Derivation of the model of elastic curved rods from three-dimensional micropolar elasticity

Abstract

In this paper we derive a model of curved elastic rods from the threedimensional linearized micropolar elasticity. Derivation is based on the asymptotic expansion method with respect to the thickness of the rod. The method is used without any a priori assumption on the scaling of the unknowns. The leading term, displacement and microrotation, is identified as the unique solution of a certain onedimensional problem. Appropriate convergence results are proved.

Evolution Model for Linearized Micropolar Plates by the Fourier Method

“…Thus s(ε)(t, ·) ∈ V 0 (Ω) 2 , s(ε) ∈ L ∞ (0, T ; V 0 (Ω) 2 ) and the series (48) converges uniformly in the strong topology of V 0 (Ω) 2 .…”

“…Let us denote by V n 00 the family of all n-dimensional subspaces of 2 holds, using the characterization (15) from Theorem 1 twice, once for the three-dimensional Problem 2 and once for the limit Problem 3, we obtain…”

“…Let us suppose that s 0 (ε) → s 0 strongly in V 0 (Ω)2 , ∇ ε s 0 (ε) → (E 0 , G 0 ) strongly in (L 2 (Ω) 9 ) 2 , s 1 (ε) → s 1 strongly in (L 2 (Ω) 3 ) 2where s 1 depends only on the first two variables, and let us suppose that p(ε) → p strongly in L 2 ([0, T ]; L 2 (Ω) 3 ). Let s(ε) be the solution of (40), (41).…”

Evolution model of linear micropolar plate

“…Asymptotic analysis has been applied on the derivation and justification of several lower dimensional models of equilibrium and dynamics in classical elasticity (rods, curved rods, plates, shells). Recently, the technique has been applied on the derivation and justification of the equilibrium models (rod [I. Aganović, J. Tambača, Z. Tutek, submitted], plate [1], curved rod [2]) starting from the micropolar elasticity. The micropolar elasticity uses two kinematical vector fields, the displacement and the microrotation of the material points; the material points are allowed to rotate without stretch [5].…”

“…)2 and ∇ ε s 0 (ε) → L = (L 1 , L 2 ) strongly in (L 2 (Ω) 9 ) 2 then all three convergences in(5.3) are strong in L 2 (0, T ; V 0 (Ω) 2 ) i.e., L 2 (0, T ; (L 2 (Ω) 3 ) 2 ) i.e., L 2 (0, T ; (L 2 (Ω) 9 ) 2 ) if and only if L = (E 0 , G 0 ),where E 0 and G 0 are given by (5.4) and (5.5) with u(t) and ω(t) replaced by u 0 and ω 0 . 6 Proof of Theorem 5.2 Lemma 6.1 Let the assumptions of Theorem 5.2 be fulfilled.…”