The effect of interfacial properties on the cohesion of highly filled composite materials

It has long been recognized that the cohesion of composite materials, in low confinement, is strongly affected by the properties of the interfacial transition zone (ITZ) between inclusions and matrix. While the effect of the ITZ on the elasticity properties of composites has been studied by many authors in the context of linear homogenization methods, the upscaling of the cohesion strength of highly filled composite materials has not been addressed. This is the focus of the non-linear homogenization procedure developed in this paper, which is based on the separation of the heterogeneous material system in phases of constant strength properties, a non-linear elastic representation of the limit stress state in each phase, and the definition of appropriate effective strain quantities that capture the morphological features of the microstructure. Applied to a three phase composite model composed of rigid inclusion, interface zone and matrix, the model provides a quantitative means of studying the effect of the interface cohesion and the interface volume fraction on the composite cohesion. In particular, we identify a critical interface-to-matrix cohesion ratio, below which the composite cohesion is smaller than the one of the matrix. Furthermore, the model lends itself readily to the study of the degradation of the interfacial properties in composite materials. This is shown for non-degraded and chemically softened cement-based materials, for which we provide conclusive evidence (1) that the interface strength properties of mortar are far more affected by chemical degradation than the one of the cement paste matrix; and (2) that chemical degradation does affect the mechanical strength performance of the cement paste not only through a change of volume proportions (i.e. increase of porosity), but as well through a pure chemical softening of the solid’s cohesion.     

Size effects on stress concentration induced by a prolate ellipsoidal particle and void nucleation mechanism

There generally exist two void nucleation mechanisms in materials, i.e. the breakage of hard second-phase particle and the separation of particle–matrix interface. The role of particle shape in governing the void nucleation mechanism has already been investigated carefully in the literatures. In this study, the coupled effects of particle size and shape on the void nucleation mechanisms, which have not yet been carefully addressed, have been paid to special attention. To this end, a wide range of particle aspect ratios (but limited to the prolate spheroidal particle) is considered to reflect the shape effect; and the size effect is captured by the Fleck–Hutchinson phenomenological strain plasticity constitutive theory (Advance in Applied Mechanics, vol. 33, Academic Press, New York, 1997, p. 295). Detailed theoretical analyses and computations on an infinite block containing an isolated elastic prolate spheroidal particle are carried out to light the features of stress concentrations and their distributions at the matrix–particle interface and within the particle. Some results different from the scale-independent case are obtained as: (1) the maximum stress concentration factor (SCF) at the particle–matrix interface is dramatically increased by the size effect especially for the slender particle. This is likely to trigger the void nucleation at the matrix–particle interface by cleavage or atomic separation. (2) At a given overall effective strain, the particle size effect significantly elevates the stress level at the matrix–particle interface. This means that the size effect is likely to advance the interface separation at a smaller overall strain. (3) For scale-independent cases, the elongated particle fracture usually takes place before the interface debonding occurs. For scale-dependent cases, although the SCF within the particle is also accentuated by the particle size effect, the SCF at the interface rises at a much faster rate. It indicates that the probability of void nucleation by the interface separation would increase.     

Bifurcation phenomena in the rigid inclusion power law matrix composite sphere

Bifurcation of interface separation related to cavity nucleation is analyzed for a radially loaded composite sphere consisting of a rigid inclusion separated from a power law matrix by a uniform, non-linear cohesive zone. Equations for the spherically symmetric and non-symmetric problems are obtained from a hyperelastic finite strain theory by a limiting process that preserves non-linear matrix and interface response at infinitesimal strain. A complete solution to the symmetric problem is presented including bifurcation load, stresses, and evolution of elasto-plastic boundary and interface separation. An analysis of non-symmetric bifurcation, under symmetric conditions of geometry and loading, yields the bifurcation load and first non-symmetric mode shape associated with rigid inclusion displacement. An energy analysis is carried out for both symmetric and non-symmetric problems in order to assess stability of spherically symmetric states to spherically symmetric and non-symmetric “rigid body mode” perturbations.Results are provided for an interface force law that captures interface failure in normal mode and linear response in shear mode. For the symmetric problem, (i) there are threshold parameter values above which bifurcation will generally not occur, (ii) threshold values below which there do not exist equilibria in the post bifurcation regime, (iii) bifurcation occurs after attainment of the maximum interface strength. For the non-symmetric problem, (i) bifurcation always occurs, although it can be delayed by interfacial shear, (ii) for the smooth interface, non-symmetric bifurcation occurs after attainment of the maximum interface strength and always precedes symmetric bifurcation.     

