A multi-scale framework for effective elastic properties of porous materials

This paper presents a statistical micromechanics-based multi-scale material modeling framework to predict the effective elastic moduli of porous materials. The present formulation differs from most of the existing theoretical models in that the interaction effects among the pores are directly accounted for by considering the pair-wise interaction and the statistical information of pore distribution is included by applying the ensemble volume averaging process. The theory of average fields is employed to derive the stress and strain concentration factor tensors that relate the local average fields to the global averages. Closed-form and analytical explicit expressions for the effective elastic moduli of porous materials are obtained in terms of the mechanical properties of the matrix material and porosity. The dependence of effective elastic properties on the porosity is investigated. Comparison of our theoretical prediction with the results of the published experimental data and other existing theoretical models is performed to illustrate the predictive capability of the proposed framework for porous materials.     

Ultrasonic wave propagation in two-phase media: Spherical inclusions

The scattering theory, recently developed via the extended method of equivalent inclusion, is used to study the propagation of time-harmonic waves in two-phase media of elastic matrix with randomly distributed elastic spherical inclusion materials. The elastic moduli and mass density of the composite medium are determined as functions of frequencies when given properties and concentration of the spheres and the matrix. Velocities and attenuation of ultrasonic waves in two-component media are determined. An averaging theorem that requires the equivalence of the strain energy and the kinetic energy between the effective medium and the original matrix with inhomogeneities is employed to derive the effective moduli and mass density. The functional dependency of these quantities upon frequencies and concentration provides a method of data analysis in ultrasonic evaluation of material properties. Numerical results for effective moduli, velocity and/or attenuation as functions of concentration of spherical inclusion material, or porosity, are graphically displayed.     

Finite-element analysis of hyperelastic thin shells with large strains

The objective of this contribution is the development of theoretical and numerical models applicable to large strain analysis of hyperelastic shells confining particular attention to incompressible materials. The theoretical model is developed on the basis of a quadratic displacement approximation in thickness coordinate by neglecting transverse shear strains. In the case of incompressible materials this leads to a three-parametric theory governed solely by mid-surface displacements. The material incompressiblity is expressed by two equivalent equation sets considered at the element level as subsidiary conditions. For the simulation of nonlinear material behaviour the Mooney-Rivlin model is adopted including neo-Hookean materials as a special case. After transformation of nonlinear relations into incremental formulation doubly curved triangular and quadrilateral elements are developed via the displacement method. Finally, examples are given to demonstrate the ability of these models in dealing with large strain as well as finite rotation shell problems.The present study is supported by a research grant of the German National Science Foundation (DFG) under Ba 969/3-1.dedicated to Prof. Dr. Dr. Erwin Stein for his 65th birthday anniversary     

Effective moduli for thermopiezoelectric materials with microcracks

Dilute, Self-Consistent (SC), Mori-Tanaka (MT) and differential micromechanics methods are developed for microcrack- weakened thermopiezoelectric solids. These methods are capable of determination of effective properties such as the conductivity, electroelastic moduli, thermal expansion and pyroelectric coefficients. The above material constants affected by the microcracks are derived by way of Stroh's formulation and some recently developed explicit solutions of a crack in an infinite piezoelectric solid subjected to remote thermal, electrical and elastic loads. In common with the corresponding uncoupled thermal, electric and elastical behavior, the dilute and Mori-Tanaka techniques give explicit estimates of the effective thermoelectroelastic moduli. The SC and differential schemes, however, give only implicit estimates, with nonlinear algebraic matrix equations, of the effective thermoelectroelastic moduli. Numerical results are given for a particular cracked material to examine the behavior of each of the four micromechanics models. This revised version was published online in August 2006 with corrections to the Cover Date.     

Propagation of low-frequency elastic waves in double-porositymedia containing viscoelastic component in the pores

A self-consistent scheme named the effective field method (EFM) is applied for the calculation of the velocities and quality factors of elastic waves propagating in double-porosity media. A double-porosity medium is considered to be a heterogeneous material composed of a matrix with primary pores and inclusions that are represent by flat (crack-like) secondary pores. The prediction of the effective viscoelastic moduli consists of two steps. First, we calculate the effective viscoelastic properties of the matrix with the primary small-scale pores (matrix homogenization). Then, the porous matrix is treated as a homogeneous isotropic host where the large-scale secondary pores are embedded. Spatial distribution of inclusions in the medium is taken into account via a special two-point correlation function. The results of the calculation of the viscoelastic properties of double-porosity media containing isotropic fields of crack-like inclusions and double-porosity media with some non-isotropic spatial distributions of crack-like inclusions are presented.     

