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RELAXATION PHENOMENA IN METALS AND ALLOYS SUBJECTED TO DEFORMATION
V. S. Postnikov
INTRODUCTION
When real solid bodies are deformed, even in the purely elastic region, phenomena are observed in their behavior that find no explanation in the ordinary theory of elasticity.
Thus, for example, if a tensile force of finite magnitude is applied to a rod, only part of the total deformation corresponding to the given force arises instantaneously (Fig. 1); the remaining part of the deformation arises gradually. When the force is removed, the deformation disappears, but again not at once, rather gradually. This phenomenon has been called elastic aftereffect.
If a constant force (even a very small one) is applied for a long time to a rod, a deformation arises that changes continuously with time, and the more noticeably the higher the temperature of the rod. This phenomenon has been called creep.
Fig. 1. Dependence of the deformation \(\varepsilon\) on time \(t\). The constant force begins to act at the moment \(t_0\) and ceases to act (is removed) at the moment \(t_1\).
If the deformation is kept constant, then a smaller force is required to maintain the prescribed deformation at each subsequent moment of time; the stress produced in the rod is said to relax or weaken.
Finally, if a periodically varying force is applied to a rod, then, owing to the resulting phase shift of the stress relative to the deformation in the specimen, an irreversible conversion of part of the mechanical energy into heat will occur,
explained by the presence in solids of “internal friction”[^1]. The elastic moduli in this case turn out to depend on the frequency of the applied force.
This group of properties (“internal friction,” elastic aftereffect, variation of elastic constants with frequency, etc.), owing to which a solid behaves in a nonelastic manner in the elastic region, has received in the foreign literature the name of “elastic imperfections” of a material, or “anelasticity.” Such a name, however, does not reflect the internal mechanism of the phenomena.
In accordance with the terminology established in our literature, it is expedient to call these properties relaxation properties, bearing in mind that they are connected with the establishment of statistical equilibrium in a body deformed at a finite rate. Such a process is called relaxation, since by relaxation is meant any approach to statistical equilibrium.
Interest in the study of these phenomena arose long ago. As early as the first quarter of our century, B. P. Weinberg and V. D. Kuznetsov carried out a number of experimental works devoted to the study of internal friction in solids[^1].
A. F. Ioffe is responsible for the classical investigation of elastic aftereffect in crystalline quartz[^2]. Elastic aftereffect in an alloy was studied by N. N. Davidenkov and G. A. Kusmirskaya[^3]. The investigations of V. S. Gorskii[^4] and E. Ya. Pumper[^5] and others[^6] shed light on phenomena connected with the disturbance of statistical equilibrium in alloys.
Alongside the experimental study of relaxation phenomena in solids, attempts were also made at a theoretical investigation of these phenomena ([^6]–[^13], etc.) by means of mechanical and electrical analogies, by formal generalization of Hooke’s law, and so on.
We shall confine ourselves to a consideration of the principal directions, without dwelling on the method of analogies, which cannot give a physical picture of the phenomena under study.
1. FORMAL GENERALIZATION OF HOOKE’S LAW
Since for real solids subjected to deformation there is a gradual change of deformation and stress, a generalization of Hooke’s law in the first approximation may consist in taking into account the dependence of the magnitude of the stress \(\sigma\) not only on the deformation \(\varepsilon\), but also on the rate of deformation \(\dot{\varepsilon}\). For the case of a linear stressed state this dependence may be represented by the relation
\[ \sigma = a \cdot \varepsilon + b \cdot \dot{\varepsilon}. \tag{1} \]
A more general dependence of this kind is the relation
\[ \sigma + a \cdot \dot{\sigma} = b \cdot \varepsilon + c \cdot \dot{\varepsilon}. \tag{2} \]
Such a generalization for the case of simple tension (or compression) has been made by a number of investigators \(^{6,8,9}\).
Equation (2) describes stress relaxation, reversible aftereffect, internal friction, and the dependence of elastic moduli on the frequency of the external force.
To clarify the physical picture in the behavior of a body obeying equation (2), the latter is rewritten in the form
\[ \sigma + \tau_{\varepsilon}\dot{\sigma} = M_{\mathrm p}\varepsilon + M_{\mathrm p}\tau_{\sigma}\dot{\varepsilon}, \tag{3} \]
where the constants \(a\), \(b\), and \(c\) are replaced by the new independent constants \(\tau_{\varepsilon}\), \(\tau_{\sigma}\), \(M_{\mathrm p}\), whose meaning will be clarified below.
Putting
\[ \varepsilon=\mathrm{const}, \qquad \dot{\varepsilon}=0, \]
we obtain:
\[ \sigma+\tau_{\varepsilon}\dot{\sigma}=0, \qquad \text{whence} \qquad \sigma(t)=\sigma_{0}\cdot e^{-t/\tau_{\varepsilon}}. \]
Consequently, \(\tau_{\varepsilon}\) is the time required for the stress (at constant strain \(\varepsilon\)) to decrease by a factor of \(e\); it is called the stress-relaxation time at constant strain.
Similarly, putting \(\sigma=\mathrm{const}\), \(\dot{\sigma}=0\), we clarify the meaning of the quantity \(\tau_{\sigma}\), which is the strain-relaxation time at constant stress. As for the constant \(M_{\mathrm p}\), its physical meaning is revealed if one assumes that at the instant \(t=0\) a strain \(\varepsilon=\varepsilon_{0}\) is created instantaneously.
Then the stress \(\sigma(t)\) will relax to the equilibrium value \(M_{\mathrm p}\varepsilon_{0}\). Indeed, for \(\varepsilon=\varepsilon_{0}\), \(\dot{\varepsilon}=0\), and \(\sigma=\sigma_{0}\) at the instant \(t=0\), from (3) we have:
\[ \sigma+\tau_{\varepsilon}\dot{\sigma}=M_{\mathrm p}\varepsilon, \]
whence
\[ \sigma(t)=M_{\mathrm p}\varepsilon_{0} + (\sigma_{0}-M_{\mathrm p}\varepsilon_{0})e^{-t/\tau_{\varepsilon}}. \tag{4} \]
For \(t=\infty\), from (4) we have \(\sigma(\infty)=M_{\mathrm p}\varepsilon_{0}\), i.e. the quantity
\[ M_{\mathrm p}=\frac{\sigma(\infty)}{\varepsilon_{0}} \]
is the “relaxed” modulus (by \(M_{\mathrm p}\) one may understand the shear modulus or Young’s modulus), in contrast to the quantity
\[ M_{\mathrm n}=\frac{\sigma(0)}{\varepsilon_{0}}, \]
which may be called the “unrelaxed” modulus. Here \(\sigma(0)\) is the initial value of the stress (at \(t=0\)).
