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hw4solution

# hw4solution - 1.033/1.57 H#4 Champagne...

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1.033/1.57: H#4: “Champagne Method” (3D-ElastoPlasticity) Due: Part 1-3 (H#4): December 1, 2003 Part 4 (Q#3): December 5, 2003 MIT 1.033/1.57 Fall 2003 Instructor: Franz-Josef ULM We consider a rigid infinite half-space with an empty hole of diameter 2 a and length L (see figure below). In this hole we want to force a deformable cylinder sample of same length L but of a greater diameter 2 R > 2 a than the bore hole. The cylinder is initially stress free, and it is entirely forced into the hole. During this processus, the cylinder preserves its length. The contact between the cylinder walls and the bore hole walls is frictionless. We want to study the stress fields and the required force F to maintain the cylinder in the bore hole. Throughout this exercise we assume isothermal and quasi-static evolutions, and body forces are disregarded. L L 2 R 2 a e z F 2 R 2 a e r e θ A A B B Rigid Halfspace Deformable Cylinder Section A Section B Deformable cylinder in a rigid halfspace.

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Due: Part 1-3 (H#4): December 1, 2003 Part 4 (Q#3): December 5, 2003 page 2 1. Deformation and Strain: We consider a displacement field of the form: ξ = u ( r ) e r + D with u ( r ) the displacement in the radial direction e r ; it is a pure radial displacement field. D is a rigid body displacement field which ensures during the process that the axis of the cylinder sample coincides with the axis of the bore hole. (a) Specify the condition which the displacement field needs to satisfy to be kinematically admissible. (b) Once the cylinder sample is in the hole, we want to deal with the problem within the hypothesis of small perturbations. For the problem in hand, specify the restriction on the considered displacement field. (c) Determine the linearized strain tensor. 2. Elasticity : We want to determine the stress field in the sample at the end of the processus, and the required force to enforce the sample in the hole. In a first approximation, we suppose a linear elastic isotropic behavior of the cylinder sample (Lamé constants λ, µ ).
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