The phenomenon of internal friction (viscosity). Friction

Viscosity called the ability of a fluid to resist shear forces. This property of a liquid manifests itself only when it moves. Let us assume that a certain amount of liquid is enclosed between two flat unlimited parallel plates (Fig. 2.1); the distance between them is P; the speed of movement of the upper plate relative to the lower one is υ.

Experience shows that the layer of liquid immediately adjacent to the wall sticks to it. It follows that the speed of movement of the liquid adjacent to the bottom wall is zero, and to the top wall – υ. The intermediate layers move at a speed that gradually increases from 0 to υ.

Rice. 2.1.

Thus, there is a difference in speed between adjacent layers, and mutual sliding of the layers occurs, which leads to the manifestation of the force of internal friction.

To move one plate relative to another, it is necessary to apply a certain force G to the moving plate, equal to the resistance force of the fluid as a result of internal friction. Newton found that this force is proportional to speed And, contact surfaces S and inversely proportional to the distance between the plates n , i.e.

where μ is the proportionality coefficient, called dynamic viscosity (or dynamic viscosity coefficient).

To further clarify this dependence, it should be related to the infinitesimal distance between the layers of liquid, then

where Δ υ is the relative speed of movement of neighboring layers; Δ P - the distance between them. Or at the limit

The last expression represents Newton's law for internal friction. The plus or minus sign is taken depending on the sign of the velocity gradient dv/dn.

Since τ = T/S there is a tangential shear stress, then Newton’s law can be given a more convenient form:

The tangential stress arising in a fluid is proportional to the velocity gradient in the direction perpendicular to the velocity vector and the area along which it acts.

The proportionality coefficient µ characterizes the physical properties of the liquid and is called dynamic viscosity. From Newton's formula it follows that

The physical meaning of the coefficient p follows from this expression: if , then µ = τ.

In hydrodynamics, the quantity

called kinematic viscosity (kinematic viscosity coefficient).

Dynamic viscosity µ decreases with increasing temperature, and increases with increasing pressure. However, the influence of pressure for dropping liquids is negligible. The dynamic viscosity of gases increases with increasing temperature, but changes only slightly with changes in pressure.

Newton's law for internal friction in liquids differs significantly from the laws of friction in solids. In solids there is static friction. In addition, the friction force is proportional to normal pressure and depends little on the relative speed of movement. In a fluid that obeys Newton's law, in the absence of a relative velocity of movement of the layers, there is no friction force. The friction force does not depend on pressure (normal stress), but depends on the relative speed of movement of the layers. Liquids that obey Newton's law are called Newtonian. However, there are liquids that do not obey this law (anomalous liquids). These include various types of emulsions, colloidal solutions, which are heterogeneous bodies consisting of two phases (solid and liquid).

Thus, clay solutions used in drilling oil wells and some types of oils do not obey Newton’s law near their pour point. Experiments have established that in such liquids movement occurs after the tangential stresses reach a certain value called initial shear stress.

For such liquids, a more general dependence for τ is valid (Bingham’s formula):

where τ0 is the initial shear stress; η – structural viscosity.

Thus, these liquids at voltage τ< τ0 ведут себя как твердые тела и начинают течь лишь при τ ≥ τ0. В дальнейшем градиент скорости пропорционален не т, а разнице τ -τ0.

Graphically, the relationship between and τ is depicted by curve 1 for Newtonian liquids and curve 2 for anomalous liquids (Fig. 2.2).

Rice. 2.2. Addictiondv/dn from shear stress

When structural fluids move through a pipeline, three modes of their movement are observed: structural, laminar, turbulent.

Structural. To start movement, a certain initial pressure drop in the pipeline Δ is required R 0, after which the liquid separates from the walls and begins to move as one whole (like a solid).

Laminar. With increasing pressure drop Δ R the speed of fluid movement will increase and a laminar flow regime will begin to develop near the walls. As the speed further increases, the region of the laminar regime will expand, then the structural regime completely turns into laminar.

Turbulent. With a further increase in speed, the laminar regime becomes turbulent (see paragraph 6.1).