Putting \(\sigma(0)=\sigma_{0}\), we obtain the equation
\[ \varepsilon(t)=M_{\mathrm p}^{-1}\cdot\sigma_{0} + \left(\varepsilon_{0}-M_{\mathrm p}^{-1}\cdot\sigma_{0}\right)e^{-t/\tau_{\sigma}}, \tag{5} \]
entirely analogous to equation (4).
If a periodically varying force is applied to a solid body, a periodically varying deformation arises. However, now the deformation \(\varepsilon\) and the stress \(\sigma\) will not coincide in phase. This means that the graphical relation between stress and deformation will no longer be a straight line, but an ellipse, whose area is proportional to the energy dissipated during a period of oscillation.
Analytically this means that the relation between stress and deformation is effected through a complex modulus, whose expression is found by substituting into equation (3)
\[ \sigma(t)=\sigma_0\cdot e^{i(\omega t+\alpha)} \quad \text{and} \quad \varepsilon(t)=\varepsilon_0\cdot e^{i\omega t}, \]
which gives
\[ (1+i\omega\cdot\tau_\varepsilon)\cdot\sigma = M_p\cdot(1+i\omega\cdot\tau_\sigma)\cdot\varepsilon, \]
or
\[ \sigma=M^*\cdot\varepsilon, \]
where the complex modulus \(M^*\) has the form
\[ M^*=\frac{1+i\omega\cdot\tau_\sigma}{1+i\omega\cdot\tau_\varepsilon}\cdot M_p . \tag{6} \]
The tangent of the angle of lag of the deformation \(\alpha\) behind the stress, equal to the ratio of the imaginary part of the complex modulus to its real part, is determined by the relation
\[ \operatorname{tg}\alpha = \frac{\omega(\tau_\sigma-\tau_\varepsilon)} {1+\omega^2\cdot\tau_\sigma\cdot\tau_\varepsilon}. \tag{7} \]
If we introduce the geometric means
\[ \tau=(\tau_\varepsilon\cdot\tau_\sigma)^{1/2}, \qquad M=(M_n\cdot M_p)^{1/2}, \]
and use the relation
\[ \frac{M_n}{M_p}=\frac{\tau_\sigma}{\tau_\varepsilon}, \]
then expression (7) assumes the form
\[ \operatorname{tg}\alpha = \frac{M_n-M_p}{M}\cdot \frac{\omega\cdot\tau}{1+(\omega\cdot\tau)^2}. \tag{8} \]
Since \(\operatorname{tg}\alpha\) is usually used as a measure of internal friction, we have therefore obtained the dependence of internal friction on frequency, or rather on the parameter \(\omega\cdot\tau\).
As the dynamic modulus one usually takes the quantity equal to the ratio of the stress to that part of the deformation which is in phase with the stress, namely:
\[ M_\omega= \frac{1+\omega^2\cdot\tau_\sigma^2} {1+\omega^2\cdot\tau_\sigma\cdot\tau_\varepsilon} \cdot M_p . \tag{9} \]
or
\[ M_{\omega}^{*}=M_{\mathrm{н}}-\frac{M_{\mathrm{н}}-M_{\mathrm{р}}}{1+(\omega\tau)^2}. \tag{10} \]
Figure 2 presents the variation of the internal friction \(Q=\operatorname{tg}\alpha\) and of the dynamic modulus \(M_{\omega}\) as a function of the parameter \(\omega\cdot\tau\). As is seen from Fig. 2, the magnitude of the internal friction reaches a maximum at \(\omega\cdot\tau=1\); at this point the dynamic modulus has the greatest rate of change as a function of the parameter \(\omega\cdot\tau\).
For \(\omega\cdot\tau>10\) the internal friction is insignificant, while the change in the modulus is practically equal to zero, i.e. the dynamic modulus is equal to the “unrelaxed” modulus \(M_{\mathrm{н}}\). Consequently, in this frequency region \((\omega\cdot\tau>10)\) there is practically no relaxation. For \(\omega\cdot\tau<0.1\) the dynamic modulus is practically equal to the “relaxation” modulus \(M_{\mathrm{р}}\), and the internal friction is again very small. In the intermediate frequency region we have partial relaxation of the dynamic modulus and considerable internal friction, which reaches a maximum at \(\omega\tau=1\).
Fig. 2. Frequency dependence of the internal friction \((I)\) and of the shear modulus \((II)\).
Summing up what has been set forth, one may say that the indicated generalization of Hooke’s law makes it possible to describe stress relaxation at constant deformation, reversible aftereffect, and the dependence of the dynamic moduli on frequency. Internal friction in a solid body obeying equation (3) follows of itself, being the result of the existence of a phase shift between stress and deformation under periodic variation of the external force.
However, equation (3) does not describe purely elastic phenomena or the phenomena of strain hardening and cold working observed in metals and alloys. Taking this into account leads to a further, but already less effective, generalization\(^9\) of equation (3). Let us also note that real solid bodies, as a rule, exhibit not one relaxation time \(\tau\), but a whole set of them or even a continuous spectrum.
2. BOLTZMANN’S THEORY OF ELASTIC AFTEREFFECT
According to the theory of elasticity, deformation and stress are related at any instant of time by the relation
\[ \sigma(t)=M\cdot\varepsilon(t). \tag{11} \]
This equation has two important properties: 1) it is linear; 2) it relates the instantaneous value of the deformation to the instantaneous value of the stress.
