6. Elasticity equations

6.1 Context

Solids, like fluids, are continuous media. This similarity meant that the corresponding equations were derived simultaneously. The equations of elasticity already appear in Euler (1755); they are non-linear hyperbolic equations that translate the conservation laws of mass, angular momentum and energy. We introduce a function u(x, t) ∊ R ^d which represents the position at time t of the volume element (in dimension d of space), which at time t = 0 is at point x. Assuming that the only forces acting on the medium depend solely on displacement (constant density and temperature), Newton's equations for this displacement reduce to the system established by Cauchy and Poisson :