Plasticity and ductile fracture of IF steels: experiments and micromechanical modeling

If one aims at the simulation of plasticity and failure of multiphase materials, the choice of an appropriate material law is of major importance. Plasticity models for porous metals contain, in addition to the yield surface and the flow potential, also functions describing the void nucleation, dependent on some macroscopically observable quantities, and the growth of these voids. In this paper, a micromechanically based method to develop a void nucleation function for porous plasticity models is proposed which is valid for all possible microstructures as long as the amount of second phase particles is low (i.e. the particles do not interact with respect to the stress and strain fields), and as long as the particles are large enough (above 0.1 μm) justifying a continuum mechanical approach. The method described consists of two stages: In the first stage, the microstructure is investigated via a finite element model. The FE model implicitly contains the effects of the shape of the precipitates, of the material parameters of both the matrix and the precipitates, of the void nucleation hypothesis (by the assumption of “nucleation limits” for characteristic damage-related quantities), and of the applied stress state. In the second stage, during postprocessing, the volume fraction of precipitates as well as the influences of the particle orientation distribution, size distribution, and size dependence of the damage-related quantities are taken into account. The model is applied to the microstructure of IF (Interstitially Free) steel, a material with a ductile matrix and rigid second phase particles of cubical shape. This microstructure is particularly suited for investigating shape and size effects. The model shows that either the size effect or the shape effect dominate the void nucleation behavior: in the case of particles of roughly the same size, the size distribution will hardly alter the nucleation strain distribution obtained by taking into account only the shape and orientation effects. For particles of very different sizes, the size effect will completely override the rather “sharp” original distribution regarding particle shape and orientation.     

A Finite Strain Analysis of Cavity Formation at a Rigid Inhomogeneity

The formation of a cavity by inclusion-matrix interfacial separation is examined by analyzing the response of a plane rigid inclusion embedded in an unbounded incompressible matrix subject to remote equibiaxial dead load traction. A vanishingly thin interfacial cohesive zone, characterized by normal and tangential interface force-separation constitutive relations, is assumed to govern separation behavior. Rotationally symmetric cavity shapes (circles) are shown to be solutions of an interfacial integral equation depending on the strain energy density of the matrix, the interface force constitutive relation and the remote loading. Nonsymmetrical cavity formation, under rotationally symmetric conditions of geometry and loading, is treated within the theory of infinitesimal strain superimposed on a given finite strain state. Rotationally symmetric and nonsymmetric bifurcations are analyzed and detailed results, for the Mooney–Rivlin strain energy density and for an exponential interface force-separation law, are presented. For the nonsymmetric rigid body displacement mode, a simple formula for the critical load is presented. The effect on bifurcation behavior of interfacial shear stiffness and other interface parameters is treated as well. In particular we demonstrate that (i) for the smooth interface nonsymmetric bifurcation always precedes rotationally symmetric bifurcation, (ii) unlike rotationally symmetric bifurcation, there is no threshold value of interface parameter for which nonsymmetric bifurcation will not occur and (iii) interfacial shear may significantly delay the onset of nonsymmetric bifurcation. Also discussed is the range of validity of a nonlinear infinitesimal strain theory previously presented by the author (Levy [1]). This revised version was published online in July 2006 with corrections to the Cover Date.     

Influences of particle size and interface energy on the stress concentration induced by the oblate spheroidal particle and the void nucleation mechanism

Separation of the particle–matrix interface and breakage of the second-phase particle are two main void nucleation mechanisms, which are directly associated with the stress concentration factors (SCFs) at the interface and within the particle, respectively. This work investigates the coupled effects of particle size and particle shape on these stress concentrations by solving an infinite solid containing an oblate spheroidal particle under remote stress boundary condition. The phenomenological strain plasticity theory by Fleck–Hutchinson [Fleck, N.A., Hutchinson, J.W., 1997. Strain gradient plasticity. In: Hutchinson, J.W., Wu, T.Y. (Eds.), Advance in Applied Mechanics, vol. 33. Academic Press, New York, pp. 295–361] is adopted to capture the size effect, various particle aspect ratios are considered to depict the particle shape effect and an interfacial energy concept is introduced to settle the double-traction equilibrium problem at the matrix–particle interface. By using a Ritz procedure, solutions about the stress concentrations are numerically achieved and three main results are found. First, the interfacial normal stress near the particle pole, the interfacial shear stress and the particle opening stress are dramatically elevated and their distributions are significantly modified by decrease in the particle size. Second, this particle size effect is influenced by the remote effective strain, remote stress triaxiality and the interfacial energy to different extent. Finally, the particle shape effect is coupled with this particle size effect, and the more oblate the particle is, the more significant the size effect on SCF elevation is. These findings are helpful for us to understand deeply the void nucleation mechanism at the micron scale.     

