The free energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis

The construction of effective models for materials that undergo martensitic phase transformations requires usable and accurate functional representations for the free energy density. The general representation of this energy is known to be highly non-convex; it even lacks the property of quasi-convexity. A quasi-convex relaxation, however, does permit one to make certain estimates and powerful conclusions regarding phase transformation. The general expression for the relaxed free energy is however not known in the n-variant case. Analytic solutions are known only for up to 3 variants, whereas cases of practical interests involve 7-13 variants. In this study we examine the n-variant case utilizing relaxation theory and produce a seemingly obvious but very powerful observation regarding a lower bound to the quasi-convex relaxation that makes practical evolutionary computations possible. We also examine in detail the 4-variant case where we explicitly show the relation between three different forms of the free energy of mixing: upper bound by lamination, the Reuß lower bound, and a lower estimate of the -measure bound. A discussion of the bounds and their utility is provided; sample computations are presented for illustrative purposes.     

RETRACTED: The free energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis

This article has been retracted at the request of the author and editor.Reason: The above paper was published without fully incorporating all the required corrections. The corrected paper is published in Journal of the Mechanics and Physics of Solids, 51/4, I-XXVI. An erratum also immediately precedes the corrected article, p. 763.     

An upper bound to the free energy of n-variant polycrystalline shape memory alloys

In the last two decades, the problem of computing the elastic energy of phase transforming materials has been studied by a variety of research groups. Due to the non-quasiconvexity of the underlying multi-well landscape, different relaxation methods have been used in order to estimate the quasiconvex envelope of the energy density, for which no explicit expression is known at present.This paper combines a recently developed lamination bound for monocrystalline shape memory alloys which relies on martensitic twinned microstructures with the work of Smyshlyaev and Willis [1998a. A ‘non-local’ variational approach to the elastic energy minimization of martensitic polycrystals. Proc. R. Soc. London A 454, 1573–1613]. As a result, a lamination upper bound for n-variant polycrystalline martensitic materials is obtained.The lamination bound is then compared with Reuß- and Taylor-type estimates. While, for given volume fractions, good agreement of lamination upper and convexification lower bounds is obtained, a comparison using energy-minimizing volume fractions computed from the various bounds yields larger differences. Finally, we also investigate the influence of the polycrystal's texture. For a strong ellipsoidal texture, we observe even better agreement of upper and lower bounds than for the case of isotropic statistics.     

A multiscale thermomechanical model for cubic to tetragonal martensitic phase transformations

We develop a multiscale thermomechanical model to analyze martensitic phase transformations from a cubic crystalline lattice to a tetragonal crystalline lattice. The model is intended for simulating the thermomechanical response of single-crystal grains of austenite. Based on the geometrically nonlinear theory of martensitic transformations, we incorporate microstructural effects from several subgrain length scales. The effective stiffness tensor at the grain level is obtained through an averaging scheme, and preserves crystallographic information from the lattice scale as well as the influence of volumetric changes due to the transformation. The model further incorporates a transformation criterion that includes a surface energy term, which takes into account the creation of interfaces between martensite and austenite. These effects, which are often neglected in martensitic transformation models, thus appear explicitly in the expression of the transformation driving force that controls the onset and evolution of the transformation. In the derivation of the transformation driving force, we clarify the relations between different combinations of thermodynamic potentials and state variables. The predictions of the model are illustrated by analyzing the response of a phase-changing material subjected to various types of deformations. Although the model is developed for cubic to tetragonal transformations, it can be adapted to simulate martensitic transformations for other crystalline structures.     

Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory

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.     

Interfacial energy and dissipation in martensitic phase transformations. Part II: Size effects in pseudoelasticity

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.     

A phase field approach for multivariant martensitic transformations of stable and metastable phases

A phase field model approach for multivariant martensitic transformations of stable and metastable phases is introduced. The evolution of the microstructure is examined with respect to elastic energy minimization in which one or two martensitic orientation variants are considered. In this context, the martensitic nucleation behavior is simulated for different activation barriers. Furthermore, the influence of time-dependent external loads on the formation of the different phases is studied. The numerical implementation is performed with finite elements and an implicit time integration scheme.     

Interface propagation and microstructure evolution in phase field models of stress-induced martensitic phase transformations

Analytical solutions for diffuse interface propagation are found for two recently developed Landau potentials that account for the phenomenology of stress-induced martensitic phase transformations. The solutions include the interface profile and velocity as a function of temperature and stress tensor. An instability in the interface propagation near lattice instability conditions is studied numerically. The effect of material inertia is approximately included. Two methods for introducing an athermal interface friction in phase field models are discussed. In the first method an analytic expression defines the location of the diffuse interface, and the rate of change of the order parameters is required to vanish if the driving force is below a threshold. As an alternative and more physical approach, we demonstrate that the introduction of spatially oscillatory stress fields due to crystal defects and the Peierls barrier, or to a jump in chemical energy, reproduces the effect of an athermal threshold. Finite element simulations of microstructure evolution with and without an athermal threshold are performed. In the presence of spatially oscillatory fields the evolution self-arrests in realistic stationary microstructures, thus the system does not converge to an unphysical single-phase final state, and rate-independent temperature- and stress-induced phase transformation hysteresis are exhibited.     