Boltzmann1, in constructing the theory of elastic aftereffect, proceeded from the principle of superposition, but supplemented it with the assumption that the deformation at a given moment depends on the behavior of the specimen at preceding moments of time. Mathematically, superposition can be formulated as follows: let a unit stress at the moment \(t=0\) produce the resulting deformation \(\varphi(t)\). If, during an earlier time interval from \(t'\) to \(t' + dt'\), the stress changed from \(\sigma(t')\) to \(\sigma(t' + dt')\), then it may be considered that during this time interval \(dt'\) there acted a stress of constant magnitude
\[ \sigma(t' + dt') - \sigma(t') \simeq \dot{\sigma}(t') \cdot dt', \]
which caused the deformation
\[ \varphi(t' + dt') \cdot \dot{\sigma}(t') \cdot dt'. \]
The total deformation by the time \(t\) will be equal to the time integral of the expression
\[ \varphi(t - t') \cdot \dot{\sigma}(t') \cdot dt', \]
i.e.,
\[ \varepsilon(t) = \int_{-\infty}^{t} \varphi(t - t') \dot{\sigma}(t')\,dt'. \tag{12} \]
Inversion of this integral gives:
\[ \sigma(t) = \int_{-\infty}^{t} f(t - t') \dot{\varepsilon}(t')\,dt'. \tag{13} \]
If the functions \(\varphi(t-t')\) or \(f(t-t')\) have once been determined, then equations (12) or (13) can be used to find the “inelastic” behavior of a solid. However, Boltzmann’s theory makes it possible to say nothing about the form of the functions \(\varphi(t-t')\) and \(f(t-t')\).
Usually equations (12) and (13) are written in a somewhat different form. Integrating the right-hand side of equation (12) by parts, we obtain:
\[ \varepsilon(t) = \varphi(0)\cdot \sigma(t) + \int_{-\infty}^{t} \dot{\varphi}(t-t') \cdot \sigma(t') \cdot dt'. \tag{14} \]
The function \(\varphi(0)\) is related to the static modulus by the relation
\[ 1 = M_{\mathrm{н}} \cdot \varphi(0). \]
Substituting \(\varphi(0)=\dfrac{1}{M_{\mathrm{н}}}\) in (14) and introducing \(\Phi(t-t')=\dot{\varphi}(t-t')\),
we obtain the equation
\[ \varepsilon(t)=\frac{1}{M_{\mathrm{H}}}\cdot\sigma(t)+\int_{-\infty}^{t}\Phi(t-t')\,\sigma(t')\,dt', \tag{15} \]
which is often encountered in the literature\(^{6,9}\). It can be shown that equation (3) is a particular case of the integral equation (15) when the kernel \(\Phi(t-t')\) is chosen in the form
\[ \Phi(t-t')=\frac{\tau_{\sigma}-\tau_{\varepsilon}}{M_{p}\cdot \tau_{\sigma}^{2}}\cdot e^{-\frac{(t-t')}{\tau_{\sigma}}}. \tag{16} \]
Thus, almost everything that was said above (concerning the formal generalization of Hooke’s equation) remains valid also for the Boltzmann theory with a kernel of the form (16).
Of all the relaxation properties of materials, the greatest attention is attracted by the change of the dynamic moduli with frequency and by internal friction. It can be shown that these two characteristics are related to the function \(\Phi(t)\) by the following equations:
\[ \Delta(\omega)=\int_{0}^{\infty}\Phi(t)\cdot \cos\omega t\,dt \tag{17} \]
and
\[ Q(\omega)=\int_{0}^{\infty}\Phi(t)\sin\omega t\,dt, \tag{18} \]
where
\[ \Delta(\omega)=\frac{M_{\infty}-M_{\omega}}{M_{0}} \]
is the degree of relaxation of the modulus (the modulus defect) at a given frequency \(\omega\) of oscillations.
From equations (17) and (18) it is clear that the unknown function \(\Phi(t)\) can in principle be determined by measuring the defect \(\Delta(\omega)\) of the dynamic modulus or from the magnitude of the internal friction. Such a calculation, however, is associated with large errors, since to determine \(\Phi(t)\) it is necessary to perform the operation of inverting the integral (17) or (18), and a small error in the measurement of \(\Delta(\omega)\) or \(Q(\omega)\) will appear as a relatively large error in the determination of \(\Phi(t)\).
Relaxation spectrum
If it is assumed that the material is characterized by a single time constant (relaxation time) \(\tau\), and the function \(\Phi(t)\) is chosen in the form
\[ \Phi(t)=\Delta_{0}\tau^{-1}\cdot e^{-t/\tau}, \tag{19} \]
then equations (17) and (18) take the familiar form (see (9) and (10)):
\[ \Delta(\omega)=\frac{\Delta_0}{1+(\omega\tau)^2} \tag{20} \]
and
\[ Q(\omega)=\frac{\Delta_0\,\omega\tau}{1+(\omega\tau)^2}. \tag{21} \]
Here \(\Delta_0=\dfrac{M_\infty-M_0}{M_0}\) is the greatest value of the modulus defect (complete relaxation of the modulus). Relations (20) and (21), as functions of the parameter \(\omega\cdot\tau\), may be represented graphically as shown in Fig. 2.
If several relaxation processes occur simultaneously in a specimen, each characterized by its own particular relaxation time \(\tau_i\), then the “contribution” of each of these processes, for example, to the relaxation of stress will constitute a definite fraction of the complete stress relaxation, determined by the ratio
\[ \frac{\sigma(t)-\sigma(\infty)}{\sigma(\infty)}. \]
The totality of all relaxation times of the individual relaxation processes forms the so-called relaxation spectrum of the given material. It may be either discrete or continuous.
