Stress Waves in Solids by H. Kolsky

j . Sound Vib. (1964) i, 88-i lo

STRESS

WAVES

IN S O L I D S

H. KOLSKY Brown University, Providence, Rhode Island, U.S.A. (Received 19June I963) A large and growing number of original papers on both the experimental and the theoretical aspects of stress wave propagation is appearing in the scientific literature, and two international conferences solely concerned with the subject have been held during the last five years. The purpose of this paper is to review recent experimental and theoretical advances in the propagation of deformation waves of arbitrary shape through elastic and anelastic solids, and also to attempt to outline the problems on which present efforts are being directed and to predict probable lines of future development. INTRODUCTION The subject of wave propagation in elastic solids has a long and distinguished history. The early theoretical work is associated with names such as Navier, Poisson, Stokes, Rayleigh and Kelvin. At the beginning of this century, when the concept of an elastic ether was finally abandoned, interest in the whole field died down; and with the exception of seismological studies, which were largely concerned with the application of earlier theory to rather complex physical situations, very little new work was done. During the last two or three decades there has been a remarkable revival of interest in the field. A large and growing number of original papers on both the experimental and the theoretical aspects of the subject is appearing in the scientific literature, and two international conferences solely concerned with stress-wave propagation have been held during the last five years (l-Z). At the end of this paper, references are given to a number of books, articles and reviews (3-17) which have appeared in the last decade and which are concerned with various aspects of the subject. There are a number of reasons for this revival of interest, of which the most important are: first, the availability of experimental apparatus with which stress waves can be produced and detected; second, the practical engineering need for data on the behaviour of materials at very high rates of loading when stress waves are inevitably generated; third, the growing interest in the structure of solids, where high frequency stress waves provide a powerful experimental tool for investigating the microscopic processes which take place in a solid when it is deformed; and last (and unfortunately, not least) the universal increase in armament research where the investigation of the response of structures to large forces maintained for very short times plays such a prominent part. One class of investigation in which there has been considerable and continuing interest is that of the propagation of pulses of high frequency sinusoidal oscillations of very small amplitude. This has in fact become a subject of its own, known as Ultrasonics, and in the present article it is not proposed to consider in any detail either the many and important results on the structure of solids and fluids which have been achieved by the use of this technique, or the use of such pulse techniques for flaw detection in engineering components. T h e purpose of this paper is instead to review the recent experimental and thcoretical advances in the propagation of waves of larger amplitude and arbitrary shape through elastic and anelastic solids, to attempt to outline the problems on which present efforts are being directed and to predict probable lines of future development. 88

STRESS WAVES IN SOLIDS

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From a theoretical standpoint the subject falls naturally into two parts which correspond to whether or not the medium through which the stress wave is being propagated can be considered to obey Hooke's Law. The experimental techniques used are generally similar whether elastic or anelastic solids are being investigated, but the purpose of carrying out experiments is often quite different in the two types of investigation. Thus, in principle, any elastic wave problem can be treated theoretically when the values of the elastic constants of the medium are known. In practice very few such problems indeed can be solved, and experiments are carried out either to test the validity of approximate theories or to obtain information about the behaviour of systems which are far too complicated for theoretical studies even to be attempted. Such experimental investigations are then the dynamic counterpart of the experimental stress analysis of engineering structures. In the second type of investigation, where the solid through which the waves are being propagated cannot be assumed to obey Hooke's Law, experiment plays a different role as in nearly every case an "exact" theoretical treatment is of very limited validity since it depends on a mathematical model of the anelastic behaviour of the solid. Now, for many materials Hooke's Law is an extremely close approximation to the actual mechanical behaviour of the solid when the deformations are sufficiently small, and the law is of the same form, i.e. a linear one, for all elastic solids. In contrast, the various mathematical representations of anelastic behaviour which have been used are approximate ones, which describe specific solids, are of very limited validity, and often would appear to have no physical validity at all. Thus experimental investigations are generally the only possible method of determining the real behaviour of the materials, and theory can, at best, act only as a general guide to the type of behaviour which may be expected. Departures from Hooke's Law may manifest themselves in two ways: first, the stressstrain relation may depart from linearity and also be different for loading and unloading; such behaviour is characteristic of metals strained" plastically" beyond their proportional limit; second, the stress-strain relation may depend on the rate of straining so that the material behaves in a manner analogous to that of a viscous fluid; such materials are called viscoelastic, and viscoelastic effects are most clearly seen in high polymers such as rubber and the many different plastics. As will be described later, many such viscoelastic materials are "linear" in the sense that although they do not obey Hooke's Law, their behaviour can be described by linear differential equations involving the stress, the strain, and their derivatives with respect to time. This division into non-linear behaviour, which does not depend on strain-rate, and linear viscoelastic behaviour, which does not depend on amplitude, is convenient although often somewhat artificial. Under some conditions many anelastic materials behave both in a non-linear and in a time-dependent manner so that the stress-strain relation, or the constitutive equation, as it is often called, must be expressed in the form of a non-linear differential equation involving stress, strain and time and is often of considerable complexity. However, since on the one hand the mechanical behaviour of metals is generally not highly rate-dependent, and on the other hand most unfilled polymers and rubbers are linearly viscoelastic for small deformations, theoretical studies in which the non-linearity and the time-dependence are dealt with separately have considerable practical justification. Waves travelling in a rate-dependent solid are termed viscoelastic waves, whilst waves travelling in a solid which shows the phenomenon of yield, so that the stress-strain curve is concave towards the strain axis, are termed plastic waves. When the stress-strain curve is non-linear but curves towards the stress axis, i.e. becomes "stiffer" for large stresses than for smaller ones, the nature of wave propagation is radically different from that of either viscoelastic waves or of plastic waves. When a mechanical pulse is propagated through such a medium, the velocity of propaga-

