Institute of Fundamental Technological Research

In metal single crystals, the observed formation of deformation banding pattern has been explained by greater latent hardening of slip systems than their self-hardening, which promotes spatial segregation of plastic slips and lamination towards single-slip domains. Numerical studies focusing on the formation of deformation bands usually involved initial imperfections, boundary-induced heterogeneity, or the postulate of minimal global energy expenditure which additionally promoted non-uniformity of deformation. This article analyses the case when no such mechanism enforcing locally non-uniform deformation is implemented in the finite element (FE) method, while the global system of equations of incremental equilibrium is solved in a standard way. The new finding in this paper is that the deformation banding pattern can appear spontaneously in FE simulations of homogeneous single crystals even in the absence of any mechanism favouring deformation banding in the numerical code. This has been demonstrated in several examples in the small strain formalism using a plane-strain model in which the twelve fcc slip systems are reduced to three effective plastic slip mechanisms. Incremental slips are determined at the Gauss-point level either by incremental work minimization in the rate-independent case or by rate-dependent regularization. In the rate-independent approach, the trust-region algorithm is developed for the selection of active slip systems with the help of the augmented Lagrangian method. Conditions under which a banding pattern appears spontaneously or is suppressed are discussed. In particular, a critical rate sensitivity exponent is identified.

Crystal plasticity, Small strain, Slip-system selection, Path instability, Microstructure formation, Finite element method

Hardness, Lattice rotation, Plasticity, Strain Gradient

The recently introduced quasi-extremal energy principle for incremental non-potential problems in rate-independent plasticity is applied to select the deformation pattern and active slip systems in single crystals. The standard crystal plasticity framework with a non-symmetric slip-system interaction matrix at finite deformation is used. The incremental work criterion for the formation of deformation bands is combined with the quasi-extremal energy principle for determining the active slip systems and slip increments in the bands. In this way, the incremental energy minimization approach has been extended to the non-potential problem of deformation banding in metal single crystals. It is shown that fulfilment of the mathematical criterion for incipient deformation banding in a homogeneous crystal in the multiple-slip case under certain conditions requires non-positive determinant of the hardening moduli matrix. Numerical examples of energetically preferable patterns of deformation bands are presented for Cu and Ni single crystals.

Solids, Plasticity, Crystal plasticity, Material stability, Slip-system selection, Energy minimization, Quasi-minimization, Time integration, Implicit, Channel-die, Deformation band

Incremental energy minimization is revisited as a method of determining an incremental solution for rate-independent dissipative solids undergoing isothermal quasi-static deformation. The incremental minimization is applied to the total internal energy of the compound thermodynamic system that consists of a deforming body with internal variables, a conservative loading device, and an ambient heat reservoir. It is shown that the difference between the virtual and actual dissipation rates plays a fundamental role in this minimization, which is related to thermodynamic extremal principles of local and global type. The analysis is carried out within the gradient plasticity framework with the energetic forces derived as the variational derivative of the Helmholtz free energy depending on the spatial gradient of arbitrary internal variables. Specifications are given for existing models of gradient plasticity.

Dissipation, internal variables, gradient, plasticity, energy, minimization, rate-independent, variational

A new model of gradient crystal plasticity is developed in which the incompatibility of plastic deformation field is simultaneously included in two different ways. The first one is well known and incorporates the gradient effect of accumulated rotation of the crystallographic lattice on the kinematic hardening in the Cosserat crystal plasticity model. The second way incorporates the effect of the current incompatibility of lattice spin on the total dislocation density rate, resulting in an additional isotropic hardening term in the gradient-enhanced hardening law. The latter effect involves a natural length scale that is fully determined in terms of standard quantities of a non-gradient hardening law and evolves during plastic deformation. The relative significance of the two effects depends on the values of material parameters, which is demonstrated by the results of calculations of 1D and 3D examples for a Cu single crystal. If the second effect is predominant, then the Cosserat formulation provides only a regularization, and then the model can be used to predict size effects. This is shown by 3D simulations of the size effect in spherical indentation of a copper single crystal. Since the gradient-enhanced hardening law does not involve any adjustable parameter, the obtained agreement of the model prediction with the experimental indentation size effect on hardness can be regarded as surprisingly good.

gradient plasticity, crystal plasticity, Cosserat continuum, length scale, indentation size effect, 3D model

The direct incremental energy minimization in rate-independent plasticity does not account for the skew-symmetric part of the tangent stiffness matrix. In crystal plasticity, this corresponds to neglecting the asymmetry of the matrix of interaction moduli for active slip-systems. This limitation has been overcome in the recently proposed quasi-extremal energy principle (QEP) applicable to nonpotential problems. In the present article it is shown how to extend QEP to finite increments in the backward-Euler computational scheme. A related constitutive algorithm is proposed which enables automatic selection of active slip systems using an energetic criterion, along any path of large deformation of a rate-independent single crystal with a nonsymmetric slip-system interaction matrix. Numerical examples have been calculated for a fcc single crystal subjected to simple shear or uniaxial tension. The slip system activity predicted by using the QEP algorithm has been found to be more reliable in describing the actual plastic response of metal crystals than conventional rate-dependent modeling in cases where the selection of active slip-systems is essential.

solids, plasticity, material stability, Lagrangian, slip-system selection, time integration, implicit

The effect of formation and evolution of stress-induced martensitic microstructures on macroscopic mechanical properties of shape memory alloys in the pseudoelastic regime is investigated with account for size-dependent energy of interfaces. A quantitative relationship is established between the changes in free energy and dissipation on the interfaces at three microstructural scales and the overall mechanical characteristic of the material under tensile loading. The multiscale analysis carried out for a polycrystalline NiTi shape memory alloy has revealed that the interfacial energy storage and dissipation can strongly affect the shape and width of the stress–strain hysteresis loop. The predicted non-monotonic stress–strain response for the material of a selected grain size shows a remarkable similarity to the experimental one extracted from a tensile test of a laminate by Hallai and Kyriakides (2013). By the classical Maxwell construction, the non-monotonic response for a material element results in a commonly observed stress plateau for a tensile specimen, which is associated with the propagation of phase transformation fronts. This behaviour is confirmed with striking accuracy by 3D finite-element computations performed for a macroscopic tensile specimen, in which propagating instability bands are treated explicitly.

microstructures, martensitic transformation, size effects, incremental energy minimization, propagating instabilities

A model of crystal plasticity is developed in which the effects of plastic flow non-uniformity are described through the full dislocation density tensor. The micromorphic approach is used in which the dislocation density tensor is represented by the curl of an independent constitutive variable called microdeformation. The microdeformation tensor is enforced by an energetic penalty term to be close to the actual plastic distortion tensor. The curl of microdeformation tensor enters the constitutive model in two independent but complementary ways. First, it is an argument of the free energy density function, which describes the kinematic-type hardening in cyclic non-uniform deformation. Second, its rate influences the rates of critical resolved shear stresses, which corresponds to additional isotropic hardening caused by incompatibility of the plastic flow rate. The latter effect, missing in the standard slip-system hardening rule, is described in a simple manner that does not require any extra parameter in comparison to the non-gradient theory. In the proposed model there are two independent internal length scales whose interplay is examined by means of 1D and 2D numerical examples of plastic shearing of a single crystal.

gradient theory, crystal plasticity, dissipation, length scale, cyclic deformation, numerical regularization

