Continuum Mechanics Through the Twentieth Century

This chapter has for object to remind the reader of the early developments of continuum mechanics-after the seminal works in mechanics by Descartes, Huygens, Newton and Leibniz-in the expert hands of the initiators of this science (the Bernoulli family, d’Alembert, Euler, Lagrange). This was rapidly followed by the foundational contributions of the first half of the Nineteenth century with Cauchy and Navier (in France), Piola (in Italy), Kirchhoff (in Germany), and those of various giants of science such as Green, Kelvin, Stokes, Maxwell, Boussinesq, Poiseuille, Clebsch, von Helmholtz, Voigt, Mohr, and Barré de Saint-Venant later in the century. The emphasis is placed on the role played by so-called “ingénieurs-savants”, many of them educated at the French Ecole Polytechnique and the engineering schools inspired by this school all over Europe. Lamé, Navier and Duhamel in France and their Italian colleagues are examples of such people who harmoniously combined works in a much wanted contribution to civil engineering and a sure mathematical expertise in analysis. In contrast, the German and English contributors were more inclined towards an emerging true mechanical engineering and sometimes a burgeoning mathematical physics. This means that various national styles were being created despite the overall solution power of analysis and the birth of linear and tensor algebras.

Chapter Two deals with the transition period between circa 1880 and 1914, which prepares the way for the Twentieth century. It also advocates an attitude towards a development that is characteristic of a period when many engineering scientists believe in a then fixed paradigm and no further evolution is thought possible in spite of a contemporary revolution in theoretical and mathematical physics. Of course this corresponds to a period of natural consolidation with the general creation of efficient engineering schools all over Europe and the appearance of newborn ones in the USA. Of particular interest in this rather quiet landscape are queries concerning going beyond the most traditional behaviours (linear elasticity and Newtonian viscous fluids). Here are distinguished the emerging attempts at the description of more involved behaviours such as viscoelasticity (Voigt, Boltzmann, Volterra), and friction and plasticity (Tresca, Barré de Saint-Venant, Lévy, Huber, Mises). In spite of the relative quietness of the period, new interests of investigation are considered, mainly in the dynamic frame, the consideration of continua with internal degrees of freedom (Duhem, the Cosserat brothers), and elements of homogenization theory. Perhaps more attractive at the time were the discussions about the general principles of mechanics by people like Hertz, Mach, Duhem (with his general energetics), Poincaré, Hamel and Hellinger. This pondering will prove extremely useful in the second half of the Twentieth century.

This first specialized chapter deals with the awaited generalization to mechanical behaviours that deviate from linear elasticity and standard Newtonian viscous fluids, that is, elasticity in large deformations and the rheology of complex fluids. These extensions were kindled by the mechanics of rubber elasticity and artificial fabrics and of fluids with high viscosity and visco-plastic response. It happens that the same scientists were involved in these two lines as a result of a required focus on the bases of continuum mechanics, in particular the theory of finite deformations in a rational geometric background, and the need to account for complex flow features in some fluids. Ronald Rivlin, with his incommensurable contributions, is the great hero in this adventure. Other scientists whose work was seminal are initially E. Bingham, M. Reiner, L.G.R. Treloar, P. J. Flory, M.A. Mooney, and F.D. Murnaghan, and more recently J.G. Oldroyd, A.E. Green, J.L. Ericksen, C.A. Truesdell, B.D. Coleman, and W. Noll. The survey includes the models of neo-Hookean materials, Mooney-Rivlin materials, Rivlin-Ericksen fluids, and unsuccessful attempts such as those of Reiner-Rivlin fluids and hypoelasticity. Appropriately introduced tools have been those of Rivlin-Ericksen tensors, Oldroyd and Jaumann time derivatives, and invariant representations of scalar and tensorvalued functions. Through Rivlin and his co-workers the whole carries a strong print of British applied mathematics although Italian and Russian contributions to nonlinear elasticity cannot be overlooked. The mechanics of soft living tissues has now become the best field of application of these developments.

Of great importance is the innovative and enduring influence that a well-organized professional society may have on the development of a science. This is the case of mechanical engineering with the American Society of Mechanical Engineers. As documented in this chapter, this society provided a specific forum to its members at a spot-on time. It brought a spirit that permeated many American works in continuum mechanics. This may be described as: good modelling (without too much abstraction and unnecessary formalism), good applied mathematics providing real applicable solutions with numbers and curves, and a specific interest in the relationship of these solutions with experimental facts. The prominent figure obviously is the founder of the Applied Mechanics Division of the ASME, Stephen P. Timoshenko. For easiness in presentation, a few most influential centres are highlighted in this chapter. These are Stanford (with Timoshenko himself), the M.I.T (with Eric Reissner), Brown (with William Prager) and Columbia (with Raymond Mindlin). Each of these is most representative of identified avenues of research: advanced strength of materials, mathematics applied to problems of engineering, tremendous and contagious developments in the theory of plasticity, and accurate dynamical theory of structural elements (e.g., plates and shells) and coupled fields (electroelasticity). This was to swarm all over the USA and then the whole world community of mechanics.

