Interface Conditions for a Phase Field Model with Anisotropic and Non-Local Interactions

An alternative formulation of the phase field method is utilized from an integral equation perspective. The technique allows one to derive macroscopic conditions at the interface from the microscopic potentials. Differential geometry and asymptotic analysis yield interface conditions, in arbitrary spatial dimension, for interactions that may include anisotropy as well as non-local potentials. The interface conditions can be expressed in various formulations, for example, in terms of the principal curvature directions of the interface, or the second order directional derivatives of the (signed) distance function and the Hessian of the surface tension.     

Comparison of Iterative Methods for Computing the Pressure Field in a Dynamic Network Model

In dynamic network models, the pressure map (the pressure in the pores) must be evaluated at each time step. This calculation involves the solution of a large number of nonlinear algebraic systems of equations and accounts for more than 80 of the total CPU–time. Each nonlinear system requires at least the partial solution of a sequence of linear systems. We present a comparative study of iterative methods for solving these systems, where we apply both standard routines from the public domain package ITPACK 2C and our own routines tailored to the network problem. The conjugate gradient method, preconditioned by symmetric successive overrelaxation, was found to be consistently faster and more robust than the other solvers tested. In particular, it was found to be much superior to the successive overrelaxation technique currently used by many researchers.     

Traveling Waves in a Convolution Model for Phase Transitions

The existence, uniqueness, stability and regularity properties of traveling-wave solutions of a bistable nonlinear integrodifferential equation are established, as well as their global asymptotic stability in the case of zero-velocity continuous waves. This equation is a direct analog of the more familiar bistable nonlinear diffusion equation, and shares many of its properties. It governs gradient flows for free-energy functionals with general nonlocal interaction integrals penalizing spatial nonuniformity. (Accepted October 16, 1995)     

Viscosity of Strain Gradient Effects on the Kinetics of Propagating Phase Boundaries in Solids

The theory of thermoelastic materials undergoing solid-solid phase transformations requires constitutive information that governs the evolution of a phase boundary. This is known as a kinetic relation which relates a driving traction to the speed of propagation of a phase boundary. The kinetic relation is prescribed in the theory from the onset. Here, though, a special kinetic relation is derived from an augmented theory that includes viscous, strain gradient and heat conduction effects. Based on a special class of solutions, namely travelling waves, the kinetic relation is inherited from the augmented theory as the viscosity, strain gradient and heat conductivity are removed by a suitable limit process.     

Kinetic Relations for a Lattice Model of Phase Transitions

The aim of this article is to analyse travelling waves for a lattice model of phase transitions, specifically the Fermi–Pasta–Ulam chain with piecewise quadratic interaction potential. First, for fixed, sufficiently large subsonic wave speeds, we rigorously prove the existence of a family of travelling wave solutions. Second, it is shown that this family of solutions gives rise to a kinetic relation which depends on the jump in the oscillatory energy in the solution tails. Third, our constructive approach provides a very good approximate travelling wave solution.     

Interfaces of Phase Transition and Disappearance and Method of Negative Saturation for Compositional Flow with Diffusion and Capillarity in Porous Media

The specific case of interfaces separating a single-phase fluid and a two-phase continuum appears in the theory of compositional flow through porous media. They are usually called the interfaces of phase transition (PT-interfaces) or the interfaces of phase disappearing (PD-interfaces). The principle of equivalence is proved which shows that a single-phase multi-component fluid may be replaced by an equivalent fictitious two-phase fluid having specific properties. The equivalent properties are such that the extended saturation of a fictitious phase is negative. This principle enables us to develop the uniform system of two-phase equations in the overall domain in terms of the extended saturation (the NegSat model), and to apply the direct numerical simulation. In the case with diffusion, the uniform NegSat model contains a new term proportional to the gradient of saturation in the relation for flow velocity. The canonical NegSat model represents a transport equation with discontinuous nonlinearities. The qualitative analysis of this model shows that the PT-interfaces represent the shocks of the extended saturation, or, in some cases, can transform into weak shocks. The diffusion and capillarity do not destroy necessarily the shocks, but change their velocity. The analytical technique is developed which allows capturing PT-shocks. The method is illustrated by several examples of miscible gas injection in oil reservoir. In two-dimensional case, the effects of multiple shock collisions in heterogeneous media are automatically modeled. In the case of immiscible fluids and a classic interface, the suggested method converges to the VOF-method.     