A consistent u‐p formulation for porous media with hysteresis

This paper presents a continuum formulation based on the theory of porous media for the mechanics of liquid unsaturated porous media. The hysteresis of the liquid retention model is carefully modelled, including the derivation of the corresponding consistent tangent moduli. The quadratic convergence of Newton's method for solving the highly nonlinear system with an implicit finite element code is demonstrated. A u‐p formulation is proposed where the time discretisation is carried out prior to the space discretisation. In this way, the derivation of all consistent moduli is fairly straightforward. Time integration is approximated with the Theta and Newmark's methods, and hence the fully coupled nonlinear dynamics of porous media is considered. It is shown that the liquid retention model requires also the consistent second‐order derivative for quadratic convergence. Some predictive simulations are presented illustrating the capabilities of the formulation, in particular to the modelling of complex porous media behaviour. Copyright © 2014 John Wiley & Sons, Ltd.     

Large deformation analysis of rubber based on a reproducing kernel particle method

 A nonlinear formulation of the Reproducing Kernel Particle Method (RKPM) is presented for the large deformation analysis of rubber materials which are considered to be hyperelastic and nearly incompressible. In this approach, the global nodal shape functions derived on␣the basis of RKPM are employed in the Galerkin approximation of the variational equation to formulate the discrete equations of a boundary-value hyperelasticity problem. Existence of a solution in RKPM discretized hyperelasticity problem is discussed. A Lagrange multiplier method and a direct transformation method are presented to impose essential boundary conditions. The characteristics of material and spatial kernel functions are discussed. In the present work, the use of a material kernel function assures reproducing kernel stability under large deformation. Several of numerical examples are presented to study the characteristics of RKPM shape functions and to demonstrate the effectiveness of this method in large deformation analysis. Since the current approach employs global shape functions, the method demonstrates a superior performance to the conventional finite element methods in dealing with large material distortions.     

A formulation for an unsaturated porous medium undergoing large inelastic strains

 This paper presents a formulation for a saturated and partially saturated porous medium undergoing large elastic or elastoplastic strains. The porous material is treated as a multiphase continuum with the pores of the solid skeleton filled by water and air, this last one at constant pressure. This pressure may either be the atmospheric pressure or the cavitation pressure. The governing equations at macroscopic level are derived in a spatial and a material setting. Solid grains and water are assumed to be incompressible at the microscopic level. The isotropic elastoplastic behaviour of the solid skeleton is described by the multiplicative decomposition of the deformation gradient into an elastic and a plastic part. The effective stress state is limited by the Drucker-Prager yield surface, for which a particular “apex formulation” is advocated. The water is assumed to obey Darcy's law. Numerical examples of strain localisation of dense and loose sand conclude the paper. Received 15 March 2001     

Simulation of penetration into porous geologic media

We present a computational study on the penetration of steel projectiles into porous geologic materials. The purpose of the study is to extend the range of applicability of a recently developed constitutive model to simulations involving projectile penetration into geologic media. The constitutive model is nonlinear, thermodynamically consistent, and properly invariant under superposed rigid body motions. The equations are valid for large deformations and they are hyperelastic in the sense that the stress tensor is related to a derivative of the Helmholtz free energy. The model uses the mathematical structure of plasticity theory to capture the basic features of the mechanical response of geological materials including the effects of bulking, yielding, damage, porous compaction and loading rate on the material response. The new constitutive model has been successfully used to simulate static laboratory tests under a wide range of triaxial loading conditions, and dynamic spherical wave propagation tests in both dry and saturated geologic media.     

