F Neesen introduced multiple exponential terms of the type now familiar to describe his after-effect experiments. We now need to look at the state of theoretical work at these times. We have already noted Kelvin's Thomson concept of 'viscosity of solids'. E is Young's modulus and A is a time constant. No real explanation was given for this equation and Maxwell used it to calculate gas viscosity, given by the product EA. He was aware of the experiments of Weber and Kohlrausch and noted that the simple exponential decay of stress implied by 1.
Born in Belfast, he was educated at home by his father a Professor of Mathematics. In , he entered the University of Glasgow and in , the University of Cambridge. In be became Professor of National Philosophy in Glasgow, where he created the first physics laboratory in a British University. He was President of the Royal Society from and was granted a peerage in , taking the title 'Baron Kelvin of Largs'. He is chiefly famous for his work on the electrical conductivity of solutions.
It is perhaps ironic that the concepts of the rivals Hooke and Newton were united forever by Maxwell in his equation 1. This expression actually describes what is now known as the Kelvin-Voigt body. Kelvin see Thomson , did experiments on the damping of metals and applied the concept implied by 1. In justice, one ought to refer to a Kelvin-Meyer body, or simply a Meyer material. An important contribution to the subject was made by L u d w i g B o l t z m a n n , who is best known for his work on the kinetic theory of matter and the concept of entropy.
His three rheological papers, published in , and , were to have a significant impact on the mathematical theory of viscoelasticity. Boltzmann's early work in , written as he attained his thirtieth year, was apparently motivated by the lack of generality in Meyer's formulation, and the paper contains a long criticism of Meyer's work. Considering the isotropic viscoelastic case, Boltzmann assumed that the stress at time t not only depended on the strains at that time, but also on those in previous times; it was explicitly assumed that the longer the interval from the present to the past time, the smaller would be the contribution to the stress resulting from a given strain.
This is an expression of the now familiar principle of 'fading memory'. The assumption of linear superposition was also made, with a footnote that states that the principle of superposition will not hold for large "starke" deformations. The shear components corresponding to 1. For these components, Boltzmann then deduced several ways of finding r from torsional experiments using various strain patterns; relaxation, free vibration and short steps of strain were among the patterns considered.
Some experiments on glass fibres in torsional vibration were shown to agree well with Boltzmann's calculations and the closing remarks in the paper concern the way the general theory collapses to the viscous case for certain forms of the memory function r Boltzmann's paper clearly deserves its reputation as a classic - it criticised an existing theory Meyer's , it gave a new formulation and tested it against new experiments on glass fibres; it finally closed with some far-sighted predictive remarks.
Meyer in turn attacked Boltzmann vigorously on all fronts, arguing that his theory was not based on an atomic hypothesis and violated ideas then current about atoms; he was also unhappy about the new experiments. We have already indicated that Meyer's theory was nothing more than that for the so called Kelvin-Voigt model, which was to be published later by Kelvin and Voigt , It is evident that, although both the Maxwell model 1. Maxwell recognized this fact and used the Boltzmann formulation in his Encyclopaedia Britannica article of From now on, will generally denote a time constant, as opposed to a modulus as in 1.
The Kelvin-Voigt-Meyer model is best regarded as a Boltzmann model with zero r and the addition of a viscous component, unless one is prepared to admit generalized functions for r The complete Boltzmann theory was fully three-dimensional and is the general form of linear viscoelasticity. We may quote and agree with Markovitz who concluded that it was the "first successful theory of rheology". Boltzmann's two later papers and were replies to criticisms by Meyer and others and essentially added nothing fundamental to the subject, although they did contain carefully worked examples.
Appraisal A widely held view marks rheology as a relatively modern science, which has really come into prominence in the latter half of the 20th century.
This we concede, but the present chapter has shown that its origins are in antiquity and that the 19th century in particular contains much research of relevance. This was carried out by a number of famous scientists and they provided a solid foundation for the mushrooming activity of the 20th century. The advances in previous centuries were not monotonic and, like in so many other fields, there were the inevitable personality clashes. The antipathy between Hooke and Newton and between Meyer and Boltzmann are prime examples of this.
What is clear is that, by the turn of the 20th century, there was a general acknowledgement of the existence of materials that could not be classified by either of the classical extremes of Hookean elastic solids and Newtonian viscous fluids. He had a great natural aptitude for making mechanical toys, and he was initially educated at home by his father, a clergyman. He went to Westminster school at the age of 13 and then to Oxford University, where he met Robert Boyle, the chemist, who was a council member of the newly-formed Royal Society.
He assisted Boyle with some of his experiments and in he was hired as Curator of Experiments by the Royal Society at the urging of Boyle. He struggled financially and in addition to the curatorship and a lectureship, he was also Professor of Geometry at Gresham College in London. During his time at the Royal Society, Hooke investigated an enormous range of subjects, including mechanics, light, chemistry, philosophy, botany and microscopy.
