Prof. Henryk Petryk, PhD, DSc 

Doctoral thesis
1978  Ustalone płaskie przepływy ośrodków idealnie plastycznych ze swobodnym brzegiem
 301 
Habilitation thesis
1988  Niejednoznaczność i niestateczność procesów deformacji plastycznych 
Professor
1995  Title of professor 
Other
2010  Corresponding Member of Polish Academy of Sciences 
2016  Secretary General of IUTAM 
Supervision of doctoral theses
1.  20101125  Kursa Michał  Modelowanie deformacji plastycznych w kryształach metali metodą przyrostowej minimalizacji energii  637  
2.  1992  Kiryk Romuald  Mikromechaniczny model lepkoplastycznych materiałów polikrystalicznych 
Recent publications
1.  Ryś M., Forest S.^{♦}, Petryk H., A micromorphic crystal plasticity model with the gradientenhanced incremental hardening law, International Journal of Plasticity, ISSN: 07496419, DOI: 10.1016/j.ijplas.2019.102655, Vol.128, pp.102655121, 2020 Abstract: A model of crystal plasticity is developed in which the effects of plastic flow nonuniformity 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 kinematictype hardening in cyclic nonuniform 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 slipsystem hardening rule, is described in a simple manner that does not require any extra parameter in comparison to the nongradient 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. Keywords:gradient theory, crystal plasticity, dissipation, length scale, cyclic deformation, numerical regularization Affiliations:
 
2.  Stupkiewicz S., RezaeeHajidehi M., Petryk H., Multiscale analysis of the effect of interfacial energy on nonmonotonic stress–strain response in shape memory alloys, INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, ISSN: 00207683, DOI: 10.1016/j.ijsolstr.2020.04.006, pp.115, 2020 Abstract: The effect of formation and evolution of stressinduced martensitic microstructures on macroscopic mechanical properties of shape memory alloys in the pseudoelastic regime is investigated with account for sizedependent 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 nonmonotonic 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 nonmonotonic 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 finiteelement computations performed for a macroscopic tensile specimen, in which propagating instability bands are treated explicitly. Keywords:microstructures, martensitic transformation, size effects, incremental energy minimization, propagating instabilities Affiliations:
 
3.  Petryk H., A quasiextremal energy principle for nonpotential problems in rateindependent plasticity, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/j.jmps.2019.103691, Vol.136, pp.103691122, 2020 Abstract: The rateproblem of continuing equilibrium is examined for a general class of rateindependent elastoplastic solids, without assuming the normality flow rule or symmetry of the tangent stiffness matrix. Accordingly, the problem addressed is of nonpotential type, for which the usual stationarity or minimum principles for a governing potential do not apply. It is shown that the rateproblem can nevertheless be formulated as a quasiextremal 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 finitedimensional setting for a spatially discretized material body with generalized velocities and a number of plastic multipliers as unknowns. If a solution is nonunique then incrementally stable solutions can be selected using the quasiextremal principle in which the minimized energy function includes the secondorder terms. Examples and extensions concern an elasticplastic continuum obeying a nonassociative plastic flow rule, without or with a higherorder gradient term in the loading function. The issue of selection of active slipsystems in a single crystal of a nonsymmetric slipsystem interaction matrix is also addressed. Keywords:nonassociative plasticity, crystal plasticity, rate problem, free energy potential, dissipation, internal variables, thermodynamic basis, variational formulation, incremental energy minimization, nonuniqueness, path stability, gradient theory Affiliations:
 
4.  Ryś M., Petryk H., Gradient crystal plasticity models with a natural length scale in the hardening law, International Journal of Plasticity, ISSN: 07496419, DOI: 10.1016/j.ijplas.2018.07.015, Vol.111, pp.168187, 2018 Abstract: A class of crystal plasticity models based on the concept of microforces conjugate to sliprate gradients is examined in the small strain framework. As an extension of the usual formulation, sliprate 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 physicallybased dislocation theory of plasticity. The condition for plastic flow on a crystallographic slip system involves other length scales, associated with the secondorder gradients of slip and slip rate in energetic and dissipative terms, respectively. The interplay between the lengthscales of physically different origin is illustrated by the examples of monotonic and cyclic deformation of one and twodimensional 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. Keywords:Gradient theory, Dissipation, Crystal plasticity, Length scale, Cyclic deformation Affiliations:
 
5.  Kursa M., KowalczykGajewska K., Lewandowski M.J., Petryk H., Elasticplastic properties of metal matrix composites: Validation of meanfield approaches, EUROPEAN JOURNAL OF MECHANICS ASOLIDS, ISSN: 09977538, DOI: 10.1016/j.euromechsol.2017.11.001, Vol.68, pp.5366, 2018 Abstract: Several micromechanical and numerical approaches to estimating the effective properties of heterogeneous media are analyzed. First, micromechanical meanfield 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 elasticplastic regime using tangent and secant linearization procedures. In the examined range of strain and ceramic volume content, both the MoriTanaka averaging scheme and the generalized selfconsistent 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 meanfield predictions for a small ceramic volume content are very close to the results obtained from the finiteelement analysis of a spherical unit cell. Keywords:Metalmatrix composites, Effective properties, Analytical estimates, Numerical homogenization, Nonlinear analysis Affiliations:
 
6.  Tůma K., Stupkiewicz S., Petryk H., Rateindependent dissipation in phasefield modelling of displacive transformations, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/j.jmps.2018.02.007, Vol.114, pp.117142, 2018 Abstract: In this paper, rateindependent dissipation is introduced into the phasefield framework for modelling of displacive transformations, such as martensitic phase transformation and twinning. The finitestrain phasefield 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 ratepotential, is enhanced by including a mixedtype dissipation potential that combines viscous and rateindependent contributions. Effective computational treatment of the resulting incremental problem of nonsmooth 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 nonsmooth problem of evolution is converted into a smooth stationarity problem. The model is implemented in a finiteelement code and applied to solve two and threedimensional boundary value problems representative for shape memory alloys Keywords:Phasefield method, Microstructure, Martensite, Twinning, Nonsmooth optimization Affiliations:
 
