By A. S. Nowick
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Additional info for Anelastic Relaxation in Crystalline Solids
W e have seen that, when 2 <^ 1, (ø) and Ji (co) are simply reciprocals of o n e another [Eq. 3-14a)] while for ˜2 <^ 1, M(t) a n d J(t) are reciprocals (Problem 2-2). 4. T h e most i m p o r t a n t result of this chapter is that a n y o n e response function m a y b e used t o characterize t h e anelastic behavior of t h e m a terial, since all other response functions m a y b e derived from it with t h e 40 2 TH E BOLTZMAN N SUPERPOSITIO N PRINCIPL E aid of t h e Boltzmann principle. Although this is a major step forward, it does n o t meet all t h e r e q u i r e m e n t s of a complete formal theory.
1-1) where dots are used to denote time derivatives, in the usual w a y . + For conciseness, an equation of this type will be referred to as a differential stress-strain equation. Since E q . 1-1) contains both the features of linearity and time-dependence, it m u s t provide the basis for a description of linear viscoelastic behavior (cf. T a b l e 1-1). T o represent only t h e more specialized case of anelasticity, where we have the additional r e quirement of a u n i q u e equilibrium relationship between stress and strain, further restrictions m u s t be placed u p o n E q .
1-8) I t is useful to obtain other forms of E q . 1-8). First let = t t’ be t h e "elapsed t i m e . 1-9) so long as σ ( — o o ) is taken to b e zero. 1-Π) Similarly, from E q s . 1-13) e T h e s u m m a t i o n s over discrete steps of stress or strain, respectively, are deliberately omitted in E q s . 1-13), since in these forms there is n o singularity in t h e integrand even w h e n a or are p e r m i t t e d to u n d e r g o discontinuous changes. 1-7) implies that there exists a relation between t h e J(t) a n d M(t) functions.