Lev Truskinovsky

Quasi–incompressible Cahn–Hilliard fluids and topological transitions

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

Mechanics of a discrete chain with bi-stable elements

It has become common to model materials supporting several crystallographic phases as elastic continua with non (quasi) convex energy. This peculiar property of the energy originates from the multi-stability of the system at the microlevel associated with the possibility of several energetically equivalent arrangements of atoms in crystal lattices. In this paper we study the simplest prototypical discrete system—a one-dimensional chain with a finite number of bi-stable elastic elements. Our main assumption is that the energy of a single spring has two convex wells separated by a spinodal region where the energy is concave. We neglect the interaction beyond nearest neighbors and explore in some detail a complicated energy landscape for this mechanical system. In particular we show that under generic loading the chain possesses a large number of metastable configurations which may contain up to one (snap) spring in the unstable (spinodal) state. As the loading parameters vary, the system undergoes a number of bifurcations and we show that the type of a bifurcation may depend crucially on the details of the concave (spinodal) part of the energy function. In special cases we obtain explicit formulas for the local and global minima and provide a quantitative description of the possible quasi-static evolution paths and of the associated hysteresis.

Ericksen's bar revisited : Energy wiggles

This paper addresses the non-uniqueness pointed out by Ericksen in his classical analysis of the equilibrium of a one-dimensional elastic bar with non-convex energy. According to Ericksen, for the bar in a hard device, the piecewise constant functions delivering the global minimum of the energy can have an arbitrary numberN of discontinuities in strain (phase-boundaries). Following some previous work in this area, we regularize the problem in order to resolve this degeneracy. We add two non-local terms to the energy density: one depends on the high (second) derivatives of the displacement, the other contains low (zero) derivatives. The low-derivative term (scaled with a constant β) introduces a strong non-locality, and simulates a three-dimensional interaction with the loading device, forcing the formation of layered microstructures in the process of energy minimization. The high-derivative (strain-gradient) term (scaled with a different constant α), represents a surface energy contribution which penalizes the formation of phase interfaces and prevents the infinite refinement of microstructures. In our description we consider the positions of interfaces as variables. This singles out in a natural way an infinite number of finite-dimensional subspaces, where all the essential nonlinearity is concentrated. In this way we can calculate explicitly the local minimizers (metastable states) and their energy, which turns out to be a multi-valued function of the interface positions and the imposed overall straind. Our approach thus gives an explicit framework for the study of the rich variety of finite-scale equilibrium microstructures for the bar and their stability properties. This allows for the study of a number of properties of phase transitions in solids; in particular their hysteretic behavior. Among our goals is the investigation of the phase diagram of the system, described by the functionN(d, α, β) giving the number of phase-boundaries in the absolute minimizer. We observe the somewhat counterintuitive effect that the energy at the global minimum, as a function of the overall strain, generically develops non-smooth oscillations (wiggles).

Finite Scale Microstructures in Nonlocal Elasticity

In this paper we develop a simple one-dimensional model accounting for the formation and growth of globally stable finite scale microstructures. We extend Ericksens model [9] of an elastic “bar” with nonconvex energy by including both oscillation-inhibiting and oscillation-forcing terms in the energy functional. The surface energy is modeled by a conventional strain gradient term. The main new ingredient in the model is a nonlocal term which is quadratic in strains and has a negative definite kernel. This term can be interpreted as an energy associated with the long-range elastic interaction of the system with the constraining loading device. We propose a scaling of the problem allowing one to represent the global minimizer as a collection of localized interfaces with explicitly known long-range interaction. In this limit the augmented Ericksens problem can be analyzed completely and the equilibrium spacing of the periodic microstructure can be expressed as a function of the prescribed average displacement. We then study the inertial dynamics of the system and demonstrate how the nucleation and growth of the microstructures result in the predicted stable pattern. Our results are particularly relevant for the modeling of twined martensite inside the austenitic matrix.

Kinetics of Martensitic Phase Transitions: Lattice model

Martensitic phase transitions are often modeled by mixed-type hyperbolic-elliptic systems. Such systems lead to ill-posed initial-value problems unless they are supplemented by an additional kinetic relation. In this paper we explicitly compute an appropriate closing relation by replacing the continuum model with its natural discrete prototype. The procedure can be viewed as either regularization by discretization or a physically motivated account of underlying discrete microstructure. We model phase boundaries by traveling wave solutions of a fully inertial discrete model for a bi-stable lattice with harmonic long-range interactions. Although the microscopic model is Hamiltonian, it generates macroscopic dissipation which can be specified in the form of a rela- tion between the velocity of the discontinuity and the conjugate configurational force. This kinetic relation respects entropy inequality but is not a consequence of the usual Rankine-Hugoniot jump conditions. According to the constructed solution, the dissipation at the macrolevel is due to the induced radiation of lattice waves carrying energy away from the propagating front. We show that sufficiently strong nonlocality of the lattice model may be responsible for the multivaluedness of the kinetic relation and can quantitatively affect kinetics in the near-sonic region. Direct numerical simulations of the transient dynamics suggest stability of at least some of the computed traveling waves.

