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The rheological properties of matter under high pressure
THE RHEOLOGICAL PROPERTIES OF MATTER UNDER HIGH PRESSURE P. W. Bridgman Harvard University, Cambridae, Mass. Received Nov. 21, I956 I NTRODUCTION
By the "rheological" properties of matter I shall understand those properties which play a role when a body receives a permanent alteration of geometrical configuration without fracture. Such permanent alterations of shape follow the application of shearing stresses. In the case of liquids, even the feeblest shearing stresses produce flow and the relevant property of the liquid is its viscosity. In the case of solids, permanent deformation or set does not occur until the shearing stress rises above some threshold value, and the relevant property of the solid may be designated by the generic name plasticity. Both viscosity and plasticity are affected by hydrostatic pressure. The effects of pressure are in many cases large--larger than for other ordinary phenomena--and we may accordingly expect a knowledge of the pressure effects to be of considerable significance in its suggestions as to the underlying mechanisms. In the following I shall be mainly concerned with either my own work or with work with which I have had more or less close contact. This will omit phenomena of undoubted significance, such, for example, as the splashing of a steel projectile on impact against an armor plate or the plastic distortion of the plate itself. M y reason for not treating such topics is both my own lack of familiarity with them and the poorly defined nature of the stress systems accompanying such phenomena. LIQUIDS This discussion of the effects in liquids will be rather academic in character, because I shM1 omit the effects which are perhaps most interesting from a practicM point of view and from the point of view of industry, namely, the effect of pressure on oils and other lubricants, and leave these to the discussion of Dr. Dow, who has experimented extensively in this field. M y own work in this field was done some years ago; discussion will be found ill my book, "The Physics of High Pressure," as well as in the original references. Some 45 different liquids were examined, usually over 7
P. W. BRIDGMAN
a pressure range of 12,000 kg./cm? and at two temperatures, 30 ° and 75°C. The method was, in most cases, a falling weight method. The time of fall of a weight through the liquid in question was determined electrically, and the apparatus was so mounted that it could be inverted and the measurement repeated a sufficient number of times to give the requisite accuracy. The liquids measured embraced the ordinary organic liquids whose other properties are commonly listed in tables, and included water. The only liquid metal was mercury. In every case, except for a temporary episode in water to be referred to later, viscosity increases with rising pressure at constant temperature. Furthermore, the increase is, in practically every case, at an accelerated rate with increasing pressure, so that the curve of viscosity against pressure is concave upward. In the case of a number of liquids sensible upward curvature does not begin until pressures of 2,000 or 3,000 kg./cm? are reached, so that the initial part of the curve is straight, as had been found by several early experimenters whose measurements were confined to this range. If the logarithm of viscosity is plotted against pressure, the curve obtained is, in the majority of cases, concave toward the pressure axis. The curvature is much the greatest at low pressures; above 2,000 or 3,000 kg./cm. ~ the curve approximates to a straight line in a little more than half the cases, while in the remaining cases it gently reverses curvature. This means that above 3,000 kg./cm. ~ viscosity either increases geometrically or else, even more rapidly as pressure increases, arithmetically. The range in numerical values for the effect of pressure is extreme; the effect of 12,000 kg./cm. ~ varies from an increase of 30% for mercury, to a two-fold increase for water, a ten-fold increase for methyl alcohol, a 100-fold increase for propyl alcohol, a thousand-fold increase for amyl alcohol, and an increase by 107-fold for eugenol. Contrasted with these figures are decreases in compressibility of not more than fifteen-fold brought about b y 12,000 kg./ era. 2, and decreases of thermal expansion by the same pressure of not more than two- or three-fold. There is a rather close correlation between the magnitude of the effect of pressure on viscosity and the complication of the molecule, the effect being least for monatomic mercury, and greatest for the complicated organic compound, eugenol. The effect of complication of the molecule is very plainly shown, for example, by the series of the alcohols above, or by the various compounds derived from benzene; the relative pressure effect is greater the more complicated the group substituted for hydrogen. There are also very marked constitutive effects, the iso- compounds, for example, having a larger pressure coefficient than normal compounds. Or a heavier atom substituted into a molecule produces, in general, a larger pressure effect, as shown by various series of halogen compounds. The effect of temperature combined with pressure is abnormal. Most
RHEOLOGICAL PROPERTIES AT HIGH PRESSURE
9
temperature coefficients become less at high pressures, but the temperature coefficient of viscosity becomes markedly greater with increasing pressure; for some substances the temperature coefficient at 12~000 kg./ cm. ~ is four-fold greater than at atmospheric pressure. The temperature coefficient of viscosity under pressure, when combined with the compressibility, gives important theoretical implications. Several of the earlier theories of the viscosity of liquids agreed in predicting that viscosity at constant volume should be constant. That is, if temperature is raised, decreasing the viscosity and increasing the volume, and if simultaneously the pressure is increased sufficiently to bring the volume back to the initial value, then the expectation was that the increase of viscosity brought about by the increase of pressure would exactly cancel the decrease brought about by the increase of temperature. This is approximately true for the first few thousand kg./cm. 2, but above that there may be significant and large departures. The variation is in such a direction that at constant volume the viscosity is less at the higher temperature. That is, the increased molecular agitation due to increasing temperature has a specific effect in decreasing viscosity apart from the effect on volume. The effects of pressure on the viscosity of water are abnormal, as are so many of its other properties. At low temperatures, between 0 ° and approximately 25°C., the initial effect of increasing pressure is to decrease viscosity instead of to increase it. At the low temperatures the curves pass through a minimum in the neighborhood of 1000 kg./cm. 2 and above this rise with upward concavity. Even at