Plastic effects on the kinetics of phase transformations

Many solid state phase transformations e.g. in steels or shape memory alloys are accompanied by severe mechanical deformations during the microstructure evolution. In contrast to elastic effects, which can nowadays be included e.g. in a phase field formulation of these processes, the proper incorporation of plastic deformation is not yet established. The reason is that different dissipative processes play a role here, which influence the motion of the interface. In this project we intend to develop novel sharp interface and phase field methods to simulate the microstructure evolution in the plastic regime.

Motivation

Phase field models are typically written in variational form starting from a free energy functional, and sharp interface descriptions for moving boundary problems can be formulated similarly. This is a common approach for applications in physics and materials science. We investiaget why and under which circumstances this postulate for deriving the equations of motion is justified, and what are limitations. We investigate this in particular for alloys, systems with elastic, viscoelastic and plastic effects, mainly based on analytical and numerical investigations in one dimension. We have found that the naturally guessed equations of motion, as derived via partial functional derivatives from a free energy, are usually reasonable, only for materials with plastic effects this assumption is more delicate due to the presence of internal variables.

Introduction

Fig. 1: The order parameter in a phase field model distinguishes between the different phases. At an interface, it changes its value continuously, and in the bulk phases it is spatially constant.

The modeling of microstructure evolution has become a central topic in materials science and physics, since these pattern formation processes are not only essential phenomena in our day-to-day life but also important for the understanding and prediction of material behavior. In this respect the development of phase field models can be considered as a milestone, which triggered an enormous intensification of research activities in this direction. Starting with pure curvature driven motion phase field modeling experienced a first `gold rush' by the investigation of diffusion limited solidification, in particular dendritic growth, and substantially contributed to a deeper understanding of this complex problem. Since then, it has become a routine toolkit for modeling interfacial pattern formation processes not only for highly idealized scientific investigations, but also for realistic simulations of kinetic processes in engineering materials. Later on, more and more phenomena have been investigated by the means of phase field modeling, and nowadays even applications in biology, medicine and soft matter science start to emerge.

With the increasing knowledge about this modeling tool also the understanding of this method has reached a significantly deeper level. Initially, the method was considered purely as a mathematical tool that avoids the complex tracking of interface during moving boundary problems. Instead, order parameters are introduced to discriminate between different `phases', and at interfaces these order parameters change only gradually. The description of interface dynamics is then reduced to partial differential equations for the order parameters. The original idea is that in the sharp interface limit, where the lengthscale over which the order parameters are smeared out at the interfaces between the `phases' is small in comparison to the relevant physical lengthscales, the dynamics effectively recovers the governing laws for the kinetics of the sharp interfaces. Although physically all interfaces have a finite thickness, this true physical width is negligibly small in comparison to the scales of the patterns, and therefore a sharp interface description is usually appropriate. The numerical lengthscale, which is introduced in phase field models, is usually a pure auxiliary parameter, and numerical efficiency demands to actually choose it much larger than the true interface thickness. For quantitative modeling it is therefore mandatory to check that the results are insensitive to a change of the numerical interface thickness. Typically, first order equations (in time) are constructed for the phase field evolution, and in this `traditional picture' the choice of the right hand side of the equations is not restricted, as long as it recovers the proper sharp interface limit.

Based on the experience that thermodynamically motivated models indeed to the correct equations of motion in the sharp interface limit, the paradigm has changed to use such extensions of free energy functionals and the derived phase field models in a predictive manner also for problems, where an established sharp interface description is not yet available. In fact, it is often easier to find proper energy functionals than to derive or guess the correct sharp interface equations.

Project description

Fig. 2: Simple stress-strain curve involving a phase transformation.

In view of the extension of phase field models by plasticity, we investigate several prototype models and inspect the equations of motion, as motivated from a thermodynamic approach without reference to a sharp interface limit, from a physical point of view. We find that an extension to plasticity, which introduces the new aspect of ``internal variables'' which are not derived from the same free energy functional as the phase field equations, brings in aspects, which make it less obvious, whether the usual approach to derive equations of motion, is appropriate. We note that this issue is also relevant for plastic deformations in other materials like ferroelectrics, magneto elastic materials, quasicrystals, polymeric bodies etc..

Recent results

Fig. 3: Asymmetric growth scenario. Starting from configuration (a), where the dashed vertical line depicts the interface between phase 1 and 2, in (b) phase 2 has grown. However, in the newly converted phase the plastic strain is adopted from the mother phase, i.e. the plastic strain is now inhomogeneous in phase 2. For growth in the other direction (c), it is assumed that phase 1 stays in a homogenous plastic strain state.

We have analyzed various phase field and sharp interface models for the description of moving boundary problems concerning the physical meaning of the driving forces. We contrasted the difference between the partial work, that is typically used in variational phase field models, where a virtual energy change is calculated while all other physical fields are kept constant, to a total work expression, where also for the trial move of the interface all physical fields are adjusted instantaneously. Although the latter is typically not used in phase field modeling, we argued that it should be the proper driving force in cases where the interface motion is slow in comparison to the kinetics of the other fields.

We demonstrated, that in cases of coupling to fast diffusion and static elasticity the two driving force expressions, which can be considered as limiting cases, coincide, and therefore the usual approach of the partial variational derivative should be appropriate.

This, however, is different for models with plasticity, because they contain additional internal variables which follow separate dynamics. These are usually not derived from the same free energy functional as the evolution equations for the other fields. Consequently, the partial and total work approach lead to different expressions for the driving forces. One central difference is that the latter contains also nonlocal terms which result from plastic bulk dissipation. The usual assumption is, however, that only the interface dissipation should play a role for the front propagation, and therefore it seems unlikely that bulk terms can be relevant. We therefore generalized the expression by proposing the true interface dissipation (where all other fields are slaved by the phase field) as driving force. It turns out, that this physically motivated expression coincides again with the partial work expression, provided that the material does not grow in different defect states than the underlying substrate phase. Otherwise, interface pinning and destabilization effects may occur, as well as the splitting of the direct and inverse transition curves, that is relevant for many solid-state transformations.

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