Simulation of local material properties based on moving-window GMC

When analyzing the behavior of composite materials under various loading conditions, the assumption is generally made that the behavior due to randomness in the material can be represented by a homogenized, or effective, set of material properties. This assumption may be valid when considering displacement, average strain, or even average stress of structures much larger than the inclusion size. The approach is less valid, however, when considering either behavior of structures of size at the scale of the inclusions or local stress of structures in general. In this paper, Monte Carlo simulation is used to assess the effects of microstructural randomness on the local stress response of composite materials. In order to achieve these stochastic simulations, the mean, variance and spectral density functions describing the randomly varying elastic properties are required as input. These are obtained here by using a technique known as moving-window generalized method of cells (moving-window GMC). This method characterizes a digitized composite material microstructure by developing fields of local effective material properties. Once these fields are generated, it is straightforward to obtain estimates of the associated probabilistic parameters required for simulation. Based on the simulated property fields, a series of local stress fields, associated with the random material sample under uniaxial tension, is calculated using finite element analysis. An estimation of the variability in the local stress response for the given random composite is obtained from consideration of these simulations.     

Analysis on the variances of material and structural properties based on random field theory

Based on the random field theory (RFT) and the stochastic finite element method (SFEM), the variances of the mechanical properties of materials and structures are studied. Manufacturing processes can easily lead to the spatial variations of the load and the material properties such as moduli and density. Characterizing the elastic moduli, load and density with one-dimensional random fields, the analytical solutions for the coefficient of variations (COVs) of effective material moduli, displacement and natural frequencies of beams are obtained. Then, with the fiber and matrix properties, volume fraction modeled by two-dimensional random fields and the fiber angle as a single random variable, a Monte Carlo simulation (MCS) is performed to generate the variances of effective modulus of fiber-reinforced composite laminar plate. Compared with the previous numerical conclusions, the present results reveal that the variances of effective material properties and structural displacement are greatly dependent on both the random fields and the sizes of structures in theory.     

Cohesive bonding interface model for the effective properties of randomly structured composites

The present paper aims at quantifying the effective behaviour of random composites at the microstructure scale using different combinations of intrinsic phase and interface properties. The composite microstructure is generated using Monte Carlo simulation. The methodology allows random spatial distribution of phases. It aims also at providing different evolutions of the interface quantity as function of phase ratio. Microstructures are then converted into finite element model. The FE model handles the interface effect using a cohesive zone model. This model represents the interface separation and traction using three parameters derived from a surface potential function. The sensitivity of the effective elastic properties to both interface and phase properties is then quantified. The predicted results show a strong non-linear dependence of the effective properties on the interface effect.     

Stochastic finite element analysis of a cable-stayed bridge system with varying material properties

Stochastic seismic finite element analysis of a cable-stayed bridge whose material properties are described by random fields is presented in this paper. The stochastic perturbation technique and Monte Carlo simulation (MCS) method are used in the analyses. A summary of MCS and perturbation based stochastic finite element dynamic analysis formulation of structural system is given. The Jindo Bridge, constructed in South Korea, is chosen as a numerical example. The Kocaeli earthquake in 1999 is considered as a ground motion. During the stochastic analysis, displacements and internal forces of the considered bridge are obtained from perturbation based stochastic finite element method (SFEM) and MCS method by changing elastic modulus and mass density as random variable. The efficiency and accuracy of the proposed SFEM algorithm are evaluated by comparison with results of MCS method. The results imply that perturbation based SFEM method gives close results to MCS method and it can be used instead of MCS method, especially, if computational cost is taken into consideration.     

Micromechanical modeling of local field distribution for a planar composite under plastic deformation

Summary Due to statistical distribution of local material property, local stress and strain fields in a composite are random in nature. Classical micromechanical methods can only predict the average value of these local fields in different phases. An analytical method, which combines the maximum entropy theory and secant moduli method, is proposed in this paper. The distribution of the local field for a planar composite with plastic deformation is examined by the proposed method. The results show that with increase of plastic deformation the stress field in the matrix becomes more and more inhomogeneous. The predicted results on the stress distribution are in reasonable agreement with finite element simulation. Some salient features near the elastic and plastic deformation transition revealed by finite element simulation are also discussed.     