Dependence of viscosity on temperature and pressure. Viscometers

The viscosity of a droplet liquid depends largely on temperature and, to a lesser extent, on pressure. The dependence of viscosity on pressure is neglected in most cases. For example, at pressures up to 50–105 Pa, the viscosity changes by no more than 8.5%. The exception is water at a temperature of 25°C - its viscosity decreases slightly with increasing pressure. Another feature of water is that its density increases with a decrease in temperature to +4°C, and with a further decrease in temperature (from +4 to 0°C) it decreases. This explains the fact that water freezes from the surface. At a temperature of about 0°C, it has the lowest density, and layers of liquid having the same temperature as the lightest float to the surface, where water freezes if its temperature is less than 0°C.

At atmospheric pressure, the viscosity of water depending on temperature is determined by the Poiseuille formula

Where v – kinematic viscosity; µ – dynamic viscosity; ρ is the density of water at a given temperature; t – water temperature.

The viscosity of a liquid is determined using instruments called viscometers. For liquids more viscous than water, an Engler viscometer is used. This device consists of a container with a hole through which, at a temperature of 20°C, the time for draining distilled water is determined. T 0 and liquid T , the viscosity of which needs to be determined. Ratio of quantities T And T 0 is the number of conventional Engler degrees:

After determining the viscosity of the liquid in conventional Engler degrees, the kinematic viscosity (cm2/s) is found using the empirical Ubellode formula

The v values ​​obtained using this formula are in good agreement with experimental data.

Internal friction occurs in a liquid due to the interaction of molecules. Unlike external friction, which occurs at the point of contact of two bodies, internal friction takes place inside a moving medium between layers with different speeds.

At speeds above the critical speed, the layers close to the walls noticeably lag behind the average due to friction, significant speed differences arise, which entails the formation of vortices.

So, viscosity, or internal friction in liquids, causes not only energy loss due to friction, but also new formations - vortices.

Newton established that the force of viscosity, or internal friction, must be proportional to the velocity gradient (a value showing how quickly the speed changes when moving from layer to layer in a direction perpendicular to the direction of movement of the layers) and the area over which the action of this force is detected. Thus, we arrive at Newton's formula:

, (I.149)

Where - viscosity coefficient, or internal friction, a constant number characterizing a given liquid or gas.

To find out the physical meaning, let us put in formula (I.149) sec –1, m 2; then numerically ; hence, the viscosity coefficient is equal to the friction force, which occurs in a liquid between two areas in m 2, if the velocity gradient between them is equal to unity.

SI unit of dynamic viscosity = pascal second (Pa s).

(Pa s) is equal to the dynamic viscosity of the medium in which, with laminar flow and a velocity gradient with a module equal to (m/s) per (m), an internal friction force in (N) appears on (m 2) the contact surface of the layers (Pa · s = N · s/m 2).

The unit allowed for use until 1980: poise (P), named after the French scientist Poiseuille, who was one of the first (1842) to begin precise studies of viscosity when liquids flow in thin tubes (the relationship between units of dynamic viscosity: 1 P = 0.1 Pa s)

Poiseuille, observing the movement of liquids in capillary tubes, deduced law , Whereby:

, (I.150)

where is the volume of liquid flowing through the tube during time;

Tube radius (with smooth walls);

Pressure difference at the ends of the tube;

Duration of fluid flow;

Tube length.

The greater the viscosity, the greater the forces of internal friction that arise in it. Viscosity depends on temperature, and the nature of this dependence is different for liquids and gases:

q the dynamic viscosity of liquids decreases sharply with increasing temperature;

q The dynamic viscosity of gases increases with increasing temperature.

In addition to the concept of dynamic viscosity, the concepts turnover And kinematic viscosity.

Fluidity is called the reciprocal of dynamic viscosity.

SI unit of fluidity = m 2 / (N s) = 1 / (Pa s).

Kinematic viscosity is called the ratio of dynamic viscosity to the density of the medium.

The SI unit of kinematic viscosity is m 2 /s.

Until 1980, the unit allowed for use was Stokes (St). The relationship between units of kinematic viscosity:

1 Stokes (St) = 10 –4 m 2 /s.

When a spherical body moves in a liquid, it has to overcome the force of friction:

. (I.153)

Formula (I.153) is Stokes law .

The determination of liquid viscosity using a Hoeppler viscometer is based on Stokes' law. A ball is lowered into a pipe of a certain diameter filled with a liquid, the viscosity of which must be determined, and the speed of its fall is measured, which is a measure of the viscosity of the liquid.

