Shocks versus kinks in a discrete model of displacive phase transitions

We consider dynamics of phase boundaries in a bistable one-dimensional lattice with harmonic long-range interactions. Using Fourier transform and Wiener–Hopf technique, we construct traveling wave solutions that represent both subsonic phase boundaries (kinks) and intersonic ones (shocks). We derive the kinetic relation for kinks that provides a needed closure for the continuum theory. We show that the different structure of the roots of the dispersion relation in the case of shocks introduces an additional free parameter in these solutions, which thus do not require a kinetic relation on the macroscopic level. The case of ferromagnetic second-neighbor interactions is analyzed in detail. We show that the model parameters have a significant effect on the existence, structure, and stability of the traveling waves, as well as their behavior near the sonic limit.     

Stability of Subsonic Planar Phase Boundaries in a van der Waals Fluid

We are concerned with the structural stability of dynamic phase changes occurring across sharp interfaces in a multidimensional van der Waals fluid. Such phase transitions can be viewed as propagating discontinuities. However, they are usually subsonic, and thus undercompressive. The lacking information lies in an additional jump condition, which may be derived from the viscosity-capillarity criterion. This condition is rather simple in the case of reversible phase transitions, since it reduces to a generalized equal area rule. In a previous work, I proved that reversible planar phase boundaries are weakly linearly stable, in the sense introduced by Majda for shock fronts. This means that they satisfy a generalized Lopatinsky condition but not a uniform one. The aim of this paper is to point out the influence of viscosity on the stability analysis, in order to deal with the more realistic case of dissipative phase transitions. The main difficulty lies in the additional jump condition, which is no longer explicit and depends on the (unknown) internal structure of the interface. We overcome it by using bifurcation arguments on the nondimensional parameter measuring the competition between viscosity and capillarity. We show by perturbation that the positivity of this parameter stabilizes the phase transitions. As a conclusion, we find that dissipative planar phase boundaries are uniformly linearly stable, in the sense of the uniform Lopatinsky condition. Accepted December 14, 1998     

Kinetic relations and the propagation of phase boundaries in solids

This paper treats the dynamics of phase transformations in elastic bars. The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition. A special elastic material with a piecewise-linear, non-monotonic stress-strain relation is considered, and the Riemann problem for this material is analyzed. For a large class of initial data, it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities.     

Propagating phase boundaries: Formulation of the problem and existence via the Glimm method

This paper treats the hyperbolic-elliptic system of two conservation laws which describes the dynamics of an elastic material having a non-monotone strain-stress function. FollowingAbeyaratne &Knowles, we propose a notion of admissible weak solution for this system in the class of functions of bounded variation. The formulation includes an entropy inequality, a kinetic relation (imposed along any subsonic phase boundary) and an initiation criterion (for the appearance of new phase boundaries). We prove theL ¹-continuous dependence of the solution to the Riemann problem. Our main result yields the existence and the stability of propagating phase boundaries. The proofs are based onGlimm's scheme and in particular on the techniques ofGlimm andLax. In order to deal with the kinetic relation, we prove a result of pointwise convergence of the phase boundary.     

A nonlinearly thermoelastic half-space under time-dependent normal and shear loading

A nonlinearly thermoelastic half-space is subjected to combined time-dependent normal and shear loading. The solution is obtained by a numerical method which is shown to yield accurate results by comparison with some known analytical solutions which can be obtained in some special cases. When shocks are involved, it is shown that the numerical results satisfy all the Rankine-Hugoniot jump conditions as well as the entropy condition across the shock.     

Kinetics of phase transformations in the peridynamic formulation of continuum mechanics

We study the kinetics of phase transformations in solids using the peridynamic formulation of continuum mechanics. The peridynamic theory is a nonlocal formulation that does not involve spatial derivatives, and is a powerful tool to study defects such as cracks and interfaces.We apply the peridynamic formulation to the motion of phase boundaries in one dimension. We show that unlike the classical continuum theory, the peridynamic formulation does not require any extraneous constitutive laws such as the kinetic relation (the relation between the velocity of the interface and the thermodynamic driving force acting across it) or the nucleation criterion (the criterion that determines whether a new phase arises from a single phase). Instead this information is obtained from inside the theory simply by specifying the inter-particle interaction. We derive a nucleation criterion by examining nucleation as a dynamic instability. We find the induced kinetic relation by analyzing the solutions of impact and release problems, and also directly by viewing phase boundaries as traveling waves.We also study the interaction of a phase boundary with an elastic non-transforming inclusion in two dimensions. We find that phase boundaries remain essentially planar with little bowing. Further, we find a new mechanism whereby acoustic waves ahead of the phase boundary nucleate new phase boundaries at the edges of the inclusion while the original phase boundary slows down or stops. Transformation proceeds as the freshly nucleated phase boundaries propagate leaving behind some untransformed martensite around the inclusion.     

