H. Petryk

Modelling of laminated microstructures in stress-induced martensitic transformations

Abstract 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.

Material instability and strain-rate discontinuities in incrementally nonlinear continua

Abstract A systematic analysis of theoretical aspects of material instability is presented for a general class of time-independent, incrementally nonlinear solids which admit a velocity-gradient potential. The second-order energy criteria of instability of uniform straining and of equilibrium are derived for a material element embedded in a quasi-statically deforming continuum. Several theorems concerning propagation of acceleration waves in an incrementally nonlinear material are proved, with particular reference to stationary discontinuities. Initiation of shear band formation either by a dynamic growth of an initial disturbance or by a quasi-static bifurcation is studied as a symptom of material instability. The derived condition for the instability of uniform quasi-static straining does not coincide with that for instability of equilibrium and both differ in general from the familiar condition of strong ellipticity loss formulated for the tangent moduli. This is illustrated on an example of a non-proportional deformation path of material obeying the J 2 corner theory of plasticity.

On discretized plasticity problems with bifurcations

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

Plastic instability: Criteria and computational approaches

SummaryGeneral 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.

On constitutive inequalities and bifurcation in elastic-plastic solids with a yield-surface vertex

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

Thermodynamic conditions for stability in materials with rate-independent dissipation

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.

Post-critical plastic deformation in incrementally nonlinear materials

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.

General conditions for uniqueness in materials with multiple mechanisms of inelastic deformation

Abstract 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.

Macroscopic rate-variables in solids undergoing phase transformation

Abstract 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.

Post-critical plastic deformation of biaxially stretched sheets

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.