Behavior Of Heterogeneous Materials Research Articles

Comparative analysis of two- and three-dimensional simulations of Al/Al 2O 3 behavior on the meso-scale level

A problem of the applicability of two-dimensional models in simulations of heterogeneous material behavior on the meso-scale level has been studied, using a metal matrix composite (MMC) structure as an example. In order to incorporate explicitly the MMC-structure in 3D calculations, a special algorithm has been applied to design an artificial structure containing an Al(6 0 6 1)-matrix and Al 2O 3-inclusions. Two- and three-dimensional simulations of the mechanical behavior of the MMC-structure under tension have been performed using both finite-element and finite-difference methods. Qualitative and quantitative analysis of different components of stress and strain tensors has been provided on the basis of mesomechanical concepts. Results of three-dimensional calculations have been compared with those obtained in 2D formulations of plane strain and plane stress conditions. Based on the comparative analysis, some conclusions regarding the applicability of 2D approximation when considering deformation behavior on the meso-scale level have been drawn.

Effects of microstructural heterogeneity on the spallation behavior of materials

It is of utmost importance to understand the spallation behaviour of heterogeneous materials. In this paper, a driven nonlinear threshold model with stress fluctuation is presented to study the effects of microstructural heterogeneity on continuum damage evolution. The spallation behavior of heterogeneity material is analyzed with this model. The heterogeniety of mesoscopic units is characterized in terms of Weibull modulus m of strength distibution and stress fluctuation parameter k. At high stress, the maximum damage increases with m; while at low stress, the maximum damage decreases. In addition, for low stress, severe stress fluctuation causes higher damage; while for high stress, causes lower damage.

Tensile fracture behavior of heterogeneous materials based on fractal geometry

Many of heterogeneous structural materials, like concrete, have different behavior under tensile stresses in comparison to their behavior under compressive stresses. The aim of this paper is to interpret behavior of such materials subjected to tensile stresses, by using newly introduced concept of fractal geometry. In the first part of this paper, tensile behavior of granular composites has been studied by using fractal geometry. It is shown that the fractality of the cross section in this kind of composites can be used to interpret the size effect on tensile strength. In fact, this work is a modification with innovations on the previous studies on fractal based size effect. This hypothesis that the fracture surfaces of quasi-brittle materials are fractals has been verified by several investigations. Accordingly, in the other part of this paper, softening process in heterogeneous materials is studied. Resulting from presented approach, a new softening curve for quasi-brittle materials is proposed. This new softening curve is denominated “Quasi-fractal softening curve” and is consisted of two parts, a linear portion in beginning part and an exponential portion in rest of the curve. This makes it very compatible to the pre-existing softening curves.

Three_Dimensional Simulation of Fracture Behavior of Elastic-Brittle Material with Initial Crack Pattern

Numerical simulation of mechanical behavior of heterogeneous materials under loading with an explicit consideration for their 3D mesoscale structure is given. A 3-D problem in the dynamic formulation is solved using the finite-difference method. As an example, a coal specimen under tension and compression is examined, with an initial crack pattern taken into account. It is been shown that the fracture criterion we used in the calculations provides adequate description of fracture behavior under tension and compression on both macro and mesoscale level.

Three-Dimensional (3D) Microstructure Visualization and Finite Element Modeling of the Mechanical Behavior of Heterogeneous Materials

A serial sectioning process was used to develop a three-dimensional (3D) representation of the microstructure of a SiC particle reinforced Al composite, for visualization and finite-element modeling (FEM). The Young’s modulus and stress-strain behavior of the composite predicted by the 3D model of microstructure correlated very well with experimental results.

Quadrilateral isoparametric finite elements for plane elastic Cosserat bodies

4-node, 8-node and 8(4)-node quadrilateral plane isoparametric elements are used for the solution of boundary value problems in linear isotropic Cosserat elasticity. The patch test is applied to validate the finite elements. Engineering problems of stress concentration around a circular hole in plane strain condition and mechanical behaviors of heterogeneous materials with rigid inclusions and pores are computed to test the accuracy and capability of these three types of finite elements.