The rate-problem of continuing equilibrium is examined for a general class of rate-independent elastoplastic solids, without assuming the normality flow rule or symmetry of the tangent stiffness matrix. Accordingly, the problem addressed is of non-potential type, for which the usual stationarity or minimum principles for a governing potential do not apply. It is shown that the rate-problem can nevertheless be formulated as a quasi-extremal energy principle. It is characterized by explicit dependence of the minimized energy function or functional not only on variables undergoing variations but also, although only in a particular way, on an unknown solution as a parameter. To enable transparent and mathematically simple presentation of the main concept, the energy function is defined in a finite-dimensional setting for a spatially discretized material body with generalized velocities and a number of plastic multipliers as unknowns. If a solution is non-unique then incrementally stable solutions can be selected using the quasi-extremal principle in which the minimized energy function includes the second-order terms. Examples and extensions concern an elastic-plastic continuum obeying a non-associative plastic flow rule, without or with a higher-order gradient term in the loading function. The issue of selection of active slip-systems in a single crystal of a non-symmetric slip-system interaction matrix is also addressed.

non-associative plasticity, crystal plasticity, rate problem, free energy potential, dissipation, internal variables, thermodynamic basis, variational formulation, incremental energy minimization, non-uniqueness, path stability, gradient theory

A class of crystal plasticity models based on the concept of microforces conjugate to slip-rate gradients is examined in the small strain framework. As an extension of the usual formulation, slip-rate gradients are introduced here into the incremental hardening law, including in this way a natural internal length scale derived recently in a closed form from relationships of the physically-based dislocation theory of plasticity. The condition for plastic flow on a crystallographic slip system involves other length scales, associated with the second-order gradients of slip and slip rate in energetic and dissipative terms, respectively. The interplay between the length-scales of physically different origin is illustrated by the examples of monotonic and cyclic deformation of one- and two-dimensional models of Cu single crystals with boundary constraints imposed on plastic slips. It is shown that selected earlier results are reproduced accurately if one or another length scale ceases to play an essential role. For cyclic deformations, the effects of the energetic length scale in the flow condition and of the natural length scale in the incremental hardening law can both be significant at the micron scale.

Gradient theory, Dissipation, Crystal plasticity, Length scale, Cyclic deformation

Several micromechanical and numerical approaches to estimating the effective properties of heterogeneous media are analyzed. First, micromechanical mean-field estimates of elastic moduli for selected metal matrix composite systems are compared with the results of finite element calculations performed for two simplified unit cells: spherical and cylindrical. Advantages and deficiencies of such numerical verification of analytical homogenization schemes are indicated. Next, predictions of both approaches are compared with available experimental data for two composite systems for tension and compression tests in the elastic-plastic regime using tangent and secant linearization procedures. In the examined range of strain and ceramic volume content, both the Mori-Tanaka averaging scheme and the generalized self-consistent scheme lead to reliable predictions when combined with the tangent linearization, while the use of secant moduli results in a too stiff response. It is also found that the mean-field predictions for a small ceramic volume content are very close to the results obtained from the finite-element analysis of a spherical unit cell.

Metal-matrix composites, Effective properties, Analytical estimates, Numerical homogenization, Nonlinear analysis

In this paper, rate-independent dissipation is introduced into the phase-field framework for modelling of displacive transformations, such as martensitic phase transformation and twinning. The finite-strain phase-field model developed recently by the present authors is here extended beyond the limitations of purely viscous dissipation. The variational formulation, in which the evolution problem is formulated as a constrained minimization problem for a global rate-potential, is enhanced by including a mixed-type dissipation potential that combines viscous and rate-independent contributions. Effective computational treatment of the resulting incremental problem of non-smooth optimization is developed by employing the augmented Lagrangian method. It is demonstrated that a single Lagrange multiplier field suffices to handle the dissipation potential vertex and simultaneously to enforce physical constraints on the order parameter. In this way, the initially non-smooth problem of evolution is converted into a smooth stationarity problem. The model is implemented in a finite-element code and applied to solve two- and three-dimensional boundary value problems representative for shape memory alloys

Phase-field method, Microstructure, Martensite, Twinning, Non-smooth optimization

A novel methodology is proposed for estimating the strain hardening exponent of a metal single crystal directly from the spherical indentation test, without the need of solving the relevant inverse problem. The attention is focused on anisotropic piling-up and sinking-in that occur simultaneously in different directions, in contrast to the standard case of axial symmetry for isotropic materials. To correlate surface topography parameters with the value of material hardening exponent, a finite-element study of spherical indentation has been performed within a selected penetration depth range using a finite-strain crystal plasticity model. It is shown how the power-law hardening exponent can be estimated from the measured pile-up/sink-in pattern around the residual impression after indentation in a (001)-oriented fcc single crystal of a small initial yield stress. For this purpose, a new parameter of surface topography is defined as the normalized material volume displaced around the nominal contact zone, calculated by integration of the local residual height (positive or negative) over a centered circular ring. That indicator can be easily determined from an experimental topography map available in a digital form. Comparison is made with the estimates based on measurements of the contact area and the slope of the load–penetration depth curve in logarithmic coordinates. The proposed methodology is extended to estimation of the hardening exponent simultaneously with the initial yield stress when the latter is not negligible. Experimental verification for a Cu single crystal leads to promising conclusions.

metal crystal, elastoplasticity, finite deformation, strain hardening, experimental identification

A finite-strain phase field model for martensitic phase transformation and twinning in shape memory alloys is developed and confronted with the corresponding sharp-interface approach extended to interfacial energy effects. The model is set in the energy framework so that the kinetic equations and conditions of mechanical equilibrium are fully defined by specifying the free energy and dissipation potentials. The free energy density involves the bulk and interfacial energy contributions, the latter describing the energy of diffuse interfaces in a manner typical for phase-field approaches. To ensure volume preservation during martensite reorientation at finite deformation within a diffuse interface, it is proposed to apply linear mixing of the logarithmic transformation strains. The physically different nature of phase interfaces and twin boundaries in the martensitic phase is reflected by introducing two order-parameters in a hierarchical manner, one as the reference volume fraction of austenite, and thus of the whole martensite, and the second as the volume fraction of one variant of martensite in the martensitic phase only. The microstructure evolution problem is given a variational formulation in terms of incremental fields of displacement and order parameters, with unilateral constraints on volume fractions explicitly enforced by applying the augmented Lagrangian method. As an application, size-dependent microstructures with diffuse interfaces are calculated for the cubic-to-orthorhombic transformation in a CuAlNi shape memory alloy and compared with the sharp-interface microstructures with interfacial energy effects.

