Boundary element analysis for effective stiffness tensors: effect of fabric tensor determination method

Second-rank fabric tensors have been extensively used to describe structural anisotropy and to predict orthotropic elastic constants. However, there are many different definitions of, and approaches to, determining the fabric tensor. Most commonly used is a fabric tensor based on mean intercept length measurements, but star volume distribution and star length distribution are commonly used, particularly in studies of trabecular bone. Here, we investigate the effect of the fabric tensor definition on elastic constant predictions using both synthetic, idealized microstructures as well as a micrograph of a porous ceramic. We use an efficient implantation of a symmetric Galerkin boundary element method to model the mechanical response of the various microstructures, and also use a boundary element approach to calculate the necessary volume averages of stress and strain to obtain the effective properties of the media.     

Investigation of the Embedded Element Technique for ModellingWavy CNT Composites

This paper presents a comparison of different finite element approaches to modelling polymers reinforced with wavy, hollow fibres with the aim of predicting the effective elastic stiffness tensors of the composites. The waviness of the tubes is described by sinusoidal models with different amplitude-to-wavelength parameters. These volume elements are discretized by structured volume meshes onto which fibres in the form of independently meshed beam, shell or volume elements are superimposed. An embedded element technique is used to link the two sets of meshes. Reference solutions are obtained from conventional three-dimensional volume models of the same phase arrangements. Periodicity boundary conditions are applied in all cases and fibre volume fractions of up to a few percent are considered. The results indicate that embedded element techniques using shell elements for discretizing the fibres may provide an attractive combination of accuracy, computational cost and flexibility for modelling composites reinforced by arbitrarily, three-dimensionally curved nanotubes.     

A new homogenization method based on a simplified strain gradient elasticity theory

Homogenization methods utilizing classical elasticity-based Eshelby tensors cannot capture the particle size effect experimentally observed in particle–matrix composites at the micron and nanometer scales. In this paper, a new homogenization method for predicting effective elastic properties of multiphase composites is developed using Eshelby tensors based on a simplified strain gradient elasticity theory (SSGET), which contains a material length scale parameter and can account for the size effect. Based on the strain energy equivalence, a homogeneous comparison material obeying the SSGET is constructed, and two sets of equations for determining an effective elastic stiffness tensor and an effective material length scale parameter for the composite are derived. By using Eshelby’s eigenstrain method and the Mori–Tanaka averaging scheme, the effective stiffness tensor based on the SSGET is analytically obtained, which depends not only on the volume fractions and shapes of the inhomogeneities (i.e., phases other than the matrix) but also on the inhomogeneity sizes, unlike what is predicted by the existing homogenization methods based on classical elasticity. To illustrate the newly developed homogenization method, sample cases are quantitatively studied for a two-phase composite filled with spherical, cylindrical, or ellipsoidal inhomogeneities (particles) using the averaged Eshelby tensors based on the SSGET that were derived earlier by the authors. Numerical results reveal that the particle size has a large influence on the effective Young’s moduli when the particles are sufficiently small. In addition, the results show that the composite becomes stiffer when the particles get smaller, thereby capturing the particle size effect.     

Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size

The main objective of this paper is to present a generic meso-scale probability model for a large class of random anisotropic elastic microstructures in order to perform a parametric analysis of the Representative Volume Element (RVE) size. This new approach can be useful for a direct experimental identification of random anisotropic elastic microstructures when the standard method cannot easily be applied to anisotropic elastic microstructures. Such a RVE is used to construct the macroscopic properties in the context of stochastic homogenization. The probability analysis is not performed as usual for a given particular random microstructure defined in terms of its constituents. Instead, it is performed for a large class of random anisotropic elastic microstructures. For this class, the probability distribution of the random effective stiffness tensor is explicitly constructed. This allows a full probability analysis of the RVE size to be carried out and its convergence to be studied. The procedure of homogenization is based on a homogeneous Dirichlet condition on the boundary of the RVE. The probability model used for the stiffness tensor-valued random field of the random anisotropic elastic microstructure is an extension of the model recently introduced by the author for elliptic stochastic partial differential operators. The stochastic boundary value problem is numerically solved by using the stochastic finite element method. The probability analysis of the RVE size is performed by studying the probability distribution of the random operator norm of the random effective stiffness tensor with respect to the spatial correlation length of the random microstructure.     

