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E. 21) ∂B where σ is the stress tensor, τi = σij nj and Tik = Σijk nj . (Please note that ej · dS(y) = nj (y)dS(y)). 17) is obtained by taking the limit ξ → x. 4). 25) 8 CHAPTER 1. 29) It is sometimes convenient, however, to write the internal stress directly. 32) = Eijmn Again, the normal n is defined at the (boundary) field point y. 2. e. as r → 0). 33). 3). 36), that only contain weakly singular integrals, are available in the literature (see, for example, Cruse and Richardson [39]). 39) satisfy certain smoothness requirements (see Martin et al.

ERROR ESTIMATION Problem 2 : Traction Boundary Conditions Solve the Navier-Cauchy equations: (λ + µ)∇(∇ · u) + µ∇2 u = 0 in B subject to the boundary conditions: t = g on ∂B where the tractions satisfy the consistency conditions of static equilibrium: t ds = 0 ∂B (r × t) ds = 0 ∂B It is known that the solution to the above problem exists, and is unique up to a rigid body motion (Fung [46]). 20) where r0 ∈ R2 is a translation, and ω = ω k is an axial vector representing a rotation. 9). 22) one obtains an integral equation of the first kind for the displacement.

Numerical results from the hypersingular BCM, collocated on edges and at corners, from Mukherjee and Mukherjee [111], are available in Chapter 4. e. that the error is evaluated at selected points) residual-based error estimates for Dirichlet, Neumann and mixed boundary value problems (BVPs) in linear elasticity are presented first in this chapter. Interesting relationships between the actual error and the hypersingular residuals are proved for the first two classes of problems, while heuristic error estimators are presented for mixed BVPs.

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