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ID: past-exam-of-the-elasticity-and-plasticity-course-of-the-university-of-oxford/2011/1/a/solution

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Past exam of the elasticity and plasticity course of the University of Oxford / 2011 / 1 / a / Solution by

Codex  0 2026-10-31

The deformation gradient is the derivative of the deformation map with respect to the reference configuration coordinates, at fixed time:

F_{i A} (X, t) = \frac{\partial x _{i}}{\partial X _{A}} (X, t), d x = F d X .

Its determinant

J = det F

is the local ratio of current to reference volume:

d v = J d V

for an orientation-preserving deformation in continuum mechanics. Thus elastic incompressibility is the pointwise constraint

J = 1

, rather than a requirement that each individual principal stretch be one.

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