Elsevier

Computational Materials Science

Volume 62, September 2012, Pages 35-46
Computational Materials Science

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

https://doi.org/10.1016/j.commatsci.2012.05.008Get rights and content

Abstract

This paper discusses a microscopic failure probability analysis of a microstructure in a composite material considering a microscopic randomness of both elastic properties and strength of component materials. Since a microscopic random variation of an elastic property of component materials will have an influence on a microscopic stress field, the influence of a microscopic random variation on a microscopic failure probability of a composite material should be investigated. For the analysis, the Monte-Carlo simulation, the perturbation-based stochastic homogenization method and a stochastic multiscale stress analysis method are employed. A random variation of the microscopic stresses and microscopic failure probability are estimated with the stochastic multiscale stress analysis methodology via the stochastic homogenization analysis.

With the numerical examples, the influence of the microscopic random variation on the microscopic failure probability thorough the multiscale stress analysis is discussed. Also, accuracy of the perturbation-based approach is investigated with comparison of the Monte-Carlo simulation.

Highlights

► We analyze failure probability of composites considering microscopic randomness. ► Both random variations of elastic properties and strength are considered. ► Accuracy of the proposed method is investigated compared with the MC simulation. ► Microscopic randomness of an elastic property causes increase of failure probability. ► The influence depends on macro and microscopic conditions and is complex.

Introduction

A microscopic random variation observed in a heterogeneous material such as a composite material will have an influence on a homogenized property of the material. Propagation of the probabilistic characteristics through the different scales will sometimes cause a serious random response of a composite structure. Therefore, it should be taken into account in a design process of a composite structure.

A problem for evaluating the influence of a microscopic random variation on a homogenized material property is called as “the stochastic homogenization problem”, and it has been noticed in both fields of computational mechanics and mechanics of materials. Kaminski and Kleiber [1] or Sakata et al. [2] discussed the stochastic homogenization problem with the numerical results of the Monte-Carlo simulation, and some computational methods have been proposed for the purpose of accurate estimation with a lower computational cost.

For instance, Kaminski and Kleiber [3] reported a basic formulation for the perturbation-based stochastic homogenization method. Sakata et al. reported the perturbation-based three-dimensional stochastic homogenization analysis for estimation of a random variation of the homogenized equivalent elastic constants of an orthotropic material [4]. Kaminski or Sakata et al. also discussed the accuracy of the perturbation-based approach with comparison of the Monte-Carlo simulation, applicability and effectiveness of two [5] or higher order perturbation based approach [6] were discussed.

Also, Xu et al. reported some computational methods for the stochastic homogenization method [7], [8], or Lombardo et al. reported a stochastic modeling of a chaotic masonry [9]. In recent, Tootkaboni and Graham-Brady proposed the stochastic homogenization method based on the spectral stochastic finite element method [10].

As one of the other approaches, a stochastic homogenization method with function approximation has been proposed. Sakata et al. proposed the Kriging-approximation-based approach [11] or Kaminski proposed a response function-based approach [12]. These methods will give a more accurate estimation especially in case that a nonlinear response between a microscopic random variable and a macroscopic response is observed, but an expansion-based approach like the perturbation-based technique will be still effective for a large number of random variables. From this viewpoint, some additional reports on the perturbation-based approach with the equivalent inclusion method have been published [13], [14]. For this background, a further study on a stochastic homogenization method will be continued.

As for the reliability evaluation of a composite structure, the multiscale stress analysis and the failure probability analysis are also very important. Actually, the failure probability analysis considering random variations of both strength of materials and observed stresses has been performed for design of a composite structure [15].

In general, the previous reports have only considered the random variations of a macroscopic quantity which takes an influence on a stress field, for example, shape or size of a structure, homogenized properties of a material, an loading or boundary conditions. However, an influence of the microscopic random variation on the microscopic stress field should be also investigated, since a heterogeneous material will be fractured resulted from a microscopic failure caused by a non-uniform microscopic stress field, and a microscopic random variation will have a certain influence on both of a homogenized material property and a microscopic stress field.

