Existence of solutions for the anti-plane stress for a new class of “strain-limiting” elastic bodies

Abstract

The main purpose of this study is to establish the existence of a weak solution to the anti-plane stress problem on V-notch domains for a class of recently proposed new models that could describe elastic materials in which the stress can increase unboundedly while the strain yet remains small. We shall also investigate the qualitative properties of the solution that is established. Although the equations governing the deformation that are being considered share certain similarities with the minimal surface problem, the boundary conditions and the presence of an additional model parameter that appears in the equation and its specific range makes the problem, as well as the result, different from those associated with the minimal surface problem. Communicated by Y. Giga. M. Bulíček and J. Málek acknowledge the support of the project LL1202 in the programme ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic. M. Bulíček is a researcher in the

On elastic solids with limiting small strain: modelling and analysis

“…The first positive result concerning the existence and uniqueness of a weak solution to the problem (45) under assumptions (A1)-(A4) was established in [8] in a very special geometry, in the case of the so-called anti-plane stress problem. To be more precise, the following theorem was proved in [8].…”

“…Despite its very special setting, it provides useful insight into the problem. Indeed, it is shown in [8], that in the anti-plane geometry, the problem (45) has a weak solution if and only if the following boundary-value problem has a weak solution:…”

“…On the other hand, the generalization by Rajagopal of the class of elastic bodies affords one the ability to describe bounded strain and stress at crack tips, at tips of notches, etc. (see Rajagopal and Walton [60], Gou et al [24], Bulíček et al [8], Kulvait et al [32]). …”

“…The rigorous mathematical analysis of boundary-value problems that arise in implicitly constituted strain-limiting models has, so far, focused on proving the existence and uniqueness of solutions for any size of data measured in suitable norms in three particular settings: (i) antiplane stress problems resulting in a two-dimensional scalar nonlinear problem (see [8] where the existence of weak solutions in nonconvex domains for values of the model parameter α in the range α ∈ (0, 2), and in convex domains for the range α ∈ (0, ∞) was established); (ii) the complete multi-dimensional system in the spatially periodic setting in any number of space dimensions (see [9]); (iii) the complete multi-dimensional system in general bounded domains (see Section 4 in this paper). We state these results in the form of theorems in Section 4.…”

On the Existence of Integrable Solutions to Nonlinear Elliptic Systems and Variational Problems with Linear Growth

“…(D3) below).1 a , a > 0.Problem (1.8) is then an almost direct analogue of problem (1.1) with N = d; the only aspect in which the latter model differs from (1.1) (and is therefore considerably more difficult) is that, in contrast with (1.1), one is forced to operate in the space of symmetric matrices and function spaces of symmetric gradients. We refer the interested reader to [17,18,19,10,9] for a detailed overview of limiting strain models, their theoretical justification stemming from implicit constitutive theory, a discussion of their importance in modeling the responses of materials near regions of stress-concentration, where |T T T| is large, and their mathematical analysis (see in particular the survey paper [9] for more details).Analogously to problem (1.1), we adopt the following natural assumptions associated with limiting strain models (see [9]): there exist constants C 0 ≥ 0 and C 1 , C 2 > 0 such that, for all T T T ∈ R d×d sym , ε ε ε *…”