His Micrographia contains a description of a new compound microscope and discussions of the physics of thin plates and views on combustion, besides impressive drawings of what he saw in the microscope. This book alone would have made his reputation as a first-rate scientist. Hooke's Micwgraphia contains comments of immediate relevance to today's experimental rheologist: "I have often thought that probably there might be a way found out to make an artificial glutinous composition, much resembling, if not fully as good, nay better, than that excrement, or whatever substance it may be out of which the silk worm wiredraws his clew.
This hint.. I suppose he will have no occasion to be displeased. He assisted with the surveying and architecture of the rebuilt London, following the Great Fire of cf. Cooper, He also did much to transform the Royal Society from an amateur's club to a professional body. To some, Hooke was sociable and respected, 'a person of great suavity and goodness' To others he was too cynical and miserly to be much liked. In his later years he was described as being 'of middling stature, something crooked, pale faced, and his face but little below, but his head is lardge; his eie full and popping, and not quick; a gray eie.
He haz a delicate head of haire browne, and of an excellent moist curle' cf. Keynes He was nevertheless combative and this led to many disputes on priority, notably with Newton, who was seven years younger than Hooke. Newton and Hooke were always at loggerheads and it seems likely that Hooke's status as a scientist was deliberately lessened by Newton, following Hooke's death in A sympathetic biography is that by Espinasse No portrait of Hooke is known to havc survived.
In its place we reproduce a famous picture Fig 1. Hooke's spiral springs, from De potentia restitutiva, Isaac N e w t o n This portrait of Newton and the accompanying text is taken from the opening pages of the first issue of the first volume of the Journal of Rheology published in His father, an illiterate but reasonably prosperous farmer, died three months before the birth. Three years later, his strong-minded mother remarried and moved to another village, leaving Isaac Newton at Woolsthorpe, to be brought up by a grandmother.
His stepfather, a wealthy clergyman, died eight years later, and Newton's mother then returned to Woolsthorpe. Some have seen this separation from his mother as an important factor in the shaping of the suspicious and neurotic personality of the adult Newton see, for example, Gjertsen He made many remarkable discoveries in mathematics at an early age. During the time he was absent from Cambridge in due to the plague, he began his work on gravitational problems and the motion of the moon. It was not until that Edmond Halley interested him in writing up his work. Eventually, it became known as the Principia; it was published in The Principia contains Newton's famous hypothesis about the response of fluids to a steady shearing motion, which assured his place in our History.
Between and , Newton was unwell, suffering from insomnia and nervous trouble. In , he was offered and accepted the Wardenship of the Mint, a post he retained until his death in ; he was buried in Westminster Abbey, London. An exceptional biography of Newton is that of Westfall , who discusses not only his scientific work, but also his philosophical and theological studies, as well as his work as Master of the Mint.
Aside from problems in his relations with Robert Hooke, which we have already alluded to in our discussion of Hooke, Newton was also in priority disputes with Leibnitz over the Calculus. As we have hinted, these difficulties with people were probably there from boyhood Westfall and cannot be overlooked, any more than can his devotion to alchemy and religious studies. The picture is of a self-sufficient, secretive personality, mellowing somewhat in old age. He never married and, given these facts, it is hard not to have some sympathy with his opponents. He remains a remote, mysterious person, one of the supreme intelligences of all time.
Augustin-Louis Cauchy Cauchy as a young academician. Half-length portrait by Boilly see Belhoste His father was a successful royalist government lawyer and he had to leave Paris at the beginning of the Revolution for the village of Arceuil. Laplace and the chemist, Berthollet, also lived in Arceuil and, during the time of Napoleon, young Cauchy came to know these famous scientists. Lagrange, when visiting Laplace's house, noticed Cauchy's mathematical ability.
In , Cauchy became a member of the Academic and held three Professorships in Paris. In , he left these positions, because he declined to swear allegiance to Louis Philippe; he was extremely pious and a strong Catholic. Cauchy was forced into exile and went to Turin. He finally returned to the Ecole Polytechnique in ; he died in Cauchy's interest in elasticity was generated by Navier's memoir of , which was presented to the Academic.
In , he fornmlated the general theory of stress, and in so doing defined the stress tensor and derived the equations of motion. These have been the cornerstone for much of theoretical rheology ever since and Truesdell in his "Essays in the history of mechanics" asserts that "Clearly this work of Cauchy marks one of the great turning points of mechanics and mathematical physics" Truesdell , p He published seven books and about papers, and has around 16 fundamental concepts and theories to his credit; certainly he was one of the greatest mathematicians in history.
Claude-Alphonse Valson was Professor of Calculus at the Universit5 de Grenoble and his volume on Cauchy devotes nearly as much space to Cauchy's work for the Catholic Church as it does to his mathematics. It is uncritical and adulatory, and it is no surprise to learn that Valson also espoused Cauchy's views on society and religion. Valson praised Cauchy's teaching, but the evidence shows he was a classroom failure Belhoste, The well-known work of Bell has a chapter on Cauchy and it is also adulatory, but concentrates on mathematics.
The more recent study of Cauchy by Belhoste is more balanced and one has to admit that Cauchy's narrow views, plus his many meanspirited actions and disputations over priority with his mathematical contemporaries, do not make him an attractive personality, despite his extraordinary contributions to mathematics and mechanics. After what to some was an unpromising beginning, he suddenly blossomed and his first scientific paper was published by the Royal Society of Edinburgh when he was only thirteen.
