Problem 4. Consider a spherical aluminium ball with radius a at initial temperature To. When the...
Problem 4. Consider a spherical aluminium ball with radius a at initial temperature To. When the temperature is raised to T such as the temperature variation is denoted AT=T- To, the ball will expand in the same way for all directions. The related eigenstrain tensor et, is the thermal strain. The coefficient of thermal expansion of AL is denoted aa. Isotropic elasticiy is assumed with Ea and Va the Young's modulus and the Poisson's ratio of aluminium, respectively. (a) Find the eigenstrain tensor em as function of AT, Qa. (b) Assuming there is no constraint in the ball such as the ball will expand as the temperature rises, find the total strain in the ball. (c) Suppose now that the ball is embedded in a rigid matrix and the matrix sustains a temperature change AT. Then, there is a constraint on the ball due to the surrounding rigid matrix which does not let the ball to expand. Calculate the stress tensor in the ball.