solids, mechanics of

▪ physics

Introduction

      science concerned with the stressing (stress), deformation (deformation and flow), and failure of solid materials and structures.

      What, then, is a solid? Any material, fluid or solid, can support normal forces. These are forces directed perpendicular, or normal, to a material plane across which they act. The force per unit of area of that plane is called the normal stress. Water at the base of a pond, air in an automobile tire, the stones of a Roman arch, rocks at the base of a mountain, the skin of a pressurized airplane cabin, a stretched rubber band, and the bones of a runner all support force in that way (some only when the force is compressive).

      A material is called solid rather than fluid if it can also support a substantial shearing (shear stress) force over the time scale of some natural process or technological application of interest. Shearing forces are directed parallel, rather than perpendicular, to the material surface on which they act; the force per unit of area is called shear stress. For example, consider a vertical metal rod that is fixed to a support at its upper end and has a weight attached at its lower end. If one considers a horizontal surface through the material of the rod, it will be evident that the rod supports normal stress. But it also supports shear stress, and this becomes evident when one considers the forces carried across a plane that is neither horizontal nor vertical through the rod. Thus, while water and air provide no long-term support of shear stress, granite, steel, and rubber normally do so and are therefore called solids. Materials with tightly bound atoms or molecules, such as the crystals formed below melting temperature by most substances or simple compounds and the amorphous structures formed in glass and many polymer substances at sufficiently low temperature, are usually considered solids.

      The distinction between solids and fluids is not precise and in many cases will depend on the time scale. Consider the hot rocks of the Earth's mantle. When a large earthquake occurs, an associated deformation disturbance called a seismic wave propagates through the adjacent rock, and the entire Earth is set into vibrations which, following a sufficiently large earthquake, may remain detectable with precise instruments for several weeks. The rocks of the mantle are then described as solid—as they would also be on the time scale of, say, tens to thousands of years, over which stresses rebuild enough in the source region to cause one or a few repetitions of the earthquake. But on a significantly longer time scale, say, on the order of a million years, the hot rocks of the mantle are unable to support shearing stresses and flow as a fluid. The substance called Silly Putty (trademark), a polymerized silicone gel familiar to many children, is another example. If a ball of it is left to sit on a table at room temperature, it flows and flattens on a time scale of a few minutes to an hour. But if picked up and tossed as a ball against a wall, so that large forces act only over the short time of the impact, the Silly Putty bounces back and retains its shape like a highly elastic solid.

      Several types of solids can be distinguished according to their mechanical behaviour. In the simple but common case when a solid material is loaded at a sufficiently low temperature or short time scale, and with sufficiently limited stress magnitude, its deformation is fully recovered upon unloading. The material is then said to be elastic. But substances can also deform permanently, so that not all the deformation is recovered. For example, if one bends a metal coat hanger substantially and then releases the loading, it springs back only partially toward its initial shape; it does not fully recover but remains bent. The metal of the coat hanger has been permanently deformed, and in this case, for which the permanent deformation is not so much a consequence of longtime loading at sufficiently high temperature but more a consequence of subjecting the material to large stresses (above the yield stress), the permanent deformation is described as a plastic deformation and the material is called elastic-plastic. Permanent deformation of a sort that depends mainly on time of exposure to a stress—and that tends to increase significantly with time of exposure—is called viscous, or creep, deformation, and materials that exhibit those characteristics, as well as tendencies for elastic response, are called viscoelastic solids (or sometimes viscoplastic solids, when the permanent strain is emphasized rather than the tendency for partial recovery of strain upon unloading).

      Solid mechanics has many applications. All those who seek to understand natural phenomena involving the stressing, deformation, flow, and fracture of solids, as well as all those who would have knowledge of such phenomena to improve living conditions and accomplish human objectives, have use for solid mechanics. The latter activities are, of course, the domain of engineering, and many important modern subfields of solid mechanics have been actively developed by engineering scientists concerned, for example, with mechanical, structural, materials, civil, or aerospace engineering. Natural phenomena involving solid mechanics are studied in geology, seismology, and tectonophysics, in materials science and the physics of condensed matter, and in some branches of biology and physiology. Furthermore, because solid mechanics poses challenging mathematical and computational problems, it (as well as fluid mechanics) has long been an important topic for applied mathematicians concerned, for example, with partial differential equations and with numerical techniques for digital computer formulations of physical problems.

      Here is a sampling of some of the issues addressed using solid mechanics concepts: How do flows develop in the Earth's mantle and cause continents to move and ocean floors to subduct (i.e., be thrust) slowly beneath them? How do mountains form? What processes take place along a fault during an earthquake, and how do the resulting disturbances propagate through the Earth as seismic waves, shaking, and perhaps collapsing, buildings and bridges? How do landslides occur? How does a structure on a clay soil settle with time, and what is the maximum bearing pressure that the footing of a building can exert on a soil or rock foundation without rupturing it? What materials should be chosen, and how should their proportion, shape, and loading be controlled, to make safe, reliable, durable, and economical structures—whether airframes, bridges, ships, buildings, chairs, artificial heart valves, or computer chips—and to make machinery such as jet engines, pumps, and bicycles? How do vehicles (cars, planes, ships) respond by vibration to the irregularity of surfaces or mediums along which they move, and how are vibrations controlled for comfort, noise reduction, and safety against fatigue failure? How rapidly does a crack grow in a cyclically loaded structure, whether a bridge, engine, or airplane wing or fuselage, and when will it propagate catastrophically? How can the deformability of structures during impact be controlled so as to design crashworthiness into vehicles? How are the materials and products of a technological civilization formed—e.g., by extruding metals or polymers through dies, rolling material into sheets, punching out complex shapes, and so on? By what microscopic processes do plastic and creep strains occur in polycrystals? How can different materials, such as fibre-reinforced composites, be fashioned together to achieve combinations of stiffness and strength needed in specific applications? What is the combination of material properties and overall response needed in downhill skis or in a tennis racket? How does the human skull respond to impact in an accident? How do heart muscles control the pumping of blood in the human body, and what goes wrong when an aneurysm develops?

History

      Solid mechanics developed in the outpouring of mathematical and physical studies following the great achievement of Newton in stating the laws of motion, although it has earlier roots. The need to understand and control the fracture of solids seems to have been a first motivation. Leonardo da Vinci sketched in his notebooks a possible test of the tensile strength of a wire. Galileo, who died in the year of Newton's birth (1642), had investigated the breaking loads of rods under tension and concluded that the load was independent of length and proportional to the cross section area, this being a first step toward a concept of stress. He also investigated the breaking loads on beams that were suspended horizontally from a wall into which they were built.

Concepts of stress, strain, and elasticity

      The English scientist Robert Hooke (Hooke, Robert) discovered in 1660, but published only in 1678, that for many materials the displacement under a load was proportional to force, thus establishing the notion of (linear) elasticity but not yet in a way that was expressible in terms of stress and strain. Edme Mariotte (Mariotte, Edme) in France published similar discoveries in 1680 and, in addition, reached an understanding of how beams like those studied by Galileo resist transverse loadings—or, more precisely, resist the torques caused by those transverse loadings—by developing extensional and compressional deformations, respectively, in material fibres along their upper and lower portions. It was for the Swiss mathematician and mechanician Jakob Bernoulli (Bernoulli, Jakob) to observe, in the final paper of his life, in 1705, that the proper way of describing deformation was to give force per unit area, or stress, as a function of the elongation per unit length, or strain, of a material fibre under tension. The Swiss mathematician and mechanician Leonhard Euler (Euler, Leonhard), who was taught mathematics by Jakob's brother Johann Bernoulli, proposed, among many contributions, a linear relation between stress σ and strain ε, in 1727, of the form σ = Eε, where the coefficient E is now generally called Young's modulus after the British naturalist Thomas Young, who developed a related idea in 1807.

      The notion that there is an internal tension acting across surfaces in a deformed solid was expressed by the German mathematician and physicist Gottfried Wilhelm Leibniz (Leibniz, Gottfried Wilhelm) in 1684 and Jakob Bernoulli in 1691. Also, Jakob Bernoulli and Euler introduced the idea that at a given section along the length of a beam there were internal tensions amounting to a net force and a net torque (see below). Euler introduced the idea of compressive normal stress as the pressure in a fluid in 1752. The French engineer and physicist Charles-Augustin Coulomb (Coulomb, Charles-Augustin de) was apparently the first to relate the theory of a beam as a bent elastic line to stress and strain in an actual beam, in a way never quite achieved by Bernoulli and, although possibly recognized, never published by Euler. He developed the famous expression σ = My/I for the stress due to the pure bending of a homogenous linear elastic beam; here M is the torque, or bending moment, y is the distance of a point from an axis that passes through the section centroid, parallel to the torque axis, and I is the integral of y² over the section area. The French mathematician Antoine Parent introduced the concept of shear stress in 1713, but Coulomb was the one who extensively developed the idea, first in connection with beams and with the stressing and failure of soil in 1773 and then in studies of frictional slip in 1779.

      It was the great French mathematician Augustin-Louis Cauchy (Cauchy, Augustin-Louis, Baron), originally educated as an engineer, who in 1822 formalized the concept of stress in the context of a generalized three-dimensional theory, showed its properties as consisting of a 3 × 3 symmetric array of numbers that transform as a tensor, derived the equations of motion for a continuum in terms of the components of stress, and developed the theory of linear elastic response for isotropic solids. As part of his work in this area, Cauchy also introduced the equations that express the six components of strain (three extensional and three shear) in terms of derivatives of displacements for the case in which all those derivatives are much smaller than unity; similar expressions had been given earlier by Euler in expressing rates of straining in terms of the derivatives of the velocity field in a fluid.

Beams (beam), columns (column), plates, and shells

      The 1700s and early 1800s were a productive period during which the mechanics of simple elastic structural elements were developed—well before the beginnings in the 1820s of the general three-dimensional theory. The development of beam theory by Euler, who generally modeled beams as elastic lines that resist bending, as well as by several members of the Bernoulli family and by Coulomb, remains among the most immediately useful aspects of solid mechanics, in part for its simplicity and in part because of the pervasiveness of beams and columns in structural technology. Jakob Bernoulli proposed in his final paper of 1705 that the curvature of a beam was proportional to its bending moment. Euler in 1744 and Johann's son, Daniel Bernoulli (Bernoulli, Daniel), in 1751 used the theory to address the transverse vibrations of beams, and in 1757 Euler gave his famous analysis of the buckling of an initially straight beam subjected to a compressive loading; such a beam is commonly called a column. Following a suggestion of Daniel Bernoulli in 1742, Euler in 1744 introduced the concept of strain energy per unit length for a beam and showed that it is proportional to the square of the beam's curvature. Euler regarded the total strain energy as the quantity analogous to the potential energy of a discrete mechanical system. By adopting procedures that were becoming familiar in analytical mechanics and following from the principle of virtual work as introduced in 1717 by Johann Bernoulli for such discrete systems as pin-connected rigid bodies, Euler rendered the energy stationary and in this way developed the calculus of variations as an approach to the equations of equilibrium and motion of elastic structures.