On beams made of a phase-transforming material

This paper presents a theory for the mechanics of beams made of single crystals of shape memory alloys. The behavior of such beams can be quite unexpected and complicated due to the presence and propagation of phase boundaries. It is shown that the usual laws of mechanics do not fully determine the propagation of phase boundaries and that there is a need for additional constitutive information in the form of a kinetic relation. A simple experiment to measure this kinetic relation is proposed. Finally, a strategy to use such beams for propulsion at small scales is presented.     

Kink Surfaces in a Directionally Reinforced neo-Hookean Material under Plane Deformation: II. Kink Band Stability and Maximally Dissipative Band Broadening

Stationary kinks (elastostatic shocks) are examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for the additional fiber reinforcing stiffness. Previous work has shown that such a transversely isotropic material can support stationary kinks in plane deformation if the reinforcing is sufficiently great. If the deformation on one side of the kink involves a load axis aligned with the fiber axis, then the more general plane deformation on the other side of the kink is characterized in terms of a one-parameter family of (kink orientation, kink strength)-pairs. Here, the ellipticity status of the two correlated deformation states is shown to span all four possible ellipticity/nonellipticity permutations. If both deformation states are elliptic, then a suitable intermediate deformation is shown to be nonelliptic. Maximally dissipative quasi-static kink motion is examined and interpreted in terms of kink band broadening in on-axis loading. Such maximally dissipative kinks nucleate only in compression as weak kinks, with subsequent motion converting nonelliptic deformation to elliptic deformation. The associated fiber rotation involves three periods: an initial period of slow rotation, a secondary period of rapid rotation, and a final period of essentially constant orienation.     

(Adiabatic) phase boundaries in a bistable chain with twist and stretch

Mass–spring chains with only extensional degrees of freedom have provided insights into the behavior of crystalline solids, including those capable of phase transitions. Here we add rotational degrees of freedom to the masses in a chain and study the dynamics of phase boundaries across which both the twist and stretch can jump. We solve impact and Riemann problems in the chain by numerical integration of the equations of motion and show that the solutions are analogous to those in a phase transforming rod whose stored energy function depends on both twist and stretch. From the dynamics of phase boundaries in the chain we extract a kinetic relation whose form is familiar from earlier studies involving chains with only extensional degrees of freedom. However, for some combinations of parameters characterizing the energy landscape of our springs we find propagating phase boundaries for which the rate of dissipation, as calculated using isothermal expressions for the driving force, is negative. This suggests that we cannot neglect the energy stored in the oscillations of the masses in the interpretation of the dynamics of mass–spring chains. Keeping this in mind we define a local temperature of our chain and show that it jumps across phase boundaries, but not across sonic waves. Hence, impact problems in our mass–spring chains are analogous to those on continuum thermoelastic bars with Mie–Gruneisen type constitutive laws. At the end of the paper we use our chain to shed some light on experiments involving yarns that couple twist and stretch to perform useful work in response to various stimuli.     

Erosion Profile by a Global Model for Granular Flow

In this paper we study an integro-differential equation that models the erosion of a mountain profile caused by small avalanches. The equation is in conservative form, with a non-local flux involving an integral of the mountain slope. Under suitable assumptions on the erosion rate, the mountain profile develops several types of singularities, which we call kinks, shocks and hyper-kinks. We study the formation of these singularities and derive admissibility conditions. Furthermore, entropy weak solutions to the Cauchy problem are constructed globally in time, taking limits of piecewise affine approximate solutions. Entropy and entropy flux functions are introduced, and a Lax entropy condition is established for the weak solutions.     

