Linear elasticity explained

Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations (or strains) and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce yielding.

These assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis.

Mathematical formulation

Equations governing a linear elastic boundary value problem are based on three tensor partial differential equations for the balance of linear momentum and six infinitesimal strain-displacement relations. The system of differential equations is completed by a set of linear algebraic constitutive relations.

Direct tensor form

In direct tensor form that is independent of the choice of coordinate system, these governing equations are:^[1]

• Cauchy momentum equation, which is an expression of Newton's second law. In convective form it is written as: $$\backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol\; +\; \backslash mathbf\; =\; \backslash rho\; \backslash ddot$$
• Strain-displacement equations: $$\backslash boldsymbol\; =\; \backslash tfrac\; \backslash left[\backslash boldsymbol\{\backslash nabla\}\backslash mathbf\{u\}\; +\; (\backslash boldsymbol\{\backslash nabla\}\backslash mathbf\{u\})^\backslash mathrm\{T\}\backslash right]$$
• Constitutive equations. For elastic materials, Hooke's law represents the material behavior and relates the unknown stresses and strains. The general equation for Hooke's law is $$\backslash boldsymbol\; =\; \backslash mathsf:\backslash boldsymbol,$$

where

\boldsymbol{\sigma} is the Cauchy stress tensor, \boldsymbol{\varepsilon} is the infinitesimal strain tensor, u is the displacement vector, C is the fourth-order stiffness tensor, F is the body force per unit volume, \rho is the mass density, \boldsymbol{\nabla} represents the nabla operator, (\bullet) ^{T} represents a transpose, \ddot{(\bullet)} represents the second material derivative with respect to time, and A : B = A _{ij} B _{ij} is the inner product of two second-order tensors (summation over repeated indices is implied).

Cartesian coordinate form

Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are:

• Equation of motion: $$\backslash sigma\_\; +\; F\_i\; =\; \backslash rho\; \backslash partial\_\; u\_i$$ where the

{(\bullet)} _{,j} subscript is a shorthand for \partial{(\bullet)} / \partial x _{j} and \partial _{tt} indicates \partial ^{2} / \partial t ^{2} , \sigma _{ij} = \sigma _{ji} is the Cauchy stress tensor, F _{i} is the body force density, \rho is the mass density, and u _{i} is the displacement.These are 3 independent equations with 6 independent unknowns (stresses). In engineering notation, they are: $$\backslash begin\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; F\_x\; =\; \backslash rho\; \backslash frac\; \backslash \backslash \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; F\_y\; =\; \backslash rho\; \backslash frac\; \backslash \backslash \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; F\_z\; =\; \backslash rho\; \backslash frac\backslash end$$

• Strain-displacement equations: $$\backslash varepsilon\_\; =\backslash frac\; (u\_\; +\; u\_)$$ where

\varepsilon _{ij} = \varepsilon _{ji} is the strain. These are 6 independent equations relating strains and displacements with 9 independent unknowns (strains and displacements). In engineering notation, they are: $$\backslash begin\backslash epsilon\_x=\backslash frac\; \backslash \backslash \backslash epsilon\_y=\backslash frac\; \backslash \backslash \backslash epsilon\_z=\backslash frac\backslash end\backslash qquad\backslash begin\backslash gamma\_=\backslash frac+\backslash frac\; \backslash \backslash \backslash gamma\_=\backslash frac+\backslash frac\; \backslash \backslash \backslash gamma\_=\backslash frac+\backslash frac\backslash end$$

• Constitutive equations. The equation for Hooke's law is: $$\backslash sigma\_\; =\; C\_\; \backslash ,\; \backslash varepsilon\_$$ where

C _{ijkl} is the stiffness tensor. These are 6 independent equations relating stresses and strains. The requirement of the symmetry of the stress and strain tensors lead to equality of many of the elastic constants, reducing the number of different elements to 21^[2] C _{ijkl} = C _{klij} = C _{jikl} = C _{ijlk} .

An elastostatic boundary value problem for an isotropic-homogeneous media is a system of 15 independent equations and equal number of unknowns (3 equilibrium equations, 6 strain-displacement equations, and 6 constitutive equations). Specifying the boundary conditions, the boundary value problem is completely defined. To solve the system two approaches can be taken according to boundary conditions of the boundary value problem: a displacement formulation, and a stress formulation.