Phase-field method, Microstructure, Martensite, Size effects, Shape memory alloys

A simple gradient-enhancement of the classical continuum theory of plasticity of single crystals deformed by multislip is proposed for incorporating size effects in a manner consistent with phenomenological laws established in materials science. Despite considerable efforts in developing gradient theories, there is no consensus regarding the minimal set of physically based assumptions needed to capture the slip-gradient effects in metal single crystals and to provide a benchmark for more refined approaches. In order to make a step towards such a reference model, the concept of the tensorial density of geometrically necessary dislocations generated by slip-rate gradients is combined with a generalized form of the classical Taylor formula for the flow stress. In the governing equations in the rate form, the derived internal length scale is expressed through the current flow stress and standard parameters so that no further assumption is needed to define a characteristic length. It is shown that this internal length scale is directly related to the mean free path of dislocations and possesses physical interpretation which is frequently missing in other gradient-plasticity models.

gradient plasticity, geometrically necessary dislocations, single crystal, strain-hardening, internal length scale, size effect

In our previous paper, a simple gradient-enhancement of the classical continuum theory of plasticity of single crystals deformed by multislip has been proposed for incorporating size effects. A single internal length scale has been derived as an explicit function of the flow stress defined as the isotropic part of critical resolved shear stresses. The present work is focused on verification whether the simplifications involved are not too severe and allow satisfactory predictions of size effects. The model has been implemented in a finite element code and applied to three-dimensional simulations of fcc single crystals. We have found that the experimentally observed indentation size effect in a Cu single crystal is captured correctly in spite of the absence of any adjustable length-scale parameter. The finite element treatment relies on introducing non-local slip rates that average and smoothen on an element scale the corresponding local quantities. Convergence of the finite element solution to the analytical one is also verified for the one-dimensional problem of a boundary layer formed at a constrained interface.

gradient plasticity, geometrically necessary dislocations, boundary layer, size effects, indentation, finite element method

A new constitutive algorithm for the rate-independent crystal plasticity is presented. It is based on asymptotically exact formulation of the set of constitutive equations and inequalities as a minimum problem for the incremental work expressed by a quadratic function of non-negative crystallographic slips. This approach requires selective symmetrization of the slip-system interaction matrix restricted to active slip-systems, while the latent hardening rule for inactive slip-systems is arbitrary. The active slip-system set and incremental slips are determined by finding a constrained minimum point of the incremental work. The solutions not associated with a local minimum of the incremental work are automatically eliminated in accord with the energy criterion of path stability. The augmented Lagrangian method is applied to convert the constrained minimization problem to a smooth unconstrained one. Effectiveness of the algorithm is demonstrated by the large deformation examples of simple shear of a face-centered cubic (fcc) crystal and rolling texture in a polycrystal. The algorithm is extended to partial kinematic constraints and applied to a uniaxial tension test in a high-symmetry direction, showing the ability of the algorithm to cope with the non-uniqueness problem and to generate experimentally observable solutions with a reduced number of active slip-systems.

solids, crystal plasticity, rate-independent constitutive equations, material stability, variational methods, incremental energy minimization, augmented Lagrangian method

Texture evolution in commercially pure titanium deformed by equal-channel angular pressing (ECAP) and extrusion with forward–backward rotating die (KoBo) is studied both experimentally and numerically. New results are provided that demonstrate the effects of distinct and complex deformation paths on the texture in the ultra-fine grained (UFG) material obtained after severe plastic deformation (SPD). The numerical simulations are based on the self-consistent viscoplastic method of grain-to-polycrystal scale transition. A recently proposed modification of the probabilistic scheme for twinning is used that provides consistent values of the twin volume fraction in grains. The basic components of the experimentally observed texture are reasonably well reproduced in the modelling. The numerical simulations provide an insight into the internal mechanisms of plastic deformation, revealing substantial activity of mechanical twinning in addition to the basal and prismatic slip in titanium processed by ECAP.

Texture evolution, UFG materials, SPD processes, Crystal plasticity, Twinning

Mathematical and physical aspects of the applicability of the Onsager, Prigogine as well as the Glansdorff and Ziegler thermodynamic extremal principles (TEPs) to non-equilibrium thermodynamics are examined for systems at fixed temperature with respect to their ability to provide kinetic equations approved in materials science. TEPs represent an alternative to the classical phenomenological equations approach. As TEPs are, more or less, a pure mathematical tool, they cannot significantly contribute to a deeper physical understanding. However, if a system can be described by discrete characteristic (thermodynamic) parameters, it is demonstrated that application of Onsager’s TEP or Ziegler’s TEP represents a systematic way to derive a set of explicit evolution equations for these parameters. This approach can significantly simplify the treatment of the problem and, thus, can also be applied to rather complex systems, for which the classical approach, involving application of phenomenological equations, fails. The application of TEPs is demonstrated on plasticity with respect to constitutive equations as well as on grain growth and coarsening with respect to evolution equations of discrete parameters. No exploitation of Prigogine’s TEP has been reported for applications in materials science. Contrarily, Prigogine’s TEP can be invalidated if the coefficients of the dissipation function depend on the evolution of state variables with time. This is demonstrated by a further practical example worked out for the solute drag phenomenon. Glansdorff’s and Prigogine’s evolution criterion, however, is always fulfilled near the equilibrium state of convex Gibbs energy. Extensions of TEPs to non-linear non-equilibrium thermodynamics are demonstrated for homogeneous and quasi-homogeneous dissipation functions.

Non-equilibrium, Thermodynamics, Entropy, Onsager’s principle, Thermodynamic extremal principles

The optimization procedure is worked out for finding an optimal content of phases in metal–ceramic composites in case of conflicting objectives regarding thermo-mechanical properties of the material for a specific target application. Relationships between the material composition and effective properties of the composite are calculated by employing several methods of continuum micromechanics. A constrained minimization problem is solved for a single objective function based on the weighted squared distances from the best available thermo-mechanical properties for the material system selected. A compound block diagram is proposed for quick assessment of the consequences of deviating from the optimal composition. The developed procedure is applied to practical examples of Al2O3–Cu composites for brake disks and Al2O3–NiAl composites for valves of potential use in automotive industry.

Metal–matrix composites (MMCs), Thermomechanical, Plastic deformation, Micro-mechanics, Multi-criteria optimization

Nano- and micro-indentation of Cu single crystals is performed in directions not aligned with crystallographic axes. Such tests correspond to mechanical characterization of incidentally oriented grains in a polycrystalline or composite material. Orientation and size dependence of complex patterns of surface piling-up and sinking-in around the imprint are investigated. Experimental observations are compared with finite element simulations based on the large deformation crystal plasticity theory.

Copper, Nanoindentation, AFM, EBSD, Crystal plasticity, Finite element method

A model of pseudoelasticity in shape memory alloys is developed within the incremental energy minimization framework. Three constitutive functions are involved: the Helmholtz free energy and rate-independent dissipation that enter incrementally the minimized energy function, and the constraint function that defines the limit transformation strains. The proposed implementation is based on a unified augmented Lagrangian treatment of both the constitutive constraints and nonsmooth dissipation function. A methodology for easy reformulation of the model from the small-strain to finite-deformation regime is presented. Finite element computations demonstrate robustness of the finite-strain version of the model and illustrate the effects of tension–compression asymmetry and transversal isotropy of the surface of limit transformation strains.

shape memory alloys (SMA), phase transformation, energy methods, finite element method, augmented Lagrangian method

The phenomenon of spontaneous formation of deformation bands in metal single crystals deformed plastically by crystallographic multislip is investigated theoretically by using the energy criterion of instability of a uniform deformation path. The second-order energy criterion for incipient deformation banding is derived in a time-continuous setting for a rate-independent elastic–plastic crystal. The need for selective symmetrization of the local interaction matrix for active slip-systems is demonstrated. A computational approach to deformation banding is developed by using non-convex constrained minimization of the incremental work with respect to increments in crystallographic shears and kinematical degrees of freedom. Calculated examples of deformation banding patterns in fcc single crystals are compared with experimental observations.

