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Anelastic Relaxation in Crystalline Solids by A. S. Nowick

By A. S. Nowick

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T h e s e forms are Λ ( « ) - Λ = | - / Γ Λ « ) ^ γ (2-4-3) 38 2 THE BOLTZMANN SUPERPOSITION PRINCIPLE A brief derivation of these relations is given in A p p e n d i x B. I t s h o u l d be noted that the integrals in E q s . 4-4) are principal values. A n interesting result is obtained by setting ø = 0 in E q . 4-3). 4-5) J 0 which states that t h e total area u n d e r t h e curve of J 2{co) plotted against In ø is simply π / 2 times t h e m a g n i t u d e of t h e relaxation of / . 4-4) with ø = 0 a n d o o gives the additional r e s u l t / 2 ( 0 ) A C 0 0) = 0, which was already noted in E q .

4-6) σ while at the other extreme, ω τ σ< ^ 1, / (ω) ~ 6J (ø ) Λ-/ι(ω)^<5/(ωτ )^ Æ 2 < 1) σ T h e s e results show t h e comparative sharpness of the change in J 1 and in J 2 in the vicinity of ø = 1, namely, that toward either low or high frequencies, J x approaches its asymptotic limits faster t h a n J 2 approaches zero. T h i s effective narrowness of the transition region of J 1 as compared to the J 2 peak is also clear from an examination of the plots of these quantities in Fig. 3-8. Alternative dynamic response functions are the internal friction (ø) and the absolute dynamic compliance |/|(ω).

W e m a y suspect that there is a more basic description of anelastic behavior in t e r m s of which all of t h e response functions can b e expressed. I t will become apparent in t h e next t w o chapters that such a description is possible, a n d involves t h e concept of a relaxation s p e c t r u m . PROBLEMS 2-1. Verify t h e equivalence of E q s . 2-2). 2-2. Show that for ` << 1, t h e following relation exists between t h e normalized creep function ł( ) a n d t h e normalized stress relaxation function ( ): y>(t) ~ 1 - cp(t) C o m b i n e this result with P r o b l e m 1-2, to show that \jM(t)=J{t) where 0(A2) + 0{A>) means t e r m s of order A 2 .

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