An inverted Rivlin-type universal relation for simple shear. (April 2022)

• Record Type: Journal Article
• Authors: Murphy, Jeremiah G.; Saccomandi, Giuseppe; Vitral, Eduardo
• Is Part Of: International journal of non-linear mechanics. Volume 140(2022)
• Publication Date: 2022-04
• Subjects: Isotropy -- Simple shear -- Pure shear stress -- Normal traction -- Inverted universal relation; Nonlinear mechanics -- Periodicals; Mécanique non linéaire -- Périodiques; Nonlinear mechanics; Periodicals; 531
• DOI: 10.1016/j.ijnonlinmec.2022.103911
• Languages: English
• ISSNs: 0020-7462

Abstract

Abstract: The pure shear stress formulation for incompressible isotropic hyperelastic materials is generalised to include an in-plane normal traction. A new inverted Rivlin-type universal relation for simple shear is then obtained in terms of the left Cauchy–Green deformation tensor, the counterpart for the stress formulation of the classical Rivlin relation for simple shear that follows from a displacement formulation of the problem. Bounds are then obtained for the corresponding amount of shear and the triaxial stretches in terms of the applied stress. The general results obtained are then illustrated for the neo-Hookean material, for which it is shown that a compressive normal stress in conjunction with a shearing traction results in a non-monotonic relationship between the amount of shear and the shear stress, with a finite amount of shear predicted in the limit of infinite shear stress. However the relative displacement of the top and bottom faces of the sheared specimen is shown to be a monotonic function of the shear stress, with an infinite relative displacement predicted for an infinite shear stress. Highlights: A new inverted Rivlin-type universal relation for simple shear is obtained. Bounds are obtained for the corresponding amount of shear and the triaxial stretches in terms of the applied stress. The mechanical response is determined for the neo-Hookean material. The relative displacement between the top and bottom faces of a sheared neo-Hookean specimen is investigated.