }_{zx}^H=-E_z^H{S}_{\mathrm{31}}^H,\hspace{1em}{\nu }_{xy}^H=-E_x^H{S}_{\mathrm{12}}^H\hfill \\ G_{yz}^H={\displaystyle \frac{\mathrm{1}}{S_{\mathrm{44}}^H}},\hspace{1em}{G}_{zx}^H={\displaystyle \frac{\mathrm{1}}{S_{\mathrm{55}}^H}},\hspace{1em}{G}_{xy}^H={\displaystyle \frac{\mathrm{1}}{S_{\mathrm{66}}^H}}\hfill \end{array}\},

where S_{ij}^H is the components of the compliance matrix computed from the homogenized elastic tensor.

Mechanical analysis of a laminated plate

Laminated composites consist of a number of laminae, and therefore the stress-strain relationship of lamina is first considered and then the relationship for a laminated plate can be obtained. In the case of an orthotropic material as a unidirectional fiber reinforced composite material, by using the laminate theory, the resultant stress-strain relationship of the laminated plate can be expressed by the following equation.

(9) \left\{\begin{array}{c}{N}_x\hfill \\ N_y\hfill \\ N_{xy}\hfill \\ M_x\hfill \\ M_y\hfill \\ M_{xy}\hfill \end{array}\right\}=\left[\begin{array}{cccccc}\overline{A_{11}}\hfill & \overline{A_{12}}\hfill & 2\overline{A_{16}}\hfill & \overline{B_{11}}\hfill & \overline{B_{12}}\hfill & 2\overline{B_{16}}\hfill \\ \overline{A_{12}}\hfill & \overline{A_{22}}\hfill & 2\overline{A_{26}}\hfill & \overline{B_{12}}\hfill & \overline{B_{22}}\hfill & 2\overline{B_{26}}\hfill \\ \overline{A_{16}}\hfill & \overline{A_{26}}\hfill & 2\overline{A_{66}}\hfill & \overline{B_{16}}\hfill & \overline{B_{26}}\hfill & 2\overline{B_{66}}\hfill \\ \overline{B_{11}}\hfill & \overline{B_{12}}\hfill & 2\overline{B_{16}}\hfill & \overline{D_{11}}\hfill & \overline{D_{12}}\hfill & 2\overline{D_{16}}\hfill \\ \overline{B_{12}}\hfill & \overline{B_{22}}\hfill & 2\overline{B_{26}}\hfill & \overline{D_{12}}\hfill & \overline{D_{22}}\hfill & 2\overline{D_{26}}\hfill \\ \overline{B_{16}}\hfill & \overline{B_{26}}\hfill & 2\overline{B_{66}}\hfill & \overline{D_{16}}\hfill & \overline{D_{26}}\hfill & 2\overline{D_{66}}\hfill \end{array}\right]\cdot \left\{\begin{array}{c}{ϵ}_{x0}\hfill \\ {ϵ}_{y0}\hfill \\ {ϵ}_{xy0}\hfill \\ {\kappa }_x\hfill \\ {\kappa }_y\hfill \\ {\kappa }_{xy}\hfill \end{array}\right\},

where

(10) \begin{array}{l}\overline{A_{ij}}={\displaystyle \sum _{k=1}^N{\left({\overline{Q}}_{ij}\right)}_k}\left[h_k-h_{k-1}\right]\hfill \\ \overline{B_{ij}}={\displaystyle \frac{1}{2}}{\displaystyle \sum _{k=1}^N{\left({\overline{Q}}_{ij}\right)}_k}\left[h_k^2-h_{k-1}^2\right]\hfill \\ \overline{D_{ij}}={\displaystyle \frac{1}{3}}{\displaystyle \sum _{k=1}^N{\left({\overline{Q}}_{ij}\right)}_k}\left[h_k^3-h_{k-1}^3\right]\hfill \end{array}\}\left[i,j=1,2,6\right],

and N is the resultant stress, M is the resultant bending moment, ϵ₀ is the in-plane strain on the neutral surface, κ is the curvature, h_k is the distance from the neutral surface to the upper surface of kth lamina. {\overline{Q}}_{ij} is the translated stiffness matrix of each lamina according to the fiber orientation. Details of the constitutive equation of lamina are provided in the appendix.

Multiscale stochastic analysis of a laminated composite plate

Monte-Carlo simulation

For estimation of the probabilistic characteristics of a homogenized material property caused by a microscopic random variation, the Monte-Carlo simulation can be usable as a simple approach.