A physically motivated anisotropic tensorial representation of damage with separate functions for void nucleation,growth, and coalescence

A phenomenological anisotropic damage progression formulation for porous ductile metals with second phases is described through mechanisms of void nucleation, growth and coalescence. The model is motivated from fracture mechanisms and microscale physical observations. To describe the creation of new pores, the decohesion at the particle–matrix interface and the fragmentation of second phase particles, the void-crack nucleation equation is related to several microstructural parameters (fracture toughness, length scale parameter, particle size, volume and fraction of second phase), the plastic strain level, and the stress state. Nucleation is represented by a general symmetric second rank tensor, and its components are proportional to the absolute value of the plastic strain rate components. Based on the Rice and Tracey model, void growth is a scalar function of the trace of damage tensor and the positive triaxiality. Like nucleation, coalescence is a second rank tensor governed by the plastic strain rate tensor and the stress state. The coalescence threshold is related to the void length scale for void impingement and void sheet mechanisms. The coupling of damage with the Bammann–Chiesa–Johnson (BCJ) plasticity model is written in the thermodynamic framework and derives from the concept of effective stress assuming the hypothesis of energy equivalence. A full-implicit algorithm is used for the stress integration and the determination of the consistent tangent operator. Finally, macroscale correlations to cast A356 AL alloy and wrought 6061-T6 AL alloy experimental data are completed with predictive void-crack evolution to illustrate the applicability of the anisotropic damage model.     

A stochastic approach to the internal damage in a structure solid: Part 1: Theoretical model

The material system is considered as heterogenous medium of actual microstructural elements. These elements exhibit random geometric and physical characteristics and are further disturbed by a latitude of randomly oriented, second phase particles. A stochastic model is presented for the occurring damage process due to the nucleation and growth of microvoids under external loading. From a micromechanical point of view, the nucleation of a void at a partile-matrix interface is considered to be associated with the cut-off of the interfacial binding potential. The growth of an elemental void is seen, then, to follow a random walk of the discrete Markov type. The latter is associated with the build-up of strain in front of the tip of the advancing void and the redistribution of local stress. As the void reaches the boundary between neighbouring elements, a dicrete inter-elemental fracture process is examined in relation to the intensities of transformation within the elemental boundary.     

Verification of the transferability of micromechanical parameters by cell model calculations with visco-plastic materials

Finite element (FE) calculations of a cylindrical cell containing a spherical hole have been performed under large strain conditions for varying triaxiality with three different constitutive models for the matrix material, i.e. rate independent plastic material with isotropic hardening, visco-plastic material under both isothermal and adiabatic conditions, and porous plastic material with a second population of voids nucleating strain controlled. The “mesoscopic” stress-strain and void growth responses of the cell are compared with predictions of the modified Gurson model in order to study the effects of varying triaxiality and strain rate on the critical void volume fraction. The interaction of two different sizes of voids was modelled by changing the strain level for nucleation and the stress triaxiality. The study confirms that the void volume fraction at void coalescence does not depend significantly on the triaxiality if the initial volume fraction of the primary voids is small and if there are no secondary voids. The strain rate does not affect f_c either. The results also indicate that a single internal variable, f, is not sufficient to characterize the fracture processes in materials containing two different size-scales of void nucleating particles.     

Effect of yield surface curvature and void nucleation on plastic flow localization

The effect of void nucleation is incorporated in a recently proposed material model that accounts for a combination of kinematic hardening and isotropic hardening of a porous ductile material. Since each of plastic dilatancy, void nucleation and yield surface curvature have a strong influence on predictions of plastic flow localization, the present material model can be used to study the interaction of these effects. Nucleation controlled by the plastic strain as well as nucleation controlled by the maximum normal stress on the particle-matrix interface are modelled. The predictions of the material model, for various combinations of parameters, are illustrated by analyses of shear band formation under plane strain or axisymmetric conditions, and by analyses of necking in biaxially stretched sheets.     

