ON OPTIMAL TEMPERATURE PATHS FOR THERMORHEOLOGICALLY SIMPLE VISCOELASTIC MATERIALS

When a fiber-reinforced viscoelastic material is cooled from cure temperature to room temperature the different coefficients of thermal expansion of the matrix and fibers lead to residual stresses. To study this phenomenon in detail, a thin, isotropic, thermorheologically simple, linearly viscoelastic plate reinforced by a random system of fibers lying in the plane of the plate is considered. The following question is asked: Of all temperature paths theta (t), 0 less than equivalent to t less than equivalent to T, which take on prescribed values at t equals 0 and t equals T, is there a path which renders the residual stress at t equals T a minimum? The Euler equation associated with the above problem is a nonlinear integral equation, which is used to show that smooth minimizers are generally not possible. For a Maxwell material with an exponential shift function, the Euler equation is solved in closed form. This solution is used to compute the optimal temperature path for polymethyl methacrylate with initial and final temperatures 90 degree C and 80 degree C, respectively, and with T equals 5 hours. The optimal path produces a residual stress of 32 psi as comapred to 220 psi for a linear path.