For example, it is assumed that an observed value of a microscopic quantity X^{*} can be expressed as a linear function of a random variable as

(11) X^{*}=X^{\mathrm{0}}\left(\mathrm{1}+\alpha \right),

where X^{\mathrm{0}} is the expected value of the quantity, \alpha is a normalized Gaussian random variable. In this case, the observed value of the homogenized elastic tensor becomes a function of the normalized random variable \alpha , and also the realization of the stochastic response of the in-plane strain or curvature is a function of \alpha . The expected value \mathrm{Exp}\left[\right] and variance \mathrm{Var}\left[\right] of the strain can be approximately computed as, for example,

(12) \mathrm{Exp}\left[{\mathit{ϵ}}^{*}\right]\approx {\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum {\mathit{ϵ}}^{*}\left(\alpha \right)},

(13) \mathrm{Var}\left[{\mathit{ϵ}}^{*}\right]\approx {\displaystyle \frac{\mathrm{1}}{n-\mathrm{1}}}{\displaystyle \sum {\left({\mathit{ϵ}}^{*}\left(\alpha \right)-\mathrm{Exp}\left[{\mathit{ϵ}}^{*}\right]\right)}^{\mathrm{2}}}.

From these quantities, the coefficient of variance \mathrm{CV}\left[\right] can be also computed as

(14) \mathrm{CV}\left[{\mathit{ϵ}}^{*}\right]=\sqrt{\mathrm{Var}\left[{\mathit{ϵ}}^{*}\right]}/\mathrm{Exp}\left[{\mathit{ϵ}}^{*}\right].

In case of assuming the Gaussian random variable, the random number is generated with the following randomization formula.

(15) \alpha =\sqrt{-\mathrm{2}{\sigma }^{\mathrm{2}}\log U_{\mathrm{1}}}\times \sin \mathrm{2}\mathrm{\pi }{U}_{\mathrm{2}},

where and \mathrm{0}\le U_{\mathrm{2}}\le \mathrm{1} are observed values of a uniform random variable.

Sensitivity of the mechanical property of laminated plate for a microscopic random variable

Since the Monte-Carlo simulation is very expensive, some efficient methods have been proposed. In this paper, an approximation-based stochastic homogenization method is adopted.

When an approximation based stochastic analysis method is employed, some order sensitivities of a mechanical response of the laminated plate for a microscopic random variable will be needed. If a single scale stochastic analysis problem is considered, for example, the first order sensitivity of the in-plane strain or curvature of the plate for an elastic property can be computed as follows.

At first, a realization of an elastic constant can be expressed as a function of a normalized random variable as Eq. (11). In this case, the first order derivative of the in plane strain for the random equivalent elastic constants can be computed as

(16) {\displaystyle \frac{\partial \left\{{ϵ}^{*}\right\}}{\partial E_x^{*}}}={\displaystyle \frac{\partial {\left[A^{*}\right]}^{-\mathrm{1}}}{\partial E_x^{*}}}\left\{N\right\}.

In addition, when a heterogeneous material is assumed, the realization of the equivalent elastic constant is a function of the normalized random variable. In this case, the sensitivity for a microscopic random variable can be expressed by a hierarchical form as

(17) {\displaystyle \frac{\partial \left\{{ϵ}^{*}\right\}}{\partial \alpha }}={\displaystyle \frac{\partial {\left[A^{*}\right]}^{-\mathrm{1}}}{\partial E_x^{*}}}{\displaystyle \frac{\partial E_x^{*}}{\partial \alpha }}\left\{N\right\}.

The term {\displaystyle \frac{\partial E_x^{*}}{\partial \alpha }} can be computed with a stochastic homogenization method, and therefore the stochastic sensitivity of a mechanical response of a laminated composite plate for a microscopic random variable can be evaluated via a stochastic homogenization analysis. If it is difficult to express the sensitivity term with an analytical form, the finite difference approach can be usable.

By using the first-order second moment method (FOSM), for example, the variance of the strain against the microscopic random variation can be computed as follows.

(18) \mathrm{Var}\left[{ϵ}^{*}\right]\approx {\displaystyle \sum _i{\displaystyle \sum _j{\displaystyle \frac{\partial \left\{{ϵ}^{*}\right\}}{\partial {\alpha }_i}}{\displaystyle \frac{\partial \left\{{ϵ}^{*}\right\}}{\partial {\alpha }_j}}}}\mathrm{cov}\left({\alpha }_i,{\alpha }_j\right),

where \mathrm{cov} is the covariance.

