The Facet method: A hierarchical multilevel modelling scheme for anisotropic convex plastic potentials

An entirely new analytical expression describing plastic anisotropy is presented. It is designed to be used in combination with multilevel models. It makes use of the theory of dual plastic potentials, which is shortly revisited. An analysis of convexity is presented. Note that the new method is not optimal when not used in combination with a multilevel model; other methods are better suited for identification on the basis of mechanical test data. Compared with already existing methods which work with multilevel models, the new method has the following advantages: (i) it is automatically convex anywhere in the six-dimensional stress or strain rate space; (ii) it can be used for materials with a stress differential effect, such as hcp metals or pre-strained cubic materials; (iii) its identification procedure is such that not only the Taylor theory, but also more advanced theories, such as the Alamel-model or self-consistent models, can be used to identify the parameters; (iv) an analytical expression of the plastic potential can be obtained in both strain rate and stress space, which is an important advantage when implementing the model in finite element codes for metal forming. Equipotential surfaces in strain rate space and corresponding yield loci obtained by the new method for four materials (one ferrite single crystal, one aluminium alloy and two types of steel) are presented and discussed.     

An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations

The initial yield surface of 2D lattice materials is investigated under biaxial loading using finite element analyses as well as by analytical means. The sensitivity of initial yield surface to the dominant deformation mode is explored by using both low- and high-connectivity topologies whose dominant deformation mode is either local bending or strut stretching, respectively. The effect of microstructural irregularity on the initial yield surface is also examined for both topologies. A pressure-dependent anisotropic yield criterion, which is based on total elastic strain energy density, is proposed for 2D lattice structures, which can be easily extended for application to 3D cellular solids. Proposed criterion uses elastic constants and yield strengths under uniaxial loading, and does not rely on any arbitrary parameter. The analytical framework developed allows the introduction of new scalar measures of characteristic stresses and strains that are capable of representing the elastic response of anisotropic materials with a single elastic master line under multiaxial loading.     

A strong contrast homogenization formulation for multi-phase anisotropic materials

Homogenization relations, linking a material's properties at the mesoscale to those at the macroscale, are fundamental tools for design and analysis of microstructure. Recent advances in this field have successfully applied spectral techniques to Kroner-type perturbation expansions for polycrystalline and composite materials to provide efficient inverse relations for materials design. These expansions have been termed ‘weak-contrast’ expansions due to the conditionally convergent integrals, and the reliance upon only small perturbations from the reference property. In 1955, Brown suggested a different expansion for electrical conductivity that resulted in absolutely convergent integrals. Torquato subsequently applied the method to elasticity, with good results even for high-contrast materials; thus it is commonly referred to as a ‘strong contrast’ expansion. The methodology has been applied to elasticity for two phases of isotropic material, generally assuming macroscopic isotropy (with noted exceptions), thus resulting in a rather elegant form of the solution.

More recently, a multi-phase form of the solution was developed for conductivity. This paper builds upon this result to apply the method to elasticity of polycrystalline materials with both local and global anisotropy. New spectral formulations are subsequently developed for both the weak and strong contrast solutions. These form the basis for efficient microstructure analysis using these frameworks, and subsequently for inverse design applications. The process is taken through to demonstration of a property closure, which acts as the basis for materials design; the closure delineates the envelope of all physically realizable property combinations for the chosen properties, based on the particular homogenization relation being used.     

Plastic yielding in cyclically loaded porous materials

The present paper focuses on plastic yielding of cyclically loaded porous materials. Unit cell models are employed to observe the evolution of the yield surface of porous materials under cyclic loading conditions. Non-linear isotropic as well as non-linear kinematic hardening matrix materials are considered. The yield surfaces computed with the unit cell models are compared to predictions of a micro-mechanical porous plasticity model that incorporates hardening. It is found that, in the case of kinematic hardening, the porous plasticity model underestimates the yield strength for larger hydrostatic stresses. An improvement of the model is proposed, so that a reasonable micro-mechanical approach to model porous materials under cyclic loadings is found.     