He entered Edinburgh University at the age of sixteen. Maxwell went up to Trinity College, Cambridge, in to embark on a distinguished scientific career. In he moved back to Scotland as Professor of Natural Philosophy at Marischal College, Aberdeen, the move being motivated at least in part by a desire to be near his ailing father, his mother having died when Maxwell was only nine. He married the daughter of the Principal of Marischal College.
In , Maxwell moved to Kings College, London, and remained there until the death of his father in There followed a lengthy stay at his family home in Scotland, where he combined the life style of a gentleman farmer with that of a research scientist. There he set up the famous Cavendish Laboratory, which was to become a unique institution, headed by a succession of men of genius. Maxwell is universally accepted as the greatest theoretical physicist of the 19th century and he is sometimes spoken of as the 'Father of modern physics'. He certainly had an immense influence on subsequent generations of scientists.
Albert Einstein, for example, concluded that "one scientific epoch ended and another began with James Clerk Maxwell" and R A Millikan, another Nobel Laureate, wrote of 'one of the most penetrating intellects of all time'. However, his contributions of immediate relevance to rheologists have been unusually influential. Best known is his work on "The Dynamic Theory of Gases" Maxwell , in which he introduced the famous linear differential equation relating stress and strain; this allowed for the first time the idea of a relaxation process in a viscous fluid.
Less well known is his experimental work on the determination of the "thickdom" or viscosity of gases and his perceptive distinction between 'solids' and 'liquids'. This is highlighted in Markovitz's historical essay on "The emergence of rheology": "In his writing on states of matter, Maxwell drew a careful distinction between solids and liquids and, in many of his publications, pointed out that materials such as pitch are fluids: If therefore, we define a fluid as a substance which cannot remain in permanent equilibrium under a stress not equal in all directions, we must call these substances cold pitch and asphalt fluids, though they are so viscous that we can walk on them without leaving any footprints.
W h a t is required to alter the form of a soft solid is a sufficient force, and this, when applied, produces its effect at once. In the case of a viscous fluid, it is time which is required, and if enough time is given, the very smallest force will produce a sensible effect" Markovitz Maxwell interacted with Lord Kelvin, who was also to leave his mark on early developments in rheology. An example of the association is contained in a letter written by Maxwell in to a relative, C H Cay see, for example, Campbell and Garnet, , the point at issue being the interpretation of experimental data from a viscometer.
The letter was written from Glenlair, the family house in the Scottish Highlands; it contains a hint of Maxwell's quaint and suggestive turn of phrase: "I set Professor W Thomson a prop. He sent me 18 pages of letter of suggestions about it, none of which would work; but on Jan 3rd, in the railway from Largs, he got the way of it, which is all right; so we are jolly, having stormed the citadel, when we only hoped to sap it by approximations".
By common consent, Maxwell was somewhat diffident, a man of quiet disposition, unassuming, devoid of any trace of pomposity, but of great warmth and sure character. He had a strong Christian faith and his obituary in Nature paid tribute to this important aspect of his life: "His simple Christian faith gave him a peace too deep to be ruffled by bodily pain or external circumstances".
In , Maxwell died of cancer at the relatively early age of Ludwig B o l t z m a n n Ludwig Eduard Boltzmann was one of the greatest theoretical physicists, famed for his work on gas kinetic theory, heat and entropy. The son of a taxation official, he was born on the evening of Mardi Gras, February 20th, just before Ash Wednesday, and he later attributed in jest his rapid changes in moods, from high to low, to this aspect of his beginning.
He received his doctorate from the University of Vienna in , with Josef Stefan as supervisor.
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He worked on the kinetic theory of gases for his PhD, a subject to which he would later contribute much more original work. After two years, at the early age of 25, he was appointed Professor of Mathematical Physics at the University of Graz. He moved to Munich in , then back to Vienna in , when a Chair became vacant on the death of Stefan. Ernst Mach was appointed as a Professor of Philosophy at the University of Vienna in , but Boltzmann was at philosophical odds with Mach, and so he moved on to Leipzig in Mach's early retirement in finally left Boltzmann able to reclaim his own chair at Vienna, since it had not yet been filled.
In his career, Boltzman defended the atomistic viewpoint vigorously. It is perhaps surprising that debate on these issues should be taking place at such a late date, but the question of the 'reality' of atoms and molecules was a very serious point of contention in the late 19th century, and controversy was only laid to rest in the early 20th century. Mach was one of the opponents of Boltzmann, together with the chemist Wilhelm Ostwald. Despite a victory for atomicism at a public meeting in , the attack on it continued, and Boltzmann began to feel insecure about the outcome. His health declined and he became depressed, possibly about the future of his work and ideas.
However, the work of 1. On September 20th, at the age of 62 years, he committed suicide by hanging, while staying at the beautiful Bay of Duino near Trieste with his wife and daughter. Soon after his suicide, his views were vindicated by the work of Perrin and others and the acceptance of the atomic viewpoint was inevitable. Boltzmann was short and stout, played the piano well, wrote humorous poems and loved nature and the countryside. He was an excellent teacher and gave very well-attended popular lectures; there is a whole volume of Populgre Schriften available.