higher temperatures the total increase is abnormally low compared with other liquids. The effects can be qualitatively explained in water by well known considerations of the changing proportions of mono- and trihydrol. Within the last few years there has been considerable theoretical activity in accounting for the viscosity of liquids in general and the pressure effects in particular, improving on t h e earlier theories which demanded that viscosity be constant at constant volume. In general comment on any theory of the viscosity of liquids it is obvious that the fundamental mechanism must be different from that of gases. This is shown, for example, by the fact that in a gas the reciprocal of viscosity and thermal conductivity change together with pressure, whereas in a liquid thermal conductivity increases with pressure but reciprocal viscosity may decrease enormously. The most important of the theoretical work has been done by the Princeton school of physical chemists (1). The basic idea of this work is that there is a similarity between the molecular process by which momentum is transferred from one layer to another in a liquid flowing viscously and the molecular processes in ordinary chemical reactions. The molecule which moves along the velocity gradient in viscous flow receives
10
P. W. BRIDGMAN
or delivers energy. When it passes from a position in one layer to one in the next it is thought of as passing over a potential barrier before it can settle down into its new position, just as a molecule taking part in a chemical reaction has to pass over a potential barrier before it finds its new position of greater stability. The same sort of exponential formula governs the velocity of both processes, and there is an activation energy. The activation energy for viscous flow can be calculated from the kinetics of the liquid, and, in general, is considerably smaller than the energy of vaporization.' Ewell and Eyring, and also Ewell, develop this point of view, and obtain, among other things, expressions for the effect of pressure on viscosity which agree with my data up to 2,000 kg./cm3 in the early paper, and which were later improved to give agreement up to 7,000. This was followed by papers by Eyring and by Frisch, Kincaid and Steam, in which the concept of a maneuver space was introduced, which had to be available to the molecule if it is to be able to slip past its neighbor into a position of greater stability with respect to the viscous motion. This maneuver space is connected with various thermodynamic parameters. With a theory of this kind it is possible to reproduce my measurements on viscosity up to 10,000 kg./cm. 2 with only small error. Qualitatively the concept of maneuver space has points of resemblance to the concept of an interlocking between the molecules which I used to explain qualitatively some of the pressure effects, such as the strong dependence on complexity of the molecule. If the maneuver space is recognized to involve the shapes of the molecules, the connection between the two concepts becomes quite close. SOLIDS There is a much richer variety of phenomena for solids than for liquids, corresponding to the much greater range of kinds of deformation to which a solid may be subjected. On the theoretical side our understanding of the nature of the phenomena is, however, in a much more rudimentary state than in the case of liquids. There is practically no theoretical work on plasticity of solids which goes beyond pure description, and even this description is confined to the simplest aspects of the phenomena. For purposes of illustration we may take the simple description of the most elementary facts, as given by the yon Mises theory. According to this theory the substance is in the plastic state if a certain function of the stresses, namely, (X. -
Y~)~ q- ( Y , -
Z , ) ~ -k ( Z , -
X . ) 2,
(1)
exceeds a critical value. If the stresses do not exceed this value, the substance is in the elastic range. In the plastic range there is a connection between the stresses and the rate at which the strain varies with the time,
RHEOLOGICAL
PROPERTIES
AT HIGH
PRESSURE
11
that is, the rate of flow, as follows: Xx - ½(Yy + Z~) = Yy -- ½(Z, + Xx) = Z , - ½(X, + Yv)
(2)
The symbols in this equation denote the principal components of stress and of strain velocity, which are assumed to be along the same directions. It is usual to integrate these equations, and write an exactly similar set in which the dots in the strains above are omitted, that is, a set of equations connecting the strains instead of the strain rates with the stresses. These equations are recognized to be valid in only a narrow range just above the point of initial flow. They leave out of consideration a great variety of complicated effects, which are of the greatest practical importance, such as strain-hardening and various time effects. The beginning of an attempt has been made to include these other effects within the theory, but the mathematical difficulties are extreme and comparatively little success has been achieved. The effects of pressure on plastic flow which I have investigated (2) are included only very roughly within the scope of the simple equations given above, because in the pressure effects we are usually concerned with large deformations, whereas the equations are expressly restricted to small plastic deformations. Perhaps the simplest of the pressure effects has to do with the variation of behavior in simple tension. In order to study these effects a miniature testing laboratory was set up within the pressure chamber, and tensile tests conducted exactly as at atmospheric pressure, with measurements of tensile load and elongation. According to the elementary equations given above, the addition of a hydrostatic pressure to a given stress system should be without effect on the plastic behavior, either on the plastic limit as specified in Eq. (1) or on the rates of flow or the strains as specified in Eq. (2). Actually, a tensile test conducted under pressure may be dramatically different from one conducted at atmospheric pressure. The greatest difference for a substance like mild steel is the enormous increase of ductility produced by pressure. In one of my experiments a mild steel when pulled under a surrounding hydrostatic pressure of 25,000 kg./cm2 permitted an elongation of 300-fold at the neck without fracture, whereas at atmospheric pressure it broke with an elongation at the neck of approximately two-fold. A convenient way of measuring the ductility is in terms of the "natural" strain at fracture, that is, as loge A o / A , where A0 and A are the initial and final cross section at the neck. The natural strain at fracture for mild steel was found to be a linear function of pressure up to the limit of my experiments, 30,000 kg./cm ~. Another effect which is very prominent under pressure is strain-hardening, the existence of which is not even recognized in the elementary equations. It is known that at atmospheric pressure the strain-hardening curve, that is, the curve of "true stress"