Analytic Approximate Solution for a Flow of a Second-Grade Viscoelastic Fluid in a Converging Porous Channel

The problem of a two-dimensional steady flow of a second-grade fluid in a converging porous channel is considered. It is assumed that the fluid is injected into the channel through one wall and sucked from the channel through the other wall at the same velocity, which is inversely proportional to the distance along the wall from the channel origin. The equations governing the flow are reduced to ordinary differential equations. The boundary-value problem described by the latter equations is solved by the homotopy perturbation method. The effects of the Reynolds and crossflow Reynolds number on the flow characteristics are examined.     

Construction of a Slip-line Field in Photoviscoplastic Model Test

Slip-line fields in plastic deformations have been theoretically obtained for contact deformations between rigid and idealized-rigid perfectly plastic media under idealized contact conditions. In these analyses, simple assumptions such as idealized perfect lubrication, Coulomb friction or uniform contact pressure have been made However, the deformation and stress states near the contact surface are seriously affected by the mechanical properties of the contact media used and the lubricant or the contact pressure on the contact surface Hence the theoretical slip-line fields are fairly different from the fields obtained from actual experiments The actual slip-line fields are easily constructed, however, by using the isoclinic- fringe pattern obtained from photoviscoplastic model tests using polymers.     

A Rigidity Result for a Reduced Model of a Cubic-to-Orthorhombic Phase Transition in the Geometrically Linear Theory of Elasticity

In this article we study a simplified two-dimensional model for a cubic-to-orthorhombic phase transition occurring in certain shape-memory-alloys. In the low temperature regime the linear theory of elasticity predicts various possible patterns of martensite arrangements: Apart from the well known laminates this phase transition displays additional structures involving four martensitic variants—so called crossing twins.Introducing a variational model including surface energy, we show that these structures are rigid under small energy perturbations. Combined with an upper bound construction this gives the optimal scaling behavior of incompatible microstructures. These results are related to papers by Capella and Otto (Commun. Pure Appl. Math. 62(12):1632–1669, 2009; Proc. R. Soc. Edinb., Sect. A, Math. 142:273–327, 2012) as well as to a paper by Dolzmann and Müller (Meccanica 30:527–539, 1995).     

Complex Dynamics in a ODE Model Related to Phase Transition

Motivated by some recent studies on the Allen–Cahn phase transition model with a periodic nonautonomous term, we prove the existence of complex dynamics for the second order equation

$$\begin{aligned} -\ddot{x} + \left( 1 + \varepsilon ^{-1} A(t)\right) G'(x) = 0, \end{aligned}$$

where A(t) is a nonnegative T-periodic function and \(\varepsilon > 0\) is sufficiently small. More precisely, we find a full symbolic dynamics made by solutions which oscillate between any two different strict local minima \(x_0\) and \(x_1\) of G(x). Such solutions stay close to \(x_0\) or \(x_1\) in some fixed intervals, according to any prescribed coin tossing sequence. For convenience in the exposition we consider (without loss of generality) the case \(x_0 =0\) and \(x_1 = 1\).

     

On the Barenblatt Model for Non-Equilibrium Two Phase Flow in Porous Media

We prove global existence and uniqueness of solutions to a quasilinear Goursat problem, which was proposed by G. I. Barenblatt to describe non-equilibrium two phase fluid flow in permeable porous media. When the equilibrium relaxation time tends to zero, the solution is shown to converge to the entropy solution of the corresponding initial-boundary value problem for the classical Buckley-Leverett equation.     