Cylindrical coordinate form

In cylindrical coordinates (

r,\theta,z ) the equations of motion areThe strain-displacement relations areand the constitutive relations are the same as in Cartesian coordinates, except that the indices 1 ,2 ,3 now stand for r ,\theta ,z , respectively.

Spherical coordinate form

In spherical coordinates (

r,\theta,\phi ) the equations of motion areThe strain tensor in spherical coordinates is

(An)isotropic (in)homogeneous media

In isotropic media, the stiffness tensor gives the relationship between the stresses (resulting internal stresses) and the strains (resulting deformations). For an isotropic medium, the stiffness tensor has no preferred direction: an applied force will give the same displacements (relative to the direction of the force) no matter the direction in which the force is applied. In the isotropic case, the stiffness tensor may be written: $$C\_=\; K\; \backslash ,\; \backslash delta\_\backslash ,\; \backslash delta\_+\; \backslash mu\backslash ,\; (\backslash delta\_\backslash delta\_+\backslash delta\_\backslash delta\_-\; \backslash tfrac\backslash ,\; \backslash delta\_\backslash ,\backslash delta\_)$$ where

\delta _{ij} is the Kronecker delta, K is the bulk modulus (or incompressibility), and \mu is the shear modulus (or rigidity), two elastic moduli. If the medium is inhomogeneous, the isotropic model is sensible if either the medium is piecewise-constant or weakly inhomogeneous; in the strongly inhomogeneous smooth model, anisotropy has to be accounted for. If the medium is homogeneous, then the elastic moduli will be independent of the position in the medium. The constitutive equation may now be written as:$$\backslash sigma\_\; =\; K\; \backslash delta\_\; \backslash varepsilon\_\; +\; 2\backslash mu\; \backslash left(\backslash varepsilon\_\; -\; \backslash tfrac\; \backslash delta\_\; \backslash varepsilon\_\backslash right).$$

This expression separates the stress into a scalar part on the left which may be associated with a scalar pressure, and a traceless part on the right which may be associated with shear forces. A simpler expression is:^{[3] [4]} $$\backslash sigma\_\; =\; \backslash lambda\; \backslash delta\_\; \backslash varepsilon\_+2\backslash mu\backslash varepsilon\_$$where λ is Lamé's first parameter. Since the constitutive equation is simply a set of linear equations, the strain may be expressed as a function of the stresses as:^[5] $$\backslash varepsilon\_\; =\; \backslash frac\; \backslash delta\_\; \backslash sigma\_\; +\; \backslash frac\; \backslash left(\backslash sigma\_\; -\; \backslash tfrac\; \backslash delta\_\; \backslash sigma\_\backslash right)$$which is again, a scalar part on the left and a traceless shear part on the right. More simply:$$\backslash varepsilon\_=\; \backslash frac\backslash sigma\_\; -\; \backslash frac\; \backslash delta\_\backslash sigma\_\; =\; \backslash frac\; [(1+\backslash nu)\; \backslash sigma\_\{ij\}-\backslash nu\backslash delta\_\{ij\}\backslash sigma\_\{kk\}]$$where

\nu is Poisson's ratio and E is Young's modulus.

Elastostatics

Elastostatics is the study of linear elasticity under the conditions of equilibrium, in which all forces on the elastic body sum to zero, and the displacements are not a function of time. The equilibrium equations are then $$\backslash sigma\_\; +\; F\_i\; =\; 0.$$In engineering notation (with tau as shear stress),

(\partial \sigma _{x})/(\partial x) + (\partial \tau _{yx})/(\partial y) + (\partial \tau _{zx})/(\partial z) + F _{x} = 0 (\partial \tau _{xy})/(\partial x) + (\partial \sigma _{y})/(\partial y) + (\partial \tau _{zy})/(\partial z) + F _{y} = 0 (\partial \tau _{xz})/(\partial x) + (\partial \tau _{yz})/(\partial y) + (\partial \sigma _{z})/(\partial z) + F _{z} = 0 This section will discuss only the isotropic homogeneous case.

