Generalized shear of a soft rectangular block

The problem of the simple shear of a block has been treated in terms of a shear displacement, applied uniformly in a lateral direction and assumed to be a linear function of the height above the base. In this paper, simple shear is generalized: the shear displacement is neither uniform in the lateral direction nor necessarily a linear function of the height. Using second-order isotropic elasticity, the analytical solutions show that the shear displacements are characterized by the product of sine and hyperbolic sine functions of the height and depth variables, respectively. The height dependence of the shear displacement is predicted to be a combination of linear and sinusoidal functions, and is verified against the test data of agar-gelatin cuboidal blocks. If the gravity effect is incorporated, a quadratic dependence on height is additionally predicted. The calculation of stresses reveals the presence of not only negative normal stresses but also sinusoidally varying shear stresses on the lateral planes tending to distort the block about the height direction. These results can be of great importance in tissue/cell mechanics.