In contradistinction with Chapter 4, the present chapter deals with a more voluntary tendency at axiomatization and abstraction, probably inherited from the early writings of Hamel in Germany and Duhem in France at the dawn of the Twentieth century. Such a program was essentially expanded under the magisterial leadership of Clifford Truesdell in the USA, after his careful historical perusal of mechanics from the origin to the 1940s. The pursued aim was a rational reconstruction of the whole of continuum mechanics in a somewhat Bourbakian style. Impressive encyclopedic contributions by Truesdell, Toupin and Noll were the lighthouses that “illuminated” the world community of mechanics. Simultaneously, a scientific journal (the A.R.M.A.) set forth standards and a definite style. A rather strict thermodynamic frame work was proposed by B.D. Coleman and W. Noll. The notions of fading memory and the required satisfaction of the Clausius-Duhem (thermodynamic) inequality are fundamental ingredients in this presentation. However, attractive as it was, some parts of this true credo imposed too much constraint on the thermomechanical modelling so that some freedom had to be granted and some generalization were necessary in a too much corseting frame work. As dutifully exposed in this chapter, this led to the conception of a rational extended thermodynamics (in particular by I. Müller) as also a less revolutionary but very efficient thermo-mechanics with well-chosen internal state variables.

Although pertaining to specific aspects of the development of continuum mechanics in the period of interest, it happens that this coincides with a technical expertise in applied mathematics particularly well cultivated in the United Kingdom, hence, an unavoidable regional bias in spite of the international nature of science. The prevailing influence of some institutions such as the University of Cambridge is obvious, while, unexpectedly, research fostered by technical problems met during the Second World War, also had a strong influence on the selection of projects. A recurring theme is a specific interest in mathematical problems posed by the theory of elasticity, no doubt a consequence of the enduring influence of past “elasticians” of great mathematical dexterity among whom A.E.H. Love must be singled out. A clear-cut emphasis was placed on problems dealing with the existence of field singularities such as happens with cracks, dislocations, and other material defects. Here great names are those of A.A. Griffith, Ian Sneddon, “Jock” Eshelby, and A.N. Stroh. Simultaneously, an “immoderate” but fruitful taste for problems of elastic wave propagation with applications in both mechanics and geophysics was demonstrated and still remains a subject of attraction. Furthermore, a geometrical approach to defect theory was proposed by a group around Bruce Bilby, while Rodney Hill produced among the most powerful results in plasticity theory and homogenisation procedure. Still it is the mathematical dexterity and elegance allied to a deep physical insight that best characterizes most of these works.

The French case is peculiar because of the well-known “French exception”. In the present case, this exception is provided—in spite of attempts at changes—by the enduring distinction between university education and the celebrated engineering schools familiarly known in French as “Grandes Ecoles” and of which the Ecole polytechnique remains the world acknowledged paragon. After an attempt at explaining this duality in higher French education as well as the ever present centralization of all things in France, of necessity the development of mechanical engineering sciences in this country through the Twentieth century is examined by schools and centres of influence with mention of the most remarkable individuals who have allowed a renascence of continuum mechanics in the country: namely, the University of Paris (Paris 6 also called UPMC to be more precise) with Paul Germain, the Ecole Polytechnique with Jean Mandel, the University of Grenoble with its polytechnic institute, and other centres which have developed in the period 1950–2000 in spite of the Parisian Jacobinism. Each centre has succeeded to develop special trends in continuum mechanics at the international level of competition, often in the fields of continuum thermomechanics, nonlinear deformations, plasticity and visco-plasticity, rheology, fracture mechanics, coupled fields, homogenization techniques, and other mathematical methods. This is presented in great detail with as much neutrality as possible from the part of a long-time Parisian.

Poland is a country that suffered much from it various neighbors for almost 200 years. Having finally reached a certain stability after World War II, but under the acute and “benevolent” control of its big eastern brother, the Polish mechanics community succeeded in developing a remarkable research activity. Such activity justifies this independent chapter, all the more that the author knows well the country, having started his friendly visits there in the 1970s. No doubt that this development, out of proportion with the size of the country, is due to the excellence and hard work of a selected group of engineers, physicists, and applied mathematicians, among them W. Olszak, W. Nowacki, S. Kaliski, A. Sawszuk, and H. Zorski, and their disciples. These people rebuilt Polish mechanics on a ground that was solidly established early in the Twentieth century (by scientists such as Huber, Zaremba, Natanson, Zorawski, and Banach) as recalled at the beginning of the chapter. As exposed next, the main subject matters entertained in the second half of the Twentieth century have been plasticity, thermoelasticity, coupled fields (electroelasticity), wave dynamics, and generalized continuum mechanics in its different avatars. This undoubtedly received world applause. The positive role played by the Polish Academy of Sciences with its research centres is emphasized.

