Components Of Effective Tensor Research Articles

Homogenization of particulate and fibrous composites with some non-Gaussian material uncertainties

A fundamental problem investigated in this paper is computational homogenization of two-component particulate and fibrous composites with material characteristics treated as symmetric but non-Gaussian random processes. Homogenization method is based on equity of deformation energies for the real multi-component and the homogenized Representative Volume Elements (RVE) of the given composite structure. The iterative generalized stochastic perturbation technique is used to provide the additional Taylor expansions about expected values and to finally derive analytical formulas for the basic probabilistic characteristics of the effective tensor components. These expansions use polynomial response functions relating this tensor components with real material characteristics determined through the series of the Finite Element Method homogenization tests and the Least Squares Method. Applicability of this stochastic strategy and its Stochastic Finite Element Method implementation are verified for two well-known probability distributions, i.e. the uniform and the triangular one, to model uncertainty in Young modulus of some composite components. Validation of this approach is carried out using two alternative methods – statistical one known as the Monte-Carlo simulation as well as the semi-analytical one using the same polynomial responses as the perturbation approach, but finally processed using classical analytical probability integrals. This study generally confirms applicability of the proposed probabilistic method to homogenize composite structures with linear elastic and isotropic components exhibiting uniform and triangular uncertainties in their elastic properties.

Modeling of Phases Adhesion in Composite Materials Based on Spring Finite Element with Zero Length

A new numerical method for homogenization of elastic properties of dispersedly-reinforced composites was presented. The method takes into account special model of adhesive contact. Homogenization of properties was performed by averaging the solutions of boundary value problems on representative volume cell (RVC) using the finite element method (FEM). A new approach of calculation of components of effective tensor of elastic moduli was proposed. A heterogeneous finite element model with elements of two types was built: three-dimensional tetrahedron elements for every phases and spring element with zero-length for adhesion layer with zero-thickness. The results of homogenization of elastic properties of dispersedly-reinforced composites with variable stiffness of the adhesive layer between phases were obtained and analyzed. The homogenization results were compared with the available experimental data.

Dual probabilistic homogenization of the rubber-based composite with random carbon black particle reinforcement

This study concerns numerical determination of the basic statistics of the effective elasticity tensor for the rubber reinforced with the carbon black particles. This goal is achieved by an application of the iterative and generalized stochastic perturbation technique implemented as the Stochastic Finite Element Method and applied to the homogenization problem of such a composite. A fundamental difference of this approach to the traditional Taylor expansion is in development of the basic equations for higher order probabilistic characteristics, where traditional linearization procedure (expectation is approximated by the zeroth order term) has been replaced by sequential (iterative) symbolic calculation of these characteristics. The radius of the carbon black particle has been chosen as the input Gaussian random parameter and it affects both FEM-based and also analytical method of the effective tensor components determination. Sensitivity analysis in addition to this radius together with the FEM computational error for the homogenization problem are carried out here prior to the principal stochastic analysis. We contrast the iterative SFEM with two other probabilistic numerical methods, namely the classical Monte-Carlo scheme and also semi-analytical probabilistic FEM strategy. Both stochastic perturbation and semi-analytical method are related to the same polynomial response functions of the input random particle radius, but the first employs Taylor series expansion while the second – symbolic integration with Gaussian PDF to calculate the final probabilistic characteristics of the effective tensor. This study shows some remarkable differences in-between numerical and analytical homogenization methods in the context of geometrical uncertainty in the RVE of such a composite.

Various approaches in probabilistic homogenization of the CFRP composites

A variety of the homogenization methods is collected here and contrasted to determine the basic probabilistic characteristics of the CFRP composite in a presence of Gaussian uncertainty in its components material characteristics. We compare algebraic approximations of the effective elasticity tensor components with these calculated by using of two different FEM strategies. One may find in this elaboration a comparison of the Voigt–Reuss versus Hashin–Shtrikmann upper and lower bounds, the results of spatial averaging of this composite RVE, Mori–Tanaka, Vanishing Fiber Diameter as well as the Self-Consistent homogenization theories together with two independent FEM solutions of the so-called cell problem. Symbolic probabilistic analysis is implemented according to the generalized tenth order stochastic perturbation technique supported by the Response Function Method. Polynomial basis with statistically optimized order enables determination of the analytical functions relating effective tensor components with material parameters of the composite constituents. The expected values series confirm fundamental tendencies observed before in deterministic comparisons of these theories, an analysis of the coefficients of variations shows the resulting uncertainty level and sensitivity of the effective characteristics, while skewness and kurtosis show that Gaussian character of the homogenized tensor prevails in this problem.