7.  Petryk H., Stupkiewicz S., Kucharski S., On direct estimation of hardening exponent in crystal plasticity from the spherical indentation test, INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, ISSN: 00207683, DOI: 10.1016/j.ijsolstr.2016.09.025, Vol.112, pp.209221, 2017 Abstract: 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 pilingup and sinkingin 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 finiteelement study of spherical indentation has been performed within a selected penetration depth range using a finitestrain crystal plasticity model. It is shown how the powerlaw hardening exponent can be estimated from the measured pileup/sinkin 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. Keywords:metal crystal, elastoplasticity, finite deformation, strain hardening, experimental identification Affiliations:
 
8.  Tůma K., Stupkiewicz S., Petryk H., Size effects in martensitic microstructures: Finitestrain phase field model versus sharpinterface approach, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/j.jmps.2016.04.013, Vol.95, pp.284307, 2016 Abstract: A finitestrain phase field model for martensitic phase transformation and twinning in shape memory alloys is developed and confronted with the corresponding sharpinterface 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 phasefield 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 orderparameters 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, sizedependent microstructures with diffuse interfaces are calculated for the cubictoorthorhombic transformation in a CuAlNi shape memory alloy and compared with the sharpinterface microstructures with interfacial energy effects. Keywords:Phasefield method, Microstructure, Martensite, Size effects, Shape memory alloys Affiliations:
 
9.  Petryk H., Stupkiewicz S., A minimal gradientenhancement of the classical continuum theory of crystal plasticity. Part I: The hardening law, ARCHIVES OF MECHANICS, ISSN: 03732029, Vol.68, No.6, pp.459485, 2016 Abstract: A simple gradientenhancement 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 slipgradient 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 sliprate 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 gradientplasticity models. Keywords:gradient plasticity, geometrically necessary dislocations, single crystal, strainhardening, internal length scale, size effect Affiliations:
 
10.  Stupkiewicz S., Petryk H., A minimal gradientenhancement of the classical continuum theory of crystal plasticity. Part II: Size effects, ARCHIVES OF MECHANICS, ISSN: 03732029, Vol.68, No.6, pp.487513, 2016 Abstract: In our previous paper, a simple gradientenhancement 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 threedimensional 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 lengthscale parameter. The finite element treatment relies on introducing nonlocal 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 onedimensional problem of a boundary layer formed at a constrained interface. gradient plasticity, geometrically necessary dislocations, boundary layer, size effects, indentation, finite element method Affiliations:
 
11.  Petryk H., Kursa M., Incremental work minimization algorithm for rateindependent plasticity of single crystals, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, ISSN: 00295981, DOI: 10.1002/nme.4925, Vol.104, No.3, pp.157184, 2015 Abstract: A new constitutive algorithm for the rateindependent 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 nonnegative crystallographic slips. This approach requires selective symmetrization of the slipsystem interaction matrix restricted to active slipsystems, while the latent hardening rule for inactive slipsystems is arbitrary. The active slipsystem 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 facecentered 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 highsymmetry direction, showing the ability of the algorithm to cope with the nonuniqueness problem and to generate experimentally observable solutions with a reduced number of active slipsystems. Keywords:solids, crystal plasticity, rateindependent constitutive equations, material stability, variational methods, incremental energy minimization, augmented Lagrangian method Affiliations:
 
12.  KowalczykGajewska K., Sztwiertnia K.^{♦}, Kawałko J.^{♦}, Wierzbanowski K.^{♦}, Wroński M.^{♦}, Frydrych K., Stupkiewicz S., Petryk H., Texture evolution in titanium on complex deformation paths: Experiment and modelling, MATERIALS SCIENCE AND ENGINEERING ASTRUCTURAL MATERIALS PROPERTIES MICROSTRUCTURE AND PROCESSING, ISSN: 09215093, DOI: 10.1016/j.msea.2015.04.040, Vol.637, pp.251263, 2015 Abstract: Texture evolution in commercially pure titanium deformed by equalchannel 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 ultrafine grained (UFG) material obtained after severe plastic deformation (SPD). The numerical simulations are based on the selfconsistent viscoplastic method of graintopolycrystal 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. Keywords:Texture evolution, UFG materials, SPD processes, Crystal plasticity, Twinning Affiliations:
 
13.  Fischer F.D.^{♦}, Svoboda J.^{♦}, Petryk H., Thermodynamic extremal principles for irreversible processes in materials science, ACTA MATERIALIA, ISSN: 13596454, DOI: 10.1016/j.actamat.2013.11.050, Vol.67, No.153, pp.120, 2014 Abstract: Mathematical and physical aspects of the applicability of the Onsager, Prigogine as well as the Glansdorff and Ziegler thermodynamic extremal principles (TEPs) to nonequilibrium 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 nonlinear nonequilibrium thermodynamics are demonstrated for homogeneous and quasihomogeneous dissipation functions. Keywords:Nonequilibrium, Thermodynamics, Entropy, Onsager’s principle, Thermodynamic extremal principles Affiliations:
 
14.  Kursa M., KowalczykGajewska K., Petryk H., Multiobjective optimization of thermomechanical properties of metalceramic composites, COMPOSITES PART BENGINEERING, ISSN: 13598368, DOI: 10.1016/j.compositesb.2014.01.009, Vol.60, pp.586596, 2014 Abstract: The optimization procedure is worked out for finding an optimal content of phases in metal–ceramic composites in case of conflicting objectives regarding thermomechanical 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 thermomechanical 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. Keywords:Metal–matrix composites (MMCs), Thermomechanical, Plastic deformation, Micromechanics, Multicriteria optimization Affiliations:
 
15.  Kucharski S., Stupkiewicz S., Petryk H., Surface PileUp Patterns in Indentation Testing of Cu Single Crystals, EXPERIMENTAL MECHANICS, ISSN: 00144851, DOI: 10.1007/s1134001498831, Vol.54, pp.957969, 2014 Abstract: Nano and microindentation 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 pilingup and sinkingin around the imprint are investigated. Experimental observations are compared with finite element simulations based on the large deformation crystal plasticity theory. Keywords:Copper, Nanoindentation, AFM, EBSD, Crystal plasticity, Finite element method Affiliations:
 
16.  Stupkiewicz S., Petryk H., A robust model of pseudoelasticity in shape memory alloys, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, ISSN: 00295981, DOI: 10.1002/nme.4405, Vol.93, No.7, pp.747769, 2013 Abstract: 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 rateindependent 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 smallstrain to finitedeformation regime is presented. Finite element computations demonstrate robustness of the finitestrain version of the model and illustrate the effects of tension–compression asymmetry and transversal isotropy of the surface of limit transformation strains. Keywords:shape memory alloys (SMA), phase transformation, energy methods, finite element method, augmented Lagrangian method Affiliations:
 