The English scientist O. Reynolds in 1883, as a result of his research, came to the conclusion that the criterion for characterizing the movement of liquids and gases can be numbers determined by a dimensionless set of quantities related to a given liquid and its given movement. The composition of these abstract numbers, called numbers Reynolds, such.

Internal friction I Internal friction II Internal friction

in solids, the property of solids to irreversibly convert mechanical energy imparted to the body during the process of deformation into heat. Voltage is associated with two different groups of phenomena—inelasticity and plastic deformation.

Inelasticity is a deviation from the properties of elasticity when a body is deformed under conditions where there is practically no residual deformation. When deforming at a finite rate, a deviation from thermal equilibrium occurs in the body. For example, when bending a uniformly heated thin plate, the material of which expands when heated, the stretched fibers will cool, the compressed fibers will heat up, resulting in a transverse temperature difference, i.e. elastic deformation will cause a violation of thermal equilibrium. Subsequent temperature equalization by thermal conduction is a process accompanied by the irreversible transition of part of the elastic energy into thermal energy. This explains the experimentally observed damping of free bending vibrations of the plate - the so-called Thermoelastic effect. This process of restoring disturbed balance is called relaxation (See Relaxation).

During elastic deformation of an alloy with a uniform distribution of atoms of various components, a redistribution of atoms in the substance may occur due to the difference in their sizes. The restoration of the equilibrium distribution of atoms by diffusion (See Diffusion) is also a relaxation process. Manifestations of inelastic, or relaxation, properties, in addition to those mentioned, are elastic aftereffect in pure metals and alloys, elastic hysteresis, etc.

The deformation that occurs in an elastic body depends not only on the external mechanical forces applied to it, but also on the temperature of the body, its chemical composition, external magnetic and electric fields (magneto- and electrostriction), grain size, etc. This leads to a variety of relaxation phenomena, each of which makes its own contribution to W. t. If several relaxation processes occur in the body simultaneously, each of which can be characterized by its own relaxation time (See Relaxation) τ i, then the totality of all relaxation times of individual relaxation processes forms the so-called relaxation spectrum of a given material ( rice. ), characterizing a given material under given conditions; Each structural change in the sample changes the relaxation spectrum.

The following methods are used for measuring voltage: studying the damping of free vibrations (longitudinal, transverse, torsional, bending); study of the resonance curve for forced oscillations (See Forced oscillations); relative dissipation of elastic energy during one period of oscillation. The study of solid state physics is a new, rapidly developing field of solid state physics and is a source of important information about the processes that occur in solids, in particular in pure metals and alloys that have been subjected to various mechanical and thermal treatments.

V. t. during plastic deformation. If the forces acting on a solid body exceed the elastic limit and plastic flow occurs, then we can talk about quasi-viscous resistance to flow (by analogy with a viscous fluid). The mechanism of high stress during plastic deformation differs significantly from the mechanism of high voltage during inelasticity (see Plasticity, Creep). The difference in energy dissipation mechanisms also determines the difference in viscosity values, which differ by 5-7 orders of magnitude (plastic flow viscosity, reaching values ​​of 10 13 -10 8 n· sec/m 2, is always significantly higher than the viscosity calculated from elastic vibrations and equal to 10 7 - 10 8 n· sec/m 2). As the amplitude of elastic vibrations increases, plastic shears begin to play an increasingly important role in the damping of these vibrations, and the value of viscosity increases, approaching the values ​​of plastic viscosity.

Lit.: Novik A.S., Internal friction in metals, in the book: Advances in metal physics. Sat. articles, trans. from English, part 1, M., 1956; Postnikov V.S., Relaxation phenomena in metals and alloys subjected to deformation, “Uspekhi Fizicheskikh Nauk”, 1954, v. 53, v. 1, p. 87; him, Temperature dependence of internal friction of pure metals and alloys, ibid., 1958, vol. 66, century. 1, p. 43.

