Deformation of solids

Type of physical science: Condensed matter physics

Field of study: Solids

Solids acted upon by external forces respond by elastic strain, plastic yield, and rupture. Knowledge of the deformation mechanics of nonrigid bodies allows scientists and engineers to predict the behavior of solids under a wide range of stress conditions.

Overview

Deformation occurs whenever eternal forces acting on a solid body exceed the internal cohesive forces that impart rigidity to the body. Such deformation may be transitory (elastic) or permanent (plastic), depending on the nature of the solid and the magnitude and mode of application of the external forces. Both elastic and plastic deformation can be illustrated with a common safety pin: Closure of the pin produces elastic strain, the original shape can be recovered completely by unclasping the pin; on the other hand, straightening the pin beyond its normal, open position creates plastic strain, which can be removed only by additional deformation. The rigidity of a solid is a relative property that depends on myriad factors, including the size and shape of the loaded body, rate of load application, temperature, prior deformation effects, presence of intergranular of pore fluid, and, in the case of crystalline solids, the crystal structure and growth history. As a result of these complexities, the study of plastic deformation is largely an empirical science, whereas the study of elasticity is firmly rooted in the laws of mechanics.

If a heavy-gauge copper wire of known length and diameter is suspended from the ceiling and weights are attached to a loop in its free end, each weight increment increases tensile stress and tends to elongate the wire. Deformation proceeds through initial elastic strain, subsequent plastic yield and, ultimately, rupture. After each weight is added, the new length and diameter are measured to calculate tensile stress (total weight divided by original cross-sectional area), axial strain (elongation divided by original length), and transverse strain (reduction in diameter divided by original diameter). Plotting tensile stresses against corresponding axial or transverse strains produces a standard stress-strain diagram similar to the one in figure. The linear portion of the curve in the figure defines the elastic range of deformation. Within this range, stress and strain are directly proportional and the deformed material obeys Hooke's law.

(The force exerted by a spring is proportional to the distance it is stretched or compressed, and is in the direction opposite to the displacement. Cooper, as well as most crystalline solids, exhibits Hookean behavior under slowly applied tension, compression, and shear. The elastic limit (yield point) marks the end of elastic deformation in the figure. Beyond this point, plastic strain occurs in the test specimen and Hooke's law no longer applies.

Energy expended in elastic deformation is almost totally recoverable by unloading the specimen prior to reaching the yield point. A minute amount of mechanical energy is converted into heat and, if the material is crystalline, addition energy is retained as strain energy in the crystals. Such losses are negligible for most solids and it is generally assumed that energy recovery in the elastic range is 100 percent efficient; in other words, elastic deformation is reversible. For most materials, elastic recovery is also virtually instantaneous. If a specimen is loaded beyond the yield point, however, it recovers its shape only partially when unloaded. The amount of permanent (plastic) strain depends on the extent to which the yield point is exceeded and the nature of the solid. In the copper wire experiment, a considerable amount of stress beyond the yield point is required to produce rupture (R in the figure). The maximum stress developed in the specimen occurs prior to failure and is known as the ultimate strength (U in the figure).

Materials such as concrete, cast iron, and stone exhibit little or no plastic strain; consequently, they rupture at or just beyond the yield point. For these brittle solids, the ultimate stress, rupture stress, and yield-point stress are practically the same. In contrast, ductile solids display extensive plastic strain, which delays rupture well beyond the yield point stress. When tensile stress exceeds the ultimate strength of a ductile solid as shown in the figure, the specimen experiences a pronounced reduction in the cross-sectional area called "necking." Necking weakens the specimen and reduces the load required to produce failure. For most materials, the elastic stress-strain relationship is identical for tension and compression; beyond the elastic limit, however, most materials deform differently under tension, compression, and shear. Extremely ductile solids, such as copper or lead, do not rupture at all under compression. Instead, they "flatten out" as long as compression increases; for these solids, the terms "ultimate strength" and "rupture strength" are meaningful only in the context of tensile deformation.

The figure is an "ordinary" stress-strain diagram rather than a "true" stress-strain diagram, since it is based on the original cross-sectional area of the test material. Most solids behave like copper wire in that they exhibit transverse strain during axial strain, known as the Poisson effect. The Poisson effect is proportional to the ductility of the test specimen. True stresses must be obtained by dividing the applied load by the actual cross-sectional area after each loading. In the figure, the effect of plotting true stress against strain is indicated by the dashed curve. True stress-strain diagrams are seldom utilized.

The slope of the elastic portion of stress-strain diagrams for tension-compression has a specific numerical value (yield point stress divided by related strain), which is known as Young's modulus or the modulus of elasticity. The corresponding value in stress-strain diagrams for shear is called the shear modulus or the modulus of rigidity. Young's modulus measures stiffness.

Aluminum, for example, has a Young's modulus value of about 703,000 kilograms per square centimeter. Steel, which is considerably stiffer, has a Young's modulus value of about 2,109,000 kilograms per square centimeter. The large size of these numbers reflects the fact that large stresses are required to produce strains in most solids. The Poisson effect is quantitatively expressed as transverse strain divided by axial strain, a value known as Poisson's ratio. For most materials, Poisson's ratio ranges from 0.25 to 0.35. Since tension, compression, and shear produce elastic dimensional changes, it follows that elastic changes in volume must also occur.

This characteristic is measured by the bulk modulus, obtained by dividing the average normal stress (hydrostatic stress) by dilation (change in volume divided by original volume).

Collectively, Young's modulus, the shear modulus, Poisson's ratio, and the bulk modulus are known as the elastic constants. Every solid is characterized, in a mechanical sense, by its own set of elastic constants. Once determined through experiment, these constants provide a quantitative means of predicting a solid's behavior during elastic deformation. In the special case of isotropic elastic solids, the four elastic constants are interrelated in such a way that if any two are known, the remaining two may be calculated through simple equations given in all texts dealing with elasticity.

Plastic strain is a permanent and irreversible condition resulting from loss of cohesion caused by rupture of atomic or molecular bonds. Plastic yielding tends to follow one of three stress-strain patterns, depending on the imposed stress conditions and the nature of the solid. In the first case, the stress-strain curve is approximately horizontal beyond the yield point. Such a solid exhibits ideal plasticity, or St. Venant behavior, which is a condition of continuous yield at constant stress. Few solids behave in this fashion. In the second case, the stress-strain curve assumes a negative slope between the yield point and rupture and lacks a maxima corresponding to the ultimate strength. This condition reflects the phenomenon of strain weakening, which is common to amorphous materials and some polycrystalline solids deformed under low confining pressure. In the third use, strain hardening features a stress-strain curve with a positive slope and maxima corresponding to ultimate strength. The latter behavior is illustrated in the figure and is typical of polycrystalline solids deformed under a wide variety of conditions. The loss of ductility associated with strain hardening results from complex grain boundary effects that inhibit slip in crystal lattices, thus requiring increased loads for each additional strain increment.

For polycrystalline solids under nonhydrostatic stress, plastic yield occurs by incremental displacements within individual crystal lattices. Such displacements are accomplished by natural, mobile defects in the lattices known as dislocations. Stress causes the dislocations to migrate to grain boundaries, where they are annihilated; as a result, the crystalline aggregate becomes more rigid than in its prestress state. Planar movement of dislocations is called glide; it involves bond rupture across the glide plane, translation, and reestablishing bonds.

Generally, glide planes are lattice planes with the most dense atomic packing; within such planes, there is usually a preferred glide direction, which is marked by the closest atomic spacing.

Strain-hardening effects in polycrystalline aggregates can produce microscopic fractures along crystal boundaries that permit rotation and translation of individual grains. This process of cataclastic flow does not involve plastic yield; nevertheless, it does lead to ductile-like behavior. Like plastic yield, cataclastic flow is cumulative, time-dependent, and irreversible; it is a common phenomenon in naturally deformed rock bodies and can produce strains of virtually any magnitude.

If deformation of a polycrystalline solid takes place at a sufficiently high temperature, defects may appear by thermal activation. Such dislocations may climb out of a glide plane and, in conjunction with normal glide, significantly enhance plastic yield. In such cases, a state of superplasticity may develop, in which grain boundary slip (not rupture) occurs between adjacent crystals if the aggregate is sufficiently fine-grained. This favorable texture may be achieved by solid-state dynamic recrystallization. Although superplasticity is rarely attained, it can produce nearly steady-state strains that approximate ideal plastic behavior.