17.  Petryk H., Kursa M., The energy criterion for deformation banding in ductile single crystals, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/j.jmps.2013.03.004, Vol.61, No.8, pp.18541875, 2013 Abstract: 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 secondorder energy criterion for incipient deformation banding is derived in a timecontinuous setting for a rateindependent elastic–plastic crystal. The need for selective symmetrization of the local interaction matrix for active slipsystems is demonstrated. A computational approach to deformation banding is developed by using nonconvex 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. Keywords:Metal crystal, Plasticity, Finite deformation, Laminate microstructure, Incremental energy minimization Affiliations:
 
18.  Petryk H., Stupkiewicz S., Instability of equilibrium of evolving laminates in pseudoelastic solids, INTERNATIONAL JOURNAL OF NONLINEAR MECHANICS, ISSN: 00207462, DOI: 10.1016/j.ijnonlinmec.2011.07.005, Vol.47, pp.317330, 2012 Abstract: This study is concerned with isothermal stability of equilibrium of evolving laminated microstructures in pseudoelastic solids with a multiwell free energy function. Several possible modes of instability associated with phase transition between energy wells are analysed. The related rateindependent dissipation is included by imposing a threshold value on the thermodynamic driving force. For a homogenized phasetransforming laminate with no length scale it is shown that localization instability is a rule in case of a nonzero interfacial jump of a directional nominal stress, irrespectively of actual boundary conditions. A stabilizing effect of elastic microstrain 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. Keywords:Microstructures, Phase transformation, Laminates, Energy methods, Stability Affiliations:
 
19.  Petryk H., Stupkiewicz S., Modelling of microstructural evolution on complex paths of large plastic deformation, INTERNATIONAL JOURNAL OF MATERIALS RESEARCH, ISSN: 18625282, DOI: 10.3139/146.110683, Vol.103, No.3, pp.271277, 2012 Abstract: A procedure for the modelling of microstructural changes induced by nonuniform 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 nonhardening 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 nonconventional 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 crosssection are calculated. Keywords:nonuniform deformation, dislocation cells, grain refinement, hardening, quantitative prediction Affiliations:
 
20.  Stupkiewicz S., Maciejewski G., Petryk H., Elastic microstrain energy of austenite–martensite interface in NiTi, MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING, ISSN: 09650393, DOI: 10.1088/09650393/20/3/035001, Vol.20, pp.0350019, 2012 Abstract: The interfacial energy due to elastic microstrains at the austenite–twinned martensite interface is calculated for the NiTi shape memory alloy undergoing cubictomonoclinic 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 finitestrain kinematics. The effect of atomicscale interfacial energy is studied. Keywords:microstructure, martensitic phase transformation, shape memory alloys (SMA), interface structure, micromechanical modelling Affiliations:
 
21.  KowalczykGajewska K., Petryk H., Sequential linearization method for viscous/elastic heterogeneous materials, EUROPEAN JOURNAL OF MECHANICS ASOLIDS, ISSN: 09977538, Vol.30, No.5, pp.650664, 2011 Abstract: 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 inhomogeneitymatrix system with viscous or elastic interaction rules are solved sequentially within one incremental step. It is demonstrated that in the case of a selfconsistent 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 twophase material it is shown that the proposed approach provides acceptable predictions in comparison with the existing models. Keywords:Micromechanics, Viscoelasticity, Viscoplasticity, Homogenization, Selfconsistent scheme Affiliations:
 
22.  Petryk H., Kursa M., Selective symmetrization of the slipsystem interaction matrix in crystal plasticity, ARCHIVES OF MECHANICS, ISSN: 03732029, Vol.63, No.3, pp.287310, 2011 Abstract: The symmetry issue of the interaction matrix between multiple slipsystems in the theory of crystal plasticity at finite deformation is revisited. By appealing to possibly nonuniform distribution of slipsystem activity in a representative spacetime element of a crystal, symmetry of the slipsystem interaction matrix for the representative element is derived under assumptions that have a physical meaning. This conclusion refers to active slipsystems only. Accordingly, for any given hardening law, a new symmetrization rule is proposed that is restricted to active slipsystems and leaves the latent hardening of inactive slipsystems unchanged. Advantages of the proposal in comparison with full symmetrization are illustrated by a simple example of uniaxial tension. Keywords:finite deformation, metal crystal, plasticity, hardening, symmetry Affiliations:
 
23.  Stupkiewicz S., Petryk H., A bicrystal aggregate model of pseudoelastic behaviour of shapememory alloy polycrystals, INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES, ISSN: 00207403, DOI: 10.1016/j.ijmecsci.2009.09.012, Vol.52, pp.219228, 2010 Abstract: A multiscale model of stressinduced phase transformation and martensite variant reorientation in shape memory alloy (SMA) polycrystals is developed. It is proposed to include neighbouringgrain interaction in a simple manner by introducing an intermediate bicrystal 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 rateindependent dissipation function is used that incorporates the dissipation due to forward and reverse austenitetomartensite 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 nonproportional loading paths. Keywords:Phase transformation, Dissipation, Microstructure, Multiscale model, Incremental energy minimization Affiliations:
 
24.  Petryk H., Stupkiewicz S., Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/j.jmps.2009.11.004, Vol.58, pp.390408, 2010 Abstract: 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 threedimensional 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 microstrain 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 rankthree laminated microstructure of finite characteristic dimensions in a pseudoelastic CuAlNi shape memory alloy is examined quantitatively. Keywords:Microstructures, Phase transformation, Grain boundaries, Energy methods, Shape memory alloys (SMA) Affiliations:
 
25.  Petryk H., Stupkiewicz S., Maciejewski G., Interfacial energy and dissipation in martensitic phase transformations. Part II: Size effects in pseudoelasticity, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/j.jmps.2009.11.003, Vol.58, pp.373389, 2010 Abstract: 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 rateindependent 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 quasistatic 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 rateindependent 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 firstorder 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 pathindependence 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 stressinduced 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. Keywords:Microstructures, Phase transformation, Grain boundaries, Energy methods, Stability Affiliations:
 
