Homogenization of fiber-reinforced composites with random properties using the weighted least squares response function approach (original) (raw)
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2009
The main issue this paper addresses is the derivation and implementation of a general homogenization method, including the simultaneous determination of sensitivity gradients and probabilistic moments of the effective elasticity tensor. This is possible with an application of the perturbation method based on Taylor expansion and with the effective modules method. The computational procedure is implemented using plane strain analysis carried out with the finite element method (program MCCEFF) and the symbolic computations system MAPLE. The sensitivity gradients and probabilistic moments are commonly determined on the basis of partial derivatives for the homogenized elasticity tensor, calculated using the response function method with respect to some composite parameters. They are subjected separately to a normalization procedure (in deterministic analysis) and the relevant algebraic combinations (for the stochastic case). This enriched homogenization procedure is tested on a periodic fiber-reinforced two component composite, where the material parameters are taken as design variables and then, the input random quantities. The results of computational analysis are compared against the results of the central finite difference approach in the case of sensitivity gradients determination as well as the direct Monte-Carlo simulation approach. This numerical methodology may be further applied not only in the context of the homogenization method, but also to extend various discrete computational techniques, such as Boundary/Finite element and finite difference together with various meshless methods.
International Journal for Numerical Methods in Engineering, 2011
The main aim of this paper is a development of the semi-analytical probabilistic version of the finite element method (FEM) related to the homogenization problem. This approach is based on the global version of the response function method and symbolic integral calculation of basic probabilistic moments of the homogenized tensor and is applied in conjunction with the effective modules method. It originates from the generalized stochastic perturbation-based FEM, where Taylor expansion with random parameters is not necessary now and is simply replaced with the integration of the response functions. The hybrid computational implementation of the system MAPLE with homogenization-oriented FEM code MCCEFF is invented to provide probabilistic analysis of the homogenized elasticity tensor for the periodic fiberreinforced composites. Although numerical illustration deals with a homogenization of a composite with material properties defined as Gaussian random variables, other composite parameters as well as other probabilistic distributions may be taken into account. The methodology is independent of the boundary value problem considered and may be useful for general numerical solutions using finite or boundary elements, finite differences or volumes as well as for meshless numerical strategies.
International Journal for Numerical Methods in Engineering, 2012
The purpose of this elaboration is to develop an efficient method for a determination of the probabilistic entropy loss in the homogenization process of the periodic composites with random material characteristics. A definition of this entropy convenient for the Gaussian continuous distribution is adopted and implemented into the computer algebra system MAPLE. Homogenization of the fiber-reinforced composite with randomly defined Young's moduli of the constituents is carried out in the FEM and homogenization-oriented code based on the four-noded plane strain elements. Probabilistic procedure has triple character and is alternatively based on the Monte Carlo simulation, on the generalized stochastic perturbation-based analysis, and on the recently developed semi-analytical determination of homogenized tensor using the response function method. Because of this triple usage of the probabilistic methods, it is possible to make a detailed comparison of all those techniques, especially in view of the entropy variation during the homogenization. This procedure may be linked with other homogenization techniques, also for various constitutive models and/or for the upper and lower bounds on the effective tensor components.
2010
The main issue that is addressed in this paper is the application of the probabilistic homogenization method to periodic composites with random geometrical parameters. It is possible: thanks to the perturbation method based on Taylor expansion technique combined with numerical determination of the response function relating the effective parameters with the random input parameter. The whole computational procedure is build on the plane strain analysis of the finite element method performing the homogenization procedure according to the effective modules method (program MCCEFF); the symbolic computations system MAPLE is employed for the stochastic perturbation analysis. Probabilistic moments are determined on the basis of partial derivatives for the homogenized elasticity tensor calculated using symbolically determined response function method with respect to the fiber radius. The numerical methodology worked out may be further successively applied not only in the context of homogenization method but also to extent of various discrete computational techniques such as boundary/finite element, finite differences or finite volume methods and the various meshless strategies, also for the structural sensitivity analysis.
Composite Structures, 2014
The main purpose of this work is computational simulation of the expectations, standard deviations, skewness and kurtosis of the homogenized tensor for some composites with metallic components. The Representative Volume Element (RVE) of this composite contains a single cylindrical fiber and their components are treated as statistically homogeneous and isotropic media uniquely defined by the Gaussian elastic modulus. Probabilistic approach is based on the generalized stochastic perturbation technique allowing for large random dispersions of the input random variables and is implemented using the polynomial response functions recovered using the Least Squares Method. Homogenization technique employed is dual and consists of (1) stress version of the effective modules method and (2) its displacements counterpart based on the deformation energies of the real and homogenized composites. The cell problem is solved for the first case by the plane strain homogenization-oriented code MCCEFF and, in the 3D case, using the system ABAQUS Ò (8-node linear brick finite elements C3D8), where the uniform deformations are imposed on specific outer surfaces of the composite cell; probabilistic part is carried out in the symbolic computations package MAPLE Ò. We compare probabilistic coefficients of the effective elasticity tensor computed in this way with the corresponding coefficients for their upper and lower bounds and this is done for the composite with small and larger contrast between Young moduli of the fiber and the matrix. The main conclusion coming from the performed numerical analysis is a very good agreement of the probabilistic moments resulting from 2 and 3D computer models; this conclusion is totally independent from the contrast between elastic moduli of both composite components.