On evolutional and catastrophic regimes of meso-macro energy exchange in dynamically loaded media

Study of the role of velocity fluctuations in processes of energy exchange between mesoscopic and macroscopic scale levels represents an important stage in the development of our knowledge about the shockwave behavior of heterogeneous materials. In the case of dynamic deformation, the concept of mesoparticle has much wider meaning as applied to quasistatic processes, in which mesoparticles are specific structural defects such as dislocation groups, vortex structures, shear bends, and other structural formations of scale 0.1‐10 μ m [1]; they represent field structures, the only distinct feature of which is the presence of a velocitycorrelated group of medium points. This means that a mesoparticle is a space formation, the internal points of which have identical or close velocities. The lifetime of these structures is determined by the duration of the dynamic deformation process. Each mesoparticle, as a separate structural formation, has its own overall velocity. In a heterogeneous medium, a dynamically deformed structure is characterized by velocity scattering. Therefore, mesoparticle velocity fluctuations can remove a considerable part of momentum and energy transferred to the medium. According to recent experiments [2], the generally accepted opinion that 90% of plastic work is immediately transformed into heat is erroneous. It is found that, in the microsecond range of dynamic loading, only 30‐ 35% of plastic work is transformed into heat, while the rest of the work is spent on the formation of the mesostructure. The main mechanism of structure formation is large-scale velocity fluctuations, the quantitative characteristic of which is the standard deviation of the mesoparticle velocities. Analysis of the experimental data on shock loading of materials reveals two regimes of energy exchange between mesoscales and macroscales of dynamic deformation—smooth or evolutional and catastrophic regimes. The latter regime implies that the mean particle velocity decreases suddenly at a certain strain rate, while the standard deviation of velocities increases. This statement was verified by impact tests of two sets of 30XH4M steel targets subjected to different thermal treatments. As an example, Fig. 1 shows the time profiles of the mean velocity and standard deviation of velocities as obtained upon plane impact loading of two targets with velocities of 324 and 320 m/s. The profiles were recorded using a two-channel velocity interferometer [3]. Although the impact velocities are close to each other in these experiments, the time profiles of the mean velocity and standard deviation of velocities are significantly different. In the first case, the mean velocity increases gradually up to 310 m/s, while in the second case, the break of the mean velocity occurs at a velocity of 160 m/s (point B in Fig. 1b). The behavior of the standard deviation of velocities is opposite: its increase rate is much higher in the second case. The second case corresponds to the catastrophic regime of energy exchange between the mesoscale and macroscale. To deduce a criterion for the transition from the evolutional to catastrophic regime of energy exchange between the mesoscopic and macroscopic scales of dynamic deformation, we consider the one-dimensional propagation of an elastoplastic wave in a medium characterized by mesoscopic velocity fluctuations. The equations of momentum and mass conservation have the form

Modeling the Behavior of Heterogeneous Materials with Non Linear Couplings using “Translated Fields”

AbstractThe determination of the behavior of heterogeneous materials with complex physical and mechanical couplings constitutes a challenge in the design of new materials and the modeling of their effective properties. In real inhomogeneous materials, the simultaneous presence of elastic mechanisms and non linear inelastic ones (viscoplastic, magnetic, ferroelectric, shape memory effect etc.) leads to a complex non linear coupling between the mechanical fields which is tricky to represent in a simple and efficient way. Hence, for many situations the effective global behavior does not follow the same structure than the local constitutive one. Regarding space-time couplings for instance, a heterogeneous material composed of phases described by Maxwell elements can not be considered as a Maxwellian solid at the macro scale.In this paper, we introduce a new micro-macro approach based on translated fields in its generalized form to be applied to different coupled phenomena. The local total strain (rate) is composed additively of an elastic strain (rate) and an inelastic one which is no more limited to be “stress free” as considered originally by Kröner. An extended (non conventional) self-consistent model is then proposed starting from the integral equation for a translated strain (rate) field and using the projection operators algebra introduced by Kunin. The chosen translated field is the compatible inelastic strain (rate) of the fictitious inelastic heterogeneous medium submitted to a uniform unknown boundary condition. The self-consistency condition amounts to define analytically these boundary conditions so that a relative simple and compact strain (rate) concentration equation is obtained.In order to illustrate the method, the case of a non linear elastic-viscoplastic coupling is developed and applied to different classes of composites and polycrystals.