      That same variational approach played a major role in the development by French mathematicians in the early 1800s of a theory of small transverse displacements and vibrations of elastic plates. This theory was developed in preliminary form by Sophie Germain (Germain, Sophie) and was also worked on by Siméon-Denis Poisson (Poisson, Siméon-Denis) in the early 1810s; they considered a flat plate as an elastic plane that resists curvature. Claude-Louis-Marie Navier gave a definitive development of the correct energy expression and governing differential equation a few years later. An uncertainty of some duration arose in the theory from the fact that the final partial differential equation for the transverse displacement is such that it is impossible to prescribe, simultaneously, along an unsupported edge of the plate, both the twisting moment per unit length of middle surface and the transverse shear force per unit length. This was finally resolved in 1850 by the Prussian physicist Gustav Robert Kirchhoff (Kirchhoff, Gustav Robert), who applied virtual work and variational calculus procedures in the framework of simplifying kinematic assumptions that fibres initially perpendicular to the plate's middle surface remain so after deformation of that surface.

      The first steps in the theory of thin shells were taken by Euler in the 1770s; he addressed the deformation of an initially curved beam as an elastic line and provided a simplified analysis of the vibration of an elastic bell as an array of annular beams. Johann's grandson, Jakob Bernoulli “the Younger,” further developed this model in the last year of his life as a two-dimensional network of elastic lines, but he could not develop an acceptable treatment. Shell theory did not attract attention again until a century after Euler's work. The first consideration of shells from a three-dimensional elastic viewpoint was advanced by Hermann Aron in 1873. Acceptable thin-shell theories for general situations, appropriate for cases of small deformation, were then developed by the British mathematician, mechanician, and geophysicist Augustus Edward Hough Love in 1888 and by the British mathematician and physicist Horace Lamb in 1890 (there is no uniquely correct theory, as the Dutch applied mechanician and engineer W.T. Koiter and the Soviet mechanician V.V. Novozhilov clarified in the 1950s; the difference between predictions of acceptable theories is small when the ratio of shell thickness to a typical length scale is small). Shell theory remained of immense interest well beyond the mid-1900s, in part because so many problems lay beyond the linear theory (rather small transverse displacements often dramatically alter the way that a shell supports load by a combination of bending and membrane action) and in part because of the interest in such lightweight structural forms for aeronautical technology.

The general theory of elasticity

      Linear elasticity as a general three-dimensional theory began to be developed in the early 1820s based on Cauchy's work. Simultaneously, Navier had developed an elasticity theory based on a simple corpuscular, or particle, model of matter in which particles interacted with their neighbours by a central force attraction between particle pairs. As was gradually realized, following work by Navier, Cauchy, and Poisson in the 1820s and '30s, the particle model is too simple and makes predictions concerning relations among elastic moduli that are not met by experiment. Most of the subsequent development of this subject was in terms of the continuum theory. Controversies concerning the maximum possible number of independent elastic moduli in the most general anisotropic solid were settled by the British mathematician George Green (Green, George) in 1837. Green pointed out that the existence of an elastic strain energy required that of the 36 elastic constants relating the 6 stress components to the 6 strains, at most 21 could be independent. The Scottish physicist Lord Kelvin (Kelvin, William Thomson, Baron) put this consideration on sounder ground in 1855 as part of his development of macroscopic thermodynamics, showing that a strain energy function must exist for reversible isothermal or adiabatic (isentropic) response and working out such results as the (very modest) temperature changes associated with isentropic elastic deformation (see below Thermodynamic considerations (solids, mechanics of)).

      The middle and late 1800s were a period in which many basic elastic solutions were derived and applied to technology and to the explanation of natural phenomena. The French mathematician Adhémar-Jean-Claude Barré de Saint-Venant derived in the 1850s solutions for the torsion of noncircular cylinders, which explained the necessity of warping displacement of the cross section in the direction parallel to the axis of twisting, and for the flexure of beams due to transverse loadings; the latter allowed understanding of approximations inherent in the simple beam theory of Jakob Bernoulli, Euler, and Coulomb. The German physicist Heinrich Rudolf Hertz (Hertz, Heinrich) developed solutions for the deformation of elastic solids as they are brought into contact and applied these to model details of impact collisions. Solutions for stress and displacement due to concentrated forces acting at an interior point of a full space were derived by Kelvin, and those on the surface of a half space by the French mathematician Joseph Valentin Boussinesq and the Italian mathematician Valentino Cerruti. The Prussian mathematician Leo August Pochhammer analyzed the vibrations of an elastic cylinder, and Lamb (Lamb, Sir Horace) and the Prussian physicist Paul Jaerisch derived the equations of general vibration of an elastic sphere in the 1880s, an effort that was continued by many seismologists in the 1900s to describe the vibrations of the Earth. In 1863 Kelvin had derived the basic form of the solution of the static elasticity equations for a spherical solid, and these were applied in following years to such problems as calculating the deformation of the Earth due to rotation and tidal forcing and measuring the effects of elastic deformability on the motions of the Earth's rotation axis.

      The classical development of elasticity never fully confronted the problem of finite elastic straining, in which material fibres change their lengths by other than very small amounts. Possibly this was because the common materials of construction would remain elastic only for very small strains before exhibiting either plastic straining or brittle failure. However, natural polymeric materials show elasticity over a far wider range (usually also with enough time or rate effects that they would more accurately be characterized as viscoelastic), and the widespread use of natural rubber and similar materials motivated the development of finite elasticity. While many roots of the subject were laid in the classical theory, especially in the work of Green, Gabrio Piola, and Kirchhoff in the mid-1800s, the development of a viable theory with forms of stress-strain relations for specific rubbery elastic materials, as well as an understanding of the physical effects of the nonlinearity in simple problems such as torsion and bending, was mainly the achievement of the British-born engineer and applied mathematician Ronald S. Rivlin in the 1940s and '50s.

Waves

      Poisson, Cauchy, and George G. Stokes showed that the equations of the general theory of elasticity predicted the existence of two types of elastic deformation waves which could propagate through isotropic elastic solids. These are called body waves. In the faster type, called longitudinal (longitudinal wave), dilational, or irrotational waves, the particle motion is in the same direction as that of wave propagation; in the slower type, called transverse (transverse wave), shear, or rotational waves, it is perpendicular to the propagation direction. No analogue of the shear wave exists for propagation through a fluid medium, and that fact led seismologists in the early 1900s to understand that the Earth has a liquid core (at the centre of which there is a solid inner core).

      Lord Rayleigh (Rayleigh, John William Strutt, 3rd Baron) showed in 1885 that there is a wave type that could propagate along surfaces, such that the motion associated with the wave decayed exponentially with distance into the material from the surface. This type of surface wave, now called a Rayleigh wave, propagates typically at slightly more than 90 percent of the shear wave speed and involves an elliptical path of particle motion that lies in planes parallel to that defined by the normal to the surface and the propagation direction. Another type of surface wave, with motion transverse to the propagation direction and parallel to the surface, was found by Love for solids in which a surface layer of material sits atop an elastically stiffer bulk solid; this defines the situation for the Earth's crust. The shaking in an earthquake is communicated first to distant places by body waves, but these spread out in three dimensions and to conserve the energy propagated by the wave field must diminish in their displacement amplitudes as r⁻¹, where r is the distance from the source. The surface waves spread out in only two dimensions and must, for the same reason, diminish only as fast as r^−1/2. Thus, the shaking effect of the surface waves from a crustal earthquake is normally felt more strongly, and is potentially more damaging, at moderate to large distances. Indeed, well before the theory of waves in solids was in hand, Thomas Young (Young, Thomas) had suggested in his 1807 lectures on natural philosophy that the shaking of an earthquake “is probably propagated through the earth in the same manner as noise is conveyed through air.” (It had been suggested by the American mathematician and astronomer John Winthrop, following his experience of the “Boston” earthquake of 1755, that the ground shaking was due to a disturbance propagated like sound through the air.)

      With the development of ultrasonic transducers operated on piezoelectric principles, the measurement of the reflection and scattering of elastic waves (elastic wave) has developed into an effective engineering technique for the nondestructive evaluation of materials for detection of such potentially dangerous defects as cracks. Also, very strong impacts, whether from meteorite collision, weaponry, or blasting and the like in technological endeavours, induce waves in which material response can be well outside the range of linear elasticity, involving any or all of finite elastic strain, plastic or viscoplastic response, and phase transformation. These are called shock waves (shock wave); they can propagate much beyond the speed of linear elastic waves and are accompanied by significant heating.

Stress concentrations and fracture

      In 1898 G. Kirsch derived the solution for the stress distribution around a circular hole in a much larger plate under remotely uniform tensile stress. The same solution can be adapted to the tunnellike cylindrical cavity of a circular section in a bulk solid. Kirsch's solution showed a significant concentration of stress at the boundary, by a factor of three when the remote stress was uniaxial tension. Then in 1907 the Russian mathematician Gury Vasilyevich Kolosov, and independently in 1914 the British engineer Charles Edward Inglis, derived the analogous solution for stresses around an elliptical hole. Their solution showed that the concentration of stress could become far greater, as the radius of curvature at an end of the hole becomes small compared with the overall length of the hole. These results provided the insight to sensitize engineers to the possibility of dangerous stress concentrations at sharp reentrant corners, notches, cutouts, keyways, screw threads, and similar openings in structures for which the nominal stresses were at otherwise safe levels. Such stress concentration sites are places from which a crack can nucleate.

      The elliptical hole of Kolosov and Inglis defines a crack in the limit when one semimajor axis goes to zero, and the Inglis solution was adopted by the British aeronautical engineer A.A. Griffith in 1921 to describe a crack in a brittle solid. In that work Griffith made his famous proposition that a spontaneous crack growth would occur when the energy released from the elastic field just balanced the work required to separate surfaces in the solid. Following a hesitant beginning, in which Griffith's work was initially regarded as important only for very brittle solids such as glass, there developed, largely under the impetus of the American engineer and physicist George R. Irwin, a major body of work on the mechanics of crack growth and fracture, including fracture by fatigue and stress corrosion cracking, starting in the late 1940s and continuing into the 1990s. This was driven initially by the cracking of a number of American Liberty ships during World War II, by the failures of the British Comet airplane, and by a host of reliability and safety issues arising in aerospace and nuclear reactor technology. The new complexion of the subject extended beyond the Griffith energy theory and, in its simplest and most widely employed version in engineering practice, used Irwin's stress intensity factor as the basis for predicting crack growth response under service loadings in terms of laboratory data that is correlated in terms of that factor. That stress intensity factor is the coefficient of a characteristic singularity in the linear elastic solution for the stress field near a crack tip; it is recognized as providing a proper characterization of crack tip stressing in many cases, even though the linear elastic solution must be wrong in detail near the crack tip owing to nonelastic material response, large strain, and discreteness of material microstructure.

Dislocations

      The Italian elastician and mathematician Vito Volterra (Volterra, Vito) introduced in 1905 the theory of the elastostatic stress and displacement fields created by dislocating solids. This involves making a cut in a solid, displacing its surfaces relative to one another by some fixed amount, and joining the sides of the cut back together, filling in with material as necessary. The initial status of this work was simply regarded as an interesting way of generating elastic fields, but, in the early 1930s, Geoffrey Ingram Taylor, Egon Orowan, and Michael Polanyi realized that just such a process could be going on in ductile crystals (crystal) and could provide an explanation of the low plastic shear strength of typical ductile solids, much as Griffith's cracks explained low fracture strength under tension. In this case, the displacement on the dislocated surface corresponds to one atomic lattice spacing in the crystal. It quickly became clear that this concept provided the correct microscopic description of metal plasticity, and, starting with Taylor in the 1930s and continuing into the 1990s, the use of solid mechanics to explore dislocation interactions and the microscopic basis of plastic flow in crystalline materials has been a major topic, with many distinguished contributors.

      The mathematical techniques advanced by Volterra are now in common use by earth scientists in explaining ground displacement and deformation induced by tectonic faulting. Also, the first elastodynamic solutions for the rapid motion of crystal dislocations, developed by South African materials scientist F.R.N. Nabarro in the early 1950s, were quickly adapted by seismologists to explain the radiation from propagating slip distributions on faults. The Japanese seismologist H. Nakano had already shown in 1923 how to represent the distant waves radiated by an earthquake as the elastodynamic response to a pair of force dipoles amounting to zero net torque. (All his manuscripts were destroyed in the fire in Tokyo associated with the great Kwanto earthquake in that same year, but copies of some had been sent to Western colleagues and the work survived.)

Continuum plasticity theory

      The macroscopic theory of plastic flow has a history nearly as old as that of elasticity. While in the microscopic theory of materials, the word “plasticity” is usually interpreted as denoting deformation by dislocation processes, in macroscopic continuum mechanics it is taken to denote any type of permanent deformation of materials, especially those of a type for which time or rate of deformation effects are not the most dominant feature of the phenomenon (the terms viscoplasticity, creep, or viscoelasticity are usually used in such cases). Coulomb's work of 1773 on the frictional yielding of soils (soil mechanics) under shear and normal stress has been mentioned; yielding denotes the occurrence of large shear deformations without significant increase in applied stress. His results were used to explain the pressure of soils against retaining walls and footings in the work of the French mathematician and engineer Jean Victor Poncelet in 1840 and the Scottish engineer and physicist William John Macquorn Rankine in 1853. The inelastic deformation of soils and rocks often takes place in situations for which the deforming mass is infiltrated by groundwater, and Austrian-American civil engineer Karl Terzaghi (Terzaghi, Karl) in the 1920s developed the concept of effective stress, whereby the stresses that enter a criterion of yielding or failure are not the total stresses applied to the saturated soil or rock mass but rather the effective stresses, which are the difference between the total stresses and those of a purely hydrostatic stress state with pressure equal to that in the pore fluid. Terzaghi also introduced the concept of consolidation, in which the compression of a fluid-saturated soil can take place only as the fluid slowly flows through the pore space under pressure gradients, according to Darcy's law; this effect accounts for the time-dependent settlement of constructions over clay soils.

      Apart from the earlier observation of plastic flow at large stresses in the tensile testing of bars, the theory of continuum plasticity for metallic (metal) materials begins with Henri Edouard Tresca in 1864. His experiments on the compression and indentation of metals led him to propose that this type of plasticity, in contrast to that in soils, was essentially independent of the average normal stress in the material and dependent only on shear stresses, a feature later rationalized by the dislocation mechanism. Tresca proposed a yield criterion for macroscopically isotropic metal polycrystals based on the maximum shear stress in the material, and that was used by Saint-Venant to solve an early elastic-plastic problem, that of the partly plastic cylinder in torsion, and also to solve for the stresses in a completely plastic tube under pressure.

      The German applied mechanician Ludwig Prandtl (Prandtl, Ludwig) developed the rudiments of the theory of plane plastic flow in 1920 and 1921, with an analysis of indentation of a ductile solid by a flat-ended rigid indenter, and the resulting theory of plastic slip lines was completed by H. Hencky in 1923 and Hilda Geiringer in 1930. Additional developments include the methods of plastic limit analysis, which allowed engineers to directly calculate upper and lower bounds to the plastic collapse loads of structures or to forces required in metal forming. Those methods developed gradually over the early 1900s on a largely intuitive basis, first for simple beam structures and later for plates, and were put on a rigorous basis within the rapidly developing mathematical theory of plasticity about 1950 by Daniel C. Drucker and William Prager in the United States and Rodney Hill in Great Britain.

      The Austrian-American applied mathematician Richard von Mises (Mises, Richard von) proposed in 1913 that a mathematically simpler theory of plasticity than that based on the Tresca yield criterion could be based on the second tensor invariant of the deviatoric stresses (i.e., of the total stresses minus those of a hydrostatic state in which pressure is equal to the average normal stress over all planes). An equivalent yield criterion had been proposed independently by the Polish engineer Maksymilian Tytus Huber. The Mises theory incorporates a proposal by M. Levy in 1871 that components of the plastic strain increment tensor are in proportion to one another just as are the components of deviatoric stress. This criterion was generally found to provide slightly better agreement with experiment than did that of Tresca, and most work on the application of plasticity theory uses this form. Following a suggestion of Prandtl, E. Reuss completed the theory in 1930 by adding an elastic component of strain increments, related to stress increments in the same way as for linear elastic response. This formulation was soon generalized to include strain hardening, whereby the value of the second invariant for continued yielding increases with ongoing plastic deformation, and was extended to high-temperature creep response in metals or other hot solids by assuming that the second invariant of the plastic (now generally called “creep”) strain rate is a function of that same invariant of the deviatoric stress, typically a power law type with Arrhenius temperature dependence.

      This formulation of plastic and viscoplastic, or creep, response has been applied to all manner of problems in materials and structural technology and in flow of geologic masses. Representative problems addressed include the growth and subsequent coalescence of microscopic voids in the ductile fracture of metals, the theory of the indentation hardness test, the extrusion of metal rods and rolling of metal sheets, design against collapse of ductile steel structures, estimation of the thickness of the Greenland Ice Sheet, and modeling the geologic evolution of the Plateau of Tibet. Other types of elastic-plastic theories intended for analysis of ductile single crystals originate from the work of G.I. Taylor and Hill and base the criterion for yielding on E. Schmid's concept from the 1920s of a critical resolved shear stress along a crystal slip plane, in the direction of an allowed slip on that plane; this sort of yield condition has approximate support from the dislocation theory of plasticity.

Viscoelasticity

      The German physicist Wilhelm Weber noticed in 1835 that a load applied to a silk thread produced not only an immediate extension but also a continuing elongation of the thread with time. This type of viscoelastic response is especially notable in polymeric solids but is present to some extent in all types of solids and often does not have a clear separation from what could be called viscoplastic, or creep, response. In general, if all of the strain is ultimately recovered when a load is removed from a body, the response is termed viscoelastic, but the term is also used in cases for which sustained loading leads to strains that are not fully recovered. The Austrian physicist Ludwig Boltzmann (Boltzmann, Ludwig Eduard) developed in 1874 the theory of linear viscoelastic stress-strain relations. In their most general form, these involve the notion that a step loading (a suddenly imposed stress that is subsequently maintained constant) causes an immediate strain followed by a time-dependent strain which, for different materials, either may have a finite limit at long time or may increase indefinitely with time. Within the assumption of linearity, the strain at time t in response to a general time-dependent stress history σ(t) can then be written as the sum (or integral) of terms that involve the step-loading strain response due to a step loading dt′dσ(t′)/dt′ at time t′. The theory of viscoelasticity is important for consideration of the attenuation of stress waves and the damping of vibrations.

      A new class of problems arose with the mechanics of very-long-molecule polymers (polymer), which do not have significant cross-linking and exist either in solution or as a melt. These are fluids in the sense that they cannot long support shear stress, but at the same time they have remarkable properties like those of finitely deformed elastic solids. A famous demonstration is to pour one of these fluids slowly from a beaker and to cut the flowing stream suddenly with scissors; if the cut is not too far below the place of exit from the beaker, the stream of falling fluid immediately contracts elastically and returns to the beaker. The molecules are elongated during flow but tend to return to their thermodynamically preferred coiled configuration when forces are removed.

      The theory of such materials came under intense development in the 1950s after the British applied mathematician James Gardner Oldroyd showed in 1950 how viscoelastic stress-strain relations of a memory type could be generalized to a flowing fluid. This requires that the constitutive relation, or rheological relation, between the stress history and the deformation history at a material “point” be properly invariant to a superposed history of rigid rotation, which should not affect the local physics determining that relation (the resulting Coriolis and centrifugal effects are quite negligible at the scale of molecular interactions). Important contributions on this issue were made by the applied mathematicians Stanisław Zaremba and Gustav Andreas Johannes Jaumann in the first decade of the 1900s; they showed how to make tensorial definitions of stress rate that were invariant to superposed spin and thus were suitable for use in constitutive relations. But it was only during the 1950s that these concepts found their way into the theory of constitutive relations for general viscoelastic materials; independently, a few years later, properly invariant stress rates were adopted in continuum formulations of elastic-plastic response.

