16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #6- Boundary value problems in linear elasticity ↑B B Figure 1: Schematic of generic problem in linear elasticity Equations of equilibrium(3 equations, 6 unknowns f1=0 Compatibility( 6 equations, 9 unknowns) 1/ dui du
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 ): σji,j + fi = 0 (1) • Compatibility ( 6 equations, 9 unknowns): �∂ui � �ij = (2) 1 2 ∂xj + ∂uj ∂xi 1
Constitutive Law(6 equations, O unknowns) C Bound dary conditions of two types Traction or natural boundary conditions: For tractions t imposed on the portion of the surface of the body aBt Displacement or essential boundary conditions: For displacement u imposed on the portion of the surface of the body aBu, this includes the supports for which we have u=0 让= (5) One can prove existence and uniqueness of the solution( the fields: ui(,), Ei(ak), oii(ak)) n B
• Constitutive Law (6 equations, 0 unknowns) : σij = Cijkl�kl (3) • Boundary conditions of two types: – Traction or natural boundary conditions: For tractions ¯t imposed on the portion of the surface of the body ∂Bt: niσij = tj = t ¯j (4) – Displacement or essential boundary conditions: For displacements u¯ imposed on the portion of the surface of the body ∂Bu, this includes the supports for which we have u¯ = 0: ui = u¯i (5) One can prove existence and uniqueness of the solution ( the fields: ui(xj ), �ij (xk), σij (xk)) in B. 2