Let us consider the case of a continuous distribution of relaxation constants. Since relaxation times often lie in a very broad interval of values, it is expedient to pass to a logarithmic scale. We introduce the distribution function \(\psi(\tau)\), which we define so that the product \(\psi(\tau)\cdot d(\ln \tau)\) represents the “contribution” to the total modulus defect \(\Delta_0\) due to processes whose relaxation times lie in the interval \(d(\ln \tau)\) situated in the neighborhood of the value \(\tau\). The totality of all the “contributions” gives the total modulus defect, i.e.
\[ \Delta_0=\int_{-\infty}^{+\infty}\psi(\tau)\cdot d(\ln \tau). \tag{22} \]
For a material with one relaxation constant the stress \(\sigma(t)\) relaxes, as shown above, according to a simple exponential law, namely:
\[ \sigma(t)=\sigma_0\cdot e^{-t/\tau}. \tag{23} \]
Consequently, in the approximation in which the superposition principle is valid, for the case considered here of a material with a continuous distribution of relaxation constants in the neighborhood of the values \(\tau\), we obtain that for the relative quantity
the relaxation of stress occurs according to the law
\[ \psi(\tau)e^{-t/\tau}\cdot d(\ln \tau). \]
The total relaxation of the relative value of the stress is determined by the relation
\[ \frac{\sigma(t)-\sigma(\infty)}{\sigma(\infty)} = \int_{-\infty}^{+\infty} \psi(\tau)e^{-t/\tau}d(\ln \tau). \tag{24} \]
On the basis of expressions (19), (20), (21) one may write:
\[ \Phi(t)= \int_{-\infty}^{+\infty} \psi(\tau)\cdot \tau^{-1}\cdot e^{-t/\tau}d(\ln \tau), \tag{25} \]
\[ \Delta(\omega)= \int_{-\infty}^{+\infty} \frac{\psi(\tau)}{1+(\omega\tau)^2}\,d(\ln \tau), \tag{26} \]
\[ Q(\omega)= \int_{-\infty}^{+\infty} \frac{\psi(\tau)\cdot \omega\tau}{1+(\omega\tau)^2}\cdot d(\ln \tau). \tag{27} \]
Consequently, if the distribution function \(\psi(\tau)\) is known, then one can easily find all the principal quantities \((\Delta, Q, \Phi)\) characterizing the relaxation properties of a solid body.
However, while the function \(\Phi(t)\) can be determined rather simply (although not quite exactly) experimentally, as yet there exist no methods for the direct determination of the function \(\psi(\tau)\). Certain simplifying assumptions can be made concerning the function \(\psi(\tau)\), but they do not lead to results that agree at all well with experiment. It should also be noted that, in order to determine the function \(\psi(\tau)\) from the known form of the functions
\[ \frac{\sigma(t)-\sigma(\infty)}{\sigma(\infty)}, \quad \Delta(\omega) \quad \text{and} \quad Q(\omega), \]
in the general case it is necessary to perform the inversion of the integral (24), (26), or (27), which introduces considerable inaccuracy into the result. Because of the extreme sensitivity of the function \(\psi(\tau)\) to errors in the determination of \(Q(\omega)\) and \(\Delta(\omega)\), for the characterization of the relaxation spectrum, instead of it one considers the quantity \(Q(\omega)=\tg\alpha\) as a function of the relaxation time or of the frequency of oscillations.
Figure 3 presents a typical relaxation spectrum (for more detail see below, § 4). Here \(I\) is the internal friction caused by the presence of pairs of atoms with different atomic radii (substitutional solutions); \(II\) is internal friction caused by viscous flow along grain boundaries; \(III\) is internal friction caused by viscous flow in “amorphous” regions introduced by plastic deformation;
deformations; IV—internal friction caused by diffusion of interstitial atoms, for example carbon or nitrogen atoms (the transition of these atoms into preferential interstitial positions); V—internal friction caused by transverse thermal conductivity during bending of the specimen; VI—internal friction caused by intercrystallite thermal conductivity.
This relaxation spectrum will characterize the given material under the given conditions. Any change in the specimen, whether
Fig. 3. Typical relaxation spectrum of a solid at room temperature.
it be a change in dimensions, composition, degree of deformation, annealing, grain size, temperature, etc., is reflected in a characteristic change of the relaxation spectrum. The study of these changes will make it possible to penetrate more deeply into the mechanism of certain processes in metals and alloys than other methods permit. It is somewhat strange that two peaks (V and VI), which are due to thermal relaxation and which have little relation (or none at all) to the plastic and strength properties of bodies, have been well studied \(^{2,6,15}\), whereas two peaks (II, III), which are directly related to the plastic properties of bodies, are poorly studied \(^{1,4,17,18}\).
3. THEORY OF RELAXATION PHENOMENA ON THE BASIS OF GENERAL THERMODYNAMIC CONSIDERATIONS
Restricting ourselves, as before, to the case of a linear stress state (for the volumetric case see \(^{7,16}\)), let us consider the isothermal (the nonisothermal case was recently considered by N. S. Fastov \(^{22}\)) deformation of a homogeneous isotropic body.
RELAXATION PHENOMENA IN METALS AND ALLOYS
When an elastic body is deformed at a finite rate, the deformed body at any given instant of time may also not be in a state of statistical equilibrium. In the classical theory of elasticity, the state of an elastically deformed body is completely determined by the value of the temperature \(T\) and the magnitude of the deformation \(\varepsilon\). In the case we are considering, however, these quantities are not sufficient, and one must introduce a new independent thermodynamic variable that characterizes the instantaneous deviation of the elastically deformed body from the state of statistical equilibrium and that may be called the relaxation parameter. (As applied to liquids, the idea of a relaxation mechanism of sound absorption was first put forward by Mandelstam and Leontovich \(^{19}\).) For the case of a volume stress state, the relaxation parameter is a tensor of second rank.
