1. | 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: 0020-7683, DOI: 10.1016/j.ijsolstr.2016.09.025, Vol.112, pp.209-221, 2017Petryk 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: 0020-7683, DOI: 10.1016/j.ijsolstr.2016.09.025, Vol.112, pp.209-221, 2017Abstract: A novel methodology is proposed for estimating the strain hardening exponent of a metal single crystal directly from the spherical indentation test, without the need of solving the relevant inverse problem. The attention is focused on anisotropic piling-up and sinking-in that occur simultaneously in different directions, in contrast to the standard case of axial symmetry for isotropic materials. To correlate surface topography parameters with the value of material hardening exponent, a finite-element study of spherical indentation has been performed within a selected penetration depth range using a finite-strain crystal plasticity model. It is shown how the power-law hardening exponent can be estimated from the measured pile-up/sink-in pattern around the residual impression after indentation in a (001)-oriented fcc single crystal of a small initial yield stress. For this purpose, a new parameter of surface topography is defined as the normalized material volume displaced around the nominal contact zone, calculated by integration of the local residual height (positive or negative) over a centered circular ring. That indicator can be easily determined from an experimental topography map available in a digital form. Comparison is made with the estimates based on measurements of the contact area and the slope of the load–penetration depth curve in logarithmic coordinates. The proposed methodology is extended to estimation of the hardening exponent simultaneously with the initial yield stress when the latter is not negligible. Experimental verification for a Cu single crystal leads to promising conclusions. Keywords: Metal crystal, Elastoplasticity, Finite deformation, Strain hardening, Experimental identification | | 35p. |
2. | Tůma K., Stupkiewicz S., Petryk H., Size effects in martensitic microstructures: Finite-strain phase field model versus sharp-interface approach, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/j.jmps.2016.04.013, Vol.95, pp.284-307, 2016Tůma K., Stupkiewicz S., Petryk H., Size effects in martensitic microstructures: Finite-strain phase field model versus sharp-interface approach, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/j.jmps.2016.04.013, Vol.95, pp.284-307, 2016Abstract: A finite-strain phase field model for martensitic phase transformation and twinning in shape memory alloys is developed and confronted with the corresponding sharp-interface approach extended to interfacial energy effects. The model is set in the energy framework so that the kinetic equations and conditions of mechanical equilibrium are fully defined by specifying the free energy and dissipation potentials. The free energy density involves the bulk and interfacial energy contributions, the latter describing the energy of diffuse interfaces in a manner typical for phase-field approaches. To ensure volume preservation during martensite reorientation at finite deformation within a diffuse interface, it is proposed to apply linear mixing of the logarithmic transformation strains. The physically different nature of phase interfaces and twin boundaries in the martensitic phase is reflected by introducing two order-parameters in a hierarchical manner, one as the reference volume fraction of austenite, and thus of the whole martensite, and the second as the volume fraction of one variant of martensite in the martensitic phase only. The microstructure evolution problem is given a variational formulation in terms of incremental fields of displacement and order parameters, with unilateral constraints on volume fractions explicitly enforced by applying the augmented Lagrangian method. As an application, size-dependent microstructures with diffuse interfaces are calculated for the cubic-to-orthorhombic transformation in a CuAlNi shape memory alloy and compared with the sharp-interface microstructures with interfacial energy effects. Keywords: Phase-field method, Microstructure, Martensite, Size effects, Shape memory alloys | | 40p. |
3. | Petryk H., Stupkiewicz S., A minimal gradient-enhancement of the classical continuum theory of crystal plasticity. Part I: The hardening law, ARCHIVES OF MECHANICS, ISSN: 0373-2029, Vol.68, No.6, pp.459-485, 2016Petryk H., Stupkiewicz S., A minimal gradient-enhancement of the classical continuum theory of crystal plasticity. Part I: The hardening law, ARCHIVES OF MECHANICS, ISSN: 0373-2029, Vol.68, No.6, pp.459-485, 2016Abstract: A simple gradient-enhancement of the classical continuum theory of plasticity of single crystals deformed by multislip is proposed for incorporating size effects in a manner consistent with phenomenological laws established in materials science. Despite considerable efforts in developing gradient theories, there is no consensus regarding the minimal set of physically based assumptions needed to capture the slip-gradient effects in metal single crystals and to provide a benchmark for more refined approaches. In order to make a step towards such a reference model, the concept of the tensorial density of geometrically necessary dislocations generated by slip-rate gradients is combined with a generalized form of the classical Taylor formula for the flow stress. In the governing equations in the rate form, the derived internal length scale is expressed through the current flow stress and standard parameters so that no further assumption is needed to define a characteristic length. It is shown that this internal length scale is directly related to the mean free path of dislocations and possesses physical interpretation which is frequently missing in other gradient-plasticity models. Keywords: gradient plasticity, geometrically necessary dislocations, single crystal, strain-hardening, internal length scale, size effect | | 25p. |
4. | Stupkiewicz S., Petryk H., A minimal gradient-enhancement of the classical continuum theory of crystal plasticity. Part II: Size effects, ARCHIVES OF MECHANICS, ISSN: 0373-2029, Vol.68, No.6, pp.487-513, 2016Stupkiewicz S., Petryk H., A minimal gradient-enhancement of the classical continuum theory of crystal plasticity. Part II: Size effects, ARCHIVES OF MECHANICS, ISSN: 0373-2029, Vol.68, No.6, pp.487-513, 2016Abstract: In our previous paper, a simple gradient-enhancement of the classical continuum theory of plasticity of single crystals deformed by multislip has been proposed for incorporating size effects. A single internal length scale has been derived as an explicit function of the flow stress defined as the isotropic part of critical resolved shear stresses. The present work is focused on verification whether the simplifications involved are not too severe and allow satisfactory predictions of size effects. The model has been implemented in a finite element code and applied to three-dimensional simulations of fcc single crystals. We have found that the experimentally observed indentation size effect in a Cu single crystal is captured correctly in spite of the absence of any adjustable length-scale parameter. The finite element treatment relies on introducing non-local slip rates that average and smoothen on an element scale the corresponding local quantities. Convergence of the finite element solution to the analytical one is also verified for the one-dimensional problem of a boundary layer formed at a constrained interface.
