# Elasticity, mathematical theory of

A part of mechanics which studies displacement, deformation and stress, arising from the rest or motion of elastic bodies under the actions of loads.

The stress at any point of the body is characterized by 6 quantities — the components of the stress: the normal stress $\sigma _ {xx} , \sigma _ {yy} , \sigma _ {zz}$ and the tangential stress $\sigma _ {xy} , \sigma _ {yz} , \sigma _ {zx}$, where $\sigma _ {xy} = \sigma _ {yx}$, etc. The deformation at any point of the body is also characterized by 6 quantities — the components of deformation: the relative extensions $\epsilon _ {xx} , \epsilon _ {yy} , \epsilon _ {zz}$ and the shifts $\epsilon _ {xy} , \epsilon _ {yz} , \epsilon _ {zx}$, where $\epsilon _ {xy} = \epsilon _ {yx}$, etc.

The basic physical law in the linear theory of elasticity is a generalization of Hooke's law, according to which the normal stress depends linearly on the deformation. For isotropic substances this dependence takes the form:

$$\tag{1 } \sigma _ {xx} = \ 3 \lambda \epsilon + 2 \mu \epsilon _ {xx} ,\ \ \sigma _ {yy} = \ 3 \lambda \epsilon + 2 \mu \epsilon _ {yy} ,$$

$$\sigma _ {zz} = 3 \lambda \epsilon + 2 \mu \epsilon _ {zz} ,$$

$$\sigma _ {xy} = 2 \mu \epsilon _ {xy} ,\ \sigma _ {yz} = 2 \mu \epsilon _ {yz} ,\ \sigma _ {zx} = 2 \mu \epsilon _ {zx} ,$$

where $\epsilon = ( \epsilon _ {xx} + \epsilon _ {yy} + \epsilon _ {zz} )/3$ is the mean of the (hydrostatic) deformation and $\lambda$ and $\mu = G$ are the Lamé constants. The equations (1) can also be written in the form

$$\tag{2 } \sigma _ {xx} - \sigma = \ 2 \mu ( \epsilon _ {xx} - \epsilon ) \dots \sigma _ {xy} = \ 2 \mu \epsilon _ {xy} \dots \sigma = 3K \epsilon ,$$

where $\sigma = ( \sigma _ {xx} + \sigma _ {yy} + \sigma _ {zz} )/3$ is the mean of the (hydrostatic) stress and $K$ is the modulus of overall compression.

For anisotropic substances the 6 relations between the components of the stress and the deformation take the form

$$\sigma _ {xx} = \ c _ {11} \epsilon _ {xx} + c _ {12} \epsilon _ {yy} + c _ {13} \epsilon _ {zz} + c _ {14} \epsilon _ {xy} + c _ {15} \epsilon _ {yz} + c _ {16} \epsilon _ {zx} ,$$

$${\dots \dots \dots \dots \dots } .$$

From the 36 coefficients $c _ {ij}$ entering here, called moduli of elasticity, 21 are independent and characterize the elastic properties of the anisotropic substance.

The mathematical theory of elasticity under equilibrium consists in this, that knowing the action of external forces (loads) and the so-called boundary conditions, it is possible to determine the values, at each point of the body, of the components of the stress and the deformation, as well as the components $u _ {x} , u _ {y} , u _ {z}$ of the displacement vector at each part of the body, i.e. to determine these 15 quantities as functions of the coordinates $x, y, z$ of a point of the body. The starting point for the solution of this problem are the differential equations for equilibrium:

$$\tag{3 } \frac{\partial \sigma _ {xx} }{\partial x } + \frac{\partial \sigma _ {xy} }{\partial y } + \frac{\partial \sigma _ {xz} }{\partial z } + \rho X = 0,$$

$$\frac{\partial \sigma _ {yx} }{\partial x } + \frac{\partial \sigma _ {yy} }{\partial y } + \frac{\partial \sigma _ {yz} }{\partial z } + \rho Y = 0,$$

$$\frac{\partial \sigma _ {zx} }{\partial x } + \frac{\partial \sigma _ {zy} }{\partial y } + \frac{\partial \sigma _ {zz} }{\partial z } + \rho Z = 0,$$

where $\rho$ is the density of the substance, and $X, Y, Z$ are the projections on the coordinate axes of the mass forces (e.g. gravity) acting on each part of the body, relative to the mass of that part.

To the 3 equations of equilibrium are adjoined the 6 equations (1) in the case of an isotropic body and 6 equations, which take in the linear theory the form

$$\tag{4 } \epsilon _ {xx} = \ \frac{\partial u _ {x} }{\partial x } \dots 2 \epsilon _ {xy} = \ \frac{\partial u _ {x} }{\partial y } + \frac{\partial u _ {y} }{\partial x } \dots$$

exhibiting the dependence between the components of the deformation and the displacement.

