Elasticity, Theory of

the branch of mechanics that studies the displacements, strains, and stresses that occur under the action of loads in elastic bodies at rest or in motion. The theory of elasticity is the basis for calculations of strength, deformability, and stability in construction, aircraft and rocket building, machine building, mining, and other fields of engineering and industry, as well as in physics, seismology, biomechanics, and other sciences. The objects studied by the methods of the theory of elasticity are various bodies acted on by, for example, forces and temperature fields or exposed to radiation. Such bodies include machines, structures, structural members, components of structural members, rock masses, dams, geological structures, and parts of living organisms.

Calculations performed by the methods of the theory of elasticity are used to determine the permissible loads under which stresses or displacements that are dangerous from the viewpoint of strength or inadmissible in terms of operating conditions occur in an object that is being designed. They are also used to determine the following: the most appropriate configurations and dimensions of structures, structural members, and components of structural members; the overloads that occur during dynamic loading, for example, during the passage of elastic waves; the amplitudes and frequencies of vibrations of structural members or their components and the dynamic stresses that occur in the members or components; and the forces under which an object being designed becomes unstable. Such calculations are also employed to determine which materials are most appropriate for fabricating an object being designed or which materials can be used to replace parts of the human body, such as bone and muscle tissues and blood vessels. The methods of the theory of elasticity are used effectively in the method of successive approximations to solve certain classes of problems in the theory of plasticity.

The physical laws governing the elasticity of materials have been reliably verified by experiment and apply to most materials, at least for small strains and sometimes for very large strains. These laws reflect the one-to-one relationships between the instantaneous values of stresses o and strains e, in contrast to the laws of plasticity. (In the laws of plasticity, the stresses depend on the process whereby the strains change; that is, when the same strains are achieved by different processes, the stresses are different.) When a cylindrical specimen of length l, radius r, and cross-sectional area F is stretched, a ratio exists between the tensile stress P, the longitudinal elongation of the specimen Δr, and the lateral elongation Δr. This ratio is given by the equations σ₁ = E∊₁ and ∊₂ = – v∊₁, where σ₁ = P/F is the normal stress in the cross section, ∊₁ = Δl/l is the relative elongation of the specimen, ∊₂ = Δr/r is the relative change in lateral size, E is Young’s modulus (the modulus of elasticity in tension), and v is Poisson’s ratio. When a thin-walled tubular specimen is twisted, the shearing stress τ in the cross section is calculated from the values of the cross-sectional area, the radius of the cross section, and the applied torque. The relationship between the shear strain γ, which is determined from the slope of the generatrices, and τ is given by the equation τ = Gγ, where G is the modulus of elasticity in shear.

When specimens cut from an isotropic material in different directions are tested, identical values of E, G, and v are obtained. On the average, many structural metals and alloys, as well as rubber, plastics, glass, ceramics, and concrete, are isotropic. For an-isotropic materials, such as wood, crystals, reinforced concrete, reinforced plastics, and laminated rocks, the elastic properties depend on direction. The stress at any point of a body is characterized by the following six quantities, called the stress components: the normal stresses σ_xx, σ_yy, and σ_zz, and the shearing stresses σ_xy, σ_yz, and σ_zx; here, σ_xy = σ_yx and so forth. The strain at any point of the body is characterized by the following six quantities, called the strain components: the relative elongations ∊_xx, ∊_yy, and ∊_zz and the shears ∊_xy, ∊_yz, and ∊_zx; here, ∊_xy = ∊_yx and so forth. The fundamental physical law of the theory of elasticity is the generalized Hooke’s law, according to which the normal stresses are linearly related to the strains. For isotropic materials, the relationships have the form

(1) σ_xx = 3 λ ∊ + 2µ∊_XX
σ_yy = 3 λ ∊ + 2µ∊_yy
σ_zz = 3 λ ∊ + 2µ∊_2z
σ_xy = 2µ∊_xy
σ_yz = 2µ∊_yz
σ_zx = 2µ∊_zx

where ∊ = ⅓(∊_xx + ∊_yy + ∊_zz) is the mean (hydrostatic) strain and λ and µ = G are the Lamé constants. Thus, the elastic properties of an isotropic material are characterized by the two constants λ and µ or by any two moduli of elasticity expressed in terms of these constants.