Modelling of dynamic ductile fracture and application to the simulation of plate impact tests on tantalum

A model of dynamic damage by void nucleation and growth is proposed for elastic-viscoplastic materials sustaining intense loading. The model is dedicated to ductile materials for which fracture is caused by microvoiding. The material contains potential nucleation sites where microvoids are generated when the local pressure overcomes the nucleation pressure. A probability density function is adopted to describe the fluctuation of the nucleation pressure within the material. The void growth is described by using a hollow sphere model where micro-inertia effects are accounted for. The matrix weakening due to void growth is also included.The model has been first tested under uniaxial deformation. When the strain rate is assumed constant, the pressure inside the material has nearly a linear response up to a maximum. An analytical expression for the maximum pressure is proposed.Finite element simulations of plate impact tests have been carried out and compared to experiments on tantalum. From simulations based on the proposed model, an increase of the spall strength is observed with higher shock intensities. Therefore, the relationship between the velocity pullback and spall strength usually assumed in the literature (based on the acoustic approach) seems to be inadequate. Velocity profiles are simulated for different flyer thicknesses and different impact velocities with close agreement with experiments.     

Three-dimensional micromechanical modeling of voided polymeric materials

A three-dimensional micromechanical unit cell model for particle-filled materials is presented. The cell model is based on a Voronoi tessellation of particles arranged on a body-centered cubic (BCC) array. The three-dimensionality of the present cell model enables the study of several deformation modes, including uniaxial, plane strain and simple shear deformations, as well as arbitrary principal stress states.The unit cell model is applied to studies on the micromechanical and macromechanical behavior of rubber-toughened polycarbonate. Different load cases are examined, including plane strain deformation, simple shear deformation and principal stress states. For a constant macroscopic strain rate, the different load cases show that the macroscopic flow strength of the blend decreases with an increase in void volume fraction, as expected. The main mechanism for plastic deformation is broad shear banding across inter-particle ligaments. The distributed nature of plastic straining acts to reduce the amount of macroscopic strain softening in the blend as the initial void volume fraction is increased. In the case of plane strain deformation, the plastic flow is observed to initiate across inter-particle ligaments in the direction of constraint. This particular mode of deformation could not have been captured using a two-dimensional, plane strain idealization of cylindrical voids in a matrix.The potential for localized crazing and/or cavitation in the matrix is addressed. It is observed that the introduction of voids acts to relieve hydrostatic stress in the matrix material, compared to the homopolymer. It is also seen that the predicted peak hydrostatic stress in the matrix is higher under plane strain deformation than under triaxial tension (with equal lateral stresses), for the same macroscopic stress triaxiality.The effect of void volume fraction on the macroscopic uniaxial tension behavior of the different blends is examined using a Considère construction for dilatant materials. The natural draw ratio was predicted to decrease with an increase in void volume fraction.     

Void growth and coalescence in ductile solids with stage III and stage IV strain hardening

State of the art ductile fracture models often rely on simple power laws to describe the strain hardening of the matrix material. Power laws do not distinguish between the two main stages of hardening observed in polycrystals, referred to as stage III and stage IV hardening, and which emerge from the evolution of the dislocation substructure. The aim of this study is to couple a physics based strain hardening law including these two stages to a micromechanics based ductile damage model. One of the main motivations is that, the stage IV constant hardening rate stage, occurring only at large strain, will be attained in most ductile failure problems if not at the overall level of deformation, at least locally around the growing voids. Furthermore, proper modelling of the stage III involving dislocation storage and recovery terms and the transition to stage IV provides a link with the underlying physical mechanisms of deformation and with the microstructure. First, in order to evaluate the effects of the stage III and stage IV hardening on void growth and coalescence, an extensive parametric study is performed on two-dimensional (2D) axisymmetric finite element (FE) unit cell calculations, using a Kocks-Mecking type hardening law. The cell calculations demonstrate that accounting for the stage IV hardening can have a profound effect on delaying void coalescence and increasing the ductility. The magnitude of the recovery term during stage III has also a significant effect on the void growth rate. Then, the Kocks-Mecking law is incorporated into the Gologanu-Leblond-Devaux (GLD) porous plasticity model supplemented by two different versions of the Thomason void coalescence criterion. The predictions of the damage model are in good agreement with the results of the FE calculations in terms of the stress-strain curves, the evolution of void shape and porosity, as well as the strain value at the onset of void coalescence.     

