18 research outputs found
Closure and commutability results for Gamma-limits and the geometric linearization and homogenization of multi-well energy functionals
Under a suitable notion of equivalence of integral densities we prove a
-closure theorem for integral functionals: The limit of a sequence of
-convergent families of such functionals is again a -convergent
family. Its -limit is the limit of the -limits of the original
problems. This result not only provides a common basic principle for a number
of linearization and homogenization results in elasticity theory. It also
allows for new applications as we exemplify by proving that geometric
linearization and homogenization of multi-well energy functionals commute
Closure result for -limits of functionals with linear growth
We consider integral functionals , doubly indexed by > 0 and , satisfying a standard linear growth condition. We investigate the question of -closure, i.e., when the -convergence of all families with finite implies -convergence of . This has already been explored for -growth with > 1. We show by an explicit counterexample that due to the differences between the spaces and with > 1, an analog cannot hold. Moreover, we find a sufficient condition for a positive answer
Homogenization and the limit of vanishing hardening in Hencky plasticity with non-convex potentials
We prove a homogenization result for Hencky plasticity functionals with
non-convex potentials. We also investigate the influence of a small hardening
parameter and show that homogenization and taking the vanishing hardening limit
commute
Geometric linearization of theories for incompressible elastic materials and applications
We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g. encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers
Geometric linearization of theories for incompressible elastic materials and applications
We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g. encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers