A bound on the strain energy for the traction problem in finite elasticity with localized non-zero surface data

For the boundary value problem in finite elasticity in which nonzero tractions are given on a connected subdomain Γ of the boundary, the rest of the boundary is stress-free, and there are no body forces, a bound is obtained for the strain energy in terms of the L₂ integral norm of the surface tractions with the constant involved depending only upon Γ and the material constants. The result is obtained in the context of finite elasticity under the assumptions that the unstressed body occupies a convex domain and the displacement gradients are sufficiently small. In the context of the linear theory, the same result is obtained without these assumptions.