Keywords: gradient plasticity, geometrically necessary dislocations, boundary layer, size effects, indentation, finite element method | | 25p. |
5. | Petryk H., Kursa M., Incremental work minimization algorithm for rate-independent plasticity of single crystals, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, ISSN: 0029-5981, DOI: 10.1002/nme.4925, Vol.104, No.3, pp.157-184, 2015Petryk H., Kursa M., Incremental work minimization algorithm for rate-independent plasticity of single crystals, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, ISSN: 0029-5981, DOI: 10.1002/nme.4925, Vol.104, No.3, pp.157-184, 2015Abstract: A new constitutive algorithm for the rate-independent crystal plasticity is presented. It is based on asymptotically exact formulation of the set of constitutive equations and inequalities as a minimum problem for the incremental work expressed by a quadratic function of non-negative crystallographic slips. This approach requires selective symmetrization of the slip-system interaction matrix restricted to active slip-systems, while the latent hardening rule for inactive slip-systems is arbitrary. The active slip-system set and incremental slips are determined by finding a constrained minimum point of the incremental work. The solutions not associated with a local minimum of the incremental work are automatically eliminated in accord with the energy criterion of path stability. The augmented Lagrangian method is applied to convert the constrained minimization problem to a smooth unconstrained one. Effectiveness of the algorithm is demonstrated by the large deformation examples of simple shear of a face-centered cubic (fcc) crystal and rolling texture in a polycrystal. The algorithm is extended to partial kinematic constraints and applied to a uniaxial tension test in a high-symmetry direction, showing the ability of the algorithm to cope with the non-uniqueness problem and to generate experimentally observable solutions with a reduced number of active slip-systems. Keywords: solids, crystal plasticity, rate-independent constitutive equations, material stability, variational methods, incremental energy minimization, augmented Lagrangian method | | 40p. |
6. | Kowalczyk-Gajewska 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 A-STRUCTURAL MATERIALS PROPERTIES MICROSTRUCTURE AND PROCESSING, ISSN: 0921-5093, DOI: 10.1016/j.msea.2015.04.040, Vol.637, pp.251-263, 2015Kowalczyk-Gajewska 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 A-STRUCTURAL MATERIALS PROPERTIES MICROSTRUCTURE AND PROCESSING, ISSN: 0921-5093, DOI: 10.1016/j.msea.2015.04.040, Vol.637, pp.251-263, 2015Abstract: Texture evolution in commercially pure titanium deformed by equal-channel angular pressing (ECAP) and extrusion with forward–backward rotating die (KoBo) is studied both experimentally and numerically. New results are provided that demonstrate the effects of distinct and complex deformation paths on the texture in the ultra-fine grained (UFG) material obtained after severe plastic deformation (SPD). The numerical simulations are based on the self-consistent viscoplastic method of grain-to-polycrystal scale transition. A recently proposed modification of the probabilistic scheme for twinning is used that provides consistent values of the twin volume fraction in grains. The basic components of the experimentally observed texture are reasonably well reproduced in the modelling. The numerical simulations provide an insight into the internal mechanisms of plastic deformation, revealing substantial activity of mechanical twinning in addition to the basal and prismatic slip in titanium processed by ECAP. Keywords: Texture evolution, UFG materials, SPD processes, Crystal plasticity, Twinning | | 35p. |
7. | Fischer F.D., Svoboda J., Petryk H., Thermodynamic extremal principles for irreversible processes in materials science, ACTA MATERIALIA, ISSN: 1359-6454, DOI: 10.1016/j.actamat.2013.11.050, Vol.67, No.153, pp.1-20, 2014Fischer F.D., Svoboda J., Petryk H., Thermodynamic extremal principles for irreversible processes in materials science, ACTA MATERIALIA, ISSN: 1359-6454, DOI: 10.1016/j.actamat.2013.11.050, Vol.67, No.153, pp.1-20, 2014Abstract: Mathematical and physical aspects of the applicability of the Onsager, Prigogine as well as the Glansdorff and Ziegler thermodynamic extremal principles (TEPs) to non-equilibrium thermodynamics are examined for systems at fixed temperature with respect to their ability to provide kinetic equations approved in materials science. TEPs represent an alternative to the classical phenomenological equations approach. As TEPs are, more or less, a pure mathematical tool, they cannot significantly contribute to a deeper physical understanding. However, if a system can be described by discrete characteristic (thermodynamic) parameters, it is demonstrated that application of Onsager’s TEP or Ziegler’s TEP represents a systematic way to derive a set of explicit evolution equations for these parameters. This approach can significantly simplify the treatment of the problem and, thus, can also be applied to rather complex systems, for which the classical approach, involving application of phenomenological equations, fails. The application of TEPs is demonstrated on plasticity with respect to constitutive equations as well as on grain growth and coarsening with respect to evolution equations of discrete parameters. No exploitation of Prigogine’s TEP has been reported for applications in materials science. Contrarily, Prigogine’s TEP can be invalidated if the coefficients of the dissipation function depend on the evolution of state variables with time. This is demonstrated by a further practical example worked out for the solute drag phenomenon. Glansdorff’s and Prigogine’s evolution criterion, however, is always fulfilled near the equilibrium state of convex Gibbs energy. Extensions of TEPs to non-linear non-equilibrium thermodynamics are demonstrated for homogeneous and quasi-homogeneous dissipation functions. Keywords: Non-equilibrium, Thermodynamics, Entropy, Onsager’s principle, Thermodynamic extremal principles | | 45p. |
8. | Kursa M., Kowalczyk-Gajewska K., Petryk H., Multi-objective optimization of thermo-mechanical properties of metal-ceramic composites, COMPOSITES PART B-ENGINEERING, ISSN: 1359-8368, DOI: 10.1016/j.compositesb.2014.01.009, Vol.60, pp.586-596, 2014Kursa M., Kowalczyk-Gajewska K., Petryk H., Multi-objective optimization of thermo-mechanical properties of metal-ceramic composites, COMPOSITES PART B-ENGINEERING, ISSN: 1359-8368, DOI: 10.1016/j.compositesb.2014.01.009, Vol.60, pp.586-596, 2014Abstract: The optimization procedure is worked out for finding an optimal content of phases in metal–ceramic composites in case of conflicting objectives regarding thermo-mechanical properties of the material for a specific target application. Relationships between the material composition and effective properties of the composite are calculated by employing several methods of continuum micromechanics. A constrained minimization problem is solved for a single objective function based on the weighted squared distances from the best available thermo-mechanical properties for the material system selected. A compound block diagram is proposed for quick assessment of the consequences of deviating from the optimal composition. The developed procedure is applied to practical examples of Al2O3–Cu composites for brake disks and Al2O3–NiAl composites for valves of potential use in automotive industry. Keywords: Metal–matrix composites (MMCs), Thermomechanical, Plastic deformation, Micro-mechanics, Multi-criteria optimization | | 40p. |