From this viewpoint, Sakata et al. reported the results of the Monte-Carlo simulation for the multiscale stochastic stress analysis of a composite material [16], [17]. The authors also discussed accuracy of the first-order perturbation-based multiscale stochastic stress analysis for a particle reinforced composite material [18]. For a multiscale stochastic stress analysis problem, Xu et al. reported a Green function-based approach for stress analysis considering a microscopic random fluctuation [19]. Kaminski discussed a failure probability analysis of a particle reinforced composite material thorough a single scale stochastic finite element analysis [20]. For the purpose of a practical application of the multiscale stochastic stress analysis for an unknown microscopic randomness, a concept of the inverse stochastic homogenization analysis for the multiscale stochastic stress analysis was proposed [21].

From the reported results, it can be recognized that a microscopic random variation in a composite material has a complex and significant influence on both the homogenized material property and the multiscale stress field. However, its influence on the failure probability of a composite material considering the multiscale stochastic stress analysis problem has not been discussed yet. Investigation of applicability of the multiscale stochastic stress analysis method to the multiscale failure probability analysis considering a microscopic uncertainty will be quite important for reliability evaluation of a composite structure.

Therefore, in this paper, a microscopic failure probability analysis with considering a random variation of an elastic property of a component material is performed.

The aims of this paper are to investigate the influence of the microscopic random variation on the failure probability of a composite material and the accuracy of the perturbation-based multiscale stress analysis method for the multiscale failure probability analysis, not a proposal of the perturbation-based stochastic homogenization or stress analysis methodology itself. For the accuracy investigation, in particular, this paper focuses on investigation of applicability and accuracy of the first-order perturbation-based approach, since a higher order perturbation-based stochastic homogenization method does not always improve accuracy of the estimation. In addition, in reliability evaluation with the failure probability analysis, a first order approximation is sometimes employed [15], and therefore the accuracy investigation of the first order-perturbation based approach for the microscopic failure probability analysis with the multiscale stochastic stress analysis considering a microscopic random variation must be carried out.

At first, the outline of the computational methodology for the multiscale stochastic stress analysis and failure probability analysis is described. The perturbation-based multiscale stochastic stress analysis has been reported, but it is limited to only a particle reinforced composite material [18], and therefore the perturbation-based multiscale stochastic stress analysis method is also extended to a macroscopically orthotropic material such as a unidirectional fiber reinforced composite in this paper.

Next, the influence of a microscopic random variation on the failure probability of a microstructure is analyzed with the Monte-Carlo simulation. The accuracy of the perturbation based method for the analysis is also investigated. Finally, a detailed numerical analysis on the microscopic failure probability considering a microscopic random variation of an elastic property of a component material is performed.

Section snippets

Outline of the problem

For reliability evaluation of a structure, a safety factor will be used empirically if random variations of both an observed maximum stress and strength of a material are unknown. However, in some cases, one or both of them will include some degree of a random variation and it can be identified. In this case, a failure probability can be adopted as one of criteria for evaluating reliability of a structure. Fig. 1 shows a typical relationship between randomly distributed maximum stress and

Problem settings

In this paper, a uniaxially loaded plate of a unidirectional glass fiber reinforced composite material is assumed. A schematic view of a finite element model of a unit cell for the composite structure is illustrated in Fig. 2. Similar to the previous reports [3], [4], a square fiber arrangement is assumed. The properties of the fiber and the matrix are employed correspond to E-glass and Epoxy resin. The expected values of the elastic properties and strength of the component materials are listed

The Monte-Carlo simulation for a single random variation of a microscopic elastic property

In this section, the influence of the microscopic random variation in the elastic properties of component materials on the failure probability of the composite material is investigated. Since the detailed multiscale stochastic analysis of a unidirectional fiber reinforced composite material has not been reported yet, the influence of the microscopic random variation on the microscopic stress field is investigated at first. In order to evaluate the influence with fewer computational assumptions,

Conclusion

In this paper, the microscopic failure probability analysis considering both random variations of the microscopic elastic properties and the strength of the component materials is performed. In particular, influence of the random variation on the microscopic failure probability of a unidirectional fiber reinforced composite material is investigated. The microscopic failure probability is estimated with the Monte-Carlo simulation or the first order perturbation-based stochastic homogenization

Acknowledgement

The first author is pleased to acknowledge support in part by Grants-in-Aid for Young Scientists (B) (No. 23760097) from the Ministry of Education, Culture, Sports Science and Technology.

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