For small deformations and small deviations from the state of statistical equilibrium, the free energy of an elastically deformed body, referred to unit volume, may be represented by the formula
\[ F = F_0(T) + a \cdot \varepsilon^2 + b \cdot \xi^2 + c \cdot \varepsilon \cdot \xi, \tag{28} \]
where \(F_0\) is the free energy of the undeformed body in a state of statistical equilibrium, and \(\xi\) is the relaxation parameter. Applying to the problem of interest to us the “thermodynamic” theory of nonequilibrium systems developed by Academician M. A. Leontovich \(^{20}\), we shall denote by \(\bar{\xi}\) the “equilibrium” value of the relaxation parameter, which, for a given deformation, is determined by the relation
\[ \left(\frac{\partial F}{\partial \xi}\right)_{T,\varepsilon}=0. \tag{29} \]
From (28) and (29) it follows that
\[ 2b \cdot \bar{\xi} + c \cdot \varepsilon = 0, \]
whence
\[ \bar{\xi} = -\frac{c}{2b}\cdot \varepsilon. \tag{30} \]
Let us introduce a new quantity, \(\zeta\), defined by the relation
\[ \zeta = \xi - \bar{\xi}. \tag{31} \]
Then condition (29) is satisfied for \(\zeta = 0\). In the variables \(\varepsilon\) and \(\zeta\), the free energy \(F\) takes the form
\[ F = F_0 + \left(a-\frac{c^2}{4b}\right)\cdot \varepsilon^2 + b \cdot \zeta^2. \tag{32} \]
On the other hand, the free energy of an elastic body deformed in a quasistatic manner and, consequently, being in...
is in a state of statistical equilibrium, is equal to
\[ F=F_0+\frac{M}{2}\varepsilon^2, \tag{33} \]
where \(M\) is the isothermal modulus of elasticity (\(E\) or \(G\)). Setting \(\zeta=0\) in (32) and comparing with (33), we obtain:
\[ a-\frac{c^2}{4b}=\frac{M}{2}, \tag{34} \]
whence
\[ F=F_0+\frac{M}{2}\varepsilon^2+b\zeta^2. \tag{35} \]
From the known minimal properties of the free energy it follows that
\[ b>0. \tag{36} \]
Defining the elastic stress by the relation
\[ \sigma=\left(\frac{\partial F}{\partial \varepsilon}\right)_{\zeta}, \tag{37} \]
we obtain:
\[ \sigma=M\varepsilon+2b\zeta\frac{\partial \zeta}{\partial \varepsilon}. \tag{38} \]
Since
\[ \frac{\partial \zeta}{\partial \varepsilon}=-\frac{\partial \bar{\zeta}}{\partial \varepsilon}=\frac{c}{2b}, \]
on the basis of expression (30), it follows that
\[ \sigma=M\varepsilon+c\zeta. \tag{39} \]
The relaxation parameter \(\xi\) is a quantity whose rate of change is a function of \(\zeta\); this rate is the smaller, the closer the body is to the state of equilibrium, and becomes zero when equilibrium is reached, i.e., when \(\zeta=0\). Considering \(\zeta\) to be small, let us put
\[ \dot{\xi}=f(\zeta) \tag{40} \]
and, expanding this function in a series, we shall restrict ourselves in the expansion to the linear term
\[ \dot{\xi}=f(0)+f'(0)\zeta. \]
In view of the fact that for \(\zeta=0\) the quantity \(\dot{\xi}\) is also equal to zero, we obtain:
\[ f(0)=0 \]
and, consequently,
\[ \dot{\xi}=f'(0)\zeta. \]
The coefficient \(f'(0)\) must be negative and have the dimension
neglecting \(c\varepsilon k^{-1}\). Assuming \(f'(0)=-\dfrac{1}{\tau}\), where \(\tau\) is a parameter characteristic of the given body (the relaxation time), we obtain:
\[ \dot{\zeta}=-\frac{1}{\tau}\zeta \quad (\tau>0). \tag{41} \]
Taking (30) and (31) into account, we obtain:
\[ \dot{\zeta}+\frac{1}{\tau}\zeta=-\frac{c}{2b}\dot{\varepsilon}. \tag{42} \]
Let us integrate this differential equation, assuming that in the infinitely remote past the body was in statistical equilibrium. Then
\[ \zeta(t)=-\frac{c}{2b}\int_{-\infty}^{t} e^{-\frac{t-t'}{\tau}} \dot{\varepsilon}(t')\,dt'. \tag{43} \]
and
\[ \sigma(t)=M\varepsilon(t)+\frac{c^2}{2b}\int_{-\infty}^{t} e^{-\frac{t-t'}{\tau}}\dot{\varepsilon}(t')\,dt'. \tag{44} \]
Introduce a new quantity
\[ \eta=\frac{c^2\tau}{2b}, \]
having the dimensions of internal friction \(\left(\dfrac{\mathrm{g}}{\mathrm{cm}\cdot\mathrm{sec}}\right)\). Then
\[ \sigma(t)=M\varepsilon+\frac{\eta}{\tau}\int_{-\infty}^{t} e^{-\frac{t-t'}{\tau}}\dot{\varepsilon}(t')\,dt'. \tag{45} \]
For the case in which the deformation is a purely periodic function of time, i.e.
\[ \varepsilon=\varepsilon_0 e^{i\omega t}, \]
we obtain the generalized Hooke’s law in the following form:
\[ \sigma=M\varepsilon+\frac{i\omega\eta\varepsilon}{1+i\omega\tau}, \]
whence
\[ \sigma=M^{*}\varepsilon, \tag{46} \]
where
\[ M^{*}=M+\frac{i\omega\eta}{1+i\omega\tau}. \tag{47} \]
is the complex modulus. The fact that the modulus is a complex quantity indicates, as we saw above, the presence of a phase shift between stress and strain, causing elastic hysteresis, damping of oscillations, and other relaxation properties.^16
V. S. POSTNIKOV
The formal generalization of Hooke’s law\(^{6,9}\) follows as a consequence of the more general theory just considered. To see this, let us find \(\Delta(\omega)\) and \(Q(\omega)\) on the basis of expression (47), which we shall transform somewhat.
