The energy functional of linear elasticity is obtained as G-limit of suitable rescalings of the energies of finite elasticity...

PB - Gauthier-Villars;Elsevier VL - 29 UR - http://hdl.handle.net/1963/4267 U1 - 3996 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Ogden-type energies for nematic elastomers JF - International Journal of Non-Linear mechanics Y1 - 2012 A1 - Virginia Agostiniani A1 - Antonio DeSimone KW - Nonlinear elasticity AB -Ogden-type extensions of the free-energy densities currently used to model the mechanical behavior of nematic elastomers are proposed and analyzed. Based on a multiplicative decomposition of the deformation gradient into an elastic and a spontaneous or remanent part, they provide a suitable framework to study the stiffening response at high imposed stretches. Geometrically linear versions of the models (Taylor expansions at order two) are provided and discussed. These small strain theories provide a clear illustration of the geometric structure of the underlying energy landscape (the energy grows quadratically with the distance from a non-convex set of spontaneous strains or energy wells). The comparison between small strain and finite deformation theories may also be useful in the opposite direction, inspiring finite deformation generalizations of small strain theories currently used in the mechanics of active and phase-transforming materials. The energy well structure makes the free-energy densities non-convex. Explicit quasi-convex envelopes are provided, and applied to compute the stiffening response of a specimen tested in plane strain extension experiments (pure shear).

PB - Elsevier VL - 47 IS - 2 U1 - 6971 U2 - Mathematics U4 - -1 ER - TY - JOUR T1 - Second order approximations of quasistatic evolution problems in finite dimension JF - Discrete & Continuous Dynamical Systems - A Y1 - 2012 A1 - Virginia Agostiniani KW - discrete approximations KW - perturbation methods KW - saddle-node bifurcation KW - Singular perturbations KW - vanishing viscosity AB -In this paper, we study the limit, as ε goes to zero, of a particular solution of the equation $\epsilon^2A\ddot u^ε(t)+εB\dot u^ε(t)+\nabla_xf(t,u^ε(t))=0$, where $f(t,x)$ is a potential satisfying suitable coerciveness conditions. The limit $u(t)$ of $u^ε(t)$ is piece-wise continuous and verifies $\nabla_xf(t,u(t))=0$. Moreover, certain jump conditions characterize the behaviour of $u(t)$ at the discontinuity times. The same limit behaviour is obtained by considering a different approximation scheme based on time discretization and on the solutions of suitable autonomous systems.

VL - 32 UR - http://aimsciences.org//article/id/560b82d9-f289-498a-a619-a4b132aaf9f8 ER - TY - JOUR T1 - Gamma-convergence of energies for nematic elastomers in the small strain limit JF - Continuum. Mech. Therm. Y1 - 2011 A1 - Virginia Agostiniani A1 - Antonio DeSimone KW - Liquid crystals AB -We study two variational models recently proposed in the literature to describe the mechanical behaviour of nematic elastomers either in the fully nonlinear regime or in the framework of a geometrically linear theory. We show that, in the small strain limit, the energy functional of the first one I\\\"-converges to the relaxation of the second one, a functional for which an explicit representation formula is available.

PB - Springer VL - 23 UR - http://hdl.handle.net/1963/4141 U1 - 3882 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER -