Sagar and Payne [SP] analyze the stress-strain relationships and material properties of

Chapter 12, Problem 12.2.19

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Sagar and Payne [SP] analyze the stress-strain relationships and material properties of a cylinder alternately subjected to heating and cooling and consider the equation 2T r 2 + 1 r T r = 1 4K T t , 1 2 < r < 1, 0 < T, where T = T (r,t) is the temperature, r is the radial distance from the center of the cylinder, t is time, and K is a diffusivity coefficient. a. Find approximations to T (r, 10) for a cylinder with outside radius 1, given the initial and boundary conditions: T (1, t) = 100 + 40t, T 1 2 , t = t, 0 t 10; T (r, 0) = 200(r 0.5), 0.5 r 1. Use a modification of the Backward-Difference method with K = 0.1, k = 0.5, and h = r = 0.1. b. Use the temperature distribution of part (a) to calculate the strain I by approximating the integral I = 1 0.5 T (r,t)r dr, where = 10.7 and t = 10. Use the Composite Trapezoidal method with n = 5.