26.  Stupkiewicz S., Petryk H., Grainsize effect in micromechanical modelling of hysteresis in shape memory alloys, ZAMMZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK, ISSN: 00442267, DOI: 10.1002/zamm.201000008, Vol.90, pp.783795, 2010 Abstract: 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 atomicscale 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 rateindependent dissipation. The sizedependent 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 forwardreverse transformation cycle is derived and combined with a micromechanical model of a polycrystalline NiTi shape memory alloy. A numerical example illustrating sizedependent hysteresis in the stressinduced martensitic transformation is presented. Keywords:interfacial energy, incremental energy minimization, dissipation, martensitic transformation Affiliations:
 
27.  Petryk H., Stupkiewicz S., Energia powierzchniowa, dyssypacja i efekty skali w modelowaniu mikrostruktur martenzytycznych, Czasopismo Techniczne. Mechanika, ISSN: 00114561, Vol.107, No.20, pp.99108, 2010 Abstract: 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 mikrostrukturalnych pomiędzy poszczególnymi wariantami martenzytu, fazami lub ziarnami. Ewolucja mikrostruktury jest wyznaczana drogą przyrostowej minimalizacji całkowitej energii dostarczanej do rozpatrywanego układu w procesie makroskopowo quasistatycznym 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. Keywords:energia powierzchniowa, minimalizacja energii, przemiana martenzytyczna Affiliations:
 
28.  Petryk H., Stupkiewicz S., Kuziak R.^{♦}, Grain refinement and strain hardening in IF steel during multiaxis compression: Experiment and modelling, JOURNAL OF MATERIALS PROCESSING TECHNOLOGY, ISSN: 09240136, DOI: 10.1016/j.jmatprotec.2007.11.068, Vol.204, pp.255263, 2008 Abstract: The effect of severe plastic deformation (SPD) during cyclic multiaxis 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 cellblock (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 strainrate 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 multiaxis compression are calculated and compared to experimental data. Keywords:Modelling, Microstructure, Hardening, Severe plastic deformation Affiliations:
 
29.  Stupkiewicz S., Maciejewski G., Petryk H., Lowenergy morphology of the interface layer between austenite and twinned martensite, ACTA MATERIALIA, ISSN: 13596454, DOI: 10.1016/j.actamat.2007.07.034, Vol.55, No.18, pp.62926306, 2007 Abstract: A micromechanical scheme is developed for predicting the morphology and interfacial energy of the interface layer between the parent phase and internally twinned martensite. Lowenergy 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 nonlinear elasticity with eigenstrains, shape sensitivity analysis with respect to general shape parametrization and minimization employing a gradientbased algorithm. As an application, lowenergy morphologies are studied for the austenite–martensite interface in the cubictoorthorhombic 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. Keywords:Microstructure, Martensitic phase transformation, Shape memory alloys (SMA), Interface structure, Micromechanical modelling Affiliations:
 
30.  Petryk H., Stupkiewicz S., A quantitative model of grain refinement and strain hardening during severe plastic deformation, MATERIALS SCIENCE AND ENGINEERING ASTRUCTURAL MATERIALS PROPERTIES MICROSTRUCTURE AND PROCESSING, ISSN: 09215093, DOI: 10.1016/j.msea.2006.08.076, Vol.444, pp.214219, 2007 Abstract: 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 cellblocks is expressed as a function of the effective plastic strain influenced by strainrate reversals. The estimated growth of the highangle boundary area fraction depends on the complexity of the threedimensional 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. Keywords:Modelling, Microstructure, Severe plastic deformation, Ultrafine grained materials Affiliations:
 
31.  Schurig M.^{♦}, Bertram A.^{♦}, Petryk H., Micromechanical analysis of the development of a yield vertex in polycrystal plasticity, ACTA MECHANICA, ISSN: 00015970, DOI: 10.1007/s0070700704628, Vol.194, pp.141158, 2007 Abstract: The TaylorLin 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. Affiliations:
 
32.  Richert M.^{♦}, Petryk H., Stupkiewicz S., Grain refinement in AlMgSi alloy during cyclic extrusioncompression: Experiment and modelling, ARCHIVES OF METALLURGY AND MATERIALS, ISSN: 17333490, Vol.52, pp.4954, 2007 Abstract: The effect of severe plastic deformation (SPD) during cyclic extrusioncompression (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 extrusioncompression are calculated and compared to experimental data. Keywords:Modelling, Microstructure, Hardening, Severe plastic deformation Affiliations:
 
33.  Maciejewski J., Kopeć H.^{♦}, Petryk H., Finite element analysis of strain nonuniformity in two processes of severe plastic deformation, ENGINEERING TRANSACTIONS (ROZPRAWY INŻYNIERSKIE), ISSN: 0867888X, Vol.55, No.3, pp.197216, 2007 Abstract: Two severe plastic deformation (SPD) processing techniques, namely equalchannel angular pressing (ECAP) and cyclic extrusioncompression (CEC), are investigated by using the finite element method. The major aspect examined is the nonuniformity 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 nonuniformity. Comparison is made of distributions of equivalent strain after two passes of ECAP for two different routes and with two die profiles. Affiliations:
 
34.  Stupkiewicz S., Petryk H., Finitestrain micromechanical model of stressinduced martensitic transformations in shape memory alloys, MATERIALS SCIENCE AND ENGINEERING ASTRUCTURAL MATERIALS PROPERTIES MICROSTRUCTURE AND PROCESSING, ISSN: 09215093, DOI: 10.1016/j.msea.2006.01.112, Vol.438440, pp.126130, 2006 Abstract: A micromechanical model of stressinduced martensitic transformation in single crystals of shape memory alloys is developed. This model is a finitestrain counterpart to the approach presented recently in the smallstrain setting [S. Stupkiewicz, H. Petryk, J. Mech. Phys. Solids 50 (2002) 2303–2331]. The stressinduced 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 rateindependent 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. Keywords:Microstructures, Laminates, Finite deformations, Micromechanics, Shape memory alloys Affiliations:
 
35.  Petryk H., Thermodynamic conditions for stability in materials with rateindependent dissipation, PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY AMATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, ISSN: 1364503X, DOI: 10.1098/rsta.2005.1584, Vol.A363, pp.24792515, 2005 Abstract: A distinctive feature of the examined class of solids is that a part of the entropy production is due to rateindependent 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 microscale instabilities in the presence of two different scales of time. Specific conditions for material stability against internal structural rearrangements under deformationsensitive loading are derived within the incremental constitutive framework of multimode inelasticity. Application to spontaneous formation of deformation bands in a continuum is presented. Conditions for stability or instability of a quasistatic 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 multislip or undergoing stressinduced martensitic transformation. Affiliations:
 