90

ii. KOLSKY

tion is greatcst h)r those parts of tile wave whosc amplitudes arc greatcst, and conscquently the front of the disturbance becomes stccper and steeper :Is the pulse progresses through thc medium. The final shape of the wave is then determincd by dissipative processes which take place because of the intensc stress gradients produced in the wave front, and often also depends on the microscopic structure of the medium. Such waves, which are called shock waves, of course also occur in fluids. In discussing and describing recent work in thc field of stress waves in this articlc, the four types of problems outlincd above will be considered separately, viz: elastic waves, viscoclastic waves, plastic waves and shock wavcs. ELASTIC WAVFS Almost the only problems for which a complete wave solution is available are the following: (a) TIlE INFINITE ISOTROPIC ELASTIC SOLID

Only two types of wave can propagate, namely, dilatational or P waves which travel with the velocity ct = ~/[(k + ~G)/p] and distortional or S waves which travel with the velocity c2 = x/(G/p), k is here the bulk modulus, G the shear modulus and p the density. (b) TIlE SEMI-INFINITESOLID In addition to the P and S waves, Rayleigh surface waves, which travel with a velocity c, a little lower than that of S waves, are propagated parallel to the free surface. The ratio G/c2 depends on the value of Poisson's ratio for the material and for real materials must lie between o.874 and o.955. (C) TIlE INFINITE PLATE

Lamb (18) has obtained complete solutions for two types of plane waves which he has termed symmetrical and antisymmetrical waves. These are longitudinal and flexural in character respectively, and unlike the other waves considered above, these waves are dispersive, i.e. their velocity of propagation depends on their wavelength A, or rather on A/d, where d is the thickness of the plate. When A/d is very much greater than unity, symmetrical waves travel with the velocity .~v/{E/p(l -v2)}, where E is Young's modulus and v is Poisson's ratio. On the other hand, antisymmetrical, or flexural, waves, travel with vanishingly small velocities as A/d tends to infinity. When A/d is very much smaller than unity, both symmetrical and antisymmetrical waves travel with velocities which asymptotically approach % the velocity of Rayleigh surface waves. (d) CYLINDRICAL BARS OF INFINITE LENGTH Three types of wave can be propagated along such bars, namely: extensional waves, torsional waves and flexural waves. Although the solution of this problem was first produced by Poehhammer (i9) and Chree (2o) during the last century, it is only during the last twenty years or so that numerical calculations based on this solution have been made, or that the experimental and theoretical problems associated with the propagation have been considered in any sort of detail. The theory shows that, in general, all these types of waves are dispersive, and in addition that waves of any one wavelength may be propagated in a number of different "modes" along the cylinder. For longitudinal and torsional waves these modes are associated with the presence of nodal cylinders where one component of the motion vanishes. For longitudinal waves the nodal cylinders correspond to

STRESS WAVES IN SOLIDS

9x

the absence of motion in the radial direction, whereas for torsional waves they correspond to no motion at all; the annuli on opposite sides of the nodal cylinder rotate in opposite directions. For the fundamental mode extensional waves travel with velocities which depend on Poisson's ratio v and on the value of A/a, where a is the radius of the cylinder andA is the wavelength. When A/a >>I, the waves travel with the velocity X/(E/p)= Co. When A/a < I, they travel with the Rayleigh surface wave velocity G. For intermediate values of A/a they travel at phase velocities between Co and c,. Flexural waves in the fundamental mode travel at vanishingly small velocities as A/a tends to infinity and at G asA/a tends to zero. Torsional waves travel at constant velocity c2 = "v/(G/p) for all wavelengths so long as the propagation is in the fundamental mode. Higher modes, whoever, show dispersion. The theory of wave propagation in solid circular cylinders was extended by Ghosh (2 x) to consider wave propagation along hollow cylinders. This problem was also considered by Glebe and Blechschmidt (22), who obtained an approximate solution which is inaccurate at high frequencies. Hermann and Mirsky (23) and Greenspan (24) have evaluated the dispersion curves resulting from Ghosh's solution. The problem is of interest in that for very long wavelengths the hollow cylinder behaves as a rod, i.e. there is no dispersion, and the velocity of propagation is ~/(E/p); whereas, for very short wavelengths it behaves as a plate, and when the wavelength becomes short compared with the wall thickness of the cylinder, the velocity of propagation approaches the Rayleigh surface wave velocity c,. The extreme complexity of exact solutions in the theory of elastic wave propagation has led to a number of "approximate" theories being developed. Instead of solving the elastic equations for the prescribed boundary conditions these theories use physical considerations to determine the likely nature of the motion. The earliest of these was that due to Rayleigh and Love, who modified the simple theory of propagation of extensional waves along a rod to allow for the radial motion associated with the lateral expansions and contractions produced by the finite value of Poisson's ratio. Other notable approximate treatments are those due to Timoshenko (25) for flexural waves in beams where rotary inertia and shear deformation are taken into account, and the theory due to Mindlin and Hermann (26) for extensional waves in rods where allowance is made not only for radial motion but also for radial shear. This theory has recently been further developed by Mindlin and McNiven (27). An excellent review of the many other recent approximate and "exact" approaches to elastic wave propagation is given in the article by Miklowitz (i6). Although considerable theoretical work has been carried out on elastic wave propagation, many important questions, even for the very simple geometries capable of analytical treatment, remain largely unanswered. Thus, in the problem of wave propagation along an infinite elastic cylinder, the Pochhammer-Chree solution shows that propagation can occur in an infinite number of modes. How the energy of an arbitrary disturbance is divided between these separate modes cannot, in general, be determined theoretically. The theory for an infinite cylinder is based on the boundary condition that at the free cylindrical surface both the normal stress and the shear stress shall vanish. For a finite cylinder, however, we have the additional boundary conditions that the ends of the cylinder shall be free from stress. As shown in Love (28) the Pochhammer-Chree solutions do not correspond to such end conditions, and no exact solution for wave propagation in an elastic cylinder of finite length has so far been obtained. Another problem to which it is difficult to find a theoretical answer is the signalvelocity of a disturbance along a cylinder of finite length. In an unbounded elastic medium waves can travel with one of two velocities, the dilatational velocity ct and the distortional velocity

