Void Shape Effects Research Articles

Constitutive behavior of porous ductile materials accounting for micro-inertia and void shape

The goal of this paper is to characterize the mechanical behavior of porous materials by taking into account the void shape and micro-inertia effects. A representative volume element (RVE) defined by two confocal prolate spheroids is used to describe the porous material. The matrix behavior is assumed to be rigid and non linear viscous. Based on the work of Molinari and Mercier (2001), the macroscopic stress is decomposed into static and dynamic parts. In the present work the static contribution is described by the Gologanu et al. model (1993). The dynamic stress is obtained by choosing the trial velocity field proposed by Gologanu et al. (1993). With the proposed modeling a link is established between the macroscopic dynamic stress, on the one hand and, on the other hand, the macroscopic strain rate tensor and its time derivative. To validate our model, finite element simulations have been performed. Two shapes of void (spherical and prolate with an aspect ratio of 5) and two volume fraction of voids (0.001 or 0.1) are considered. The influence of micro-inertia on the macroscopic flow stress surface is analyzed and it is shown that the flow surface obtained by the analytical approach is in good agreement with finite element computations.

Heterogenous void growth revealed by in situ 3-D X-ray microtomography using automatic cavity tracking

Ductile fracture by nucleation, growth and coalescence of internal voids is the dominant fracture mechanism in metals at ambient temperature. Micromechanics-based models for each elementary mechanism have been developed and enhanced over the past 40years, allowing microstructure-informed failure predictions essentially assuming homogeneous damage evolution. In situ 3-D microtomography has been instrumental to assess these models from experimental evolution of average void density and size. Nevertheless, statistical effects on the damage evolution associated with microstructure distribution heterogeneities have not yet been addressed quantitatively due to the difficulty in tracking individual cavities. Here, we show, from 3-D in situ microtomography characterization of a Ti–6Al–4V alloy, factor ∼4 variations among the growth rate of individual cavities undergoing the same stress triaxiality and same plastic deformation. This statistical analysis has been made possible owing to an advanced tracking algorithm relying on a graph-based data association approach initially developed for the field of computer vision to track target motions. The measured variations originate from void shape and crystal orientation effects, as well as from local constraints changes due to the presence of two phases with different strengths.

Numerical studies of porous ductile materials containing arbitrary ellipsoidal voids – I: Yield surfaces of representative cells

Abstract This work is devoted to some numerical limit-analyses, performed by the finite element method, of elementary cells of arbitrary ellipsoidal shape and containing confocal ellipsoidal voids. The aim is essentially, in the present Part I, to validate an approximate analytic criterion for such cells recently proposed by Madou and Leblond, 2012a , Madou and Leblond, 2012b , and in Part II, to supplement the model by proposing reasonable evolution equations for the length and orientation of the axes of the voids. The results obtained are however not specifically attached to this particular model and could assist in the definition of any similar one for plastic porous materials incorporating void shape effects. The present Part I is devoted to the numerical determination of the yield surfaces corresponding to eight different void geometries (prolate and oblate spheroids, circular and elliptic cylinders, circular and elliptic cracks, two general ellipsoids). The results obtained confirm the general validity of Madou and Leblond, 2012a , Madou and Leblond, 2012b 's proposed criterion, although slight corners not accounted for in this criterion are apparent on the numerical yield surfaces of cylindrical cells.

Numerical studies of porous ductile materials containing arbitrary ellipsoidal voids – II: Evolution of the length and orientation of the void axes

In Part I, Madou and Leblond, 2012a, Madou and Leblond, 2012b's criterion for plastic porous materials containing arbitrary ellipsoidal voids was validated by comparing its predictions with the results of some numerical limit-analyses of elementary cells containing such voids. In the present Part II, our aim is now to complete the model by proposing reasonable evolution equations for the length and orientation of the axes of the voids. Again, however, the equations proposed are not attached to this specific model and could be used in conjunction with any similar criterion accounting for void shape effects. In the definition of the evolution equations looked for, a central role is played by “elastic” expressions for the strain and rotation rates of the voids proposed by Ponte-Castaneda and Zaidman (1994) and Kailasam and Ponte-Castaneda (1998) from homogenization theory. The importance of plastic effects however makes it necessary to modify these expressions; this is done heuristically by introducing stress-dependent correction factors determined numerically in a number of reference cases and suitably interpolated between these cases.