Discretization and hysteresis

Abstract This paper presents a simple and explicit mathematical example of the effects of discretization on a nonconvex variational problem. We describe a one-dimensional model which we call the Ericksen-Timoshenko bar. The energy includes a term that is nonconvex in the strain, quadratic terms in an internal variable and its derivatives, and the simplest quadratic coupling. In the framework of classical elasticity theory, the model has a strong integral nonlocality. Under special constitutive hypotheses, one can construct a collection of stationary points with an arbitrary number of interfaces between phases. We show that solutions with more than one interface are saddle points of the energy, unstable with respect to motion of the interface. We then discretize the energy and show that the saddle points of the continuum problem all correspond to local minimizers of the discrete problem. Thus, the “energy landscape” of the continuum problem is essentially smooth, while the landscape of the discretization is bumpy. This result, which is due to the constraints imposed by the discretization, is independent of mesh size.

About the “normal growth” approximation in the dynamical theory of phase transitions

Nonequilibrium phase transitions can often be modeled by a surface of discontinuity propagating into a metastable region. The physical hypothesis of “normal growth” presumes a linear relation between the velocity of the phase boundary and the degree of metastability. The phenomenological coefficient, which measures the “mobility” of the phase boundary, can either be taken from experiment or obtained from an appropriate physical model. This linear approximation is equivalent to assuming the surface entropy production (caused by the kinetic dissipation in a transition layer) to be quadratic in a mass flux.In this paper we investigate the possibility of deducing the “normal growth” approximation from the viscosity-capillarity model which incorporates both strain rates and strain gradients into constitutive functions. Since this model is capable of describing fine structure of a “thick” advancing phase boundary, one can derive, rather than postulate, a kinetic relation governing the mobility of the phase boundary and check the validity of the “normal growth” approximation.We show that this approximation is always justified for sufficiently slow phase boundaries and calculate explicitly the mobility coefficient. By using two exact solutions of the structure problem we obtained unrestricted kinetic equations for the cases of piecewise linear and cubic stress-strain relations. As we show, the domain of applicability of the “normal growth” approximation can be infinitely small when the effective viscosity is close to zero or the internal capillary length scale tends to infinity. This singular behavior is related to the existence of two regimes for the propagation of the phase boundary — dissipation dominated and inertia dominated.

Mobility of lattice defects: discrete and continuum approaches

Abstract In this paper, we study a highly idealized model of a moving lattice defect allowing for an explicit, “first principles” computation of a functional relation between the macroscopic configurational force and the velocity of the defect. The discrete model is purely conservative and contains information only about elasticities of the constitutive elements. The apparent dissipation is due to the presence of microinstabilities and the nonlinearity-induced tunneling of the energy from long to short wavelengths. This type of “radiative damping” is believed to be generic and accounting for a considerable fraction of inelastic irreversibility associated with fracture, plasticity and phase transitions. The paper contains direct comparison of the exact lattice solution with various continuum and quasicontinuum approximations. Despite its simplicity, the model can be used directly for the description of dynamic phase transitions in thin films.

Macro- and micro-cracking in one-dimensional elasticity

Abstract In classical fracture mechanics, the equilibrium configurations of an elastic body are obtained by minimizing an energy functional containing two contributions, bulk and surface. Usually, the bulk energy is convex and the surface energy is concave. While this type of minimization successfully describes macroscopic cracks, it fails to model micro-defects forming a so-called process zone. To describe this phenomenon, we consider, in this paper, a model with a non-concave, “bi-modal” surface energy, which allows the formation of both macro- and micro-cracks. Specifically, we consider the simplest one-dimensional problem for a bar in a hard device and show that if the surface energy is not subadditive, the solution exhibits a new mode of failure with a finite number of micro-cracks coexisting with one fully developed macro-crack. We present an explicit example of a “quantized” micro-cracking with a subsequent development into a single macro-crack.

Rate independent hysteresis in a bi-stable chain

Abstract The nontrivial behavior of an elastic chain with identical bi-stable elements may be considered prototypical for a large number of nonlinear processes in solids ranging from phase transitions to fracture. The energy landscape of such a chain is extremely wiggly which gives rise to multiple equilibrium configurations and results in a hysteretic evolution and a possibility of trapping. In the present paper, which extends our previous study of the static equilibria in this system (Puglisi and Truskinovsky, J. Mech. Phys. Solids (2000) 1), we analyze the behavior of a bi-stable chain in a soft device under quasi-static loading. We assume that the system is over-damped and explore the variety of available nonequilibrium transformation paths. In particular, we show that the “minimal barrier” strategy leads to the localization of the transformation in a single spring. Loaded periodically, our bi-stable chain exhibits finite hysteresis which depends on the height of the admissible barrier; the cold work/heat ratio in this model is a fixed constant, proportional to the Maxwell stress. Comparison of the computed inner and outer hysteresis loops with recent experiments on shape memory wires demonstrates good qualitative agreement. Finally we discuss a relation between the present model and the Preisach model which is a formal interpolation scheme for hysteresis, also founded on the idea of bi-stability.