Two Effective Thermal Conductivity Models for Porous Media with Hollow Spherical Agglomerates

Based on the microstructure features of xonotlite-type micro-pore calcium silicate, two unit cell models, the point-contact hollow spherical model and the surface-contact hollow cubic model, are developed. As one of several excellent insulation materials, xonotlite is represented as porous media with hollow spherical agglomerates. By one-dimensional heat conduction analysis using theunit cell, the effective thermal conductivity of xonotlite is determined. The results show that both of the models are in agreement with experimental data. The algebraic expressions based on the unit cell models can be used to calculate the effective thermal conductivity of porous media that have similar structure features as xonotlite.Paper presented at the Seventh Asian Thermophysical Properties Conference, August 23–28, 2004, Hefei and Huangshan, Anhui, P. R. China     

A three‐scale compressible microsphere model for hyperelastic materials

This paper presents a seminumerical homogenization framework for porous hyperelastic materials that is open for any hyperelastic microresponse. The conventional analytical homogenization schemes do apply to a limited number of elementary hyperelastic constitutive models. Within this context, we propose a general numerical scheme based on the homogenization of a spherical cavity in an incompressible unit hyperelastic solid sphere, which is denoted as the mesoscopic representative volume element (mRVE). The approach is applicable to any hyperelastic micromechanical response. The deformation field in the sphere is approximated via nonaffine kinematics proposed by Hou and Abeyaratne (JMPS 40:571‐592,1992). Symmetric displacement boundary conditions driven by the principal stretches of the deformation gradient are applied on the outer boundary of the mRVE. The macroscopic quantities, eg, stress and moduli expressions, are obtained by analytically derived pointwise geometric transformations. The macroscopic expressions are then computed numerically through quadrature rules applied in the radial and surface directions of the sphere. A three‐scale compressible microsphere model is derived from the developed seminumerical homogenization framework where the micro‐meso transition is based on the nonaffine microsphere model at every point of the mRVE. The numerical scheme developed for the derivation of macroscopic homogenized stresses and moduli terms as well as the modeling capability of the three‐scale microsphere model is investigated through representative boundary value problems.     

Solving hyperelastic material problems by asymptotic numerical method

This paper presents a numerical algorithm based on a perturbation technique named asymptotic numerical method (ANM) to solve nonlinear problems with hyperelastic constitutive behaviors. The main advantages of this technique compared to Newton–Raphson are: (a) a large reduction of the number of tangent matrix decompositions; (b) in presence of instabilities or limit points no special treatment such as arc-length algorithms is necessary. The ANM uses high order series approximation with auto-adaptive step length and without need of any iteration. Introduction of this expansion into the set of nonlinear equations results into a sequence of linear problems with the same linear operator. The present work aims at providing algorithms for applying the ANM to the special case of compressible and incompressible hyperelastic materials. The efficiency and accuracy of the method are examined by comparing this algorithm with Newton–Raphson method for problems involving hyperelastic structures with large strains and instabilities.     

T-stress in orthotropic functionally graded materials: Lekhnitskii and Stroh formalisms

A new interaction integral formulation is developed for evaluating the elastic T-stress for mixed-mode crack problems with arbitrarily oriented straight or curved cracks in orthotropic nonhomogeneous materials. The development includes both the Lekhnitskii and Stroh formalisms. The former is physical and relatively simple, and the latter is mathematically elegant. The gradation of orthotropic material properties is integrated into the element stiffness matrix using a “generalized isoparametric formulation” and (special) graded elements. The specific types of material gradation considered include exponential and hyperbolic-tangent functions, but micromechanics models can also be considered within the scope of the present formulation. This paper investigates several fracture problems to validate the proposed method and also provides numerical solutions, which can be used as benchmark results (e.g. investigation of fracture specimens). The accuracy of results is verified by comparison with analytical solutions.     

复合材料粘弹性本构关系与热应力松弛规律研究Ⅰ:理论分析

基于均匀化理论研究了复合材料粘弹性分析的多尺度方法, 以及复合材料等效热应力松弛规律。引入了等效粘弹性热应力系数张量和等效时变热膨胀系数的概念, 建立了含温度变化的复合材料热粘弹性本构关系, 并给出了基于均匀化理论的复合材料粘弹性松弛模量、等效热应力松弛系数和等效时变热膨胀系数的预测方法。对特殊复合材料的粘弹性性质进行了分析, 结果表明: (1) 复合材料的粘弹性本构关系具有与常规材料的本构关系类似的形式, 但一般复合材料的热应力松弛规律与常规材料不同, 其热膨胀不能瞬时完成, 而具有明显的时变性质;(2) 空心材料的热膨胀具有瞬时性质, 其等效时变热膨胀系数与基体材料的热膨胀系数相同, 其热应力松弛规律与基体材料的松弛规律相同;(3) 当各组分材料的松弛模量的各分量可分解成不同的系数与相同的时间函数的乘积时, 复合材料的等效时变热膨胀系数与时间无关, 其松弛规律与常规材料的松弛规律完全相同。     