Assessment of hollow spheroid models for ductile failure prediction by limit analysis and conic programming

Abstract Several extensions of the Gurson model have been proposed such as to account for void shape effects (e.g. Gologanu, M., Leblond, J., 1993. Approximate models for ductile metals containing non-spherical voids – case of axisymmetric prolate ellipsoidal cavities. Journal of the Mechanics and Physics of Solids 41 (11), 1723–1754; Gologanu, M., Leblond, J., Perrin, G., Devaux, J., 1994. Approximate models for ductile metals containing non-spherical voids – case of axisymmetric oblate ellipsoidal cavities. Journal of Engineering Materials and Technology 116, 290–297; Gologanu, M., Leblond, J., Perrin, G., Devaux, J., 1997. Recent extensions of Gurson's model for porous ductile metals. In: Suquet, P. (Ed.), Continuum Micromechanics, Springer Verlag; Garajeu, M., Suquet, P., 1997. Effective properties of porous ideally plastic or viscoplastic materials containing rigid particles. Journal of the Mechanics and Physics of Solids 45, 873–902 and more recently by Monchiet, V., Charkaluk, E., Kondo, D., 2007. An improvement of Gurson-type models of porous materials by using Eshelby-like trial velocity fields. Comptes Rendus Mecanique 335, 32–41). The main goal of this study is to assess the latter models by establishing relevant numerical LA-type lower and upper bounds to the exact solutions for 3D stress and strain conditions. Numerical limit analysis techniques are extended to the case of spheroid cavities under various 3D loading conditions. First numerical tests for the hollow sphere (Gurson) model were performed to verify the computational efficiency of the new codes. A second series of tests, performed for uniform strain rate boundary conditions and axisymmetric loadings, allowed to assess the criteria proposed by Gologanu et al. in the case of prolate voids as well as oblate ones. It is shown that the 1997-Gologanu et al. criterion appears to be very accurate for the above boundary conditions. For the assessment of a recent criterion derived by Monchiet et al. (2007), we performed other computations corresponding to uniform stress boundary conditions for both oblate and prolate cavities. It is shown that the Monchiet et al. (2007) criterion appears to be an estimate which may be improved. However, this criterion is very accurate when compared to the numerical results with strain rate boundary conditions. Finally a first attempt with 3D-loadings seems to confirm the above conclusions.

Coupling effects of void shape and void size on the growth of an elliptic void in a fiber-reinforced hyper-elastic thin plate

The growth of a prolate or oblate elliptic micro-void in a fiber reinforced anisotropic incompressible hyper-elastic rectangular thin plate subjected to uniaxial extensions is studied within the framework of finite elasticity. Coupling effects of void shape and void size on the growth of the void are paid special attention to. The deformation function of the plate with an isolated elliptic void is given, which is expressed by two parameters to solve the differential equation. The solution is approximately obtained from the minimum potential energy principle. Deformation curves for the void with a wide range of void aspect ratios and the stress distributions on the surface of the void have been obtained by numerical computation. The growth behavior of the void and the characteristics of stress distributions on the surface of the void are captured. The combined effects of void size and void shape on the growth of the void in the thin plate are discussed. The maximum stresses for the void with different sizes and different void aspect ratios are compared.

Numerical modeling of elasto-plastic porous materials with void shape effects at finite deformations

A new constitutive model for elasto-plastic (rate-independent) porous materials subjected to general three-dimensional finite deformations is presented. The new model results from simple modifications of an earlier model of Kailasam and Ponte Castañeda (1997, 1998) [40,41] so that it reproduces the exact spherical and cylindrical shell solution (composite sphere and composite cylinder assemblage) under purely hydrostatic loadings, while predicting (by calibration) accurately the void shape evolution according to the recent “second-order” model of Danas and Ponte Castañeda [17]. Furthermore, the present model is based on a rigorous homogenization method which is capable of predicting both the constitutive behavior and the microstructure evolution of porous materials. The microstructure is described by voids of arbitrary ellipsoidal shapes and orientations and as a result the material exhibits deformation-induced (or morphological) anisotropy at finite deformations. This is in contrast with the well-known Gurson [32] model which assumes that the voids remain spherical during the deformation process and thus the material remains always isotropic. The present model is implemented numerically in a finite element program where a three-dimensional thin-sheet (butterfly) specimen is subjected to a combination of shear and traction loading conditions in order to examine the effect of stress triaxiality and shearing upon material failure. The ability of the present model to take into account the nontrivial evolution of the microstructure and especially void shape effects leads to the prediction of material failure even at low stress triaxialities and small porosities without the use of additional phenomenological damage criteria. At high stress triaxialities, the present model gives similar predictions as the Gurson model.

