SOLIDS MECHANICS

science concerned with the stressing, deformation, 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 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 solidas 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 stressand that tends to increase significantly with time of exposureis 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 structureswhether airframes, bridges, ships, buildings, chairs, artificial heart valves, or computer chipsand 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 formede.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? 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 (16541705), 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. (198991); 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 Zdenek P. Bazant and Luigi Cedolin, Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories (1991). James Robert Rice Basic principles In addressing any problem in continuum or solid mechanics, three factors must be considered: (1) the Newtonian equations of motion, in the more general form recognized by Euler, expressing conservation of linear and angular momentum 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 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 of diffusing constituents, (5) the first law of thermodynamics, which introduces the concept of heat flux and relates changes in energy 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 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 molecularor possibly other larger microscopicscales far below those of the intended application. Linear and angular momentum principles: stress and equations of motion Figure 1: The position vector x and the velocity vector v of a 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 r = r(x, t) be the mass density of the material. Here v and r 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, r corresponds to an average of mass per unit of volume, and rv 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 rvdV and H = V rx 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.