Dynamics of steps along a martensitic phase boundary I: Semi-analytical solution

Abstract

We study the motion of steps along a martensitic phase boundary in a cubic lattice. To enable analytical calculations, we assume antiplane shear deformation and consider a phase transforming material with a stress-strain law that is piecewise linear with respect to one component of shear strain and linear with respect to another. Under these assumptions we derive a semi-analytical solution describing a steady sequential motion of the steps under an external loading. Our analysis yields kinetic relations between the driving force, the velocity of the steps and other characteristic parameters of the motion. These are studied in detail for the two-step and three-step configurations. We show that the kinetic relations are significantly affected by the material anisotropy. Our results indicate the existence of multiple solutions exhibiting sequential step motion.

Kinetics of a twinning step

“…This suggests non-existence of traveling wave solutions with velocity lower than the threshold value in the z = 0 case, in agreement with the conjectures made in [13, 16]. Meanwhile, solutions at sufficiently high velocities ( V ≥ V h ≈ 0.9908 at χ = 1) are also inadmissible, because the large amplitude of waves propagating behind the step front causes the n = 2 bonds directly above the step to switch from variant I to variant II, which violates the first inequality in (30) (see also [8]).…”

“…We remark that the kinetic relations obtained here are in general quantitatively different from the ones obtained in [23, 27] for the closely related one-dimensional FK model due to the different kernel (23) in the integral equation (25). In particular, the amplitude of the oscillations in our kernel decays at infinity [8], while in the one-dimensional case it remains constant. Note also that the one-dimensional problem has stronger singularities at the resonance velocities in the z = 0 case [16].…”

“…[17]). Analyzing the asymptotic strain behavior as in [8, 15, 33], one can show that the amplitude of the lattice waves slowly decays as | ξ | increases, so that the strains tend to constant values at infinity, as assumed in (5).…”

“…where we recall that λ ( k ξ ) is given by (21). The reader is referred to [8, 13, 16] for more details.…”

“…The two wells represent two different twin variants and are separated by a spinodal region where the potential is non-convex. This is an extension of the model with bilinear interactions that was used in [8, 9] to study high-velocity dynamics of steps along a phase boundary and in [10], where their quasistatic evolution was considered. Piecewise linear interactions were assumed by many authors to describe propagation of phase boundaries, fracture and dislocations in a crystal lattice (see [11–24] and references therein).…”