36.  Maciejewski G., Stupkiewicz S., Petryk H., Elastic microstrain energy at the austenitetwinned martensite interface, ARCHIVES OF MECHANICS, ISSN: 03732029, Vol.57, No.4, pp.277297, 2005 Abstract: A micromechanical scheme is developed for the analysis of elastic microstrains induced by local incompatibilities at the austenitetwinned martensite interface. The aim of the paper is to estimate the elastic microstrain 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 cubictoorthorhombic transformation in CuAlNi shape memory alloy are analyzed by the finite element method for the assumed class of zigzag shapes of the austenitemartensite interface at the microlevel. Finally, the effect of the interphase boundary energy on the microstructure of the transition layer is studied. Keywords:microstructure, martensitic phase transformation, shape memory alloys (SMA), interface structure, micromechanical modelling Affiliations:
 
37.  Petryk H., Stupkiewicz S., Micromechanical modelling of stressinduced phase transition in shape memory alloys, ARCHIVES OF METALLURGY AND MATERIALS, ISSN: 17333490, Vol.49, pp.765777, 2004 Abstract: A micromechanical model of stressinduced martensitic transformation in shape memory alloys is presented. A laminated microstructure of austenite and martensite phases is assumed along with a timeindependent 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. Affiliations:
 
38.  Stupkiewicz S., Petryk H., Micromechanical modelling of stressinduced martensitic transformation and detwinning in shape memory alloys, JOURNAL DE PHYSIQUE IV, ISSN: 11554339, DOI: 10.1051/jp4:2004115017, Vol.115, pp.141149, 2004 Abstract: The paper is concerned with modelling of stressinduced 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 timeindependent 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. Affiliations:
 
39.  Petryk H., Incremental energy minimization in dissipative solids, Comptes Rendus Mécanique, ISSN: 16310721, DOI: 10.1016/S16310721(03)001098, Vol.331, No.7, pp.469474, 2003 Abstract: The incremental energy minimization is examined as a method of determining solution paths for timeindependent dissipative solids. Isothermal quasistatic 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. Keywords:Solids and structures, Dissipative materials, Plasticity, Energy, Path stability Affiliations:
 
40.  Petryk H., Fischer F.D.^{♦}, Marketz W.^{♦}, Clemens H.^{♦}, Appel F.^{♦}, An energy approach to the formation of twins in TiAl, METALLURGICAL AND MATERIALS TRANSACTIONS APHYSICAL METALLURGY AND MATERIALS SCIENCE, ISSN: 10735623, DOI: 10.1007/s116610030184z, Vol.34, No.12, pp.28272836, 2003 Abstract: 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. Affiliations:
 
41.  Bigoni D.^{♦}, Petryk H., A note on divergence and flutter instabilities in elastic–plastic materials, INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, ISSN: 00207683, DOI: 10.1016/S00207683(01)002487, Vol.39, No.4, pp.911926, 2002 Abstract: 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 timeindependent, 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. Keywords:Instability of plastic flow, Elastic–plastic material, Material stability, Flutter Affiliations:
 
42.  Petryk H., Thermann K.^{♦}, Postcritical plastic deformation in incrementally nonlinear materials, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/S00225096(01)001314, Vol.50, No.5, pp.925954, 2002 Abstract: 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 postcritical plastic deformation is based on the concept of a representative nonuniform solution in a homogeneous material. The indeterminacy of a postcritical 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 timeindependent plasticity leads then to gradual concentration of postcritical plastic deformation. The volume fraction occupied by shear bands is found to have initially a welldefined, 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 yieldvertex 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. Keywords:Plasticity, Shear bands, Material instability, Energy criterion, Bifurcation Affiliations:
 
43.  Stupkiewicz S., Petryk H., Modelling of laminated microstructures in stressinduced martensitic transformations, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/S00225096(02)000297, Vol.50, No.11, pp.23032331, 2002 Abstract: This paper is concerned with micromechanical modelling of stressinduced 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 timeindependent thermodynamic criterion. Plasticitylike 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. Keywords:Phase transformation, Microstructures, Layered material, Constitutive behaviour, Shape memory alloy Affiliations:
 
44.  Petryk H., Secondorder work and dissipation on indirect paths, Comptes Rendus Mécanique, ISSN: 16310721, Vol.330, pp.121126, 2002 Abstract: The concept of the secondorder dissipation on arbitrary complex paths in the space of internal state variables is developed. A general framework for timeindependent dissipative solids is adopted which encompasses plasticity, microcracking 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. Affiliations:
 
45.  Petryk H., Thermann K.^{♦}, Postcritical deformation pattern in plane strain plastic flow with yieldsurface vertex effect, INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES, ISSN: 00207403, DOI: 10.1016/S00207403(00)000102, Vol.42, No.11, pp.21332146, 2000 Abstract: 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 timeindependent material with a yieldsurface vertex effect is investigated numerically in plane strain beyond the critical instant of ellipticity loss under quasistatic loading. The energy criterion of path instability applied to a family of postcritical 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 welldefined 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. Keywords:Plasticity, Shear bands, Material instability, Energy criterion, Bifurcation Affiliations:
 
46.  Petryk H., General conditions for uniqueness in materials with multiple mechanisms of inelastic deformation, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/S00225096(99)000368, Vol.48, No.2, pp.367396, 2000 Abstract: This study is concerned with multimode inelastic behaviour at macroscopically uniform deformation. The material is assumed to be timeindependent; the physical origin of inelasticity may be otherwise arbitrary, including plasticity of crystals and polycrystals, microcracking, phase transformation, etc. A nonlinear rateproblem of continuing mechanical equilibrium at finite strain is examined for a material element subject to deformationsensitive 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 shearband bifurcation along a smooth straining path. Keywords:Microstructures, Multimode inelasticity, Constitutive behaviour, Finite strain, Bifurcation Affiliations:
 