Multiplying the numerator and denominator of the second term on the right-hand side of (47) by \((1-i\omega\tau)\), we obtain:
\[ M^{*}=M+\frac{\omega^{2}\tau^{2}}{1+\omega^{2}\tau^{2}}\frac{\eta}{\tau} +\frac{i\omega\tau}{1+\omega^{2}\tau^{2}}\frac{\eta}{\tau}. \]
As the dynamic modulus one usually takes the real part of the complex modulus, i.e.,
\[ M_{\omega}=M+\frac{\omega^{2}\tau^{2}}{1+\omega^{2}\tau^{2}}\frac{\eta}{\tau}. \tag{48} \]
From expression (48) we find:
\[ \begin{aligned} M_{\infty}&=M+\frac{\eta}{\tau},\\ M_{0}&=M. \end{aligned} \tag{49} \]
Taking (48) and (49) into account, we obtain:
\[ \Delta(\omega)=\frac{M_{\infty}-M_{\omega}}{M} =\frac{\Delta_{0}}{1+(\omega\tau)^{2}}, \tag{50} \]
where
\[ \Delta_{0}=\frac{M_{\infty}-M}{M}. \]
The expression obtained coincides completely with expression (20). If, as the measure of internal friction, one takes \(\operatorname{tg}\alpha\), equal to the ratio of the imaginary part of the complex modulus to its real part, then one can verify that
\[ Q(\omega)=\frac{\Delta_{0}\cdot\omega\cdot\tau} {1+\frac{M_{\infty}}{M}\cdot(\omega\tau)^{2}}. \tag{51} \]
The expression obtained differs from expression (21) by the factor \(\dfrac{M_{\infty}}{M}\) (of order unity) in the denominator.
4. VARIOUS MECHANISMS OF RELAXATION PHENOMENA IN SOLIDS
The deformation arising in an elastic body is determined not only by the external mechanical forces applied to it, but also by the temperature of the body, its chemical composition, external magnetic and electric fields (magnetostriction and electrostriction), a change in the degree of order in an alloy, etc. This dependence of the total deformation on the indicated variables leads—
leads to a diversity of relaxation phenomena, each of which makes its own contribution to the internal friction \(Q(\omega)\), to the modulus defect \(\Delta(\omega)\), and to other relaxation characteristics of the material.
Let us briefly consider the various sources of “elastic imperfections” of a material.
Thermal conductivity
Thermal conductivity is one of the sources of relaxation phenomena in metals. An increase in temperature at constant external pressure is usually accompanied by an increase in volume. Conversely, the adiabatic application of an all-round or linear tensile stress (a decrease in “external pressure”) leads to a lowering of temperature and, consequently, causes a flow of heat into the specimen from the surrounding medium. Since the temperature difference gradually decreases, the specimen undergoes an insignificant increase in length.
The relaxation time for equalization of temperature throughout the specimen is equal to\(^{6,15,21}\)
\[ \tau \simeq \frac{d^{2}}{D}, \tag{52} \]
where \(d\) is the linear dimension of the specimen, and \(D\) is the coefficient of thermal diffusivity
\[ D=\frac{\lambda}{c_{v}}, \tag{53} \]
\(\lambda\) is the specific thermal conductivity, and \(c_{v}\) is the specific heat at constant volume.
The degree of relaxation in this case is
\[ \Delta_{0}^{(T)}=T\cdot E_{s}\frac{a^{2}}{\rho c_{p}}, \tag{54} \]
where \(T\) is the absolute temperature, \(E_{s}\) is the adiabatic modulus of elasticity, \(a\) is the linear coefficient of thermal expansion, \(\rho\) is the density, and \(c_{p}\) is the specific heat at constant pressure.
Temperature equalization may take place not only between the specimen and the medium, but also between different parts of the specimen. For this purpose it is necessary to create an inhomogeneous stress in the specimen, which may be macroscopic or microscopic. The simplest case of an inhomogeneous macroscopic stress occurs in bending of a specimen. Here the material on one side of the neutral plane is subjected to tension, and on the other to compression. For a specimen of circular cross section the relaxation time \(\tau\) is determined\(^{6}\) by the formula
\[ \tau=\frac{d^{2}}{4.3\cdot 2\pi D}, \tag{55} \]
where \(d\) is the diameter of the specimen, and \(D\) is the coefficient of thermal diffusivity—
...ness. The degree of relaxation in this case is also determined by expression (54).
Microscopic inhomogeneous stress arises owing to the elastic anisotropy of the grains or because of the random orientation of individual crystals. A macroscopically homogeneous force applied to the specimen will result in microscopic inhomogeneity of the stress from crystal to crystal, which will lead to a change in temperature in the crystals and, as a consequence of this (the presence of a temperature gradient in the grains), to the occurrence of heat flows.
The relaxation time in this case is
\[ \tau \simeq \frac{d^2}{D}, \tag{56} \]
where \(d\) is the mean linear size of a grain. The degree of relaxation is equal to\(^{6}\):
\[ \Delta_0 = R \cdot \Delta_0^{(T)} . \tag{57} \]
Here
\[ R = \frac{\overline{(E-2)}-\overline{(E-1)^2}} {\overline{(E-2)}} \]
is the relative value of the mean-square deviation of the elastic modulus of neighboring crystals. The factor \(R\) indicates that the degree of relaxation (and the maximum of internal friction) will be the greater, the greater the elastic anisotropy of the individual crystallites.
Diffusion
In an unstressed solid solution, a state of statistical equilibrium corresponds to a certain (generally speaking, disordered) distribution of the dissolved atoms. If the alloy is deformed, then a redistribution of the dissolved atoms arises in it, proceeding at a certain rate (with its own relaxation time \(\tau\)). In the case of a periodically varying stress, this redistribution will lead to additional scattering of the elastic energy of the oscillations.
The fact that diffusion will lead to inelastic phenomena was first recognized by V. S. Gorskii\(^{4}\), who theoretically considered this effect and carried out experiments confirming his theory.