Void growth and coalescence in anisotropic plastic solids

Large strain finite element calculations of unit cells subjected to triaxial axisymmetric loadings are presented for plastically orthotropic materials containing a periodic distribution of aligned spheroidal voids. The spatial distribution of voids and the plastic flow properties of the matrix are assumed to respect transverse isotropy about the axis of symmetry of the imposed loading so that a two-dimensional axisymmetric analysis is adequate. The parameters varied pertain to load triaxiality, matrix anisotropy, initial porosity and initial void shape so as to include the limiting case of penny-shaped cracks. Attention is focussed on comparing the individual and coupled effects of void shape and material anisotropy on the effective stress–strain response and on the evolution of microstructural variables. In addition, the effect of matrix anisotropy on the mode of plastic flow localization is discussed. From the results, two distinct regimes of behavior are identified: (i) at high triaxialities, the effect of material anisotropy is found to be persistent, unlike that of initial void shape and (ii) at moderate triaxialities the influence of void shape is found to depend strongly on matrix anisotropy. The findings are interpreted in light of recent, microscopically informed models of porous metal plasticity. Conversely, observations are made in relation to the relevance of these results in the development and calibration of a broader set of continuum damage mechanics models.

Void shape effects and voids starting from cracked inclusion

Numerical, axisymmetric cell model analyses are used to study the growth of voids in ductile metals, until the mechanism of coalescence with neighbouring voids sets in. A special feature of the present analyses is that extremely small values of the initial void volume fraction are considered, down to 10 −10, which means that the metal undergoes huge strains before coalescence. This is accounted for in the present analyses by using remeshing techniques. The evolution of the void shape during the large deformations is a natural outcome of the numerical analysis. Also the effect of different initial void shapes is considered, as well as the effect of different spacings between the voids in the axial and transverse directions. While these first analyses are carried out for voids in a homogeneous metal, a second set of cell model studies are carried out for voids that initiate from a crack in a hard second phase particle. As the particle deforms relatively little the void growth is here dominated by strong blunting of the metal at the tip of the initial penny-shaped crack. These analyses are used to estimate how well the void shape evolution would be approximated by assuming that the presence of the particle in the material adjacent to the void can be neglected.

The growth and coalescence of ellipsoidal voids in plane strain under combined shear and tension

New extensions of a model for the growth and coalescence of ellipsoidal voids based on the Gurson formalism are proposed in order to treat problems involving shear and/or voids axis not necessarily aligned with the main loading direction, under plane strain loading conditions. These extensions are motivated and validated using 3D finite element void cell calculations with overall plane strain enforced in one direction. The starting point is the Gologanu model dealing with spheroidal void shape. A void rotation law based on homogenization theory is coupled to this damage model. The predictions of the model closely agree with the 3D cell calculations, capturing the effect of the initial void shape and orientation on the void rotation rate. An empirical correction is also introduced for the change of the void aspect ratio in the plane transverse to the main axis of the void departing from its initially circular shape. This correction is needed for an accurate prediction of the onset of coalescence. Next, a new approach is proposed to take strain hardening into account within the Thomason criterion for internal necking, avoiding the use of strain hardening-dependent fitting parameters. The coalescence criterion is generalized to any possible direction of the coalescence plane and void orientation. Finally, the model is supplemented by a mathematical description of the final drop of the stress carrying capacity during coalescence. The entire model is developed for plane strain conditions, setting the path to a 3D extension. After validation of the model, a parametric study addresses the effect of shear on the ductility of metallic alloys for a range of microstructural and flow parameters, under different stress states. In general, the presence of shear, for identical stress triaxiality, decreases the ductility, partly explaining recent experimental results obtained in the low stress triaxiality regime.

Micro-mechanical modelling of ductile failure in 6005A aluminium using a physics based strain hardening law including stage IV

The strain hardening and damage behaviour of isothermally heat treated 6005A aluminium is investigated in order to link the thermal treatment conditions, microstructure and fracture strain. The need for a plastic flow rule involving a stage IV hardening at large strain was found essential to generate quantitative predictions when using an enhanced Gurson type damage model for the material behaviour. This model relies on an explicit description of the three stages of nucleation, growth and coalescence of voids, involving void shape effects. An implementation within a 3D finite element code allows for the simulation of the full tensile response curves up to cracking initiation and final failure. Reasonable agreement is found when comparing the predicted and experimentally measured fracture strains for a wide range of heat treatment conditions using real microstructure based parameters. (C) 2010 Elsevier Ltd. All rights reserved.

