Interface kinetics in phase field models: isothermal transformations in binary alloys and steps dynamics in molecular-beam-epitaxy

We present a unified description of interface kinetic effects in phase field models for isothermal transformations in binary alloys and steps dynamics in molecular-beam-epitaxy. The phase field equations of motion incorporate a kinetic cross-coupling between the phase field and the concentration field. This cross coupling generalizes the phenomenology of kinetic effects and was omitted until recently in classical phase field models. We derive general expressions (independent of the details of the phase field model) for the kinetic coefficients within the corresponding macroscopic approach using a physically motivated reduction procedure. The latter is equivalent to the so-called thin interface limit but is technically simpler. It involves the calculation of the effective dissipation that can be ascribed to the interface in the phase field model. We discuss in details the possibility of a non positive definite matrix of kinetic coefficients, i.e. a negative effective interface dissipation, although being in the range of stability of the underlying phase field model. Numerically, we study the step-bunching instability in molecular-beam-epitaxy due to the Ehrlich-Schwoebel effect, present in our model due to the cross-coupling. Using the reduction procedure we compare the results of the phase field simulations with the analytical predictions of the macroscopic approach

Fast crack propagation by surface diffusion

We present a continuum theory which describes the fast growth of a crack by surface diffusion. This mechanism overcomes the usual cusp singularity by a self-consistent selection of the crack tip radius. It predicts the saturation of the steady state crack velocity appreciably below the Rayleigh speed and tip blunting. Furthermore, it includes the possibility of a tip splitting instability for high applied tensions

Viscoelastic Fracture of Biological Composites

Soft constituent materials endow biological composites, such as bone, dentin and nacre, with viscoelastic properties that may play an important role in their remarkable fracture resistance. In this paper we calculate the scaling properties of the quasi-static energy release rate and the viscoelastic contribution to the fracture energy of various biological composites, using both perturbative and non-perturbative approaches. We consider coarse-grained descriptions of three types of anisotropic structures: (i) Liquid-crystal-like composites (ii) Stratified composites (iii) Staggered composites, for different crack orientations. In addition, we briefly discuss the implications of anisotropy for fracture criteria. Our analysis highlights the dominant lengthscales and scaling properties of viscoelastic fracture of biological composites. It may be useful for evaluating crack velocity toughening effects and structure-dissipation relations in these materials.Comment: 18 pages, 3 figure

Achieving realistic interface kinetics in phase field models with a diffusional contrast

Phase field models are powerful tools to tackle free boundary problems. For phase transformations involving diffusion, the evolution of the non conserved phase field is coupled to the evolution of the conserved diffusion field. Introducing the kinetic cross coupling between these two fields [E. A. Brener, G. Boussinot, Phys. Rev. E {\bf 86}, 060601(R) (2012)], we solve the long-standing problem of a realistic description of interface kinetics when a diffusional contrast between the phases is taken into account. Using the case of the solidification of a pure substance, we show how to eliminate the temperature jump at the interface and to recover full equilibrium boundary conditions. We confirm our results by numerical simulations

Melting of alloys along the inter-phase boundaries in eutectic and peritectic systems

We discuss a simple model of the melting kinetics along the solid-solid interface in eutectic and peritectic systems. The process is controlled by the diffusion inside the liquid phase and the existence of a triple junction is crucial for the velocity selection problem. Using the lubrication approximation for the diffusion field in the liquid phase we obtain scaling results for the steady-state velocity of the moving pattern depending on the overheating above the equilibrium temperature and on the material parameters of the system, including the dependences on the angles at the triple junction

Frictional shear cracks

We discuss crack propagation along the interface between two dissimilar materials. The crack edge separates two states of the interface, ``stick'' and ``slip''. In the slip region we assume that the shear stress is proportional to the sliding velocity, i.e. the linear viscous friction law. In this picture the static friction appears as the Griffith threshold for crack propagation. We calculate the crack velocity as a function of the applied shear stress and find that the main dissipation comes from the macroscopic region and is mainly due to the friction at the interface. The relevance of our results to recent experiments, Baumberger et al, Phys. Rev. Lett. 88, 075509 (2002), is discussed

Theory of unconventional singularities of frictional shear cracks

Crack-like objects that propagate along frictional interfaces, i.e.~frictional shear cracks, play a major role in a broad range of frictional phenomena. Such frictional cracks are commonly assumed to feature the universal square root near-edge singularity of ideal shear cracks, as predicted by Linear Elastic Fracture Mechanics. Here we show that this is not the generic case due to the intrinsic dependence of the frictional strength on the slip rate, even if the bodies forming the frictional interface are identical and predominantly linear elastic. Instead, frictional shear cracks feature unconventional singularities characterized by a singularity order $\xi$ that differs from the conventional $-\tfrac{1}{2}$ one. It is shown that $\xi$ depends on the friction law, on the propagation speed and on the symmetry mode of loading. We discuss the general structure of a theory of unconventional singularities, along with their implications for the energy balance and dynamics of frictional cracks. Finally, we present explicit calculations of $\xi$ and the associated near-edge fields for linear viscous-friction -- which may emerge as a perturbative approximation of nonlinear friction laws or on its own -- for both in-plane (mode-II) and anti-plane (mode-III) shear loadings.Comment: 15 pages, 2 figure

Fracture and Friction: Stick-Slip Motion

We discuss the stick-slip motion of an elastic block sliding along a rigid substrate. We argue that for a given external shear stress this system shows a discontinuous nonequilibrium transition from a uniform stick state to uniform sliding at some critical stress which is nothing but the Griffith threshold for crack propagation. An inhomogeneous mode of sliding occurs, when the driving velocity is prescribed instead of the external stress. A transition to homogeneous sliding occurs at a critical velocity, which is related to the critical stress. We solve the elastic problem for a steady-state motion of a periodic stick-slip pattern and derive equations of motion for the tip and resticking end of the slip pulses. In the slip regions we use the linear viscous friction law and do not assume any intrinsic instabilities even at small sliding velocities. We find that, as in many other pattern forming system, the steady-state analysis itself does not select uniquely all the internal parameters of the pattern, especially the primary wavelength. Using some plausible analogy to first order phase transitions we discuss a ``soft'' selection mechanism. This allows to estimate internal parameters such as crack velocities, primary wavelength and relative fraction of the slip phase as function of the driving velocity. The relevance of our results to recent experiments is discussed.Comment: 12 pages, 7 figure