In one of my recent posts I noted the peridynamics formulation for the study of phase transformations and wondered how it might be related to phase field models. Over at iMechanica, in the comments section, there are plenty of discussions on the formulation; specifically, here are the sections contrasting the method with phase field methods:

Peridynamics is, very loosely speaking, a continuum analog of pair-potentials in molecular dynamics … It may be a good way to model long-range effects that are important at small scales. We found that the long-range effects provide some interesting mechanics during nucleation that one doesn’t see in phase-field type models (Section 5 and Appendix B of [Journal of the Mechanics and Physics of
Solids, 54 (2006) 1811–1842, and is available at doi:10.1016/j.jmps.2006.04.001] ) …The dispersion relation is similar to atomistic (discrete) models of solids (Section 12 in [1]). In peridynamics, at short wavelengths, the velocity of the waves goes to zero. In atomistic phonon calculations as well, the velocity goes to zero when the wavelength approaches the lattice spacing. This is in sharp contrast to classical continuum (sharp-interface) and phase-field models. In MD models, one doesn’t put dissipation into the atomic potentials, but one has short waves with small velocity generated as a phase boundary (or other defect) moves. These are the source of dissipation when one examines the motion at macroscopic levels and are lumped as heat, roughly. We find a very similar mechanism in peridynamics, where we have dissipation at macroscopic scales without putting any dissipation in at the microscopic scale. This is again unlike phase-field modeling where one needs to explicitly put in viscosity at the microscopic level to obtain dissipation at the macroscopic scale. All of this seems to make peridynamics a interesting tool to examine dynamics processes.

Also, in Stewart Silling’s original paper formulating peridynamics [1], his introduction and discussion sections give a perspective of the theory.

[1] Silling, S.A., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys.Solids 48, 175–209.

Finally, here are the relevant sections of the paper itself on the question of phase field versus peridynamics:

In classical continuum theory, these phase transforming materials have been modeled using an energy that has multiple minima, each minimum corresponding to a particular phase or variant of martensite. In a dynamic, or even quasistatic, setting, the usual constitutive information, strain energy density as a function of strain, is insufficient to determine a unique solution. For example, even simple Riemann problems with a single phase or twin boundary in the initial conditions allow a one-parameter family of solutions. Therefore, we require further material information to pick the physically correct solution. Abeyaratne and Knowles, 1990 and Abeyaratne and Knowles, 1991b have proposed that this extra information may be specified in the form of a nucleation criterion and a kinetic relation.

The nucleation criterion determines whether a new phase will nucleate from a single phase. The kinetic relation determines the kinetics or the rules that govern the evolution of the phase boundary. It relates the velocity to a thermodynamic driving force, these being conjugate variables in the dissipation (or entropy) inequality. The driving force is related to Eshelby’s idea of the force acting on a defect (Eshelby, 1956 and Eshelby, 1975). The nucleation criterion and the kinetic relation provide uniqueness and well-posedness to initial-boundary value problems. Physically, they can be thought of as a macroscopic remnant of the lattice level atomic motion from one energy well to another that is lost in the continuum theory. However, a systematic derivation from a microscopic theory as well as experimental confirmation remain a topic of active research.

Another approach to overcome the inability of classical continuum mechanics to model the kinetics of phase transformations is to regularize or augment the theory, notably by adding a strain gradient (capillarity) and viscosity to the constitutive relation. This augmented theory leads to a unique solution for the motion of phase boundaries (Abeyaratne and Knowles, 1991a and Truskinovsky, 1993). Further, Abeyaratne and Knowles (1991a) have shown a correspondence between such methods and the kinetic relation. However, nucleation is incompletely explored in this theory, and computational evidence suggests that it is in fact quite difficult. Further, this theory leads to fourth-order equations which are difficult to deal with computationally: they are stiff and one needs smooth elements in the finite element method (see for example, Kloucek and Luskin, 1994 and Dondl and Zimmer, 2004).

There is a closely related phase-field approach (see for example, Artemev et al., 2001 and Wang et al., 1994) in the infinitesimal strain setting. Here, one uses the transformation strain as an internal variable or order parameter, considers the free energy density as a function of this order parameter and uses linear elasticity to penalize the incompatibility in this internal variable field. This leads to a second-order equation which is computationally attractive. However, the equilibrium and the dynamics can be different from that of the regularized theories described earlier (Bhattacharya, 2003). The connection between this theory and kinetic relations remains unexplored (Killough, 1998 has some discussion on this question), nucleation remains difficult and most studies are quasistatic.

Time for some more reading!

Update: A link to peridynamic based fracture code; via Biswajit.

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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!