A micromechanical analysis of elastoplastic behavior of porous materials

In this paper, we present a micromechanical analysis of elastoplastic behavior of porous materials. The non-uniform transformation field analysis (NTFA) is used and the non-uniform distribution of local plastic strain in the solid matrix is taken into account. Comparisons with the classical Gurson's model and standard FEM solution are presented.     

A consistent nonlocal scheme based on filters for the homogenization of heterogeneous linear materials with non-separated scales

In this work, the question of homogenizing linear elastic, heterogeneous materials with periodic microstructures in the case of non-separated scales is addressed. A framework if proposed, where the notion of mesoscopic strain and stress fields are defined by appropriate integral operators which act as low-pass filters on the fine scale fluctuations. The present theory extends the classical linear homogenization by substituting averaging operators by integral operators, and localization tensors by nonlocal operators involving appropriate Green functions. As a result, the obtained constitutive relationship at the mesoscale appears to be nonlocal. Compared to nonlocal elastic models introduced from a phenomenological point of view, the nonlocal behavior has been fully derived from the study of the microstructure. A discrete version of the theory is presented, where the mesoscopic strain field is approximated as a linear combination of basis functions. It allows computing the mesoscopic nonlocal operator by means of a finite number of transformation tensors, which can be computed numerically on the unit cell.     

A new micromechanical approach of micropolar continuum modeling for 2-D periodic cellular material

In this paper, we present a new united approach to formulate the equivalent micropolar constitutive relation of two-dimensional (2-D) periodic cellular material to capture its non-local properties and to explain the size effects in its structural analysis. The new united approach takes both the displacement compatibility and the equilibrium of forces and moments into consideration, where Taylor series expansion of the displacement and rotation fields and the extended aver-aging procedure with an explicit enforcement of equilibrium are adopted in the micromechanical analysis of a unit cell. In numerical examples, the effective micropolar constants obtained in this paper and others derived in the literature are used for the equivalent micropolar continuum simulation of cellular solids. The solutions from the equivalent analysis are compared with the discrete simulation solutions of the cellu-lar solids. It is found that the micropolar constants developed in this paper give satisfying results of equivalent analysis for the periodic cellular material.     

A micromechanical model for porous materials with a reinforced matrix

In this study a micromechanical model is proposed for ductile porous material whose matrix is reinforced by small inclusions. The solid phase is described by a pressure sensitive plastic model. Based on works of Maghous et al. [6], a macroscopic plastic criterion is firstly obtained by using a two-step homogenization procedure. The effect of porosity at the mesoscale and the influence of inclusions at the microscale are taken into account simultaneously by this criterion. With a non-associated plastic flow rule, the micro-macro model is applied to modeling of mechanical behavior of a cement paste. In particular, we have considered at the microscopic scale the formation of calcite grains by carbonation process in the solid matrix. The studied cement paste is then seen as a reinforced matrix–pore system. Comparisons between numerical results and experimental data show that the proposed model is able to capture the main features of the mechanical behavior of the studied material.     

On asymptotic elastodynamic homogenization approaches for periodic media

A fairly large family of asymptotic elastodynamic homogenization methods is shown to be derivable from Willis exact elastodynamic homogenization theory for periodic media under appropriate approximation assumptions about, for example, frequencies, wavelengths and phase contrast. In light of this result, two long-wavelength and low-frequency asymptotic elastodynamic approaches are carefully analyzed and compared in connection with higher-order strain-gradient media. In particular, these approaches are proved to be unable to capture, at least in the one-dimensional setting, the optical branches of the dispersion curve. As an example, a two-phase string is thoroughly studied so as to illustrate the main results of the present work.     