9. | Kucharski S., Stupkiewicz S., Petryk H., Surface Pile-Up Patterns in Indentation Testing of Cu Single Crystals, EXPERIMENTAL MECHANICS, ISSN: 0014-4851, DOI: 10.1007/s11340-014-9883-1, Vol.54, pp.957-969, 2014Kucharski S., Stupkiewicz S., Petryk H., Surface Pile-Up Patterns in Indentation Testing of Cu Single Crystals, EXPERIMENTAL MECHANICS, ISSN: 0014-4851, DOI: 10.1007/s11340-014-9883-1, Vol.54, pp.957-969, 2014Abstract: Nano- and micro-indentation of Cu single crystals is performed in directions not aligned with crystallographic axes. Such tests correspond to mechanical characterization of incidentally oriented grains in a polycrystalline or composite material. Orientation and size dependence of complex patterns of surface piling-up and sinking-in around the imprint are investigated. Experimental observations are compared with finite element simulations based on the large deformation crystal plasticity theory. Keywords: Copper, Nanoindentation, AFM, EBSD, Crystal plasticity, Finite element method | | 35p. |
10. | Stupkiewicz S., Petryk H., A robust model of pseudoelasticity in shape memory alloys, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, ISSN: 0029-5981, DOI: 10.1002/nme.4405, Vol.93, No.7, pp.747-769, 2013Stupkiewicz S., Petryk H., A robust model of pseudoelasticity in shape memory alloys, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, ISSN: 0029-5981, DOI: 10.1002/nme.4405, Vol.93, No.7, pp.747-769, 2013Abstract: A model of pseudoelasticity in shape memory alloys is developed within the incremental energy minimization framework. Three constitutive functions are involved: the Helmholtz free energy and rate-independent dissipation that enter incrementally the minimized energy function, and the constraint function that defines the limit transformation strains. The proposed implementation is based on a unified augmented Lagrangian treatment of both the constitutive constraints and nonsmooth dissipation function. A methodology for easy reformulation of the model from the small-strain to finite-deformation regime is presented. Finite element computations demonstrate robustness of the finite-strain version of the model and illustrate the effects of tension–compression asymmetry and transversal isotropy of the surface of limit transformation strains. Keywords: shape memory alloys (SMA), phase transformation, energy methods, finite element method, augmented Lagrangian method | | 40p. |
11. | Petryk H., Kursa M., The energy criterion for deformation banding in ductile single crystals, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/j.jmps.2013.03.004, Vol.61, No.8, pp.1854-1875, 2013Petryk H., Kursa M., The energy criterion for deformation banding in ductile single crystals, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/j.jmps.2013.03.004, Vol.61, No.8, pp.1854-1875, 2013Abstract: The phenomenon of spontaneous formation of deformation bands in metal single crystals deformed plastically by crystallographic multislip is investigated theoretically by using the energy criterion of instability of a uniform deformation path. The second-order energy criterion for incipient deformation banding is derived in a time-continuous setting for a rate-independent elastic–plastic crystal. The need for selective symmetrization of the local interaction matrix for active slip-systems is demonstrated. A computational approach to deformation banding is developed by using non-convex constrained minimization of the incremental work with respect to increments in crystallographic shears and kinematical degrees of freedom. Calculated examples of deformation banding patterns in fcc single crystals are compared with experimental observations. Keywords: Metal crystal, Plasticity, Finite deformation, Laminate microstructure, Incremental energy minimization | | 40p. |
12. | Petryk H., Stupkiewicz S., Instability of equilibrium of evolving laminates in pseudo-elastic solids, INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS, ISSN: 0020-7462, DOI: 10.1016/j.ijnonlinmec.2011.07.005, Vol.47, pp.317-330, 2012Petryk H., Stupkiewicz S., Instability of equilibrium of evolving laminates in pseudo-elastic solids, INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS, ISSN: 0020-7462, DOI: 10.1016/j.ijnonlinmec.2011.07.005, Vol.47, pp.317-330, 2012Abstract: This study is concerned with isothermal stability of equilibrium of evolving laminated microstructures in pseudo-elastic solids with a multi-well free energy function. Several possible modes of instability associated with phase transition between energy wells are analysed. The related rate-independent dissipation is included by imposing a threshold value on the thermodynamic driving force. For a homogenized phase-transforming laminate with no length scale it is shown that localization instability is a rule in case of a non-zero interfacial jump of a directional nominal stress, irrespectively of actual boundary conditions. A stabilizing effect of elastic micro-strain energy at the boundary of the localization zone is demonstrated for laminates of finite spacing. Illustrative numerical examples are given for an evolving austenite–martensite laminate in a crystal of CuZnAl shape memory alloy. Keywords: Microstructures, Phase transformation, Laminates, Energy methods, Stability | | 35p. |
13. | Petryk H., Stupkiewicz S., Modelling of microstructural evolution on complex paths of large plastic deformation, INTERNATIONAL JOURNAL OF MATERIALS RESEARCH, ISSN: 1862-5282, DOI: 10.3139/146.110683, Vol.103, No.3, pp.271-277, 2012Petryk H., Stupkiewicz S., Modelling of microstructural evolution on complex paths of large plastic deformation, INTERNATIONAL JOURNAL OF MATERIALS RESEARCH, ISSN: 1862-5282, DOI: 10.3139/146.110683, Vol.103, No.3, pp.271-277, 2012Abstract: A procedure for the modelling of microstructural changes induced by non-uniform large plastic deformation of metals is developed. For a given plastic working process, a set of deformation paths for different initial locations of a material element is generated first for a non-hardening material by using the finite element method. Next, changes of a cellular microstructure and related hardening effects along each path are calculated by using a recently proposed model. The procedure is applied to a non-conventional process of cold extrusion assisted by cyclic rotation of the die. The evolution of microstructural parameters, their effect on strain hardening and a distribution map over the specimen cross-section are calculated. Keywords: non-uniform deformation, dislocation cells, grain refinement, hardening, quantitative prediction | | 30p. |
14. | Stupkiewicz S., Maciejewski G., Petryk H., Elastic micro-strain energy of austenite–martensite interface in NiTi, MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING, ISSN: 0965-0393, DOI: 10.1088/0965-0393/20/3/035001, Vol.20, pp.035001-9, 2012Stupkiewicz S., Maciejewski G., Petryk H., Elastic micro-strain energy of austenite–martensite interface in NiTi, MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING, ISSN: 0965-0393, DOI: 10.1088/0965-0393/20/3/035001, Vol.20, pp.035001-9, 2012Abstract: The interfacial energy due to elastic micro-strains at the austenite–twinned martensite interface is calculated for the NiTi shape memory alloy undergoing cubic-to-monoclinic B2 ↔ B19' transformation. For each crystallographically distinct microstructure, an energetically favourable local shape of the interface is determined. The approach employs finite element computations and energy minimization with respect to shape parameters, taking into account elastic anisotropy of the phases and finite-strain kinematics. The effect of atomic-scale interfacial energy is studied. Keywords: microstructure, martensitic phase transformation, shape memory alloys (SMA), interface structure, micromechanical modelling | | 30p. |