Fabric tensor based boundary element analysis of porous solids

In this paper, the boundary element analysis of porous solids (sintered materials, foams, etc.) is studied utilizing a fabric tensor. The fabric tensor provides a measure of anisotropy in the solid, as well as information concerning the geometry and distribution of the pores. The homogenized, orthotropic elastic properties of a porous solid can then be predicted with the fabric tensor. To illustrate the analysis, the effect of porosity on a trabecular bone-titanium bimaterial is studied. The boundary element analysis uses an anisotropic, bimaterial Green's function so the interface does not require discretization. It is shown that the anisotropic Stroh variables are independent of the structural density and dependent on the eigenvalues of the fabric tensor. An example calculation is presented where the effect of porosity on the in-plane maximum shear stress in a trabecular bone-titanium bimaterial is substantial.     

Stiffness tensor random fields through upscaling of planar random materials

Unique effective material properties are not possible for random heterogeneous materials at intermediate length scales, which is to say at some mesoscale above the microscale yet prior to the attainment of the representative volume element (RVE). Focusing on elastic moduli in particular, a micromechanical analysis based on the Hill–Mandel condition leads to the conclusion that two fields, stiffness and compliance, are required to bound the response of the material. In particular, we analyze means and correlation coefficients of a random planar material with a two-phase microstructure of random checkerboard type. We employ micromechanics, which can be viewed as an upscaling, smoothing procedure using the concept of a mesoscale “window”, and random field theory to compute the correlation structure of 4th-rank tensor fields of stiffness and compliance for a given mesoscale. Results are presented for various correlation distances, volume fractions, and contrasts in stiffness between phases. The main contribution of this research is to provide the data for developing analytical correlation functions, which can then be used at any mesoscale to generate micromechanically based inputs into analytical and computational mechanics models.     

A boundary condition in Padé series for frequency‐domain solution of wave propagation in unbounded domains

A boundary condition satisfying the radiation condition at infinity is frequently required in the numerical simulation of wave propagation in an unbounded domain. In a frequency domain analysis using finite elements, this boundary condition can be represented by the dynamic stiffness matrix of the unbounded domain defined on its boundary. A method for determining a Padé series of the dynamic stiffness matrix is proposed in this paper. This method starts from the scaled boundary finite‐element equation, which is a system of ordinary differential equations obtained by discretizing the boundary only. The coefficients of the Padé series are obtained directly from the ordinary differential equations, which are not actually solved for the dynamic stiffness matrix. The high rate of convergence of the Padé series with increasing order is demonstrated numerically. This technique is applicable to scalar waves and elastic vector waves propagating in anisotropic unbounded domains of irregular geometry. It can be combined seamlessly with standard finite elements. Copyright © 2006 John Wiley & Sons, Ltd.     

Temperature-dependent multi-scale modeling of surface effects on nano-materials

In this paper, a novel temperature-dependent multi-scale method is developed to investigate the role of temperature on surface effects in the analysis of nano-scale materials. In order to evaluate the temperature effect in the micro-scale (atomic) level, the temperature related Cauchy–Born hypothesis is implemented by employing the Helmholtz free energy, as the energy density of equivalent continua relating to the inter-atomic potential. The multi-scale technique is applied in atomistic level (nano-scale) to exhibit the temperature related characteristics. The first Piola–Kirchhoff stress and tangential stiffness tensor are computed, as the first and second derivatives of the free energy density to the deformation gradient, which are transferred to the macro-scale level. The Lagrangian finite element formulation is incorporated into the heat transfer analysis to develop the thermo-mechanical finite element model, and an intrinsic function is employed to model the surface and temperature effects in macro-scale level. The stress and tangential stiffness tensors are derived at each quadrature point by interpolating the data from nearby representative atom. The boundary Cauchy–Born (BCB) elements are introduced to capture the surface, edge and corner effects. Finally, the numerical simulation of proposed model together with the direct comparison with fully atomistic model illustrates that the technique provides promising results for facile modeling of boundary effect on thermo-mechanical behavior of metallic nano-scale devices.     

