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# ha3 - 16.225 Computational Mechanics of Materials Homework...

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16.225 - Computational Mechanics of Materials Homework assignment # 3 Handed out: 10/15/03 Due: 10/29/03 1. Behavior of the 3-node simplex element in the incompressible limit. Using the 3-node simplex element, repeat the plane-strain, plate-with- hole calculations of the previous assignment using traction boundary conditions for the case h 0 . 125 and for the following values of the Poisson ratio: ν = 0 . 49 , 0 . 499 , 0 . 49999. Comment on the results. 2. Behavior of the 3-node simplex element in problems with singularities. Consider the case of a semi-infinite crack coincident with the half axis x 2 = 0, x 1 < 0 subjected to mode I remote loading. Restrict the analysis to the rectangular domain [ L, L ] × [0 , L ]. The boundary conditions on the line x 2 = 0 are: traction-free for x 1 < 0, u 2 = 0 (symmetry) for x 1 0. On the remainder of the boundary the displacements are constrained to equal the K -field: K r u 1 = cos( θ/ 2)[ κ 1 + 2 sin 2 ( θ/ 2)] 2 µ 2 π K r u 2 = sin( θ/ 2)[ κ + 1 2 cos 2 ( θ/ 2)] 2 µ 2 π (1) where κ = 3 4 ν , µ is the shear modulus, ν Poisson’s ratio, r is the distance to the origin and the polar angle θ is measured from the x 1 - axis. Compute the finite

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ha3 - 16.225 Computational Mechanics of Materials Homework...

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