92

it. KOLSKV

C2. Consequently, when the end of a cylindrical elastic rod is struck, some energy" would be expected to travel along it at the velocity c~ since a dilatational wave will travel out from the point of impact, and a portion of the wavefr,nt (albeit a very small portion if the length of the cylinder is very much greater than its radius) will reach the opposite end without being reflected at the cylindrical boundary. Now the Pochhalnmer theory as normally presented shows that the group velocity of longitudinal waves is never greater than Co = ~/(E/p); and since co is considerably less than Q, the theory clearly, doesnot take into account such precursor waves. Adem (29) would appear to have been the first to point out that in addition to the progressive waves treated by Pochhammer, the equation can be satisfied by solutions inw)lving complex wave numbers. Such solutions correspond to waves which are attenuated exponentially with distance of travel. Although such solutions will have no relevance for an infinite rod, they mav well make it possible to satisfy, the condition of zero end stresses in a finite rod, and to account for the propagation of disturbances at velocities greater than co. Experimental work on testing the predictions of the Pochhammer-Chree theory and seeing to what extent modes higher than the first are excited in practice, has been carried out by' Davies (3 o) and subsequently by a number of other workers (31-33). In these investigations pulses produced by impacts or by the detonation of explosives at the end 120 I O. lOOt

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36

Time (microsec)

F i g u r e I. C o m p a r i s o n betnveen o b s e r v e d d i s p l a c e m e n t - t i m e curve in steel rod and ordinates o b t a i n e d by F o u r i e r s y n t h e s i s . E x p e r i m e n t a l c u r v e ; :- calculated o r d i n a t e s ; - - u n d i s t o r t e d pulse.

of elastic cylinders have been observed after they have travelled various distances. Various types of pulse-shape, and cylinders covering a range of diameters have been used, and the observations have all shown that, in practice, the fundamental mode is the one normally excited and the predictions of pulse shape based on this mode are found to be in excellent agreement with the observed results. Figure I shows a comparison, carried out by Hsieh and the author (33), between the predicted and observed pulse shape in an experiment in which a small lead azide charge was detonated at one end of a steel rod 0-2 5 in. in diameter and 9 in. long; the observations of displacement were made at the opposite end of the rod. On the same figure, the displacement-time curve corresponding to the initial pulse is shown. It may be seen that the theory predicts, and experiment bears out, a considerable lengthening of the pulse during propagation and the appearance of a series of high frequency oscillations at the end of the pulse. These result from the lower group velocities of high

STRESS WAVES IN SOLIDS

93

frequency components in the fundamental mode. No evidence of propagation in the higher modes was found in the experiments listed above. Some workers (34-36 ) claim to have found definite evidence for higher mode propagation, particularly at points close to the source of the stress pulse. This is where such effects are to be expected, since the group velocities of low frequency waves in the second and higher modes are very much smaller than for the fundamental mode. Experiments designed to measure the signal velocity for extensional waves in cylindrical bars have been carried out by Hughes, Pondrom and Mims (38), by Kolsky (39) and by Miklowitz and Nisewanger (34). These experiments have all shown that a small part of the pulse does travel with the dilatational velocity cl as would be expected on physical grounds. For cylinders where the ratio of length to radius is reasonably large (of the order of xo or more), however, the amplitudes of the precursor waves are extremely small. A considerable amount of experimental work has also been carried out on the propagation of flexural pulses along elastic bars (4o, 4Q. The results were found to be in good agreement with the predictions of the exact Pochhammer theory and of the Timoshenko (25) approximate treatment; the latter corresponds very closely to the exact theory for the fundamental mode. Ripperger and Abramson (4 I) claim to have found a signal velocity of cl for the propagation of flexural waves although, here again, the amplitude of the precursor was found to be very small indeed. Very little experimental work appears to have been carried out on torsional wave propagation in rods. Experimental work of Owen and Davies (42) and of Krafft (43) shows that no dispersion occurs and presumably all the energy travels in the fundamental mode. In these experiments Owen and Davies (42) produced the torsional waves by transverse impact of bullets on the side of the bar whilst Krafft (43) used axial impact with a rifled bullet; torsional waves were produced as a result of the frictional resistance of the bar to the rotational motion of the bullet. The physical reason for the extreme complexity of elastic wave propagation in a bounded elastic solid as compared with the relative simplicity of propagation in an unbounded medium, is associated with the nature of the reflection of an elastic wave at a free boundary. It may be shown--see for example (3)--that when a dilatational (P) wave is obliquely incident on a free surface, the conditions for vanishing normal and shear stresses at the boundary require that both a dilatational and a distortional (S) wave be reflected. The direction in which these two reflected waves travel obeys Snell's Law, i.e. the reflected P wave makes an angle equal to the angle of incidence; whereas the "refracted" S wave travels in a direction such that the ratio of the sines of the angles to the normal is equal to the ratio of the velocities of propagation of S and P waves in the solid, i.e. the "refractive index" is cl/c2. The amplitudes of the reflected P wave and refracted S wave depend on the angle of incidence and on the value of Poisson's ratio of the elastic medium. Similarly, when an S wave is incident on a free boundary, a reflected S wave and a refracted P wave are generated. If a disturbance is set up at a point in a bounded elastic solid, in general both a P and an S wave travel out from the point. As each of these waves reaches a free boundary, it is reflected and produces two waves; and these, in turn, travel through the solid and are reflected at its free boundaries to produce more waves. It may thus be seen that in any bounded solid an extremely complicated wave pattern is generated even after very few reflections have taken place. The simple theory of reflection at a free boundary is adequate at all angles of incidence less than ~r/2, although for wave trains of finite breadth, diffraction effects must be taken into account. For glancing incidence of a P wave, however, the theory breaks down, since the simple treatment predicts a reflected P wave, travelling parallel to the surface, which

