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.
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
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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 (Equation58) at the node I is
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