Phoebus Rosakis

A Thermodynamic Internal Variable Model for the Partition of Plastic Work into Heat and Stored Energy in Metals

The energy balance equation for elastoplastic solids includes heat source terms that govern the conversion of some of the plastic work into heat. The remainder contributes to the stored energy of cold work due to the creation of crystal defects. This paper is concerned with the fraction β of the rate of plastic work converted into heating. We examine the status of the common assumption that β is a constant with regard to the thermodynamic foundations of thermoplasticity and experiments. A general internal-variable theory is introduced and restricted to abide by the second law of thermodynamics. Experimentally motivated assumptions reduce this theory to a special model of classical thermoplasticity. The only part of the internal energy not determined from the isothermal response is the stored energy of cold work, a function only of the internal variables. We show that this function can be inferred from stress and temperature data from a single adiabatic straining experiment. Experimental data from dynamic Kolsky-bar tests at various strain rates yield a unique stored energy function. Its knowledge is crucial for the determination of the thermomechanical response in non-isothermal processes. Such a prediction agrees well with results from dynamic tests at different rates. In these experiments, β is found to depend strongly on both strain and strain rate for various engineering materials. The model is successful in predicting this dependence. Requiring β to be constant is thus an approximation of dubious validity.

Ellipticity and deformations with discontinuous gradients in finite elastostatics

Loss of ellipticity of the equilibrium equations of finite elastostatics is closely related to the possible emergence of elastostatic shocks, i.e., deformations with discontinuous gradients. In certain situations where constitutive response functions are essentially one-dimentional, such as anti-plane shear or bar theories, strong ellipticity is closely related to convexity of the elastic potential and invertibility of certain constitutive response functions. The present work addresses the analogous issues within the context of three dimensional elastostatics of compressible but not necessarily isotropic hyperelastic materials. A certain direction-dependent resolution of the deformation gradient is introduced and its existence and uniqueness for a given direction are established. The elastic potential is expressed as a function of kinematic variables arising from this resolution. Strong ellipticity is shown to be equivalent to the positive definiteness of the Hessian matrix of this function, thus sufficing for its strict convexity. The underlying variables are interpretable physically as simple shears and extensions. Their work-conjugates define a traction response mapping. It is shown that discontinuous deformation gradients are sustainable if and only if this mapping fails to be invertible. This result is explicit, in the sense that it characterizes the set of all possible piecewise homogeneous deformations given the elastic potential function.

Unstable kinetic relations and the dynamics of solid-solid phase transitions

In recent continuum-mechanical models of phase transitions in solids, the kinetic relation for a transition is usually assumed to be such that the driving force acting on a phase boundary is a monotonically increasing function of phase boundary velocity. The present paper explores the implications of relinquishing this assumption in the dynamics of one-dimensional elastic bars undergoing stress-induced transitions. Among other results, it is found that, for a class of non-monotonic kinetic relations, models of the kind discussed here permit stick-slip motions of a phase boundary, as observed in certain experiments.

Dynamic twinning processes in crystals

This paper describes recent work by the authors on a continuum model for crystal twinning. Twinning is described as an anti-plane shear deformation with discontinuous strains, governed by an elastic potential with multiple wells. Possible shapes of twin lamellae and twinning steps and various regimes of their steady dynamic growth are studied. The model includes a kinetic relation governing anisotropic twin boundary motion in two dimensions under applied stress.

Compact zones of shear transformation in an anisotropic solid

Abstract The possible configurations of transformed regions in a material that undergoes mechanical phase change are investigated. Anti-plane shear kinematics are assumed. A constitutive law is proposed for a cubic crystal which can sustain deformations with strain discontinuities, such as twinning. It is characterized by an anisotropic stored energy function which loses ellipticity and has multiple potential wells with corresponding stress-free states, connected by transformation strains which are simple shears. The problem of a bounded transformed high-strain zone of unknown shape in an infinite body is studied. The strains must take values within different phases, i.e. disjoint subsets of the domain of strong ellipticity. This condition imposes severe restrictions on the shape of the inclusion ; its boundary cannot be smooth and cannot have corners. Only sufficiently slender regions, terminating in cusps and oriented along special directions are possible. Qualitative geometrical features of twin lamellae are thus predicted. Integral representations of the deformation and estimates of the strains in such zones are obtained.

The role of the spinodal region in one-dimensional martensitic phase transitions

Abstract A common approach in modeling martensitic phase transitions in the framework of continuum mechanics involves a nonconvex energy. This paper analyzes the influence of the spinodal region, or the region where the energy density is concave, on the resulting equilibria. We compare a one-dimensional model with a degenerate spinodal region to models with a finite spinodal region. In all models we consider an elastic bar with a nonconvex energy placed on a rigid elastic foundation, to mimic elastic interactions between different phases in higher dimensions. Interfacial energy is modeled by a strain-gradient term. We find that when the spinodal region is small, global minima are not affected, and the minimum energy as a function of the overall strain exhibits nonsmooth oscillations associated with sudden finite phase nucleation. However, a sufficiently wide spinodal region results in the partial smoothening of the global minimum energy and infinitesimal phase nucleation in the interior of the bar. This involves gradual growth of a pretransitional nucleus with strain in the spinodal region. We show a hysteresis path using an energetic strategy of switching between branches of local minima.

Characterization of Convex Isotropic Functions

Necessary and sufficient conditions are given for the convexity of a scalar valued function of tensors that is proper isotropic, or invariant under rotations. These conditions are also appropriate for functions defined only for orientation preserving tensors. They are weaker than Balls convexity conditions for fully isotropic functions (invariant under all orthogonal tensors) [B1]. The results are applied in obtaining polyconvexity conditions for the stored energy function.

On the morphology of ferroelectric domains

Abstract A theoretical model is proposed in order to explain the lamellar or spiked morphology of domains of opposite polarization observed in ferroelectric crystals in their polar phase. A nonconvex free energy is constructed, which has two isolated minima corresponding to states of opposite spontaneous polarization. The jump conditions for the electric field and polarization vector across domain walls assume a special form for this free energy. The electrostatic problem for a single domain of unknown shape and possibly curved walls is analyzed in two dimensions. The fields inside and outside are found explicitly. Metasbility then restricts the shape of the domain to be lamellar, i.e. slender, with small wall curvature, and terminating in sharp cusped ends along the polar axis. Thermodynamic considerations allow evaluation of the driving force at every wall point. This shows that such domains may elongate, but cannot thicken in the presence of moderate applied electric fields.

Microbuckling of fibrin provides a mechanism for cell mechanosensing

Biological cells sense and respond to mechanical forces, but how such a mechanosensing process takes place in a nonlinear inhomogeneous fibrous matrix remains unknown. We show that cells in a fibrous matrix induce deformation fields that propagate over a longer range than predicted by linear elasticity. Synthetic, linear elastic hydrogels used in many mechanotransduction studies fail to capture this effect. We develop a nonlinear microstructural finite-element model for a fibre network to simulate localized deformations induced by cells. The model captures measured cell-induced matrix displacements from experiments and identifies an important mechanism for long-range cell mechanosensing: loss of compression stiffness owing to microbuckling of individual fibres. We show evidence that cells sense each other through the formation of localized intercellular bands of tensile deformations caused by this mechanism.

Bifurcation and metastability in a new one-dimensional model for martensitic phase transitions

Abstract Materials undergoing stress-induced martensitic phase transitions often form complex twinned microstructures with multiple phase boundaries. They also exhibit hysteretic mechanical behavior. We propose and analyze a one-dimensional model for twinning. We consider two elastic bars coupled by a system of continuously distributed linear springs. One of the bars has a two-well nonconvex elastic energy density that models a two-variant martensitic phase. The other bar is linearly elastic and is meant to model the parent austenite phase. Interfacial energy is modeled by a strain-gradient term. Various types of boundary conditions model parameter-dependent loading. A local bifurcation analysis shows that local energy minima (metastable states) often involve a large number of phase boundaries. This is confirmed by the global-bifurcation diagrams obtained numerically. We observe that this microstructure emerges via both sudden (finite) and gradual (infinitesimal) phase nucleation. We propose an energetic argument that predicts hysteresis in overall load-deformation behavior due to metastability of multiple equilibria. A limiting case with zero interfacial energy is treated analytically, yielding global solution diagrams.