15. | Kowalczyk-Gajewska K., Petryk H., Sequential linearization method for viscous/elastic heterogeneous materials, EUROPEAN JOURNAL OF MECHANICS A-SOLIDS, ISSN: 0997-7538, Vol.30, No.5, pp.650-664, 2011Kowalczyk-Gajewska K., Petryk H., Sequential linearization method for viscous/elastic heterogeneous materials, EUROPEAN JOURNAL OF MECHANICS A-SOLIDS, ISSN: 0997-7538, Vol.30, No.5, pp.650-664, 2011Abstract: The paper addresses the problem of suitable approximation of the interaction between phases in heterogeneous materials that exhibit both viscous and elastic properties. A novel approach is proposed in which linearized subproblems for an inhomogeneity-matrix system with viscous or elastic interaction rules are solved sequentially within one incremental step. It is demonstrated that in the case of a self-consistent averaging scheme, an additional accommodation subproblem, besides purely viscous and elastic subproblems, is to be solved in order to estimate the material response satisfactorily. By examples of an isotropic two-phase material it is shown that the proposed approach provides acceptable predictions in comparison with the existing models. Keywords: Micromechanics, Viscoelasticity, Viscoplasticity, Homogenization, Self-consistent scheme | | 30p. |
16. | Petryk H., Kursa M., Selective symmetrization of the slip-system interaction matrix in crystal plasticity, ARCHIVES OF MECHANICS, ISSN: 0373-2029, Vol.63, No.3, pp.287-310, 2011Petryk H., Kursa M., Selective symmetrization of the slip-system interaction matrix in crystal plasticity, ARCHIVES OF MECHANICS, ISSN: 0373-2029, Vol.63, No.3, pp.287-310, 2011Abstract: The symmetry issue of the interaction matrix between multiple slip-systems in the theory of crystal plasticity at finite deformation is revisited. By appealing to possibly non-uniform distribution of slip-system activity in a representative space-time element of a crystal, symmetry of the slip-system interaction matrix for the representative element is derived under assumptions that have a physical meaning. This conclusion refers to active slip-systems only. Accordingly, for any given hardening law, a new symmetrization rule is proposed that is restricted to active slip-systems and leaves the latent hardening of inactive slip-systems unchanged. Advantages of the proposal in comparison with full symmetrization are illustrated by a simple example of uniaxial tension. Keywords: finite deformation, metal crystal, plasticity, hardening, symmetry | | 20p. |
17. | Stupkiewicz S., Petryk H., A bi-crystal aggregate model of pseudoelastic behaviour of shape-memory alloy polycrystals, INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES, ISSN: 0020-7403, DOI: 10.1016/j.ijmecsci.2009.09.012, Vol.52, pp.219-228, 2010Stupkiewicz S., Petryk H., A bi-crystal aggregate model of pseudoelastic behaviour of shape-memory alloy polycrystals, INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES, ISSN: 0020-7403, DOI: 10.1016/j.ijmecsci.2009.09.012, Vol.52, pp.219-228, 2010Abstract: A multi-scale model of stress-induced phase transformation and martensite variant reorientation in shape memory alloy (SMA) polycrystals is developed. It is proposed to include neighbouring-grain interaction in a simple manner by introducing an intermediate bi-crystal level into the sequential averaging scheme for SMA. The constitutive relationships are defined by specifying the free energy and dissipation functions. At the level of a single grain, the rate-independent dissipation function is used that incorporates the dissipation due to forward and reverse austenite-to-martensite transformation as well as reorientation of martensite variants. The global response of the model is simulated numerically by minimizing the total incremental energy supply. Specific examples are calculated for a NiTi polycrystal for proportional and non-proportional loading paths. Keywords: Phase transformation, Dissipation, Microstructure, Multi-scale model, Incremental energy minimization | | 32p. |
18. | Petryk H., Stupkiewicz S., Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/j.jmps.2009.11.004, Vol.58, pp.390-408, 2010Petryk H., Stupkiewicz S., Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/j.jmps.2009.11.004, Vol.58, pp.390-408, 2010Abstract: This paper is a continuation of the Part I (H. Petryk, S. Stupkiewicz, Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory. J. Mech. Phys. Solids, 2010, doi:10.1016/j.jmps.2009.11.003). A fully three-dimensional model of an evolving martensitic microstructure is examined, taking into account size effects due to the interfacial energy and also dissipation related to annihilation of interfaces. The elastic micro-strain energy at microstructured interfaces is determined with the help of finite element computations and is approximated analytically. Three interface levels are examined: of grain boundaries attained by parallel martensite plates, of interfaces between austenite and twinned martensite, and of twin interfaces within the martensite phase. Minimization of the incremental energy supply, being the sum of the increments in the free energy and dissipation of the bulk and interfacial type at all levels, is used as the evolution rule, based on the theory presented in Part I. An example of the formation and evolution of a rank-three laminated microstructure of finite characteristic dimensions in a pseudoelastic CuAlNi shape memory alloy is examined quantitatively. Keywords: Microstructures, Phase transformation, Grain boundaries, Energy methods, Shape memory alloys (SMA) | | 32p. |
19. | 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: 0022-5096, DOI: 10.1016/j.jmps.2009.11.003, Vol.58, pp.373-389, 2010Petryk 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: 0022-5096, DOI: 10.1016/j.jmps.2009.11.003, Vol.58, pp.373-389, 2010Abstract: A model of evolving martensitic microstructures is formulated that incorporates the interfacial energy and dissipation on three different scales corresponding to the grain boundaries attained by martensite plates, the interfaces between austenite and martensite plates, and the twin interfaces within martensite plates. Three different time scales are also considered in order to clarify the meaning of rate-independent dissipation related to instabilities at more refined temporal and spatial scales. On the slowest time scale, the process of deformation and martensitic phase transformation is investigated as being composed of segments of smooth quasi-static evolution separated by sudden jumps associated with creation or annihilation of interfaces. A general evolution rule is used in the form of minimization of the incremental energy supply to the whole compound thermodynamic system, including the rate-independent dissipation. Close relationship is shown between the evolution rule and the thermodynamic condition for stability of equilibrium, with no a priori assumption on convexity of the dissipation function. It is demonstrated that the extension of the minimum principle from the first-order rates to small but finite increments requires a separate symmetry restriction imposed on the state derivative of the dissipation function. Formulae for the dissipation associated with annihilation of interfaces are proposed that exhibit limited path-independence and satisfy that symmetry requirement. By exploiting the incremental energy minimization rule with the help of the transport theorems, local propagation conditions are derived for both planar and curved phase transformation fronts. The theory serves as a basis for the algorithm for calculation of the stress-induced evolution of martensitic microstructures along with their characteristic dimensions and related hysteresis loops in shape memory alloys; the examples are given in Part II of the paper. Keywords: Microstructures, Phase transformation, Grain boundaries, Energy methods, Stability | | 32p. |
