Shear, pure and simple

Abstract

In a 2012 article in the International Journal of Non-Linear Mechanics, Destrade et al. showed that for nonlinear elastic materials satisfying Truesdell's so-called empirical inequalities, the deformation corresponding to a Cauchy pure shear stress is not a simple shear. Similar results can be found in a 2011 article of L. A. Mihai and A. Goriely. We confirm their results under weakened assumptions and consider the case of a shear load, i.e. a Biot pure shear stress. In addition, conditions under which Cauchy pure shear stresses correspond to (idealized) pure shear stretch tensors are stated and a new notion of idealized finite simple shear is introduced, showing that for certain classes of nonlinear materials, the results by Destrade et al. can be simplified considerably.

Development of a shear device for precise simple shear measurements of rubber mats

“…For material characterisation and model parameter identification, simple shear is a very important deformation state (cf. other works [ 1‐11 ] and Upadhyay et al [ 12 ] ). Especially for engineering components like rubber bushings or earthquake isolation bearings, simple shear plays a very important role.…”

“…To get an understanding of error measures, the deviation between current σ FE ( s ) and theoretical σ th ( s ) stress state (cf. Thiel et al [ 11 ] ) is calculated. Using the definition of the von Mises equivalent stress, a scalar stress value can be obtained.…”

Do we need Truesdell’s empirical inequalities? On the coaxiality of stress and stretch

“…In a recent contribution[47], this consequence of the distinctness of principal stresses is further discussed and utilized to verify and generalize a statement by Destrade et al[13] on the deformations corresponding to simple shear Cauchy stresses. Similar work can be found in an early paper by Moon and Truesdell[30] as well as in a more recent article by Mihai and Goriely[25].…”

Remarks on pure and simple shear

“…In particular, for a given stress response, a finite pure shear stretch always induces a pure shear stress if and only if for all λ ∈ R + , there exists s ∈ R such that λ 1 = 1 λ2 = λ and λ 3 = 1 imply σ 1 = −σ 2 = s and σ 3 = 0, where σ i denotes the i-th eigenvalue of σ(F F T ) (the principle stresses) for F = diag(λ 1 , λ 2 , λ 3 ). It follows [4]:…”

“…For α ∈ R, we call F ∈ GL + (3) an (idealized) left finite simple shear deformation gradient [4] and V ∈ Sym + (3) a finite pure shear stretch if F = F α and V = V α have the form…”