94

11. KOLSKY

is of equal amplitude and opposite sign to the incident wave. Thus all motion in the medium ceases, and we have the trivial solution of zero stress and zero particle velocity everwvhere. The theory of reflection is based on the concept of a train of waves of infinite breadth and of infinite duration, and equilibrium between the incident and reflectcd waves is assumed to have been set up. The practical problem of reflection at glancing incidence, however, is concerned with the transient conditions which obtain when a dilatational wave reaches a boundary parallel to its direction of travel, and this problem has been considered by Sauter (44), Goodier and Bishop (45) and Rocslcr (46). Sauter's analysis is the most thorough and takes into account diffraction effects, which depend on the wavelength, as well as on the geometrical conditions of reflection. The expressions he derives are, however, extremely complicated and difficult to apply to real physical situations. Goodier and Bishop (45) and Roesler (46) have treated the problem by using a limiting process for the reflection of waves as the angle of incidence approaches 7r/2. These analyses show that an S wave is generated, and the amplitude of the incident P wave is attenuated as it progresses along the boundary. Schardin (47) and Christie (48) have shown experimentally that such a trailing S wave is in fact produced. The experimental technique employed by these investigators was to propagate waves in plates of glass and other transparent elastic materials, and observe the reflection at glancing incidence by means of a photoelastic technique; a series of highspeed photographs of the photoelastic pattern was taken while the pulse was being reflected. Christie (48 ) was able to make some quantitative measurements of the intensity of the reflected S wave at very large angles of incidence and at glancing incidence, and he found his experimental results were in good agreement with the numerical predictions of the treatment developed by Roesler. Other experimental investigations on elastic wave propagation which should be mentioned are those carried out to simulate seismological situations. The specimens employed have been either large blocks of concrete and granite (49-5o), where the threedimensional problems could be tackled directly, or plate specimens (49--52)- As shown by Bishop (53), the propagation of waves in plates and two-dimensional propagation in infinite solids are similar except for the numerical values of the elastic constants. Most of this model seismological work has been carried out by the use of piezoelectric or ferroelectric crystals as generators and receivers. The technique has differed from that normally used in ultrasonics in that the crystals are pulsed by a high transient electrical d.c. potential (> mooV) for a few microseconds, and individual pulses rather than sinusoidal wave trains are generated. To ensure this, the crystals employed are ones whose natural frequencies are remote from the predominant Fourier components in the voltage pulse, so that "ringing" does not occur. Sherwood (52), in model seismological studies with plates, used small explosive charges to generate the stress pulses, and condenserunits to detect and measure them. It will be seen that most of the experimental work carried out on elastic waves hitherto has been concerned with studying propagation in specimens of comparatively simple geometrical shape, where the experimental results could be compared directly with exact or "approximate" theoretical predictions. With increasing confidence in the experimental techniques and in the interpretation of the observations, it should be possible to study more complicated problems. One such problem that has been tackled expcrimentally by Ripperger and Abramson (54) is the passage of longitudinal and flexural elastic pulses across the discontinuity in cross-sectional area between two cylindrical bars of different diameters. This problem, which involves discontinuous boundary conditions in stress and displacement at the junction between the two bars, cannot be treated theorctically with any pretence of rigour. It is a good example of the type of

STRESSWAWS m SOLIDS

95

problem, the solution of which is likely to be of practical importance, but which only experiment can hope to elucidate. In addition to studying more complicated geometries, future work in elastic wave propagation is likely to be concerned both theoretically and experimentally with the diffraction phenomena. These occur when a mechanical pulse starts to be propagated in a specimen, the lateral dimensions of which are of the same order of magnitude as the wavelengths of the dominant components of the Fourier spectrum of the pulse. As in optics or acoustics, a ray theory then becomes inadequate, and we enter the area which for light is termed physical optics in contrast with geometrical optics. In the propagation of elastic waves this generally involves the transition between conditions of plane strain and conditions of plane stress; and when neither of these can be expected to apply, the problem is one of considerable complexity even for specimens of very simple geometrical shape. VISCOELASTIC W A V E S