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what “Internal friction” is in other dictionaries:

1) the property of solids to irreversibly absorb mechanical energy received by the body during its deformation. Internal friction manifests itself, for example, in the damping of free vibrations.2) In liquids and gases, the same as viscosity ... Big Encyclopedic Dictionary

INTERNAL FRICTION is the same as viscosity... Modern encyclopedia

In solids, the property of solids is irreversibly converted into mechanical heat. energy imparted to a body during the process of its deformation. V. t. is associated with two different. groups of phenomena of inelasticity and plasticity. deformation. Inelasticity represents... ... Physical encyclopedia- 1) the property of solids to irreversibly convert mechanical energy received by the body during its deformation into heat. Internal friction manifests itself, for example, in the damping of free vibrations. 2) In liquids and gases the same as viscosity. * * *… … encyclopedic Dictionary

Internal friction Internal friction. Conversion of energy into heat under the influence of oscillatory stress of a material. (Source: “Metals and alloys. Directory.” Edited by Yu.P. Solntsev; NPO Professional, NPO Mir and Family; St. Petersburg ... Dictionary of metallurgical terms

Viscosity (internal friction) is a property of solutions that characterizes the resistance to external forces that cause their flow. (See: SP 82 101 98. Preparation and use of construction mortars.)

INTERNAL FRICTION in solids - the property of solids to be irreversibly converted into mechanical heat.

energy imparted to the body during the processes of its deformation, accompanied by a violation of thermodynamics in it. balance. Voltage is one of the inelastic, or relaxation, properties (see.), which are not described by the theory of elasticity. The latter is based on the hidden assumption of quasi-static. the nature (infinitesimal speed) of elastic deformation, when the thermodynamics in the deformed body are not violated. equilibrium. At the same time, in the kl. a moment in time is determined by the value of the deformation at the same moment. For linear stress state. A body that obeys this law is called. perfectly elastic, M0

- static ideally elastic body corresponding to the type of deformation under consideration (tension, torsion). With periodic deformation of an ideally elastic body are in the same phase. When deforming at a finite rate, a deviation from the thermodynamic occurs in the body. balance, causing appropriate relaxation. a process (return to an equilibrium state), accompanied by dissipation (dissipation) of elastic energy, i.e., its irreversible transition into heat. For example, when bending a uniformly heated plate, the material expands when heated, stretched fibers cool, compressed fibers heat up, as a result of which a transverse temperature gradient arises, i.e. elastic deformation will cause failure. Equalization of temperature through thermal conductivity represents relaxation. a process accompanied by the irreversible transition of part of the elastic energy into thermal energy, which explains the experimentally observed damping of free bending vibrations of the plate. During elastic deformation of an alloy with a uniform distribution of component atoms, a redistribution of the latter may occur due to the difference in their sizes. Restoring the equilibrium distribution by also representing a relaxation. process. Manifestations of inelastic, or relaxation, properties, in addition to those mentioned, are elastic aftereffects in pure metals and alloys,

elastic hysteresis

and etc. I The deformation that occurs in an elastic body is determined not only by the external mechanical forces applied to it. II forces, but also changes in body temperature, its chemical. composition, external magnet. and electric fields (magneto- and electrostriction), grain sizes, etc. Rice. 1. Typical relaxation spectrum of a solid at room temperature associated with the processes: - anisotropic distribution of dissolved atoms under the influence of external stresses;- in the boundary layers of polycrystal grains; III- at the boundaries between twins;

This leads to a variety of relaxation options. phenomena, each of which makes its own contribution to V. t. If several relaxations occur in the body at the same time. processes, each of which can be characterized by its own relaxation time, then the totality of all relaxation times dep. relaxation processes forms the so-called. relaxation the spectrum of a given material (Fig. 1), which characterizes a given material under given conditions;

Each structural change in the sample is reflected by a characteristic change in relaxation.

spectrum

There are several phenomenological 1 theories of inelastic, or relaxation, properties, which include: a) the Boltzmann-Volterra theory of elastic aftereffect, which seeks such a connection between stress and deformation, which reflects the previous history of the deformable body: where the type of “memory function” remains unknown; b) the method of rheology, models, which leads to relationships like: 2 This linear differential deformation characterizes the time dependence and is the basis for describing the linear viscoelastic behavior of a solid body.

Rice. 2. Vocht's mechanical model, consisting of springs connected in parallel and piston in the cylinder filled with viscous liquid. 2 .

Rice. 3. Maxwell model with series spring connection 1 to

Where piston in cylinder The phenomena described by equations (1) are modeled mechanically. and electric diagrams representing the serial and parallel connection of elastic (springs) and viscous (piston in a cylinder with a viscous liquid) elements or containers and active resistances. Naib. simple models: parallel connection of elements, leading to dependence (the so-called Vocht solid body - Fig. 2), and sequential. connection of elements

(so-called Maxwell's solid body - Fig. 3). The path followed. and parallel connections of several. Vocht and Maxwell models with different values ​​of spring stiffness and coefficient.