12
P. w .
BRIDGMAN
against the equilibrium strain, which is the strain reached asymptotically after flow has ceased, starts out by rising with concave curvature t o w a r d the axis of strain, but after passing the point of maximum load where "necking" starts becomes straight and continues straight until the fracture point is reached at a strain of the general order of magnitude of unity. If the tension specimen is pulled under pressure a similar strain hardening curve is found, except that now the linear portion of the curve is much prolonged, out to natural strains of four or five, corresponding to the enhanced ductility imparted by the pressure. Furthermore, within the range of strain common to atmospheric and higher pressures, the strain-hardening curve at high pressure coincides with that at atmospheric. To a second approximation, however, strain-hardening at a higher pressure is greater than at a lower pressure. That is, for a given strain the true stress for the pulling under higher pressure is somewhat greater than for the pulling at lower pressure. At the lower end of the strain-hardening curve, before the curve becomes linear, it is possible to establish an effect of pressure. At high pressure a somewhat greater tensile stress is required to exceed the elastic limit and enter the plastic range than at lower pressures. That is, the "constant" to which the stress function in expression (1) is equal is an increasing function of pressure. This effect of pressure is not large compared with its effect on ductility, the general order of magnitude of the effect being a raising of the elastic limit by from 5 to 10% for a pressure of 10,000 kg./cm. 2. If a tension specimen is pulled under pressure to an elongation greater than the elongation at fracture at atmospheric pressure, but not to the fracture point under pressure, and the pressure is then released and the specimen pulled again at atmospheric pressure, it will be found that fracture does not occur at once, but further elongation is permitted, so that the strain at which a tensile fracture occurs is a function of the previous history and, in particular, a function of the pressure at which it was previously pulled. Theoretically this has most important implications, because it shows that the properties of a substance are not functions only of the state of stress and strain, but involve also the history. It is difficult to formulate this sort of thing in mathematical terms. It is obvious that a complete mathematical theory, even a purely descriptive theory, must be of a very considerable degree of complexity. The stress at fracture of a specimen previously pulled under high pressure and then broken at atmospheric pressure may be considerably higher than that of a virgin specimen broken a t atmospheric pressure. In this way an enhancement of properties may be produced which may be of some practical interest. The numerical parameters which determine the effect of pressure on the plastic properties of steel are a strong function of the condition of the
R H E O L O G I C A L P R O P E R T I E S AT H I G H P R E S S U R E
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steel as determined by the heat treatment. If the steel is heat treated so as to become harder, the slope of the line of strain at fracture as a function of hydrostatic pressure increases markedly, so that the ductility imparted by a given hydrostatic pressure becomes less the harder the steel. Even with the hardest steels a perceptible amount of ductility was imparted by the highest pressures (30,000 kg./cm.2), as contrasted with a completely brittle fracture at atmospheric pressure. The slope of the strain hardening line also increases markedly as the hardness of the steel is increased by heat treatment. Certain substances which are normally completely brittle to tensile stresses at atmospheric pressure may have a measurable degree of ductility imparted to them by high pressure. Thus Balsley (3), working with apparatus of Griggs, has produced a tensile elongation of 40% in limestone pulled under 10,000 kg./cm.2. In work as yet unpublished, I have found that ordinary glass tolerates no measureable permanent tensile stretch even up to 30,000 kg./cm. 2, but does exhibit a great increase of tensile strength. Thus under a hydrostatic pressure of 27,000 kg./cm. 2 pyrex glass supported up to 24,500 kg./em. 2 superposed tensile stress, whereas the tensile strength under normal conditions at atmospheric pressure is of the order of 500 kg./em. 2. Hydrostatic pressure has an effect on flow under simple compressive stresses as well as under tensile stresses. On mild steel these effects are not as dramatic as the tensile effects, because even at atmospheric pressure very large compressive strains may be supported without fracture if the strain is kept homogeneous. There is an effect of hydrostatic pressure on the stress at initial yield under simple compressive stress; this effect is similar to the corresponding tensile effect. The initial flow stress is raised by something of the order of 5 to 10% for 10,000 kg./cm. ~. Other substances, normally brittle for simple compression, may show much larger effects than steel. Thus carboloy, which is normally as brittle as glass, will support plastic shortening up to at least 10% when exposed to simple compressive stress superposed on a pressure of 30,000 kg./cm. 2. Glass may become enormously strong in compression. I have measured a compressive strength of 45,000 kg./cm. 2 in pyrex glass when immersed in a liquid carrying 25,000 kg./cm. 2, against a normal compressive strength of not more than 2,000 kg./cm. 2. The most striking effect was obtained with single crystal sapphire, made by the Linde Air Products Company by their new technique for producing long single crystal rods. Normally these single crystals break brittlely like glass under simple compressive stresses. One crystal was observed in which failure under simple compression took the form of slip without fracture along the basal plane of the crystal, exactly as in single crystals of a ductile metal. The conditions in this experiment were not altogether simple or well defined, there probably being microscopic internal imperfections in the crystal, but there would
14
P. W. BRIDGMAN
seem to be no reason for doubting the qualitative results that a degree of plasticity has been imparted by pressure to a crystal normally brittle. Other effects of hydrostatic pressure under somewhat more complicated conditions are qualitatively not different from what might be expected from the simple tension and compression experiments. Thus the Brinell hardness of steel, as measured by the penetration of a ball driven by a known load, increases under pressure by something between 5 and 10% for a pressure increment of 10,000 kg./cm?. Or, if a plate of mild steel is penetrated by a punch, the ductility may be enormously increased by the pressure. Thus, in one experiment the punch broke completely through the plate after a penetration of perhaps 20% of the thickness at atmospheric pressure, whereas under 25,000 kg./cm. 2 the punch was driven 95% through the plate with no loss of cohesion and with a degree of strain-hardening that allowed the support of a greater load than a piece of virgin metal of the same geometrical configuration. Another type of experiment under conditions which are not quite as well defined as in those just mentioned duplicates more nearly the conditions of pure laminar flow obtaining in liquids during viscous flow and permits examination of a wide range of materials. In these experiments (4) the material is in the form of a thin wafer a few