The effect of crack face contact on the anisotropic effective moduli of microcrack damaged media

The generalized self-consistent method (GSCM) in conjunction with a computational finite element method is used to calculate the anisotropic effective moduli of a medium containing damage consisting of microcracks with an arbitrary degree of alignment. Since cracks respond differently under different external loads, the moduli of the medium subjected to tension, compression and an initially stress-free state are evaluated and shown to be significantly different, which will further affect the wave speed inside the damaged media. There are four independent material moduli for a 2-D plane stress orthotropic medium in tension or compression, and seven independent material moduli for a 2-D plane stress orthotropic cracked medium, which is initially stress free. When friction exists, it further changes the effective moduli. Numerical methods are used to take into account crack face contact and friction. The wave slowness profiles for microcrack damaged media are plotted using the predicted effective material moduli.     

Elastoplasticity theory for porous materials

A continuum theory is derived for the modeling of elastoplastic work-hardening porous materials. The theory provides a set of constitutive relations which, using the properties of the inelastic matrix, determines by an incremental procedure the overall response of the porous solid to various types of loading. In the elastic region, effective elastic moduli of the porous material are obtained. Comparison with theoretical and experimental results are given.     

On the Relationship Between Microstructure of the Cortical Bone and its Overall Elastic Properties

The relationship between microstructure of the cortical bone and its effective elastic properties is discussed. We utilize results of Kachanov et al (1994) on materials with cracks/pores of diverse shapes. Bone's microstructure is modeled using available micrographs. The calculated anisotropic elastic constants for porous cortical bone are compared with available experimental data. For Young's moduli and shear moduli the agreement is good, whereas Poisson's ratios differ significantly. Possible reasons for this difference are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Parallel 3-d simulations for porous media models in soil mechanics

 Numerical simulations in 3-d for porous media models in soil mechanics are a difficult task for the engineering modelling as well as for the numerical realization. Here, we present a general numerical scheme for the simulation of two-phase models in combination with an abstract material model via the stress response with a specialized parallel saddle point solver. Therefore, we give a brief introduction into the theoretical background of the Theory of Porous Media and constitute a two-phase model consisting of a porous solid skeleton saturated by a viscous pore-fluid. The material behaviour of the skeleton is assumed to be elasto-viscoplastic. The governing equations are transfered to a weak formulation suitable for the application of the finite element method. Introducing an abstract formulation in terms of the stress response, we define a clear interface between the assembling process and the parallel solver modules. We demonstrate the efficiency of this approach by challenging numerical experiments realized on the Linux Cluster in Chemnitz. Received 15 February 2002 / Accepted 12 April 2002     

硬化水泥浆体弹性模量细观力学模型

应用复合材料力学理论和有孔介质力学(Poromechanics)理论建立了一个描述硬化硅酸盐水泥浆体弹性模量的细观力学模型, 将硬化水泥浆体从不同尺度上划分为4个层次, 即C-S-H凝胶、 水泥水化产物、 水泥浆体骨架和水泥浆体, 分别应用不同的细观力学模型予以描述: 将C-S-H视为饱和的有孔介质; 应用Mori-Tanaka模型描述水泥水化产物的弹性性质; 应用三相模型(Three-phase model)模拟水泥浆体骨架的有效弹性模量; 最后, 再次应用Mori-Tanaka模型和有孔介质理论, 计算水泥浆体的排水和不排水弹性模量(Drained and undrained elastic moduli)。该模型所需要的参数为水泥浆体各个组成部分的自身弹性性质, 使用方便。通过预测文献中的实测结果, 证明了该模型的有效性。      

The Analytical and Numerical Study on the Nanoindentation of Nonlinear Elastic Materials

In nanoindentation testing of materials, the analytical/numerical models to connect the indentation load, indentation depth and material properties are crucial for the extraction of mechanical properties. This paper studied the methods of extracting the mechanical properties of nonlinear elastic materials and built general relationships of the indentation load and depth of hyperelastic materials combined with the dimensional analysis and finite element method (FEM). Compared with the elastic contact models and other nonlinear elastic contact models, the proposed models can extract the mechanical properties of nonlinear elastic materials under large deformation simply and effectively.