Metal crystal, Plasticity, Finite deformation, Laminate microstructure, Incremental energy minimization

This study is concerned with isothermal stability of equilibrium of evolving laminated microstructures in pseudo-elastic solids with a multi-well free energy function. Several possible modes of instability associated with phase transition between energy wells are analysed. The related rate-independent dissipation is included by imposing a threshold value on the thermodynamic driving force. For a homogenized phase-transforming laminate with no length scale it is shown that localization instability is a rule in case of a non-zero interfacial jump of a directional nominal stress, irrespectively of actual boundary conditions. A stabilizing effect of elastic micro-strain energy at the boundary of the localization zone is demonstrated for laminates of finite spacing. Illustrative numerical examples are given for an evolving austenite–martensite laminate in a crystal of CuZnAl shape memory alloy.

Microstructures, Phase transformation, Laminates, Energy methods, Stability

A procedure for the modelling of microstructural changes induced by non-uniform large plastic deformation of metals is developed. For a given plastic working process, a set of deformation paths for different initial locations of a material element is generated first for a non-hardening material by using the finite element method. Next, changes of a cellular microstructure and related hardening effects along each path are calculated by using a recently proposed model. The procedure is applied to a non-conventional process of cold extrusion assisted by cyclic rotation of the die. The evolution of microstructural parameters, their effect on strain hardening and a distribution map over the specimen cross-section are calculated.

non-uniform deformation, dislocation cells, grain refinement, hardening, quantitative prediction

The interfacial energy due to elastic micro-strains at the austenite–twinned martensite interface is calculated for the NiTi shape memory alloy undergoing cubic-to-monoclinic B2 ↔ B19' transformation. For each crystallographically distinct microstructure, an energetically favourable local shape of the interface is determined. The approach employs finite element computations and energy minimization with respect to shape parameters, taking into account elastic anisotropy of the phases and finite-strain kinematics. The effect of atomic-scale interfacial energy is studied.

microstructure, martensitic phase transformation, shape memory alloys (SMA), interface structure, micromechanical modelling

The paper addresses the problem of suitable approximation of the interaction between phases in heterogeneous materials that exhibit both viscous and elastic properties. A novel approach is proposed in which linearized subproblems for an inhomogeneity-matrix system with viscous or elastic interaction rules are solved sequentially within one incremental step. It is demonstrated that in the case of a self-consistent averaging scheme, an additional accommodation subproblem, besides purely viscous and elastic subproblems, is to be solved in order to estimate the material response satisfactorily. By examples of an isotropic two-phase material it is shown that the proposed approach provides acceptable predictions in comparison with the existing models.

Micromechanics, Viscoelasticity, Viscoplasticity, Homogenization, Self-consistent scheme

The symmetry issue of the interaction matrix between multiple slip-systems in the theory of crystal plasticity at finite deformation is revisited. By appealing to possibly non-uniform distribution of slip-system activity in a representative space-time element of a crystal, symmetry of the slip-system interaction matrix for the representative element is derived under assumptions that have a physical meaning. This conclusion refers to active slip-systems only. Accordingly, for any given hardening law, a new symmetrization rule is proposed that is restricted to active slip-systems and leaves the latent hardening of inactive slip-systems unchanged. Advantages of the proposal in comparison with full symmetrization are illustrated by a simple example of uniaxial tension.

finite deformation, metal crystal, plasticity, hardening, symmetry

A multi-scale model of stress-induced phase transformation and martensite variant reorientation in shape memory alloy (SMA) polycrystals is developed. It is proposed to include neighbouring-grain interaction in a simple manner by introducing an intermediate bi-crystal level into the sequential averaging scheme for SMA. The constitutive relationships are defined by specifying the free energy and dissipation functions. At the level of a single grain, the rate-independent dissipation function is used that incorporates the dissipation due to forward and reverse austenite-to-martensite transformation as well as reorientation of martensite variants. The global response of the model is simulated numerically by minimizing the total incremental energy supply. Specific examples are calculated for a NiTi polycrystal for proportional and non-proportional loading paths.

Phase transformation, Dissipation, Microstructure, Multi-scale model, Incremental energy minimization

This paper is a continuation of the Part I (H. Petryk, S. Stupkiewicz, Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory. J. Mech. Phys. Solids, 2010, doi:10.1016/j.jmps.2009.11.003). A fully three-dimensional model of an evolving martensitic microstructure is examined, taking into account size effects due to the interfacial energy and also dissipation related to annihilation of interfaces. The elastic micro-strain energy at microstructured interfaces is determined with the help of finite element computations and is approximated analytically. Three interface levels are examined: of grain boundaries attained by parallel martensite plates, of interfaces between austenite and twinned martensite, and of twin interfaces within the martensite phase. Minimization of the incremental energy supply, being the sum of the increments in the free energy and dissipation of the bulk and interfacial type at all levels, is used as the evolution rule, based on the theory presented in Part I. An example of the formation and evolution of a rank-three laminated microstructure of finite characteristic dimensions in a pseudoelastic CuAlNi shape memory alloy is examined quantitatively.

Microstructures, Phase transformation, Grain boundaries, Energy methods, Shape memory alloys (SMA)

A model of evolving martensitic microstructures is formulated that incorporates the interfacial energy and dissipation on three different scales corresponding to the grain boundaries attained by martensite plates, the interfaces between austenite and martensite plates, and the twin interfaces within martensite plates. Three different time scales are also considered in order to clarify the meaning of rate-independent dissipation related to instabilities at more refined temporal and spatial scales. On the slowest time scale, the process of deformation and martensitic phase transformation is investigated as being composed of segments of smooth quasi-static evolution separated by sudden jumps associated with creation or annihilation of interfaces. A general evolution rule is used in the form of minimization of the incremental energy supply to the whole compound thermodynamic system, including the rate-independent dissipation. Close relationship is shown between the evolution rule and the thermodynamic condition for stability of equilibrium, with no a priori assumption on convexity of the dissipation function. It is demonstrated that the extension of the minimum principle from the first-order rates to small but finite increments requires a separate symmetry restriction imposed on the state derivative of the dissipation function. Formulae for the dissipation associated with annihilation of interfaces are proposed that exhibit limited path-independence and satisfy that symmetry requirement. By exploiting the incremental energy minimization rule with the help of the transport theorems, local propagation conditions are derived for both planar and curved phase transformation fronts. The theory serves as a basis for the algorithm for calculation of the stress-induced evolution of martensitic microstructures along with their characteristic dimensions and related hysteresis loops in shape memory alloys; the examples are given in Part II of the paper.

