Hysteretic Behavior of Random Particulate Composites by the Stochastic Finite Element Method (original) (raw)

Random Stiffness Tensor of Particulate Composites with Hyper-Elastic Matrix and Imperfect Interface

Materials

The main aim of this study is determination of the basic probabilistic characteristics of the effective stiffness for inelastic particulate composites with spherical reinforcement and an uncertain Gaussian volume fraction of the interphase defects. This is determined using a homogenization method with a cubic single-particle representative volume element (RVE) of such a composite and the finite element method solution. A reinforcing particle is spherical, located centrally in the RVE, surrounded by the thin interphase of constant thickness, and remains in an elastic reversible regime opposite to the matrix, which is hyper-elastic. The interphase defects are represented as semi-spherical voids, which are placed on the outer surface of this particle. The interphase is modeled as hyper-elastic and isotropic, whose effective stiffness is calculated by the spatial averaging of hyper-elastic parameters of the matrix and of the defects. A constitutive relation of the matrix is recovered ex...

Energy Fluctuations in the Homogenized Hyper-Elastic Particulate Composites with Stochastic Interface Defects

Energies

The principle aim of this study is to analyze deformation energy of hyper-elastic particulate composites, which is the basis for their further probabilistic homogenization. These composites have some uncertain interface defects, which are modelled as small semi-spheres with random radius and with bases positioned on the particle-matrix interface. These defects are smeared into thin layer of the interphase surrounding the reinforcing particle introduced as the third component of this composite. Matrix properties are determined from the experimental tests of Laripur LPR 5020 High Density Polyurethane (HDPU). It is strengthened with the Carbon Black particles of spherical shape. The Arruda–Boyce potential has been selected for numerical experiments as fitting the best stress-strain curves for the matrix behavior. A homogenization procedure is numerically implemented using the cubic Representative Volume Element (RVE). Spherical particle is located centrally, and computations of deforma...

2D versus 3D probabilistic homogenization of the metallic fiber-reinforced composites by the perturbation-based stochastic Finite Element Method

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.

Perturbation-based stochastic multi-scale computational homogenization method for the determination of the effective properties of composite materials with random properties

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

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.

Numerical and statistical estimates of the representative volume element of elastoplastic random composites

European Journal of Mechanics - A/Solids, 2012

In many applications elastoplastic composites are used in limited amounts, therefore it is important to have estimates of the size of their representative volume element both for modeling and experimental purposes. In this work the tensile response of particle reinforced random composites is simulated by microstructural finite element models. Several microstructural realizations are considered for each composition and volume, and the scatter in the response is used as representativeness metric. The microstructural morphology is characterized using methods and statistical descriptors that can be employed with micrographs of real materials. Numerical results show that the representative volume element dimensions can be estimated by verifying either the consistency of the stressestrain curve for single microstructural realizations and increasing material volume sizes or the convergence of the response of several microstructural realizations at the same material volume size. The analysis of the stressestrain state at the microstructural level shows that the plastic strain and the hydrostatic pressure in the matrix material depend hyperbolically on the interparticle distance. Microstructural analyses show that the matrix coarseness is correlated to the scatter in the mechanical response and therefore can be used to have approximate estimates of the representative volume element size.

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

Archives of Mechanics, 2011

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.

Towards designing composites with stochastic composition: Effect of fluctuations in local material properties

Mechanics of Materials, 2016

This article presents a numerical study of the mechanical behavior of particulate composites with stochastic composition. Two types of such materials are considered: composites with homogeneous elastic-plastic matrix and randomly distributed inclusions of stiffness sampled from a distribution function, and composites with matrix having spatially varying elastic-plastic material parameters with no inclusions as well as with randomly distributed identical inclusions. We observe that the presence of fluctuations in either inclusions or matrix material properties leads to smaller effective modulus, smaller strain hardening and a reduction of the yield stress of the composite. Fluctuations of the yield stress of the matrix leads to a significant reduction of the mean yield stress of the composite. Fluctuations of the elastic modulus and of the strain hardening are associated with the reduction of the mean of the distributions of elastic modulus and strain hardening of the composite. For the range of parameters considered, fluctuations lead to maximum principal stress fields with narrow distribution of values, which implies enhanced resistance to damage initiation. Increasing the variance of the distribution functions from which local material properties are sampled, while keeping the mean constant, renders these effects more pronounced. This study is motivated by the growing interest in additive manufacturing technologies which open new possibilities for designing composite materials.

Stochastic Finite Element Analysis of Composites

Composite materials can have a large variation in material properties, especially in the transverse direction. This can cause difficulty when designing with these materials and may lead to overdesigning the material. Using finite element software, a model was created to simulate failure of composite materials in the transverse direction. Using micrograph images, a simulation of microstructures was generated based on fiber volume fraction and special positioning of fibers. Using a multi-level approach, the material properties were calculated in the micromechanical model. The comparison of the obtained elastic properties with rule of mixture revealed that the transverse properties are poorly predicted by rule of mixture. A mesomechanical model was then developed based on the material properties obtained from the micromechanics model. Using a progressive failure approach, each element could independently be disabled when the element fails, causing a simulated propagation of failed material. The results showed that failure follows a Weibull distribution consistent with experimental observation. The developed stochastic model will allow for infinite possibilities of failure replicating experimental findings.

An inverse micro-mechanical analysis toward the stochastic homogenization of nonlinear random composites

Computer Methods in Applied Mechanics and Engineering, 2019

An inverse Mean-Field Homogenization (MFH) process is developed to improve the computational efficiency of non-linear stochastic multiscale analyzes by relying on a micro-mechanics model. First full-field simulations of composite Stochastic Volume Element (SVE) realizations are performed to characterize the homogenized stochastic behavior. The uncertainties observed in the non-linear homogenized response, which result from the uncertainties of their micro-structures, are then translated to an incrementalsecant MFH formulation by defining the MFH input parameters as random effective properties. These effective input parameters, which correspond to the micro-structure geometrical information and to the material phases model parameters, are identified by conducting an inverse analysis from the full-field homogenized responses. Compared to the direct finite element analyzes on SVEs, the resulting stochastic MFH process reduces not only the computational cost, but also the order of uncertain parameters in the composite micro-structures, leading to a stochastic Mean-Field Reduced Order Model (MF-ROM). A data-driven stochastic model is then built in order to generate the random effective properties under the form of a random field used