W h e n a specimen of any real solid is taken round a stresscycle, some of the mechanical energy is lost as a result of dissipative mechanisms in the material. T o this extent the concept of elasticwave propagation, as considered in the previous section isan idealization, and the equations of motion should always include terms corresponding to dissipative processes in the material. In practice, however, for most crystalline materials and for many amorphous materials, such as glass, when the deformations are sufficientlysmall the losses due to internal friction are extremely small; and the experimental results on wave propagation are found to be in excellent agreement with the assumption of linear elastic behaviour. In contrast, many of the materials which are composed of giant organic molecules and are known as high polymers, show very large internal dissipation. With the growing engineering importance of these materials, which include plastics and rubber, considerable effort has been directed in recent years to the study of the dynamic behaviour of structures composed of such materials, and also to the problems of stress wave propagation through such viscoelastic solids. In addition to the high mechanical dissipation that these materials exhibit, they also show a very marked dependence of the values of the elastic" constants" on the rate of application of stress. Thus, Young's modulus for a metal like steel will be found to have substantially the same value, whether the measurements are carried out in a conventional testing machine, or are obtained by measuring the velocity c of an elastic pulse along a thin wire of the material and using the relation E = pc2. A similar comparison for a filament of natural rubber will show the value of E in the wave experiment to be considerably greater than that found in the testing machine. Under some conditions the "dynamic" value obtained from the wave propagation experiment may be as much as a thousand times the "static" value obtained in the testing machine. This dependence of elastic modulus on time of loading means that from the point of view of wave propagation, the material is dispersive, i.e. sinusoidal waves of high frequency have higher velocities of propagation than low frequency waves. It is also found that the internal friction results in high frequency waves being attenuated more rapidly than waves of lower frequency. As a result of these two effects, i.e. velocity dispersion and frequency-dependent attenuation, the shape of a mechanical pulse as it is propagated through a viscoelastic medium changes rapidly even in the absence of the type of boundary effects discussed in the previous section. As mentioned in the introduction to this article, the type of viscoelastic response on which most theoretical and experimental effort has hitherto been expended is that termed linear viscoelastic behaviour. A linear viscoelastic solid may be defined in a number of

96

ii. KoI.SKY

different ways which can be shown to be mathematically equivalent (55). One definition is that it is a material for which the stress components and tile strain components are related by linear differential equations which involve tile stress, the strain, and their derivatives with respcct to time. Thus when we consider uniaxial deformations, so that only one component of stress a, and one component of strain E, are relevant, the stress-strain relation may be written as Po-=

QE

(I)

where P and Q are linear differential operators so that

i' = ao+al D + a 2 D 2 . . . . and

Q = bo+blD+b2D 2.... D is here the operator d/dt and the a's and b's are numerical constants. Historically, a linear viscoelastic solid was first defined by Boltzmann (56), in terms of his principle of linear superposition of time-dependent mechanical response. In Boltzmann's treatment it is assumed that the mechanical response of a material depends on its entire previous loading history ; and that if a specimen is subjected to a number of separate deformations, the stresses at subsequent times will be the simple algebraic sum of those which would have been found to be acting if the deformations had been carried out singly. Alternatively, if the specimen is subjected to a series of stress cycles, the total resulting deformations will be simply the sum of the individual deformations produced by the separate stress cycles. Mathematically, the principle can be written for uniaxial stress and strain as either t

= .I f ( t - r) --

dr,

(2)

ts3

w h e r e f ( t - r ) is called the stress relaxation function, and a and E are the stress and strain respectively, or as t

E=

f g(t-r)(do/dr)dr,

(3)

• zo

where g ( t - r) is termed the creep function. Boltzmann's treatment is physically perhaps the most satisfactory approach to linear viscoelastic behaviour, and the stress-relaxation function can be measured experimentally by observing how the stress varies with time at constant extension, whilst the creep function can be measured by observing the extension at constant stress. This is in contrast with the operator functions P a n d Q in equation (i), which have to be fitted to experimental results, a sufficient number of a and b terms being included to give a reasonable fit. It may be shown that where there are a finite number of terms in the operator equation, the behaviour can be represented by a mechanical model consisting of springs which are perfectly elastic, and viscous elements, called dashpots, for which the stress is proportional to the time rate of change of the extension, i.e. springs which are Hookean and dashpots which obey Newton's Law of Viscosity. One simple model of this type is the Maxwell model, which consists of a spring in series with a dashpot. For this model P = a0 + al D (and all the other a's are zero) and Q = bl D (and all the other b's are zero). Another simple model is the Kelvin-Voigt model which consists of a spring connected across a dashpot. For this model P is a constant and Q = bo+biD. It is found (57) that real viscoelastic solids, such as rubbers and plastics, behave in a much more complicated manner and can be described by such simple models only when

STRESSWAVESIN SOLIDS

97

a narrow spectrum of loading times is involved, i.e. when the ratio between the longest time and the shortest time that is relevant is not more than, say, so. The disadvantage of the integral representation as derived from the Boltzmann superposition principle is that it leads to difficult integral equations when k is applied to specific problems. The difficulties with the use of the operator method are first, that finding the appropriate operators involves considerable computation and curve fitting, and second, that if a large range of times has to be covered, the P and Q operators must each contain a considerable number of terms. This means that in trying to tackle a specific problem one may be faced with differential equations of the fifth, sixth or higher order, and these may prove quite intractable. There is another approach to linear viscoelastic behaviour, namely the use of complex elastic moduli. This again mathematically is equivalent to the methods outlined above and hence inherently involves many of the same difficulties, but for many purposes, and particularly wave propagation, it is extremely convenient. The computations which are required when this method is applied to specific problems, are generally in the form of the summation of Fourier series, and electronic computer programmes, as well as numerical methods which use hand machines, are readily available for carrying out computations of this type. When a linear viscoelastic solid is subjected to a stress o that varies sinusoidaIIy with time at an angular frequency p, the strain • produced will vary sinusoidally with time at the same frequencyp; but in general there will be a phase lag ~ between the strain and the stress. At any one frequency the phase lag 8, and the ratio between the stress amplitude o0, and the strain amplitude •0, will be constants independent of the absolute values of the amplitude. Using complex notation we may write = o0 exp (ipt), and so that or

e = •0 exp (ipt- i8), o/• = (o0/e0) exp (i~) = E1 + iE2, a = (Et + iE2)•.