Thermodynamic and relaxation-based modeling of the interaction between martensitic phase transformations and plasticity

This paper focuses on the issue plasticity within the framework of a micromechanical model for single-crystal shape-memory alloys. As a first step towards a complete micromechanical formulation of such models, we work with classical J₂-von Mises-type plasticity for simplicity. The modeling of martensitic phase transitions is based on the concept of energy relaxation (quasiconvexification) in connection with evolution equations derived from inelastic potentials. Crystallographic considerations lead to the derivation of Bain strains characterizing the transformation kinematics. The model is derived for arbitrary numbers of martensite variants and thus can be applied to any shape-memory material such as CuAlNi or NiTi. The phase transition model captures effects like tension/compression asymmetry and transformation induced anisotropy. Additionally, attention is focused on the interaction between phase transformations and plasticity in terms of the inheritance of plastic strain. The effect of such interaction is demonstrated by elementary numerical studies.     

An upper bound solution for the force of combined backward-torward extrusion

A rigid-triangle velocity field for combined backward-forward extrusion based on the experiments and the slip-line field is proposed in this paper. The flow separation point in the rigid-triangle velocity field is defined in accordance with the slip-line theory. A formula of minimum upper bound solution for the punch pressure of the combined extrusion is derived. The values from this formula are compared with those from the slip-line solution and with experimental results. The formula of upper bound solution can be used in practice.     

An upper and lower bound for the die swell of the plane jet

Summary A lower and upper bound for the die swell of a plane jet of a viscoelastic liquid is obtained and applied to a Newtonian jet. Extension of the bounds for other fluids is suggested.

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Zusammenfassung Eine untere und obere Grenze für die Aufweitung eines ebenen viskoelastischen Flüssigkeitsstrahls werden angegeben und auf einen newtonschen Freistrahl angewandt. Die Anwendung der angegebenen Kriterien auf andere Flüssigkeiten wird angeregt.

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With 2 tables     

A phase field approach with a reaction pathways-based potential to model reconstructive martensitic transformations with a large number of variants

A pathway tree is constructed by recursively duplicating a single reconstructive martensitic transformation path with respect to lattice symmetries and point-group rotations. An energy potential built on this pathway is implemented in a phase-field technique in large strain framework, with the transformational strain as the order parameter. A specific splitting between non-dissipative elastic behavior and the dissipative evolution of the order parameter allows for the modeling of acoustic waves during rapid transformations. A simple toy-model transition from hexa- to square-lattice successfully demonstrates the possibility to model reconstructive martensitic transformations for a large number of variants (more than one hundred). Pure traction applied to our toy-model shows that variants can nucleate into previously created variants, with a hierarchical nucleation of variants spanning over five levels of transformation.     

An upper bound approach on yield surfaces of porous materials

Summary Triaxial deformation of a porous material is analyzed. A material model is proposed where the material consists of spherical cells each of which has a spherical pore in the center of the matrix. The velocity field in the matrix is assumed and the upper bound approach is attempted. The yield surfaces of the porous materials with various volume fractions of pores v _f are obtained as ellipsoids whose axes become smaller with increasing v _f.They coincide with the yield surfaces which have been proposed by the authors earlier.

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Ein Obere-Schranken-Ansatz für den Fließort von porösem Material

Übersicht Analysiert wird die dreiachsige Deformation eines porösen Werkstoffs, der durch kugelförmige Zellen mit sphärischen Löchern im Zentrum der Matrix modelliert wird. Für die Anwendung des Obere-Schranken-Satzes wird ein Geschwindigkeitsfeld in der Matrix angesetzt. Damit erhält man als Fließort von porösen Werkstoffen mit verschiedenen Porenvolumen-Anteilen v _fEllipsoide, deren Hauptachsen mit zunehmendem v _fkleiner werden. Sie stimmen mit den Fließorten, welche die Autoren schon früher halbexperimentell und halb-analytisch ermittelt haben, überein.

     

An integral bound for the strain energy in nonlinear elasticity in terms of the boundary displacements

For the displacement boundary value problem in nonlinear elastostatics with zero body force, an integral bound for the strain energy is obtained in terms of theL ₂-norms of the given boundary displacements and their tangential derivatives (assumed sufficiently small). The constants involved depend upon the strain energy density function and upon the geometry of the domain.     

An upper bound for validity limits of asymptotic analytical approaches based on normal form theory

Perturbation methods are routinely used in all fields of applied mathematics where analytical solutions for nonlinear dynamical systems are searched. Among them, normal form theory provides a reliable method for systematically simplifying dynamical systems via nonlinear change of coordinates, and is also used in a mechanical context to define Nonlinear Normal Modes (NNMs). The main recognized drawback of perturbation methods is the absence of a criterion establishing their range of validity in terms of amplitude. In this paper, we propose a method to obtain upper bounds for amplitudes of changes of variables in normal form transformations. The criterion is tested on simple mechanical systems with one and two degrees-of-freedom, and for complex as well as real normal form. Its behavior with increasing order in the normal transform is established, and comparisons are drawn between exact solutions and normal form computations for increasing levels of amplitudes. The results clearly establish that the criterion gives an upper bound for validity limit of normal transforms.