The indentation modulus of elastically anisotropic materials for indenters of arbitrary shape

The contact of an indenter of arbitrary shape on an elastically anisotropic half space is considered. It is demonstrated in a theorem that the solution of the contact problem is the one that maximizes the load on the indenter for a given indentation depth. The theorem can be used to derive the best approximate solution in the Rayleigh-Ritz sense if the contact area is a priori assumed to have a certain shape. This approach is used to analyze the contact of a sphere and an axisymmetric cone on an anisotropic half space. The contact area is assumed to be elliptical, which is exact for the sphere and an approximation for the cone. It is further shown that the contact area is exactly elliptical even for conical indenters when a limited class of Green's functions is considered. If only the first term of the surface Green's function Fourier expansion is retained in the solution of the axisymmetric contact problem, a simpler solution is obtained, referred to as the equivalent isotropic solution. For most anisotropic materials, the contact stiffness determined using this approach is very close to the value obtained for both conical and spherical indenters by means of the theorem. Therefore, it is suggested that the equivalent isotropic solution provides a quick and efficient estimate for quantities such as the elastic compliance or stiffness of the contact. The “equivalent indentation modulus”, which depends on material and orientation, is computed for sapphire and diamond single crystals.     

A yield function for single crystals containing voids

A yield function for single crystals containing voids has been developed based on a variational approach. This first yield function is phenomenologically extended by modifying the dependence on the mean stress and introducing three adjustable parameters. Unit cell finite element calculations are performed for various stress triaxiality ratios, main loading directions and porosity levels in the case of a perfectly plastic FCC single crystal. The three model parameters are adjusted on the unit cell calculations so that a very good agreement between simulation results and the proposed model is obtained.     

Micromechanical analysis of kinematic hardening in natural clay

This paper presents a micromechanical analysis of the macroscopic behaviour of natural clay. A microstructural stress–strain model for clayey material has been developed which considers clay as a collection of clusters. The deformation of a representative volume of the material is generated by mobilizing and compressing all the clusters along their contact planes. Numerical simulations of multistage drained triaxial stress paths on Otaniemi clay have been performed and compared the numerical results to the experimental ones in order to validate the modelling approach. Then, the numerical results obtained at the microscopic level were analysed in order to explain the induced anisotropy observed in the clay behaviour at the macroscopic level. The evolution of the state variables at each contact plane during loading can explain the changes in shape and position in the stress space of the yield surface at the macroscopic level, as well as the rotation of the axes of anisotropy of the material.     

Experimental observations of evolving yield loci in biaxially strained AA5754-O

Experimental measurement of the plastic biaxial mechanical response for an aluminum alloy (AA5754-O) sheet metal is presented. Traditional methods of multiaxial sheet metal testing require the use of finite element analysis (FEA) or other assumptions of material response to determine the multiaxial true stress versus true strain behavior of the as-received sheet material. The method used here strives to produce less ambiguous measurements of data for a larger strain range than previously possible, through a combination of the Marciniak flat bottom ram test and an X-ray diffraction technique for stress measurement. The study is performed in conjunction with a study of the microstructural changes that occur during deformation, and these microstructural results are briefly mentioned in this work. Issues of calibration and applicability are discussed, and results are presented for uniaxial (U), plane strain (PS), and balanced biaxial (BB) extension. The results show repeatable behavior (within quantified uncertainties) for U to 20%, PS to almost 15%, and BB to above 20% in-plane strains. The results are first compared with three common yield locus models (von Mises’, Hill’48, and Hosford’79), and show some unexpected results in the shape change of the yield locus at high strain levels (>5% strain). These changes include the rotation of the locus toward the von Mises surface and elongation in the balanced biaxial direction. Comparison with a more complex yield locus model (Yld2000-2d with eight adjustable parameters) showed that the locus elongation in the biaxial direction could be fit well (for a specific level of work), but at the detriment of fit to the plane strain data. Artificially large plastic strain ratios would be needed to match both the biaxial and plane strain behavior even with this more complex model.     

Constitutive behaviour of anisotropic materials under shock loading

Thermodynamically and mathematically consistent constitutive equations suitable for shock wave propagation in an anisotropic material are presented in this paper. Two fundamental tensors α_ij and β_ij which represent anisotropic material properties are defined and can be considered as generalisations of the Kronecker delta symbol, which plays the main role in the theory of isotropic materials. Using two fundamental tensors α_ij and β_ij, the concept of total generalised “pressure” and pressure corresponding to the thermodynamic (equation of state) response are redefined. The equation of state represents mathematical and physical generalisation of the classical Mie–Grüneisen equation of state for isotropic material and reduces to the Mie–Grüneisen equation of state in the limit of isotropy. Based on the generalised decomposition of the stress tensor, the modified equation of state for anisotropic materials, and the modified Hill criteria, combined with the associated flow rule, a system of constitutive equations suitable for shock wave propagation is formulated. The behaviour of aluminium alloy 7010-T6 under shock loading conditions is considered. A comparison of numerical simulations with existing experimental data shows good agreement of the general pulse shape, Hugoniot Elastic Limits (HELs), and Hugoniot stress levels, and suggests that the constitutive equations are performing satisfactorily. The results are presented and discussed, and future studies are outlined.     