(4)

Here E, and E2 are termed the real and imaginaryparts of the complex modulus E. It may be seen that gz/g, = tan ~ and that o0/•0 = ~/(E~ + E~); this ratio is often denoted by E*. Now E* and tan8 are quantities which can be obtained directly from vibration experiments and, if their values are known over a sufficiently wide range of frequencies, the mechanical response of the solid to any uniaxial mechanical deformation can be calculated by Fourier synthesis. We now return to the problem of wave propagation in such a linear viscoelastic solid. If we consider the simplest type of propagation, namely a longitudinal wave travelling in the x direction along a thin rod, where the effects of lateral inertia may be neglected, Newton's second law of motion gives the relation

ao/ax

=

;(a2u/at2),

(5)

where p is the density of the rod and u is the displacement in the x direction. Equation (5) will hold whatever the stress-strain relation may be. For a stress which is varying sinusoidally with time in a linear viscoelastic solid, we may use relation (4), so that (5) becomes

(Et + iE2) (0 2 u/ax 2) = p(a 2 u/Ot2), since

• = au/ax.

(6)

98

H. KOLSKY

The solution of (6) for a sinusoidal displacement u ~ u . e x p ( i p t ) at the origin x = o and a wave travelling in the direction of increasing ,v is u = u,,exp[-oa'+ip(t

- x c)],

where

,"-=

(E*;>)'Zsec(8/2), !

and

~

(p/c) tan

:=

[(E2/E , = tan8

,

(a/z);

and

E*=

(7) (s)

i

v'(E~+E~)].

Thus, if we know the value of E I and E2, or of E* and tall 8 at any frequency p, equations (7) and (8) give us the phase velocity c, with which such a sinusoidal wave is propagated, and the coefficient ~, with which the amplitude of the wave is attenuated with increasing x as the wave progresses along the rod. Now since the principle of superposition is applicable, any disturbance travelling along the rod may be considered as the sum of a spectrum of Fourier components, and the propagation of the pulse can be represented as a Fourier integral. Thus we can express the displacement pulse produced at the origin, by the relation u = j Ap exp (ipt) dp, O

where Ap is in general complex; then at any distance x along the rod we have u = j A, exp [ - c~x + ip(t - x/c)] dp.

(9)

O

and c are here of course functions o f p as given by equation (8). In order to solve the specific problem of a mechanical pulse propagated along a viscoelastic rod, where the shape of the pulse is known at one end of the rod, we must first make a Fourier analysis of the initial pulse. The cosine and sine terms in this analysis give the real and imaginary parts respectively of Ap in equation (9). We must also know the values of c and ~, or E* and tan8, over the frequency range covered by the relevant Fourier components (for a reasonably smooth pulse fifty Fourier components give an extremely close fit, and the viscoelastic properties need to be known over a range of frequencies of this order). The Fourier integral in equation (9) can then be approximated by a Fourier sum, and the shape of the pulse at any distance x along the rod can be computed. The author (58) has carried out a number of such computations, and the results of these were compared with experimental measurements of pulse shapes in viscoelastic rods. Figure 2 shows such a comparison between theory and experiment. In this experiment the pulse was produced by a small explosive charge detonated at one end of a polyethylene rod 6o cm long and 1.25 cm in diameter. The shape of the pulse arriving at the other end of the rod was measured with a condenser unit. It may be seen that the shape computed from the response of the material to stresses which vary sinusoidally with time agrees well with that observed in this experiment. In the experiment a certain amount of geometrical dispersion was produced as a result of lateral inertia effects in the bar. These were discussed in the previous section. It was found, however, that this could be allowed for by carrying out the computations for a value of x slightly greater than the actual length of the bar. In the treatment described above, the experimentally measured values of c and c~, or of E* and tan3, have to be inserted in the computation, so that the measured properties of the specific solid under investigation have to be used. Now, for a large class of viscoelastic solids over a considerable range of frequencies and for a range of temperatures

99

STRESS W A V E S I N SOLIDS

thc value of tan3 isfound to be comparatively independent of frequency. This isadmittedly only an approximation, the validityof which is more justifiedfor some solids than for others. If, however, we assume that tan3 is in fact independent of frequency and is small compared with unity, it may be shown (I I, 5 8) that the modulus E* at frequency p can be approximated to by the relation E'=

log,(p/po)],

E~[z4 2ta__n3zr

where Eg is the value of E* at some reference frequency P0. From equation (8) we then have

,,o, and ~ -- P [I - tan ~r 3 l°g* (P/P°)] tan ~ ' and, since tan3 < I, "~"

Kp,

(II)

where K = tan3/2Co.

950

900

850

'+l o

-6

IrO Log(#po)

I 20

30

po= 2 ~ x 103 cydes/sec

Figure 2. Velocity and attenuation for polyethylene rods. Curve I: Phase velocity (m/see); Curve 2 : Log ct. Thus to a first approximation we should expect to find that c plotted against logp should give a straight line of gradient (tanS)fir, and log~ plotted against logp should give a straight line of gradient unity. Figure 2 shows such plots for the experimental results obtained by Hillier (59) for sinusoidal waves of different frequencies travelling along filaments of polyethylene, and it may be seen that these results agree very well with the predictions of the above approximations. If we take equations (IO) and (i x) to represent the propagation constants of a class of viscoelastic solids, it is possible to treat in non-dimensional form a number of problems concerned with the propagation and reflection of waves in viscoelastic solids. Such calculations have been shown to lead to predictions which are borne out by experiment

(58).

