Denis Pilipenko

Brittle fracture in viscoelastic materials as a pattern-formation process.

A continuum model of crack propagation is presented and discussed. We obtain steady state solutions with a self-consistently selected propagation velocity and shape of the crack, provided that elastodynamic and viscoelastic effects are taken into account. Two different mechanism of crack propagation, a first order phase transition and surface diffusion are considered, and we discuss different loading modes. The arising free boundary problems are solved by phase field methods and a sharp interface approach using a multipole expansion technique.

Crack propagation as a free boundary problem.

A sharp interface model of crack propagation as a phase transition process is discussed. We develop a multipole expansion technique to solve this free boundary problem numerically. We obtain steady state solutions with a self-consistently selected propagation velocity and shape of the crack, provided that elastodynamic effects are taken into account. Also, we find a saturation of the steady state crack velocity below the Rayleigh speed, tip blunting with increasing driving force, and a tip splitting instability above a critical driving force.

Pattern formation during diffusion limited transformations in solids

We develop a description of diffusion limited growth in solid-solid transformations, which are strongly influenced by elastic effects. Density differences and structural transformations provoke stresses at interfaces, which affect the phase equilibrium conditions. We formulate equations for the interface kinetics similar to dendritic growth and study the growth of a stable phase from a metastable solid in both a channel geometry and in free space. We perform sharp interface calculations based on Greens function methods and phase field simulations, supplemented by analytical investigations. For pure dilatational transformations we find a single growing finger with symmetry breaking at higher driving forces, whereas for shear transformations the emergence of twin structures can be favorable. We predict the steady state shapes and propagation velocities, which can be higher than in conventional dendritic growth.

Pattern formation during diffusional transformations in the presence of triple junctions and elastic effects

We compare different scenarios for dendritic melting of alloys with respect to the front propagation velocity. In contrast to conventional dendritic growth, selection can here be also due to the presence of a grain boundary or coherence strains, and the propagation speed is higher. The most favorable situation is partial melting, where two parabolic fronts, one melting and one solidifying interface, are moving together, since the process is then determined by diffusion in the thin liquid layer. There, and also in phase field simulations of melting in peritectic and eutectic systems, we observe a rotation of the triple junction relative to the growth direction. Finally, we discuss the role of elastic effects due to density and structural differences on solid-state phase transformations, and we find that they significantly alter the selection principles. In particular, we obtain free dendritic growth even with isotropic surface tension. This is investigated by Greens function methods and a phase field approach for growth in a channel and illustrated for the formation of a twin phase.

Crack growth by surface diffusion in viscoelastic media.

We discuss steady state crack growth in the spirit of a free boundary problem. It turns out that mode I and mode III situations are very different from each other: In particular, mode III exhibits a pronounced transition towards unstable crack growth at higher driving forces, and the behavior close to the Griffith point is determined entirely through crack surface dissipation, whereas in mode I the fracture energy is renormalized due to a remaining finite viscous dissipation. Intermediate mixed-mode scenarios allow steady state crack growth with higher velocities than for pure mode I.

Theory of dendritic growth in the presence of lattice strain

We discuss elastic effects due to lattice strain which are a new key ingredient in the theory of dendritic growth for solid-solid transformations. Both thermal and elastic fields are eliminated by Greens function techniques, and a closed nonlinear integro-differential equation for the evolution of the interface is derived. We find dendritic patterns even without the anisotropy of the surface energy required by classical dendritic growth theory. In this sense, elastic effects serve as a new selection mechanism.

Numerical solution of transport equations for plasmas with transport barriers

An approach to solve numerically transport equations for plasmas with spontaneously arising and arbitrarily located transport barriers, regions with a strongly reduced transfer of energy, is proposed. The transport equations are written in a form conserving heat flux and solved numerically by using piecewisely exact analytical solutions of linear differential equations. Compared to standard methods, this approach allows to reduce significantly the number of operations required to obtain a converged solution with a heat conductivity changing abruptly at a critical temperature gradient and to use large time steps in modeling the formation and dynamics of transport barriers. Computations for the tokamak JET are done.

Crack Propagation as a Free Boundary Problem

We present a continuum theory which predicts the steady state propagation of cracks. The theory overcomes the usual problem of a finite time cusp singularity of the Grinfeld instability by the inclusion of elastodynamic effects which restore selection of the steady state tip radius and velocity. We developed a sharp interface method and a phase field approach to investigate the model. The simulations confirm analytical predictions for fast crack propagation.