20. | Stupkiewicz S., Petryk H., Grain-size effect in micromechanical modelling of hysteresis in shape memory alloys, ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK, ISSN: 0044-2267, DOI: 10.1002/zamm.201000008, Vol.90, pp.783-795, 2010Stupkiewicz S., Petryk H., Grain-size effect in micromechanical modelling of hysteresis in shape memory alloys, ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK, ISSN: 0044-2267, DOI: 10.1002/zamm.201000008, Vol.90, pp.783-795, 2010Abstract: Size effects in pseudoelastic polycrystalline shape memory alloys are studied by considering a representative spherical laminated domain (subgrain) and its interfacial energy at three scales: at the subgrain boundaries, at the austenite–martensite interfaces, and at the twin boundaries. Two sources of interfacial energy are accounted for, namely the atomic-scale energy of twin and phase boundaries and the elastic strain energy at microstructured interfaces, the latter being predicted theoretically. The evolution of microstructure of the representative domain is determined using the incremental energy minimization rule applied to the sum of the increments in the Helmholtz free energy and rate-independent dissipation. The size-dependent part of dissipation is estimated by assuming that negative increments in interfacial energy, associated with annihilation of interfaces, cannot be reverted back into the bulk free energy and are thus dissipated. Simple analytic formula for the interfacial energy dissipated in a complete forward-reverse transformation cycle is derived and combined with a micromechanical model of a polycrystalline NiTi shape memory alloy. A numerical example illustrating size-dependent hysteresis in the stress-induced martensitic transformation is presented. Keywords: interfacial energy, incremental energy minimization, dissipation, martensitic transformation | | 20p. |
21. | Petryk H., Stupkiewicz S., Energia powierzchniowa, dyssypacja i efekty skali w modelowaniu mikrostruktur martenzytycznych, Czasopismo Techniczne. Mechanika, ISSN: 0011-4561, Vol.107, No.20, pp.99-108, 2010Petryk H., Stupkiewicz S., Energia powierzchniowa, dyssypacja i efekty skali w modelowaniu mikrostruktur martenzytycznych, Czasopismo Techniczne. Mechanika, ISSN: 0011-4561, Vol.107, No.20, pp.99-108, 2010Abstract: W niniejszym artykule przedstawiono energetyczne podejście do wieloskalowego modelowania ewolucji mikrostruktur martenzytycznych w stopach z pamięcią kształtu. Energia swobodna Helmholtza i energia dyssypowana w układzie reprezentowane są przez sumy członów odpowiadających energii objętościowej oraz energii powierzchniowej na granicach mikro-strukturalnych pomiędzy poszczególnymi wariantami martenzytu, fazami lub ziarnami. Ewolucja mikrostruktury jest wyznaczana drogą przyrostowej minimalizacji całkowitej energii dostarczanej do rozpatrywanego układu w procesie makroskopowo quasi-statycznym i izotermicznym. Ogólną procedurę zastosowano do numerycznych symulacji powstawania i ewolucji warstwowych struktur martenzytycznych indukowanych naprężeniowo w stopach z pamięcią kształtu. W energii powierzchniowej uwzględniono energię mikroodkształceń sprężystych w otoczeniu granic mikrostrukturalnych, wyznaczoną przy użyciu metody elementów skończonych. Policzone przykłady opisują ewolucję mikrostruktury martenzytycznej w formie laminatu trzeciego rzędu w stopie CuAlNi dla przemiany β1→γ1′ oraz jej zależność od sposobu uwzględnienia dyssypacji energii powierzchniowej. Keywords: energia powierzchniowa, minimalizacja energii, przemiana martenzytyczna | | 6p. |
22. | Petryk H., Stupkiewicz S., Kuziak R., Grain refinement and strain hardening in IF steel during multi-axis compression: Experiment and modelling, JOURNAL OF MATERIALS PROCESSING TECHNOLOGY, ISSN: 0924-0136, DOI: 10.1016/j.jmatprotec.2007.11.068, Vol.204, pp.255-263, 2008Petryk H., Stupkiewicz S., Kuziak R., Grain refinement and strain hardening in IF steel during multi-axis compression: Experiment and modelling, JOURNAL OF MATERIALS PROCESSING TECHNOLOGY, ISSN: 0924-0136, DOI: 10.1016/j.jmatprotec.2007.11.068, Vol.204, pp.255-263, 2008Abstract: The effect of severe plastic deformation (SPD) during cyclic multi-axis compression on grain refinement and strain hardening in interstitial free (IF) steel is studied quantitatively. In the experimental part, the material samples were cold deformed in the MAXStrain(R) system by successive compression in two mutually orthogonal directions. The electron backscatter diffraction (EBSD) technique was used to measure the average spacing of the dislocation cell (low angle) and cell-block (high angle) boundaries. In the modelling part, the decrease in size of dislocation cells and cell blocks was expressed in terms of the effective plastic strain defined such that strain-rate reversals slow down its accumulation. The strengthening effect of microstructural evolution was included in the continuum mechanics framework of finite strain plasticity. Examples of simulation of the behaviour of IF steel severely deformed by multi-axis compression are calculated and compared to experimental data. Keywords: Modelling, Microstructure, Hardening, Severe plastic deformation | |
23. | Stupkiewicz S., Maciejewski G., Petryk H., Low-energy morphology of the interface layer between austenite and twinned martensite, ACTA MATERIALIA, ISSN: 1359-6454, DOI: 10.1016/j.actamat.2007.07.034, Vol.55, No.18, pp.6292-6306, 2007Stupkiewicz S., Maciejewski G., Petryk H., Low-energy morphology of the interface layer between austenite and twinned martensite, ACTA MATERIALIA, ISSN: 1359-6454, DOI: 10.1016/j.actamat.2007.07.034, Vol.55, No.18, pp.6292-6306, 2007Abstract: A micromechanical scheme is developed for predicting the morphology and interfacial energy of the interface layer between the parent phase and internally twinned martensite. Low-energy morphologies are determined by minimizing, with respect to shape parameters, the elastic microstrain energy associated with local incompatibility of transformation strains. The computational scheme involves a finite element solution to a problem of non-linear elasticity with eigenstrains, shape sensitivity analysis with respect to general shape parametrization and minimization employing a gradient-based algorithm. As an application, low-energy morphologies are studied for the austenite–martensite interface in the cubic-to-orthorhombic transformation in a CuAlNi shape memory alloy. Discussion of the results of the analysis includes comparison to alternative simplified methods in terms of the predicted morphologies and the corresponding interfacial energies. Keywords: Microstructure, Martensitic phase transformation, Shape memory alloys (SMA), Interface structure, Micromechanical modelling | |
