Anelastic Relaxation in Crystalline Solids

Abstract

We are espe cially grateful for the support of much of the research leading to this book, as well as partial support of the writing itself, by the U. S. Atom ic Energy Commission. We thank Miss Maureen Weaver for her skill ful typing of a substantial fraction of the manuscript. Finally, we wish to express our deepest gratitude to our wives for their encouragement and understanding over the several years during which this book was being written.XV Dielectric relaxation has been the subject of a recent book by Daniel (1967). * It should be remembered in what follows that the terminology "force," "mass," and "displacement" is symbolic and would actually become "torque," "moment of inertia," and "angle of twist" in the case of the torsion pendulum. * Actually, an exponential decay with time is a direct consequence of linearity. De parture from exponential decay indicates that the material is not behaving anelastically, in that the internal friction is a function of strain amplitude. t These quantities are now total energies rather than energies per unit volume. 40 2 THE BOLTZMANN SUPERPOSITION PRINCIPLE aid of the Boltzmann principle. Although this is a major step forward, it does not meet all the requirements of a complete formal theory. Since a response function only describes the behavior of a material under given experimental conditions, the use of such a function to define the anelasticproperties of a solid is not a completely satisfactory one. We may suspect that there is a more basic description of anelastic behavior in terms of which all of the response functions can be expressed. It will become apparent in the next two chapters that such a description is possible, and involves the concept of a relaxation spectrum. PROBLEMS 2-1. Verify the equivalence of Eqs. (2.2-1) and (2.2-2). 2-2. Show that for A <^1 , the following relation exists between the normalized creep function ip(t) and the normalized stress relaxation function <p(t):v(0 « i -9>( 0Combine this result with Problem 1-2, to show that l/A/(0=/(*) +°(zl2) where 0(zl2) means terms of order A"1.General References . "Mathematical Structure of the Theories of Viscoelasticity." Her mann, Paris.

Dynamics of 2D material membranes

“…A difficult question is: what mechanisms determine the loss modulus E 2 in 2D materials? A number of mechanisms is known to cause anelastic relaxation losses in solids [139], most of which might play a role in 2D materials. Information about the underlying mechanism can to some extent be obtained by measurements as a function of frequency and temperature.…”

Effect of substitutional Ni atoms on the Snoek relaxation in ferrite and martensite Fe-C alloys: An atomisitic investigation

“…1). We convert the internal friction profile computed for a single crystal to the one for polycrystal via the Reuss averaging method [38,32]. The profile given by AKMC simulation is in excellent agreement with the one computed from TK model.…”

“…where ∆W = σ : d is the energy loss per cycle and W max is the maximum elastically stored energy per unit volume during a cycle. Note that if carbon atoms are diffusing in low-carbon Fe, Q −1 follows a Debye curve [32]:…”

Effect of Thermo-Elastic Damping on Transverse Waves Propagating in a Single-Wall Carbon Nanotube

“…Zener [12] has explored the results of thermo-elastic damping from the irreversible flow of heat driven by local temperature gradients that through the coupling accompany the strain field. Since the early work by Zener, quite a few * e-mail: m.selim@psau.edu.sa studies relating to thermo-elastic vibration analysis for rod, beam, plate, and composite laminate structures have been reported [13][14][15][16][17].…”