Thermomechanical simulation of shape memory alloys structures: variational methods and associated numerial tools.

Michaël Peigney (1)

1 IFSTTAR - Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux

Shape Memory Alloys (SMA) o ffer new perspectives in various fi elds such as aeronautics, robotics, biomedicals, or civil engineering. E ffi cient design of such innovative systems requires both adequate material models and numerical methods for simulating the response of SMA structures. Whereas much e ffort has been devoted to developing constitutive laws for describing the behaviour of SMAs, the structural problem (i.e. the simulation of a three-dimensional SMA structure) has received far less attention, in spite of substantial di culties notably due to the strong thermomechanical coupling and the presence of physical constraints on the internal variables. The time-discretization of the evolution problem obtained is not obvious, and special care must be taken to avoid convergence di culties and ensure robustness of the numerical schemes. Computation time and ease of implementation (for instance in an existing finite element code) also are major issues that need to be addressed. In this communication are presented some recent results in that direction. A central result is a recent time-discretization scheme for the thermomechanical problem. A variational formulation is attached to the corresponding incremental problem, allowing one to prove the existence of solutions for a large class of usual SMA models. The variational nature of the problem at hand also calls for an easy implementation in an existing fi nite element code, building on well-established descend algorithms. Using that approach, the solution of the thermomechanical incremental problem is typically obtained by solving a sequence of linear thermal problems and purely mechanical (i.e. at prescribed temperature) nonlinear problems. That approach is fairly general and applies for a wide range of SMA models. The numerical scheme for solving the purely mechanical problem, however, strongly depends on the particular model that is used. In a micromechanical modelling of SMAs, the phase transformation is described locally by an internal vectorial variable which is physically constrained to satisfy a set of inequalities at each point. We show that the corresponding incremental problem can be recast as a linear complementarity problem, for which e ffcient algorithms (such as interior-point methods) are available. That reformulation essentially consists in a change of variables. In terms of variational formulation, that approach amounts to replace a convex but non-quadratic minimization problem with an equivalent quadratic minimization problem.

Mots clés: Computational Plasticity; Shape memory alloys; Thermomechanical coupling; Linear complementarity problem

Domaines: Mécanique des solides [physics.class-ph]; Mécanique des solides [physics.class-ph]

Michaël Peigney. Thermomechanical simulation of shape memory alloys structures: variational methods and associated numerial tools.. XI International Conference on Computational Plasticity. Fundamentals and Applications, Sep 2011, Barcelone, Spain. 10p. ⟨hal-00833385⟩