24. | Petryk H., Stupkiewicz S., A quantitative model of grain refinement and strain hardening during severe plastic deformation, MATERIALS SCIENCE AND ENGINEERING A-STRUCTURAL MATERIALS PROPERTIES MICROSTRUCTURE AND PROCESSING, ISSN: 0921-5093, DOI: 10.1016/j.msea.2006.08.076, Vol.444, pp.214-219, 2007Petryk H., Stupkiewicz S., A quantitative model of grain refinement and strain hardening during severe plastic deformation, MATERIALS SCIENCE AND ENGINEERING A-STRUCTURAL MATERIALS PROPERTIES MICROSTRUCTURE AND PROCESSING, ISSN: 0921-5093, DOI: 10.1016/j.msea.2006.08.076, Vol.444, pp.214-219, 2007Abstract: The effect of severe plastic deformation (SPD) on grain refinement and strain hardening in polycrystalline metals is studied quantitatively. The decrease in size of dislocation cells and cell-blocks is expressed as a function of the effective plastic strain influenced by strain-rate reversals. The estimated growth of the high-angle boundary area fraction depends on the complexity of the three-dimensional deformation path. The strain hardening due to both dislocation and boundary strengthening is described in terms of microstructural parameters and incorporated in the continuum mechanics framework of finite strain plasticity. The proposed model provides a tool for quantitative comparison of different SPD processes. Examples of simulation of the behaviour of pure aluminium deformed by equal channel angular pressing (ECAP) and cyclic extrusion–compression are calculated. Keywords: Modelling, Microstructure, Severe plastic deformation, Ultra-fine grained materials | |
25. | Schurig M., Bertram A., Petryk H., Micromechanical analysis of the development of a yield vertex in polycrystal plasticity, ACTA MECHANICA, ISSN: 0001-5970, DOI: 10.1007/s00707-007-0462-8, Vol.194, pp.141-158, 2007Schurig M., Bertram A., Petryk H., Micromechanical analysis of the development of a yield vertex in polycrystal plasticity, ACTA MECHANICA, ISSN: 0001-5970, DOI: 10.1007/s00707-007-0462-8, Vol.194, pp.141-158, 2007Abstract: The Taylor-Lin polycrystal model is used to simulate plastic deformations of a polycrystal. These propagate a corner in the subsequent yield surface due to the intersection of the yield loci of a number of slip mechanisms. Two approaches to identify subsequent yield surfaces and the development of the yield vertex at the applied stress are discussed. A linear regression analysis of the rounded corner is used to identify the corner angle and its development for different strain processes. On deformation paths composed of two aligned segments, the development of a secondary vertex is shown to follow similar laws. | |
26. | Richert M., Petryk H., Stupkiewicz S., Grain refinement in AlMgSi alloy during cyclic extrusion-compression: Experiment and modelling, ARCHIVES OF METALLURGY AND MATERIALS, ISSN: 1733-3490, Vol.52, pp.49-54, 2007Richert M., Petryk H., Stupkiewicz S., Grain refinement in AlMgSi alloy during cyclic extrusion-compression: Experiment and modelling, ARCHIVES OF METALLURGY AND MATERIALS, ISSN: 1733-3490, Vol.52, pp.49-54, 2007Abstract: The effect of severe plastic deformation (SPD) during cyclic extrusion-compression (CEC) on grain refinement and strain hardening in AlMgSi alloy is studied quantitatively. New experimental results are presented showing that the average microband thickness and grain size decrease below 100 nm, i.e. a nanocrystalline material is obtained. In the modelling part, the decrease in size of dislocation cells and microbands is expressed in terms of the effective plastic strain defined such that strain rate reversals slow down its accumulation. Examples of simulation of the behaviour of AlMgSi alloy severely deformed by cyclic extrusion-compression are calculated and compared to experimental data. Keywords: Modelling, Microstructure, Hardening, Severe plastic deformation | |
27. | Maciejewski J., Kopeć H., Petryk H., Finite element analysis of strain non-uniformity in two processes of severe plastic deformation, ENGINEERING TRANSACTIONS (ROZPRAWY INŻYNIERSKIE), ISSN: 0867-888X, Vol.55, No.3, pp.197-216, 2007Maciejewski J., Kopeć H., Petryk H., Finite element analysis of strain non-uniformity in two processes of severe plastic deformation, ENGINEERING TRANSACTIONS (ROZPRAWY INŻYNIERSKIE), ISSN: 0867-888X, Vol.55, No.3, pp.197-216, 2007Abstract: Two severe plastic deformation (SPD) processing techniques, namely equal-channel angular pressing (ECAP) and cyclic extrusion-compression (CEC), are investigated by using the finite element method. The major aspect examined is the non-uniformity of the accumulated, equivalent plastic strain after processing with the use of different shapes of the die. The quantitative effect of several parameters on the plastic flow is determined. It is found that the diameter ratio of the chambers and narrower channel in the CEC method, and also the inclination angle of connecting conical parts, can affect strongly the degree of strain non-uniformity. Comparison is made of distributions of equivalent strain after two passes of ECAP for two different routes and with two die profiles. | |
28. | Stupkiewicz S., Petryk H., Finite-strain micromechanical model of stress-induced martensitic transformations in shape memory alloys, MATERIALS SCIENCE AND ENGINEERING A-STRUCTURAL MATERIALS PROPERTIES MICROSTRUCTURE AND PROCESSING, ISSN: 0921-5093, DOI: 10.1016/j.msea.2006.01.112, Vol.438-440, pp.126-130, 2006Stupkiewicz S., Petryk H., Finite-strain micromechanical model of stress-induced martensitic transformations in shape memory alloys, MATERIALS SCIENCE AND ENGINEERING A-STRUCTURAL MATERIALS PROPERTIES MICROSTRUCTURE AND PROCESSING, ISSN: 0921-5093, DOI: 10.1016/j.msea.2006.01.112, Vol.438-440, pp.126-130, 2006Abstract: A micromechanical model of stress-induced martensitic transformation in single crystals of shape memory alloys is developed. This model is a finite-strain counterpart to the approach presented recently in the small-strain setting [S. Stupkiewicz, H. Petryk, J. Mech. Phys. Solids 50 (2002) 2303–2331]. The stress-induced transformation is assumed to proceed by the formation and growth of parallel martensite plates within the austenite matrix. Propagation of phase transformation fronts is governed by a rate-independent thermodynamic criterion with a threshold value for the thermodynamic driving force, including in this way the intrinsic dissipation due to phase transition. This criterion selects the initial microstructure at the onset of transformation and governs the evolution of the laminated microstructure at the macroscopic level. A multiplicative decomposition of the deformation gradient into elastic and transformation parts is assumed, with full account for the elastic anisotropy of the phases. The pseudoelastic behavior of Cu–Zn–Al single crystal in tension and compression is studied as an application of the model. Keywords: Microstructures, Laminates, Finite deformations, Micromechanics, Shape memory alloys | |