Microstructures, Phase transformation, Grain boundaries, Energy methods, Stability

Size effects in pseudoelastic polycrystalline shape memory alloys are studied by considering a representative spherical laminated domain (subgrain) and its interfacial energy at three scales: at the subgrain boundaries, at the austenite–martensite interfaces, and at the twin boundaries. Two sources of interfacial energy are accounted for, namely the atomic-scale energy of twin and phase boundaries and the elastic strain energy at microstructured interfaces, the latter being predicted theoretically. The evolution of microstructure of the representative domain is determined using the incremental energy minimization rule applied to the sum of the increments in the Helmholtz free energy and rate-independent dissipation. The size-dependent part of dissipation is estimated by assuming that negative increments in interfacial energy, associated with annihilation of interfaces, cannot be reverted back into the bulk free energy and are thus dissipated. Simple analytic formula for the interfacial energy dissipated in a complete forward-reverse transformation cycle is derived and combined with a micromechanical model of a polycrystalline NiTi shape memory alloy. A numerical example illustrating size-dependent hysteresis in the stress-induced martensitic transformation is presented.

interfacial energy, incremental energy minimization, dissipation, martensitic transformation

W niniejszym artykule przedstawiono energetyczne podejście do wieloskalowego modelowania ewolucji mikrostruktur martenzytycznych w stopach z pamięcią kształtu. Energia swobodna Helmholtza i energia dyssypowana w układzie reprezentowane są przez sumy członów odpowiadających energii objętościowej oraz energii powierzchniowej na granicach mikro-strukturalnych pomiędzy poszczególnymi wariantami martenzytu, fazami lub ziarnami. Ewolucja mikrostruktury jest wyznaczana drogą przyrostowej minimalizacji całkowitej energii dostarczanej do rozpatrywanego układu w procesie makroskopowo quasi-statycznym i izotermicznym. Ogólną procedurę zastosowano do numerycznych symulacji powstawania i ewolucji warstwowych struktur martenzytycznych indukowanych naprężeniowo w stopach z pamięcią kształtu. W energii powierzchniowej uwzględniono energię mikroodkształceń sprężystych w otoczeniu granic mikrostrukturalnych, wyznaczoną przy użyciu metody elementów skończonych. Policzone przykłady opisują ewolucję mikrostruktury martenzytycznej w formie laminatu trzeciego rzędu w stopie CuAlNi dla przemiany β1→γ1′ oraz jej zależność od sposobu uwzględnienia dyssypacji energii powierzchniowej.

energia powierzchniowa, minimalizacja energii, przemiana martenzytyczna

The effect of severe plastic deformation (SPD) during cyclic multi-axis compression on grain refinement and strain hardening in interstitial free (IF) steel is studied quantitatively. In the experimental part, the material samples were cold deformed in the MAXStrain(R) system by successive compression in two mutually orthogonal directions. The electron backscatter diffraction (EBSD) technique was used to measure the average spacing of the dislocation cell (low angle) and cell-block (high angle) boundaries. In the modelling part, the decrease in size of dislocation cells and cell blocks was expressed in terms of the effective plastic strain defined such that strain-rate reversals slow down its accumulation. The strengthening effect of microstructural evolution was included in the continuum mechanics framework of finite strain plasticity. Examples of simulation of the behaviour of IF steel severely deformed by multi-axis compression are calculated and compared to experimental data.

Modelling, Microstructure, Hardening, Severe plastic deformation

A micromechanical scheme is developed for predicting the morphology and interfacial energy of the interface layer between the parent phase and internally twinned martensite. Low-energy morphologies are determined by minimizing, with respect to shape parameters, the elastic microstrain energy associated with local incompatibility of transformation strains. The computational scheme involves a finite element solution to a problem of non-linear elasticity with eigenstrains, shape sensitivity analysis with respect to general shape parametrization and minimization employing a gradient-based algorithm. As an application, low-energy morphologies are studied for the austenite–martensite interface in the cubic-to-orthorhombic transformation in a CuAlNi shape memory alloy. Discussion of the results of the analysis includes comparison to alternative simplified methods in terms of the predicted morphologies and the corresponding interfacial energies.

Microstructure, Martensitic phase transformation, Shape memory alloys (SMA), Interface structure, Micromechanical modelling

The effect of severe plastic deformation (SPD) on grain refinement and strain hardening in polycrystalline metals is studied quantitatively. The decrease in size of dislocation cells and cell-blocks is expressed as a function of the effective plastic strain influenced by strain-rate reversals. The estimated growth of the high-angle boundary area fraction depends on the complexity of the three-dimensional deformation path. The strain hardening due to both dislocation and boundary strengthening is described in terms of microstructural parameters and incorporated in the continuum mechanics framework of finite strain plasticity. The proposed model provides a tool for quantitative comparison of different SPD processes. Examples of simulation of the behaviour of pure aluminium deformed by equal channel angular pressing (ECAP) and cyclic extrusion–compression are calculated.

Modelling, Microstructure, Severe plastic deformation, Ultra-fine grained materials

The Taylor-Lin polycrystal model is used to simulate plastic deformations of a polycrystal. These propagate a corner in the subsequent yield surface due to the intersection of the yield loci of a number of slip mechanisms. Two approaches to identify subsequent yield surfaces and the development of the yield vertex at the applied stress are discussed. A linear regression analysis of the rounded corner is used to identify the corner angle and its development for different strain processes. On deformation paths composed of two aligned segments, the development of a secondary vertex is shown to follow similar laws.

The effect of severe plastic deformation (SPD) during cyclic extrusion-compression (CEC) on grain refinement and strain hardening in AlMgSi alloy is studied quantitatively. New experimental results are presented showing that the average microband thickness and grain size decrease below 100 nm, i.e. a nanocrystalline material is obtained. In the modelling part, the decrease in size of dislocation cells and microbands is expressed in terms of the effective plastic strain defined such that strain rate reversals slow down its accumulation. Examples of simulation of the behaviour of AlMgSi alloy severely deformed by cyclic extrusion-compression are calculated and compared to experimental data.

Modelling, Microstructure, Hardening, Severe plastic deformation

Two severe plastic deformation (SPD) processing techniques, namely equal-channel angular pressing (ECAP) and cyclic extrusion-compression (CEC), are investigated by using the finite element method. The major aspect examined is the non-uniformity of the accumulated, equivalent plastic strain after processing with the use of different shapes of the die. The quantitative effect of several parameters on the plastic flow is determined. It is found that the diameter ratio of the chambers and narrower channel in the CEC method, and also the inclination angle of connecting conical parts, can affect strongly the degree of strain non-uniformity. Comparison is made of distributions of equivalent strain after two passes of ECAP for two different routes and with two die profiles.

A micromechanical model of stress-induced martensitic transformation in single crystals of shape memory alloys is developed. This model is a finite-strain counterpart to the approach presented recently in the small-strain setting [S. Stupkiewicz, H. Petryk, J. Mech. Phys. Solids 50 (2002) 2303–2331]. The stress-induced transformation is assumed to proceed by the formation and growth of parallel martensite plates within the austenite matrix. Propagation of phase transformation fronts is governed by a rate-independent thermodynamic criterion with a threshold value for the thermodynamic driving force, including in this way the intrinsic dissipation due to phase transition. This criterion selects the initial microstructure at the onset of transformation and governs the evolution of the laminated microstructure at the macroscopic level. A multiplicative decomposition of the deformation gradient into elastic and transformation parts is assumed, with full account for the elastic anisotropy of the phases. The pseudoelastic behavior of Cu–Zn–Al single crystal in tension and compression is studied as an application of the model.

Microstructures, Laminates, Finite deformations, Micromechanics, Shape memory alloys

A distinctive feature of the examined class of solids is that a part of the entropy production is due to rate-independent dissipation, as in models of plasticity, damage or martensitic transformations. The standard condition for thermodynamic stability is shown to be too restrictive for such solids and, therefore, an extended condition for stability of equilibrium is developed. The classical thermodynamic theory of irreversible processes is used along with the internal variable approach, with the emphasis on the macroscopic effects of micro-scale instabilities in the presence of two different scales of time. Specific conditions for material stability against internal structural rearrangements under deformation-sensitive loading are derived within the incremental constitutive framework of multi-mode inelasticity. Application to spontaneous formation of deformation bands in a continuum is presented. Conditions for stability or instability of a quasi-static process induced by varying loading are given under additional constitutive postulates of normality and symmetry. As illustration of the theory, the stability of equilibrium or a deformation path under uniaxial tension is analysed for a class of inelastic constitutive laws for a metal crystal deformed plastically by multi-slip or undergoing stress-induced martensitic transformation.

