A stochastic theory of anelastic creep

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Abstract
Creep (the time-dependent yielding of a material under stress) arises from the stress-modulated motion of defects. We develop, for the case of linear anelasticity, a fundamental theory that exploits the randomness of defect motion leading to creep. The random motion of defects causes fluctuations in the strain, and the formalism of linear response theory for classical variables is brought to bear on the problem to obtain the fluctuation-dissipation (FD) theorem for anelasticity: formulae are derived for the complex compliance and the creep function in terms of the autocorrelation of the strain fluctuations in zero external stress. These formulae are also expressed in terms of the power spectrum of the fluctuating strain, as this is the quantity of potential experimental interest. Moment theorems are derived for the power spectrum, to constrain and to give physical meaning to the parameters introduced in empirical network models of anelasticity. A generalized stochastic equation is derived connecting the stress and strain, and a representation for the compliance that models the dissipation in the system with the help of a memory function is obtained. This function is related to the internal stress autocorrelation by a corollary of the FD theorem (the generalized Nyquist theorem), in order to facilitate its modelling in any given physical situation. The ‘softening’ of anelastic relaxation as a consequence of the incorporation of memory effects is demonstrated. The formalism is illustrated and the theorems verified by a direct calculation of the creep function for Snoek relaxation in a cubic crystal. A stochastic method is used to evaluate the required statistical averages. The theory developed can be extended, with appropriate modifications, to other problems of non-elastic behaviour.