He was said to be kind towards students and never to have failed anyone taking his courses in his later years. He also gave a theoretical proof of Stefan's fourth-power radiation law for a black body. The Encyclopaedia Britannica entry does not mention his work on viscoelasticity, nor does the Broda biography, but the three papers written in , and are his fundamental contribution to our subject. Despite their importance to rheology, they appear almost as a continuum aside in a life devoted to atomic and molecular dynamics.
T h e B e g i n n i n g of E x p e r i m e n t a l Fluid R h e o l o g y Much of the previous chapter has dealt with problems arising from solid theology, in particular from fibre mechanics. We now need to address the beginnings of serious experimental fluid theology. Some of the relevant background in Newtonian flow has already been discussed in the previous chapter, but it is important for us to highlight the innovative experimental work of Hagen , a civil engineer, and Poiseuille , , a physician, both of whom studied flow in small-diameter tubes for different reasons; Hagen was looking for basic hydraulic information and Poiseuille was interested in blood flow.
The experimental technique of Poiseuille was exemplary, and is still worthy of study. The same can be said of the seminal work of Couette 1 The work of Hagen, Poiseuille and Couette set the stage for the 20th century preoccupation with non-Newtonian liquids. At this time, non-classical behaviour was most clearly observed through the variation of viscosity with shear rate.
The first experiments that unequivocally demonstrated a non-Newtonian viscosity varying with shear rate seem to be those of Theodore Schwedoff, who was Dean of Sciences in Odessa in the late s. He used a thin-gap Couette-type apparatus, and was therefore able to infer shear stress and shear rate fairly accurately. Two of his papers appeared in and in the Journal de Physique. In the first, he described concepts of viscosity and springiness that he sought to measure; he also dealt with fluids t h a t showed a yield point, such as weak gelatine gels.
He found t h a t the material relaxed after a step strain, and he tried to use Maxwell's ideas to describe what he saw. In his second paper, Schwedoff concentrated on viscosity.
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He concluded: " the viscosity is not constant, as one usually supposes: it varies with the speed of shearing". Since he was dealing with gel materials t h a t had a yield stress, 1Maurice Marie Alfred Couette was born in in Tours, France, the only son of a cloth merchant. In , he obtained a bachelor's degree in mathematics from the Faculty of Sciences in Poitiers. He then enrolled in the newly opened Free Faculty of Sciences in Angers and, in , he received a degree in physics.
After completing one year's voluntary military service, Couette settled in Paris. From , he studied at the Physical Research Laboratory at the Sorbonne, completing his doctoral thesis on "Studies in liquid friction" in In , at the invitation of the Chancellor, Couette returned to the Catholic University of Angers, where he was to spend the next 43 years as Professor of Physical Sciences. He gave up university teaching in and died ten years later. A full account of Couette's life and scientific achievements is provided by Piau et al We note that J G Butcher invented the terms "elastico-viscous" and "elastico-plastico-viscous", of which 2.
The paper of Schwedoff was the forerunner to a multitude of papers on variable viscosity effects in a plethora of materials which were to occupy much of the literature of the first half of the 20th century e. Hatschek According to Scott Blair , there was a tendency to label all anomalous behaviour as manifestations of 'plasticity', with no clear idea as to what that meant.
In , Bingham published an important book entitled "Fluidity and Plasticity". It contained a m a m m o t h 82 page bibliography and provided a great deal of information on measurements for various systems, including gel-like materials with a yield stress. It also included an important discussion on 'wall-slip' in viscometers. Strangely, the book does not have a very lengthy discussion on variable-viscosity systems, which probably reflects the mood at the time of writing. This mood had to change, and it did so as a result of rheometrical experiments on a multitude of different materials.
So, by , Scott Blair, in the March issue of 'British Plastics', was able to report that "Rheological methods are in practice applied to an enormous variety of materials. Among the more obvious applications are the following: metals elastic and plastic properties , oils, paints and varnishes, bitumens, tars and pitch, gums, inks and starches, flours for bread and biscuit making, cheese, butter, cream and milk, tooth pastes, soaps and toilet creams, ceramic clays, dyestuffs and fibres of all kinds In order to classify the plethora of possibilities in a simple way, the British Rheologists Club published a working chart on rheological behaviour Nature , , p and invited further comments.
Figure 2. References to many of the more relevant contributions to the expanding literature are provided in the books by Houwink , Scott Blair and PhilippotF The book is an important source of reference on the state of colloidal rheology in the early s. We note that the often misunderstood term 'thixotropy', which we discuss in Chapter 7, 2Wladimir Philippoff was born in Peterhof, Russia in and moved to Germany in , receiving his Dr Eng Elec Eng degree in From he worked at the Kaiser Wilhelm Institute of Chemistry on dynamic testing in viscoelasticity.
He was awarded the Bingham medal in Fig 2. With the increasing interest in fluid rheology came the need to describe nonlinear fluids in an appropriate manner and a large number of empirical models describing the observed shear stress-shear rate behaviour were invented. This was essentially a curvefitting exercise, and, to be attractive, the models had to contain as few constants as possible.