A characteristic feature of relaxation caused by diffusion is the relatively large value of \(\tau\) and the strong dependence of this quantity on temperature. The relaxation time in this case is also determined by the relation
\[ \tau \simeq \frac{d^2}{D}, \tag{58} \]
where \(d\) is the distance over which diffusion can take place, \(D\) is the diffusion coefficient. This diffusion coefficient can be written as
\[ D \simeq \frac{a^2}{\theta}, \tag{59} \]
where \(a\) is the distance over which an atom moves in an elementary diffusion act, \(\theta\) is the mean time interval between elementary diffusion acts\(^{23}\). Combining (58) and (59), we obtain:
\[ \tau \simeq \left(\frac{d}{a}\right)^2 \cdot \theta . \tag{60} \]
The strong temperature dependence of the relaxation time \(\tau\) is due to the rapid temperature variation of \(\theta\). The mean time interval, as is known\(^{24}\), is determined as follows:
\[ \theta \simeq \frac{h \cdot N}{H} e^{H/RT}, \tag{61} \]
where \(h\) is Planck’s constant, \(N\) is Avogadro’s number, \(R\) is the gas constant, \(T\) is the absolute temperature, \(H\) is the so-called heat of activation, referred to one mole of substance. Combining (60) and (61), we obtain:
\[ \tau \simeq \left(\frac{d}{a}\right)^2 \frac{h \cdot N}{H} e^{H/RT}. \tag{62} \]
The degree of relaxation for the case of dilute solid solutions\(^{15}\) in this case is equal to
\[ \Delta^{(D)}_0 = \frac{E_{\mathrm{H}} \cdot w}{\rho \cdot R \cdot T} \cdot \left(\frac{\delta \varepsilon}{\delta c}\right)_{\sigma} \cdot (1-c) \cdot c, \tag{63} \]
where \(c\) is the atomic concentration of the solid solution, \(\rho\) is the density, \(w\) is the atomic weight of the solvent, \(E_{\mathrm{H}}\) is the unrelaxed modulus of elasticity, \(R\) is the gas constant, \(T\) is the absolute temperature.
Relaxation caused by diffusion reaches a large magnitude. For example, for \(\alpha\)-brass (70% copper, 30% zinc), from equation (63) we have for \(\Delta^{(D)}_0\) a value of 0.5, whereas for thermoelastic relaxation the value is \(\Delta^{(T)}_0 = 0.0036\). This means that the internal friction caused by relaxation of the concentration flow will be 140 times greater than the internal friction caused by relaxation of the heat flow (see Fig. 3). However, relaxation phenomena connected with diffusion are difficult to observe because of the very long relaxation time. Since the relaxation time is proportional to \(\dfrac{d^2}{D}\), a short relaxation time can be obtained by decreasing the grain size and increasing the measurement temperature (increasing \(D\)). With increasing temperature not only \(D\) increases, but also the grain size, so that the relaxation time cannot become less than \(10^6\) sec at any temperature.
V. S. POSTNIKOV
Preferential distribution of atoms in a stress field
In a stress-free disordered solid solution, the distribution of dissolved atoms is isotropic with respect to each individual atom of the basic lattice. This means that the nearest dissolved atom has the same probability of being situated along any possible crystallographic direction. For example, in a body-centered cubic lattice, interstitial atoms may occupy positions of the type \((0, 1/2, 1/2)\), or \((1/2, 0, 1/2)\), or \((1/2, 1/2, 0)\), which have tetragonal symmetry with tetragonal axes along the principal axes.
If a tensile force is applied, for example along the \(z\)-axis, then the isotropic distribution of atoms is disturbed; the equilibrium distribution will now be one in which a larger number of dissolved atoms are found in positions with tetragonal axis \(z\). The transition to the new equilibrium position is associated with a certain relaxation time \(\tau\).
In a face-centered lattice, the axis connecting nearest neighbors lies in one of the \(\langle 110\rangle\) directions. If the dissolved atom (of substitutional type) is larger than the solvent atom, then two neighboring atoms along one of these \(\langle 110\rangle\) directions will produce a local stretching of the lattice as compared with the direction perpendicular to the line joining these atoms.
If a tensile force is applied, for example along the \(\langle 110\rangle\) direction, then this direction will become preferential. The transition of other pairs of atoms into this preferential position will here also be associated with a certain relaxation time \(\tau\).
Relaxation phenomena associated with the atom-redistribution process indicated here have been observed by various investigators in \(\alpha\)-brass \(^{13}\), in Fe-C, Fe-N \(^{23}\), and in Ti-O, Ti-C, Ti-N \(^{17}\). Since the redistribution of atoms in the lattice under the influence of stresses occurs by diffusion, then for the degree of relaxation \(\Delta_{0}\) formula (63) will, with some approximation, be applicable if in it the quantity \(\left(\dfrac{\partial \varepsilon}{\partial c}\right)_{\sigma}\) is replaced by the quantity \(\left(\dfrac{\partial \varepsilon}{\partial c_{1}} - \dfrac{\partial \varepsilon}{\partial c_{2}}\right)_{\tau}\), where \(c_{1}\) and \(c_{2}\) are the atomic concentrations of the preferential and non-preferential positions, respectively, while the quantity \(c\) is replaced by some mean of these two atomic concentrations \(^{6}\).
For an interstitial-type solid solution, the relaxation time should be comparable with the lifetime of the interstitial atom in the interstitial position. This lifetime, in turn, is determined \(^{23}\) by the expression
\[ \tau \cong \frac{1}{\nu}\cdot e^{H/RT}, \tag{64} \]
where \(\nu\) is the frequency of vibrations of the interstitial atom in the interstitial position.
...tion, \(H\) is the heat of activation per mole of interstitial atoms, determined by the type of interstitial position and by the nature of both the interstitial atoms and the atoms of the solvent.
For substitutional solid solutions the relaxation time is determined by a formula analogous to formula (62).
The Zener theory developed here has recently met with an objection raised by Nowick\(^{31}\), who showed that its application is limited to the region of very small concentrations.
Ordered Distribution of Atoms in the Lattice
In ordering alloys (Cu, Zn, AuCu, etc.), after the critical temperature \(T_k\) a certain degree of order is established in the distribution of atoms, quite definite for each temperature. As was first indicated by V. S. Gorsky\(^{4}\), the creation of a stressed state in an alloy changes the equilibrium degree of order. The new equilibrium distribution is established not instantaneously, but after some time \(\tau\). The only investigation carried out to determine the relaxation time belongs to V. S. Gorsky\(^{4}\), who found for the alloy CuAu that
\[ \tau \sim e^{H/RT}, \tag{65} \]
where \(H\) is the heat of activation, having the same value as for macroscopic diffusion, namely \(27\,400\ \text{cal/mol}\). At a temperature of \(270^\circ\), which is \(100^\circ\) below \(T_k\), this relaxation time is equal to \(10\) sec.