On micro–macro interface conditions for micro scale based FEM for inelastic behavior of heterogeneous materials

In this paper we introduce a two scale computational strategy for modeling the inelastic behavior of heterogeneous materials. We apply the FEM not only on the structural scale but also on the lower, so called micro scale, where it replaces the standard procedure, in which we define the material behavior through the constitutive relations. The special care is given to handling the interface condition between micro- and macro scales. The proposed approach is based on the localized Lagrange multiplier method and can lead to the realization of either displacement based or force based formulation. Numerical tests were done on porous and composite materials. We investigate the influence of the boundary conditions, micro scale representation window size and phase contrast on the heterogeneous material response of a single FE as well as of the whole structure.

Study of the mechanical behaviour of a macroscopic glass–polyester composite by ESPI method and numerical simulations

This paper presents results of an experimental and numerical investigation concerning the mechanical behaviour of macromodels for fibre-reinforced composites. Cubic specimens containing an increasing number of glass fibres in a polyester matrix have been produced; the specimens have a 30 mm long side and the fibres a 2 mm long diameter. The displacement field on the surface orthogonal to the fibre direction has been experimentally determined by means of the non-destructive electronic speckle pattern interferometry (ESPI) method for specimens subject to transversal compression. The experimental results have been compared with those of numerical finite element (FE) simulations. The quality of the experimental results and their agreement with the numerical analyses show that the use of macromodels combined with ESPI method could be a valuable guide for a deeper understanding of the mechanical behaviour of heterogeneous materials and for the formulation of models.

Machining Simulation of Ductile Iron and Its Constituents, Part 1: Estimation of Material Model Parameters and Their Validation

A microstructure-level simulation model was recently developed to characterize machining behavior of heterogeneous materials. During machining of heterogeneous materials such as cast iron, the material around the machining-affected zone undergoes reverse loading, which manifests itself in permanent material softening. In addition, cracks are formed below and ahead of the tool. To accurately simulate machining of heterogeneous materials the microstructure-level model has to reproduce the effect of material softening on reverse loading (MSRL effect) and material damage. This paper describes procedures used to calculate the material behavior parameters for the aforementioned phenomena. To calculate the parameters associated with the MSRL effect, uniaxial reverse loading experiments and simulations were conducted using individual constituents of ductile iron. The material model was validated with reverse loading experiments of ductile iron specimens. To determine the parameters associated with fracture of each constituent, experiments and simulation of notched specimens are performed. During the validation stage, response of simulated ductile iron was in good agreement with the experimental data.

The modified affine homogenisation scheme for nonlinear composites: Comparison with variational bounds and estimates

The effective behaviour of heterogeneous materials exhibiting a random microstructure, such as composites and polycrystals, can be efficiently estimated or bounded with homogenisation methods. For materials with a nonlinear behaviour, use is classically made of a Linear Comparison Composite (LCC) presenting generally the same microstructure as the nonlinear material, but for which the local (linearised) compliance is uniform per phase. Homogenisation schemes, developed for linear behaviour, can thus be applied to the LCC. The way the LCC is defined has a great influence on the estimated effective behaviour and local mechanical fields, in particular for large nonlinearities and contrasts between the phases. For few years, it has become clear that not only the phase averages of strain and stress must be used for defining the LCC, but also their intraphase fluctuations. In that framework, the present authors have recently proposed a modified affine procedure [1] which presents the advantage over other formulations to be applicable to any type of behaviour, including nonlinear viscoelasticity. In this paper, we present a comparison, for simpler behaviour (nonlinear elasticity), between the proposed approach and several formulations previously proposed in the literature.

