Some Stochastic Problems of Thermoviscoelasticity
Two problem groups are discussed. First, a straight bar is considered, with both ends fixed and exposed, from time t = 0 on, to a uniform temperature fluctuating in a random manner about mean value zero. The material of the bar is assumed to obey Norton’s law of nonlinear viscoelasticity. Viscosity is supposed to be temperature-independent. The Fokker-Planck equation is set up for two different temperature processes, and is solved numerically. The problem of first-passage time is discussed.
Second, viscosity-dependence on temperature is introduced in the form of thermorheologically simple behavior. Reduced time is then a stochastic process. Some basic properties of this process are discussed. A Fokker-Planck equation for the joint conditional probability density of reduced time and temperature is set up for the two temperature processes. Moments for the first process are given. For the second process solution in terms of Hermite polynomials is indicated.
KeywordsHermite Polynomial Stochastic Problem White Noise Excitation Nonlinear Viscoelasticity Linear Viscoelastic Material
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