Quantitative In Situ Studies of Dynamic Fracture in Brittle Solids Using Dynamic X-ray Phase Contrast Imaging

We demonstrate the use of X-ray phase contrast imaging with sub-microsecond temporal resolution to obtain quantitative visualization of dynamic fracture processes in brittle solids. We examine an amorphous solid (fused silica), a ceramic single crystal (single-crystal quartz), and a polycrystalline ceramic (boron carbide), in the form of single-edge notched specimens loaded using a three-point apparatus at nominal strain rates up to \(\sim \)800 s^?1. We observe that the crack tip speed for boron carbide is relatively independent of mode I stress intensity factor rate (\(\dot {K}_{\mathrm {I}}\)) for these rates of loading, while that of fused silica and single-crystal quartz increases with \(\dot {K}_{\mathrm {I}}\). Further, for the amorphous and single crystal cases, we observe the development of a crack tip instability in the form of crack branching as the crack tip speed approaches 45% of the Rayleigh wave speed. This suggests that strain-rate-dependent mechanisms contribute to crack branching. Such mechanisms may, in turn, affect the macroscopic fracture properties of these materials.     

A Sharp Interface Model for the Propagation of Martensitic Phase Boundaries

A model for the quasistatic evolution of martensitic phase boundaries is presented. The model is essentially the gradient flow of an energy that can contain elastic energy due to the underlying change in crystal structure in the course of the phase transformation and surface energy penalizing the area of the phase boundary. This leads to a free boundary problem with a nonlocal velocity that arises from a coupling to the elasticity equation. We show existence of solutions under a technical convergence condition using an implicit time-discretization.     

Converging Bounds for the Effective Shear Speed in 2D Phononic Crystals

Calculation of the effective quasistatic shear speed c in 2D solid phononic crystals is analyzed. The plane-wave expansion (PWE) and the monodromy-matrix (MM) methods are considered. For each method, the stepwise sequence of upper and lower bounds is obtained which monotonically converges to the exact value of c. It is proved that the two-sided MM bounds of c are tighter and their convergence to c is uniformly faster than that of the PWE bounds. Examples of the PWE and MM bounds of effective speed versus concentration of high-contrast inclusions are demonstrated.     

Differentiation of Energy Functionals in Two-Dimensional Elasticity Theory for Solids with Curvilinear Cracks

This paper considers the equations of two-dimensional elasticity theory in nonsmooth domains. The domains contain curvilinear cracks of variable length. On the crack faces, conditions are specified in the form of inequalities describing mutual nonpenetration of the crack faces. It is proved that the solutions of equilibrium problems with a perturbed crack converge to the solution of the equilibrium problem with an unperturbed crack in the corresponding space. The derivative of the energy functional with respect to the length of a curvilinear crack is obtained.     

Wave Penetration through Elastic Solids with a Periodic Array of Rectangular Flaws

In the context of wave propagation in damaged (elastic) solids, an analytical approach for anti-plane normal penetration of a plane wave through a periodic array of rectangular flaws is developed. Reduced the problem to integral equations holding over the openings, an approximation of one-mode type will lead to explicit analytical formulas for the scattering parameters. Numerical resolution of the relevant equations will finally provide some graphs to be compared. Sommario. Nellambito della propagazione ondosa in solidi elastici danneggiati, si sviluppa un approccio analitico per studiare la penetrazione normale di unonda piana attraverso una fila periodica di difetti di forma rettangolare. Si considera il caso (scalare) della propagazione anti-piana. Ridotto il problema a due equazioni integrali basate sulla distanza fra difetti adiacenti, unapprossimazione del tipo one-mode condurrà a formule analitiche esplicite (rispetto alla frequenza) per i coefficienti di diffrazione. La risoluzione numerica delle equazioni integrali esatte ed approssimate fornirà infine alcuni grafici di confronto.