Computational mechanics

      The digital computer revolutionized the practice of many areas of engineering and science, and solid mechanics was among the first fields to benefit from its impact. Many computational techniques have been used in this field, but the one that emerged by the end of 1970s as, by far, the most widely adopted is the finite-element method. This method was outlined by the mathematician Richard Courant in 1943 and was developed independently, and put to practical use on computers, in the mid-1950s by the aeronautical structures engineers M.J. Turner, Ray W. Clough, Harold Clifford Martin, and LeRoy J. Topp in the United States and J.H. Argyris and Sydney Kelsey in Britain. Their work grew out of earlier attempts at systematic structural analysis for complex frameworks of beam elements. The method was soon recast in a variational framework and related to earlier efforts at deriving approximate solutions of problems described by variational principles. The new technique involved substituting trial functions of unknown amplitude into the variational functional, which is then rendered stationary as an algebraic function of the amplitude coefficients. In the most common version of the finite-element method, the domain to be analyzed is divided into cells, or elements, and the displacement field within each element is interpolated in terms of displacements at a few points around the element boundary (and sometimes within it) called nodes. The interpolation is done so that the displacement field is continuous across element boundaries for any choice of the nodal displacements. The strain at every point can thus be expressed in terms of nodal displacements, and it is then required that the stresses associated with these strains, through the stress-strain relations of the material, satisfy the principle of virtual work for arbitrary variation of the nodal displacements. This generates as many simultaneous equations as there are degrees of freedom in the finite element model, and numerical techniques for solving such systems of equations are programmed for computer solution.

Basic principles

      In addressing any problem in continuum or solid mechanics, three factors must be considered: (1) the Newtonian equations of motion (Newton's laws of motion), in the more general form recognized by Euler, expressing conservation of linear and angular momentum (momentum, conservation of) for finite bodies (rather than just for point particles), and the related concept of stress, as formalized by Cauchy, (2) the geometry of deformation and thus the expression of strains (strain) in terms of gradients in the displacement field, and (3) the relations between stress and strain that are characteristic of the material in question, as well as of the stress level, temperature, and time scale of the problem considered.

      These three considerations suffice for most problems. They must be supplemented, however, for solids undergoing diffusion processes in which one material constituent moves relative to another (which may be the case for fluid-infiltrated soils or petroleum reservoir rocks) and in cases for which the induction of a temperature field by deformation processes and the related heat transfer cannot be neglected. These cases require that the following also be considered: (4) equations for conservation of mass (mass, conservation of) of diffusing constituents, (5) the first law of thermodynamics, which introduces the concept of heat flux and relates changes in energy (energy, conservation of) to work and heat supply, and (6) relations that express the diffusive fluxes and heat flow in terms of spatial gradients of appropriate chemical potentials and of temperature. In many important technological devices, electric and magnetic fields affect the stressing, deformation, and motion of matter. Examples are provided by piezoelectric crystals and other ceramics for electric or magnetic actuators and by the coils and supporting structures of powerful electromagnets. In these cases, two more considerations must be added: (7) James Clerk Maxwell's (Maxwell's equations) set of equations interrelating electric and magnetic fields to polarization and magnetization of material media and to the density and motion of electric charge, and (8) augmented relations between stress and strain, which now, for example, express all of stress, polarization, and magnetization in terms of strain, electric field, magnetic intensity, and temperature. The second law of thermodynamics, combined with the above-mentioned principles, serves to constrain physically allowed relations between stress, strain, and temperature in (3) and also constrains the other types of relations described in (6) and (8) above. Such expressions, which give the relationships between stress, deformation, and other variables, are commonly referred to as constitutive relations.

      In general, the stress-strain relations are to be determined by experiment. A variety of mechanical testing machines and geometric configurations of material specimens have been devised to measure them. These allow, in different cases, simple tensile, compressive, or shear stressing, and sometimes combined stressing with several different components of stress, as well as the determination of material response over a range of temperatures, strain rates, and loading histories. The testing of round bars under tensile stress, with precise measurement of their extension to obtain the strain, is common for metals and for technological ceramics and polymers. For rocks and soils, which generally carry load in compression, the most common test involves a round cylinder that is compressed along its axis, often while being subjected to confining pressure on its curved face. Frequently, a measurement interpreted by solid mechanics theory is used to determine some of the properties entering stress-strain relations. For example, measuring the speed of deformation waves or the natural frequencies of vibration of structures can be used to extract the elastic moduli of materials of known mass density, and measurement of indentation hardness of a metal can be used to estimate its plastic shear strength.

      In some favourable cases, stress-strain relations can be calculated approximately by applying principles of mechanics at the microscale of the material considered. In a composite material, the microscale could be regarded as the scale of the separate materials making up the reinforcing fibres and matrix. When their individual stress-strain relations are known from experiment, continuum mechanics principles applied at the scale of the individual constituents can be used to predict the overall stress-strain relations for the composite. For rubbery polymer materials, made up of long chain molecules that randomly configure themselves into coillike shapes, some aspects of the elastic stress-strain response can be obtained by applying principles of statistical thermodynamics to the partial uncoiling of the array of molecules by imposed strain. For a single crystallite of an element such as silicon or aluminum or for a simple compound like silicon carbide, the relevant microscale is that of the atomic spacing in the crystals; quantum mechanical principles governing atomic force laws at that scale can be used to estimate elastic constants. In the case of plastic flow processes in metals and in sufficiently hot ceramics, the relevant microscale involves the network of dislocation lines that move within crystals. These lines shift atom positions relative to one another by one atomic spacing as they move along slip planes. Important features of elastic-plastic and viscoplastic stress-strain relations can be understood by modeling the stress dependence of dislocation generation and motion and the resulting dislocation entanglement and immobilization processes that account for strain hardening.

      To examine the mathematical structure of the theory, considerations (1) to (3) above will now be further developed. For this purpose, a continuum model of matter will be used, with no detailed reference to its discrete structure at molecular—or possibly other larger microscopic—scales far below those of the intended application.

Linear and angular momentum principles: stress and equations of motion

 Let x denote the position vector of a point in space as measured relative to the origin of a Newtonian reference frame; x has the components (x₁, x₂, x₃) relative to a Cartesian set of axes, which is fixed in the reference frame and denoted as the 1, 2, and 3 axes in Figure 1—>. Suppose that a material occupies the part of space considered, and let v = v(x, t) be the velocity vector of the material point that occupies position x at time t; that same material point will be at position x + vdt an infinitesimal interval dt later. Let ρ = ρ(x, t) be the mass density of the material. Here v and ρ are macroscopic variables. What is idealized in the continuum model as a material point, moving as a smooth function of time, will correspond on molecular-length (or larger but still “microscopic”) scales to a region with strong fluctuations of density and velocity. In terms of phenomena at such scales, ρ corresponds to an average of mass per unit of volume, and ρv to an average of linear momentum per unit volume, as taken over spatial and temporal scales that are large compared to those of the microscale processes but still small compared to those of the intended application or phenomenon under study. Thus, from the microscopic viewpoint, v of the continuum theory is a mass-weighted average velocity.

      The linear momentum P and angular momentum H (relative to the coordinate origin) of the matter instantaneously occupying any volume V of space are then given by summing up the linear and angular momentum vectors of each element of material. Such summation over infinitesimal elements is represented mathematically by the integrals P = ∫_V ρvdV and H = ∫_V ρx × vdV. In this discussion attention is limited to situations in which relativistic effects can be ignored. Let F denote the total force and M the total torque, or moment (relative to the coordinate origin), acting instantaneously on the material occupying any arbitrary volume V. The basic laws of Newtonian mechanics are the linear and angular momentum principles that F = dP/dt and M = dH/dt, where time derivatives of P and H are calculated following the motion of the matter that occupies V at time t. When either F or M vanishes, these equations of motion correspond to conservation of linear or angular momentum.

      An important, very common, and nontrivial class of problems in solid mechanics involves determining the deformed and stressed configuration of solids or structures that are in static equilibrium; in that case the relevant basic equations are F = 0 and M = 0. The understanding of such conditions for equilibrium, at least in a rudimentary form, long predates Newton. Indeed, Archimedes of Syracuse (3rd century BC), the great Greek mathematician and arguably the first theoretically and experimentally minded physical scientist, understood these equations at least in a nonvectorial form appropriate for systems of parallel forces. This is shown by his treatment of the hydrostatic equilibrium of a partially submerged body and by his establishment of the principle of the lever (torques about the fulcrum sum to zero) and the concept of centre of gravity.

Stress

  Assume that F and M derive from two types of forces, namely, body forces f, such as gravitational attractions—defined such that force fdV acts on volume element dV (see Figure 1—>)—and surface forces, which represent the mechanical effect of matter immediately adjoining that along the surface S of the volume V being considered. Cauchy formalized in 1822 a basic assumption of continuum mechanics that such surface forces could be represented as a stress vector T, defined so that TdS is an element of force acting over the area dS of the surface (Figure 1—>). Hence, the principles of linear and angular momentum take the forms

      which are now assumed to hold good for every conceivable choice of region V. In calculating the right-hand sides, which come from dP/dt and dH/dt, it has been noted that ρdV is an element of mass and is therefore time-invariant; also, a = a(x, t) = dv/dt is the acceleration, where the time derivative of v is taken following the motion of a material point so that a(x, t)dt corresponds to the difference between v(x + vdt, t + dt) and v(x, t). A more detailed analysis of this step shows that the understanding of what TdS denotes must now be adjusted to include averages, over temporal and spatial scales that are large compared to those of microscale fluctuations, of transfers of momentum across the surface S due to the microscopic fluctuations about the motion described by the macroscopic velocity v.

  The nine quantities σ_ij(i, j = 1, 2, 3) are called stress components; these will vary with position and time—i.e., σ_ij = σ_ij(x, t)—and have the following interpretation. Consider an element of surface dS through a point x with dS oriented so that its outer normal (pointing away from the region V, bounded by S) points in the positive x_i direction, where i is any of 1, 2, or 3. Then σ_i1, σ_i2, and σ_i3 at x are defined as the Cartesian components of the stress vector T (called T⁽ⁱ⁾) acting on this dS. Figure 2—> shows the components of such stress vectors for faces in each of the three coordinate directions. To use a vector notation with e₁, e₂, and e₃ denoting unit vectors along the coordinate axes (Figure 2—>), T⁽ⁱ⁾ = σ_i1e₁ + σ_i2e₂ + σ_i3e₃. Thus, the stress σ_ij at x is the stress in the j direction associated with an i-oriented face through point x; the physical dimension of the σ_ij is [force]/[length]². The components σ₁₁, σ₂₂, and σ₃₃ are stresses directed perpendicular, or normal, to the face on which they act and are normal stresses; the σ_ij with i ≠ j are directed parallel to the face on which they act and are shear stresses.

 By hypothesis, the linear momentum principle applies for any volume V. Consider a small tetrahedron (Figure 3—>) at x with an inclined face having an outward unit normal vector n and its other three faces oriented perpendicular to the three coordinate axes. Letting the size of the tetrahedron shrink to zero, the linear momentum principle requires that the stress vector T on a surface element with outward normal n be expressed as a linear function of the σ_ij at x. The relation is such that the j component of the stress vector T is T_j = n₁σ_1j + n₂σ_2j + n₃σ_3j for (j = 1, 2, 3). This relation for T (or T_j) also demonstrates that the σ_ij have the mathematical property of being the components of a second-rank tensor.

