More Examples on the Derivation of the Direct Integral Forms

and objectives In tutorial 2, we demonstrated the basic steps for deriving the direct boundary integral equation. In this tutorial we will explain, with more examples, the derivation of the direct boundary integral equation for systems of partial differential equations. We will consider two examples: elasticity problems and shear-deformable plate bending problems. 1 Elasticity equations In this section, we will derive the direct boundary integral equation form for elasticity problems (see Ref. [1] for more details about the theory). 1.1 Governing equations 0 b , i j ij = + σ (1) where (σ) denotes the stresses and (b) are the body forces. Equation (1) is called the Navier equation in terms of stresses. j ij i n p σ = (2) in which (n) is the normal to the boundary (Γ). () i j j i ij , u , u 2 1 + = ε (3) and i i ii , u = ε (4) where (ε) denotes the strains.