Where - so-called module defect, or complete degree of relaxation; G) . The theory of high voltage, according to which the source of high voltage is the movement of dislocations, explains, for example, the decrease in high voltage with the introduction of impurities by the fact that the latter impede the movement of dislocations. This resistance to the movement of dislocations is often (by analogy with the viscosity of liquids) called. viscous. Voltage resistance in highly deformed materials is explained by the mutual braking of dislocations, etc. The following methods are used for measuring voltage resistance: a) study of the damping of free vibrations (longitudinal, transverse, torsional, bending); b) study of the resonance curve for forced ones; c) study of the attenuation of an ultrasonic pulse with wavelength . The measures of high voltage are: a) vibration decrement, where is the phase shift between stress and strain during elastic vibrations, the value Q

similar to electric oscillatory circuit; c) relative dissipation of elastic energy during one period of oscillation; d) width, where is the deviation from the resonant frequency, at which the square of the amplitude of forced oscillations decreases by 2 times. Diff. V.T. measures for small values ​​of attenuation () are related to each other: To exclude plastic.

deformation, the vibration amplitude during measurements should be so small that Q -1 did not depend on her.

The relaxation spectrum can be obtained by changing the cyclic frequency. fluctuations, and temp. In the absence of relaxation processes in the temperature range under study, the current temperature increases monotonically, and if such a process takes place, then a maximum (peak) of the current temperature appears on the temperature dependence curve at a temperature where

H-relaxation activation energy. process, - material constant, - cyclic. oscillation frequency. Internal friction in metals, 2nd ed., M., 1974; Physical acoustics, ed. W. Mason, trans. from English, vol. 3, part A - The influence of defects on the properties of solids, M., 1969; Novik A.S., Berry B., Relaxation phenomena in crystals, trans.

from English, M., 1975..

B. N. Finkelshtein

Internal friction in solids can be caused by several different mechanisms, and although they all ultimately result in the conversion of mechanical energy to heat, these

the mechanisms involve two different dissipative processes. These two processes are, roughly speaking, analogues of viscous losses and losses by thermal conduction during the propagation of sound waves in liquids.

The first type of process depends directly on the inelastic behavior of the body. If the stress-strain curve for a single vibration cycle has the form of a hysteresis loop, then the area contained within this loop represents the mechanical energy that is lost in the form of heat. When a sample undergoes a closed stress cycle "statically", a certain amount of energy is dissipated and these losses represent part of the specific dissipation due to vibration of the sample. As Jemant and Jackson showed, even in the case when the hysteresis loop is so narrow that it cannot be measured statically, it has a significant effect on the damping of oscillations, since in an oscillation experiment the sample can perform a large number of closed hysteresis cycles. The energy loss per cycle is constant, so the specific dissipation and, therefore, the logarithmic decrement does not depend on frequency. Jemant and Jackson found that for many materials the logarithmic decrement is indeed constant over a fairly wide frequency range, and concluded that the main cause of internal friction in these cases may simply be due to the "static" nonlinearity of the stress-strain relationship of the material. Similar results were obtained by Wegel and Walter at high frequencies.

It is possible to distinguish two types of viscous losses in solids, which qualitatively corresponds to the behavior of the Maxwell and Vocht models described in the previous paragraphs. Thus, when the load is held constant, it can lead to irreversible deformation, as in the Maxwell model, or the deformation can asymptotically tend to some constant value over time and slowly disappear when the load is removed, as occurs in the Vocht model. The latter type of viscosity is sometimes called internal viscosity, and the mechanical behavior of such bodies is referred to as retarded elasticity.