      Suppose that a different set of Cartesian reference axes 1′, 2′, and 3′ have been chosen. Let x₁′, x₂′, and x₃′ denote the components of the position vector of point x and let σ_kl′(k, l = 1, 2, 3) denote the nine stress components relative to that coordinate system. The σ_kl′ can be written as the 3 × 3 matrix [σ′], and the σ_ij as the matrix [σ], where the first index is the matrix row number and the second is the column number. Then the expression for T_j implies that [σ′] = [α][σ][α]^T, which is the defining equation of a second-rank tensor. Here [α] is the orthogonal transformation matrix, having components α_pq = e_p′ · e_q for p, q = 1, 2, 3 and satisfying [α]^T[α] = [α][α]^T = [I], where the superscript T denotes transpose (interchange rows and columns) and [I] denotes the unit matrix, a 3 × 3 matrix with unity for every diagonal element and zero elsewhere; also, the matrix multiplications are such that if [A] = [B][C], then A_ij = B_i1C_1j + B_i2C_2j + B_i3C_3j.

Equations of motion (motion, equation of)

      Now the linear momentum principle may be applied to an arbitrary finite body. Using the expression for T_j above and the divergence theorem of multivariable calculus, which states that integrals over the area of a closed surface S, with integrand n_i f (x), may be rewritten as integrals over the volume V enclosed by S, with integrand ∂f (x)/∂x_i; when f (x) is a differentiable function, one may derive that

      at least when the σ_ij are continuous and differentiable, which is the typical case. These are the equations of motion for a continuum. Once the above consequences of the linear momentum principle are accepted, the only further result that can be derived from the angular momentum principle is that σ_ij = σ_ji (i, j = 1, 2, 3). Thus, the stress tensor is symmetric.

Principal stresses

 Symmetry of the stress tensor has the important consequence that, at each point x, there exist three mutually perpendicular directions along which there are no shear stresses. These directions are called the principal stress directions, and the corresponding normal stresses are called the principal stresses. If the principal stresses are ordered algebraically as σ_I, σ_II, and σ_III (Figure 4—>), then the normal stress on any face (given as σ_n = n · T) satisfies σ_I ≤ σ_n ≤ σ_III. The principal stresses are the eigenvalues (or characteristic values) s, and the principal directions the eigenvectors n, of the problem T = sn, or [σ]{n} = s{n} in matrix notation with the 3-column {n} representing n. It has solutions when det ([σ] − s[I ]) = −s³ + I₁s² + I₂s + I₃ = 0, with I₁ = tr[σ], I₂ = −(1/2)I ²/₁ + (1/2)tr([σ][σ]), and I₃ = det [σ]. Here “det” denotes determinant and “tr” denotes trace, or sum of diagonal elements, of a matrix. Since the principal stresses are determined by I₁, I₂, and I₃ and can have no dependence on how one chooses the coordinate system with respect to which the components of stress are referred, I₁, I₂, and I₃ must be independent of that choice and are therefore called stress invariants. One may readily verify that they have the same values when evaluated in terms of σ_ij′ above as in terms of σ_ij by using the tensor transformation law and properties noted for the orthogonal transformation matrix.

      Very often, in both nature and technology, there is interest in structural elements in forms that might be identified as strings, wires, rods, bars, beams, or columns, or as membranes, plates, or shells. These are usually idealized as, respectively, one- or two-dimensional continua. One possible approach is then to develop the consequences of the linear and angular momentum principles entirely within that idealization, working in terms of net axial and shear forces and bending and twisting torques at each point along a one-dimensional continuum, or in terms of forces and torques per unit length of surface in a two-dimensional continuum.

Geometry of deformation

Strain and strain-displacement relations

      The shape of a solid or structure changes with time during a deformation process. To characterize deformation, or strain, a certain reference configuration (deformation and flow) is adopted and called undeformed. Often, that reference configuration is chosen as an unstressed state, but such is neither necessary nor always convenient. If time is measured from zero at a moment when the body exists in that reference configuration, then the upper case X may be used to denote the position vectors of material points when t = 0. At some other time t, a material point that was at X will have moved to some spatial position x. The deformation (deformation and flow) is thus described as the mapping x = x(X, t), with x = x(X, 0) = X. The displacement vector u is then u = x(X, t) − X; also, v = ∂x(X, t)/∂t and a = ∂²x(X, t)/∂t².

      It is simplest to write equations for strain in a form that, while approximate in general, is suitable for the case when any infinitesimal line element dX of the reference configuration undergoes extremely small rotations and fractional change in length, in deforming to the corresponding line element dx. These conditions are met when |∂u_i/∂X_j| 1. Many solids are often sufficiently rigid, at least under the loadings typically applied to them, that these conditions are realized in practice. Linearized expressions for strain in terms of [∂u/∂X], appropriate to this situation, are called small strain or infinitesimal strain. Expressions for strain will also be given that are valid for rotations and fractional length changes of arbitrary magnitude; such expressions are called finite strain.

 Two simple types of strain are extensional strain and shear strain. Consider a rectangular parallelepiped, a bricklike block of material with mutually perpendicular planar faces, and let the edges of the block be parallel to the 1, 2, and 3 axes. If the block is deformed homogeneously, so that each planar face moves perpendicular to itself and so that the faces remain orthogonal (i.e., the parallelepiped is deformed into another rectangular parallelepiped), then the block is said to have undergone extensional strain relative to each of the 1, 2, and 3 axes but no shear strain relative to these axes. If the edge lengths of the undeformed parallelepiped are denoted as ΔX₁, ΔX₂, and ΔX₃, and those of the deformed parallelepiped as Δx₁, Δx₂, and Δx₃ (see Figure 5A—>, where the dashed-line figure represents the reference configuration and the solid-line figure the deformed configuration), then the quantities λ₁ = Δx₁/ΔX₁, λ₂ = Δx₂/ΔX₂, and λ = Δx₃/ΔX₃ are called stretch ratios. There are various ways that extensional strain can be defined in terms of them. Note that the change in displacement in, say, the x₁ direction between points at one end of the block and those at the other is Δu₁ = (λ₁ − 1)ΔX₁. For example, if E₁₁ denotes the extensional strain along the x₁ direction, then the most commonly understood definition of strain is E₁₁ = (change in length)/(initial length) = (Δx₁ − ΔX₁)/ΔX₁ = Δu₁/ΔX₁ = λ₁ − 1. A variety of other measures of extensional strain can be defined by E₁₁ = g(λ₁), where the function g(λ) satisfies g(1) = 0 and g′(1) = 1, so as to agree with the above definition when λ₁ is very near 1. Two such measures in common use are the strain E M = (λ ²/₁ − 1)/2, based on the change of metric tensor, and the logarithmic strain E L = ln(λ₁).

 To define a simple shear strain, consider the same rectangular parallelepiped, but now deform it so that every point on a plane of type X₂ = constant moves only in the x₁ direction by an amount that increases linearly with X₂. Thus, the deformation x₁ = γX₂ + X₁, x₂ = X₂, x₃ = X₃ defines a homogeneous simple shear strain of amount γ and is illustrated in Figure 5B—>. Note that this strain causes no change of volume. For small strain, the shear strain γ can be identified as the reduction in angle between two initially perpendicular lines.

Small-strain tensor

 The small strains, or infinitesimal strains, ε_ij are appropriate for situations with |∂u_k/∂X_l| 1 for all k and l. Two infinitesimal material fibres, one initially in the 1 direction and the other in the 2 direction, are shown in Figure 6—> as dashed lines in the reference configuration and as solid lines in the deformed configuration. To first-order accuracy in components of [∂u/∂X], the extensional strains of these fibres are ε₁₁ = ∂u₁/∂X₁ and ε₂₂ = ∂u₂/∂X₂, and the reduction of the angle between them is γ₁₂ = ∂u₂/∂X₁ + ∂u₁/∂X₂. For the shear strain denoted ε₁₂, however, half of γ₁₂ is used. Thus, considering all extensional and shear strains associated with infinitesimal fibres in the 1, 2, and 3 directions at a point of the material, the set of strains is given by

      The ε_ij are symmetric—i.e., ε_ij = ε_ji—and form a second-rank tensor (that is, if Cartesian reference axes 1′, 2′, and 3′ were chosen instead and the ε_kl′ were determined, then the ε_kl′ are related to the ε_ij by the same equations that relate the stresses σ_kl′ to the σ_ij). These mathematical features require that there exist principal strain directions; at every point of the continuum it is possible to identify three mutually perpendicular directions along which there is purely extensional strain, with no shear strain between these special directions. The directions are the principal strain directions, and the corresponding strains include the least and greatest extensional strains experienced by fibres through the material point considered. Invariants of the strain tensor may be defined in a way paralleling those for the stress tensor.

      An important fact to note is that the strains cannot vary in an arbitrary manner from point to point in the body. This is because the six strain components are all derivable from three displacement components. Restrictions on strain resulting from such considerations are called compatibility relations; the body would not fit together after deformation unless they were satisfied. Consider, for example, a state of plane strain in the 1, 2 plane (so that ε₃₃ = ε₂₃ = ε₃₁ = 0). The nonzero strains ε₁₁, ε₂₂, and ε₁₂ cannot vary arbitrarily from point to point but must satisfy

      ∂²ε₂₂/∂X ²/₁ + ∂²ε₁₁/∂X ²/₂ = 2∂²ε₁₂/∂X₁∂X₂,

      as may be verified by directly inserting the relations for strains in terms of displacements.

      When the smallness of stretch and rotation of line elements allows use of the infinitesimal strain tensor, a derivative ∂/∂X_i will be very nearly identical to ∂/∂x_i. Frequently, but not always, it will then be acceptable to ignore the distinction between the deformed and undeformed configurations in writing the governing equations of solid mechanics. For example, the differential equations of motion in terms of stress are rigorously correct only with derivatives relative to the deformed configuration, but, in the circumstances considered, the equations of motion can be written relative to the undeformed configuration. This is what is done in the most widely used variant of solid mechanics, in the form of the theory of linear elasticity. The procedure can be unsatisfactory and go badly wrong in some important cases, however, such as for columns that buckle under compressive loadings or for elastic-plastic materials when the slope of the stress versus strain relation is of the same order as existing stresses. Cases such as these are instead best approached through finite deformation theory.