Stochastic post buckling analysis of laminated composite cylindrical shell panel subjected to hygrothermomechanical loading

In this paper, the effect of random system properties on the post buckling load of geometrically nonlinear laminated composite cylindrical shell panel subjected to hygrothermomechanical loading is investigated. System parameters are assumed as independent random variables. The higher order shear deformation theory and von-Karman nonlinear kinematics are used for basic formulation. The elastic and hygrothermal properties of the composite material are considered to be dependent on temperature and moisture concentration using micromechanical approach. A direct iterative based C⁰ nonlinear finite element method in conjunction with first-order perturbation technique proposed by present author for the plate is extended for shell panel subjected to hygrothermomechanical loading to compute the second-order statistics (mean and variances) of laminated composite cylindrical shell panel. The effect of random system properties, plate geometry, stacking sequences, support conditions, fiber volume fractions and temperature and moisture distributions on hygrothermomechanical post-buckling load of the laminated cylindrical shell panel are presented. The performance of outlined stochastic approach has been validated by comparing the present results with those available in the literature and independent Monte Carlo simulation.     

Variability response functions for effective material properties

The variability response function (VRF) is a well-established concept for efficient evaluation of the variance and sensitivity of the response of stochastic systems where properties are modeled by random fields that circumvents the need for computationally expensive Monte Carlo (MC) simulations. Homogenization of material properties is an important procedure in the analysis of structural mechanics problems in which the material properties fluctuate randomly, yet no method other than MC simulation exists for evaluating the variability of the effective material properties. The concept of a VRF for effective material properties is introduced in this paper based on the equivalence of elastic strain energy in the heterogeneous and equivalent homogeneous bodies. It is shown that such a VRF exists for the effective material properties of statically determinate structures. The VRF for effective material properties can be calculated exactly or by Fast MC simulation and depends on extending the classical displacement VRF to consider the covariance of the response displacement at two points in a statically determinate beam with randomly fluctuating material properties modeled using random fields. Two numerical examples are presented that demonstrate the character of the VRF for effective material properties, the method of calculation, and results that can be obtained from it.     

Analysis of one-dimensional stochastic finite elements using neural networks

This paper explores the applicability of neural networks for analyzing the uncertainty spread of structural responses under the presence of one-dimensional random fields. Specifically, the neural network is intended to be a partial surrogate of the structural model needed in a Monte Carlo simulation, due to its associative memory properties. The network is trained with some pairs of input and output data obtained by some Monte Carlo simulations and then used in substitution of the finite element solver. In order to minimize the size of the networks, and hence the number of training pairs, the Karhunen–Loéve decomposition is applied as an optimal feature extraction tool. The Monte Carlo samples for training and validation are also generated using this decomposition. The Nyström technique is employed for the numerical solution of the Fredholm integral equation. The radial basis function (RBF) network was selected as the neural device for learning the input/output relationship due to its high accuracy and fast training speed. The analysis shows that this approach constitutes a promising method for stochastic finite element analysis inasmuch as the error with respect to the Monte Carlo simulation is negligible.     

Uncertainty quantification in homogenization of heterogeneous microstructures modeled by XFEM

An extended finite element method (XFEM) coupled with a Monte Carlo approach is proposed to quantify the uncertainty in the homogenized effective elastic properties of multiphase materials. The methodology allows for an arbitrary number, aspect ratio, location and orientation of elliptic inclusions within a matrix, without the need for fine meshes in the vicinity of tightly packed inclusions and especially without the need to remesh for every different generated realization of the microstructure. Moreover, the number of degrees of freedom in the enriched elements is dynamically reallocated for each Monte Carlo sample run based on the given volume fraction. The main advantage of the proposed XFEM‐based methodology is a major reduction in the computational effort in extensive Monte Carlo simulations compared with the standard FEM approach. Monte Carlo and XFEM appear to work extremely efficiently together. The Monte Carlo approach allows for the modeling of the size, aspect ratios, orientations, and spatial distribution of the elliptical inclusions as random variables with any prescribed probability distributions. Numerical results are presented and the uncertainty of the homogenized elastic properties is discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

Transverse stiffness properties of unidirectional fiber composites containing debonded fibers

First-principles micromechanics modeling for the determination of transverse stiffness properties of a unidirectional fiber composite with fiber–matrix interfacial debonding is presented. The composite has a packing arrangement of a periodic square array of fibers, but contains randomly distributed debonded fibers. The finite element method is employed for the exact treatment of local microscopic stress and strain fields in a representative volume element of the composite material, and of the nonlinear problem of separation and contact of fiber and matrix at debonded interface. The randomness of the distribution of debonded fibers is dealt with by means of the Monte Carlo method, and the composite stiffness properties are found as ensemble average properties over a large number of representative volume elements. Bimodular behavior of the composite under transverse loading, i.e., different stiffnesses in tension and compression, is accurately captured.     