Combining nonhydrostatic stress and high temperatures may lead to an additional process known as solid-state diffusion. If atoms or atomic vacancies diffuse through a crystal, the resulting net deformation is "Nabarro-Herring creep"; if, on the other hand, diffusion is restricted to grain boundaries, the result is "Cobble creep." In either case, thermally activated diffusion distorts grain shapes and the crystalline aggregate as a whole. High-temperature plastic yield more or less mimics the flow of viscous liquids. In the case of a porous, polycrystalline material, an additional complexity is introduced by the presence of an intergranular fluid. Pore fluids may dissolve material selectively at high-pressure grain contacts and redeposit it in adjacent low-pressure voids. These pressure-solution effects produce changes in grain shape analogous to those of solid-state diffusion. If the pores happen to be interconnected, mass transport may take place well beyond the nearest voids; large-scale effects of this type are well known in many bodies of sedimentary and metamorphic rock.

Applications

The chief distinction between the three states of matter derives from the fact that solids resist shearing forces whereas liquids and gases do not. The essence of a solid is the property of rigidity, or strength. The branch of mechanics concerned with the deformation of solids is commonly known as "strength of materials." The goals of strength studies are to understand the mechanical behavior of solids under stress at all scales of matter; and to determine stress-strain, stress-time, and strain-time relationships of technologically important materials under a wide range of load conditions. The methods employed in this pursuit are theoretical analysis, laboratory experimentation, and descriptive field observation. Because of the universal nature of mechanical principles, the "strength of materials" approach transcends the specific nature of solids and the loads acting on them. Thus, it is not surprising to find that the methods of strength analysis form an integral part of all the scientific and engineering disciplines dealing with solids, including seismology, geodesy, geology, rock mechanics, structural engineering, glaciology, soil mechanics, metallurgy, ceramics, and solid-state physics.

On the largest scale, seismologists have studied elastic deformation of the earth by earthquake waves since the beginning of the twentieth century. Among other things, they discovered that shock wave velocities depend on the elastic constants of the conducting material.

This discovery led to determination of the strength properties of rocks between the surface and the center of the core and a fairly detailed picture of the earth's internal structure. On a more practical front, seismic studies of the earth's elasticity have provided the major exploration technique for petroleum, delineated geographic regions of high earthquake risk, and contributed to substantial progress toward scientific prediction of earthquakes.

On the continental-subcontinental scale, geologists have found that brittle and ductile deformation features preserved in rocks of present and former mountain ranges record collisions between gigantic rigid slabs of crust and mantle material. Geological evidence demonstrates that the continents, which are riding on huge rock plates 100 to 150 kilometers thick, have been transported tremendous distances over the planet's surface by heat-driven creep in the earth's mantle. These startling revelations have led to major new exploration strategies for fuels, metals, and other mineral commodities demanded by a resource-hungry world.

The disciplines of rock mechanics and soil mechanics apply strength analysis to natural materials at or near the earth's surface. Civil engineers employ these concepts in the design and construction of tunnels; mining engineers use them to improve safety and efficiency in the increasingly deeper and larger mines required to keep pace with demand for mineral resources.

The same approach is utilized by soil engineers to stabilize excavations and slopes in unconsolidated materials and in the design of earthquake-resistant foundations for dams, bridges, canals, highways, and buildings.

Nowhere is the use of strength analysis and experimental data more fundamental than in the design, construction, and operation of power plants, dams, buildings, factories, and machines. Civil and mechanical engineers employ strength concepts on a daily basis to select materials and to design structural members and machine parts. These designs must take into account load resistance, fatigue potential, operating conditions, safety, failure consequences, and aesthetics, as well as requirements imposed by other engineers responsible for electrical, fuel, cooling, hydraulic, pneumatic, and lubrication systems. Additional factors such as climate, thermal insulation, and environmental considerations may also influence design. Ultimately, load-bearing members must be designed to function at stresses below their elastic limits with a suitably chosen factor of safety as protection against design errors, material fatigue, fabrication flaws, assembly errors, and misuse. Since these requirements apply to every part of every machine and man-made structure, one may begin to appreciate the role that "strength of materials" plays in everyday life.