Plasticity-triggered architectural effects in periodic multilayers with wavy microstructures

Despite considerable presence of periodic multilayers with wavy architectures in nature and technology, little simulation data is available on their response. A recent investigation of wavy multilayers comprised of alternating elastic and elastic-perfectly plastic plies has revealed the important role that plasticity plays on their post-yield response relative to the corresponding flat configurations [Khatam, H., Pindera, M.-J., 2009b. Parametric finite-volume micromechanics of periodic materials with elastoplastic phases. Int. J. Plasticity 25 (7), 1386–1411]. Herein, we extend this investigation by considering the effect of elastic layer thickness on the post-yield response at several fixed elastic phase volume fractions using the parametric finite-volume direct averaging micromechanics (FVDAM) theory. The layer thickness is shown to have a substantial impact on the post-yield response whose extent depends on the loading mode and waviness amplitude, in contrast with the minimal impact on the homogenized elastic moduli. Decreasing the layer thickness at sufficiently high fixed volume fractions decreases the extent of strain hardening under transverse normal loading, as well as the maximum normal and shear stresses in the stiff layers, reducing the possibility of failure and thus potentially enhancing durability. The opposite holds under transverse shear loading for low waviness amplitudes. The presented results provide a framework for tailoring the elastoplastic response of multilayers with sinusoidally varying plies under different loading modes.     

Some micromechanical aspects of failure in granular materials based on second-order work

This paper discusses the notion of failure in a granular assembly by examining the key microstructural mechanisms which are most likely to trigger the nucleation and propagation of instabilities within a granular material. For this purpose, the key variable to predict the occurrence of failure, known as second-order work, is expressed from variables on the grain scale. The local behaviour incidents (where contacts may open or slide), compared to the global response of the assembly, are analysed by two approaches. First, numerical computations made by a discrete element model confirm the microscopic definition of the second-order work. Secondly, a micromechanical model, based on a homogenization procedure, relating the macroscopic behaviour to microscopic ingredients, namely contact planes, points to a close link between the occurrence of failure on the macroscopic scale as well as on the contact planes.     

Multi-level modeling of woven glass/epoxy composite for multilayer printed circuit board applications

The objective of this paper is to develop a hybrid homogenization method to predict the elastic properties of a common woven glass/epoxy composite substrate for multilayer circuit board applications. Comprehensive high resolution 3D finite element (FE) models of a quarter of the repeated unit cell (RUC) for the woven glass/epoxy composite were developed based on different micromechanical schemes. . Specifically, four different micromechanics schemes were investigated: self-consistent, Mori–Tanaka, three-phase approach and composite cylinder assemblage (CCA). The element based strain concentration matrices were determined and used to obtain the homogenized woven glass/epoxy composite properties via a specially developed MATLAB code. Attention was further devoted to the predictions of the homogenized elastic moduli of the multilayer printed circuit board (PCB). The results from our simulations, based on Mori–Tanaka and CCA, are in good agreement with existing experimental results, indicating that the newly proposed homogenization scheme can be used as a design tool to predict the overall properties of woven composite materials typically used in multilayer PCB applications.     

A micromechanical model for polycrystalline creep with grain boundary cavitation

Summary  A micromechanical model is developed to describe effects such as combined power-law creep and diffusion, grain boundary sliding and cavitation in polycrystals. Several aspects of creep-constrained cavitation are taken into account such as diffusion in a cage of creeping matrix material and cavitating facets in a cage of creeping grains. Grain boundary sliding is modelled by distributed micro-shearcracks. It is shown that the different physical mechanisms and their interactions are functions of a well-defined material parameter λ, which can be related to the material length scale L introduced by Rice. Received 18 January 2000; accepted for publication 17 May 2000     

Parametric finite-volume micromechanics of periodic materials with elastoplastic phases

