Inhomogeneous shearing deformation of a rubber-like slab within the context of finite thermoelasticity with entropic origin for the stress

Abstract

This paper deals with inhomogeneous shearing deformation of an infinite rubber-like slab under a temperature difference across its thickness. The deformation is analyzed within the context of finite thermoelasticity with entropic origin for the stress. The isothermal strain energy functions of Mooney and of Yeoh are generalized by incorporating them into the thermoelastic model. The energy balance equation, which is decoupled from the linear momentum balance equations, is solved for the temperature field using the Fourier's law of heat conduction. The derived linear temperature field and the equations of the thermoelastic model are inserted into the linear momentum balance equations, thus yielding an ordinary differential equation for the shear strain. The shear strain, the degree of inhomogeneity in the shear strain, the stress field, and the principal stresses are then determined using a finite difference method. The Mooney slab undergoes markedly greater shear strain near the colder boundary, thus exhibiting a boundary layer-like structure at higher temperature differences across the thickness. However, the Yeoh slab exhibits a relatively smaller variation of the shear strain even at the highest temperature difference. This difference in the behavior of the two slabs is attributed to the shear stiffening of the Yeoh slab, which counteracts the formation of a boundary layer-like structure. Both slabs experience constant shear stresses across the thickness, whereas they experience much greater normal stress differences varying across the thickness. The values of the major principal stress turn out to be close to those of the first normal stress difference. The presence of a large normal stress difference compared with a relatively smaller shear stress in the slabs creates the possibility for Mode I type fracture propagation.