## Continuum theory of kinetics of phase transformations

### May 30, 2007

**Title**: Kinetics of phase transformations in the peridynamic formulation of continuum mechanics

**Auhors**: Kaushik Dayal and Kaushik Bhattacharya

**Source**: Journal of Mechanics and Physics of Solids, 54, 9, September 2006, pp. 1811-1842.

**Abstract**:

We study the kinetics of phase transformations in solids using the peridynamic formulation of continuum mechanics. The peridynamic theory is a nonlocal formulation that does not involve spatial derivatives, and is a powerful tool to study defects such as cracks and interfaces.

We apply the peridynamic formulation to the motion of phase boundaries in one dimension. We show that unlike the classical continuum theory, the peridynamic formulation does not require any extraneous constitutive laws such as the kinetic relation (the relation between the velocity of the interface and the thermodynamic driving force acting across it) or the nucleation criterion (the criterion that determines whether a new phase arises from a single phase). Instead this information is obtained from inside the theory simply by specifying the inter-particle interaction. We derive a nucleation criterion by examining nucleation as a dynamic instability. We find the induced kinetic relation by analyzing the solutions of impact and release problems, and also directly by viewing phase boundaries as traveling waves.

We also study the interaction of a phase boundary with an elastic non-transforming inclusion in two dimensions. We find that phase boundaries remain essentially planar with little bowing. Further, we find a new mechanism whereby acoustic waves ahead of the phase boundary nucleate new phase boundaries at the edges of the inclusion while the original phase boundary slows down or stops. Transformation proceeds as the freshly nucleated phase boundaries propagate leaving behind some untransformed martensite around the inclusion.

**Notes**: Via iMechanica.

What is peridynamics? Here is an answer (that I got by googling):

Peridynamics is a theory of continuum mechanics that is formulated in terms of integral equations rather than partial differential equations. It assumes that particles in a continuum interact across a finite distance as in molecular dynamics. The integral equations remain valid regardless of any fractures or other discontinuities that may emerge due to loading. In contrast, the differential equations of classical theory break down when a discontinuity appears. Peridynamics predicts the deformation and failure of structures under dynamic loading, especially failure due to fracture. Cracks emerge spontaneously as a result of the equations of motion and material model and grow in whatever direction is energetically favorable. The implementation does not require a separate law that tells cracks when and where to grow.

I know that in the pre-spinodal decomposition days, there was some talk about how spinodal region is actually a region of instability, since the barrier to nucleation goes to zero as one approaches the spinodal points resulting in infinite nuclei. However, it is not clear to me if something similar is meant by “nucleation as dynamic instability” in this paper. On the other hand, the phase boundary as a travelling wave sounds similar to what one finds in phase field models. A paper that I have to read more carefully and understand!

June 2, 2007 at 8:44 pm

[…] 2nd, 2007 In one of my recent posts I noted the peridynamics formulation for the study of phase transformation…. Over at iMechanica, in the comments section, there are plenty of discussions on the formulation; […]