The materials chosen by engineers to withstand high loads are usually polycrystalline aggregates such as metals, ceramics (excluding glasses), or high-strength polymers. The strength of these solids derives from a dense, orderly atomic or molecular structure. During deformation, the crystal lattice attempts to maintain its low-energy state through the propagation of dislocations that tend to minimize the volume of severe strain. Physicists, crystallographers, and especially metallurgists have led the way in research on the role of atomic-scale phenomena in deformation. Their persistence, fueled by major technological applications for solid-state electronic components, synthetic crystals, and space-age metallurgical and ceramic materials, has led to greater understanding of the solid state and the behavior of atoms and molecules under stress. Progress in strength of materials, now largely an empirical science, will continue with advances in understanding atomic-scale phenomena in deformed solids.

Context

Discovery of the general equations governing elastic deformation and their application to engineering problems was one of the major achievements of the first half of the nineteenth century. The initial stage of this development actually began in 1638 with Galileo's attempts to analyze the deflection of cantilevered beams under load. Robert Hooke made the first major discovery in this emerging science when he established Hooke's law in 1660 (unfortunately not published until 1678); the second discovery was formulation of the general equations of elasticity by Claude-Louis-Marie-Henri Navier in 1821. Developments from 1638 to 1821 were concerned primarily with the solution of "Galileo's problem" and the related problem of column stability.

Once Hooke's law became known, both experimental and pure mathematical methods were widely applied to the solution of deformation and vibration problems. Major contributors during this period were Charles-Augustin de Coulomb, Leonhard Euler, Augustin-Louis Cauchy, and Daniel Bernoulli.

Navier's monumental discovery in 1821 was based on the unprovable assumption that elastic reactions in a body derive from changes in molecular structure produced by externally applied forces; in other words, Navier guessed that elastic restoration was caused by the repulsive action of intermolecular forces or, in modern terms, chemical bonds. Navier's contribution immediately attracted the attention of Cauchy and Simeon-Denis Poisson, two of the foremost mathematicians of the period. Both were interested in the general problem of wave propagation through elastic solids. Cauchy, in 1822, and then Poisson, in 1828, independently derived the general equations of elasticity through approaches quite different from that employed by Navier.

Arguments about the validity of assumptions leading to the general equations ended in 1837 when George Green derived them from the conservation of energy principle.

The nineteenth century was a period of rapid technological expansion. The resulting demand for new machines and structural designs with lower costs, greater strength, and more efficiency spurred engineers and scientists to develop systematic methods to analyze the strength properties of a wide array of solid components. As a result, the science of strength of materials emerged as the applied branch of elastic theory.

In 1828, Poisson made the important discovery that elastic solids transmit both longitudinal and transverse vibrations with a velocity ratio of √3:1; this finding is the foundation of modern seismology and led to the discovery of the earth's internal structure. In 1833, Gabriel Lame and an associate used elastic theory to solve a number of long-standing vibration problems, thereby providing a measure of validation of the new theory. The general theory was employed to solve problems involving flexure and torsion in 1855 and contributed to the ingenious "semi-inverse" method of analysis.

The study of plastic deformation is attended by such high-order mathematical complexities that progress has been possible only through the empirical approach, including a variety of laboratory studies, field description of geological structures, and computer modeling.

Development of polarized-light microscopy in the nineteenth century led to the discovery that plastic strain in crystals depends on lattice-scale phenomena. The glide mechanisms were discovered in 1876; by 1898, the specific glide elements for a number of mineral crystals were determined and it was found that the fundamental aspect of plastic strain is preservation of lattice structure. Sir Alfred Ewing and an associate came to the same conclusion at the turn of the century from their studies of plastic strain in synthetic metal crystals. These findings, and the numerous contributions of early twentieth century crystallographers, physicists, and metallurgists led to the modern dislocation theory, which is the basis for explaining plastic deformation in crystalline solids.

In its various forms, the study of deformation of solids has made immense contributions to science and humankind as a whole. The most obvious contributions are major developments in pure mathematical analysis; discovery of transmission of transverse waves through elastic media; discovery of the earth's internal structure and earthquake wave behavior; understanding of molecular structure and interactions; understanding of crystal lattice dynamics; and engineering applications that play a transcendent role in the design, safety, efficiency, cost, and aesthetics of machines and structures.

