Simple proof of two-well rigidity

Abstract

Abstract. We give a short proof of the rigidity estimate of Müller and Chaudhuri [3] for two strongly incompatible wells. Making strong use of the arguments of Ball and James our approach shows that incompatibility for gradient Young measures can be used to reduce rigidity estimates for several wells to one-well rigidity.

Explicit energy-minimizers of incompressible elastic brittle bars under uniaxial extension

“…The result of Chaudhuri and Müller [5] provides, in particular, a rigidity estimate for Sobolev deformations with gradient in K . They proved an L 2 estimate, but a similar proof shows an L p estimate for p ∈ (1, ∞) (see also [6], where an alternative proof of the L p rigidity estimate is given). Therefore, we can apply the SBV rigidity result of [4, Th.…”

Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions

“…If the two wells are strongly incompatible in the sense of [47], it was proven in [15,24] that there exist R ∈ SO(d) and M ∈ {A, B} such that…”

Piecewise rigidity

“…Theorem 1.2 can be used to infer a similar result in the framework of the discontinuous deformations of class SBV. The quantitative rigidity estimate we need to apply our arguments has been recently provided by De Lellis and Székelyhidi [12]: they prove in particular that if K ⊆ M N ×N is a finite set of matrices which is rigid for approximate solutions, then the quantitative rigidity estimate (1.2) holds for any p ∈ (1, +∞) provided that Ω is Lipschitz-regular: Theorem 1.2 hence applies to K = {K 1 , K 2 } as above, and, thanks to an extension by Šverák of Ball and James' result, to K consisting of three matrices without any rank-1 connection [20]. (The result of De Lellis and Székelyhidi actually extends our thesis to any finite union of compact sets, each satisfying a quantitative rigidity estimate, and such that any gradient Young measure supported by the union is, in fact, supported by only one of the sets.…”