The role of spinodal region in the kinetics of lattice phase transitions

We consider a one-dimensional chain of phase-transforming springs with harmonic long-range interactions. The nearest-neighbor interactions are assumed to be trilinear, with a spinodal region separating two material phases. We derive the traveling wave solutions governing the motion of an isolated phase boundary through the chain and obtain the functional relation between the driving force and the velocity of a phase boundary which can be used as the closing kinetic relation for the classical continuum theory. We show that a sufficiently wide spinodal region substantially alters the phase boundary kinetics at low velocities and results in a richer solution structure, with phase boundaries emitting short-length lattice waves in both direction. Numerical simulations suggest that solutions of the Riemann problem for the discrete system converge to the obtained traveling waves near the phase boundary.     

An effective approach for modeling fluid flow in heterogeneous media using numerical manifold method

A major challenge of modeling fluid flow in heterogeneous media is to model the material interfaces, which may be arbitrarily oriented or intersected with Dirichlet, Neumann, or other boundaries, making it difficult to mesh and accurately satisfy the boundary constraints. In order to solve these problems, we derived a new continuous approach in the numerical manifold method (NMM). NMM is an ideal method to handle boundaries, considering its flexibility and efficiency with fixed mathematical mesh and its integration precision. With the two‐cover‐meshing system, we construct physical covers containing gradient jump terms defined as extended degrees of freedom to realize the refraction law across material interfaces. In the global equilibrium equations, the jump terms are naturally considered with the energy‐work seepage model. In this approach, high accuracy is expected from the newly constructed jump function together with simplex integration. Moreover, high mesh efficiency is realized by fixed triangular mathematical mesh with algorithms fully considering interfaces intersecting with Dirichlet, Neumann, or other boundaries and simplex integration on elements in arbitrary shapes. The new approach was coded into our NMM fluid flow model. We calculated examples involving fluid flow through a domain including (1) a single interface, (2) an idealized fault represented by multiple material interfaces, (3) intersected interfaces, and (4) an octagonal inclusion. We compared the simulated results to analytical solutions or results with denser mesh to test precision and efficiency and thereby proved that the new approach is accurate, efficient, and flexible, especially when considering intense geometric change or intersections. Copyright © 2015 John Wiley & Sons, Ltd.     

Multifluid flows with weak and strong discontinuous interfaces using an elemental enriched space

In a previous paper, the authors presented an elemental enriched space to be used in a finite‐element framework (EFEM) capable of reproducing kinks and jumps in an unknown function using a fixed mesh in which the jumps and kinks do not coincide with the interelement boundaries. In this previous publication, only scalar transport problems were solved (thermal problems). In the present work, these ideas are generalized to vectorial unknowns, in particular, the incompressible Navier‐Stokes equations for multifluid flows presenting internal moving interfaces. The advantage of the EFEM compared with global enrichment is the significant reduction in computing time when the internal interface is moving. In the EFEM, the matrix to be solved at each time step has not only the same amount of degrees of freedom (DOFs) but also the same connectivity between the DOFs. This frozen matrix graph enormously improves the efficiency of the solver. Another characteristic of the elemental enriched space presented here is that it allows a linear variation of the jump, thus improving the convergence rate, compared with other enriched spaces that have a constant variation of the jump. Furthermore, the implementation in any existing finite‐element code is extremely easy with the version presented here because the new shape functions are based on the usual finite‐element method shape functions for triangles or tetrahedrals, and once the internal DOFs are statically condensed, the resulting elements have exactly the same number of unknowns as the nonenriched finite elements.     

Normal Modes and Nonlinear Stability Behaviour of Dynamic Phase Boundaries in Elastic Materials

This paper considers an ideal nonthermal elastic medium described by a stored-energy function W. It studies time-dependent configurations with subsonically moving phase boundaries across which, in addition to the jump relations (of Rankine–Hugoniot type) expressing conservation, some kinetic rule g acts as a two-sided boundary condition. The paper establishes a concise version of a normal-modes determinant that characterizes the local-in-time linear and nonlinear (in)stability of such patterns. Specific attention is given to the case where W has two local minimizers U ^A ,U ^B which can coexist via a static planar phase boundary. Being dynamic perturbations of such interesting configurations, this paper shows that the stability behaviour of corresponding almost-static phase boundaries is uniformly controlled by an explicit expression that can be determined from derivatives of W and g at U ^A and U ^B .     