考虑周期微结构分布特征的Mori-Tanaka方法

利用两点间应变Green函数张量概念所建立的应变场积分方程, 推导了两相复合材料中夹杂的应变集中张量。该张量较之传统Mori-Tanaka (MT)法采用的由稀疏法导出的应变集中张量, 增加了一个与夹杂体积分数和分布相关的项, 并由此发展了考虑周期微结构分布特征的MT法。传统的MT法虽然能很好地预测正六角形分布圆截面纤维增强复合材料等的有效模量, 但不能反映正方形分布时的四方对称性特征, 本文作者所发展的方法弥补了这方面的不足, 并且所预报的有效刚度和柔度仍然保持了原MT方法所具有的自洽特性。最后通过与双周期有限元计算结果的对照验证了本文方法的精度。      

Laguerre tessellations for elastic stiffness simulations of closed foams with strongly varying cell sizes

The dependency of the elastic stiffness, i.e., Young’s modulus, of isotropic closed-cell foams on the cell size variation is studied by microstructural simulation. For this purpose, we use random Laguerre tessellations which, unlike classical Voronoi models, allow to generate model foams with strongly varying cell sizes. The elastic stiffness of the model realizations is computed by micro finite element analysis using shell elements. The main result is a moderate decrease of the effective elastic stiffness for increasing cell size variations if the solid volume fraction is assumed to be constant.     

Boundary element analysis of a plate on elastic foundations

The objective of this study is to examine the applicability of the boundary element method to analysing a plate on elastic foundation. The fundamental solution of the problem is presented as a Fourier-Bessel integral. For the computation of the values of the fundamental solution an algorithm was developed in which the Fourier-Bessel integral was decomposed into an alternative convergent sequence. Equations based on the direct and indirect boundary element method were derived for a plate situated on a one- or two-parametric elastic foundation. According to the theory presented, computer programs based on the direct and indirect boundary element method were developed. These programs can be used for examining the behaviour of a smooth-boundary plate on a one- or two-parametric elastic foundation. The computer programs were tested by several examples. The results obtained by using a small number of boundary elements compared favourably to the results obtained by a fine finite element mesh. The study shows that the boundary element method is applicable to the analysis of a plate on elastic foundation.     

Probabilistic modeling of apparent tensors in elastostatics: A MaxEnt approach under material symmetry and stochastic boundedness constraints

In this work, we address the stochastic modeling of apparent elasticity tensors, for which both material symmetry and stochastic boundedness constraints have to be taken into account, in addition to the classical constraint of invertibility. We first introduce a stochastic measure of anisotropy, which is defined using metrics in the set of elasticity tensors and used for quantitatively characterizing the fulfillment of material symmetry constraints. After having defined a numerical approximation for the stochastic boundedness constraint, we then propose a methodology allowing one to unify maximum entropy based models that have been previously derived by considering some of these constraints and which consists in constructing a probabilistic model for an auxiliary random variable. The latter can be interpreted as a stochastic compliance tensor, for which the available information to be used in the maximum entropy formulation can be readily deduced from the one considered for the elasticity tensor. A numerical illustration of the approach to an elastic microstructure is finally provided.     

On Bhattacharyya relative entropy in a homogenization of composite materials

The main idea of this work is an application of relative entropy in the numerical analysis of probabilistic divergence between original material tensors of the composite constituents and its effective tensor in the presence of material uncertainties. The homogenization method is based upon the deformation energy of the representative volume elements for the fiber-reinforced and particulate composites and uncertainty propagation begins with elastic moduli of the fibers, particles, and composite matrices. Relative entropy follows a mathematical model originating from Bhattacharyya probabilistic divergence and has been applied here for Gaussian distributions. The semi-analytical probabilistic method based on analytical integration of polynomial bases obtained via the least squares method fittings enables for determination of the basic probabilistic characteristics of the effective tensor and the relative entropies. The methodology invented in this work may be extended toward other probability distributions and relative entropies, for homogenization of nonlinear composites and also accounting for some structural interface defects.     

