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
ˆ‚2T / ˆ‚r2 + 1 / r ˆ‚T / ˆ‚r = 1 / 4K ˆ‚T / ˆ‚t, 1/2 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 ˆ’ 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

where α = 10.7 and t = 10. Use the Composite Trapezoidal method with n = 5.

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0.5