Finite deformation and strain tensors

      In the theory of finite deformations, extension and rotations of line elements are unrestricted as to size. For an infinitesimal fibre that deforms from an initial point given by the vector dX to the vector dx in the time t, the deformation gradient is defined by F_ij = ∂x_i(X, t)/∂X_j; the 3 × 3 matrix [F], with components F_ij, may be represented as a pure deformation, characterized by a symmetric matrix [U], followed by a rigid rotation [R]. This result is called the polar decomposition theorem and takes the form, in matrix notation, [F] = [R][U]. For an arbitrary deformation, there exist three mutually orthogonal principal stretch directions at each point of the material; call these directions in the reference configuration N^(I), N^(II), N^(III), and let the stretch ratios be λ_I, λ_II, λ_III. Fibres in these three principal strain directions undergo extensional strain but have no shearing between them. Those three fibres in the deformed configuration remain orthogonal but are rotated by the operation [R].

      As noted earlier, an extensional strain may be defined by E = g(λ), where g(1) = 0 and g′(1) = 1, with examples for g(λ) given above. A finite strain tensor E_ij may then be defined based on any particular function g(λ) by E_ij = g(λ_I)N_i^(I)N_j^(I) + g(λ_II)N_i^(II)N_j^(II) + g(λ_III)N_i^(III)N_j^(III). Usually, it is rather difficult to actually solve for the λ's and N's associated with any general [F], so it is not easy to use this strain definition. However, for the special choice identified as g^M(λ) = (λ² − 1)/2 above, it may be shown that

      which, like the finite strain generated by any other g(λ), reduces to ε_ij when linearized in [∂u/∂X].

Stress-strain relations

Linear elastic isotropic solid

      The simplest type of stress-strain relation is that of the linear elastic solid, considered in circumstances for which |∂u_i/∂X_j| 1 and for isotropic materials, whose mechanical response is independent of the direction of stressing. If a material point sustains a stress state σ₁₁ = σ, with all other σ_ij = 0, it is subjected to uniaxial tensile stress. This can be realized in a homogeneous bar loaded by an axial force. The resulting strain may be rewritten as ε₁₁ = σ/E, ε₂₂ = ε₃₃ = −νε₁₁ = −νσ/E, ε₁₂ = ε₂₃ = ε₃₁ = 0. Two new parameters have been introduced here, E and ν. E is called Young's (Young's modulus) modulus, and it has dimensions of [force]/[length]² and is measured in units such as the pascal (1 Pa = 1 N/m²), dyne/cm², or pounds per square inch (psi); ν, which equals the ratio of lateral strain to axial strain, is dimensionless and is called the Poisson ratio.

      If the isotropic solid is subjected only to shear stress τ—i.e., σ₁₂ = σ₂₁ = τ, with all other σ_ij = 0—then the response is shearing strain of the same type, ε₁₂ = τ/2G, ε₂₃ = ε₃₁ = ε₁₁ = ε₂₂ = ε₃₃ = 0. Notice that because 2ε₁₂ = γ₁₂, this is equivalent to γ₁₂ = τ/G. The constant G introduced is called the shear modulus. (Frequently, the symbol μ is used instead of G.) The shear modulus G is not independent of E and ν but is related to them by G = E/2(1 + ν), as follows from the tensor nature of stress and strain. The general stress-strain relations are then

      where δ_ij is defined as 1 when its indices agree and 0 otherwise.

      These relations can be inverted to read σ_ij = λδ_ij(ε₁₁ + ε₂₂ + ε₃₃) + 2με_ij, where μ has been used rather than G as the notation for the shear modulus, following convention, and where λ = 2νμ/(1 − 2ν). The elastic constants λ and μ are sometimes called the Lamé constants. Since ν is typically in the range ¹/₄ to ¹/₃ for hard polycrystalline solids, λ falls often in the range between μ and 2μ. (Navier's particle model with central forces leads to λ = μ for an isotropic solid.)

      Another elastic modulus often cited is the bulk modulus K, defined for a linear solid under pressure p(σ₁₁ = σ₂₂ = σ₃₃ = −p) such that the fractional decrease in volume is p/K. For example, consider a small cube of side length L in the reference state. If the length along, say, the 1 direction changes to (1 + ε₁₁)L, the fractional change of volume is (1 + ε₁₁)(1 + ε₂₂)(1 + ε₃₃) − 1 = ε₁₁ + ε₂₂ + ε₃₃, neglecting quadratic and cubic order terms in the ε_ij compared to the linear, as is appropriate when using linear elasticity. Thus, K = E/3(1 − 2ν) = λ + 2μ/3.

Thermal strains

      Temperature change can also cause strain. In an isotropic material the thermally induced extensional strains are equal in all directions, and there are no shear strains. In the simplest cases, these thermal strains can be treated as being linear in the temperature change θ − θ₀ (where θ₀ is the temperature of the reference state), writing ε_ij^thermal = δ_ijα(θ − θ₀) for the strain produced by temperature change in the absence of stress. Here α is called the coefficient of thermal expansion. Thus, in cases of temperature change, ε_ij is replaced in the stress-strain relations above with ε_ij − ε_ij^thermal, with the thermal part given as a function of temperature. Typically, when temperature changes are modest, the small dependence of E and ν on temperature can be neglected.

anisotropy

      Anisotropic solids also are common in nature and technology. Examples are single crystals; polycrystals in which the grains are not completely random in their crystallographic orientation but have a “texture,” typically owing to some plastic or creep flow process that has left a preferred grain orientation; fibrous biological materials such as wood or bone; and composite materials that, on a microscale, either have the structure of reinforcing fibres in a matrix, with fibres oriented in a single direction or in multiple directions (e.g., to ensure strength along more than a single direction), or have the structure of a lamination of thin layers of separate materials. In the most general case, the application of any of the six components of stress induces all six components of strain, and there is no shortage of elastic constants. There would seem to be 6 × 6 = 36 in the most general case, but, as a consequence of the laws of thermodynamics, the maximum number of independent elastic constants is 21 (compared with 2 for isotropic solids). In many cases of practical interest, symmetry considerations reduce the number to far below 21. For example, crystals of cubic symmetry, such as rock salt (NaCl); face-centred cubic metals, such as aluminum, copper, or gold; body-centred cubic metals, such as iron at low temperatures or tungsten; and such nonmetals as diamond, germanium, or silicon have only three independent elastic constants. Solids with a special direction, and with identical properties along any direction perpendicular to that direction, are called transversely isotropic; they have five independent elastic constants. Examples are provided by fibre-reinforced composite materials, with fibres that are randomly emplaced but aligned in a single direction in an isotropic or transversely isotropic matrix, and by single crystals of hexagonal close packing such as zinc.

      General linear elastic stress-strain relations have the form

where the coefficients C_ijkl are known as the tensor elastic moduli. Because the ε_kl are symmetric, one may choose C_ijkl = C_ijlk, and, because the σ_ij are symmetric, C_ijkl = C_jikl. Hence the 3 × 3 × 3 × 3 = 81 components of C_ijkl reduce to the 6 × 6 = 36 mentioned. In cases of temperature change, the ε_ij above is replaced by ε_ij − ε_ij^thermal, where ε_ij^thermal = α_ij(θ − θ₀) and α_ij is the set of thermal strain coefficients, with α_ij = α_ji. An alternative matrix notation is sometimes employed, especially in the literature on single crystals. That approach introduces 6-element columns of stress and strain {σ} and {ε}, defined so that the columns, when transposed (superscript T) or laid out as rows, are {σ}^T = (σ₁₁, σ₂₂, σ₃₃, σ₁₂, σ₂₃, σ₃₁) and {ε}^T = (ε₁₁, ε₂₂, ε₃₃, 2ε₁₂, 2ε₂₃, 2ε₃₁). These forms assure that the scalar {σ}^T{dε} ≡ tr([σ][dε]) is an increment of stress working per unit volume. The stress-strain relations are then written {σ} = [c]{ε}, where [c] is the 6 × 6 matrix of elastic moduli. Thus, c₁₃ = C₁₁₃₃, c₁₅ = C₁₁₂₃, c₄₄ = C₁₂₁₂, and so on.

Thermodynamic considerations

      In thermodynamic terminology, a state of purely elastic material response corresponds to an equilibrium state, and a process during which there is purely elastic response corresponds to a sequence of equilibrium states and hence to a reversible process. The second law of thermodynamics assures that the heat absorbed per unit mass can be written θds, where θ is the thermodynamic (absolute) temperature and s is the entropy per unit mass. Hence, writing the work per unit volume of reference configuration in a manner appropriate to cases when infinitesimal strain can be used, and letting ρ₀ be the density in that configuration, from the first law of thermodynamics it can be stated that ρ₀θds + tr([σ][dε]) = ρ₀de, where e is the internal energy per unit mass. This relation shows that if e is expressed as a function of entropy s and strains [ε], and if e is written so as to depend identically on ε_ij and ε_ji, then σ_ij = ρ₀∂e([ε], s)/∂ε_ij.

      Alternatively, one may introduce the Helmholtz free energy f per unit mass, where f = e − θs = f([ε], θ), and show that σ_ij = ρ₀∂f([ε], θ)/∂ε_ij. The latter form corresponds to the variables with which the stress-strain relations were written above. Sometimes ρ₀f is called the strain energy for states of isothermal (constant θ) elastic deformation; ρ₀e has the same interpretation for adiabatic (adiabatic process) (s = constant) elastic deformation, achieved when the time scale is too short to allow heat transfer to or from a deforming element. Since the mixed partial derivatives must be independent of order, a consequence of the last equation is that ∂σ_ij([ε], θ)/∂ε_kl = ∂σ_kl([ε], θ)/∂ε_ij, which requires that C_ijkl = C_klij, or equivalently that the matrix [c] be symmetric, [c] = [c]^T, reducing the maximum possible number of independent elastic constraints from 36 to 21. The strain energy W([ε]) at constant temperature θ₀ is W([ε]) ≡ ρ₀f([ε], θ₀) = (¹/₂){ε}^T[c]{ε}.

      The elastic moduli for adiabatic response are slightly different from those for isothermal response. In the case of the isotropic material, it is convenient to give results in terms of G and K, the isothermal shear and bulk moduli. The adiabatic moduli G and  are then G = G and  = K(1 + 9θ₀K α²/ρ₀c_ε), where c_ε = θ₀∂s([ε],θ)/∂θ, evaluated at θ = θ₀ and [ε] = [0], is the specific heat at constant strain. The fractional change in the bulk modulus, given by the second term in the parentheses, is very small, typically on the order of 1 percent or less, even for metals and ceramics of relatively high α, on the order of 10⁻⁵/kelvin.

