Stress softening, strain localization and permanent set in the circumferential shear of an incompressible elastomeric cylinder
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A constitutive theory for elastomeric materials has recently been developed according to which stress is generated by different micromechanisms at different levels of deformation. When the deformation is small, the stress is given by the usual theory of rubber elasticity. As the deformation increases, some junctions of the macromolecular microstructure rupture. Junctions then re-form to generate a new microstructure. The constitutive equation allows for continuous scission of the original junctions and formation of new ones as de-formation increases. The macromolecular scission causes stress reduction. The formation of new microstructures results in permanent set on release of external load. The present work considers a hollow circular cylinder composed of such a material, also assumed to be incompressible and isotropic. The cylinder is fixed rigidly at its inner surface and undergoes axisymmetric deformation due to a uniform axial moment applied at the outer surface. There develops an outer zone of material with the original microstructure and an inner zone of material having undergone macromolecular scission, separated by a cylindrical interface, the radius of which increases with the rotation of the outer surface. The shear deformation distribution, moment-rotation response and permanent set on release of moment are determined. It is found that microstructural scission can lead to higher levels of shear deformation near the inner surface of the cylinder than in the case of purely elastic response. It is also seen that a residual state of high shear deformation can arise in a thin layer of material at the inner boundary of the cylinder.
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