The recently incorporated parametric mapping capability into the finite-volume direct averaging micromechanics (FVDAM) theory has produced a paradigm shift in the theory’s development. The use of quadrilateral subvolumes made possible by the mapping facilitates efficient modeling of microstructures with arbitrarily shaped heterogeneities, and eliminates artificial stress concentrations produced by the rectangular subvolumes employed in the standard version. Herein, the parametric FVDAM theory is extended to the inelastic domain by implementing additional formulation required to accommodate plastic and thermal loading. Two different approaches of implementing plasticity have been investigated. The first approach is based on the treatment employed in previous versions of the theory wherein plastic strain fields are represented by a series expansion in Legendre polynomials. The second approach is based on direct surface-averaging of plastic strains calculated at a number of collocation points along the quadrilateral subvolumes’ surfaces, and offers substantial simplification in the parametric finite-volume theory’s elastic–plastic framework. Moreover, substantial reductions in execution times without loss of accuracy are realized due to the elimination of redundant plastic strain calculations in the subvolumes’ interiors employed in the evaluation of the Legendre polynomial coefficients. Numerical studies demonstrate the advantages of the parametric FVDAM theory relative to the standard version, together with new results that highlight its modeling capabilities vis-a-vis an emerging class of periodic lamellar materials with wavy microstructures and the thus-far undocumented architectural effects amplified by plasticity.     

A phase-field modeling approach of hydraulic fracture in saturated porous media

Continuum porous media theories, extended by a diffusive phase-field modeling (PFM) approach, introduce a convenient and efficient tool to the simulation of hydraulic fracture in fluid-saturated heterogeneous materials. In this, hydraulic- or tension-induced fracture occurs in the solid phase. This leads to permanent local changes in the permeability, the volume fractions of the constituents as well as the interstitial-fluid flow. In this work, the mechanical behaviors of the multi-field, multi-phase problem of saturated porous media, such as the pore-fluid flow and the solid-skeleton deformation, are described using the macroscopic Theory of Porous Media (TPM). To account for crack nucleation and propagation in the sense of brittle fracture, the energy-minimization-based PFM procedure is applied, which approximates the sharp edges of the crack by a diffusive transition zone using an auxiliary phase-field variable. Furthermore, the PFM can be implemented in usual continuum finite element packages, allowing for a robust solution of initial-boundary-value problems (IBVP). For the purpose of validation and comparison, simulations of a two-dimensional IBVP of hydraulic fracture are introduced at the end of this research paper.     

Analysis of micro-structural effects on phononic waves in layered elastic media with periodic nonsmooth coordinates

Propagation of elastic phononic waves in layered composite materials is analyzed by introducing nonsmooth periodic coordinates associated with structural specifics of the materials. Spatial scales of the original (smooth) coordinates are estimated by the wave lengths. In terms of the new coordinates, the homogenization procedure occurs naturally from the continuity conditions imposed on elastic displacements and forces at layer interfaces. As a result, higher-order asymptotic approximations describing spatiotemporal ‘macro’- and ‘micro’-effects of wave propagation are obtained in closed form. Such solutions provide visualizations for the wave shapes illustrating their structure induced local details. In particular, beat-wise mode shapes and effective anisotropy of acoustic wave propagation are revealed. The subharmonic beating in wave modes occur when wave lengths orthogonal to layers is about to ‘resonate’ with layer’ thickness. If the wave speed has a non-zero projection along the layers, then phase shifts between the beats are observed in different cross sections perpendicular to the layers.     

Homogenization for wave propagation in periodic fibre-reinforced media with complex microstructure. I—Theory

The effective elastic properties of periodic fibre-reinforced media with complex microstructure are determined by the method of asymptotic homogenization via a novel solution to the cell problem. The solution scheme is ideally suited to materials with many fibres in the periodic cell. In this first part of the paper we discuss the theory for the most general situation—N arbitrarily anisotropic fibres within the periodic cell. For ease of exposition we then restrict attention to isotropic phases which results in a monoclinic composite material with 13 effective moduli and expressions for each of these are determined. In the second part of this paper we shall discuss results for a variety of specific microstructures.     