47.  Petryk H., Macroscopic ratevariables in solids undergoing phase transformation, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/S00225096(97)000999, Vol.46, No.5, pp.873894, 1998 Abstract: Averaging rules are derived for the rates of deformation gradient and nominal stress in heterogeneous solids undergoing quasistatic 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 nonsmooth 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 mesolevel 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. Keywords:Phase transformation, Finite strain, Inhomogeneous material, Strain compatibility, Asymptotic analysis Affiliations:
 
48.  Kiryk R., Petryk H., A selfconsistent model of ratedependent plastic polycrystals, ARCHIVES OF MECHANICS, ISSN: 03732029, Vol.50, No.2, pp.247263, 1998  
49.  Petryk H., Plastic instability: Criteria and computational approaches, ARCHIVES OF COMPUTATIONAL METHODS IN ENGINEERING, ISSN: 11343060, DOI: 10.1007/BF03020127, Vol.4, No.2, pp.111151, 1997 Abstract: General criteria of instability in timeindependent elasticplastic 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 yieldsurface vertex, on instability predictions is examined. A survey of the methods of postcritical analysis is presented. Affiliations:
 
50.  Petryk H., Thermann K.^{♦}, A yieldvertex modification of twosurface models in plasticity, ARCHIVES OF MECHANICS, ISSN: 03732029, Vol.49, No.5, pp.847863, 1997  
51.  Petryk H., Thermann K.^{♦}, Postcritical plastic deformation of biaxially stretched sheets, INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, ISSN: 00207683, DOI: 10.1016/00207683(95)00061E, Vol.33, No.5, pp.689705, 1996 Abstract: A theoretical and numerical analysis of the formation of a localized neck in a biaxially stretched sheet is presented. A timeindependent constitutive law is assumed to be incrementally nonlinear 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 timeindependent material is found here to have a welldefined initial value, proportional to the inplane sheet dimension. During subsequent postcritical 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 postbifurcation deformation process and are confirmed by the numerical analysis performed for a sheet of finite thickness. Affiliations:
 
52.  Petryk H., Thermodynamic Stability of Equilibrium in Plasticity, Journal of NonEquilibrium Thermodynamics, ISSN: 03400204, DOI: 10.1515/jnet.1995.20.2.132, Vol.20, pp.132149, 1995 Abstract: Thermodynamic stability of equilibrium in solids with intrinsic dissipation of plastic type is investigated. Ratedependent plastic behaviour and its rateindependent 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 secondorder stability condition in timeindependent plasticity is derived in various forms, depending on the class of disturbances under consideration. Affiliations:
 
53.  Petryk H., Thermann K.^{♦}, On discretized plasticity problems with bifurcations, INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, ISSN: 00207683, DOI: 10.1016/00207683(92)90125D, Vol.29, No.6, pp.745765, 1992 Abstract: A spatially discretized nonlinear rate problem for a timeindependent plastic solid is examined with particular reference to bifurcation. Constitutive nonlinearity in a general form encompassing the yieldsurface 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. Affiliations:
 
54.  Petryk H., Material instability and strainrate discontinuities in incrementally nonlinear continua, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/00225096(92)90014S, Vol.40, No.6, pp.12271250, 1992 Abstract: A systematic analysis of theoretical aspects of material instability is presented for a general class of timeindependent, incrementally nonlinear solids which admit a velocitygradient potential. The secondorder energy criteria of instability of uniform straining and of equilibrium are derived for a material element embedded in a quasistatically 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 quasistatic bifurcation is studied as a symptom of material instability. The derived condition for the instability of uniform quasistatic 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 nonproportional deformation path of material obeying the J2 corner theory of plasticity. Affiliations:
 
55.  Petryk H., The energy criteria of instability in timeindependent inelastic solids, ARCHIVES OF MECHANICS, ISSN: 03732029, Vol.43, No.4, pp.519545, 1991 Abstract: CONDITIONS sufficient for instability of deformation are examined for a class of incrementally nonlinear, time independent inelastic solids. Instability of a quasistatic 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 quasistatic bifurcation at varying loading. For a continuum, instability at varying loading is interpreted as sensitivity of the incremental deformation to arbitrarily small perturbing forces. Affiliations:
 
56.  Petryk H., On the secondorder work in plasticity, ARCHIVES OF MECHANICS, ISSN: 03732029, Vol.43, pp.377397, 1991  
57.  Petryk H., The Energy Criteria of instability of Equilibrium and of Quasistatic Deformations in Timeindependent Inelastic Solids, Prace IPPT  IFTR Reports, ISSN: 22993657, No.1, pp.167, 1991  
58.  Nguyen Q.S.^{♦}, Petryk H., A constitutive inequality for timeindependent dissipative solids, Comptes Rendus de l’Académie des Sciences. Série II, ISSN: 07644450, Vol.312, No.1, pp.712, 1991 Abstract: A general constitutive inequality for timeindependent solids has been recently derived by Petryk from the normality and symmetry assumptions adopted at a microlevel of an elasticplastic 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 microcracks is considered. Keywords:Elastoplasticity, Griffith crack, Variational inequality, Heterogeneous material, Internal variable material, Microcrack, Thermodynamics Affiliations:
 
59.  Petryk H., On constitutive inequalities and bifurcation in elasticplastic solids with a yieldsurface vertex, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/00225096(89)900124, Vol.37, No.2, pp.265299, 1989 Abstract: A constitutive inequality for timeindependent elasticplastic 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 yieldsurface vertex. It is demonstrated that existence of a strainrate 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. Micromechanicallybased 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 yieldsurface vertex are derived. Affiliations:
 
60.  Petryk H., Niejednoznaczność i niestateczność procesów deformacji plastycznych (Praca habilitacyjna), Prace IPPT  IFTR Reports, ISSN: 22993657, No.7, pp.1220, 1987  
61.  Petryk H., Mróz Z., Time derivatives of integrals and functionals defined on varying volume and surface domains, ARCHIWUM MECHANIKI STOSOWANEJ, ISSN: 03732029, Vol.38, No.56, pp.697724, 1986 Abstract: 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 nonlinear 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. Affiliations:
 
62.  Petryk H., Thermann K.^{♦}, Secondorder bifurcation in elasticplastic solids, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/00225096(85)900043, Vol.33, No.6, pp.577593, 1985 Abstract: Secondorder rate constitutive equations are formulated for a timeindependent elasticplastic material, obeying the normality flow rule with a smooth yield surface. Under specified regularity restrictions imposed on the involved fields, the regular secondorder rate boundary value problem with quasistatic accelerations as unknowns is posed. It is shown that every solution of this generally nonlinear 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 secondorder bifurcation. The criticality of the exclusion condition is discussed and conditions are indicated under which a secondorder bifurcation becomes possible, while the firstorder rate problem is still uniquely solvable. Affiliations:
 