IOO

11. KOLSKY

T h e velocity dispersion produced by the viscoelastic properties of solids leads to a nuxnber of interesting phenomena which are observed when mechanical pulses are propagated through them. Thus, as seen above, both the velocity of propagation and the attenuation increase with increasing frequency, so that the high-frequency components of a pulse travel faster and are attenuated more rapidly than those of lower frequency. As a result, a pulse, which is initially symmetrical in shape, becomes asymmetrical (Figure 3) and at the same time becomes broader and flatter as it travels through the solid. !

;

. . . . . . . . . . . . . . . . . . . . . .

c >.

b

~

~5.-v

_,

o -~

.

E <

0

I00

200

300

400

50C

Line (microsec)

Figure 3. Comparison between observed pulse ill polyethylene rod and ordinates calculated by Fourier synthesis. - - observed pulse ; o points obtained by Fourier synthesis. Another point of interest concerns the " s h a p e " of a pulse travelling in a viscoelastic medium. In an elastic solid the displacement in a plane wave travelling in the direction of decreasing x can be written as u = f(x + ct), (z 2) w h e r e f is an arbitrary function of the argument (x + ct). T h e particle velocity v is then given by v =

(I3)

~- c / ' ( x + c t ) ,

~u/~t

where the dash indicates differentiation with respect to the argument (x + ct); the strain E is given by =

I-O I-

,

~u/'?:x

,

. . . .

=/'(x-'-ct), _._~_.

11

-400

-200

0

.

.

(z4) ,

-=.

7o

200

400

Time (mlcrosec)

I'igure 4. Comparison of the pulse shapes of displacement, particle velocity, stress and strain in a viscoelastic rod.

STRESS WAVES IN SOLIDS

IOI

and the stress by o = E,, where E is the appropriate elastic modulus, so that o

(IS)

= Ef'(x+ct).

From (13), (14) and (iS) it may be seen that v, c, and a are allof the same pulse shape, and the displacement is proportional to the integral of these pulses. For viscoelasticsolids this is no longer true since c and E are time-dependent, and the pulse shapes will not be the same for stress,strain,and particlevelocity. From an experimental point of view this can lead to difficultiessince it is often possible to measure only strain or particle displacement when, for example, a knowledge of the stress is required. Furthermore, in some experimental techniques, notably the photoelastic technique, it is not quite clear whether the pulse shape observed is that of stress or that of strain. For elastic studies this does not matter once the detectors have been calibrated; but when viscoelastic materials are used, and much photoelastic work is done with such materials, it is not only the amplitude but the shape of the pulse which is suspect. Fortunately, for smooth pulses the differences in pulse shapes are generally not too large, as may be seen from Figure 4, which shows the results of a recent calculation carried out by S. S. Lee and the author (6o) on pulse propagation in polyethylene. The

g >. o 35

2

;~

ioo

~

/~' 20~, t-

s

Time (microsec)

Figure 5. Reflection of pulse (from polymethylmethacrylate rod to elastic rod). o r incident pulse ; o, reflected pulse.

pulse shapes for v, Eand o are calculated for a prescribed distribution of the displacement u. In the Figure the shapes are given non-dimensionally; the displacement is given as u/u ~o, where u~o is the value of the particle displacement after the pulse has passed, and the particle velocity, strain, and stress are given as V/Vo, c/¢0, and ~/o0 respectively, where the zero subscript denotes the maximum value of each quantity. Reflection of a pulse at the interface between two viscoelastic solids, or between a viscoelastic and an elastic solid, is also of interest, in that the reflection coefficient depends on the values of pc where p is the density and c is the velocity of propagation; and since 8

102

H. K O L S K Y

the latter varies with the frequency for a viscoelastic solid, the reflected and transmitted pulses can have shapes quite different from that of thc incident pulse. Thus, consider the reflection of a pulse travelling from a viscoelastic solid into an elastic solid which has a value of pc independent of frequency; let this value be poCo. Assume that the value of pc for the viscoelastic solid is less than poco for low frequencies and greater than poco for high frequencies. When a pulse is reflected at the boundary between two such solids, the low frequency components will be reflected in one phase and the high frequency components in exactly the opposite phase. The results of such reflection are shown in Figure 5, where it may be seen that the reflected pulse is S-shaped although the incident pulse was unidirectional. 'l'his calculation was carried out bv S. S. Lee and the author (6o) for the reflection of the pulse produced in a rod of polymethylmethacrylate (Perspex, Lucite) I ao cm in length, when a very sharp blow was applied to one end of it. The resultant pulse was reflected at the other end, which was in contact with an elastic rod of the appropriate Young's moduh, s. In the Figure the reflected pulse is shown at twenty times its actual size. All the problems of viscoelastic wave propagation that have been considered here were concerned with uniaxial waves so that only one component of stress and one component of strain were relevant. The extension of the treatment to three dimensions is, in theory, quite straightforward, and has an exact parallel in the development of threedimensional elasticity from the uniaxial treatment. However, instead of having two elastic constants, e.g. the bulk and shear moduli, we must now have two time-dependent relations. Thus, in the Bohzmann treatment, we would have two stress-relaxation functions or two creep functions, and in the differential operator treatment we would need a P and Q for shear and a P ' and Q' for dilatation. In fact, it seems highly probable that volume changes in all homogeneous viscoelastic solids are substantially elastic, and viscoelastic behaviour is confined to shear deformation. The experimental evidence on which this conclusion is based is, however, somewhat sparse, and more experimental work must be done before the assumption of perfectly elastic dilatational response can be regarded as universally valid. Wave propagation in non-linearly viscoelastic materials has hitherto received very little attention either theoretically or experimentally. Some efforts in this direction will be described in the next section, which discusses plastic waves, and also in the concluding section of this article. PLASTIC WAVFS Plastic waves are associated with the phenomenon of yield in solids, and the theory of their propagation has been developed chiefly in connection with the response of metal structures to large transient stresses which produce plastic deformation. If the stresses are applied very rapidly, inertia effects cannot bc neglected; and the problem has to be treated in terms of wave propagation. The first treatment of this type of problem was published by Donnell (6I) in i93o. Donnell considered the propagation of a longitudinal disturbance along a thin rod of a plastic strain-hardening material. The disturbance was produced by suddenly applying a constant stress a to one end of the rod and maintaining this stress. Donnell assumed for simplicity that the material had a bi-linear stress-strain curve, i.e. that the material was elastic for stresses up to the yield point %, the value of Young's modulus being El, and that for stresses greater than % the stress-strain curve had a second linear portion, the gradient of which was EE(E2 < EO. If a is less than the yield point %, a steep-fronted elastic pulse of constant amplitude a0 is propagated along the rod at constant velocity co = ~/(E1/p). If, however, a > a0, there

