Sharp conditions for the linearization of finite elasticity

We consider the topic of linearization of finite elasticity for pure traction problems. We characterize the variational limit for the approximating sequence of rescaled nonlinear elastic energies. We show that the limiting minimal value can be strictly lower than the minimal value of the standard linear elastic energy if a strict compatibility condition for external loads does not hold. The results are provided for both the compressible and the incompressible case.

MEASURES OF THE BASIS EFFICIENCY IN THE INTERNAL ROTATION PROBLEM

Рассмотрено понятие вариационного предела при численном решении торсионного уравнения Шрёдингера и определены значения энергий состояний и переходов между ними на вариационном пределе в зависимости от размера базиса. На большом числе модельных потенциалов показано, что количество достоверно вычисляемых уровней линейно зависит от числа базисных функций. Данное число уровней (номер наибольшего достоверного состояния) рекомендовано в качестве меры эффективности базиса. Для базиса плоских волн найдено, что коэффициент углового наклона линейной зависимости составляет ≈ 1,03. Definition of the variational limit for the Schrodinger torsion equation solution was considered. Energies of levels and transitions at the variational limits were calculated for different basis sets. With many modeling potentials it was shown that the quantity of reliable calculated levels linearly depends on the number of the basis functions. This number (the number of the maximum reliable state) was recommended as a measure of the basis efficiency. The slope of the straight line for the planar wave basis dependence is approximately equal to 1,03.

Variational linearization of pure traction problems in incompressible elasticity

We consider pure traction problems, and we show that incompressible linearized elasticity can be obtained as variational limit of incompressible finite elasticity under suitable conditions on external loads.

Convergence of equilibria for bending-torsion models of rods with inhomogeneities

We prove that, in the limit of vanishing thickness, equilibrium configurations of inhomogeneous, three-dimensional non-linearly elastic rods converge to equilibrium configurations of the variational limit theory. More precisely, we show that, as $h\searrow 0$, stationary points of the energy , for a rod $\Omega _h\subset {\open R}^3$ with cross-sectional diameter h, subconverge to stationary points of the Γ-limit of , provided that the bending energy of the sequence scales appropriately. This generalizes earlier results for homogeneous materials to the case of materials with (not necessarily periodic) inhomogeneities.

Plate theory as the variational limit of the complementary energy functionals of inhomogeneous anisotropic linearly elastic bodies

We consider a sequence of linear hyper-elastic, inhomogeneous and fully anisotropic bodies in a reference configuration occupying a cylindrical region of height [Formula: see text]. We study, by means of Γ-convergence, the asymptotic behavior as [Formula: see text] goes to zero of the sequence of complementary energies. The limit functional is identified as a dual problem for a two-dimensional plate. Our approach gives a direct characterization of the convergence of the equilibrating stress fields.

QUASI-STATIC EVOLUTIONS IN LINEAR PERFECT PLASTICITY AS A VARIATIONAL LIMIT OF FINITE PLASTICITY: A ONE-DIMENSIONAL CASE

In the framework of the energetic approach to rate-independent evolutions, we show that one-dimensional linear perfect plasticity can be obtained by linearization as a variational limit of a finite plasticity model with hardening proposed by A. Mielke (SIAM J. Math. Anal., 2004).