Abstract
The description of equilibria of shape memory alloys or other ordered materials gives rise to nonconvex variational problems. In this paper, a two-dimensional model of such materials is studied. Due to the fact that the corresponding functional has two symmetry-related (martensitic) energy wells, the numerical approximation of the deformation gradient does not converge, but tends to oscillate between the two wells, as the size of the mesh is refined. These oscillations may be interpreted in terms of microstructures. Using a nonconforming $\mathcal{P}_1 $ finite element, an estimate is given for the rate of convergence of the probability for the approximated deformation to have its gradient “near” one of the two (martensitic) wells.
Submitted Version (
Free)
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call