Equations (1) may also be written in the form

(2) σ_xx – σ = 2µ(∊_XX – ∊), . . .
σ_xy = 2µ∊_xy, . . .
σ = 3K ∊

where σ = ⅓ (σ_xx + σ_yy + σ_zz) is the mean (hydrostatic) stress and K is the bulk modulus.

For an anisotropic material, the six relationships between the stress and strain components have the form

(3) σ_xx = C₁₁ ∊_XX + C₁₂ ∊_yy + C₁₃ ∊_zz + C₁₄ ∊_xy + C₁₅ ∊_yz + C₁₆ ∊_zx

. . . . . . . . . . . . . .

The 36 constants c_ij that enter the relationships are called the moduli of elasticity; 21 are independent of one another and characterize the elastic properties of an anisotropic material.

For a nonlinear elastic isotropic material, the coefficient Φ(∊_u)/3 ∊_u is substituted for μ, everywhere in equations (2), and the equation σ = 3K ∊ is replaced by the equation σ = f(∊). Here, the quantity ∊_u is called the strain rate, and the functions Ф and f, which are universal for a given material, are determined experimentally. When Ф(∊_u) attains some critical value, plastic deformation occurs. In the case of simple stress, where the loads or stresses increase proportionally, the laws of plasticity have the same form but different values of the functions Ф and f; that is, the laws of the theory of small elastoplastic deformation hold. Also, when the stresses are reduced, that is, during unloading, equations (1) or (2) apply, but increments, or differences between two instantaneous values, are substituted for σ_ij, and ∊_ij.

In the equilibrium case, the mathematical problem of the theory of elasticity may be stated as follows: given the net externally applied forces or loads and the boundary conditions, determine the stress and strain components at any point of the body and determine the components u_x, u_y, and u_z of the displacement vector of each particle of the body; that is, determine these 15 quantities as functions of the coordinates x, y, and z of the points of the body. The initial equations for the solution of this problem are the differential equations of equilibrium

where ρ is the density of the material and X, Y, and Z are the projections, onto the coordinate axes, of the body force—for example, the force of gravity—acting on each particle of the body; the projections represent the body force per particle mass.

Equations (1) in the case of an isotropic body are combined with the three equations of equilibrium, as are six other equations of the type

which establish the relationships between the strain and displacement components.

When a part S₁ of the bounding surface of a body is acted on by given surface forces—such as the forces of contact interaction—whose projections referred to a unit area are equal to F_x, F_y, and F_z and when the displacements of the surface’s points φ_x, φ_y, and φ_z are given for a part S₂ of the surface, the boundary conditions have the form

(6) σ_xxl₁ + σ_xyl₂ + σ_xzl₃ = F_x (on S₁)
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .

(7) u_x = φ_x, u_y = φ_y, u_z = φ_z (on S₂)

where l₁,l₂, and l₃ are the cosines of the angles between the normal to the surface and the coordinate axes. The first set of conditions signifies that the stresses sought must satisfy the three equations (6) at the boundary of S₁, and the second set indicates that the displacements sought must satisfy equations (7) at the boundary of S₂. In the particular case where the part of the surface S₂ is tightly secured, it is possible that φ_x = φ_y = φ_z = 0. For example, in the problem of the equilibrium of a dam, the body force is the force of gravity, the surface S₂ of the base of the dam is rigid, and such forces as the hydrostatic head and the pressure of various superstructures and means of transportation act on the remaining surface S₁.