A micromechanical scheme is developed for the analysis of elastic micro-strains induced by local incompatibilities at the austenite-twinned martensite interface. The aim of the paper is to estimate the elastic micro-strain energy which is an important factor in the formation of microstructures during the martensitic transformation. The finite deformation framework is applied, consistent with the crystallographic theory of martensite, and full account is taken for elastic anisotropy of the phases. As an example, the microstructures in the cubic-to-orthorhombic transformation in CuAlNi shape memory alloy are analyzed by the finite element method for the assumed class of zigzag shapes of the austenite-martensite interface at the micro-level. Finally, the effect of the interphase boundary energy on the microstructure of the transition layer is studied.

microstructure, martensitic phase transformation, shape memory alloys (SMA), interface structure, micromechanical modelling

A micro-mechanical model of stress-induced martensitic transformation in shape memory alloys is presented. A laminated microstructure of austenite and martensite phases is assumed along with a time-independent thermodynamic criterion for phase transformation. In numerical examples, the pseudoelastic behaviour of single crystals of CuZnAl and CuAlNi shape memory alloys is investigated. Several aspects are examined, including the effects of the loading direction, external constraints, detwinning, and instability of macroscopically uniform transformation.

The paper is concerned with modelling of stress-induced martensitic transformations in single crystals of shape memory alloys. The transformation is assumed to proceed by the formation and growth of parallel martensitic plates within an austenite matrix, as commonly observed in experiments. Phase transition is governed by a time-independent thermodynamic criterion. Martensite variant rearrangement (detwinning) is accounted for in case of internally twinned martensites. The examples illustrate the effect of deformation constraints on the microstructure evolution and overall response. Instability of macroscopically uniform transformation is predicted due to the softening behaviour at the material point scale.

The incremental energy minimization is examined as a method of determining solution paths for time-independent dissipative solids. Isothermal quasi-static deformations are considered, and the deformation work is locally decomposed into the increments in free energy and intrinsic dissipation. General conditions necessary for the applicability of the minimization procedure are derived and discussed.

Solids and structures, Dissipative materials, Plasticity, Energy, Path stability

The energy criterion of stability of equilibrium is used to predict the formation of discrete twin bands running across the grains of polycrystalline solids. An uncontrolled rapid development of a twin band is assumed to occur when the associated release of elastic energy compensates the interfacial energy and intrinsic dissipation due to twinning. A simplified analysis is performed that leads to compact analytic formulae for the thickness of a twin band and for the critical resolved stress. The creation of subsequent discrete twin bands in a finite solid can be separated by the periods of quasistatic growth of existing twins under varying loads. An application of the energy criterion to twinning in TiAl intermetallics and its verification by observed data for the thickness of a twin band are presented.

Dynamic stability of uniform straining of a nonlinear elastic solid is known to require that all eigenvalues of the acoustic tensor associated with the tangent elastic moduli be real and nonnegative. The focus of this note is to what extent this conclusion applies to time-independent, elastoplastic materials. Nonlinearity of the elastic–plastic constitutive law imposes limits on validity of a solution to the linear problem for which the acoustic tensor is determined. The effect of those limits on the conclusions about instability is examined.

Instability of plastic flow, Elastic–plastic material, Material stability, Flutter

The formation of multiple macroscopic shear bands is investigated as a mechanism of advanced plastic flow of polycrystalline metals. The overall deformation pattern and material characteristics are determined beyond the critical instant of ellipticity loss, without the need of introducing an internal length scale. This novel approach to the modelling of post-critical plastic deformation is based on the concept of a representative nonuniform solution in a homogeneous material. The indeterminacy of a post-critical representative solution is removed by eliminating unstable solution paths with the help of the energy criterion of path instability. It is shown that the use of micromechanically based, incrementally nonlinear corner theories of time-independent plasticity leads then to gradual concentration of post-critical plastic deformation. The volume fraction occupied by shear bands is found to have initially a well-defined, finite value insensitive to the mesh size in finite element calculations. Further deformation depends qualitatively on details of the constitutive law. In certain cases, the volume fraction of active bands decreases rapidly to zero, leading to material instability of dynamic type. However, for physically hardening materials with the yield-vertex effect, the localization volume typically remains finite over a considerable deformation range. At later stages of the plane strain simulation, differently aligned secondary bands are formed in a series of bifurcations.

Plasticity, Shear bands, Material instability, Energy criterion, Bifurcation

This paper is concerned with micromechanical modelling of stress-induced martensitic transformations in crystalline solids, with the focus on distinct elastic anisotropy of the phases and the associated redistribution of internal stresses. Micro–macro transition in stresses and strains is analysed for a laminated microstructure of austenite and martensite phases. Propagation of a phase transformation front is governed by a time-independent thermodynamic criterion. Plasticity-like macroscopic constitutive rate equations are derived in which the transformed volume fraction is incrementally related to the overall strain or stress. As an application, numerical simulations are performed for cubic β1 (austenite) to orthorhombic γ1′ (martensite) phase transformation in a single crystal of Cu–Al–Ni shape memory alloy. The pseudoelasticity effect in tension and compression is investigated along with the corresponding evolution of internal stresses and microstructure.

Phase transformation, Microstructures, Layered material, Constitutive behaviour, Shape memory alloy

The concept of the second-order dissipation on arbitrary complex paths in the space of internal state variables is developed. A general framework for time-independent dissipative solids is adopted which encompasses plasticity, micro-cracking or martensitic phase transformation. Circumstances are established in which the dissipation evaluated to the second order is minimized on a radial path. As an application, this minimum property is used to simplify a sufficiency condition for stability of equilibrium.

This work is concerned with the formation of multiple macroscopic shear bands viewed as a mechanism of large plastic deformation of polycrystalline metals. The plastic deformation pattern in a time-independent material with a yield-surface vertex effect is investigated numerically in plane strain beyond the critical instant of ellipticity loss under quasi-static loading. The energy criterion of path instability applied to a family of post-critical solutions eliminates unstable paths and enables the overall deformation pattern to be determined, although the solutions remain locally indeterminate due to the absence of an internal length scale. In particular, the volume fraction of incipient shear bands is found to have a well-defined value irrespective of the mesh size in finite element calculations. As an apparently novel qualitative result, the formation of coarse, differently aligned secondary bands is observed at later stages of simulation.

Plasticity, Shear bands, Material instability, Energy criterion, Bifurcation

This study is concerned with multi-mode inelastic behaviour at macroscopically uniform deformation. The material is assumed to be time-independent; the physical origin of inelasticity may be otherwise arbitrary, including plasticity of crystals and polycrystals, micro-cracking, phase transformation, etc. A non-linear rate-problem of continuing mechanical equilibrium at finite strain is examined for a material element subject to deformation-sensitive loading under partial kinematic constraints. General conditions for uniqueness of the material response are established. As an application to predicting the onset of strain localization or failure, the condition is derived that excludes the bifurcation in a band from homogeneous deformation. In contrast to the usual requirement of ellipticity of the tangent stiffness moduli, the present condition for uniqueness takes into account any possible unloading and is directly imposed on the matrix of interaction moduli of internal mechanisms. Lower and upper bounds are established for the primary shear-band bifurcation along a smooth straining path.

