Characteristic Times of Nonhomogeneous Maxwell Materials

Abstract

This paper considers the transient stress distribution which exists in bars of nonlinear and nonhomogeneous viscoelastic (Maxwell) materials.The particular nonhomogeneity considered is an exponential variation of the fluidity in the thickness direction which is independent of time. It has previously been shown that this variation could be caused by a steady linear temperature gradient. Emphasis is given to the evaluation of the characteristic time of the stress-redistribution process.Five specific problems are solved by the Galerkin technique. In all of these problems the load remains constant. It is found that the characteristic time depends on the average stress for nonlinear problems and in a complicated way on the magnitude of the nonhomogeneity. Results are given in graphical form for a range of the nonhomogeneity parameter. The principal purpose of this paper is the qualitative determination of the time after which a Maxwell material can be considered as a viscous fluid for the determination of the state of stress. Symbols b = width of cross section, Fig. 1 h = thickness of cross section, Fig. 1 k = exponent in stress-strain-time relation, Eq. (1) n = center of rotation parameter t = time A = area of cross section E = Young's modulus or equation M = moment TV = temperature gradient (nonhomogeneity) parameter P = load e = strain f = nondimensional thickness coordinate, Fig. 1 f * = element which is characteristic of the cross section k = curvature rate A = fluidity Xo = fluidity at the cold face £ = 0 rj = characteristic parameter, Eq. (11) <r = stress o-i = initial stress (t = 0) o-oo = viscous stress (t = °° ) (To = P/A -average stress r = E\ 0 t = nondimensional time (') = d/dr( ) differentiation with respect to r