63.  Petryk H., On stability and symmetry conditions in time independent plasticity, ARCHIVES OF MECHANICS, ISSN: 03732029, Vol.37, pp.503520, 1985  
64.  Petryk H., A slipline field analysis of the rolling contact problem at high loads, INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES, ISSN: 00207403, DOI: 10.1016/00207403(83)900309, Vol.25, No.4, pp.265275, 1983 Abstract: An analysis of the steady rolling of a rough rigid cylinder over the surface of a rigidperfectly plastic solid is presented. An exact slipline 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. Affiliations:
 
65.  Petryk H., Zagadnienie przesuwania obciążonego klina po powierzchni idealnie plastycznego ośrodka, Prace IPPT  IFTR Reports, ISSN: 22993657, No.38, pp.127, 1981  
66.  Petryk H., Nonunique slipline field solutions for the wedge indentation problem, Journal de mecanique appliquee, ISSN: 03990842, Vol.4, No.3, pp.255282, 1980 Abstract: A systematic analysis of the oblique indentation of a rigid wedge into a semiinfinite block of rigidperfectly plastic material is presented. It is shown that infinitely many solutions exist for given boundary conditions. Three types of slipline fields corresponding to three distinct deformation modes are examined, two of which represent a generalization of the wellknown classical solutions. A number of numerical results are given. The completeness of the solutions, as well as their connection with the solutions for steadystate machining, is discussed. Affiliations:
 
67.  Petryk H., On slipline field solutions for steadystate and selfsimilar problems with stressfree boundaries, ARCHIVES OF MECHANICS, ISSN: 03732029, Vol.31, pp.861874, 1979  
68.  Petryk H., Ustalone płaskie przepływy ośrodków idealnie plastycznych ze swobodnym brzegiem (Praca doktorska), Prace IPPT  IFTR Reports, ISSN: 22993657, No.46, pp.175, 1977 
List of chapters in recent monographs
1. 251  Petryk H., Material Instabilities in Elastic and Plastic Solids, rozdział: Theory of Material Instability in Incrementally Nonlinear Plasticity, CISM Courses and Lectures, Springer, Vienna, 414, pp.261331, 2000  
2. 212  Petryk H., Bifurcation and stability od dissipative systems, rozdział: Theory of bifurcation and instability in timeindependent plasticity, CISM Courses and Lectures, Ed. Nguyen Q.S., Springer, Vienna, 327, pp.95152, 1993  
3. 31  Petryk H., NonEquilibrium Thermodynamics with Application to Solids, rozdział: Stability and Constitutive Inequalities in Plasticity, CISM Courses and Lectures, Muschik W. (eds), Springer, Vienna, 336, pp.259329, 1993 
Conference papers
1.  KowalczykGajewska K., Stupkiewicz S., Frydrych K., Petryk H., Modelling of Texture Evolution and Grain Refinement on Complex SPD Paths, JOURNAL OF PHYSICS: CONFERENCE SERIES, ISSN: 17426588, DOI: 10.1088/1757899X/63/1/012040, No.63, pp.012040110, 2014 Abstract: 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 twostep procedure. In the first step, motivated by saturation of material hardening at large accumulated strains, the steadystate kinematics of the process is generated for a nonhardening viscoplastic model by using the standard finite element method for a specified SPD scheme. In the second step, microstructural changes are investigated along the deformationgradient 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 nonconventional process of cold extrusion assisted by cyclic rotation of the die, called KOBO process, which leads to an ultrafine grain structure. The texture evolution is calculated for fcc and hcp metals by applying crystal viscoplasticity combined with the selfconsistent 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. Keywords:SPD processes, Texture evolution, UFG materials, Crystal plasticity, Grain refinement Affiliations:
 
2.  Kursa M., KowalczykGajewska K., Petryk H., Multiobjective optimization of effective thermomechanical properties of metalceramic composites, ECCOMAS 2012, 6th European Congress on Computational Methods in Applied Sciences and Engineering, 20120910/0914, Wiedeń (AT), Vol.1, pp.19, 2012 Abstract: Micromechanical modelling of metalceramic composites has been carried out to obtain a material of required thermomechanical properties. Quantitative transition from phase properties and morphology to macroscopic properties of a composite has been modelled by meanfield approaches, including the selfconsistent scheme. An optimization method has been developed for the objective function that expresses a distance between the required values of macrovariables and those determined for a given set of microstructural parameters. The presented example concerns application of Al2O3Cu composite to brake disks. Keywords:multiobjective optimization, composite selection, metal matrix composites Affiliations:
 
3.  Petryk H., Stupkiewicz S., Modelling of strain hardening and grain refinement during severe plastic deformation, Dynamic Behaviour of Materials : A Workshop in Memory of Prof. J.R. Klepaczko, 20090513/0515, Metz (FR), pp.4350, 2009  
4.  Petryk H., Instability and energy minimization in solids: formation of microstructures, CMDS 11, 11th International Symposium on Continuum Models and Discrete Systems, 20070730/0802, Paris (FR), pp.113118, 2008  
5.  Petryk H., Stupkiewicz S., Maciejewski G., Modeling of Austenite/Martensite Laminates with Interfacial Energy Effect, IUTAM Symposium on Size Effects on Material and Structural Behavior at Micron and NanoScales, 20040531/0604, Hong Kong (CN), pp.151162, 2004 Abstract: Stressinduced 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 ranktwo laminate are derived by minimizing the sizedependent contribution of the interfaces to the total free energy. Specific examples have been calculated for the cubictoorthorhombic (β1 to γ1) stressinduced transformation in a CuAlNi shape memory alloy. Affiliations:
 
6.  Petryk H., On the Micromacro Transition and Hardening Moduli in Plasticity, IUTAM Symposium on Micro and Macrostructural Aspects of Thermoplasticity, 19970825/0829, Bochum (DE), DOI: 10.1007/0306469367_21, pp.219230, 1999 Abstract: This paper focuses upon the theoretical foundations of the modeling of elastoplastic polycrystalline aggregates. As a starting point, the known constitutive framework for a timeindependent elasticplastic 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 macrolevel 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 fourthorder tensors of instantaneous moduli or compliances. A known difficulty in the modelling of timeindependent elasticplastic 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 nonuniquely defined. Throughout the paper, boldface small letters (Roman or Greek) denote secondorder symmetric tensors, and boldface capital letters denote fourthorder 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 timelike parameter t. Keywords:Material Element, Thermodynamic Force, Internal State Variable, Mesoscopic Level, Incremental Problem Affiliations:
 