Microstructures, Multi-mode inelasticity, Constitutive behaviour, Finite strain, Bifurcation

Averaging rules are derived for the rates of deformation gradient and nominal stress in heterogeneous solids undergoing quasi-static deformation and displacive phase transformation with coherent interfaces. Infinitesimal increments in strain and stress in the bulk material are accompanied by the finite increments in growing layers of a transformed phase. Expressions for the rates of the macroscopic variables and their products are given in several equivalent forms. The transport theorem and rate compatibility conditions for moving interfaces are extended to the initial instant of non-smooth transformation when the standard kinematical condition of compatibility is not satisfied. As an application of the averaging formulae, it is shown that the continuous growth of parallel planar layers of a transformed phase at a meso-level results in macroscopic constitutive rate equations analogous to the theory of plasticity. The normality law is obtained if the propagation of a phase transformation front in an elastic material takes place at a prescribed value of the thermodynamic driving force.

Phase transformation, Finite strain, Inhomogeneous material, Strain compatibility, Asymptotic analysis

General criteria of instability in time-independent elastic-plastic solids and the related computational approaches are reviewed. The distinction between instability of equilibrium and instability of a deformation process is discussed with reference to instabilities of dynamic, geometric or material type. Comparison is made between the bifurcation, energy and initial imperfection approaches. The effect of incremental nonlinearity of the constitutive law, associated with formation of a yield-surface vertex, on instability predictions is examined. A survey of the methods of post-critical analysis is presented.

A theoretical and numerical analysis of the formation of a localized neck in a biaxially stretched sheet is presented. A time-independent constitutive law is assumed to be incrementally non-linear as suggested by micromechanical studies of the elastoplastic deformation of polycrystalline metals. The incipient width of a necking band in an infinitely thin perfect sheet of a time-independent material is found here to have a well-defined initial value, proportional to the in-plane sheet dimension. During subsequent post-critical deformation the boundary of the necking band moves with respect to the material until the transition to localized necking is completed. These conclusions are derived on a theoretical route from the condition of stability of the post-bifurcation deformation process and are confirmed by the numerical analysis performed for a sheet of finite thickness.

Thermodynamic stability of equilibrium in solids with intrinsic dissipation of plastic type is investigated. Rate-dependent plastic behaviour and its rate-independent limit are described with the help of internal variables of averaging type. A general condition sufficient for the stability of equilibrium in the sense of Lyapunov is formulated for a continuous plastic body interacting with a reversible work source and a heat reservoir. The relation to the classical Gibbs condition is discussed. As an application, the second-order stability condition in time-independent plasticity is derived in various forms, depending on the class of disturbances under consideration.

A spatially discretized non-linear rate problem for a time-independent plastic solid is examined with particular reference to bifurcation. Constitutive non-linearity in a general form encompassing the yield-surface vertex effect is considered under the restriction that the tangent stiffness matrix for the whole system is symmetric. Theorems concerning existence, uniqueness and stability of solutions are presented. As an outcome of the theoretical analysis, a computational method is proposed for crossing bifurcation points with automatic rejection of an unstable postbifurcation branch. An illustrative example of plane strain tension is calculated by using the finite element method.

A systematic analysis of theoretical aspects of material instability is presented for a general class of time-independent, incrementally nonlinear solids which admit a velocity-gradient potential. The second-order energy criteria of instability of uniform straining and of equilibrium are derived for a material element embedded in a quasi-statically deforming continuum. Several theorems concerning propagation of acceleration waves in an incrementally nonlinear material are proved, with particular reference to stationary discontinuities. Initiation of shear band formation either by a dynamic growth of an initial disturbance or by a quasi-static bifurcation is studied as a symptom of material instability. The derived condition for the instability of uniform quasi-static straining does not coincide with that for instability of equilibrium and both differ in general from the familiar condition of strong ellipticity loss formulated for the tangent moduli. This is illustrated on an example of a non-proportional deformation path of material obeying the J2 corner theory of plasticity.

CONDITIONS sufficient for instability of deformation are examined for a class of incrementally nonlinear, time independent inelastic solids. Instability of a quasi-static deformation process (path) at varying loading is distinguished from a narrower concept of an unstable equilibrium state. The energy criterion is extended to deformation paths under generał assumptions which ensure that the incremental boundary value problem can be given a variational formulation. For a discretized problem, fulfilłment of the instability condition along a path is shown to imply either instability of the traversed equilibrium states in a dynamic sense or persisting possibility of quasi-static bifurcation at varying loading. For a continuum, instability at varying loading is interpreted as sensitivity of the incremental deformation to arbitrarily small perturbing forces.

A general constitutive inequality for time-independent solids has been recently derived by Petryk from the normality and symmetry assumptions adopted at a micro-level of an elastic-plastic heterogeneous aggregate. The objective of this communication is to discuss its validity for a class of inelastic solids characterized by the existence of a thermodynamic potential and by the maximum dissipation principle. As an example, a material with a number of Griffith's micro-cracks is considered.

Elastoplasticity, Griffith crack, Variational inequality, Heterogeneous material, Internal variable material, Microcrack, Thermodynamics

A constitutive inequality for time-independent elastic-plastic models of polycrystalline metals at finite strain is derived from the known constitutive framework for single metal crystals obeying the normality and symmetry postulates. Consequences of the inequality are examined for a general class of nonlinear constitutive rate equations at a yield-surface vertex. It is demonstrated that existence of a strain-rate potential and the normality flow rule can be inferred from the inequality alone. With the help of the constitutive inequality, it is proved that bifurcation in velocities is ruled out by positive definiteness of the quadratic functional based on the tangent moduli. Micromechanically-based justification for linearization of the bifurcation problem along a regular deformation path is obtained in that way, without the need of specifying the full constitutive law. As another example of application of the constitutive inequality, explicit restrictions on phenomenological models of material behaviour at a yield-surface vertex are derived.

THE EXPRESSIONS are derived for the first and second time derivatives of integrals and functionals whose volume or surface domains of integration vary in time. As an example, the time derivative of the potential energy in non-linear elasticity in the case of varying body domain is determined. A moving strain and stress discontinuity surface is also considered and the associated energy derivatives are obtained. The derivatives of functionals with additional constraint conditions are finally discussed by using the primary and adjoint state fields.

Second-order rate constitutive equations are formulated for a time-independent elastic-plastic material, obeying the normality flow rule with a smooth yield surface. Under specified regularity restrictions imposed on the involved fields, the regular second-order rate boundary value problem with quasistatic accelerations as unknowns is posed. It is shown that every solution of this generally non-linear rate problem is governed by a variational principle and that the corresponding functional reaches a strict absolute minimum, provided the solution satisfies a sufficient uniqueness condition. With the same incrementally linear comparison solid, Hill's exclusion condition rules out not only a first- but also a second-order bifurcation. The criticality of the exclusion condition is discussed and conditions are indicated under which a second-order bifurcation becomes possible, while the first-order rate problem is still uniquely solvable.

An analysis of the steady rolling of a rough rigid cylinder over the surface of a rigid-perfectly plastic solid is presented. An exact slip-line field solution is investigated by using some recently developed techniques. It is shown that the solution is not uniquely defined and infinitely many complete solutions can be constructed for given boundary conditions. The numerical results show interesting variations of the deformation pattern and of rolling resistance at high loads. The limits are predicted to the magnitudes of the applied load and braking torque for which the steady rolling is possible.