Experimental characterization and micromechanical modeling of superelastic response of a porous NiTi shape-memory alloy

Porous shape-memory alloys are usually brittle due to the presence of various nickel-titanium intermetallic compounds that are produced in the course of most commonly used synthesizing techniques. We consider here a porous NiTi shape-memory alloy (SMA), synthesized by spark-plasma sintering, that is ductile and displays full shape-memory effects over the entire appropriate range of strains. The porosity however is only 12% but the basic synthesizing technique has potential for producing shape-memory alloys with greater porosity that still are expected to display superelasticity and shape-memory effects. The current material has been characterized experimentally using quasi-static and dynamic tests at various initial temperatures, mostly within the superelastic strain range, but also into the plastic deformation regime of the stress-induced martensite phase. To obtain a relatively constant strain rate in the high strain-rate tests, a novel pulse-shaping technique is introduced. The results of the quasi-static experiments are compared with the predictions by a model that can be used to calculate the stress-strain response of porous NiTi shape-memory alloys during the austenite-to-martensite and reverse phase transformations in uniaxial quasi-static loading and unloading at constant temperatures. In the austenite-to-martensite transformation, the porous shape-memory alloy is modeled as a three-phase composite with the parent phase (austenite) as the matrix and the product phase (martensite) and the voids as the embedded inclusions, reversing the roles of austenite and martensite during the reverse transformation from fully martensite to fully austenite phase. The criterion of the stress-induced martensitic transformation and its reversal is based on equilibrium thermodynamics, balancing the thermodynamic driving force for the phase transformation, associated with the reduction of Gibbs’ free energy, with the resistive force corresponding to the required energy to create new interface surfaces and to overcome the energy barriers posed by various microstructural obstacles. The change in Gibbs’ free energy that produces the driving thermodynamic force for phase transformation is assumed to be due to the reduction of mechanical potential energy corresponding to the applied stress, and the reduction of the chemical energy corresponding to the imposed temperature. The energy required to overcome the resistance imposed by various nano- and subnano-scale defects and like barriers, is modeled empirically, involving three constitutive constants that are then fixed based on the experimental data. Reasonably good correlation is obtained between the experimental and model predictions.     

A micromechanical explanation of the mean stress effect in high cycle fatigue

A micro–macro approach of multiaxial fatigue in unlimited endurance is presented in this study, as an extension of a previous model recently proposed by the authors [Monchiet, V., Charkaluk, E., Kondo, D., 2006. A plasticity–damage based micromechanical modelling in high cycle fatigue. C.R. Mécanique 334 (2), 129–136]. It allows to take into account coupling between polycrystalline plasticity and damage mechanisms which occur at the scale of persistent slip bands (PSB) during cyclic deformation. The plasticity–damage coupled model is obtained by adapting the Gurson [Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth: part I – yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol. 99, 2–15] limit analysis to polycrystalline materials to take into account microvoids growth along PSBs. The macroscopic fatigue criterion corresponds to microcracks nucleation at the PSB–matrix interface. It is shown that this criterion accounts for the effect of the mean stress and of the hydrostatic pressure in high cycle fatigue. Such features of HCF are related to the damage micro-mechanisms. Finally, some illustrations concerning the particular case of cyclic affine loadings are presented and comparisons of the predictions of the fatigue criterion with experimental data show the relevance of this new approach.     

Novel variational formulations for nonlocal plasticity

A nonlocal structural model of softening plasticity is considered in the framework of the internal variable theories of inelastic behaviours of associative type. The finite-step nonlocal structural problem in a geometrically linear range is formulated according to a backward difference scheme for time integration of the flow rule. The related finite-step variational formulation in the complete set of local and nonlocal state variables is recovered. A family of mixed nonlocal variational formulations, with different combinations of state variables, is provided starting from the general variational formulation. The specialization of a mixed variational formulation to existing nonlocal models of softening plasticity, assuming both linear and nonlinear constitutive behaviour, is provided to show the effectiveness of the theory.     

A general theory for nonlocal softening plasticity of integral-type

This paper deals with a comparison of several models, proposed in the literature, of softening plasticity with internal variables regularized by nonlocal averaging of integral type.