Abstract
Treloar carried out experiments on square rubber sheets that were loaded by equal in-plane forces along their edges. Kearsley observed from Treloar’s data that the deformations were equal biaxial extensions for smaller forces, but became unequal biaxial extensions beyond a larger force. By means of a simple non-linear elasticity problem, Kearsley showed that the governing equations had multiple solutions that allowed for bifurcation from an equal biaxial to an unequal biaxial deformation.
The present work considers this bifurcation phenomenon if Treloar’s experiments were to be carried out in the temperature range, say, of 100–150°C. Tobolsky showed that rubber in this temperature range undergoes microstructural evolution attributed to a chemical process of scission and crosslinking of macromolecular junctions. When loaded by equal in-plane forces along its edges, the rubber sheet can be expected to undergo equal biaxial extensional creep. The possibility is considered that there may be a time when the equal biaxial extensional creep history branches (bifurcates) into an unequal biaxial extensional creep history.
Kearsley’s analysis is revisited using a constitutive theory developed for rubber undergoing thermally induced microstructural changes. It is assumed that there is a time when the equal biaxial extension history branches into an unequal biaxial extension history. A branching condition for determining this time is established. A numerical example shows the existence of a branching time and the subsequent branching of an equal biaxial extension history into an unequal biaxial extension history. After branching, the governing equations allow both equal biaxial and unequal biaxial extension histories. It is assumed that the sheet follows the deformation history that maximizes the rate of dissipation, which turns out to be the unequal biaxial extension history.
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