thousandths of an inch thick pressed between a very short flat faced boss on a heavy block of steel and a plane steel block. Pressure is applied between boss and plate, forcing out the material of the wafer until an equilibrium thickness is reached corresponding to the pressure. Except at the very edges the material of the wafer is in a state of approximate hydrostatic pressure. The block and boss are then rotated relatively to each other, subjecting the material of the wafer to a shearing force at its faces. At low pressures, there is surface slip between material and steel, but when the pressure has reached a value so high that the friction at the interface equals the internal flow stress, surface slip ceases, and at higher pressures the material yields by internal laminar motion, exactly as in a liquid. By measuring the turning force as a function of pressure, it is possible to determine the stress for plastic flow as a function of hydrostatic pressure. These measurements can be carried up to pressures of 50,000 kg./cm. ~ with apparatus constructed of steel, and with carboloy apparatus have been carried to 100,000 kg./cm. 2. These high stresses are made possible by the geometrical design, in which the highly stressed region is highly localized, and surrounded and supported by much larger regions comparatively free from stress. A very wide variety of materials may be studied with this simple apparatus. Under these conditions there is no substance which does not differ very markedly from the idealized yon Mises plastic solid to which Eq. (1) applies. In all cases the shearing stress at which plastic flow occurs is a rising function of pressure. In the pressure range up to 50,000 kg./cm. ~
RHEOLOGICAL PROPERTIES AT HIGH PRESSURE
15
the rise may be by a factor no more than two in the case of some of the metals or may be thousands of fold in the case of substances like paraffine. Very large deformations may be required to reach a steady state, that is, a constant invariable ratio between shearing stress and pressure; shearing deformations of as much as 100 radians sometimes were necessary. Under these very large deformations the original crystal structure of many of the materials is so highly broken down that very little remnant of structure is detectable by X-ray analysis, and, in fact, the material in some cases appears to be reduced to a true glass (5). It is evident from these results that ordinary materials are capable of existing in an almost infinite number of states of aggregation intermediate between fully crystalline and amorphous, and that, accordingly, a great variety of physical properties is to be expected, depending on the degree of work hardening. The conditions of these shearing experiments are different in an important respect from those of ordinary testing, in that, when fracture occurs, the fractured surfaces are not separated but are maintained in close contact by the pressure. Fracture under these conditions is not a run-away phenomenon, but self-healing often occurs. A very common pattern of fracture is for the material t o suddenly let go, after a critical amount of shearing distortion, with catastrophic decrease of shearing stress. The material then self heals, and as shearing distortion continues, the shearing stress builds up again to its former value, and the cycle of fracture and recovery may be repeated indefinitely. There are sometimes other patterns of fracture. Thus, beyond a certain degree of distortion fracture and self healing may be a practically continuous process, the shearing stress remaining at a constant value while the shearing process is accompanied by chattering or by grinding noises. Internal fracture is confined almost exclusively to non-metallic substances and among them to those which do not crystallize in the cubic system. Cubic systems are in general richer in cleavage or shear planes than other systems, so that there is more opportunity for slip on one plane or another even after extreme distortion. Not only is there self healing after internal fracture, but there is often cold welding between the substance and the steel parts. Thus borax glass is as tightly welded to the steel parts after shearing in this way as if it had been fused to the steel. This type of shearing experiment shows well what is perhaps the most important difference between the viscous flow of a liquid and the plastic flow of a solid. The shearing force required to maintain a liquid in laminar shearing motion is proportional to the velocity of shear, whereas in the case of a plastic solid the shearing stress varies little with the velocity of shear. Different solids vary in this respect over a wide range. Thus, in the cases of tin and lead, the two metals which show the greatest de-
16
P. W. BRIDGMAN
pendence of force on speed, a ten-fold increase in force produced a thousand-fold increase in speed. With mica, on the other hand, an increase in speed of ten-thousand-fold was produced by an increase in force too small to measure. With a substance like mica, the apparatus would obviously be unstable if one attempted to impose a constant force during the shearing process. Measurements on such substances must be made by imposing a constant velocity and then allowing the force to come automatically to its equilibrium value. It is the first task of a physical theory of the process of plastic flow, as distinguished from the purely descriptive theory, which is all we have at present, to account for this striking lack of dependence of velocity on stress. It would appear that in broad outline the forces which are active during the plastic flow of a solid are configurational in character, depending on the relative locations of atoms and molecules, as contrasted with forces which are dynamic in origin, arising from transfer of momentum on a macroscopic scale, which are primarily concerned in the viscous motion of a liquid. I have made a suggestion along these lines (6), which may express the germs of the essential situation. In an idealized lattice structure a steady "state of slip" is a possibility, in which one part of the crystal moves relatively to another by slip along one of the lattice planes. Once this motion is es~blished it would persist with no force to maintain it, since, on the average, the net work done by the atomic fortes during a relative motion by one atomic step would vanish. In actuality, of course, lattices are not perfect, and furthermore are disarranged by relative slip, but in any actual lattice there must be a very appreciable idealized component which may well be the determining factor in the relative independence of velocity of force. A somewhat similar consideration applies also to amorphous substances, for even in a noncrystalline substance such as glass, there is evidence in the details of the shearing experiments that some sort of structure is imposed by the process of shearing itself, and this structure may well function on the average sufficiently like the perfect lattice to account for the small effect of velocity on force. REFERENCES I. ]~WELI~R. H., AND :EYRING,H., J. Phys. Chem. 5, 571, 726 (1937);Ewv.LL, R. H., J. Applied Phys. 9, 257 (1938); FRISCH, D., EYRINO, H., AND KINC~U, J. F., J. Applied Phys. 11, 75 (1940); KINC.~ID,J. F., EraI~G, H., ~ D STEARN, A. E., Chem. Rev. 28, 301 (1941). 2. BRIDGM~, P. W., Rev. Modern Phys. 17, 3 (1945); J. Applied Phys. 17, 201, 225, 692 (1946). 3. B~si~Y, J. R., Trans. Am. Geophys. Union 1941, Part II, 519. 4. BR1DGMAN,P. W., Proc. Am. Acad. Arts Sci. 71, 387 (1937). 5. LARSEI~,E. S., AND BRIDGMAN,P. W., Am. J. Scl. 36, 81 (1938). 6. BRmGM~, P. W., J. Applied Phys. 8, 332 (1937).