Most notable amongst the growing list of available models were the Bingham model, the Ostwald -de Waele power-law model, the Herschel-Bulkley model and the Ellis and Williamson models see Houwink , Tanner These are still important today in giving a compact, manageable description of shearing flow.
One could say that if variable viscosity and linear viscoelasticity were the sum total of rheology, then it had all been done by The empirical shear stress-shear rate models were used to derive the relevant flow rate-pressure drop relationships for capillary flow, and these could then be compared with experiment. However, it was soon realized that a procedure was needed to solve the important inverse problem, i. This question was successfully addressed by W e i s s e n b e r g , the provenance of the solution being unexpected, since, at the time , Weissenberg was working at the Berlin crystallographic laboratory of R O Herzog at the Kaiser-Wilhelm Institut fiir Faserstoffchemie Berlin-Dahlem, which was not primarily interested in fluid mechanics.
Weissenberg's interest in these matters can be traced to a paper with Herzog entitled "On the thermal, mechanical and x-ray analysis of swelling" which was published in Kolloid-Zeitschrift. It is a remarkably wide-ranging paper and sets out a programme of research on the relations of material and molecular structures to thermal and mechanical properties.
Whilst this was clearly only a preliminary statement of a vast programme of work, it was recognized that material behaviour might be described as an interplay of kinetic, thermal and elastic energy, and a triangular diagram with vertices labelled with these three forms of energy was displayed. In a surprising switch from this very general scene to a particular problem, on their p, the first general analysis of the inverse capillary flow begins. The correct inversion formula was published twice in The first publication, submitted on March 14, was authored by Eisenschitz, Rabinowitsch and Weissenberg , but before it appeared Rabinowitsch published again August in the Zeitschrift ffir Physikalische Chemie, a more accessible journal.
However, both Rabinowitsch, in his paper, and Eisenschitz, in a paper, acknowledge unequivocally that the inversion method was due to Weissenberg. This brilliant analysis will bear repeating. Later, Mooney generalized the analysis to allow for slip. This formula has been the basis for all reliable capillary viscosity measurements since The extension to plane slit flows follows immediately; a factor 2 appears on the right hand side of 2. Many further inversion formulae for other viscometric flows are available see Coleman et al , Walters and Tanner The correct application of formulae like 2.
This was and is not always available and convenient empirical means have been devised to overcome the problem. The first was proposed by Couette and has been called the Couette method. This involves using two capillaries of the same diameter but different lengths L1 and L2, the flow condition being the same in both. A modification of the Couette method was proposed much later by Bagley and the use of the so called 'Bagley correction' has become standard practice in much modern capillary and slit rheometry.
At this point, we need to return to the work of Weissenberg. Following his contributions to crystallography, with the invention of the Goniometer in , see, for example, Harris , we see his attention turning to the general problem of material behaviour. The capturing of the angles of x-ray diffraction no doubt sharpened his appreciation of threedimensional states, with great advantage to rheology. A year later, he co-authored a paper which described an oscillating viscometer see Harris Two long papers followed, one written in German Weissenberg and one written in French, while he was in Southampton, UK Weissenberg These discuss the application of tensor methods to material problems.
In his paper, the idea that the extra stress tensor is determined by the flow history is expressed. The impact of these long papers was less than it might have been. Nevertheless the stirring of an interest in three-dimensional rheology can be seen. Weissenberg published no more work until and we shall discuss this in Chapters 6 and 7. Linear Viscoelasticity Linear viscoelasticity is a subject which stretches in its influence and importance from the early days of rheology to the present day. Certainly, the work of Maxwell, Boltzmann, Voigt, Kelvin and others, already discussed in Chapter 1, falls within the restricted area which is now commonly referred to as 'linear viscoelasticity'.
It remains an important area of research and most modern characterization studies involving viscoelastic materials are likely to involve the measurement and interpretation of data arising out of appropriate experiments, be they creep, stress relaxation or small amplitude shear flow experiments. Indeed, one could say that the bulk of the fundamental theoretical research in linear viscoelasticity was completed by the close of the 19th century. The early years of the 20th century witnessed the introduction of 'mechanical models' and these have proved to be a popular means of characterizing linear-viscoelastic behaviour.
In these models, Hookean elastic deformation is represented by a spring and Newtonian flow by a dashpot. Sometimes, a 'slider' is introduced to represent a yield criterion see, for example, Reiner a. The mechanical-model analogy involves the association of force, extension and time in the models with stress, strain and time in the material.
The Maxwell model, described mathematically by the differential equation 1. These more complex models were independently suggested by J J Thomson and Wiechert Typical of this attitude is: "While Maxwell left it for later generations to obscure his straightforward notions of continuum mechanics by the intermediary of springs and dashpots, his approach to the subject may fairly be said to have dominated it until " Truesdell b.
