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By William Walton

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Can be nonuniform, then it will be equivalent both to the force LlF! and the moment LIM!. We assume that lim LlMdLlS! = 0 ASI-O 40 3. General Theory of Stress and Strain Fig. 1. To determine the force acting across an arbitrarily oriented area To prove the last statement, consider an infinitesimal volume in the form of a tetrahedron separated from the body and shown in Fig. 1. It is bounded by i) the face ABC with an area LlS and normal n. The force LlF acting across this face is to be determined.

22), find for the trace of the stress tensor hence, emm (Jik = (Jmm/(3A + 2}1,). 4. Using the generalized Hooke's law, calculate the unilateral extension of a rod by the action of force F; I is the rod's length and S is its cross-sectional area. Find also the relationship between the constants E, v and A and J1. Solution: In the case under consideration, only one stress tensor component differs from zero. 1, + 2J1' But (Jll = FIS, ell = LJlll, e22 = LJhlh, h being the rod's dimension in one of the transverse directions.

Determine the deformation of a cylinder of radius R rotating about its axis with frequency w. Solution: In cylindrical coordinates we have the body (centrifugal) force fb = pw 2 r. The equilibrium equation assumes the form (A. 1 curl curl u = - pw 2r. As above, u = U(r)r/r, curl u = 0, div u = (rU)'/r, grad div u = [(rU)'/r ]'r/r so that Hence, Furthermore, setting a rr / R = 0 we find and finally, . _ ~ pw 2r (2A. U - 8 A. 1 A. 1 R r . =- pw 2r/(A.

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