Continuum Micromechanics: Survey

The foundations of classical homogenization techniques, which aim at predicting the overall behavior of heterogeneous materials from that of their constituents, are reviewed. After introductory definitions and a methodological preamble, attention is focused on linear elasticity, for which the basic principles of estimating and bounding the overall properties are introduced and illustrated. In this context, special recourse is made for that to the solution of the inclusion and inhomogeneity problems as reported by Eshelby in 1957. Approaches proposed recently to account in a better way for the structural morphology of the considered materials are briefly mentioned. The case of linear elasticity with eigenstrains is then discussed: several applications, including heterogeneous thermoelasticity, can be investigated within this framework. Finally, outlines of nonlinear micromechanics are briefly reported from a historical point of view: from rate-independent elastoplasticity to nonlinear elasticity and viscoplasticity, examples of a fruitful interaction between the search for new estimates and the derivation of rigorous bounds are given and the crucial question of the description of intraphase heterogeneity is emphasized. Viscoelastic coupling and rate-dependent effects are briefly discussed in conclusion.

A new class of micro–macro models for elastic–viscoplastic heterogeneous materials

Abstract The determination of the effective behavior of heterogeneous materials from the properties of the components and the microstructure constitutes a major task in the design of new materials and the modeling of their mechanical behavior. In real heterogeneous materials, the simultaneous presence of instantaneous mechanisms (elasticity) and time dependent ones (non-linear viscoplasticity) leads to a complex space–time coupling between the mechanical fields, difficult to represent in a simple and efficient way. In this work, a new self-consistent model is proposed, starting from the integral equation for a translated strain rate field. The chosen translated field is the (compatible) viscoplastic strain rate of the (fictitious) viscoplastic heterogeneous medium submitted to a uniform (unknown) boundary condition. The self-consistency condition allows to define these boundary conditions so that a relative simple and compact strain rate concentration equation is obtained. This equation is explained in terms of interactions between an inclusion and a matrix, which lead to interesting conclusions. The model is first applied to the case of two-phase composites with isotropic, linear and incompressible viscoelastic properties. In that case, an exact self-consistent solution using the Laplace–Carson transform is available. The agreement between both approaches appears quite good. Results for elastic–viscoplastic BCC polycrystals are also presented and compared with results obtained from Kroner–Weng's and Paquin et al. (Arch. Appl. Mech. 69 (1999) 14)'s model.

Micromechanical modeling of the elastic–viscoplastic behavior of polycrystalline steels

A self-consistent model developed to describe the elastic–viscoplastic behavior of heterogeneous materials is applied to low carbon steels to simulate tensile tests at various strain rates in the low temperature range. The choice of crystalline laws implemented in the model is discussed through the viscoplastic flow rule and several strain-hardening laws. Comparisons between three work-hardening models show that the account of dislocation annihilation improves the results on simulations at large strains. The evolution of the Lankford coefficients and texture development are also successfully simulated. Some microstructural aspects of deformation such as the stored energy and the evolution of the flow rates are discussed. By including the dislocation density on each slip system as internal variable, intragranular heterogeneities are underscored.

A fractional calculus approach to the description of stress and strain localization in fractal media

Evidence of fractal patterns in materials with disordered microstructure under tensile loads is undeniable. Unfortunately fractal functions cannot be solution of classical differential equations. Hence a new calculus must be developed to handle fractal processes. In this paper, we use the local fractional calculus operators recently introduced by K.M. Kolwankar [Studies of fractal structures and processes using methods of fractional calculus. PhD thesis, University of Pune, India, 1998]. Through these new mathematical tools we get the static and kinematic equations that model the uniaxial tensile behavior of heterogeneous materials. The fractional operators respect the non-integer (fractal) physical dimensions of the quantities involved in the governing equations, while the virtual work principle highlights the static-kinematic duality among them. The solutions obtained from the model are fractal and yield to scaling power laws characteristic of the nominal quantities, i.e., they reproduce the size effects due to stress and strain localization.