The degree of relaxation for ordering alloys may be determined approximately\(^{6}\) as
\[ \Delta_0 = E_{\text{н}}\mu_c\left(\frac{\Delta \varepsilon}{\Delta S_c}\right)^2, \tag{66} \]
where \(E_{\text{н}}\) is the unrelaxed elastic modulus, \(\mu_c=\left(\dfrac{\partial S_c}{\partial T_c}\right)_\sigma\) is the derivative of the mixing entropy \(S_c\) with respect to the mixing temperature \(T_c=\left(\dfrac{\partial H}{\partial S_c}\right)_\sigma\) at constant stress, \(\Delta \varepsilon\) is the change in strain associated with the change in the degree of order, \(\Delta S\) is the change in the entropy of mixing associated with the same change, and \(H\) is the heat function of a unit volume of the alloy. For example, for an alloy of the type Cu\(_3\)Au, \(\Delta_0=0.004\) at the critical temperature (\(\sim 400^\circ\)) and \(\Delta_0=0.1\) at a temperature somewhat below the critical one.
Slip Bands and Grain Boundaries
Polycrystalline bodies have a complex relaxation spectrum, which includes both sharp maxima caused by a phenomenon with a single (for example, thermoelastic relaxation)
or several close relaxation times (for example, interstitial positions), as well as maxima “smeared” over a considerable interval of times \(\tau\), which can be described by an entire set of relaxation times.
To explain the smearing of relaxation maxima in the relaxation spectrum of polycrystalline bodies, the idea of a “two-component system”\({}^{6,23}\) was introduced. This idea consists in the fact that a solid is represented as consisting of two phases, one of which is amorphous and the other perfectly elastic. The elastic phase obeys Hooke’s law:
\[ \varphi = G^{-1}\cdot P_{xy}, \]
while the amorphous phase obeys Maxwell’s relaxation equation:
\[ \frac{d\varphi}{dt} = \frac{1}{G}\cdot \frac{dP_{xy}}{dt} + \frac{P_{xy}}{\eta}, \tag{67} \]
where \(\varphi\) is the angle of shear characterizing the deformation occurring under the influence of the shearing stress \(P_{xy}\), \(G\) is the shear modulus, and \(\eta\) is the viscosity coefficient of the amorphous body. For the amorphous phase, in the case of a sudden stop of the deformation at the value \(\varphi_0\), the shearing stress \(P_{xy}\) does not disappear instantaneously, but according to the law
\[ P_{xy}=P_{xy}^{0}\cdot e^{-\frac{G\cdot t}{\eta}}, \tag{68} \]
which is obtained by integration of the differential equation (67) for \(\frac{d\varphi_0}{dt}=0\).
As is seen from (68), the stress gradually weakens (relaxes) with the relaxation time
\[ \tau=\frac{\eta}{G}. \tag{69} \]
This relaxation time will depend on the physical nature of the amorphous region, as well as on its dimensions and shape. It is not difficult to show that if the amorphous region has width \(c\) and length \(l\), then the viscosity coefficient of this region is related to the relaxation time by the equation\({}^{17}\):
\[ \eta=\frac{G\cdot \tau}{l}\cdot c, \tag{70} \]
from which it is seen that the relaxation time depends on the dimensions of the amorphous region (and its shape).
If amorphous regions are embedded in the elastic matrix (the basis of the solid), then each region will relax the stress with its own relaxation time \(\tau_i\), and the behavior of such a solid
of the body will be determined by the entire set of relaxation times \(\tau_i\), which will lead to a “smearing” of the maximum of the complete relaxation spectrum of the solid.
Internal friction caused by a phenomenon with a single relaxation time is determined by formula (51). For a polycrystalline body, the internal friction will evidently be determined by the equation
\[ Q(\omega)=\sum_{i=1}^{n}\frac{\Delta(\tau_i)\cdot\omega\cdot\tau_i}{1+\frac{M_\infty}{M}\cdot(\omega\cdot\tau_i)^2}, \tag{71} \]
where \(\Delta(\tau_i)\) is the degree of relaxation caused by the presence of a relaxation phenomenon with relaxation time \(\tau_i\).
A two-component system has a number of distinctive features. The first feature of a two-component system is the considerable magnitude of the relaxation effects, which may be caused by quite small amounts of amorphous material (in the form of disks, bands, or a network).
The second noteworthy feature is the creation of a high concentration of stresses inside the elastic matrix during relaxation of the stress inside the amorphous region.
The third feature is the great variety of types of relaxation spectrum depending on the shape and size of the amorphous regions and their distribution in the elastic matrix.
This variety of possible types of relaxation spectrum does not make it possible, in the general case, to solve the problem of the magnitude of the relaxation times and the degree of relaxation for an arbitrary solid.
The role of amorphous regions, as experiments show, may be played by slip bands\(^{6,17}\), formed in the process of plastic deformation, or by grain boundaries\(^{6,17,18,26,27,28}\). The idea of amorphous slip bands is similar to the so-called idea of dislocations\(^{6,17,29}\), on the basis of which attempts are often made to explain inhomogeneous (several relaxation times) relaxation phenomena in solids\(^{17,30}\) and other phenomena. A gratifying fact is the appearance in our press of just criticism directed against this theory\(^{32}\).
Summing up briefly what has been set forth, one may say that, although the physical picture of relaxation phenomena in metals and alloys is more or less clear, the theoretical explanation of the available data is still not entirely satisfactory.
Generalizations of Hooke’s law and the method of mechanical and electrical analogies lead to a formal construction of a theory which cannot provide a physical picture of the phenomena being studied.
A theory based on general thermodynamic considerations is the most satisfactory. But it too is limited in its possibilities, since it proceeds from the consideration only of the case of a homogeneous isotropic body. None of the essen-
existing theories does not consider the dependence of the relaxation characteristics on temperature, residual deformation (preliminary plastic deformation), and other factors. Finally, it should be noted that our scientists devote too little attention to this very important and promising problem in solid-state physics.
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