Non-quadratic Kelvin modes based plasticity criteria for anisotropic materials

Novel (non-quadratic) plasticity criteria based on Kelvin modes are formulated here for anisotropic materials. As an example, such a macroscopic criterion is applied with success to the case of FCC nickel-base single crystals. Indeed, relying on the cubic symmetry of the material, the Kelvin decomposition of elasticity tensor easily allows for the definition of an objective and loading independent criterion. The criterion identification is performed from different loading cases for CMSX2 single crystal superalloy. Tension-torsion yield surfaces at room temperature and yield stress dependence on crystal orientation are modeled. The Kelvin modes based criterion is compared to experimental data, to Hill and Barlat and coworkers macroscopic criteria and to Schmid law predictions. The results show that a simple three-parameter yield function built thanks to von Mises equivalent Kelvin stresses accounts for a satisfying plasticity criterion for such alloys.Non-quadratic norm ∥·∥_a plasticity framework is addressed. Intrinsic generalizations of Hershey-Hosford criterion are proposed for cubic material symmetry.     

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.     

Exactly solvable spherically anisotropic thermoelastic microstructures

Microstructures possessing local spherical anisotropy are considered in this paper. An example is a spherulitic polymer which can be modelled by an assemblage of spheres of all sizes in which a radial direction in every sphere is an axis of local transverse isotropy. Our purpose is to construct effectively isotropic microstructures, with spherically anisotropic and thermoelastic constituents, whose effective bulk modulus, thermal stress and specific heat can be exactly determined. The basic microstructure for which this is achieved in the present paper is the nested composite sphere assemblage of Milgrom and Shtrikman (J. Appl. Phys. 66 (1989) 3429) which was originally formulated for isotropic constituents, in the settings of conductivity and coupled fields with scalar potentials. Here, we allow the phases of this microstructure to be spherically thermoelastic with a symmetry class which can be trigonal, tetragonal, transversely isotropic, cubic or isotropic with respect to a local spherical coordinate system. A rich class of new exact results for two-phase composites and polycrystals are obtained.     

On the r-value of textured sheet metals

The r-value of a sheet metal is a measure of plastic anisotropy frequently used for prediction of performance in deep-drawing. It has also figured prominently in the literature for validation of theories where the predicted angular dependence of r is compared with the measured dependence. As plastic anisotropy in sheet metals is caused mainly by the preferred orientations of grains within the polycrystalline metal, it is natural to ask how r would depend on the orientation distribution function (ODF) w which defines the crystallographic texture of the polycrystal. In this paper a general formula relating r to w is derived for textured sheet metals whose plastic flow behavior is governed by a plastic potential f(σ, w), the anisotropic part of which depends linearly on the texture coefficients; here σ denotes the deviator of the Cauchy stress. Specific forms of this formula for orthorhombic sheets of cubic and of hexagonal metals are explicitly given.     

Effective pressure-sensitive elastoplastic behavior of particle-reinforced composites and porous media under isotropic loading

The composite under investigation consists of an elastoplastic matrix reinforced by elastic particles or weakened by pores. The material forming the matrix is pressure-sensitive. The Drucker–Prager yield criterion and a one-parameter non-associated flow rule are employed to formulate the yield behavior of the matrix. The objective of this work is to estimate the effective elastoplastic behavior of the composite under isotropic tensile and compressive loadings. To achieve this objective, the composite sphere assemblage model of Hashin [Z. Hashin, The elastic moduli of heterogeneous materials, ASME J. Appl. Mech. 29 (1962) 143–150] is used. Exact solutions are thus derived as estimations for the effective secant and tangent bulk moduli of the composite. The effects of the loading modes and phase properties on the effective elastoplastic behavior of the composite are analytically and numerically evaluated.     

