16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #6 - Boundary value problems in linear elasticity Figure 1: Schematic of generic problem in linear elasticity ? Equations of equilibrium ( 3 equations, 6 unknowns ): σ ji,j + f i = 0 (1) ? Compatibility ( 6 equations, 9 unknowns): ? ?u i ? (2)? ij = 1 2 ?x j + ?u j ?x i 1 ? Constitutive Law (6 equations, 0 unknowns) : σ ij = C ijkl ? kl (3) ? Boundary conditions of two types: ˉ – Traction or natural boundary conditions: For tractions t imposed on the portion of the surface of the body ?B t : n i σ ij = t j = t ˉ j (4) – Displacement or essential boundary conditions: For displacements ˉu imposed on the portion of the surface of the body ?B u , this includes the supports for which we have ˉu = 0: ˉu i = u i (5) One can prove existence and uniqueness of the solution ( the ?elds: u i (x j ), ? ij (x k ), σ ij (x k )) in B. 2