The Complementing Condition in Elasticity
Lavanya Ramanan · 2014
We consider a boundary value problem of nonlinear elasticity on a domain Ω [omega] in R3 [3-dimensional space] and compute the Complementing Condition for the linearized equations at a point x0 [x zero] on ∂Ω [boundary of omega]. We assume a stored energy function W (F ) depending on the first, second and third invariants of the deformation F and that the strong-ellipticity condition holds in Ω. A surface traction boundary condition is imposed at x0. The Complementing Condition is calculated from a system of 3 second-order ordinary differential equations (0 ≤ t <∞[infinity]) with boundary conditions at t = 0 and constant coefficients; the condition is satisfied if and only if the only bounded exponential solution is trivial. We compute the Complementing Condition in terms of parameters in W .