Contrary to France, Germany is not a centralized country, having lastly taken a federal form but having in the past been made of a variety of smaller states. A consequence of this mosaiclike structure is the multiplicity of scientific and engineering strongholds in friendly competition. This, together with the traditional strength of the German mechanical industry and the success of German scientific giants in the Nineteenth century, explains the status of continuum mechanics in Germany in the second half of the Twentieth century when a revival was necessary after World War II. Before WWII, the strength of the German mechanical community had materialized in a well-organized scientific society (GAMM) and journal (ZAMM) and influential textbooks (Föppl, Hamel). After WWII, the network of celebrated Technical universities was successfully revived and extended, while the Journal known as Ingenieur Archiv won prominence. The chapter exposes the role played by various centres (Munich, Bochum, Hannover, and also Berlin, Darmstadt, Aachen, etc) with the corresponding strong personality of the local leaders. Rather typical interests of German institutions are reported involving problems of plasticity, generalized continuum mechanics, fracture mechanics, and more recently the continuum thermodynamics of complex materials and computational mechanics. A successful blend of modern continuum mechanics and numerical techniques justified in a rigorous mathematical frame has thus emerged.

This chapter mostly concerns European countries that do not receive a separate focus in specific chapters. In spite of the tentative construction of a united Europe, the offered presentation still reflects the print left by History in the Nineteenth century and various zones of influence. Thus apart from the originality of Switzerland, the following large regions are identified: the Benelux with a prevailing role played by the Netherlands, Scandinavia considered as a historical and cultural linguistic region with special strength in Sweden and Denmark, the former Austro-Hungarian Empire, and southern European countries. A case at point is that of the former Austro-Hungarian Empire because this well-organized political structure - doomed to disappear with the two world conflicts - succeeded in building a network of efficient polytechnic schools in its various “provinces”. Strong individual personalities could emerge including in former Yugoslavia and Romania. The geometrical theory of dislocations in Serbia and a specific strength in applied mathematics in Romania are witness of this trend. Italy, adorned by a long section, continues to demonstrate its traditional strength in civil engineering and the allied mathematical analysis. India and China receive but a cursory treatment, while immense expectations are to materialize soon. In Japan, two original characters are singled out, K. Kondo and T. Tokuoka. With time, most countries perused have fit in an international view of continuum mechanics that shares similar subjects of interests (e.g. complex mechanical behaviour, plasticity, numerics, thermomechanics, and coupled fields).

It is remarked that essentially for ideological reasons and the use of an original language and alphabet, contributions from this immense and powerful country have often been belittled or altogether neglected. This chapter tries to correct this misconception and biased treatment. In particular, one cannot discard some original facts, among them the general high quality of teaching at high-school and university levels, the essential role played by the Academy of Sciences and its various branches, and the friendly rivalry between Moscow and Leningrad/St Petersburg. That is why, after briefly recalling the role of some precursors, attention is focused on these two main centres that host a multiplicity of competing institutions. The former Soviet Union had the chance to foster strong personalities in continuum mechanics, e.g., L.I. Sedov, A.A. Ilyushin, A.Y. Ishlinsky, G.I. Barenblatt, V.V. Novozhilov, Y.N. Rabotnov, L.M. Kachanov, A.I. Lurie, I.A. Kunin, N.I. Mushkeshisvili, S.A. Amsbartsumian and many others. Their contributions in all fields of continuum mechanics and those of their disciples are surveyed albeit much too briefly. Their books, in contrast to their unevenly translated papers, had a world wide influence in the field. Some of the now much cultivated research fields find their origin in this country that experienced different political schemes (Russian Empire, Soviet Union, Russia, and the New Independent States) and went through difficult times.

The combination of pure continuum mechanics and electromagnetism cannot be a simple linear superimposition. That explains why it took some time to arrive at a rational formulation of this exemplary coupled-field theory. In spite of the experimental discovery of simple coupled effects in the Nineteenth century (e.g., magnetostriction, piezoelectricity), one practically had to await the second half of the Twentieth century to find a rational theory of deformable magnetized, electrically polarisable and electricity conducting continua. This is due to a small group of mechanicians who possessed a good apprehending of electromagnetic theory. The role of scientists such as R.A. Toupin, R.D. Mindlin, A.C. Eringen, W.F. Brown, H.F. Tiersten, M. Lax, D.F. Nelson, K. Hutter and the author of this book was instrumental in this intellectual construct. This is reported in a vivid manner, without neglecting the constructive works of electrical engineers and some mathematical physicists. After a brief survey of Nineteenth-century developments in electromagnetism the emphasis is placed on the seminal role played by Toupin in the 1950s and 1960s and on the author’s own contributions in the period 1970–1990 concerning the fundamentals and the formulation of nonlinear electro- and magneto-elasticity often in the footsteps of H.F. Tiersten. A particular attention is paid to the evolution of the notions of electromagnetic force, momentum and stress tensor, and electro-magneto-mechanical couplings at the energy level.

