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Unformatted text preview: 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-withhole calculations of the previous assignment using traction boundary conditions for the case -h 0.125 and for the following values of the Poisson ratio: = 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 x2 = 0, x1 < 0 subjected to mode I remote loading. Restrict the analysis to the rectangular domain [-L, L] [0, L]. The boundary conditions on the line x2 = 0 are: traction-free for x1 < 0, u2 = 0 (symmetry) for x1 0. On the remainder of the boundary the displacements are constrained to equal the K-field: K r cos(/2)[ - 1 + 2 sin2 (/2)] u1 = 2 2 K r sin(/2)[ + 1 - 2 cos2 (/2)] u2 = 2 2 (1) where = 3 - 4, is the shear modulus, Poisson's ratio, r is the distance to the origin and the polar angle is measured from the x1 axis. Compute the finite element solution for meshes corresponding to 1 values of the mesh scaling factor h = 1/2p with p = 0, . . . , 5. Plot the deformed mesh and the contours of 11 , 22 . Also plot the energy-norm error uh - u E , computed by numerical quadrature, as a function of the number of elements E in log - log scale. Comment on the convergence properties of this finite element approximation. 3. Write a 6-node simplex element elmt01 to do plane stress/strain linear elasticity. Verify that your element and, thus, the implementation, satisfies the patch test. 4. Repeat the calculations in 1. using the 6-node element. Comment on the results. 5. Repeat the calculations in 2. using the 6-node element. Comment on the results. 2 ...
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- Fall '03