By the "rheological" properties of matter I shall understand those properties which play a role when a body receives a permanent alteration of geometrical configuration without fracture. Such permanent alterations of shape follow the application of shearing stresses. In the case of liquids, even the feeblest shearing stresses produce flow and the relevant property of the liquid is its viscosity. In the case of solids, permanent deformation or set does not occur until the shearing stress rises above some threshold value, and the relevant property of the solid may be designated by the generic name plasticity. Both viscosity and plasticity are affected by hydrostatic pressure. The effects of pressure are in many cases large--larger than for other ordinary phenomena--and we may accordingly expect a knowledge of the pressure effects to be of considerable significance in its suggestions as to the underlying mechanisms. In the following I shall be mainly concerned with either my own work or with work with which I have had more or less close contact. This will omit phenomena of undoubted significance, such, for example, as the splashing of a steel projectile on impact against an armor plate or the plastic distortion of the plate itself. M y reason for not treating such topics is both my own lack of familiarity with them and the poorly defined nature of the stress systems accompanying such phenomena. LIQUIDS This discussion of the effects in liquids will be rather academic in character, because I shM1 omit the effects which are perhaps most interesting from a practicM point of view and from the point of view of industry, namely, the effect of pressure on oils and other lubricants, and leave these to the discussion of Dr. Dow, who has experimented extensively in this field. M y own work in this field was done some years ago; discussion will be found ill my book, "The Physics of High Pressure," as well as in the original references. Some 45 different liquids were examined, usually over 7
P. W. BRIDGMAN
a pressure range of 12,000 kg./cm? and at two temperatures, 30 ° and 75°C. The method was, in most cases, a falling weight method. The time of fall of a weight through the liquid in question was determined electrically, and the apparatus was so mounted that it could be inverted and the measurement repeated a sufficient number of times to give the requisite accuracy. The liquids measured embraced the ordinary organic liquids whose other properties are commonly listed in tables, and included water. The only liquid metal was mercury. In every case, except for a temporary episode in water to be referred to later, viscosity increases with rising pressure at constant temperature. Furthermore, the increase is, in practically every case, at an accelerated rate with increasing pressure, so that the curve of viscosity against pressure is concave upward. In the case of a number of liquids sensible upward curvature does not begin until pressures of 2,000 or 3,000 kg./cm? are reached, so that the initial part of the curve is straight, as had been found by several early experimenters whose measurements were confined to this range. If the logarithm of viscosity is plotted against pressure, the curve obtained is, in the majority of cases, concave toward the pressure axis. The curvature is much the greatest at low pressures; above 2,000 or 3,000 kg./cm. ~ the curve approximates to a straight line in a little more than half the cases, while in the remaining cases it gently reverses curvature. This means that above 3,000 kg./cm. ~ viscosity either increases geometrically or else, even more rapidly as pressure increases, arithmetically. The range in numerical values for the effect of pressure is extreme; the effect of 12,000 kg./cm. ~ varies from an increase of 30% for mercury, to a two-fold increase for water, a ten-fold increase for methyl alcohol, a 100-fold increase for propyl alcohol, a thousand-fold increase for amyl alcohol, and an increase by 107-fold for eugenol. Contrasted with these figures are decreases in compressibility of not more than fifteen-fold brought about b y 12,000 kg./ era. 2, and decreases of thermal expansion by the same pressure of not more than two- or three-fold. There is a rather close correlation between the magnitude of the effect of pressure on viscosity and the complication of the molecule, the effect being least for monatomic mercury, and greatest for the complicated organic compound, eugenol. The effect of complication of the molecule is very plainly shown, for example, by the series of the alcohols above, or by the various compounds derived from benzene; the relative pressure effect is greater the more complicated the group substituted for hydrogen. There are also very marked constitutive effects, the iso- compounds, for example, having a larger pressure coefficient than normal compounds. Or a heavier atom substituted into a molecule produces, in general, a larger pressure effect, as shown by various series of halogen compounds. The effect of temperature combined with pressure is abnormal. Most
RHEOLOGICAL PROPERTIES AT HIGH PRESSURE
9
temperature coefficients become less at high pressures, but the temperature coefficient of viscosity becomes markedly greater with increasing pressure; for some substances the temperature coefficient at 12~000 kg./ cm. ~ is four-fold greater than at atmospheric pressure. The temperature coefficient of viscosity under pressure, when combined with the compressibility, gives important theoretical implications. Several of the earlier theories of the viscosity of liquids agreed in predicting that viscosity at constant volume should be constant. That is, if temperature is raised, decreasing the viscosity and increasing the volume, and if simultaneously the pressure is increased sufficiently to bring the volume back to the initial value, then the expectation was that the increase of viscosity brought about by the increase of pressure would exactly cancel the decrease brought about by the increase of temperature. This is approximately true for the first few thousand kg./cm. 2, but above that there may be significant and large departures. The variation is in such a direction that at constant volume the viscosity is less at the higher temperature. That is, the increased molecular agitation due to increasing temperature has a specific effect in decreasing viscosity apart from the effect on volume. The effects of pressure on the viscosity of water are abnormal, as are so many of its other properties. At low temperatures, between 0 ° and approximately 25°C., the initial effect of increasing pressure is to decrease viscosity instead of to increase it. At the low temperatures the curves pass through a minimum in the neighborhood of 1000 kg./cm. 2 and above this rise with upward concavity. Even at higher temperatures the total increase is abnormally low compared with other liquids. The effects can be qualitatively explained