Gaussian uncertainty in homogenization of rubber–carbon black nanocomposites

The objective of this work is Gaussian uncertainty in material parameters of the rubber–carbon black particle reinforced nano-composite in the framework of determination of its effective elastic parameters. The analytical 3D micro-mechanical model is contrasted here with the Finite Element Method one realized with the use of the system ABAQUS® and its tetrahedral finite elements C3D4. They discretize cubic Representative Volume Element (RVE) with a centrally located spherical carbon black particle, whose deformation energies accumulated during the uniaxial and biaxial uniform tension are computed. The polynomial-based Response Function Method allows to determine analytical functions relating effective elasticity tensor with Young moduli and Poisson ratios of this composite original components via the Weighted Least Squares Method; this is done in the symbolic environment of MAPLE. These functions serve for the initial sensitivity analysis, where we detect Poisson ratio of the rubber matrix as the crucial design parameter and its Young modulus as having secondary importance, while particle Young modulus and Poisson ratio are totally irrelevant. Next, the most influential parameters are randomized according to the Gaussian distribution and are employed in the dual probabilistic scheme – by using the stochastic perturbation and, independently, semi-analytical techniques to determine up to the fourth order probabilistic characteristics of the effective tensor components. Deterministic and probabilistic hypersensitivity of the effective constitutive tensor, expected according to the rubber matrix presence, is confirmed in all computational experiments. Such a probabilistic energy-related FEM approach will allow for future applications of more advanced constitutive models for the carbon black nano-composites, for the RVEs of larger sizes – containing large agglomerations of these particles and for the imperfect interfaces simulation.

Probabilistic entropy in homogenization of the periodic fiber‐reinforced composites with random elastic parameters

SUMMARYThe 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. Copyright © 2012 John Wiley & Sons, Ltd.

Homogenization of fiber-reinforced composites with random properties using the weighted least squares response function approach

The main aim of the paper is a determination of the basic probabilistic characteristics for the effective elasticity tensor of the periodic fiber-reinforced composites, using the generalized stochastic perturbation technique. An evaluation of the generalized stochastic perturbation method of the analytical formulas and the Monte-Carlo simulation technique is provided for the 1D periodic structure with random material parameters. The higher-order terms are determined using numerical determination of the response functions between the effective tensor components and the given random input variables. It is carried out with the use of the Least Squares Method (LSM), applied for the series of computational experiments consisting of the Finite Element Method (FEM) solutions to the cell problems for the randomized input parameters. The key problem is the weighting LSM procedure worked out to speed up the probabilistic convergence of the homogenization results.

Tenseurs effectifs de conductivité thermique des composites à texture arbitraire

This work propose a homogenization technique to obtain the effective thermal conductivity tensor of heterogeneous solids. These solids are composite materials containing anisotropic inclusions with different form and arbitrary repartition. The work concern “small scale” composites with good contact between matrix and inclusions so that there is no temperature discontinuity across interfaces. Simple expressions between the effective tensor components of the thermal conductivity and the respective components of the composite phases have been obtained. We consider the inclusions repartition and geometry with the help of texture and form tensors. The occurrence of pores is also taken into account. The theoretical results are compared with experimental data and other theories.

Perturbation based stochastic finite element method for homogenization of two-phase elastic composites

The main idea of the paper is to apply the second order perturbation and stochastic second central moment technique to solve the homogenization problem. In order to determine the effective elasticity tensor, the prevailing computational methodology discussed in the literature so far was the Monte-Carlo simulation providing appropriate expected values and higher order probabilistic moments of the effective tensor components. The technique applied in this paper aims at significantly reducing the computational cost of the simulation without sacrificing the solution accuracy. The numerical example substantiates this claim in the case of a periodic fiber-reinforced plane strain composite with random fiber and matrix Young’s moduli.