Asymptotic fields ahead a crack for a class of non linear materials under mode III

• The purpose of the paper is to described the mechanical fields near the tip of a crack deformed by an anti-plane shear at infinity for a class non linear elastic brittle or non brittle materials. For brittle materials after a critical value of strain, the material cannot suffer any tension, it is totally broken.
• For brittle material the crack is replaced by a totally damaged zone of finite thickness, a quasi-crack. Inside the damaged zone, the stress is identically zero and the shape of the separation between damaged and undamaged body is found analytically, generally it is a cycloid.
• The uniqueness is not discussed at this stage, it seems relevant from the property that the domain, where the equation is hyperbolic is embedded, in the domain of ellipticity. In this case the solution is unique like in fluid dynamics.
• I will emphazise new results: the connection between crack, quasi-crack and flux of energy (5-6) (5-9).
• I have discuss the case with α⩽0α⩽0. The influence of a critical stretch is presented Fig. 4. The case α=-.5α=-.5 is used by Knowles and Sternberg (see Ref) and Abeyaratne (see Ref). I have added new results and properties: a general Eq. (5-17) for the shock curve and Eq. (5-22) for the continuity of the stress vector, which permits to take into account the discontinuity of the behaviour for ρ=1ρ=1.

The paper considers the mechanical fields near the tip of a crack deformed by an anti-plane shear at infinity for a class of non linear elastic materials. For brittle material rupture occurs when a maximal stretch is reached. Taking account of this critical value, the crack is replaced by a totally damaged zone of finite thickness named a quasicrack. Inside this domain, the stress is identically zero and the shape of the boundary between damaged and undamaged body is found analytically.