Problem settings

In this paper, a composite laminated plate illustrated in Fig. 2 is considered as the target structure. One edge is fixed and the other edge is free. For this structure, a stochastic mechanical response such as in-plane strain, curvature or deflection against a random variation of microscopic quantities or fiber orientation is investigated. Volume fraction of fiber in each lamina is 0.3, and elastic properties of fiber and matrix are employed correspond to E-glass and Epoxy resin. The expected values of elastic properties used in this analysis are listed in Table 1. In case of bending, M= 100Nm is applied.

As the microscopic random variables, Young’s modulus of fiber and resin ( E_f , E_m ), and Poisson’s ratio of fiber and resin ( {\nu }_f , {\nu }_m ) are taken into account. Influence of these microscopic random variables on the mechanical properties of the laminated plate is analyzed, and it is compared with that of random variation of fiber orientation in each lamina. In this paper, CV (coefficient of variance) of each microscopic random variable is assumed to be 0.05. The number of trials for the Monte-Carlo simulation is 1,000.

Numerical results

Multiscale stochastic analysis of lamina

At first, a multiscale stochastic analysis of a lamina is performed. In this case, the Halpin-Tsai formula and the homogenization theory-based finite element analysis are employed, and the influences of the microscopic random variation on the homogenized elastic properties of lamina obtained from those methods are compared.

Figures 3 and 4 shows the expectation and CV of the equivalent elastic constants of lamina for each microscopic random variation. Figures 3(a) and 4(a) are the results with Halpin-Tsai formula, and Figs. 3(b) and 4(b) are the results with the homogenization theory.

It is recognized that the CV of the equivalent elastic properties of the unidirectional fiber reinforced lamina can be similarly estimated with the Halpin-Tsai and the Homogenization theory, except to the influence of {\nu }_m variation and CV estimation of {\nu }_{xy} for Young’s modulus variation. Since the equivalent Young’s modulus and shearing modulus will be dominant to uniaxial in-plane strain and bending deformation, which are considered as loading conditions in this paper, the Halpin-Tsai formula can be usable for the analysis. Of course, for more complex condition or microstructure, use of the homogenization-theory based approach is recommended.

Next, influence of the microscopic random variation on the components of the resultant stress-strain matrix of lamina is investigated. Figure 4(a) shows the CV of the A_{ij} for the microscopic random variation of elastic properties of component materials, and Fig. 4(b) shows that of D_{ij} . From this figure, it is recognized that each microscopic random variation has different influence on the mechanical properties of lamina, for example E_f has a large influence on A_{\mathrm{11}} while E_m has large influence on others. Also, it is confirmed that CV of D_{ij} is same as that of A_{ij} . From these results, it can be confirmed from the result of Fig. 5 that the stochastic response of lamina can be evaluated with the result of the previously reported stochastic homogenization analysis.

Influence of stacking sequence on stochastic response

In this section, influence of the microscopic random variation on mechanical properties of a laminated composite plate is discussed. The mechanical behavior of a laminated plate consisting of anisotropic materials such as a unidirectional fiber reinforced composite material strongly depends on the stacking sequence and its fiber orientation. At first, a symmetric orthogonal laminated plate is assumed as a typical laminated plate, and CVs of the in-plane strain of the plate for the microscopic random variation are computed. In this case, the expected value of the deflection for in-plane loading is 0, and therefore the result of CV for the deflection is omitted.

Figure 6 shows the stochastic characteristics of the CV of the in-plane strain along the loading direction for each microscopic random variation in case of the stacking sequence [0/90/90/0] and [90/0/0/90]. Figure (a) shows the estimated expectation and Fig. 6(b) shows the estimated CVs. From this figure, the random variation of E_f has the largest influence on the in-plane strain, and it is about 2.5 times of E_m variation. It is confirmed that the CVs are almost same as each other in case of the above stacking sequence and in-plane loading.

