Diffusional Anelasticity Internal Friction

In this section, pedagogical models for the time dependence of mechanical response are developed. Elastic stress and strain are rank-two tensors, and the compliance (or stiffness) are rank-four material property tensors that connect them. In this section, a simple spring and dashpot analog is used to model the mechanical response of anelastic materials. Scalar forces in the spring and dashpot model become analogs for a more complex stress tensor in materials. To enforce this analogy, we use the terms stress and strain below, but we do not treat them as tensors. 

For an ideally elastic material, the stress is linearly related to the strain by 

Anelasticity therefore affects the mechanical properties of materials. As seen below, its study yields unique information about a number of kinetic processes in materials, such as diffusion coefficients, especially at relatively low temperatures. 

The overall response of the crystal to such a stress cycle is shown in Fig. 8.16. When the stress a0 is applied suddenly, the crystal instantaneously undergoes an ideally elastic strain following Eq. 8.62. As the stress is maintained, the crystal undergoes further time-dependent strain due to the re-population of the interstitials. When the stress is released, the ideally elastic strain is recovered instantaneously and the remaining anelastic strain will be recovered in a time-dependent fashion as the interstitials regain their random distribution. 

General Formulation of Anelastic Behavior. Anelastic behavior where the strain is a function of both stress and time may be described by generalizing Eq. 8.62 and expressing the compliance in the more general form 

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