Finite element methods in probabilistic mechanics

Probabilistic methods, synthesizing the power of finite element methods with second-order perturbation techniques, are formulated for linear and nonlinear problems. Random material, geometric properties and loads can be incorporated in these methods, in terms of their fundamental statistics. By construction, these methods are applicable when the scale of randomness is not too large and when the probabilistic density functions have decaying tails. By incorporating certain computational techniques, these methods are shown to be capable of handling large systems with many sources of uncertainties. Applications showing the effects of combined random fields and cyclic loading/stress reversal are studied and compared with Monte Carlo simulation results.     

Stochastic finite element analysis of the free vibration of laminated composite plates

Using the Stochastic Finite Element Method (SFEM) to perform reliability analysis of the free vibration of composite plates with material and fabrication uncertainties has received much attention lately. In this work the stochastic analysis is performed using the First-Order Reliability Method (FORM-method 2) and the Second-Order Reliability Method (SORM). The basic random variables include laminae stiffness properties and material density, as well as the randomness in ply orientation angles. Modeling of the composite behavior utilizes a nine-noded isoparametric Lagrangian element based on the third-order shear deformation theory. Calculating the eigenvectors at the mean values of the variables proves to be a reasonable simplification which significantly increases solution speed. The stochastic finite element code is validated using comparisons with results of Monte Carlo simulation technique with importance sampling. Results show that SORM is an excellent rapid tool in the stochastic analysis of free vibration of composite plates, when compared to the slower Monte Carlo simulation techniques.     

A finite element method for the probabilistic creep of solids

We outline here a finite element technique for the creep of solids whose constitutive equation contains one or more random parameters. In contrast to other finite element techniques for the prediction of random structural response, the present method is based upon exact relations from the theory of probability. It yields, at a given value of time, the probability density function for the field variable of interest, e.g. stress or displacement components. The method is illustrated by a simple creeping beam problem, using a power-law creep constitutive equation. The calculated distributions are found to be highly skewed, and in excellent agreement with the results of Monte Carlo simulation.     

A numerical approach for determining the effective elastic symmetries of particulate-polymer composites

In this paper we deal with the problem of determining on the one hand the effective elastic properties of particulate-polymer composite materials and on the other hand the actual degree of symmetry of the resulting homogenised material. This twofold purpose has been accomplished by building a 2D as well as a 3D finite element model of the heterogeneous material and by using the strain-energy based numerical homogenisation technique. Both finite element models are able to reproduce with a good level of accuracy the real microstructure of the composite material by considering a random distribution of both particles and air bubbles (that are generated by the fabrication process). To assess the effectiveness of the proposed models, we present a numerical study to determine the effective elastic properties of the composite along with a comparison with the existing analytical and experimental results taken from literature and a sensitivity analysis in terms of the spatial distribution of the particles of the unit cell. Numerical results show that both models are able to provide the equivalent elastic properties with a very good level of accuracy when compared to experimental results and that the particulate-reinforced polymer composite could show, depending on the particles volume fraction and arrangement, an isotropic or a cubic elastic symmetry.     

2D finite element analysis of thermally bonded nonwoven materials: Continuous and discontinuous models

Due to a random structure of nonwoven materials, their non-uniform local material properties and nonlinear properties of single fibres, it is difficult to develop a numerical model that adequately accounts for these features and properly describes their performance. Two different finite element (FE) models – continuous and discontinuous – are developed here to describe the tensile behaviour of nonwoven materials. A macro-level continuum finite element model is developed based on the classic composite theory by treating the fibrous network as orthotropic material. This model is used to analyse the effect of thermally bonding points on the deformational behaviour and deformation mechanisms of thermally bonded nonwoven materials at macro-scale. To describe the effects of discontinuous microstructure of the fabric and implement the properties of polypropylene fibres, a micro-level discontinuous finite element model is developed. Applicability of both models to describe various deformational features observed in experiments with a real thermally bonded nonwoven is discussed.     