Others have appreciated the didactic role of the spring-dashpot representation cf Reiner a, Tschoegl seeing it as a way of studying viscoelastic response without the need to go into mathematical detail, and the comments of Ferry would be typical of many exponents of mechanical models. Two such models are illustrated in Figs 1. To simulate a real material, the model may require an infinite number of units with different spring constants and flow constants The model represents only the macroscopic behaviour and does not necessarily provide any insight into the molecular basis of viscoelastic phenomena; its elements should not be thought of as corresponding directly to any molecular processes".
The first reference to mechanical models is in the text of Poynting and Thomson and a full descriptive account is in the 'First Report' by Burgers It contains a number of influential simpler models as special cases. For example, Jeffreys applied the equation to interpret problems associated with the crust of the earth. Later, Fr6hlich and Sack showed that, within the linearity constraint, a theoretical analysis for a very dilute suspension of elastic solid spheres in a viscous liquid resulted in an equation like 2. To accommodate 'interfacial slipping', Oldroyd showed that equation 2.
The four-element mechanical model reproduced from Burgers Roscoe also showed that a saving in the number of elements often results from the use of canonical forms and he gave rules for deducing the number of elements in the canonical form from the arrangement of elements in any complex model.
By a suitable choice of the model parameters, the canonical forms themselves can be shown to be mechanically equivalent and Alfrey gave methods for computing the parameters of one canonical form from those of the other. In the same paper, he showed how the differential equation relating stress and strain could be obtained quite generally from either of the canonical forms and vice versa. The mechanical-model representation of linear viscoelastic behaviour results in at best an enumerable infinity of relaxation times, but the extension to a continuous distribution of relaxation times is accomplished with mathematical ease.
Numerous influential texts in the post Second World War years discuss the procedures involved, most notably Gross , Staverman and Schwarzl , Alfrey and Gurnee and Ferry A careful comparison of 2. Solution of boundary value problems for linear viscoelastic materials, including temperature variations via the time-temperature superposition principle see Chapter 6 , made great strides in the post Second World War years.
The time-temperature superposition idea was extended to moving media by Morland and Lee This key paper introduced the idea of a local "material clock time", which varies at a given particle as that particle encounters variable temperatures during its kinematic history; it still remains the basis of many calculations and has been extended to the nonlinear case. Further extensions of the variable clock time to include pressure and stress effects have also been considered. Further descriptions of these principles are given by Tanner and Huilgol and Phan-Thien One may refer to the book of Bland for a discussion of the correspondence principles linking the solution of linear viscoelastic problems to 'corresponding' linear elastic solutions.
The No-Slip Boundary Condition In Newtonian fluid mechanics, it is now widely accepted that the no-slip boundary condition holds; i. Interest in the subject goes back at least to Coulomb , who carried out some viscosity-measuring experiments using an oscillating disc in water. He tried coating the disc with tallow and also sprinkling on sandstone, neither of which made any differences to the observations.
He therefore concluded that no slip was occurring and that a true fluid property was measured. Meyer confirmed these results. Poiseuille's seminal , work actually implied the no slip boundary condition, since eventually Hagenbach and Wiedemann were able to show that there was concordance with Stokes's solution for laminar flow, which used the no slip condition. Some thought that mercury should slip at a glass wall, because of its non-wetting nature. Warburg investigated this case and found no slip; his results were later confirmed. Whetham carried out a very extensive set of experiments with various surfaces, as did Couette between and see Piau et al ; Couette also investigated slip in the turbulent regime.
In all cases, the conclusion was that, within experimental error, no slip was seen. Interestingly, Bingham , p devoted considerable space to the problem of slip and concluded: "These results seem to make it quite certain that, whether the liquid wets the solid or not, there is no measurable difference between the velocity of the solid and the liquid in contact with it, so long as the flow is linear".
The last phrase is intriguing and presumably meant that Bingham had some doubts about the non-Newtonian case; but for Newtonian liquids he saw no slip, and this view has been accepted in fluid mechanics and has even extended to non-rarefied gases. The situation in non-Newtonian fluid mechanics is more complex, and, by , rheologists were actively considering slip, especially for rubbery materials. Mooney extended Weissenberg's capillary-flow relation to include slip, since he knew that rubbers did not adhere fully to solid walls.
Much later, Oldroyd appealed to 'slip at the wall' in an attempt to explain Toms's turbulent drag reduction results. Similarly, in plasticity theory see Hill , slip between metal and die wall is commonplace. Pearson and Petrie devised a stability theory of polymer flow which depended on wall slip. Further, there were numerous papers showing jumps in discharge rates in capillaries see, for example Vinogradov and Malkin Thus the question of slipping or not continued to arise. It is now convenient to survey the various possibilities concerning slip. This possibility is discussed in detail in the recent survey of Barnes Effect i is relatively well u n d e r s t o o d and need not be discussed further; in blood flow it has a n a m e - the Fs effect ; see also Goldsmith and Mason Effect ii must certainly occur, but so far only simulations can detect it and the altered layer seems generally not to p e n e t r a t e far into the bulk material.
In connection with effect iii , Pearson and Petrie m a d e a basic contribution to the subject, although the work was not widely noticed, and the subject languished until R a m a m u r t h y showed, unequivocally, t h a t real slip can occur in extrusion. Hatzakiriakos and Dealy then followed with a set of revealing experiments using a novel sliding viscometer.