Formulation and numerical treatment of incompressibility constraints in large strain elastic–plastic analysis

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the rate‐independent finite strain analysis of solids undergoing large elastic‐isochoric plastic deformations. The formulation relies on the introduction of a mixed‐variant metric deformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure which can be additively decomposed into the exact isochoric (deviatoric) and volumetric (spheric) strain measures. This fact may be seen as the basic idea in the formulation of appropriate mixed finite elements which guarantee the accurate computation of isochoric strains. The mixed‐variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response whereas the plastic material behavior is assumed to be governed by a generalized J₂ yield criterion and rate‐independent isochoric plastic strain rates are computed using an associated flow rule. On the numerical side, the computation of the logarithmic strain tensors is based on higher‐order Padé approximations. To be able to take into account the plastic incompressibility constraint a modified mixed variational principle is considered which leads to a quasi‐displacement finite element procedure. Finally, the numerical solution of finite strain elastic‐plastic problems is presented to demonstrate the efficiency and the accuracy of the algorithm. Copyright © 1999 John Wiley & Sons, Ltd.     

Micromechanics as a basis of random elastic continuum approximations

The problem of the determination of stochastic constitutive laws for input to continuum-type boundary value problems is analyzed from the standpoint of the micromechanics of polycrystals and matrix-inclusion composites. Passage to a sought-for random continuum is based on a scale dependent window playing the role of a Representative Volume Element (RVE). It turns out that an elastic microstructure with piecewise continuous realizations of random tensor fields of stiffness cannot be uniquely approximated by a random field of stiffness with continuous realizations. Rather, two random continuum fields may be introduced to bound the material response from above and from below. As the size of the RVE relative to the crystal size increases to infinity, both fields converge to a deterministic continuum with a progressively decreasing strength of fluctuations. Since the RVE corresponds to a single finite element cell not infinitely larger than the crystal size, two random fields are to be used to bound the solution of a given boundary value problem at a given scale of resolution. The method applies to a number of other elastic microstructures, and provides the basis for stochastic finite differences and elements. The latter point is illustrated by an example of a stochastic boundary value problem of a heterogeneous membrane.     

Sensitivity analysis on the effective stiffness properties of 3-D orthotropic honeycomb cores

The present study investigates the influences of representative volume element RVE mesh and material parameters, here cell wall elastic moduli, on the effective stiffness properties of three dimensional orthotropic honeycomb cores through strain driven computational homogenization in the finite element framework. For this purpose, case studies were carried out, for which hexagonal cellular RVEs were generated, meshed with eight node linear brick finite elements of varying numbers. Periodic boundary conditions were then implemented on the RVE boundaries by using one-to-one nodal match for the corresponding corners, edges and surfaces for the imposed macroscopic strains. As a novelty, orthotropic material properties were assigned for each cell wall by means of the transformation matrices following the cell wall orientations. Thereafter, simulations were conducted and volume averaged macroscopic stresses were obtained. Eventually, effective stiffness properties were obtained, through which RVE sensitivity analysis was carried out. The investigations indicate that there is a strong relation between number of finite elements and most of the effective stiffness parameters. In addition to this, cell wall elastic moduli also play critical role on the effective properties of the investigated materials.     

Three-dimensional structural vibration analysis by the Dual Reciprocity BEM

The dual reciprocity boundary element method (DR/BEM) is employed for the analysis of free and forced vibrations of three-dimensional elastic solids. Use of the elastostatic fundamental solution in the integral formulation of elastodynamics creates an inertial volume integral in addition to the boundary ones. This volume integral is transformed into a surface integral by invoking the reciprocal theorem. A general analytical method is described for the closed form determination of the particular solutions of the displacement and traction tensors corresponding to any radial basis function employed in the transformation process. The simple but effective 1+r radial basis function is used in the applications of this paper. Quadratic continuous and discontinuous 9-noded boundary elements are used in the analysis. Free vibrations are studied by solving the corresponding eigenvalue problem iteratively. Harmonic forced vibration problems are solved directly in the frequency domain. Transient forced vibration problems are solved by integrating the equations of motion stepwise with the aid of various algorithms. Interior collection points are also used for assessing the accuracy of the method. Two numerical examples involving free and forced vibrations of a sphere and a cube are presented in detail.     