Predicting laser weld reliability with stochastic reduced‐order models

Laser welds are prevalent in complex engineering systems and they frequently govern failure. The weld process often results in partial penetration of the base metals, leaving sharp crack‐like features with a high degree of variability in the geometry and material properties of the welded structure. Accurate finite element predictions of the structural reliability of components containing laser welds requires the analysis of a large number of finite element meshes with very fine spatial resolution, where each mesh has different geometry and/or material properties in the welded region to address variability. Traditional modeling approaches cannot be efficiently employed. To this end, a method is presented for constructing a surrogate model, based on stochastic reduced‐order models, and is proposed to represent the laser welds within the component. Here, the uncertainty in weld microstructure and geometry is captured by calibrating plasticity parameters to experimental observations of necking as, because of the ductility of the welds, necking – and thus peak load – plays the pivotal role in structural failure. The proposed method is exercised for a simplified verification problem and compared with the traditional Monte Carlo simulation with rather remarkable results. Copyright © 2015 John Wiley & Sons, Ltd.     

Three‐dimensional stochastic finite element method for elasto‐plastic bodies

A new stochastic finite element method (SFEM) is formulated for three‐dimensional softening elasto‐plastic bodies with random material properties. The method is based on the Karhunen–Loeve and polynomial chaos expansions, and able to efficiently estimate complete probabilistic characteristics of the response, such as moments or PDFs. To reduce the computational complexity in the three‐dimensional setting, two alterations are made with respect to the two‐dimensional SFEM proposed earlier by the authors. First, a variability preserving modification of the Karhunen–Loeve expansion is rigorously derived and applied in the stochastic discretization of random fields representing material properties. Second, an efficient algorithm for parallel processing is developed, with time consumption being the same order as for an ordinary FEM, rendering the proposed SFEM an effective alternative to Monte‐Carlo simulation. The applicability of the proposed method to stochastic analysis of strain localization is examined using Monte‐Carlo simulation. Then, it is applied to a fault formation problem which is a recent concern of earthquake engineering. Ground surface layers are modelled by a softening elasto‐plastic body, and the evolution of probabilistic characteristics of the rupture process is analysed in detail. Some practical observations are made regarding the nature of the fault formation from the stochastic viewpoint. Copyright © 2001 John Wiley & Sons, Ltd.     

Stochastic critical stress intensity factor response of single edge notched laminated composite plate using displacement correlation method

Abstract

The second-order statistics of critical stress intensity factor (SIF) of single edge notched fiber reinforced composite plates with random system properties and subjected to uniaxial tensile loadings is investigated. This paper is an extension of reference (Lal and Kapania, 2013) by the present authors by considering more number of input random system parameters for higher accuracy. A C⁰ finite element method based on a higher-order shear deformation plate theory using displacement correlation method via isoparametric quarter point element is proposed for basic formulation. A stochastic finite element method using first-order perturbation technique and Monte Carlo simulation (MCS) is employed to examine the mean, coefficient of variance, and probability density faction of critical first mode SIF. The effect of different fiber orientations, crack length, plate thickness, a number of layers, and the lamination schemes with random system properties on the statistics of SIF of single edge crack laminated composite plate is evaluated. The tensile failure load is predicted using Hashin’s failure criteria. The present approach is validated with results available in literature and by employing independent MCS.     

Stochastic finite element analysis of layered composite beams with spatially varying non-Gaussian inhomogeneities

This study focuses on the development of a stochastic finite element-based methodology for failure assessment of composite beams with spatially varying non-Gaussian distributed inhomogeneities. The material properties in the individual laminae are modeled as non-Gaussian random fields, whose probability density functions and the correlations are estimated from the test data. The non-Gaussian random fields are discretized into a vector of correlated non-Gaussian random variables using the optimal linear expansion scheme that preserves the second-order non-Gaussian characteristics of the fields. Subsequently, the estimates of the failure probability are obtained from Monte Carlo simulations carried out on the vector of correlated random variables. Issues related to the computational efficiency of the proposed framework and the variabilities in the material properties are discussed. Numerical examples are presented, which highlight the salient features of the proposed method.