Next, the stochastic characteristics of the deflection for bending moment are analyzed. Figure 7 shows the CVs of the deflection for the microscopic random variation. It can be recognized that E_f has a larger influence on the deflection in case of the laminate [0/90/90/0], but a random variation of E_m will be more critical than E_f in case of [90/0/0/90]. This result shows that the influence of the microscopic random variation depends on not only the varied material property, but also the stacking sequence. From this result, complexity of the multiscale stochastic analysis of a laminated composite structure is recognized, and it implies importance of the presented analysis.

As an example of an asymmetric laminated plate, two types of the laminated plates [0/0/90/90] and [45/45/90/90] are considered. The estimated results for the microscopic random variations are illustrated in Figs. 8 and 9. From these figures, it is recognized that both CVs of the in-plane strain for tensile load and the deflection for the bending moment depend on the type of microscopic random variation, and their influences also depend on the stacking sequence and loading condition. In particular, CV of the deflection for E_m variation reaches about 0.045. On the other hand, the Poisson’s ratio variation has less influence on the mechanical responses of the laminated plate.

This consideration can be obtained from the presented stochastic analysis considering a multiscale stochastic problem like the stochastic homogenization analysis, and this analysis should be performed for each case of reliability-based design of a laminated composite structure.

Influence of the Fiber Orientation in Lamina on the Stochastic Responses

In the previous section, it is shown that influence of the microscopic random variations on the mechanical responses of the laminated plate depends on the stacking sequence and it is different from that of the lamina. Further, importance of the multiscale stochastic analysis of a laminated composite plate for a microscopic random variation is discussed. In this section, the stochastic analysis is performed for laminated plates having several stacking sequences.

As an example, stacking sequence [0/θ]s,[90/θ]s, [0/θ/-θ/0], and [90/θ/-θ/0]s are considered. Figures 10-13 shows the result of the CV of the in-plane strain and deflection for each stacking sequence under several values of θ. Since the CV for {\nu }_f variation is very small, CV for the Vf (volume fraction of fiber) variation is illustrated instead of it.

From Fig. 10, CV of the in-plane strain and deflection slightly change according to the value of θ, and the influences of the microscopic random variation and the fiber orientation for the in-plane strain is different from that of the deflection. The maximum CV for E_m variation is observed when \theta =\mathrm{60}\mathrm{deg}. , but the maximum CV for \theta variation is observed at \theta =\mathrm{15}\mathrm{deg}. in the result of the in-plane strain. On the other hand, influence of the fiber orientation \theta in case of the deflection is less than that of the in-plane strain. This result shows that influence of not only the microscopic random variation but also the fiber orientation of the laminate depends on the macroscopic loading condition.

Next, from Fig. 11, it is recognized that the influence of the variation in case of [90/θ]s is quite different from the case of [0/θ]s. When θ is less than 20 deg., CV of the in-plane strain for E_f variation is larger than that of E_m variation. For the deflection, E_m is still most sensitive even if the stacking sequence is [90/θ]s, and it is confirmed that the most sensitive microscopic random variable under each loading case may be different from each stacking sequence.

Figure 12 is the result for antisymmetrical laminate, the surface layers are 0 degree, and the center layers are θ/-θ. In this case, dependency on the fiber orientation is very similar to the case of [0/θ]s. When a laminated plate has same dominant layers to the mechanical response, this tendency seems to be similar.

As the final example of the several stacking sequences, the result of an asymmetric laminated plate is shown in Fig. 13. In this case, the CVs of not only the in-plane strain but also the deflection have strong dependency on the fiber orientation. In particular, in case of the deflection for the bending load, E_f variation is the most sensitive for the deflection when \theta is less than 40 degree, but E_m is the most sensitive in the other case. From these results, it can be recognized that the stochastic influence depends on the type of microscopic random variation, stacking sequence and loading condition, and it will become complex when an asymmetrical laminated composite is assumed.

Comparison between the influences of randomness in fiber orientation and microscopic material properties

When a laminated composite material is fabricated from a set of prepreg unidirectional fiber reinforced laminae, a certain degree of misalignment of fiber orientation in each lamina is sometimes observed. From this reason, in a conventional design process for a laminated composite material, small random variation of the fiber orientation is generally taken into account.

Similar to this fact, influence of a microscopic random variation should be also considered if it is not negligible.

In this section, in order to discuss importance of the stochastic homogenization-based analysis of a laminated composite material, the coefficient of variance of a microscopic elastic property, which causes a same degree random variation of the mechanical responses against an acceptable random variation of fiber orientation, is investigated.