      The fractional change in absolute temperature during an adiabatic deformation is found to involve the same small parameter: [(θ − θ₀)/θ₀]_{s = const} = −(9θ₀Kα²/ρ₀c_ε) [(ε₁₁ + ε₂₂ + ε₃₃)/3αθ₀]. Values of α for most solid elements and inorganic compounds are in the range of 10⁻⁶ to 4 × 10⁻⁵/kelvin; room temperature is about 300 kelvins, so 3αθ₀ is typically in the range 10⁻³ to 4 × 10⁻². Thus, if the fractional change in volume is on the order of 1 percent, which is quite large for a metal or ceramic deforming in its elastic range, the fractional change in absolute temperature is also on the order of 1 percent. For those reasons, it is usually appropriate to neglect the alteration of the temperature field due to elastic deformation and hence to use purely mechanical formulations of elasticity in which distinctions between adiabatic and isothermal response are neglected.

Finite elastic deformations

      When elastic response under arbitrary deformation gradients is considered—because rotations, if not strains, are large or, in a material such as rubber, because the strains are large too—it is necessary to dispense with the infinitesimal strain theory. In such cases, the combined first and second laws of thermodynamics have the form ρ₀θds + det[F]tr([F]⁻¹[σ][dF]) = ρ₀de, where [F]⁻¹ is the matrix inverse of the deformation gradient [F]. If a parcel of material is deformed by [F] and then given some additional rigid rotation, the free energy f must be unchanged in that rotation. In terms of the polar decomposition [F] = [R][U], this is equivalent to saying that f is independent of the rotation part [R] of [F], which is then equivalent to saying that f is a function of the finite strain measure [E^M] = (¹/₂)([F]^T[F] − [I]) based on change of metric or, for that matter, on any member of the family of material strain tensors. Thus,

is sometimes called the second Piola-Kirchhoff stress and is given by S_kl = ρ₀∂f([E^M],θ)/∂E^M/_kl, it being assumed that f has been written so as to have identical dependence on E^M/_kl and E^M/_lk.

Inelastic response

      The above mode of expressing [σ] in terms of [S] is valid for solids showing viscoelastic or plastic response as well, except that [S] is then to be regarded not only as a function of the present [E^M] and θ but also as dependent on the prior history of both. Assuming that such materials show elastic response to sudden stress changes or to small unloading from a plastically deforming state, [S] may still be expressed as a derivative of f, as above, but the derivative is understood as being taken with respect to an elastic variation of strain and is to be taken at fixed θ and with fixed prior inelastic deformation and temperature history. Such dependence on history is sometimes represented as a dependence of f on internal state variables whose laws of evolution are part of the inelastic constitutive description. There are also simpler models of inelastic response, and the most commonly employed forms for plasticity and creep in isotropic solids are presented next.

      To a good approximation, plastic deformation of crystalline solids causes no change in volume; and hydrostatic changes in stress, amounting to equal change of all normal stresses, have no effect on plastic flow, at least for changes that are of the same order or magnitude as the strength of the solid in shear. Thus, plastic response is usually formulated in terms of deviatoric stress, which is defined by τ_ij = σ_ij − δ_ij(σ₁₁ + σ₂₂ + σ₃₃)/3. Following Richard von Mises, in a procedure that is found to agree moderately well with experiment, the plastic flow relation is formulated in terms of the second invariant of deviatoric stress, commonly rewritten as

and called the equivalent tensile stress. The definition is made so that, for a state of uniaxial tension, σ equals the tensile stress, and the stress-strain relation for general stress states is formulated in terms of data from the tensile test. In particular, a plastic strain ε^p in a uniaxial tension test is defined from ε^p = ε − σ/E, where ε is interpreted as the strain in the tensile test according to the logarithmic definition ε = lnλ, the elastic modulus E is assumed to remain unchanged with deformation, and σ/E 1.

      Thus, in the rate-independent plasticity version of the theory, tensile data (or compressive, with appropriate sign reversals) from a monotonic load test is assumed to define a function ε^p(σ). In the viscoplastic or high-temperature creep versions of the theory, tensile data is interpreted to define dε^P/dt as a function of σ in the simplest case, representing, for example, secondary creep, and as a function of σ and ε ^p in theories intended to represent transient creep effects or rate-sensitive response at lower temperatures. Consider first the rigid-plastic material model in which elastic deformability is ignored altogether, as is sometimes appropriate for problems of large plastic flow, as in metal forming or long-term creep in the Earth's mantle or for analysis of plastic collapse loads on structures. The rate of deformation tensor D_ij is defined by 2D_ij = ∂v_i/∂x_j + ∂v_j/∂x_i, and in the rigid-plastic case [D] can be equated to what may be considered its plastic part [D^p], given as D^p/_ij = 3(dε^p/dt)τ_ij/2σ. The numerical factors secure agreement between D^p/₁₁ and dε^p/dt for uniaxial tension in the 1direction. Also, the equation implies that

which must be integrated over previous history to get ε^p as required for viscoplastic models in which dε^p/dt is a function of σ and ε^p. In the rate-independent version, [D^p] is defined as zero whenever σ is less than the highest value that it has attained in the previous history or when the current value of σ is the highest value but dσ/dt < 0. (In the elastic-plastic context, this means that “unloading” involves only elastic response.) For the ideally plastic solid, which is idealized to be able to flow without increase of stress when σ equals the yield strength level, dε^p/dt is regarded as an undetermined but necessarily nonnegative parameter, which can be determined (sometimes not uniquely) only through the complete solution of a solid mechanics boundary-value problem.

      The elastic-plastic material model is then formulated by writing D_ij = D^e/_ij + D^p/_ij, where D^p/_ij is given in terms of stress and possibly stress rate as above and where the elastic deformation rates [D^e] are related to stresses by the usual linear elastic expression D^e/_ij = (1 + ν)σ_ij^*/E − νδ_ij(σ₁₁^*+ σ₂₂^*+ σ₃₃^*)/E. Here the stress rates are expressed as the Jaumann co-rotational rates

is a derivative following the motion of a material point and where the spin Ω_ij is defined by 2Ω_ij = ∂v_i/∂x_j − ∂v_j/∂x_i. The co-rotational stress rates are those calculated by an observer who spins with the average angular velocity of a material element. The elastic part of the stress-strain relation should be consistent with the existence of a free energy f, as discussed above. This is not strictly satisfied by the form just given, but the differences between it and one which is consistent in that way involves additional terms that are on the order of σ/E² times the σ_kl^* and are negligible in typical cases in which the theory is used, since σ/E is usually an extremely small fraction of unity, say, 10⁻⁴ to 10⁻². A small-strain version of the theory is in common use for purposes of elastic-plastic stress analysis. In these cases, [D] is replaced with ∂[ε(X, t)]/∂t, where [ε] is the small-strain tensor, ∂/∂x with ∂/∂X in all equations, and [σ^*] with ∂[σ(X, t)]/∂t. The last two steps cannot always be justified, even in cases of very small strain when, for example, in a rate-independent material, dσ/dε^p is not large compared to σ or when rates of rotation of material fibres can become much larger than rates of stretching, which is a concern for buckling problems even in purely elastic solids.

Problems involving elastic response

Equations of motion of linear elastic bodies.

      The final equations of the purely mechanical theory of linear elasticity (i.e., when coupling with the temperature field is neglected, or when either isothermal or isentropic response is assumed) are obtained as follows. The stress-strain relations are used, and the strains are written in terms of displacement gradients. The final expressions for stress are inserted into the equations of motion, replacing ∂/∂x with ∂/∂X in those equations. In the case of an isotropic and homogenous solid, these reduce to

      known as the Navier equations (here, ∇ = e₁∂/∂X₁ + e₂∂/∂X₂ + e₃∂/∂X₃, and ∇² is the Laplacian operator defined by ∇·∇, or ∂²/∂x₁² + ∂²/∂x₂² + ∂²/∂x₃², and, as described earlier, λ and μ are the Lamé constants, u the displacement, f the body force, and ρ the density of the material). Such equations hold in the region V occupied by the solid; on the surface S one prescribes each component of u, or each component of the stress vector T (expressed in terms of [∂u/∂X]), or sometimes mixtures of components or relations between them. For example, along a freely slipping planar interface with a rigid solid, the normal component of u and the two tangential components of T would be prescribed, all as zero.

Body wave solutions

      By looking for body wave solutions in the form u(X, t) = pf (n · X − ct), where unit vector n is the propagation direction, p is the polarization, or direction of particle motion, and c is the wave speed, one may show for the isotropic material that solutions exist for arbitrary functions f if either

The first case, with particle displacements in the propagation direction, describes longitudinal, or dilatational, waves; and the latter case, which corresponds to two linearly independent displacement directions, both transverse to the propagation direction, describes transverse, or shear, waves.

Linear elastic beam

 The case of a beam treated as a linear elastic line may also be considered. Let the line along the 1-axis (see Figure 7—>), have properties that are uniform along its length and have sufficient symmetry that bending it by applying a torque about the 3-direction causes the line to deform into an arc lying in the 1,2-plane. Make an imaginary cut through the line, and let the forces and torque acting at that section on the part lying in the direction of decreasing X₁ be denoted as a shear force V in the positive 2-direction, an axial force P in the positive 1-direction, and torque M, commonly called a bending moment, about the positive 3-direction. The linear and angular momentum principles then require that the actions at that section on the part of the line lying along the direction of increasing X₁ be of equal magnitude but opposite sign.

      Now let the line be loaded by transverse force F per unit length, directed in the 2-direction, and make assumptions on the smallness of deformation consistent with those of linear elasticity. Let ρA be the mass per unit length (so that A can be interpreted as the cross-sectional area of a homogeneous beam of density ρ) and let u be the transverse displacement in the 2-direction. Then, writing X for X₁, the linear and angular momentum principles require that ∂V/∂X + F = ρA ∂²u/∂t² and ∂M/∂X + V = 0, where rotary inertia has been neglected in the second equation, as is appropriate for disturbances which are of a wavelength that is long compared to cross-sectional dimensions. The curvature κ of the elastic line can be approximated by κ = ∂²u/∂X² for the small deformation situation considered, and the equivalent of the stress-strain relation is to assume that κ is a function of M at each point along the line. The function can be derived by the analysis of stress and strain in pure bending and is M = EIκ, with the moment of inertia I = ∫_A(X₂)²dA for uniform elastic properties over all the cross section and with the 1-axis passing through the section centroid. Hence, the equation relating transverse load and displacement of a linear elastic beam is −∂²(EI∂²u/∂X²)/∂X² + F = ρA∂²u/∂t², and this is to be solved subject to two boundary conditions at each end of the elastic line. Examples are u = ∂u/∂X = 0 at a completely restrained (“built in”) end, u = M = 0 at an end that is restrained against displacement but not rotation, and V = M = 0 at a completely unrestrained (free) end. The beam will be reconsidered later in an analysis of response with initial stress present.

