Asymptotic models for a long, elastic cylinder

Abstract

We study asymptotic expansions for the displacement field of a long elastic cyfinder under various constitutive assumptions. We show that under simple hypotheses it is possible to derive from the equations of continuum mechanics two known beam equations and several different string models. Some of the string models correspond to those studied by S. Antman and R. Dickey. We also show that under our assumptions the problem of asymptotic expansion can be reduced to that of algebraic geometry.

Dynamical models for plates and membranes. An asymptotic approach

“…The rest of the calculations is essentially the same as for the plates. It can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14] for the case of equilibrium.…”

“…Particularly we would like to mention the work of Ciarlet, Destuynder and Raoult on plates [see [3], [11], [18]), the papers of Rigolot (see [20], [21]) and paper of Cimetiere, Geymonat, LeDret, Raoult, Tutek (see [9]) on beams. We also remain in debt to the work of Davet (see [10]), Trabucho, Viano (see [22]) and others whose names we do not mention but whose contribution play a part in our work (see 14] for the full list of references).…”

Untitled

“…The expansion for w, w (x,e) = w0 (x) + ewi (x) + e2 w2 (x) + ..., (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) implies that…”

“…We notice that since expansion (4.11) is our hypothetical p, the outer expansion u must be contained in w when both written in terms of x = r]~1^. Consequently we should expect to find unknown ^ 's from Equations (4.16) while searching for unknown tjk s. The sequence of equations for tjk s has the following form (see Equations .., (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) with the successive equations containing other Aj's which enter the formulas for PjkS. At first glance, the task of finding Aj's and tjk s using Equations (4.19) seems to be daunting.…”

Remark on indirect matching of singularly perturbed boundary value problems

“…Moreover, when studying the linear case one already wonders whether it is possible to recover the formal simplicity of Ciarlet's original arguments when dealing with the boundary layer corrections of the plate or the beam models (see, for example, [8]). Thus a question appears: How to check that the outer and the boundary layer expansions match without solving the boundary layer equations?…”