Local Minimizers and the Schmid Law in Corner-Shaped Domains

This paper focuses on the mathematical analysis of biaxial loading experiments in martensite, more particularly on how hysteresis relates to metastability. These experiments were carried out by Chu and James and their mathematical treatment was initiated by Ball, Chu and James. Experimentally it is observed that a homogeneous deformation y ₁ is the stable state for “small” loads while y ₂ is stable for “large” loads. A model was proposed by Ball, Chu and James which, for a certain intermediate range of loads, predicts crucially that y ₁ remains metastable (that is, a local—as opposed to global—minimiser of the energy). This result explains convincingly the hysteresis that is observed experimentally. It is easy to get an upper bound on the load at which metastability finishes. However, it was also noticed that this bound (the Schmid Law) may not be sharp, though this required some geometric conditions on the sample. In this research, we rigorously justify the Ball–Chu–James model by means of De Giorgi’s Γ-convergence, establish some properties of local minimisers of the (limiting) energy and prove the metastability result mentioned above. An important part of the paper is then devoted to establishing which geometric conditions are necessary and sufficient for the counter-example to the Schmid Law to apply, namely, the presence of sharp corners in the sample.