This chapter focuses on a field of continuum mechanics that belongs almost entirely to the twentieth century, so called generalized continuum mechanics. First, a special effort is produced to define this term which essentially means going beyond the traditional view of Cauchy—with the notion of stress introduced by this early nineteenth-century scientist. Three possible paths to such a generalization are discussed with the related mention of main scientific contributors: involving an additional microstructure at each material point in addition to the traditional translational degree of freedom (e.g., micromorphic media, Cosserat continua, in modern times works by Eringen and others), or a better analytic description of the displacement field at each material point by introducing higher order gradients of this displacement in the energy density (e.g., in a theory mostly expanded by Mindlin), or else calling for a truly nonlocal theory that leads to considering spatial functionals for the constitutive equations—this follows contributors such as Kröner, Rogula, Kunin, and Eringen. A more drastic “generalization” started in the mid 1950s involves a loss of the Euclidean nature of the material manifold, as may apply in a densely defective crystal. In each case, the pioneers are mentioned and the most recent formulations are briefly sketched out.

Starting with pioneering works by Peach, Koehler and Eshelby, an original branch of continuum physics has developed in the period 1950—2010 that consists in providing means of evaluating the evolution of particular material zones of bodies under the action of external loadings. These zones are essentially more or less localized regions of the bodies in which irreversible changes of properties occur through a reorganization of material components of which fracture is the most drastic form. This is interpreted as changes of local configuration in the accepted view of the continuum mechanics of deformable solids. The present conspectus reviews the formidable progress achieved in this “configurational mechanics” from an historical and somewhat personal perspective. In this general view phenomena such as fracture, phase transformations, the presence of material heterogeneities, and more generally the expansion of structural defects of different types find a natural unified frame work. Here the emphasis is placed on the original works, the various breakthroughs and their contributors, the connection with the notion of “material” force, the modern —but often unfamiliar —concept of mechanics on the material manifold, a strategy of post-processing to evaluate driving forces or to improve numerical schemes, and a methodology imported from mathematical physics. Unavoidable ingredients are those of Eshelby stress tensor, material momentum in dynamics, and material forces of inhomogeneity.

Relativity theory as understood by Einstein is a true Twentieth century development. After the introduction of the four-dimensional version of special relativity by Minkowski and that of energy-moment tensor, to which must be added the fact that general relativity is per se a continuum theory, there was need for a true relativistic theory of the continuum. The present chapter reports in a critical manner the progress made in this theory in two distinct periods, one extending before World War II, and the second in the rough time interval 1950–1980, when solutions were finally proposed in an inclusive way. The first period dealt with attempts at discussing the ad hoc introduction of classical concepts in this new landscape. This included the notion of perfect fluids and a debated discussion of the possible generalization of the notion of rigid-body motion—without which the notion of elasticity could not be introduced. A breakthrough is represented by Eckart’s introduction of a systematic covariant space-and-time resolution of four-dimensional objects and of early elements of continuum thermodynamics. This, combined with the natural influence of the then new trends in classical continuum mechanics (rationalization à la Truesdell), then led to a modern, more axiomatic, formulation that allowed a rational construct of relativistic elasticity, and its generalization to more complex thermomechanical schemes (including generalized continua) and electromagnetic deformable bodies, a development in which the author has been more than a passive witness.

This concluding chapter first summarizes the historical developments exposed in a critical manner in all preceding chapters. It emphasizes the various nonlinear generalizations proposed in the Twentieth century as also the role played by remarkable schools and individuals in the fantastic progress reached in this period. This concerns more realistic material behaviors (accounting for microstructures, involving coupled fields), a more axiomatic and thermodynamically justified approach, and a clear internationalization of engineering science. Simultaneously, progress in other collateral branches of sciences, both theoretical and experimental, has fostered a rapid, sometimes unexpected, progress in the science of continuum mechanics. The latter has become more a mechanics of materials while developing tremendously its applicable side with performing numerical schemes and requiring new developments in applied analysis and the interpretation in terms of advanced geometrical concepts. Final remarks points at the new marked interest of continuum mechanics for living matter and the unavoidable relationship, both intellectually and numerically, between different scales of description, a trademark at the dawn of the Twenty first century.