A constitutive model for plastically anisotropic solids with non-spherical voids

Plastic constitutive relations are derived for a class of anisotropic porous materials consisting of coaxial spheroidal voids, arbitrarily oriented relative to the embedding orthotropic matrix. The derivations are based on nonlinear homogenization, limit analysis and micromechanics. A variational principle is formulated for the yield criterion of the effective medium and specialized to a spheroidal representative volume element containing a confocal spheroidal void and subjected to uniform boundary deformation. To obtain closed form equations for the effective yield locus, approximations are introduced in the limit-analysis based on a restricted set of admissible microscopic velocity fields. Evolution laws are also derived for the microstructure, defined in terms of void volume fraction, aspect ratio and orientation, using material incompressibility and Eshelby-like concentration tensors. The new yield criterion is an extension of the well known isotropic Gurson model. It also extends previous analyses of uncoupled effects of void shape and material anisotropy on the effective plastic behavior of solids containing voids. Preliminary comparisons with finite element calculations of voided cells show that the model captures non-trivial effects of anisotropy heretofore not picked up by void growth models.

Application of a micromechanics model to the overall properties of heterogeneous graphite

This paper deals with the overall properties of polycrystalline graphite, a material mainly composed of voids and dense inhomogeneities embedded in a less dense matrix. First, we examine the overall average elastic properties and conductivities of such a material. Second, we evaluate the void shape effects on the overall Young’s modulus. Finally, we compare the results obtained from the analytical model with experimental data from radiolytic oxidation of graphite.

Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids

The combined effects of void shape and matrix anisotropy on the macroscopic response of ductile porous solids is investigated. The Gologanu–Leblond–Devaux’s (GLD) analysis of an rigid-ideal plastic (von Mises) spheroidal volume containing a confocal spheroidal cavity loaded axisymmetrically is extended to the case when the matrix is anisotropic (obeying Hill’s [Hill, R., 1948. A theory of yielding and plastic flow of anisotropic solids. Proc. Roy. Soc. London A 193, 281–297] anisotropic yield criterion) and the representative volume element is subjected to arbitrary deformation. To derive the overall anisotropic yield criterion, a limit analysis approach is used. Conditions of homogeneous boundary strain rate are imposed on every ellipsoidal confocal with the cavity. A two-field trial velocity satisfying these boundary conditions are considered. It is shown that for cylindrical and spherical void geometries, the proposed criterion reduces to existing anisotropic Gurson-like yield criteria. Furthermore, it is shown that for the case when the matrix is considered isotropic, the new results provide a rigorous generalization to the GLD model. Finally, the accuracy of the proposed approximate yield criterion for plastic anisotropic media containing non-spherical voids is assessed through comparison with numerical results.

A constitutive model for elastoplastic solids containing primary and secondary voids

In many ductile metallic alloys, the damage process controlled by the growth and coalescence of primary voids nucleated on particles with a size varying typically between 1 and 100 μm, is affected by the growth of much smaller secondary voids nucleated on inclusions with a size varying typically between 0.1 and 3 μm. The goal of this work is first to quantify the potential effect of the growth of these secondary voids on the coalescence of primary voids using finite element (FE) unit cell calculations and second to formulate a new constitutive model incorporating this effect. The nucleation and growth of secondary voids do essentially not affect the growth of the primary voids but mainly accelerate the void coalescence process. The drop of the ductility caused by the presence of secondary voids increases if the nucleation strain decreases and/or if their volume fraction increases and/or if the primary voids are flat. A strong coupling is indeed observed between the shape of the primary voids and the growth of the second population enhancing the anisotropy of the ductility induced by void shape effects. The new micromechanics-based coalescence condition for internal necking introduces the softening induced by secondary voids growing in the ligament between two primary voids. The FE cell calculations were used to guide and assess the development of this model. The use of the coalescence condition relies on a closed-form model for estimating the evolution of the secondary voids in the vicinity of a primary cavity. This coalescence criterion is connected to an extended Gurson model for the first population including the effect of the void aspect ratio. With respect to classical models for single void population, this new constitutive model improves the predictive potential of damage constitutive models devoted to ductile metal while requiring only two new parameters, i.e. the initial porosity of second population and a void nucleation stress, without any additional adjustment.