Figure 14 shows the CVs of the microscopic random variables which yields the same degree random variation of the in-plane stress and deflection for the random variation of fiber orientation. In this case, the acceptable random variation of the fiber orientation \mathrm{\Delta }\theta =\pm \mathrm{2} is assumed. V_f in Fig. 14 shows the random variation of the volume fraction of fiber.

It is noted that the CV indicated in Fig. 14 shows the required CV of the microscopic random variable for causing the same CV of the mechanical responses of the laminated plate against the allowed fiber orientation variation. Namely, the smaller CV shows the larger influence of the stochastic response of the plate.

From Fig. 14, it is recognized that a small microscopic random variation yields a same degree of random variation of the mechanical response of the laminated composite plate for fiber orientation variation. In particular, random variation of Young’s modulus of fiber and resin has a large influence on the stochastic response of the laminated plate, and 0.003 CV of E_f or 0.002 CV of E_m causes the same degree of CV of the in-plane strain or deflection of the plate. This result suggests that a microscopic randomness has a large influence on the stochastic mechanical response of the laminated composite plate, even though CVs of the in-plane strain and deflection are not so amplified comparing with the assumed CVs of the inputs in this paper. It shows importance of the presented stochastic analysis considering the stochastic homogenization analysis for reliability evaluation of a laminated composite structure.

Applicability of FOSM

Since the Monte-Carlo simulation involves higher computational cost, and also includes a bias, a more stable and efficient stochastic analysis method is encouraged to be developed. In this paper, applicability of the first order second moment method (FOSM) with the first order sensitivity analysis for the multiscale stochastic analysis of the laminated plate is investigated.

As an example, the FOSM is applied to the multiscale stochastic analysis of an orthogonal symmetric laminated plate [0/90/90/0], and the relative error between the estimated CV with the FOSM and the Monte-Carlo simulation is investigated.

Figure 15 shows the relative estimation error for each microscopic random variation. CVs of the in-plane strain for the tensile load and the deflection for the bending load are estimated. From this figure, it is recognized that FOSM gives good estimations of CVs, the estimation errors are less than about 5% except to the case of V_f variation. Though the estimated CV for V_f variation includes about 20% error, the absolute value of the CV is very small, and the absolute difference is 0.00029. From these results, it can be concluded that FOSM can be applicable for the presented multiscale analysis of the symmetric orthogonal laminated plate, and it shows potential of the approximation-based stochastic analysis method to the stochastic analysis problem.

Conclusions

This paper discusses multiscale stochastic analysis of a laminate composite plate consisting of unidirectional fiber reinforced composite laminae against a microscopic random variation of elastic properties of component materials.

Halpin-Tsai formula and the homogenization-theory based finite element method are used for estimation of effective elastic properties of lamina. The classical laminate theory is employed for evaluation of the mechanical properties of a laminated composite plate, and the Monte-Carlo simulation is employed for the stochastic analysis.

The influence of the microscopic random variations on the mechanical response as the in-plane strain for the tensile load and deflection for the bending moment is investigated assuming several types of the stacking sequence, and propagation of the stochastic characteristics and influence of the stacking sequence are discussed.

In addition, the CVs of the mechanical responses caused from the microscopic random variations are compared with that from misorientation of the fiber angle in the laminated plate. As an example, a random misorientation \mathrm{\Delta }\theta =\pm \mathrm{2} is assumed as an allowable random variation of the fiber orientation, and the CVs of the microscopic random variables, which cause the same degree CV of the mechanical response for the fiber orientation variation, is investigated. From the results, the influence of the random variation of the Young’s modulus of component materials is negligible, and importance of the presented multiscale stochastic analysis considering a microscopic random variation of the material property is confirmed.

From the numerical results, it is recognized that influence of the microscopic random variation on the mechanical responses of the laminated plate depends on the type of microscopic random variation, stacking sequence, loading condition and so on. This influence is complex and not negligible, and the multiscale stochastic analysis is very important for the reliability assessment of a laminated composite structure. In the next step of this study, development of an efficient analysis method will be needed. Since it is reported that the same degree of the CVs of the microscopic quantities cause high amplification of the microscopic failure probability in the case of a particle-reinforced composite material [ 16], such analysis should be performed for a laminated composite structure.

In this paper, the Gaussian microscopic random variable is considered. For a more general purpose, an efficient method for non-Gaussian variable should be discussed. Also, non-uniform random variation [ 17] should be considered as a more general problem.

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