Implementation of the transformation field analysis for inelastic composite materials

Conclusions The transformation field analysis is a general method for solving inelastic deformation and other incremental problems in heterogeneous media with many interacting inhomogeneities. The various unit cell models, or the corrected inelastic self-consistent or Mori-Tanaka fomulations, the so-called Eshelby method, and the Eshelby tensor itself are all seen as special cases of this more general approach. The method easily accommodates any uniform overall loading path, inelastic constitutive equation and micromechanical model. The model geometries are incorporated through the mechanical transformation influence functions or concentration factor tensors which are derived from elastic solutions for the chosen model and phase elastic moduli. Thus, there is no need to solve inelastic boundary value or inclusion problems, indeed such solutions are typically associated with erroneous procedures that violate (6₂); this was discussed by Dvorak (1992). In comparison with the finite element method in unit cell model solutions, the present method is more efficient for cruder mesches. Moreover, there is no need to implement inelastic constitutive equations into a finite element program. An addition to the examples shown herein, the method can be applied to many other problems, such as those arising in active materials with eigenstrains induced by components made of shape memory alloys or other actuators. Progress has also been made in applications to electroelastic composites, and to problems involving damage development in multiphase solids. Finally, there is no conceptural obstacle to extending the approach beyond the analysis of representative volumes of composite materials, to arbitrarily loaded structures.This work was supported by the Air Force Office of Scienctific Research, and by the Office of Naval Research     

Strongly orthotropic continuum mechanics and finite element treatment

Despite successful application to orthotropic analysis, any Lagrangian strain tensor that is symmetric can be classified as an isotropic metric, while the infinitely orthotropic case can be accurately dealt with using one‐dimensional elements, structural tensors or kinematic constraints. In this paper, we present a strongly orthotropic continuum mechanics basis that models the exact kinematic behavior of the intermediary class of materials and also show its application to multi‐axial media and treatment using the finite element method. By asserting that mechanistic strain metrics must be material property dependent and satisfy equilibrium, we are able to derive a novel orthotropic linear strain tensor that is asymmetric and thus capable of describing all levels of orthotropy, while maintaining generality to the well‐established isotropic approach. Subsequently formulated are a material principal rotation tensor, extended orthotropic compliance tensor and an extended Mohr's plot for strain relying on an additional metric denoted as aspectual strain. Using the developed finite element formulation, it is shown that identical stress results to conventional theory for an orthotropic linear problem are predicted, while offering a more informative analysis. A second numerical example demonstrates the unique capability of this approach to solve the erroneous response of strongly orthotropic materials under trellis shear as compared with a number of conventional and contemporary approaches and thus its ability to produce kinematically exact results. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite element analysis of viscoelastic composite structures based on a micromechanical material model

The stress and creep analysis of structures made of micro-heterogeneous composite materials is treated as a two-scale problem, defined as a mechanical investigation on different length scales. Reinforced composites show by definition a heterogeneous texture on the microlevel, determined by the constitutive behaviour of the matrix material and the embedded fibres as well as the characteristics of the bonding properties in the interphase. All these heterogeneities are neglected by the finite element analysis of structural elements on the macroscale, since a ficticious and homogeneous continuum with averaged properties is assumed. Therefore, the constitutive equations of the substitute material should well reflect the mechanical behaviour of the existing micro-heterogeneous composite in an average sense.The paper at hand starts with the brief outline of a micromechanical model, named generalized method of cells (GMC), which provides the macrostress responses due to macrostrain processes as well as the homogenised constitutive tensor of the substitute material. The macroscopic stresses and strains are obtained as volume averages of the corresponding microfields within a representative volume element. The effective material tensor constitutes the mapping between the macro-strains and the macro-stresses. The cells method is used for the homogenisation of the unidirectionally reinforced single layers of laminates made of viscoelastic resins and flexibly embedded elastic fibres. The algorithm for the homogenisation of the constitutive properties runs simultaneously to the finite element analysis at each point of numerical integration and provides the macro-stresses and the homogenised constitutive properties. The validity of the proposed two-scale simulation is investigated by solving boundary value problems and comparing the numerical results for the structures to the experimental data of creep and relaxation tests or analytical solutions.