In the general case, this problem is a three-dimensional problem of the theory of elasticity and is difficult to solve. Exact analytical solutions are available only for certain particular problems, such as the bending and twisting of a beam, the contact interaction of two bodies, stress concentration, and the action of a force on the apex of a conical body. Since the equations of the theory of elasticity are linear, the problem of the combined action of two systems of forces is solved by summing the solutions for each of the systems of forces acting separately, that is, by using the principle of linear superposition. In particular, if a solution for any body is found in the case where a concentrated force acts at some arbitrary point of the body, the solution of the problem with an arbitrary load distribution is obtained by summation, that is, by integration. Such solutions, which are called Green’s functions, have been obtained only for a small number of bodies, such as an unbounded space or a half space bounded by a plane. A number of analytical methods have been proposed for the solution of the three-dimensional problem of the theory of elasticity. These include variational methods (such as the Ritz, Bubnov-Galerkin, and Castigliano methods), the method of elastic potentials, and Betti’s method. Numerical methods, such as finite-difference techniques and the method of finite elements, are being intensively developed. The development of general methods for the solution of the three-dimensional problem of the theory of elasticity is one of the most pressing tasks of the theory.

The methods of the theory of the functions of a complex variable are widely used to solve two-dimensional problems of the theory of elasticity. In such problems, one of the displacement components is equal to zero, and the other two depend only on two coordinates. On the basis of certain simplifying assumptions, approximate solutions to many problems of practical importance have been found for rods, plates, and shells, which are often used in engineering. With respect to rods, plates, and shells, problems of equilibrium stability are of specific interest (see).

In the problem of thermoelasticity, the stresses and strains that occur as a result of a nonuniform temperature distribution are determined. In the mathematical formulation of this problem, the term – (3λ + 2μ)αT, where α is the linear thermal expansion coefficient and T(x₁, x₂, x₃) is the given temperature field, is added to the right-hand side of the first three equations in (1). The theory of electromagnetic elasticity and of the elasticity of bodies exposed to radiation is constructed in an analogous way.

The problems of the theory of elasticity for inhomogeneous bodies are of great practical interest. In these problems, the coefficients λ and µ in equations (1) are not constants but functions of the coordinates that define the field of the elastic properties of the body; this field is sometimes specified statistically as certain distribution functions. Apropos of such problems, statistical methods of the theory of elasticity that reflect the statistical nature of the properties of polycrystalline bodies are being developed.

In dynamic problems of the theory of elasticity, the quantities sought are functions of the coordinates and time. The initial equations for the mathematical solution of such problems are differential equations of motion that differ from equations (4) in that the right-hand sides contain the inertial terms ρ∂²u_x/∂t², ρ∂²u_y/∂t², and ρ∂²u_z/∂t² instead of zero. Equations (1) and (5) must also be combined with the initial equations. In addition, besides the boundary conditions (6) and (7), initial conditions that give, for example, the distribution of the displacements of the particles of the body at the initial moment must also be specified. The dynamic problems include problems of the vibrations of structural members and structures and problems of the propagation of elastic waves. In the vibration problems, we may determine, for example, the vibrational modes, possible mode shifts, the vibration amplitudes, the growth or attenuation of the amplitudes with time, resonance modes, dynamic stresses, and methods for the excitation and damping of vibrations. The elastic-wave problems deal with such matters as seismic waves and their effects on structures, waves generated during explosions and impacts, and thermoelastic waves.

The mathematical formulation of problems and the development of methods of solution in the case of finite, that is, large, elastic deformation constitute one of the current problems of the theory of elasticity.

The experimental methods of the theory of elasticity include the method of multipoint strain measurement, photoelastic testing, and the moiré method. In some cases, such methods make it possible to determine directly the distribution of stresses and strains in the object being studied or on its surface. These methods are also used to check solutions obtained by analytical and numerical methods, especially when the solutions are found under any simplifying assumptions. Combined experimental and theoretical methods, in which some information on unknown functions is obtained from experiments, are sometimes effective.