Reflection and Refraction of Anti-Plane Shear Waves from a Moving Phase Boundary

The reflection and refraction of anti-plane shear waves from an interface separating half-spaces with different moduli is well understood in the linear theory of elasticity. Namely, an oblique incident wave gives rise to a reflected wave that departs at the same angle and to a refracted wave that, after transmission through the interface, departs at a possibly different angle. Here we study similar issues for a material that admits mobile elastic phase boundaries in anti-plane shear. We consider an energy minimal equilibrium state in anti-plane shear involving a planar phase boundary that is perturbed due to an incident wave of small magnitude. The phase boundary is allowed to move under this perturbation. As in the linear theory, the perturbation gives rise to a reflected and a refracted wave. The orientation of these waves is independent of the phase boundary motion and determined as in the linear theory. However, the phase boundary motion affects the amplitudes of the departing waves. Perturbation analysis gives these amplitudes for general small phase boundary motion, and also permits the specification of the phase boundary motion on the basis of additional criteria such as a kinetic relation. A standard kinetic relation is studied to quantify the subsequent energy partitioning and dissipation on the basis of the properties of the incident wave.     

On the derivation of an admissibility condition for phase boundary propagation in an SMA bar based on a 3-D formulation

There is some considerable difficulty in determining the solution uniquely for a propagating phase boundary in shape memory alloy (SMA) bar. In this paper, we establish an admissibility condition starting from a three-dimensional (3-D) internal-variable formulation to resolve this issue. We adopt a 3-D formulation in literature which is based on a constitutive model with specific forms of the Helmholtz free energy and dissipation rate. Then the 3-D dynamical equations are reduced to the 1-D rod equations for three phase regions (coupled with the radial effect and surface condition) by using two small parameters. Connection conditions at the phase interfaces are determined. By considering the traveling-wave solution for the rod system, we eventually derive three conditions across a sharp phase boundary corresponding to the 1-D sharp-interface model, including the two usual jump conditions and an additional condition. The third condition is then used to supplement the 1-D sharp-interface model to study an impact problem. The unique solution is constructed analytically for all possible impact velocity, including three kinds of wave patterns according to different levels of the impact velocity. The results are compared with those obtained by the maximal dissipation rate criterion.     

The concept of material forces in phase transition problems within the level-set framework

The dynamics of a phase transition front in solids using the level set method is examined in this paper. Introducing an implicit representation of singular surfaces, a regularized version of the sharp interface model arises. The interface transforms into a thin transition layer of nonzero thickness where all quantities take inhomogeneous expressions within the body. It is proved that the existence of an inhomogeneous energy of the material predicts inhomogeneity forces that drive the singularity. The driving force is a material force entering the canonical momentum equation (pseudo-momentum) in a natural way. The evolution problem requires a kinetic relation that determines the velocity of the phase transition as a function of the driving force. Here, the kinetic relation is produced by invoking relations that can be considered as the regularized versions of the Rankine–Hugoniot jump conditions. The effectiveness of the method is illustrated in a shape memory alloy bar.     

Phase boundaries as agents of structural change in macromolecules

We model long rod-like molecules, such as DNA and coiled-coil proteins, as one-dimensional continua with a multi-well stored energy function. These molecules suffer a structural change in response to large forces, characterized by highly typical force-extension behavior. We assume that the structural change proceeds via a moving folded/unfolded interface, or phase boundary, that represents a jump in strain and is governed by the Abeyaratne–Knowles theory of phase transitions. We solve the governing equations using a finite difference method with moving nodes to represent phase boundaries. Our model can reproduce the experimental observations on the overstretching transition in DNA and coiled-coils and makes predictions for the speed at which the interface moves. We employ different types of kinetic relations to describe the mobility of the interface and show that this leads to different classes of experimentally observed force-extension curves. We make connections with several existing theories, experiments and simulation studies, thus demonstrating the effectiveness of the phase transitions-based approach in a biological setting.     

Global Continuous Riemann Solver for Nonlinear Elasticity

We consider in this paper an isothermal model of nonlinear elasticity. This model is described by two conservation laws that define a problem of mixed type, both elliptic and hyperbolic. We restrict ourselves to the linearly degenerate case, and consider Riemann data that lies in the hyperbolic regions. The lack of uniqueness of the Riemann problem is solved by the introduction of a so-called kinetic relation, used to narrow the set of admissible subsonic phase transitions. In this situation, we consider the Riemann problem for any data lying in the hyperbolic region, using either explicit computations or geometric arguments. This construction allows us to give sufficient conditions on the kinetic relation in order that the generated Riemann solver possesses properties of uniqueness, globality, and continuous dependence on the initial data in the L ¹ distance. Accepted October 1, 2000?Published online January 22, 2001