in water by well known considerations of the changing proportions of mono- and trihydrol. Within the last few years there has been considerable theoretical activity in accounting for the viscosity of liquids in general and the pressure effects in particular, improving on t h e earlier theories which demanded that viscosity be constant at constant volume. In general comment on any theory of the viscosity of liquids it is obvious that the fundamental mechanism must be different from that of gases. This is shown, for example, by the fact that in a gas the reciprocal of viscosity and thermal conductivity change together with pressure, whereas in a liquid thermal conductivity increases with pressure but reciprocal viscosity may decrease enormously. The most important of the theoretical work has been done by the Princeton school of physical chemists (1). The basic idea of this work is that there is a similarity between the molecular process by which momentum is transferred from one layer to another in a liquid flowing viscously and the molecular processes in ordinary chemical reactions. The molecule which moves along the velocity gradient in viscous flow receives
10
P. W. BRIDGMAN
or delivers energy. When it passes from a position in one layer to one in the next it is thought of as passing over a potential barrier before it can settle down into its new position, just as a molecule taking part in a chemical reaction has to pass over a potential barrier before it finds its new position of greater stability. The same sort of exponential formula governs the velocity of both processes, and there is an activation energy. The activation energy for viscous flow can be calculated from the kinetics of the liquid, and, in general, is considerably smaller than the energy of vaporization.' Ewell and Eyring, and also Ewell, develop this point of view, and obtain, among other things, expressions for the effect of pressure on viscosity which agree with my data up to 2,000 kg./cm3 in the early paper, and which were later improved to give agreement up to 7,000. This was followed by papers by Eyring and by Frisch, Kincaid and Steam, in which the concept of a maneuver space was introduced, which had to be available to the molecule if it is to be able to slip past its neighbor into a position of greater stability with respect to the viscous motion. This maneuver space is connected with various thermodynamic parameters. With a theory of this kind it is possible to reproduce my measurements on viscosity up to 10,000 kg./cm. 2 with only small error. Qualitatively the concept of maneuver space has points of resemblance to the concept of an interlocking between the molecules which I used to explain qualitatively some of the pressure effects, such as the strong dependence on complexity of the molecule. If the maneuver space is recognized to involve the shapes of the molecules, the connection between the two concepts becomes quite close. SOLIDS There is a much richer variety of phenomena for solids than for liquids, corresponding to the much greater range of kinds of deformation to which a solid may be subjected. On the theoretical side our understanding of the nature of the phenomena is, however, in a much more rudimentary state than in the case of liquids. There is practically no theoretical work on plasticity of solids which goes beyond pure description, and even this description is confined to the simplest aspects of the phenomena. For purposes of illustration we may take the simple description of the most elementary facts, as given by the yon Mises theory. According to this theory the substance is in the plastic state if a certain function of the stresses, namely, (X. -
Y~)~ q- ( Y , -
Z , ) ~ -k ( Z , -
X . ) 2,
(1)
exceeds a critical value. If the stresses do not exceed this value, the substance is in the elastic range. In the plastic range there is a connection between the stresses and the rate at which the strain varies with the time,
RHEOLOGICAL
PROPERTIES
AT HIGH
PRESSURE
11
that is, the rate of flow, as follows: Xx - ½(Yy + Z~) = Yy -- ½(Z, + Xx) = Z , - ½(X, + Yv)
(2)
The symbols in this equation denote the principal components of stress and of strain velocity, which are assumed to be along the same directions. It is usual to integrate these equations, and write an exactly similar set in which the dots in the strains above are omitted, that is, a set of equations connecting the strains instead of the strain rates with the stresses. These equations are recognized to be valid in only a narrow range just above the point of initial flow. They leave out of consideration a great variety of complicated effects, which are of the greatest practical importance, such as strain-hardening and various time effects. The beginning of an attempt has been made to include these other effects within the theory, but the mathematical difficulties are extreme and comparatively little success has been achieved. The effects of pressure on plastic flow which I have investigated (2) are included only very roughly within the scope of the simple equations given above, because in the pressure effects we are usually concerned with large deformations, whereas the equations are expressly restricted to small plastic deformations. Perhaps the simplest of the pressure effects has to do with the variation of behavior in simple tension. In order to study these effects a miniature testing laboratory was set up within the pressure chamber, and tensile tests conducted exactly as at atmospheric pressure, with measurements of tensile load and elongation. According to the elementary equations given above, the addition of a hydrostatic pressure to a given stress system should be without effect on the plastic behavior, either on the plastic limit as specified in Eq. (1) or on the rates of flow or the strains as specified in Eq. (2). Actually, a tensile test conducted under pressure may be dramatically different from one conducted at atmospheric pressure. The greatest difference for a substance like mild steel is the enormous increase of ductility produced by pressure. In one of my experiments a mild steel when pulled under a surrounding hydrostatic pressure of 25,000 kg./cm2 permitted an elongation of 300-fold at the neck without fracture, whereas at atmospheric pressure it broke with an elongation at the neck of approximately two-fold. A convenient way of measuring the ductility is in terms of the "natural" strain at fracture, that is, as loge A o / A , where A0 and A are the initial and final cross section at the neck. The natural strain at fracture for mild steel was found to be a linear function of pressure up to the limit of my experiments, 30,000 kg./cm ~. Another effect which is very prominent under pressure is strain-hardening, the existence of which is not even recognized in the elementary equations. It is known that at atmospheric pressure the strain-hardening curve, that is, the curve of "true stress"