      The preceding derivation was presented in the spirit of the model of a beam as the elastic line of Euler. The same equations of motion may be obtained by the following five steps: (1) integrate the three-dimensional equations of motion over a section, writing V = ∫_Aσ₁₂dA; (2) integrate the product of X₂ and those equations over a section, writing M = −∫_AX₂σ₁₁dA; (3) assume that planes initially perpendicular to fibres lying along the 1-axis remain perpendicular during deformation, so that ε₁₁ = ε₀(X, t) − X₂κ(X, t), where X ≡ X₁, ε₀(X, t) is the strain of the fibre along the 1-axis, and κ(X, t) = ∂²u/∂X², where u(X, t) is u₂ for the fibre initially along the 1-axis; (4) assume that the stress σ₁₁ relates to strain as if each point were under uniaxial tension, so that σ₁₁ = Eε₁₁; and (5) neglect terms of order h²/L² compared to unity, where h is a typical cross-section dimension and L is a scale length for variations along the direction of the 1-axis. In step (1) the average of u₂ over area A enters but may be interpreted as the displacement u of step (3) to the order retained in (5). The kinematic assumption (3) together with (5), if implemented under conditions such that there are no loadings to generate a net axial force P, requires that ε₀(X, t) = 0 and that κ(X, t) = M(X, t)/EI when the 1-axis has been chosen to pass through the centroid of the cross section. Hence, according to these approximations, σ₁₁ = −X₂M(X, t)/I = −X₂E∂²u(X, t)/∂X². The expression for σ₁₁ is exact for static equilibrium under pure bending, since assumptions (3) and (4) are exact and (5) is then irrelevant. This motivates the use of assumptions (3) and (4) in a situation that does not correspond to pure bending.

      Sometimes it is necessary to deal with solids that are already under stress in the reference configuration that is chosen for measuring strain. As a simple example, suppose that the beam just discussed is under an initial uniform tensile stress σ₁₁ = σ⁰—that is, the axial force P = σ⁰A. If σ⁰ is negative and of significant magnitude, one generally refers to the beam as a column; if it is large and positive, the beam might respond more like a taut string. The initial stress σ⁰ contributes a term to the equations of small transverse motion, which now becomes −∂²(EI∂²u/∂X²)/∂X² + σ⁰A∂²u/∂X² + F = ρA∂²u/∂t².

Free vibrations (vibration)

      Suppose that the beam is of length L, is of uniform properties, and is hinge-supported at its ends at X = 0 and X = L so that u = M = 0 there. Then free transverse motions of the beam, solving the above equation with F = 0, are described by any linear combination of the real part of solutions that have the form u = C_n exp (iω_nt)sin(nπX/L), where n is any positive integer, C_n is an arbitrary complex constant, and where

      defines the angular vibration frequency ω_n associated with the nth mode, in units of radians per unit time. The number of vibration cycles per unit time is ω_n/2π. Equation (117—>) is arranged so that the term in the brackets shows the correction, from unity, of what would be the expression giving the frequencies of free vibration for a beam when there is no σ⁰. The correction from unity can be quite significant, even though σ⁰/E is always much smaller than unity (for interesting cases, 10⁻⁶ to, say, 10⁻³ would be a representative range; few materials in bulk form would remain elastic or resist fracture at higher σ⁰/E, although good piano wire could reach about 10⁻²). The correction term's significance results because σ⁰/E is multiplied by a term that can become enormous for a beam that is long compared to its thickness; for a square section of side length h, that term (at its largest, when n = 1) is AL²/π²I ≈ 1.2L²/h², which can combine with a small σ⁰/E to produce a correction term within the brackets that is quite non-negligible compared to unity. When σ⁰ > 0 and L is large enough to make the bracketed expression much larger than unity, the EI term cancels out and the beam simply responds like a stretched string (here, string denotes an object that is unable to support a bending moment). When the vibration mode number n is large enough, however, the stringlike effects become negligible and beamlike response takes over; at sufficiently high n that L/n is reduced to the same order as h, the simple beam theory becomes inaccurate and should be replaced by three-dimensional elasticity or, at least, an improved beam theory that takes into account rotary inertia and shear deformability. (While the option of using three-dimensional elasticity for such a problem posed an insurmountable obstacle over most of the history of the subject, by 1990 the availability of computing power and easily used software reduced it to a routine problem that could be studied by an undergraduate engineer or physicist using the finite-element method or some other computational mechanics technique.)

Buckling

      An important case of compressive loading is that in which σ⁰ < 0, which can lead to buckling. Indeed, if σ⁰A < −π²EI/L², then the ω²_n is negative, at least for n = 1, which means that the corresponding ω_n is of the form ± ib, where b is a positive real number, so that the exp(iω_nt) term has a time dependence of a type that no longer involves oscillation but, rather, exponential growth, exp(bt). The critical compressive force, π²EI/L², that causes this type of behaviour is called the Euler buckling load; different numerical factors are obtained for different end conditions. The acceleration associated with the n = 1 mode becomes small in the vicinity of the critical load and vanishes at that load. Thus solutions are possible, at the buckling load, for which the column takes a deformed shape without acceleration; for that reason, an approach to buckling problems that is equivalent for what, in dynamic terminology, are called conservative systems is to seek the first load at which an alternate equilibrium solution u = u(X), other than u = 0, may exist.

      Instability by divergence—that is, with growth of displacement in the form exp(bt)—is representative of conservative systems. Columns under nonconservative loadings by, for example, a follower force, which has the property that its line of action rotates so as to be always tangent to the beam centreline at its place of application, can exhibit a flutter instability in which the dynamic response is proportional to the real or imaginary part of a term such as exp(iat)exp(bt)—i.e., an oscillation with exponentially growing amplitude. Such instabilities also arise in the coupling between fluid flow and elastic structural response, as in the subfield called aeroelasticity. The prototype is the flutter of an airplane wing—that is, a torsional oscillation of the wing, of growing amplitude, which is driven by the coupling between rotation of the wing and the development of aerodynamic forces related to the angle of attack; the coupling feeds more energy into the structure with each cycle.

      Of course, instability models that are based on linearized theories and predicting exponential growth in time actually reveal no more than that the system is deforming out of the range for which the mathematical model applies. Proper nonlinear theories that take account of the finiteness of rotation, and sometimes the large and possibly nonelastic strain of material fibres, are necessary to really understand the phenomena. An important subclass of such nonlinear analyses for conservative systems involves the static post-buckling response of a perfect structure, such as a perfectly straight column or perfectly spherical shell. That post-buckling analysis allows one to determine if increasing force is required for very large displacement to develop during the buckle or whether the buckling is of a more highly unstable type for which the load must diminish with buckling amplitude in order to still satisfy the equilibrium equations. The latter type of behaviour describes a structure whose maximum load (that is, the largest load it can support without collapsing) shows strong sensitivity to very small imperfections of material or geometry, as is the case with many shell structures.

James Robert Rice

Additional Reading

There are a number of works on the history of the subject. A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th ed. (1927, reprinted 1944), has a well-researched chapter on the origin of elasticity up to the early 1900s. Stephen P. Timoshenko, History of Strength of Materials: With a Brief Account of the History of Theory of Elasticity and Theory of Structures (1953, reprinted 1983), provides good coverage of most subfields of solid mechanics up to the period around 1940, including in some cases detailed but quite readable accounts of specific developments and capsule biographies of major figures. C. Truesdell, Essays in the History of Mechanics (1968), summarizes his studies of original source materials on Jakob Bernoulli (1654–1705), Leonhard Euler, Leonardo da Vinci, and others and connects those contributions to some of the developments in what he calls “rational mechanics” as of the middle 1900s. Two articles in Handbuch der Physik provide historical background: C. Truesdell and R.A. Toupin, “The Classical Field Theories,” vol. 3, pt. 1 (1960); and C. Truesdell and W. Noll, “The Nonlinear Field Theories of Mechanics,” vol. 3, pt. 3 (1965).There are many good books for beginners on the subject, intended for the education of engineers; one that stands out for its coverage of inelastic solid mechanics as well as the more conventional topics on elementary elasticity and structures is Stephen H. Crandall, Norman C. Dahl, and Thomas J. Lardner (eds.), An Introduction to the Mechanics of Solids, 2nd ed., with SI units (1978). Those with an interest in the physics of materials might begin with A.H. Cottrell, The Mechanical Properties of Matter (1964, reprinted 1981). Some books for beginners aim for a more general introduction to continuum mechanics, including solids and fluids; one such text is Y.C. Fung, A First Course in Continuum Mechanics, 2nd ed. (1977). A readable introduction to continuum mechanics at a more advanced level, such as might be used by scientists and engineers from other fields or by first-year graduate students, is Lawrence E. Malvern, Introduction to the Mechanics of a Continuous Medium (1969). The article by Truesdell and Toupin, mentioned above, provides a comprehensive, perhaps overwhelming, treatment of continuum mechanics fundamentals.For more specialized treatment of linear elasticity, the classics are the work by Love, mentioned above; Stephen P. Timoshenko and J.N. Goodier, Theory of Elasticity, 3rd ed. (1970); and N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, 2nd ed. (1963, reprinted 1977; originally published in Russian, 4th corrected and augmented ed., 1954). The article by Truesdell and Noll noted above is a good source on finite elasticity and also on viscoelastic fluids; a standard reference on the latter is R. Byron Bird et al., Dynamics of Polymeric Liquids, vol. 1, Fluid Mechanics, 2nd ed. (1987). Other books generally regarded as classics in their subfields are R. Hill, The Mathematical Theory of Plasticity (1950, reissued 1983); J.C. Jaeger and N.G. Cook, Fundamentals of Rock Mechanics, 3rd ed. (1979). John Price Hirth and Jens Lothe, Theory of Dislocations, 2nd ed. (1982); and Keiiti Aki and Paul G. Richards, Quantitative Seismology, 2 vol. (1980). Other aspects of stress waves in solids are covered by J.D. Achenbach, Wave Propagation in Elastic Solids (1973). In addition, the scope of finite element analysis in solid mechanics and many other areas can be gleaned from O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, 4th ed., 2 vol. (1989–91); and that of fracture mechanics from Melvin F. Kanninen and Carl H. Popelar, Advanced Fracture Mechanics (1985). Structural mechanics and issues relating to stability and elastic-plastic stress-strain relations in a way that updates the book by Hill are presented by Zdeňek P. Bǎzant and Luigi Cedolin, Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories (1991).James Robert Rice

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