29. | Petryk H., Thermodynamic conditions for stability in materials with rate-independent dissipation, PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, ISSN: 1364-503X, DOI: 10.1098/rsta.2005.1584, Vol.A363, pp.2479-2515, 2005Petryk H., Thermodynamic conditions for stability in materials with rate-independent dissipation, PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, ISSN: 1364-503X, DOI: 10.1098/rsta.2005.1584, Vol.A363, pp.2479-2515, 2005Abstract: A distinctive feature of the examined class of solids is that a part of the entropy production is due to rate-independent dissipation, as in models of plasticity, damage or martensitic transformations. The standard condition for thermodynamic stability is shown to be too restrictive for such solids and, therefore, an extended condition for stability of equilibrium is developed. The classical thermodynamic theory of irreversible processes is used along with the internal variable approach, with the emphasis on the macroscopic effects of micro-scale instabilities in the presence of two different scales of time. Specific conditions for material stability against internal structural rearrangements under deformation-sensitive loading are derived within the incremental constitutive framework of multi-mode inelasticity. Application to spontaneous formation of deformation bands in a continuum is presented. Conditions for stability or instability of a quasi-static process induced by varying loading are given under additional constitutive postulates of normality and symmetry. As illustration of the theory, the stability of equilibrium or a deformation path under uniaxial tension is analysed for a class of inelastic constitutive laws for a metal crystal deformed plastically by multi-slip or undergoing stress-induced martensitic transformation. | |
30. | Maciejewski G., Stupkiewicz S., Petryk H., Elastic micro-strain energy at the austenite-twinned martensite interface, ARCHIVES OF MECHANICS, ISSN: 0373-2029, Vol.57, No.4, pp.277-297, 2005Maciejewski G., Stupkiewicz S., Petryk H., Elastic micro-strain energy at the austenite-twinned martensite interface, ARCHIVES OF MECHANICS, ISSN: 0373-2029, Vol.57, No.4, pp.277-297, 2005Abstract: A micromechanical scheme is developed for the analysis of elastic micro-strains induced by local incompatibilities at the austenite-twinned martensite interface. The aim of the paper is to estimate the elastic micro-strain energy which is an important factor in the formation of microstructures during the martensitic transformation. The finite deformation framework is applied, consistent with the crystallographic theory of martensite, and full account is taken for elastic anisotropy of the phases. As an example, the microstructures in the cubic-to-orthorhombic transformation in CuAlNi shape memory alloy are analyzed by the finite element method for the assumed class of zigzag shapes of the austenite-martensite interface at the micro-level. Finally, the effect of the interphase boundary energy on the microstructure of the transition layer is studied. Keywords: microstructure, martensitic phase transformation, shape memory alloys (SMA), interface structure, micromechanical modelling | |
31. | Petryk H., Incremental energy minimization in dissipative solids, Comptes Rendus Mécanique, ISSN: 1631-0721, DOI: 10.1016/S1631-0721(03)00109-8, Vol.331, No.7, pp.469-474, 2003Petryk H., Incremental energy minimization in dissipative solids, Comptes Rendus Mécanique, ISSN: 1631-0721, DOI: 10.1016/S1631-0721(03)00109-8, Vol.331, No.7, pp.469-474, 2003Abstract: The incremental energy minimization is examined as a method of determining solution paths for time-independent dissipative solids. Isothermal quasi-static deformations are considered, and the deformation work is locally decomposed into the increments in free energy and intrinsic dissipation. General conditions necessary for the applicability of the minimization procedure are derived and discussed. Keywords: Solids and structures, Dissipative materials, Plasticity, Energy, Path stability | |
32. | Bigoni D., Petryk H., A note on divergence and flutter instabilities in elastic–plastic materials, INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, ISSN: 0020-7683, DOI: 10.1016/S0020-7683(01)00248-7, Vol.39, No.4, pp.911-926, 2002Bigoni D., Petryk H., A note on divergence and flutter instabilities in elastic–plastic materials, INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, ISSN: 0020-7683, DOI: 10.1016/S0020-7683(01)00248-7, Vol.39, No.4, pp.911-926, 2002Abstract: Dynamic stability of uniform straining of a nonlinear elastic solid is known to require that all eigenvalues of the acoustic tensor associated with the tangent elastic moduli be real and nonnegative. The focus of this note is to what extent this conclusion applies to time-independent, elastoplastic materials. Nonlinearity of the elastic–plastic constitutive law imposes limits on validity of a solution to the linear problem for which the acoustic tensor is determined. The effect of those limits on the conclusions about instability is examined. Keywords: Instability of plastic flow, Elastic–plastic material, Material stability, Flutter | |
33. | Stupkiewicz S., Petryk H., Modelling of laminated microstructures in stress-induced martensitic transformations, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/S0022-5096(02)00029-7, Vol.50, No.11, pp.2303-2331, 2002Stupkiewicz S., Petryk H., Modelling of laminated microstructures in stress-induced martensitic transformations, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/S0022-5096(02)00029-7, Vol.50, No.11, pp.2303-2331, 2002Abstract: This paper is concerned with micromechanical modelling of stress-induced martensitic transformations in crystalline solids, with the focus on distinct elastic anisotropy of the phases and the associated redistribution of internal stresses. Micro–macro transition in stresses and strains is analysed for a laminated microstructure of austenite and martensite phases. Propagation of a phase transformation front is governed by a time-independent thermodynamic criterion. Plasticity-like macroscopic constitutive rate equations are derived in which the transformed volume fraction is incrementally related to the overall strain or stress. As an application, numerical simulations are performed for cubic β1 (austenite) to orthorhombic γ1′ (martensite) phase transformation in a single crystal of Cu–Al–Ni shape memory alloy. The pseudoelasticity effect in tension and compression is investigated along with the corresponding evolution of internal stresses and microstructure. Keywords: Phase transformation, Microstructures, Layered material, Constitutive behaviour, Shape memory alloy | |
34. | Petryk H., Thermann K., Post-critical plastic deformation in incrementally nonlinear materials, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/S0022-5096(01)00131-4, Vol.50, No.5, pp.925-954, 2002Petryk H., Thermann K., Post-critical plastic deformation in incrementally nonlinear materials, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/S0022-5096(01)00131-4, Vol.50, No.5, pp.925-954, 2002Abstract: The formation of multiple macroscopic shear bands is investigated as a mechanism of advanced plastic flow of polycrystalline metals. The overall deformation pattern and material characteristics are determined beyond the critical instant of ellipticity loss, without the need of introducing an internal length scale. This novel approach to the modelling of post-critical plastic deformation is based on the concept of a representative nonuniform solution in a homogeneous material. The indeterminacy of a post-critical representative solution is removed by eliminating unstable solution paths with the help of the energy criterion of path instability. It is shown that the use of micromechanically based, incrementally nonlinear corner theories of time-independent plasticity leads then to gradual concentration of post-critical plastic deformation. The volume fraction occupied by shear bands is found to have initially a well-defined, finite value insensitive to the mesh size in finite element calculations. Further deformation depends qualitatively on details of the constitutive law. In certain cases, the volume fraction of active bands decreases rapidly to zero, leading to material instability of dynamic type. However, for physically hardening materials with the yield-vertex effect, the localization volume typically remains finite over a considerable deformation range. At later stages of the plane strain simulation, differently aligned secondary bands are formed in a series of bifurcations. Keywords: Plasticity, Shear bands, Material instability, Energy criterion, Bifurcation | |