7.  Petryk H., Instability of plastic deformation processes, ICTAM Kyoto 1996, XIXth International Congress of Theoretical and Applied Mechanics, 19960825/0831, Kyoto (JP), pp.497516, 1997 Abstract: The energy criterion of instability of equilibrium has been extended to quasistatic deformation processes in timeindependent 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 nonunique 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. Affiliations:
 
8.  Petryk H., Instability in timeindependent plastic materials, AEPA '96, The AsiaPacific Symposium on Advances in Engineering Plasticity and its Application, 19960821/0824, Hiroshima (JP), DOI: 10.1016/B9780080428246.500325, pp.689705, 1996 Abstract: The energy criterion of material instability in timeindependent 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 nonuniform 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 postcritical deformation paths are rejected. Keywords:Material instability, energy criterion, timeindependent plasticity, incremental nonlinearity Affiliations:
 
9.  Petryk H., Thermann K.^{♦}, On plastic strain localisation in the nonelliptic range under plane stress, COMPLAS, 4th International Conference on Computational Plasticity. Fundamentals and Applications, 19950403/0406, Barcelona (ES), pp.647658, 1995  
10.  Petryk H., Some constitutive inequalities for incrementally nonlinear models in plasticity, MODERN APPROACHES TO PLASTICITY, 19920612/0616, Horton (GR), DOI: 10.1016/B9780444899705.50008X, pp.5970, 1993  
11.  Petryk H., Slip line field solutions for sliding contact, International Conference on Tribology  friction, lubrication and wear, fifty years on, 19870701/0703, London (GB), Vol.II, pp.987994, 1987 Abstract: New, nonunique, complete slip line field solutions for the steadystate sliding of a rigid wedgeshaped 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. Affiliations:
 
12.  Petryk H., On energy criteria of plastic instability, International Seminar on Plastic Instability, 19850909/0913, Paris (FR), pp.215226, 1985  
13.  Petryk H., On the onset of instability in elasticplastic solids, Plasticity Today: Modelling, Methods and Applications, International Symposium on Current Trends and Results in Plasticity, 19830627/0630, Udine (IT), 1985  
14.  Petryk H., A consistent energy approach to defining stability of plastic deformation processes, 2nd IUTAM Symposium, Stability in the Mechanics of Continua, 19810831/0904, Nümbrecht (DE), pp.262272, 1981 
Conference abstracts
1.  Kursa M., KowalczykGajewska K., Lewandowski M.J., Petryk H., Validation of meanfield approaches for the description of elasticplastic twophase composites, SolMech 2018, 41st SOLID MECHANICS CONFERENCE, 20180827/0831, Warszawa (PL), pp.7071, 2018  
2.  Kucharski S., Stupkiewicz S., Petryk H., Size effect in indentation tests: experimental and numerical investigations, EUROMAT 2017, European Congress and Exhibition on Advanced Materials and Processes, 20170917/0922, Thessaloniki (GR), No.D4HTUEPM1, pp.12, 2017  
3.  Kucharski S., Stupkiewicz S., Petryk H., Size effect observed in spherical indentation test of single crystal copper, Nanomechanical Testing in Materials Research and Development VI, 20171001/1006, Dubrovnik (HR), pp.1, 2017 Keywords: indentation size effect, single crystal, spherical indentation, numerical simulation Affiliations:
 
4.  Tůma K., Stupkiewicz S., Petryk H., The effect of twin spacing on the morphology of austenitetwinned martensite interface, SolMech 2016, 40th Solid Mechanics Conference, 20160829/0902, Warszawa (PL), No.P069, pp.1, 2016  
5.  Kursa M., Petryk H., Algorithm for rateindependent plasticity of single crystals based on incremental work minimization, SolMech 2016, 40th Solid Mechanics Conference, 20160829/0902, Warszawa (PL), No.P180, pp.12, 2016  
6.  KowalczykGajewska K., Petryk H., The sequential selfconsistent scheme for modelling elasticviscoplastic polycrystals, ESMC 2015, 9th European Solid Mechanics Conference, 20150706/0710, LeganésMadrid (ES), pp.#21612, 2015 Abstract: The application of the sequential method for estimating the mechanical response of elasticviscoplastic 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. Keywords:Micromechanics, Sequential linearization, SelfConsistent Scheme, Polycrystals Affiliations:
 
7.  Petryk H., Kursa M., Constitutive and material instabilities in rateindependent single crystals deformed by multiple slip, ESMC 2015, 9th European Solid Mechanics Conference, 20150706/0710, LeganésMadrid (ES), pp.#28112, 2015 Keywords: Crystal Plasticity, Slipsystems Selection, Incremental Energy Minimization Affiliations:
 
8.  Tůma K., Stupkiewicz S., Petryk H., Phasefield modelling of twinning and martensitic transformation at finite strain, PCMCMM 2015, 3rd Polish Congress of Mechanics and 21st Computer Methods in Mechanics, 20150908/0911, Gdańsk (PL), pp.815816, 2015 Abstract: We develop a micromechanical phasefield 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 austenitemartensite interfaces in the cubictoorthorhombic transformation in a CuAlNi shape memory alloy and estimated the elastic part of the interfacial energy. Affiliations:
 
9.  Petryk H., Kursa M., The energy approach to rateindependent plasticity of metal single crystals, PCMCMM 2015, 3rd Polish Congress of Mechanics and 21st Computer Methods in Mechanics, 20150908/0911, Gdańsk (PL), pp.683684, 2015 Abstract: In the modelling of metal single crystals in the framework of rateindependent plasticity, there are known difficulties caused by nonuniqueness in selection of active slipsystems. A related challenge is to predict emergence of nonuniform 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 nonnegative crystallographic slips. The calculated examples of deformation banding patterns and of reduction of multiplicity of active slipsystems in fcc single crystals are compared with the experimental observations. Keywords:crystal plasticity, slipsystems selection, incremental energy minimization Affiliations:
