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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 3node simplex element in the incompressible limit. Using the 3node simplex element, repeat the planestrain, platewithhole 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 3node simplex element in problems with singularities. Consider the case of a semiinfinite 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: tractionfree for x1 < 0, u2 = 0 (symmetry) for x1 0. On the remainder of the boundary the displacements are constrained to equal the Kfield: 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 energynorm 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 6node 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 6node element. Comment on the results. 5. Repeat the calculations in 2. using the 6node element. Comment on the results. 2 ...
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 Fall '03
 RaulRadovitzky
 #, µ, 2 K, 3node simplex element

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