Principal terms:

COMPRESSIVE STRAIN: normal stress produced by a pair of oppositely directed, collinear forces that tend to shorten, or compress, a body; such a body is in a state of compression

ELASTIC STRAIN: recoverable deformation; elastically deformed solids revert to their original size and shape when a load is removed

HOMOGENEOUS SOLID: an ideal material, all samples of which exhibit the same mechanical properties; some fabricated solids, such as stainless and structural steel, are approximately homogeneous; natural solids, such as slate or red oak, are extremely nonhomogeneous

ISOTROPIC SOLID: an ideal solid in which the mechanical properties are the same in every direction; some solids approximate this state under certain conditions, but most do not (that is, they are anisotropic)

PLASTIC STRAIN: permanent deformation; a plastically deformed solid retains some degree of deformation after unloading

SHEAR STRESS: tangential stress produced by a force acting along a surface of a solid body (rather than normal to it); shear stress is produced by parallel pairs of oppositely directed forces

STRAIN: the change in form produced by external forces acting on a nonrigid, solid body; strains may lengthen (tensile strain), shorten (compressive strain), or change the angle between adjacent faces (shear strain) of a body

STRESS: the net internal mechanical reaction of a solid undergoing deformation; this reaction consists of resisting or cohesive forces derived from molecular or chemical bonds that impart rigidity to a solid.

TENSILE STRESS: normal stress produced by a pair of oppositely directed, collinear forces that tend to elongate a body; such a body is in a state of tension

UNIT STRESS: the intensity of stress, or the amount of force divided by the surface area acted upon by the force; in stress analysis , forces always are resolved mathematically into normal and tangential components acting on the surface of interest and assumed to be uniformly distributed on this surface

Bibliography

Gilman, John J. MICROMECHANICS OF FLOW IN SOLIDS. New York: McGraw-Hill, 1969. A well-written overview of the field of deformation. Excellent bibliography and diagrams. Utilizes mathematics.

Handlin, John. "Strength and Ductility." In HANDBOOK OF PHYSICAL CONSTANTS, by Sydney P. Clark. Rev. ed. New York: Geological Society of America, 1966. Good introduction to laboratory experimental methods. Deformation mechanisms and the general effects of confining pressure, temperature, pore pressure, and time are described without mathematics. Extensive tables list strength properties of many rocks and minerals. Recommended for all readers.

Hull, Derek. INTRODUCTION TO DISLOCATIONS. New York: Pergamon Press, 1975. A well-illustrated account of crystal defects and dislocation phenomena, with minimal reliance on mathematics.

Love, A. E. H. A TREATISE ON THE MATHEMATICAL THEORY OF ELASTICITY. 4th ed. New York: Dover, 1944. The classic twentieth century work on the theory of elasticity. The historical introduction may be useful to readers without a mathematics background. Complete references are provided for every major contribution to the development of elastic theory. Contains advanced mathematics.

Poirier, J. P. "The Rheology of Ices: A Key to Tectonics of the Ice Moons of Jupiter and Saturn." NATURE 299 (1982): 683-688. Good nonmathematical discussion of the creep mechanism. An excellent example of the wide range of applicability of strength concepts.

Schedl, Andrew, and B. A. van der Pluijm. "A Review of Deformation Microstructures." JOURNAL OF GEOLOGICAL EDUCATION 36 (1988): 111-120. Well-illustrated, nonmathematical introduction to crystal defects. Suitable for readers with a modest technical background.

Todhunter, Isaac, and Karl Pearson. A HISTORY OF THE THEORY OF ELASTICITY AND THE STRENGTH OF MATERIALS FROM GALILEI TO LORD KELVIN. 2 vols. Reprint. New York: Dover, 1960. Exhaustive historical and critical study of the contributions by the nineteenth century mathematicians who founded elastic theory and the science of strength of materials. Some calculus needed to understand critical arguments; many quotations in French, with some in Latin, German, and Italian.

Weertmann, Johannes. "Creep Deformation of Ice." ANNUAL REVIEW OF EARTH AND PLANETARY SYSTEMS 11 (1983): 215-240. Comprehensive review of ice deformation based on field, laboratory, and theoretical studies. Because of its isotropic polycrystalline nature, ice deformation is a fundamental aspect of crystal plasticity. Somewhat technical. Extensive bibliography.

Ductile polycrystalline deformed by slowly applied tension

Defects in Solids