弱界面异质球形夹杂本征应变问题的解析解

本文研究了具有线弹簧弱界面的异质球形夹杂的本征应变问题，所采用的线弹簧界面模型既能界面的切线方向滑动，又能考虑界面的法线方向张开，根据叠加原理、原问题的弹性场可分成三部分；二部分由真实均匀本征应变所引起，另一部分由等效的非均匀本征应变所引起，后一部分则由虚拟的Ｓｏｍｉｇｌｉａｎａ位错场所产生。本文求得了等效非均匀本征应变和虚拟位错场的Ｂｕｒｇｅｒ矢量的解析表达式，进而确定的问题的弹性场。     

SURFACE EFFECT ON NANOSIZED VOID GROWTH IN A RIGID-PERFECTLY PLASTIC MATERIAL

The influence of the surface effect on the nanosized spherical void growth in a rigidperfectly plastic material is analyzed and the mechanism of the nanosized void growth with high triaxiality is given. Based on the Rice and Tracey model for a macro void growth, the present model is proposed to account for the nanosized void growth under a uniform remote strain rate field with consideration on the surface effect. It is concluded that the surface effect yields an evident resistant influence on the nanosized void growth. That is, this influence decays as the void radius increases. With high triaxiality, the nanosized void growth is divided into two stages： the initial stage and the mature stage. At the first stage, the void grows slowly and the influence of surface effect is relatively weak, whereas at the second stage, the influence is significant and the void grows drastically.     

A physical model for nucleation and early growth of voids in ductile materials under dynamic loading

Spall fracture and other rapid tensile failures in ductile materials are often dominated by the rapid growth of voids. Recent research on the mechanics of void growth clearly shows that void nucleation may be represented as a bifurcation phenomenon, wherein a void forms spontaneously followed by highly localized plastic flow around the new void. Although thermal, viscoplastic, and work hardening effects all play an essential role in the earliest stages of nucleation and growth, the flow becomes dominated by spherical radial inertia, which soon causes all voids to grow asymptotically at the same rate, regardless of differences in initial conditions or constitutive details, provided only that there is the same density of matrix material and the same excess loading history beyond the cavitation stress.These two facts, initiation by bifurcation at a cavitation stress, at which a void first appears, and rapid domination by inertia, are used to postulate a simple, but physically realistic, model for nucleation and early growth of voids in a ductile material under rapid tensile loading. A reasonable statistical distribution for the cavitation stress at various nucleation sites and a simple similarity solution for inertially dominated void growth permit a simple calculation of the initiation and early growth of porosity in the material.Parametric analyses are presented to show the effect that loading rate, peak loading stress, density of nucleation sites, physical properties of the material, etc. have on the applied pressure and distribution of void sizes when a critical porosity is reached.     

An analysis of ductile rupture modes at a crack tip

An elastic-Viscoplastic model of a ductile, porous solid is used to study the influence of the nucleation and growth of micro-voids in the material near the tip of a crack. Conditions of small scale yielding are assumed, and the numerical analyses of the stress and strain fields are based on finite strain theory, so that crack tip blunting is fully accounted for. An array of large inclusions or inclusion colonies, with a relatively low strength, results in large voids near the crack tip at a rather early stage, whereas small second phase particles in the matrix material between the inclusions require large strains before cavities nucleate. Various distributions of the large inclusions, and various critical strains for nucleation of the small scale voids between the inclusions, are considered. Localization of plastic flow plays an important role in determining the failure path between the crack tip and the nearest larger void, and the path is strongly sensitive to the distribution of the large inclusions. Values of the J-integral and the crack opening displacement at fracture initiation are estimated, together with values of the tearing modulus during crack growth, and these values are related to experimental results.     

Damage initiation and growth in metals. Comparison between modelling and tomography experiments

Damage in heterogeneous model materials was measured using high-resolution X-ray absorption tomography. The material consisted of an aluminium matrix containing 1% and 4% of spherical ceramic particles acting as nucleation sites for an interface decohesion mechanism of damage. The damage initiation stage was quantified using the global population of particles in the 4% material. A strain path change experiment was then applied to the 1% material. The sample was first deformed in tension in order to create elongated cavities and then compressed at 45^° to rotate and close these cavities. The results of a model based on the Rice and Tracey approach accounting for the presence of particles inside the cavities and calculating their rotation with assuming a linear hardening plastic behaviour of the matrix were compared with the observations. The model was modified to account for the damage initiation phase. It was shown to give a good global prediction of the void volume fraction provided that the physical, mechanical and morphological information are corresponding in the experimental and the model cases. The cavity rotation experiment was also shown to compare well with the calculation although only one cavity was sufficiently opened after compression to allow the comparison.