A Numerical Investigation of the Effect of Induced Porosity on the Electromechanical Switching of Ferroelectric Ceramics

Ferroelectric ceramics have many applications ranging from microelectromechanical systems (MEMS) to explosively driven power supplies. In addition to chemical compositions and processing methods, porosity is an important material parameter that can affect both the electrical and mechanical responses of a ferroelectric. The main objective of the current study is to gain preliminary insight on the possible effect of porosity on the switching behavior of ferroelectrics. Numerical simulation was used to address the research objective. The numerical code used is an arbitrary Lagrangian-Eulerian, multi-material, multi-physics finite element code developed by Sandia National Laboratories. To accomplish the research objective, a phenomenological electromechanical model developed by Landis was first implemented in the code. The effects of void density, which ranges from 0 to 11%, and shape, which includes sphere and cylinder, were then investigated through a parametric study. The study indicates that the remanent polarization decreases with increased porosity density. For a given density, the porous solid that contains the cylindrical voids whose longitudinal axis is perpendicular to the applied electric field possesses the largest amount of the remanent polarization. The solid that contains the cylindrical voids whose longitudinal axis is parallel to the applied electric field has the least while the one containing spherical voids is intermediate. It is conjectured that the void shape effect is mainly due to the fact that the void perturbs the distribution of electric field and polarization with respect to from the applied electric field direction and different shapes of void result in different degrees of perturbation. The limitation of the current numerical simulation and possible future work are also discussed.

Anisotropic ductile fracture of Al 2024 alloys

The anisotropic fracture of the 2024-T351 aluminium alloy is investigated using a micromechanics-based damage model accounting for the effect of the void aspect ratio and void distribution. The 2024-T351 Al alloy contains precipitation free bands in which most void nucleating particles are located. The presence of these bands, which are parallel to the rolling direction, primarily controls the distribution of damage and overall fracture anisotropy. The primary void nucleating particles also present a preferential elongation in the rolling direction. These key microstructural features have been determined using quantitative characterisation methods. The effects of void shape and void spacing on the fracture behaviour are elucidated by means of FE cell calculations. FE simulations of cylindrical notched round bars loaded in different orientations are made and compared with experimental data, allowing a better understanding of the damage process as well as the limitations of the modelling approach.

Numerical simulation of low stress triaxiality ductile fracture

One of the main shortcoming of existing damage models when applied to the simulation of metal forming operations is their limited validity under low stress triaxiality conditions. An extended Gurson model incorporating the effect of void shape and relative void spacing on the growth and coalescence has been developed in order to encompass both low and large stress triaxiality regimes. The constitutive model has been implemented into an implicit finite element code within a finite strain set up. Identification procedures from experimental data are proposed and illustrated in an application on copper bars exhibiting different strain hardening capacity. A parametric analysis of the damage evolution in specimens deformed under uniaxial tension with necking demonstrates the importance of properly accounting for void shape and void coalescence in the case of large strains problems.

Approximate yield criteria for anisotropic metals with prolate or oblate voids

Following the study of Gologanu et al. (1997) which has extended the well-known approach of Gurson (1975), we propose approximate yield criteria for anisotropic plastic voided metals containing non spherical cavities. The plastic anisotropy of the matrix is described by means of Hill's quadratic criterion. The procedure to establish the closed form expression of approximate macroscopic criteria, in which void shape and plastic anisotropic effects are included, is detailed. The new criteria allow us to recover existing results in the cases of spherical and cylindrical voids in an Hill type plastic matrix. Moreover, they agree with previous criteria for non spherical voids in an isotropic plastic matrix. Finally, for validation purposes, we provide, in the general case of non spherical cavities in the anisotropic matrix, a comparison with the numerical exact two field criteria. To cite this article: V. Monchiet et al., C. R. Mecanique 334 (2006).

Effects of pressure-sensitivity and plastic dilatancy on void growth and interaction

Hydrostatic stress can affect the non-elastic deformation and flow stress of polymeric materials and certain metallic alloys. This sensitivity to hydrostatic stress can also influence the fracture toughness of ductile materials, which fail by void growth and coalescence. These materials typically contain a non-uniform distribution of voids of varying size-scales and void shapes. In this work, the effects of void shape and microvoid interaction in pressure-sensitive materials are examined via a two-prong approach: (i) an axisymmetric unit-cell containing a single ellipsoidal void and (ii) a plane-strain unit-cell consisting of a single large void and a population of discrete microvoids. The representative material volume in both cases is subjected to physical stress states similar to highly stressed regions ahead of a crack. Results show that oblate voids and microvoid cavitation can severely reduce the critical stress of the material. These effects can be compounded under high levels of pressure-sensitivity. In some cases, the critical stress responsible for rapid void growth is reduced to levels comparable to the yield strength of the material. The contribution of void shape and pressure-sensitivity to the thermal- and moisture-induced voiding phenomenon in IC packages is also discussed.