The interpretation of the effects of viscosity in solids at the molecular scale is not entirely clear, mainly because the types of microscopic processes that lead to mechanical dissipation

energy in the form of heat are still largely in the realm of guesswork. Tobolsky, Powell and Ehring and Alfrey studied viscoelastic behavior using rate process theory. This approach makes the assumption that each molecule (or each link in a molecular chain in the case of polymers with long molecular chains) undergoes thermal vibrations in an “energy well” formed by its neighbors. As a result of thermal fluctuations, from time to time there appears sufficient energy for a molecule to escape from the well, and in the presence of external forces, diffusion takes place, equal in all directions. The rate of diffusion depends on the probability of the molecule receiving sufficient energy to escape the well, and therefore on the absolute temperature of the body. If hydrostatic pressure is applied to a body, the height of the energy well changes, the rate of diffusion becomes different, but remains the same in all directions. Under uniaxial tension, the height of the well in the direction of the tensile stress becomes lower than in the direction perpendicular to it. Therefore, molecules are more likely to propagate parallel to the tensile stress than in the direction perpendicular to it. This flow leads to the transformation of the elastic energy accumulated by the body into random thermal motion, which on a macroscopic scale is perceived as internal friction. Where the molecules move as a whole, the flow will be irreversible and the behavior will be similar to the Maxwell model, whereas where the molecular links are entangled, the material behaves like the Vocht model and exhibits delayed elasticity.

If certain assumptions are made regarding the shape of the well of potential energy and the nature of the molecular groups that vibrate in it, it can be shown (Tobolsky, Powell, Ehring, p. 125) that the theory leads to mechanical behavior of the body similar to that described by spring-models. shock absorber discussed earlier in this chapter. This interpretation of the issue emphasizes the dependence of viscoelastic properties on temperature; Thermodynamic relationships can be derived from this dependence. The main disadvantage in applying the theory to real bodies in a quantitative sense is that the nature of the potential well for the bodies is largely a matter of conjecture and that often several different processes can occur simultaneously. However, this is still almost the only serious approach to a molecular explanation of the observed effects, and it provides a reliable basis for future development.

Losses occur in homogeneous non-metallic bodies largely as described above, and internal friction is related to the inelastic behavior of the material rather than to its macroscopic thermal properties. In metals, however, there are

losses of a thermal nature, which are generally more significant, and Zener considered several different thermal mechanisms leading to the dissipation of mechanical energy in the form of heat.

Changes in body volume must be accompanied by changes in temperature; Thus, when a body contracts, its temperature increases, and when it expands, its temperature decreases. For simplicity, we will consider the bending vibrations of the cantilever plate (tongue). Each time the tongue is bent, the inside heats up and the outside cools, so that there is a continuous flow of heat back and forth across the tongue as it oscillates. If the movement is very slow, then the heat transfer is isothermal and therefore reversible, and therefore no losses should occur at very low oscillation frequencies. If the oscillations occur so quickly that heat does not have time to flow across the tongue, then conditions become adiabatic and still no losses occur. During bending vibrations, the periods of which are comparable to the time required for heat to flow across the tongue, an irreversible conversion of mechanical energy into heat occurs, observed in the form of internal friction. Zener showed that for a vibrating reed the specific scattering is given by

And - adiabatic and isothermal values ​​of the Young's modulus of the material, - vibration frequency, - relaxation frequency, which for a tongue of rectangular cross section has the expression

here K is thermal conductivity, specific heat at constant pressure, density, thickness of the reed in the plane of vibration.

Bennewitz and Rötger measured the internal friction in German silver tongues during transverse vibrations. The results of their experiments are shown in Fig. 29 along with the theoretical curve obtained using equation (5.60). No arbitrary parameters were used in constructing this curve, and the agreement between theory and experiment is remarkably good. It is clear that in the frequency region around (approximately 10 Hz) thermal conduction in the reed is the main cause of internal friction. It can also be seen that at frequencies far from the experimental values ​​of internal friction are higher than those predicted by theory, and this indicates that other influences become relatively more important here. The longitudinal stress will be

produce similar effects, since part of the sample is compressed while the other is stretched, in which case the heat flow is parallel to the direction of propagation. Since the distance between the areas of compression and rarefaction in this case is equal to half the wavelength, the losses caused by this reason will be small at ordinary frequencies.

Fig. 29. Comparisons of internal friction values ​​for German silver plates during transverse vibrations, measured by Bennewitz and Roetger and obtained from the theoretical Zener relations.