Pearson 5 and Petrie consider the length scales involved near the wall, nominating L as the scale of the a p p a r a t u s e. It is the relative scales of these lengths, they said, t h a t determines, to a large extent, the b o u n d a r y conditions. In the various cases: 1. In this case the small scale of the asperities means t h a t any flow over t h e m is at very low Reynolds number, and viscous forces d o m i n a t e near the boundary.
Here, the argument of Richardson shows t h a t actual adhesion to the wall is not necessary in order to see, on the gross scale L , an apparent no-slip condition, even if slip occurs on the gp scale. This condition is typical of coarse powders in smooth containers, where slip at the wall is often seen. In summary, with the benefit of decades of b o t h theoretical and experimental interest, we can now conclude t h a t at least three factors are of importance: 5John Richard Anthony Pearson was born on September 18th, His early schooling took place in Egypt and then England.
After two years military service, he entered Trinity College, Cambridge in and he obtained a BA in the Mathematical and Mechanical Sciences in There followed a year at Harvard University, where he obtained an AM. He returned to Trinity in and was awarded a PhD in This really takes care of all the classical evidence on liquids and non-rarefied gases and shows that a no-slip condition is appropriate in these cases; chemical adhesion is not needed to avoid slip; see also recent work by Sarkar and Prosperetti We have chosen to locate our discussion of slip within the context of "The growing years before ", but it is clear that interest in the subject stretches beyond that time frame.
Indeed, its study is still in vogue. Theoretical Non-Linear D e v e l o p m e n t s In one sense the present chapter has already addressed a number of topics of a theoretical nature, mostly in the linear regime. The inverse problem for capillary flow and the general question of slip clearly involve a significant theoretical component. But we now want to concentrate on the distinct theoretical problems involved in 'constitutive equations' Developments in the period were sparse but significant, and they have not received the attention they merit.
We begin with a few remarks about the generalization of classical infinitesimal-strain elasticity theory to include finite strains. The slow developments in this area can be traced via the works of Truesdell , Truesdell and Noll and Murnaghan The effective solutions to some key problems was achieved by Rivlin b ; see also Rivlin and the books by Treloar and Ogden A noteworthy constitutive model for rubber was introduced at this time by Mooney ; it is now often referred to as the Mooney-Rivlin equation.
The early experiments of Poynting showed that a rubber rod lengthened when it was twisted, against all expectations of small-strain theory. This clue was not followed up until the work of Rivlin and Saunders on torsion was published in The theory of plasticity also made great strides following work by Hencky, yon Mises, Geiringer and Prager.
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Hill provides a convenient history of developments in this area. We shall now consider developments in non-Newtonian fluid mechanics in more detail. V is the usual material derivative; A and G are moduli, A is a relaxation time, p is the pressure and cr is the stress tensor. Stanislaus Zaremba , also from Cracow, disagreed with Natanson over the formulation, primarily because 2. As one m i g h t expect, several more defending a n d a t t a c k i n g papers were p u b l i s h e d in in the Bulletin of the A c a d e m y of Sciences of Cracow, b u t there were few followers and little external interest.
T h e next scientist to consider Z a r e m b a ' s work seriously seems to have been Heinrich Hencky. Covariant suffices are written below and contravariant suffices above, and the usual summation convention for repeated suffices is assumed; usually we simply use Cartesian tensors.
He spent a few years with the Alsatian railways and then went to Russia just before the First World War. He was taken prisoner in Kharkov Charkow and interned in the Urals when the war broke out. After the war, he taught at Darmstadt, Dresden and Delft; while at the Technical University in Delft Holland , he did the work on slip-line theory, plasticity and basic rheology, for which he is best known.
In , he went to the Massachusetts Institute of Technology as Associate Professor for one year, and in he taught what must have been the first regular course entitled "Rheology". After that, he returned to Delft and then to Germany and seems to have published less. Interestingly, during the Second World War, he travelled from Germany to an ASME meeting in Philadelphia in and gave a paper on plate and shell theory, which was printed in the Journal of Applied Mechanics in He was described only as "Mechanical Engineer, Mainz", and was probably engaged in the war effort.
After the second World-War, he worked in German industry. He died in a mountain-sports accident on 6 July These ideas were noted by Oldroyd and further developed by Lodge Hencky also invented the logarithmic measure which is now known as 'the Hencky strain' a and, in a paper devoted to the foundations of hydrodynamics, he also considered rates of change of stress Hencky b.
He produced a Zaremba-like constitutive equation: l?
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This was another step in the path towards invariant constitutive relations, but no problems were solved by Hencky; he does not refer to Zaremba's papers. Hencky b; Weissenberg Since often only rectilinear shearing flows were studied, the consequences of this assumption about pressure were not noticed. For example, Fromm assumed a bulk elastic law to find the mean pressure, but in simple shearing this does not determine p.
