Abstract
AbstractThrough suitable changes of variables for a typical problem in hyper-elasticity, either in the reference or deformed configurations, one can setup and analyze versions of the same problem in terms of inner or outer maps or variations. Though such kind of transformations are part of the classical background in the Calculus of Variations, we explore under what sets of hypotheses such versions can be shown to have minimizers and be equivalent to the standard form of the problem. Such sets of hypotheses lead naturally to some distinct poly-convex energy densities for hyper-elasticity. Likewise we explore optimality in either of the two forms through a special way to generate one-parameter families of feasible deformations, feasibility including injectivity and non-interpenetration of matter.
Funder
Agencia Estatal de Investigación
Junta de Comunidades de Castilla-La Mancha
Universidad de Castilla la Mancha
Publisher
Springer Science and Business Media LLC
Reference19 articles.
1. Bandeira, L., Pedregal, P.: $\mathcal {A}$-variational principles. Milan J. Math. 91, 293–314 (2023)
2. Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63(4), 337–403 (1976)
3. Ball, J.M.: Global invertibility of Sobolev functions and the interpenetration of matter. Proc. R. Soc. Edinb., Sect. A 88(3–4), 315–328 (1981)
4. Ball, J.M.: Minimizers and the Euler-Lagrange equations. In: Lecture Notes in Phys., vol. 195, pp. 1–4. Springer, Berlin (1984)
5. Ball, J.M.: Some open problems in elasticity. In: Geometry, Mechanics, and Dynamics, pp. 3–59. Springer, New York (2002)