The described type of heat loss occurs regardless of whether the body is homogeneous or not. If the material is heterogeneous, there are additional mechanisms leading to heat losses. Thus, in a polycrystalline material, neighboring grains can have different crystallographic directions with respect to the direction of deformation and, as a result, receive stresses of different magnitudes when the sample is deformed. Therefore, the temperature will vary from crystallite to crystallite, as a result of which minute heat flows will occur across the grain boundaries. As with the losses due to conduction during cantilever oscillations, there is a lower frequency limit when the deformations occur so slowly that the volume changes occur isothermally without any loss of energy, and there is also an upper frequency limit when the deformations occur adiabatically, so again no losses occur. The greatest losses occur when the applied frequency hits

between these two limits; the value of this frequency depends on the size of the crystal grain and on the thermal conductivity of the medium. Zener derived an expression for the frequency at which losses of this kind are maximum. This equation is similar to (5.61) and has the form

where a is the average linear grain size.

Randall, Rose, and Zener measured internal friction in brass specimens of various grain sizes and found that, at the frequencies used, maximum damping occurred when the grain size was very close to that given by equation (5.62). The amount of internal friction caused by these microscopic heat flows depends on the type of crystal structure as well as on the grain size, and increases with increasing elastic anisotropy of individual crystallites. Zener (, pp. 89-90) proposed that at very high frequencies, heat flow is almost entirely limited to the immediate vicinity of the grain boundary; this leads to a relationship according to which the specific scattering is proportional to the square root of the oscillation frequency. This result was confirmed experimentally for brass by Randal, Rose and Zener. At very low frequencies, on the other hand, heat flow occurs throughout the material; hence the relation is obtained according to which internal friction is proportional to the first power of frequency. The experimental results of Zener and Randal are in agreement with this conclusion.

There are two other types of heat loss that need to be mentioned. The first is associated with heat dissipation into the surrounding air; the rate of loss for this reason, however, is so small that it affects only at very low oscillation frequencies. Another type of loss may arise from a lack of thermal equilibrium between the normal Debye modes; these losses are similar to the damping of ultrasound in gases, caused by the finite time required for thermal energy to be redistributed between the various degrees of freedom of gas molecules. However, in solids the equilibrium between the different modes of vibration is established so quickly that internal friction caused by such a cause would be expected to be noticeable only at frequencies of the order of 1000 MHz. The theory of the phenomenon described above was considered by Landau and Rumer and later by Gurevich.

For polycrystalline metals, he studied internal friction caused by “viscous slip” at crystal boundaries. He conducted experiments on the damping of torsional vibrations in pure aluminum and showed that internal friction in this case

can be accurately calculated under the assumption that the metal at the crystal boundaries behaves in a viscous manner.

There are two other processes that occur in crystalline bodies during their deformations, which could lead to internal friction. The first of these is the movement of regions of disorder in crystals, which are called dislocations. The second process is the ordering of dissolved atoms when a voltage is applied; the latter occurs in cases where there are impurities dissolved in the crystal lattice. The role of dislocations in the plastic deformation of crystals was first considered by Oroven, Palaney and Taylor, and although it seems likely that the movement of these dislocations may often be a significant cause of internal friction especially at large strains, the exact mechanism by which elastic energy is dissipated is currently unclear (see Bradfield). The influence of impurities dissolved in the crystal lattice on internal friction was first considered by Gorsky and later by Snoek. The reason that the presence of such dissolved atoms leads to internal friction is that their equilibrium distribution in a stressed crystal differs from the equilibrium distribution when the crystal is unstressed. When stress is applied, the establishment of a new equilibrium takes time, so that the deformation lags behind the stress. This introduces a relaxation process, which plays an important role for oscillating stresses, the period of which is comparable to the relaxation time. The rate at which equilibrium is established depends very markedly on temperature, so this type of internal friction must be very sensitive to temperature.

A special case of internal friction has been discovered in ferromagnetic materials. Becker and Döring gave a comprehensive review of experimental and theoretical studies for materials of this type on the important application problem of the magnetostrictive effect in ultrasound excitation. It has been found that internal friction in ferromagnetic materials is much greater than in other metals, and it increases when they are magnetized; it also increases rapidly with temperature when reaching the Curie point.

A mechanism that weakens stress waves in solids, but which is not strictly speaking internal friction, is dissipation. This phenomenon occurs in polycrystalline metals when the wavelength becomes comparable to the grain size; Meson and McSkimin measured the scattering effect in aluminum rods and showed that when the wavelength is comparable to the grain size, the attenuation is inversely proportional to the fourth power of the wavelength. This dependence coincides with the one given by Rayleigh (Vol. II, p. 194) for the scattering of sound in gases.