12
P. w .
BRIDGMAN
against the equilibrium strain, which is the strain reached asymptotically after flow has ceased, starts out by rising with concave curvature t o w a r d the axis of strain, but after passing the point of maximum load where "necking" starts becomes straight and continues straight until the fracture point is reached at a strain of the general order of magnitude of unity. If the tension specimen is pulled under pressure a similar strain hardening curve is found, except that now the linear portion of the curve is much prolonged, out to natural strains of four or five, corresponding to the enhanced ductility imparted by the pressure. Furthermore, within the range of strain common to atmospheric and higher pressures, the strain-hardening curve at high pressure coincides with that at atmospheric. To a second approximation, however, strain-hardening at a higher pressure is greater than at a lower pressure. That is, for a given strain the true stress for the pulling under higher pressure is somewhat greater than for the pulling at lower pressure. At the lower end of the strain-hardening curve, before the curve becomes linear, it is possible to establish an effect of pressure. At high pressure a somewhat greater tensile stress is required to exceed the elastic limit and enter the plastic range than at lower pressures. That is, the "constant" to which the stress function in expression (1) is equal is an increasing function of pressure. This effect of pressure is not large compared with its effect on ductility, the general order of magnitude of the effect being a raising of the elastic limit by from 5 to 10% for a pressure of 10,000 kg./cm. 2. If a tension specimen is pulled under pressure to an elongation greater than the elongation at fracture at atmospheric pressure, but not to the fracture point under pressure, and the pressure is then released and the specimen pulled again at atmospheric pressure, it will be found that fracture does not occur at once, but further elongation is permitted, so that the strain at which a tensile fracture occurs is a function of the previous history and, in particular, a function of the pressure at which it was previously pulled. Theoretically this has most important implications, because it shows that the properties of a substance are not functions only of the state of stress and strain, but involve also the history. It is difficult to formulate this sort of thing in mathematical terms. It is obvious that a complete mathematical theory, even a purely descriptive theory, must be of a very considerable degree of complexity. The stress at fracture of a specimen previously pulled under high pressure and then broken at atmospheric pressure may be considerably higher than that of a virgin specimen broken a t atmospheric pressure. In this way an enhancement of properties may be produced which may be of some practical interest. The numerical parameters which determine the effect of pressure on the plastic properties of steel are a strong function of the condition of the
R H E O L O G I C A L P R O P E R T I E S AT H I G H P R E S S U R E
13
steel as determined by the heat treatment. If the steel is heat treated so as to become harder, the slope of the line of strain at fracture as a function of hydrostatic pressure increases markedly, so that the ductility imparted by a given hydrostatic pressure becomes less the harder the steel. Even with the hardest steels a perceptible amount of ductility was imparted by the highest pressures (30,000 kg./cm.2), as contrasted with a completely brittle fracture at atmospheric pressure. The slope of the strain hardening line also increases markedly as the hardness of the steel is increased by heat treatment. Certain substances which are normally completely brittle to tensile stresses at atmospheric pressure may have a measurable degree of ductility imparted to them by high pressure. Thus Balsley (3), working with apparatus of Griggs, has produced a tensile elongation of 40% in limestone pulled under 10,000 kg./cm.2. In work as yet unpublished, I have found that ordinary glass tolerates no measureable permanent tensile stretch even up to 30,000 kg./cm. 2, but does exhibit a great increase of tensile strength. Thus under a hydrostatic pressure of 27,000 kg./cm. 2 pyrex glass supported up to 24,500 kg./em. 2 superposed tensile stress, whereas the tensile strength under normal conditions at atmospheric pressure is of the order of 500 kg./em. 2. Hydrostatic pressure has an effect on flow under simple compressive stresses as well as under tensile stresses. On mild steel these effects are not as dramatic as the tensile effects, because even at atmospheric pressure very large compressive strains may be supported without fracture if the strain is kept homogeneous. There is an effect of hydrostatic pressure on the stress at initial yield under simple compressive stress; this effect is similar to the corresponding tensile effect. The initial flow stress is raised by something of the order of 5 to 10% for 10,000 kg./cm. ~. Other substances, normally brittle for simple compression, may show much larger effects than steel. Thus carboloy, which is normally as brittle as glass, will support plastic shortening up to at least 10% when exposed to simple compressive stress superposed on a pressure of 30,000 kg./cm. 2. Glass may become enormously strong in compression. I have measured a compressive strength of 45,000 kg./cm. 2 in pyrex glass when immersed in a liquid carrying 25,000 kg./cm. 2, against a normal compressive strength of not more than 2,000 kg./cm. 2. The most striking effect was obtained with single crystal sapphire, made by the Linde Air Products Company by their new technique for producing long single crystal rods. Normally these single crystals break brittlely like glass under simple compressive stresses. One crystal was observed in which failure under simple compression took the form of slip without fracture along the basal plane of the crystal, exactly as in single crystals of a ductile metal. The conditions in this experiment were not altogether simple or well defined, there probably being microscopic internal imperfections in the crystal, but there would
14
P. W. BRIDGMAN
seem to be no reason for doubting the qualitative results that a degree of plasticity has been imparted by pressure to a crystal normally brittle. Other effects of hydrostatic pressure under somewhat more complicated conditions are qualitatively not different from what might be expected from the simple tension and compression experiments. Thus the Brinell hardness of steel, as measured by the penetration of a ball driven by a known load, increases under pressure by something between 5 and 10% for a pressure increment of 10,000 kg./cm?. Or, if a plate of mild steel is penetrated by a punch, the ductility may be enormously increased by the pressure. Thus, in one experiment the punch broke completely through the plate after a penetration of perhaps 20% of the thickness at atmospheric pressure, whereas under 25,000 kg./cm. 2 the punch was driven 95% through the plate with no loss of cohesion and with a degree of strain-hardening that allowed the support of a greater load than a piece of virgin metal of the same geometrical configuration. Another type of experiment under conditions which are not quite as well defined as in those just mentioned duplicates more nearly the conditions of pure laminar flow obtaining in liquids during viscous flow and permits examination of a wide range of materials. In these experiments (4) the material is in the form of a thin wafer a few thousandths of an inch thick