A systematic analysis of the oblique indentation of a rigid wedge into a semi-infinite block of rigid-perfectly plastic material is presented. It is shown that infinitely many solutions exist for given boundary conditions. Three types of slip-line fields corresponding to three distinct deformation modes are examined, two of which represent a generalization of the well-known classical solutions. A number of numerical results are given. The completeness of the solutions, as well as their connection with the solutions for steady-state machining, is discussed.

A computationally efficient procedure for modelling of microstructural changes on complex and spatially nonuniform deformation paths of severe plastic deformation (SPD) is presented. The analysis follows a two-step procedure. In the first step, motivated by saturation of material hardening at large accumulated strains, the steady-state kinematics of the process is generated for a non-hardening viscoplastic model by using the standard finite element method for a specified SPD scheme. In the second step, microstructural changes are investigated along the deformation-gradient trajectories determined in the first step for different initial locations of a material element. The aim of this study is to predict texture evolution and grain refinement in a non-conventional process of cold extrusion assisted by cyclic rotation of the die, called KOBO process, which leads to an ultra-fine grain structure. The texture evolution is calculated for fcc and hcp metals by applying crystal visco-plasticity combined with the self-consistent scale transition scheme. In parallel, by applying the simplified phenomenological model of microstructure evolution along the trajectories, grain refinement is modelled. The results are compared with available experimental data.

SPD processes, Texture evolution, UFG materials, Crystal plasticity, Grain refinement

Micromechanical modelling of metal-ceramic composites has been carried out to obtain a material of required thermo-mechanical properties. Quantitative transition from phase properties and morphology to macroscopic properties of a composite has been modelled by mean-field approaches, including the self-consistent scheme. An optimization method has been developed for the objective function that expresses a distance between the required values of macro-variables and those determined for a given set of microstructural parameters. The presented example concerns application of Al2O3-Cu composite to brake disks.

multi-objective optimization, composite selection, metal matrix composites

Stress-induced formation and growth of internally twinned martensite plates within the austenite matrix in crystals of shape memory alloys is investigated. The recently developed model of evolving laminates is extended by taking into account the interfacial energy at three different scales. The interfacial energy density due to elastic strains at a twinned martensite boundary is calculated by the finite element method. Analytic relationships between characteristic dimensions of the rank-two laminate are derived by minimizing the size-dependent contribution of the interfaces to the total free energy. Specific examples have been calculated for the cubic-to-orthorhombic (β1 to γ1) stress-induced transformation in a CuAlNi shape memory alloy.

This paper focuses upon the theoretical foundations of the modeling of elastoplastic polycrystalline aggregates. As a starting point, the known constitutive framework for a time-independent elastic-plastic material with a finite number of internal mechanisms of plastic deformation obeying the normality flow rule is adopted at a mesoscopic level (Subsection 2.1). Implications of that assumption for the macroscopic properties of a heterogeneous material have been studied theoretically and numerically in a number of papers. No survey of previous results is attempted here, rather, the aim of this article is to present nowel qualitative conclusions. In Subsection 2.2 the transmissibility of the structure of incremental constitutive relationships from the meso- to macro-level is proved for a spatially discretized aggregate, and expressions are derived for macroscopic effective hardening moduli. This goes beyond earlier results concerning the transition rules expressed in terms of the strain or stress rates and of the fourth-order tensors of instantaneous moduli or compliances. A known difficulty in the modelling of time-independent elastic-plastic crystalline solids is that the set of incremental constitutive relationships need not have a unique solution. This problem is addressed here in relation to the question of stability of an equilibrium state and of a solution path for a material element. With the thermodynamic framework presented in Subsection 2.3 as a basis, in Section 3 the stability conditions are formulated in terms of the energy function and of the hardening moduli. The requirement of the stability of a deformation process within the material element is introduced in Subsection 3.2 and used to generate criteria for selection of a solution when the material response is non-uniquely defined. Throughout the paper, bold-face small letters (Roman or Greek) denote secondorder symmetric tensors, and bold-face capital letters denote fourth-order tensors with the minor symmetries, with I as the respective unit tensor and a superscript T indicating a transpose. Juxtaposition of two tensor symbols denotes double contraction, and a superimposed dot denotes the forward rate with respect to a time-like parameter t.

Material Element, Thermodynamic Force, Internal State Variable, Mesoscopic Level, Incremental Problem

The energy criterion of instability of equilibrium has been extended to quasi-static deformation processes in time-independent plastic solids under general assumptions that result in a symmetric tangent stiffness matrix. The consistency with the bifurcation approach is demonstrated along with the additional possibility of eliminating unstable paths in non-unique deformation processes. As an application, the computational algorithm for passing bifurcation points with automatic branch switching has been developed and used in finite element computations. Consequences of the material stability requirement are studied. In particular, an analytic formula is derived for the incipient volume fraction of a localization zone in an incrementally nonlinear material.

The energy criterion of material instability in time-independent plastic solids can be obtained by examining whether energy could be extracted from a material element embedded in a uniformly deforming continuum. This approach is used here to examine stability of non-uniform deformation in initially homogeneous or heterogeneous materials. It is shown that the incipient volume fraction of the localization zone in a homogeneous material can have a uniquely defined, finite value if unstable post-critical deformation paths are rejected.

Material instability, energy criterion, time-independent plasticity, incremental nonlinearity

New, non-unique, complete slip line field solutions for the steady-state sliding of a rigid wedge-shaped asperity over a flat surface of ductile material are presented. The plastic flow mechanism map in the wedge angle - adhesion strength parameter plane is determined. Relevance of these deformation mechanisms to friction and wear of metal surfaces is discussed.

Crystal plasticity, Material instability, Automatic selection of active slip-systems

indentation size effect, single crystal, spherical indentation, numerical simulation

The application of the sequential method for estimating the mechanical response of elastic-viscoplastic polycrystals of high viscous anisotropy is discussed. The results are compared with other averaging schemes. Since the anisotropy of viscous response is high the estimated overall response is dramatically different for different averaging schemes. Additionally the effect of different linearization procedure for the viscous part is studied, denoted as secant, affine and tangent. The results are compared to the recent FFT analysis available in the literature. For the studied example the tangent variant provides the overall response that agrees best with the FFT predictions.

Micromechanics, Sequential linearization, Self-Consistent Scheme, Polycrystals

Crystal Plasticity, Slip-systems Selection, Incremental Energy Minimization

We develop a micromechanical phase-field model that describes the transformation between the austenite and twinned martensites. The new model constrains the volume fractions of both parent and internally twinned phases such that the physically motivated bounds are not violated. As an application, we studied the twinned martensite and austenite-martensite interfaces in the cubic-to-orthorhombic transformation in a CuAlNi shape memory alloy and estimated the elastic part of the interfacial energy.

In the modelling of metal single crystals in the framework of rate-independent plasticity, there are known difficulties caused by non-uniqueness in selection of active slip-systems. A related challenge is to predict emergence of non-uniform deformation patterns and the formation and evolution of experimentally observed microstructures. A new constitutive algorithm is presented that tackles those problems using the energy approach. It is based on asymptotically exact formulation of the set of constitutive equations and inequalities as a minimum problem for the incremental work expressed by a quadratic function of non-negative crystallographic slips. The calculated examples of deformation banding patterns and of reduction of multiplicity of active slip-systems in fcc single crystals are compared with the experimental observations.

crystal plasticity, slip-systems selection, incremental energy minimization