Again anti-plane shear

Shear, pure and simple

“…In the context of anti-plane shear deformations (cf. Appendix C), it was recently shown [41] that the so-called "condition K1 with b = 1 2 ", which is exactly the equality ∂W ∂I1 = ∂W ∂I2 given in eq. (4.10), is always satisfied by tension-compression symmetric energies.…”

“…Although simple shear is generally not a suitable concept for nonlinear elasticity if (Cauchy) pure shear stresses are concerned, it does occur in finite deformations under certain displacement boundary conditions in the context of so-called anti-plane shear deformations of the form ϕ(x 1 , x 2 , x 3 ) = (x 1 , x 2 , x 3 + u(x 1 , x 2 )). In the following, we discuss the concept of shear monotonicity, which has been the subject of a recent contribution by Voss et al [41].…”

“…Now, consider the entry σ 12 (γ) = (β 1 − β −1 )γ as the amount of stress in e 1 -direction acting on a surface perpendicular to e 2 . From a physical point of view, increasing the amount of shear γ should clearly lead to an increase in the amount of Cauchy shear stress; this material property is also known as shear monotonicity [41]. Since which is equivalent to the convexity of the mapping γ → W (3 + γ 2 , 3 + γ 2 , 1).…”

Global bifurcation of anti-plane shear fronts

“…From a purely mathematical perspective, this model is interesting as it is known to support a rich variety of nontrivial equilibria [14]. The existence of such solutions on bounded domains has been established by several authors [23,26] through variational methods.…”

“…It is also pointed out in [26] that (1.1) is equivalent to the convexity of the mapping ξ ∈ R 2 → W (3 + |ξ| 2 , 3 + |ξ| 2 ), which is assumed in most variational treatments of the problem. We restrict attention to generalized neo-Hookean materials [14] with…”