Open Access Publications from the University of California

## Complex Boundary Integral Equation Formulation and Stability Analysis of a Maxwell Model and of an Elastic Model of Solid-Solid Phase Transformations

• Author(s): Greengard, Daniel Bijan
• et al.
Abstract

We study a viscoelastic model of the solid-solid phase change of

olivine to its denser $\beta$-spinel state at high pressures and

temperatures reachable in laboratory experiments matching conditions

typical of Earth's mantle. Using a previously unknown technique, the

equations are transformed to the problem of finding two complex

analytic functions in the sample satisfying certain conditions on the

outer boundary. The Sherman-Lauricella boundary integral equation is

used in a numerical algorithm that eliminates the bottleneck of having

to solve a large matrix equation at every timestep. The method is

implemented and used to compute the solution of a number of

non-axisymmetric test problems, some static and some dynamic in time.

Next we develop an alternative formulation in which the Lam e

equations of linearized elasticity are used to model the deformation

of the two phases, and we allow for compressibility. The formulation

is novel in that separate reference configurations are maintained for

the core and shell regions of the sample that grow or shrink in time

by accretion or removal at the boundary, one at the expense of the

other. We then compare the behavior of the evolution of this system

to the incompressible viscoelastic case and to an alternative elastic

model. Finally, we study the stability of circular interfaces with

axisymmetric initial data under the evolution equations. For various

parameter values of the circular interface evolution, we find families

of small perturbations of the circular interface and radial interface

velocity jump that either grow or decay exponentially in time. In

unstable cases, the growth rate increases without bound as the wave

number of the perturbation increases. In stable cases, the evolution

equations are well-posed until the interface leaves the stability

regime, at which point the numerical solutions blow up in an

oscillatory manner. Examples of stable and unstable behavior are

presented.