pressed between a very short flat faced boss on a heavy block of steel and a plane steel block. Pressure is applied between boss and plate, forcing out the material of the wafer until an equilibrium thickness is reached corresponding to the pressure. Except at the very edges the material of the wafer is in a state of approximate hydrostatic pressure. The block and boss are then rotated relatively to each other, subjecting the material of the wafer to a shearing force at its faces. At low pressures, there is surface slip between material and steel, but when the pressure has reached a value so high that the friction at the interface equals the internal flow stress, surface slip ceases, and at higher pressures the material yields by internal laminar motion, exactly as in a liquid. By measuring the turning force as a function of pressure, it is possible to determine the stress for plastic flow as a function of hydrostatic pressure. These measurements can be carried up to pressures of 50,000 kg./cm. ~ with apparatus constructed of steel, and with carboloy apparatus have been carried to 100,000 kg./cm. 2. These high stresses are made possible by the geometrical design, in which the highly stressed region is highly localized, and surrounded and supported by much larger regions comparatively free from stress. A very wide variety of materials may be studied with this simple apparatus. Under these conditions there is no substance which does not differ very markedly from the idealized yon Mises plastic solid to which Eq. (1) applies. In all cases the shearing stress at which plastic flow occurs is a rising function of pressure. In the pressure range up to 50,000 kg./cm. ~
RHEOLOGICAL PROPERTIES AT HIGH PRESSURE
15
the rise may be by a factor no more than two in the case of some of the metals or may be thousands of fold in the case of substances like paraffine. Very large deformations may be required to reach a steady state, that is, a constant invariable ratio between shearing stress and pressure; shearing deformations of as much as 100 radians sometimes were necessary. Under these very large deformations the original crystal structure of many of the materials is so highly broken down that very little remnant of structure is detectable by X-ray analysis, and, in fact, the material in some cases appears to be reduced to a true glass (5). It is evident from these results that ordinary materials are capable of existing in an almost infinite number of states of aggregation intermediate between fully crystalline and amorphous, and that, accordingly, a great variety of physical properties is to be expected, depending on the degree of work hardening. The conditions of these shearing experiments are different in an important respect from those of ordinary testing, in that, when fracture occurs, the fractured surfaces are not separated but are maintained in close contact by the pressure. Fracture under these conditions is not a run-away phenomenon, but self-healing often occurs. A very common pattern of fracture is for the material t o suddenly let go, after a critical amount of shearing distortion, with catastrophic decrease of shearing stress. The material then self heals, and as shearing distortion continues, the shearing stress builds up again to its former value, and the cycle of fracture and recovery may be repeated indefinitely. There are sometimes other patterns of fracture. Thus, beyond a certain degree of distortion fracture and self healing may be a practically continuous process, the shearing stress remaining at a constant value while the shearing process is accompanied by chattering or by grinding noises. Internal fracture is confined almost exclusively to non-metallic substances and among them to those which do not crystallize in the cubic system. Cubic systems are in general richer in cleavage or shear planes than other systems, so that there is more opportunity for slip on one plane or another even after extreme distortion. Not only is there self healing after internal fracture, but there is often cold welding between the substance and the steel parts. Thus borax glass is as tightly welded to the steel parts after shearing in this way as if it had been fused to the steel. This type of shearing experiment shows well what is perhaps the most important difference between the viscous flow of a liquid and the plastic flow of a solid. The shearing force required to maintain a liquid in laminar shearing motion is proportional to the velocity of shear, whereas in the case of a plastic solid the shearing stress varies little with the velocity of shear. Different solids vary in this respect over a wide range. Thus, in the cases of tin and lead, the two metals which show the greatest de-
16
P. W. BRIDGMAN
pendence of force on speed, a ten-fold increase in force produced a thousand-fold increase in speed. With mica, on the other hand, an increase in speed of ten-thousand-fold was produced by an increase in force too small to measure. With a substance like mica, the apparatus would obviously be unstable if one attempted to impose a constant force during the shearing process. Measurements on such substances must be made by imposing a constant velocity and then allowing the force to come automatically to its equilibrium value. It is the first task of a physical theory of the process of plastic flow, as distinguished from the purely descriptive theory, which is all we have at present, to account for this striking lack of dependence of velocity on stress. It would appear that in broad outline the forces which are active during the plastic flow of a solid are configurational in character, depending on the relative locations of atoms and molecules, as contrasted with forces which are dynamic in origin, arising from transfer of momentum on a macroscopic scale, which are primarily concerned in the viscous motion of a liquid. I have made a suggestion along these lines (6), which may express the germs of the essential situation. In an idealized lattice structure a steady "state of slip" is a possibility, in which one part of the crystal moves relatively to another by slip along one of the lattice planes. Once this motion is es~blished it would persist with no force to maintain it, since, on the average, the net work done by the atomic fortes during a relative motion by one atomic step would vanish. In actuality, of course, lattices are not perfect, and furthermore are disarranged by relative slip, but in any actual lattice there must be a very appreciable idealized component which may well be the determining factor in the relative independence of velocity of force. A somewhat similar consideration applies also to amorphous substances, for even in a noncrystalline substance such as glass, there is evidence in the details of the shearing experiments that some sort of structure is imposed by the process of shearing itself, and this structure may well function on the average sufficiently like the perfect lattice to account for the small effect of velocity on force. REFERENCES I. ]~WELI~R. H., AND :EYRING,H., J. Phys. Chem. 5, 571, 726 (1937);Ewv.LL, R. H., J. Applied Phys. 9, 257 (1938); FRISCH, D., EYRINO, H., AND KINC~U, J. F., J. Applied Phys. 11, 75 (1940); KINC.~ID,J. F., EraI~G, H., ~ D STEARN, A. E., Chem. Rev. 28, 301 (1941). 2. BRIDGM~, P. W., Rev. Modern Phys. 17, 3 (1945); J. Applied Phys. 17, 201, 225, 692 (1946). 3. B~si~Y, J. R., Trans. Am. Geophys. Union 1941, Part II, 519. 4. BR1DGMAN,P. W., Proc. Am. Acad. Arts Sci. 71, 387 (1937). 5. LARSEI~,E. S., AND BRIDGMAN,P. W., Am. J. Scl. 36, 81 (1938). 6. BRmGM~, P. W., J. Applied Phys. 8, 332 (1937).