Boundary layers occur in nonlinear elasticity when there is a gradient in a field variable, such as stress or temperature, and the response of the material has certain nonlinear behavior. Circular shear of a hollow rubber cylinder is particularly convenient for illustrating this effect because it is known from fundamental equilibrium considerations that the shear stress varies inversely as the square of the radius. Rajagopal and co-workers showed that shear boundary layers form when the shear stress-shear response becomes sufficiently soft. They also showed that shear boundary layers form when the temperature varies monotonically with radius and the shear modulus increases with temperature.
Softening can occur in polymers due to viscoelasticity and/or microstructural changes. The degree of softening depends on the deformation or temperature history. When combined with gradients in stress or temperature, these effects can lead to the onset and growth of boundary layers. Examples are presented in the context of circular shear for two different types of softening: (1) viscoelastic materials with strain dependent stress relaxation and (2) materials undergoing deformation or temperature induced microstructural changes.
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