ON OPTIMAL TEMPERATURE PATHS FOR THERMORHEOLOGICALLY SIMPLE VISCOELASTIC MATERIALS* By MORTON E. GURTIN and LEA
نویسندگان
چکیده
Introduction. 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, we consider a thin, isotropic, thermorheologically simple, linearly viscoelastic plate reinforced by a random system of fibers lying in the plane of the plate. We ask the following question: of all temperature paths 9(t), 0 < t < T, which take on prescribed values at t = 0 and t = T, is there a path which renders the residual stress at t = T a minimum?1 The Euler equation associated with the above problem is a nonlinear integral equation, which we use to show that smooth minimizers are generally not possible (cf. [1]). Indeed, solutions of the Euler equation exhibit a jump discontinuity at t = 0, are smooth during (0, T), and generally suffer a second jump discontinuity at t = T.2 For a Maxwell material with an exponential shift function we are able to solve the Euler equation in closed form. We use this solution to compute the optimal temperature path for polymethyl methacrylate with initial and final temperatures 90°C and 80°C, respectively, and with T = 5 hours. The optimal path produces a residual stress of 32 psi as compared to 220 psi for a linear path.
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