The Complementing Condition in Elasticity

We consider a boundary value problem of nonlinear elasticity on a domain [omega] in R³ [3-dimensional space] and compute the Complementing Condition for the linearized equations at a point X₀ [x zero] on boundary of omega. We assume a stored energy function depending on the first and third invariants of the deformation F and that the strong-ellipticity condition holds in [omega] . A surface traction boundary condition is imposed at X_0.
The Complementing Condition is calculated from a system of 3 second-order ordinary differential equations (0 less than and equal to t less than 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.