Experimental investigation of initial and subsequent yield surfaces for laminated metal matrix composites

The aim of this work is to construct yield surfaces to describe initial yielding and characterize hardening behavior of a highly anisotropic material. A methodology for constructing yield surfaces for isotropic materials using axial–torsion loading is extended to highly anisotropic materials. The technique uses a sensitive definition of yielding based on permanent strain rather than offset strain, and enables multiple yield points and multiple yield surfaces to be conducted on a single specimen. A target value of 20 × 10⁻⁶ is used for Al₂O₃ fiber reinforced aluminum laminates having a fiber volume fraction of 0.55. Sixteen radial probes are used to define the yield locus in the axial–shear stress plane. Initial yield surfaces for [0₄], [90₄], and [0/90]₂ fibrous aluminum laminates are well described by ellipses in the axial–shear stress plane having aspect ratios of 10, 2.5, and 3.3, respectively. For reference, the aspect ratio of the Mises ellipse for an isotropic material is 1.73. Initial yield surfaces do not have a tension–compression asymmetry. Four overload profiles (plus, ex, hourglass, and zee) are applied to characterize hardening of a [0/90]₂ laminate by constructing 30 subsequent yield surfaces. Parameters to describe the center and axes of an ellipse are regressed to the yield points. The results clearly indicate that kinematic hardening dominates so that material state evolution can be described by tracking the center of the yield locus. For a nonproportional overload of (σ, τ) = (500, 70) MPa, the center of the yield locus translated to (σ, τ) = (430, 37) MPa and the ellipse major axis was only 110 MPa.     

Constitutive framework optimized for myocardium and other high-strain, laminar materials with one fiber family

Central to this analysis is the identification of six rotation invariant scalars α_1-6 that succinctly define the strain in materials that have one family of parallel fibers arranged in laminae. These scalars were chosen so as to minimize covariance amongst the response terms in the hyperelastic limit, and they are termed strain attributes because it is necessary to distinguish them from strain invariants. The Cauchy stress t is expressed as the sum of six response terms, almost all of which are mutually orthogonal for finite strain (i.e. 14 of the 15 inner products vanish). For small deformations, the response terms are entirely orthogonal (i.e. all 15 inner products vanish). A response term is the product of a response function with its associated kinematic tensor. Each response function is a scalar partial derivative of the strain energy W with respect to a strain attribute. Applications for this theory presently include myocardium (heart muscle) which is often modeled as having muscle fibers arranged in sheets. Utility for experimental identification of strain energy functions is demonstrated by showing that common tests on incompressible materials can directly determine terms in W. Since the described set of strain attributes reduces the covariance amongst response terms, this approach may enhance the speed and precision of inverse finite element methods.     

Potential flow model of cavitation-induced interfacial fracture in a confined ductile layer

Fracture of a thin ductile layer sandwiched between stiff substrates often results from growth and coalescence of microscopic cavities ahead of an extending crack. Cavitation induced by plastic flow in a confined, ductile layer is analyzed here to evaluate the interfacial fracture toughness of such sandwich structures. For rigid-plastic materials, a new method is proposed in which the potential flow field of a fluid is used to approximate the plastic deformation. The principle of virtual work rate is applied to determine the equivalent traction-separation law. The method is demonstrated and validated for spherically symmetric cavity growth, for which an exact solution exists. We then study in detail the growth of an initially spherical cavity in a cylindrical bar of finite length subject to uniform traction at its ends. The results show that the stress-separation curves depend strongly on initial cavity size and the strain-hardening exponent, and weakly on the nominal strain. The method has clear advantages over numerical methods, such as finite-element analysis, for parametric study of cavity growth with large plastic deformation.     

A three-dimensional crystal plasticity model for duplex Ti–6Al–4V

A rate dependent crystal plasticity model for the α/β Ti–Al alloy Ti–6Al–4V with duplex microstructure is developed and presented herein. Duplex Ti–6Al–4V is a dual-phase alloy consisting of an hcp structured matrix primary α-phase and secondary lamellar α + β domains that are composed of alternating layers of secondary α laths and bcc structured residual β laths. The model accounts for distinct three-dimensional slip geometry for each phase, anisotropic and length scale dependent slip system strengths, the non-planar dislocation core structure of prismatic screw dislocations in the primary α-phase, and crystallographic texture. The model is implemented in the general purpose finite element code (ABAQUS, 2005. Ver 6.5, Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket, RI) via a UMAT subroutine.