35. | Petryk H., Thermann K., Post-critical deformation pattern in plane strain plastic flow with yield-surface vertex effect, INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES, ISSN: 0020-7403, DOI: 10.1016/S0020-7403(00)00010-2, Vol.42, No.11, pp.2133-2146, 2000Petryk H., Thermann K., Post-critical deformation pattern in plane strain plastic flow with yield-surface vertex effect, INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES, ISSN: 0020-7403, DOI: 10.1016/S0020-7403(00)00010-2, Vol.42, No.11, pp.2133-2146, 2000Abstract: This work is concerned with the formation of multiple macroscopic shear bands viewed as a mechanism of large plastic deformation of polycrystalline metals. The plastic deformation pattern in a time-independent material with a yield-surface vertex effect is investigated numerically in plane strain beyond the critical instant of ellipticity loss under quasi-static loading. The energy criterion of path instability applied to a family of post-critical solutions eliminates unstable paths and enables the overall deformation pattern to be determined, although the solutions remain locally indeterminate due to the absence of an internal length scale. In particular, the volume fraction of incipient shear bands is found to have a well-defined value irrespective of the mesh size in finite element calculations. As an apparently novel qualitative result, the formation of coarse, differently aligned secondary bands is observed at later stages of simulation. Keywords: Plasticity, Shear bands, Material instability, Energy criterion, Bifurcation | |
36. | Petryk H., General conditions for uniqueness in materials with multiple mechanisms of inelastic deformation, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/S0022-5096(99)00036-8, Vol.48, No.2, pp.367-396, 2000Petryk H., General conditions for uniqueness in materials with multiple mechanisms of inelastic deformation, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/S0022-5096(99)00036-8, Vol.48, No.2, pp.367-396, 2000Abstract: This study is concerned with multi-mode inelastic behaviour at macroscopically uniform deformation. The material is assumed to be time-independent; the physical origin of inelasticity may be otherwise arbitrary, including plasticity of crystals and polycrystals, micro-cracking, phase transformation, etc. A non-linear rate-problem of continuing mechanical equilibrium at finite strain is examined for a material element subject to deformation-sensitive loading under partial kinematic constraints. General conditions for uniqueness of the material response are established. As an application to predicting the onset of strain localization or failure, the condition is derived that excludes the bifurcation in a band from homogeneous deformation. In contrast to the usual requirement of ellipticity of the tangent stiffness moduli, the present condition for uniqueness takes into account any possible unloading and is directly imposed on the matrix of interaction moduli of internal mechanisms. Lower and upper bounds are established for the primary shear-band bifurcation along a smooth straining path. Keywords: Microstructures, Multi-mode inelasticity, Constitutive behaviour, Finite strain, Bifurcation | |
37. | Petryk H., Macroscopic rate-variables in solids undergoing phase transformation, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/S0022-5096(97)00099-9, Vol.46, No.5, pp.873-894, 1998Petryk H., Macroscopic rate-variables in solids undergoing phase transformation, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/S0022-5096(97)00099-9, Vol.46, No.5, pp.873-894, 1998Abstract: Averaging rules are derived for the rates of deformation gradient and nominal stress in heterogeneous solids undergoing quasi-static deformation and displacive phase transformation with coherent interfaces. Infinitesimal increments in strain and stress in the bulk material are accompanied by the finite increments in growing layers of a transformed phase. Expressions for the rates of the macroscopic variables and their products are given in several equivalent forms. The transport theorem and rate compatibility conditions for moving interfaces are extended to the initial instant of non-smooth transformation when the standard kinematical condition of compatibility is not satisfied. As an application of the averaging formulae, it is shown that the continuous growth of parallel planar layers of a transformed phase at a meso-level results in macroscopic constitutive rate equations analogous to the theory of plasticity. The normality law is obtained if the propagation of a phase transformation front in an elastic material takes place at a prescribed value of the thermodynamic driving force. Keywords: Phase transformation, Finite strain, Inhomogeneous material, Strain compatibility, Asymptotic analysis | |
38. | Petryk H., Plastic instability: Criteria and computational approaches, ARCHIVES OF COMPUTATIONAL METHODS IN ENGINEERING, ISSN: 1134-3060, DOI: 10.1007/BF03020127, Vol.4, No.2, pp.111-151, 1997Petryk H., Plastic instability: Criteria and computational approaches, ARCHIVES OF COMPUTATIONAL METHODS IN ENGINEERING, ISSN: 1134-3060, DOI: 10.1007/BF03020127, Vol.4, No.2, pp.111-151, 1997Abstract: General criteria of instability in time-independent elastic-plastic solids and the related computational approaches are reviewed. The distinction between instability of equilibrium and instability of a deformation process is discussed with reference to instabilities of dynamic, geometric or material type. Comparison is made between the bifurcation, energy and initial imperfection approaches. The effect of incremental nonlinearity of the constitutive law, associated with formation of a yield-surface vertex, on instability predictions is examined. A survey of the methods of post-critical analysis is presented. | |
39. | Petryk H., Post-critical plastic deformation of biaxially stretched sheets, INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, ISSN: 0020-7683, DOI: 10.1016/0020-7683(95)00061-E, Vol.33, No.5, pp.689-705, 1996Petryk H., Post-critical plastic deformation of biaxially stretched sheets, INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, ISSN: 0020-7683, DOI: 10.1016/0020-7683(95)00061-E, Vol.33, No.5, pp.689-705, 1996Abstract: A theoretical and numerical analysis of the formation of a localized neck in a biaxially stretched sheet is presented. A time-independent constitutive law is assumed to be incrementally non-linear as suggested by micromechanical studies of the elastoplastic deformation of polycrystalline metals. The incipient width of a necking band in an infinitely thin perfect sheet of a time-independent material is found here to have a well-defined initial value, proportional to the in-plane sheet dimension. During subsequent post-critical deformation the boundary of the necking band moves with respect to the material until the transition to localized necking is completed. These conclusions are derived on a theoretical route from the condition of stability of the post-bifurcation deformation process and are confirmed by the numerical analysis performed for a sheet of finite thickness. | |
40. | Petryk H., Thermodynamic Stability of Equilibrium in Plasticity, Journal of Non-Equilibrium Thermodynamics, ISSN: 0340-0204, DOI: 10.1515/jnet.1995.20.2.132, Vol.20, pp.132-149, 1995 | |