Simulated Micromechanical Models Using Artificial Neural Networks

A new method, termed simulated micromechanical models using artificial neural networks (MMANN), is proposed to generate micromechanical material models for nonlinear and damage behavior of heterogeneous materials. Artificial neural networks (ANN) are trained with results from detailed nonlinear finite-element (FE) analyses of a repeating unit cell (UC), with and without induced damage, e.g., voids or cracks between the fiber and matrix phases. The FE simulations are used to form the effective stress-strain response for a unit cell with different geometry and damage parameters. The FE analyses are performed for a relatively small number of applied strain paths and damage parameters. It is shown that MMANN material models of this type exhibit many interesting features, including different tension and compression response, that are usually difficult to model by conventional micromechanical approaches. MMANN material models can be easily applied in a displacement-based FE for nonlinear analysis of composite structures. Application examples are shown where micromodels are generated to represent the homogenized nonlinear multiaxial response of a unidirectional composite with and without damage. In the case of analysis with damage growth, thermodynamics with irreversible processes (TIP) is used to derive the response of an equivalent homogenized damage medium with evolution equations for damage. The proposed damage formulation incorporates the generalizations generated by the MMANN method for stresses and other possible responses from analysis results of unit cells with fixed levels of damage.

Cosserat modelling of size effects in the mechanical behaviour of polycrystals and multi-phase materials

Classical homogenization techniques are not designed to predict the effect of the size of the constituents on the effective mechanical behaviour of heterogeneous materials. They usually take the volume fraction and, in some cases, the morphology of phase distribution into account. This shortcoming is related to the fact that, in crystals, the elastoviscoplastic behaviour of each constituent within the aggregate may be different from that observed on the constituent alone (say the single crystal). Cosserat single crystal plasticity is used in this work to describe the influence of grain size on the effective hardening behaviour of polycrystals. For that purpose, three-dimensional finite element calculations of periodic Cosserat multi-crystalline aggregates of different grain sizes are provided. The polycrystal is regarded as a heterogeneous Cosserat medium and specific techniques for the estimation of the effective properties are presented. The approach is then applied to the case of two-phase single crystal materials for which the behaviour of one phase as a matrix turns out to be much harder than the isolated phase.

Overall behaviour of heterogeneous elastoviscoplastic materials: effect of microstructural modelling

Homogenisation methods provide an efficient way to model the mechanical behaviour of heterogeneous materials. In this paper, a homogenisation procedure is adopted that allows to determine apparent properties for Perzyna's elastoviscoplastic constitutive law for arbitrary microstructures. Numerical simulations on a representative volume element (RVE) are performed, from which the volume averaged state variables are acquired, necessary to establish the constitutive equations for the equivalent homogeneous medium. The applicability of mixed and periodic boundary conditions has been assessed. In addition, the difference between uniform and irregular distributions of the microstructural constituents is discussed. To substantiate our findings, a comparison is made between the global response of a heterogeneous and the corresponding homogenised structure.

An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals

Variational approaches for nonlinear elasticity show that Hill’s incremental formulation for the prediction of the overall behaviour of heterogeneous materials yields estimates which are too stiff and may even violate rigorous bounds. This paper aims at proposing an alternative ‘affine’ formulation, based on a linear thermoelastic comparison medium, which could yield softer estimates. It is first described for nonlinear elasticity and specified by making use of Hashin–Shtrikman estimates for the linear comparison composite; the associated affine self-consistent predictions are satisfactorily compared with incremental and tangent ones for power-law creeping polycrystals. Comparison is then made with the second-order procedure (Ponte Castañeda, P., 1996. Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids, 44 (6), 827–862) and some limitations of the affine method are pointed out; explicit comparisons between different procedures are performed for isotropic, two-phase materials. Finally, the affine formulation is extended to history-dependent behaviours; application to the self-consistent modelling of the elastoplastic behaviour of polycrystals shows that it offers an improved alternative to Hill’s incremental formulation.