STRESS WAVES I N SOLIDS

IO 3

will be two wave fronts. The elastic wave of amplitude a0, the front of which is travelling at velocity Co, will be followed by a "plastic" wave of amplitude a which travels at the lower velocity .v/(E2/p); the stress distribution along the rod is thus in the form of two steps. The height of the first step is a0 and that of the second, a. After Donnell's paper, the subject of plastic wave propagation appears to have received little further attention until the beginning of the Second World War when formal solutions of the problem for more general stress-strain relations were obtained independently in England, the United States and the Soviet Union. (See references 3, 6, 9, [3, 62, 7I.) The problem can be treated in either Lagrangian or Eulerian co-ordinates, and both have been used in the various derivations of the solutions. We will here outline the problem as treated by yon Karman (62), who used Lagrangian co-ordinates. Von Karman considered an infinitely long wire, one end of which was suddenly given a velocity V at time t = o, and this velocity was maintained for all t > o. As in Donnell's treatment, it is assumed that the wire has a linear stress-strain relation up to a critical value of the stress ao, the yield point; for stresses a > a0, the stress o is assumed to be a univalued function of the strain E so long as a is increasing, and this function is assumed to be independent of the rate of straining. It is also assumed that the effects of radial motion of the wire can be neglected so that geometrical dispersion effects (of the type discussed in an earlier section of this article) could be ignored. The solution of this problem which satisfies the equation of motion and the prescribed boundary conditions is that, at any time t, the distribution of strain along the wire can be divided into three distinct regions. (i) From the end of the wire x = o to the "plastic wave front", x = C1 t, the strain has a constant value ~1. Here C1 = % / ( S 1 / p ) , where St is the tangent modulus of the engineering stress-strain curve at a strain cl. It is thus the quantity do'/dE at E = ~,, where a' is the nominal stress, i.e. the force applied divided by the cross-sectional area of the undeformed wire. (ii) From x = C1 t to x = Cot, the strain ~ decreases monotonically in such a way that x = Ct at every point where pC 2 = da'/d~( = S), i.e. each value of the strain travels down the wire with a velocity corresponding to the tangent modulus of the stressstrain curve at that value of the strain. Here Co= a/(E/p), where E is Young's modulus for elastic deformations. (iii) At x = cot the strain has a value c0. This is the maximum elastic strain that the wire can maintain and corresponds to the yield stress a0. For x > Cot, the wire is undisturbed. The relation between the constant value of the strain ~I between the end of the wire and the plastic wave front, and the velocity V at which the end of the wire is being extended is "t

v

=

--

f( \Pil

d,.

(I6)

O

S is the gradient of the stress-strain curve do/d~ and is a function of ~. The type of strain distribution described above is shown by the broken curve in Figure 6. The earliest experimental verification of the theory was carried out by Duwez and Clark (63), who measured the distribution of permanent strain in a copper wire which had been impulsively stretched by a falling weight. They found that the relation between the permanent strain E1 and velocity of stretch 1/"given by equation (]6) was well borne out by their experimental results, and the general distribution of strain along the wire was in reasonably good agreement with the theory. A more detailed examination showed,

I0 4

II. KOLSKY

however, some serious discrepancies between theory and experiment, and ira particular the length of region of constant strain, behind the plastic wave front, was found to be shorter than predicted, and even within this region the strain was not quite constant. These discrepancies were at first attributed to the complex wave patterns which arc set up during the unloading of the wire. A careful analysis showed, however, that the unloading waves could only account in part for these discrepancies, and that the inadequacy of the theory in explaining these results lay in the assumption that the stress-strain curve was independent of the rate of straining, and the use of the " s t a t i c " stress-strain curve of the wire was unjustified. The applicability of a theory of plastic wave propagation based on a rate-independent stress-strain curve was further placed in doubt by tile experiments of Bell (64), Sternglass and Stuart (65), and Alter and Curtis (66). These workers investigated the propagation of a small amplitude longitudinal disturbance along a bar which was already stressed beyond its elastic limit. According to the rate-independent theory as outlined above, such a disturbance should travel at the velocity v'(S/p), where S is the tangent modulus of the stress-strain curve at the pre-stress of the bar. Since the yield point had been exceeded, S was considerablv less than E, Young's modulus for the bar, and the superimposed disturbance would be expected to travel at a velocity considerably less than co, the velocity of elastic waves. Experimentally they found however, that the disturbance travelled in all oooa,