EXISTENCE OF SOLUTIONS FOR THE ANTI-PLANE STRESS FOR A NEW CLASS OF ”STRAIN-LIMITING” ELASTIC BODIES MIROSLAV BUL ´ I ˇ CEK, JOSEF M ´ ALEK, K. R. RAJAGOPAL, AND JAY R. WALTON Abstract. The main purpose of this study is to establish the existence of a weak solution to the anti-plane stress problem for a class of recently pro- posed new models that could describe elastic materials in which the stress can increase unboundedly while the strain yet remains small. We shall also investi- gate 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 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. 1. Introduction Few models within the context of continuum mechanics have had the success of the linearized elastic solid model. Its great success notwithstanding, there are classes of problems to which the solutions provided by the linearized theory of elasticity is far from satisfactory, namely in the prediction of the strains and stresses at and near the tip of a crack and the propagation of cracks. The problem of fracture engaged the attention of Galileo [14] who in fact states that the problem had attracted attention much before his studies into fracture began 1 . Ever since then the problem of fracture has held the attention of physicists, engineers and applied mathematicians, but despite all this interest important open issues remain unresolved. The problem with using linearized elasticity to study the problems of stresses and strains around cracks stems from the fact that the relationship between the stress and the strain is linear and thus as the stress increases and becomes unbounded, the strain also increases and becomes unbounded, thereby violating the starting assumption in the linearization procedure that the strains are sufficiently small so that higher order terms in the strain can be neglected. In order to avoid the singularity that arises in the strains, a variety of ad hoc approaches 2 have been used 2000 Mathematics Subject Classification. 35Q74,74B20,49Q05. Key words and phrases. Plane stress, V-notch, Linearized strain, weak solution, minimal surface. M. Bul´ ıˇ cek and J. M´alek 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. K. R. Ra- jagopal thanks the National Science Foundation for its support of his work. He and J. R. Walton acknowledge support by Award No. KUS-C1-016-04 from King Abdullah University of Science and Technology. 1 Galileo refers to Aristotle’s interest in the problem. 2 There have been additional approaches to modeling fracture by introducing additional forces and additional balance laws. Unfortunately, many of the quantities introduced in such theories 1