A phase field model for the formation and evolution of martensitic laminate microstructure at finite strains

Abstract

The extraordinary properties of shape-memory alloys stem from the formation and evolution of their complex microstructure. At lower temperatures, this microstructure typically consists of martensitic laminates with coherent twin boundaries. We suggest a variational-based phase field model at finite strains for the formation and dissipative evolution of such two-variant martensitic twinned laminate microstructures. The starting point is a geometric discussion of the link between sharp interface topologies and their regularisation, which is connected to the notion of Γ-convergence. To model the energy storage in the two-phase laminates, we propose an interface energy that is coherence-dependent and a bulk energy that vanishes in the interface region, thus allowing for a clear separation of the two contributions. The dissipation related to phase transformation is modelled by use of a dissipation potential that leads to a Ginzburg–Landau type evolution equation for the phase field. We construct distinct rate-type continuous and finite-step-sized incremental variational principles for the proposed dissipative material and demonstrate its modelling capabilities by means of finite element simulations of laminate formation and evolution in martensitic CuAlNi.

Keywords:

• phase transformations
• twinned laminates
• shape-memory alloys
• martensitic microstructure
• phase field models
• variational principles

Acknowledgements

Support for this research was provided by the German Research Foundation (DFG) within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart. Furthermore, fruitful discussions with Eric Jägle are gratefully acknowledged.

Notes

Notes

1. Associated with a node I of a standard finite element e, the interpolation matrix for two-dimensional problems d = 2 has the form

in terms of the nodal shape function N and its derivatives.

2. Alternatively, one can derive the micro-balance directly from the global dissipation inequality given by . Reducing this to the local statement , introducing a dissipation potential φ and enforcing then yields which is always satisfied for a positive, convex φ with φ = 0 for .

3. For the example of a two-dimensional problem d = 2 with the scalar micro-variable p, the nodal state vector in (58) at the node I is

The constitutive state vector (27) reads . Then, associated with node I of a standard finite element e, the finite element interpolation matrices in (58) have the form in terms of the shape function N _I , M _I at note I and their derivatives. For identical interpolations of micro- and macro-motions, we set M _I  = N _I .