Abstract We present an analytical treatment of the problem pattern selection in a fully non-local symmetrical model dendritic crystal growth. Simplifications mathematical equations are based on assumption that anisotropies surface energy and kinetic effects small. Selection rules for growth velocity instability increments derived at arbitrary Peclet numbers. For dendrite growing channel, double-valued versus undercooling dependence is obtained. The upper branch solution stable changes into...

We study crack propagation in a viscoelastic solid. Using simple arguments, we derive equations for the velocity dependence of crack-tip radius, $a(v)$, and energy per unit area to propagate crack, $G(v)$. For modulus $E(\ensuremath{\omega})$ which increases as ${\ensuremath{\omega}}^{1\ensuremath{-}s}$ $(0<s<1)$ transition region between rubbery glassy region, find that $a(v)\ensuremath{\sim}G(v)\ensuremath{\sim}{v}^{\ensuremath{\alpha}}$ with...

The morphology diagram of possible structures in two-dimensional diffusional growth is given the parameter space undercooling \ensuremath{\Delta} versus anisotropy surface tension \ensuremath{\epsilon}. building block dendritic structure a dendrite with parabolic tip, and basic element seaweed doublon. transition between these shows jump velocity. We also describe velocities fractal dendrites doublons destroyed by noise. introduce renormalized capillary length density solid phase use scaling...

We study the selection of shape and growth velocity three-dimensional dendritic crystals in cubically anisotropic materials. In framework asymptotics beyond all orders we derive inner boundary-layer equation for nonaxisymmetric correction to Ivantsov paraboloid shape. The solvability condition this provides both comparison with available numerical experimental results is reasonably good.

The whole needle-crystal solution for three-dimensional (3D) dendritic growth is described analytically. Its construction involves the existing 3D selection theory tip of dendrite (M. Ben Amar and E. A. Brener [Phys. Rev. Lett. 71, 589 (1993]) plus matching tail to this tip. This exhibited here. Both intermediate final asymptotics shape are given. shape, which deviates strongly from an Ivantsov paraboloid, in qualitative agreement with experiental observations.

We propose a phase diagram for the selection of growth patterns in systems with conserved quantity which evolve at asymptotically constant rate. The occurrence different forms like fractal, compact or dendritic, and various transitions between them are characterized by scaling relations.

Fracture is a fundamental mechanism of materials failure. Propagating cracks can exhibit rich dynamical behavior controlled by subtle interplay between microscopic failure processes in the crack tip region and macroscopic elasticity. We review recent approaches to understand dynamics using phase field method. This method, developed originally for transformations, has well-known advantage avoiding explicit front tracking making material interfaces spatially diffuse. In fracture context, this...

We consider the time-dependent behavior of sidebranching deformations taking into account actual nonaxisymmetric shape needle crystal. The Green's function linearized problem is presented by a functional integral with help Mullins-Sekerka local spectrum. For short-wavelength perturbations can be calculated steepest descent method, where determined extremal trajectories governed Hamilton equations. spectrum plays role Hamilton's function. As in axisymmetric approach [J.S. Langer, Phys. Rev. A...

We present a continuum theory which predicts the steady state propagation of cracks. The overcomes usual problem finite time cusp singularity Grinfeld instability by inclusion elastodynamic effects restore selection tip radius and velocity. developed phase field model for elastically induced transitions; in limit small or vanishing elastic coefficients new phase, fracture can be studied. simulations confirm analytical predictions fast crack propagation.

We present a continuum theory to describe elastically induced phase transitions between coherent solid phases. In the limit of vanishing elastic constants in one phases, model can be used fracture on basis late stage Asaro-Tiller-Grinfeld instability. Starting from sharp interface formulation we derive equations and dissipative kinetics. develop field simulate these processes numerically; limit, it reproduces desired motion boundary conditions. perform large scale simulations eliminate...

We propose a friction model which incorporates interfacial elasticity and whose steady state sliding relation is characterized by generic nonmonotonic behavior, including both velocity weakening strengthening branches. In 1D upon the application of sideway loading, we demonstrate existence transient cracklike fronts independent sound speed, to be analogous recently discovered slow rupture fronts. Most importantly, properties these inhomogeneously loaded are determined front solutions at...

The onset of frictional instabilities, e.g. earthquakes nucleation, is intimately related to velocity-weakening friction, in which the resistance interfaces decreases with increasing slip velocity. While this response has been studied extensively, less attention given steady-state velocity-strengthening spite its potential importance for various aspects phenomena such as propagation speed interfacial rupture fronts and amount stored energy released by them. In note we suggest that a...

The strength and stability of frictional interfaces, ranging from tribological systems to earthquake faults, are intimately related the underlying spatially extended dynamics. Here we provide a comprehensive theoretical account, both analytic numeric, spatiotemporal interfacial dynamics in realistic rate-and-state friction model, featuring velocity-weakening velocity-strengthening behaviors. Slowly extending, loading-rate-dependent creep patches undergo linear instability at critical...

The failure of frictional interfaces and the spatiotemporal structures that accompany it are central to a wide range geophysical, physical engineering systems. Recent geophysical laboratory observations indicated interfacial can be mediated by slow slip rupture phenomena which distinct from ordinary, earthquake-like, fast rupture. These discoveries have influenced way we think about motion, yet nature properties not completely understood. We show is an intrinsic robust property simple...

The stress left behind in a system after frictional rupture depends not only on the physics of interface between interacting bodies but also, as new theory shows, deformation those bodies.

We present a phase field model for isothermal transformations of two-component alloys that includes Onsager kinetic cross coupling between the nonconserved $\ensuremath{\phi}$ and conserved concentration $C$. also provide reduction to corresponding macroscopic description free boundary problem. The is given in general form. Additionally we use an explicit example check reduced description, range its applicability, excellent agreement with direct simulations. relevance newly introduced terms...

A widespread framework for understanding frictional rupture, such as earthquakes along geological faults, invokes an analogy to ordinary cracks. distinct feature of cracks is that their near edge fields are characterized by a square root singularity, which intimately related the existence strict dissipation-related lengthscale separation and edge-localized energy balance. Yet, interrelations between singularity order, balance in rupture not fully understood, even physical situations...

We present a new mean-field theory of diffusion-limited aggregation (DLA). apply our approach to calculate the ensemble-averaged structure seen in recent experiments two-dimensional channel geometry. Our method explains similarity between average DLA occupancy and Saffman-Taylor finger pattern.

The onset of rapid slip along initially quiescent frictional interfaces, the process "earthquake nucleation," and dissipative spatiotemporal slippage dynamics play important roles in a broad range physical systems. Here we first show that interfaces described by generic friction laws feature stress-dependent steady-state pulse solutions, which are unstable quasi-1D approximation thin elastic bodies. We propose such pulses linear size L^{*} characteristic amplitude "critical nuclei" for...