In , Weissenberg noted some of Hencky's remarks about the difference between shearing and irrotational kinematics. It has to be admitted that Weissenberg's presentation was not as clear as it might have been. As will become clear later, especially following the work of Rivlin a , equations like 2. In this period, one also saw a full three-dimensional tensor description by Hohenemser and Prager of the inelastic, variable viscosity concept depending on the three invariants of the rate of deformation.
In summary, we see, by , many of the essential ideas of a rate-dependent fluid, and the beginning of the systematic use of tensor notation in rheology by Hencky, Weissenberg, Prager and others. This predates Reiner's b paper, which is sometimes credited with the introduction of tensorial methods in rheology see, for example, Truesdell, , p Finally, the use of the deviator c r - 5l trer I for the stress T is clearly not satisfactory in general flows, since a further isotropic pressure will still be needed for momentum balance for an incompressible fluid.
In , Hans Fromm in Berlin took up the problem again and in an engineering journal Ingenieur-Archiv he set about considering and applying the corotational ideas of Hencky. Fromm also tried to solve the capillary flow problem for these viscoelastic constitutive equations and, in later publications , , he attempted to interpret theoretically some of Philippoff's experimental results, this being the first recorded attempt to compare nonlinear viscoelastic theory and experiment for a polymeric liquid.
Fromm's paper also considered pure irrotational shearing, which yielded a constant viscosity for the model. Many of these ideas would be rediscovered more or less painfully in the next forty or fifty years. One must also admire Fromm's persistence in persevering with his pioneering ideas through fifteen years, which included the war.
For completeness, we note that Eisenschitz and Zaremba also solved some simple flow problems, but the work attracted little interest at the time. While it is now conceded that simple corotational derivative models are not very suitable for polymeric liquids, the apparent indifference of later workers to Fromm's work is surprising.
Truesdell in his survey of elasticity and fluid dynamics does footnote the work and gives references, but Noll only refers to Zaremba explicitly and to Truesdell Oldroyd refers to Hencky's paper on convected coordinates, but not to more recent papers. Weissenberg did note the Hencky distinction between irrotational and shear flows, but did not refer to the corotational derivative papers of Fromm in any of his papers.
In conclusion, we see that, between to , the idea of complex stress derivatives occurred to a number of distinguished scientists, but there was a general reluctance to cross-reference earlier work by other authors. The foregoing discussion has been essentially restricted to d i f f e r e n t i a l constitutive models, reflecting the subject matter in the papers of relevance. However, it would be remiss of us if we did not draw attention to the excellent paper of Herbert Leaderman , who studied the response of textile fibres and in the process used what appears to be 40 CHAPTER 2.
He studied at the Universities of Vienna, Berlin and Jena, majoring in mathematics, with physics and chemistry as subsidiary subjects. Over the years, Weissenberg worked with success in a wide range of disciplines, including mathematics, medical X-rays, X-ray crystallography and, of course, rheology. In , he joined the team of M Polyani at the Kaiser-Wilhelm Institut fiir Faserstoffchemie in Berlin-Dahlem and over a six year period worked with distinction on X-ray crystallography.
He built up an impressive reputation and his efforts, both theoretical and experimental, resulted in the design of an instrument which became known as the 'Weissenberg X-Ray Goniometer'. In , Weissenberg became a refugee and took up residence in the UK, where he concentrated on his rheological interests. The present historical text gives ample proof of his achievements in the field.
In addition to his own work, Weissenberg was also a motivator of research amongst colleagues and friends, as is clear from the discussion concerning his association with J E Roberts and others during the Second World War. Karl Weissenberg had the reputation of 'being an entirely engaging and unselfish person', of 'being a delightful companion, an ever helpful friend and also an excellent tennis player'. An obituary, written at the time of his death in , concluded that 'he was notable in his scientific achievements and noble in his personal qualities'.
I n t r o d u c t i o n was a vintage in the history of rheology. It saw the formal introduction of the term rheology, witnessed the founding of the first national Society of Rheology in the USA and the creation of the first scientific journal devoted exclusively to rheology the Journal of Rheology. That ts not to say that no rheological research was done before But the time did come to recognize the distinctive character of the evolving field. So, at a Plastics Symposium in the US, a committee was formed to investigate the formation of a permanent organization to look after "Rheology".
The committee met on April 29th, , and the first meeting of the Society took place on December 19th and 20th, It was further agreed to hold meetings annually. The formal definition of rheology is invariably attributed to B i n g h a m , who was apparently assisted by a colleague who was a classics scholar.
The first issue of the Journal of Rheology contains the relevant background and definition: THE N A M E The term deformation and flow of matter is a rather cumbersome one to cover the subjects of elasticity, viscosity and plasticity. The Greek roots to flow p gcz and science AdTos , already familiar in numerous words such as rheostat and geology, made the term theology appear to be at the same time distinctive and self explanatory.
By common consent, rheology excludes such subjects as pure hydrodynamics and the classical theory of elasticity. Interestingly, it was attended by two of the three scientists who, each in their own way, were to play a significant role in the evolving science. Unfortunately, the third member of the influential triumvirate, Professor M a r c u s R e i n e r , who had figured prominently in early discussions with Bingham, had to be in Palestine at the time, although he did publish two papers in the first issue of the Journal of Rheology Reiner a,b.
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