Computer Methods in Applied Mechanics and Engineering, 2016
Quantifying uncertainty in the overall elastic properties of composite materials arising from randomness in the material properties and geometry of composites at microscopic level is crucial in the stochastic analysis of composites. In this paper, a stochastic multi-scale finite element method, which couples the multi-scale computational homogenization method with the second-order perturbation technique, is proposed to calculate the statistics of the overall elasticity properties of composite materials in terms of the mean value and standard deviation. The uncertainties associated with the material properties of the constituents are considered. Performance of the proposed method is evaluated by comparing mean values and coefficients of variation for components of the effective elastic tensor against corresponding values calculated using Monte Carlo simulation for three numerical examples. Results demonstrate that the proposed method has sufficient accuracy to capture the variability in effective elastic properties of the composite induced by randomness in the constituent material properties. c
Probabilistic Elastic Characteristics of the Periodic Fiber Composites
Science and Engineering of Composite Materials
The article presented is devoted to computational estimation of the probabilistic effective properties for random fiber-reinforced composites by the Monte-Carlo simulation (MCS) technique. The composite analyzed has randomly defined Young's moduli and Poisson's ratios being uncorrelated Gaussian variables while the heterogeneous medium microgeometry is defined deterministically. Using MCS method linked with the Finite Element procedure, the upper and lower bounds of the fiber composite effective properties as well as direct approximation of the effective characteristics is carried out. The probabilistic sensitivity of the effective properties is analyzed with respect to deterministically varying reinforcement volume as well as to random elastic characteristics of composite constituents. Thanks to such analysis it is observed that even small random fluctuations in all the elastic properties of the fiber and matrix cause significant increase of the second-order probabilistic moments of the effective characteristics; the matrix Poisson's ratio has been detected as the crucial parameter for the effective tensor probabilistic moments. component the coefficient of variation is even
Journal of Solid Mechanics and Materials Engineering, 2010
This paper describes stochastic homogenization analysis of a uni-directionally aligned short fiber reinforced composite material. In case of a short fiber reinforced composite material, length or orientation of the short fiber will be a random variable in addition to elastic properties of component materials, volume fraction or cross-sectional shape of fiber. Especially, a stochastic homogenization problem considering the length or fiber orientation variation is analyzed in this paper. For the purpose of this analysis, the first-order perturbation-based stochastic homogenization method is employed. Since the perturbation term with respect to the length or the orientation variation cannot be explicitly obtained in using the homogenization method, a finite differential technique is used for approximation of the perturbation term. From the numerical results, validity, effectiveness and problem of the perturbation-based analysis are discussed. Also, influence of the length variation on a stochastic characteristic of a homogenized elastic property of a short fiber reinforced composite material is discussed.
Stochastic finite element analysis of composite structures based on material microstructure
Composite Structures, 2015
The linking of microstructure uncertainty with the random variation of material properties at the macroscale is particularly needed in the framework of the stochastic finite element method (SFEM) where arbitrary assumptions are usually made regarding the probability distribution and correlation structure of the macroscopic mechanical properties. This linking can be accomplished in an efficient manner by exploiting the excellent synergy of the extended finite element method (XFEM) and Monte Carlo simulation (MCS) for the computation of the effective properties of random two-phase composites. The homogenization is based on Hill's energy condition and involves the generation of a large number of random realizations of the microstructure geometry based on a given volume fraction of the inclusions and other parameters (shape, number, spatial distribution and orientation). In this paper, the mean value, coefficient of variation and probability distribution of the effective elastic modulus and Poisson ratio are computed taking into account the material microstructure. The effective properties are used in the framework of SFEM to obtain the response of a composite structure and it is shown that the response variability can be significantly affected by the random microstructure.
On probabilistic sensitivity of homogenized composite materials
Probabilistic and stochastic sensitivity of composite materials is still the unrecognized area of computational mechanics, especially in the context of the homogenization theories application. That is why the sensitivity analysis implemented before in deterministic homogenization methods is now examined in terms of random elastic properties and interface defects for periodic fiber-reinforced composite structures. The Monte-Carlo simulation technique is engaged in computational statistical estimation of the sensitivity gradients of up to the fourth order probabilistic characteristics of the homogenized elasticity tensor components with respect to expectations and variances of some composite input random parameters. Computational strategy is implemented in the Finite Element Method (FEM) based homogenization oriented code MCCEFF adopted for deterministic sensitivity analysis before. Thanks to such computations one can verify whether and which effective composite characteristics are more sensitive to mean values of its parameters or to the corresponding standard deviations. Presented computational implementation makes it possible to verify the type of the probability density function for output homogenized tensor components and it can be extended to other composite structures.