When on a part $S _ {1}$ of the boundary surface of the body given surface forces (e.g. forces of contact interaction) act the projections of which per unit area are equal to $F _ {x} , F _ {y} , F _ {z}$, while for a part $S _ {2}$ of the surface there is a given displacement of $\phi _ {x} , \phi _ {y} , \phi _ {z}$, then the boundary conditions have the form:

$$\tag{5 } \sigma _ {xx} l _ {1} + \sigma _ {xy} l _ {2} + \sigma _ {xz} l _ {3} = \ F _ {x} \ \ ( \mathop{\rm on} S _ {1} ),$$

$$\tag{6 } u _ {x} = \phi _ {x} , u _ {y} = \phi _ {y} , u _ {z} = \phi _ {z} \ ( \mathop{\rm on} S _ {2} ),$$

where $l _ {1} , l _ {2} , l _ {3}$ are the cosines of the angles between the normal to the surface and coordinate axes. The first condition means that the desired stress must satisfy the three equations (5) on the boundary $S _ {1}$, and the second that the desired displacement must satisfy the equations (6) on the boundary $S _ {2}$; in special cases one may have $\phi _ {x} = \phi _ {y} = \phi _ {z} = 0$( the part $S _ {2}$ of the surface is fixed).

In the general case the problem posed appears as a problem in space, the solution of which is difficult to realize. An exact analytical solution has been found only for certain partial problems: the bending and torsion of beams, the contact interaction of two bodies, concentrated stress, the action of forces on the vertex of a conical body, and others. Since the equations of the mathematical theory of elasticity are linear, to solve a problem on the simultaneous action of two systems of forces one may form the sum of the solutions for each system acting separately (the principle of linear superposition). In particular, if for some body one has found the solution for the action of the force concentrated at an arbitrary point of the body, then one can solve the problem for any distribution of loads by summation (integration). Such a solution, called a Green function, can be obtained only for a small number of bodies (an unbounded space, a half-space, a bounded plane surface, and a few others). A number of analytic methods have been proposed for solving problems of the mathematical theory of elasticity in space: variational methods (Ritz, Bubnov–Galerkin, etc. cf. Galerkin method; Ritz method), the method of elastic potentials (cf. Potentials, method of) and others. Computational methods have been intensively developed (finite-differences, the method of finite elements, etc.).

For the solution of a problem in a plane (when one of the components of the displacement is zero and two others depend only on two coordinates) methods of the theory of functions of a complex variable have found wide application. For rods, plates and shells, often used in technical problems, approximate solutions to many problems of practical importance have been found on the basis of certain simplifying assumptions (cf. Elasticity theory, planar problem of; Shell theory).

In problems of thermo-elasticity the determination of the stress and the deformation arises as a consequence of non-homogeneous distribution of temperature. To state this problem one adds to the right-hand side of the first three equations (1) the supplementary term $- ( 3 \lambda + 2 \mu ) \alpha T$, where $\alpha$ is the coefficient of linear thermal expansion and $T ( x _ {1} , x _ {2} , x _ {3} )$ is the given temperature field. In an analogous fashion one can construct a theory of electromagneto-elasticity and elasticity of bodies subject to radiation. The greatest practical interest resides in questions of the mathematical theory of elasticity for non-homogeneous bodies. For such bodies the coefficients $\lambda$ and $\mu$ in (1) are no longer constant, but are functions of the coordinates determining the field of elastic properties of the body, which may sometimes be given statistically (in the form of certain distribution functions). In connection with these problems, statistical methods of the mathematical theory of elasticity have been developed, reflecting the statistical approach to properties of polycrystalline bodies.

In dynamical questions of the theory of elasticity the stresses and displacements are functions of the coordinates and the time. The starting point for a mathematical solution of these problems is formed by the differential equations of motion, which differ from the equations (3) in that the right-hand side, instead of being zero, contains inertial terms, etc.

The mathematical theory of linear elasticity, in which only (formally) infinitesimal displacements and deformations occur, can be generalized to the theory of non-linear elasticity, where (1) and/or (4) are non-linear. In this theory methods are developed for the solution of finite (large) elastic deformations.

#### References

 [1] A.E.H. Love, "A treatise on the mathematical theory of elasticity" , Dover, reprint (1944) [2] L.S. Leibenzon, "A course on the theory of elasticity" , Moscow-Leningrad (1947) (In Russian) [3] N.I. Muskhelishvili, "Some basic problems of the mathematical theory of elasticity" , Noordhoff (1953) (Translated from Russian) [4] , Three-dimensional problems in the mathematical theory of elasticity , Tbilisi (1968) (In Russian) [5] A.I. Lur'e, "The theory of elasticity" , Moscow (1970) (In Russian) [6] J.W. [Lord Rayleigh] Strutt, "The theory of sound" , Dover, reprint (1945) [7] , The theory of temperature displacement , Moscow (1964) (In Russian; translated from English) [8] I.N. Sneddon, D.S. Berry, "Classical theory of elasticity" , Handbuch der Physik , 6 , Springer (1958) [9] S.P. Timoshenko, J.N. Goodier, "Theory of elasticity" , McGraw-Hill (1970)