Displacement formulation

In this case, the displacements are prescribed everywhere in the boundary. In this approach, the strains and stresses are eliminated from the formulation, leaving the displacements as the unknowns to be solved for in the governing equations.First, the strain-displacement equations are substituted into the constitutive equations (Hooke's Law), eliminating the strains as unknowns:$$\backslash sigma\_\; =\; \backslash lambda\; \backslash delta\_\; \backslash varepsilon\_+2\backslash mu\backslash varepsilon\_=\; \backslash lambda\backslash delta\_u\_+\backslash mu\backslash left(u\_+u\_\backslash right).$$Differentiating (assuming

λ and \mu are spatially uniform) yields:$$\backslash sigma\_\; =\; \backslash lambda\; u\_+\backslash mu\backslash left(u\_+u\_\backslash right).$$Substituting into the equilibrium equation yields:$$\backslash lambda\; u\_+\backslash mu\backslash left(u\_\; +\; u\_\backslash right)\; +\; F\_i\; =\; 0$$or (replacing double (dummy) (=summation) indices k,k by j,j and interchanging indices, ij to, ji after the, by virtue of Schwarz' theorem)$$\backslash mu\; u\_\; +\; (\backslash mu+\backslash lambda)\; u\_\; +\; F\_i\; =\; 0$$where λ and \mu are Lamé parameters.In this way, the only unknowns left are the displacements, hence the name for this formulation. The governing equations obtained in this manner are called the elastostatic equations, the special case of the steady Navier–Cauchy equations given below.

Once the displacement field has been calculated, the displacements can be replaced into the strain-displacement equations to solve for strains, which later are used in the constitutive equations to solve for stresses.

The biharmonic equation

The elastostatic equation may be written:$$(\backslash alpha^2-\backslash beta^2)\; u\_\; +\; \backslash beta^2\; u\_\; =\; -F\_i.$$

Taking the divergence of both sides of the elastostatic equation and assuming the body forces has zero divergence (homogeneous in domain) (

F _{i,i} =0 ) we have$$(\backslash alpha^2-\backslash beta^2)\; u\_\; +\; \backslash beta^2u\_\; =\; 0.$$

Noting that summed indices need not match, and that the partial derivatives commute, the two differential terms are seen to be the same and we have: $$\backslash alpha^2\; u\_\; =\; 0$$ from which we conclude that: $$u\_\; =\; 0.$$

Taking the Laplacian of both sides of the elastostatic equation, and assuming in addition

F _{i,kk} =0 , we have$$(\backslash alpha^2-\backslash beta^2)\; u\_\; +\; \backslash beta^2u\_\; =\; 0.$$

From the divergence equation, the first term on the left is zero (Note: again, the summed indices need not match) and we have:$$\backslash beta^2\; u\_\; =\; 0$$from which we conclude that:$$u\_\; =\; 0$$or, in coordinate free notation

\nabla ^{4} u = 0 which is just the biharmonic equation in u .

Stress formulation

In this case, the surface tractions are prescribed everywhere on the surface boundary. In this approach, the strains and displacements are eliminated leaving the stresses as the unknowns to be solved for in the governing equations. Once the stress field is found, the strains are then found using the constitutive equations.

There are six independent components of the stress tensor which need to be determined, yet in the displacement formulation, there are only three components of the displacement vector which need to be determined. This means that there are some constraints which must be placed upon the stress tensor, to reduce the number of degrees of freedom to three. Using the constitutive equations, these constraints are derived directly from corresponding constraints which must hold for the strain tensor, which also has six independent components. The constraints on the strain tensor are derivable directly from the definition of the strain tensor as a function of the displacement vector field, which means that these constraints introduce no new concepts or information. It is the constraints on the strain tensor that are most easily understood. If the elastic medium is visualized as a set of infinitesimal cubes in the unstrained state, then after the medium is strained, an arbitrary strain tensor must yield a situation in which the distorted cubes still fit together without overlapping. In other words, for a given strain, there must exist a continuous vector field (the displacement) from which that strain tensor can be derived. The constraints on the strain tensor that are required to assure that this is the case were discovered by Saint Venant, and are called the "Saint Venant compatibility equations". These are 81 equations, 6 of which are independent non-trivial equations, which relate the different strain components. These are expressed in index notation as:$$\backslash varepsilon\_+\backslash varepsilon\_-\backslash varepsilon\_-\backslash varepsilon\_=0.$$In engineering notation, they are:

The strains in this equation are then expressed in terms of the stresses using the constitutive equations, which yields the corresponding constraints on the stress tensor. These constraints on the stress tensor are known as the Beltrami-Michell equations of compatibility:$$\backslash sigma\_\; +\; \backslash frac\backslash sigma\_\; +\; F\_\; +\; F\_\; +\; \backslash frac\backslash delta\_\; F\_\; =\; 0.$$In the special situation where the body force is homogeneous, the above equations reduce to^[6] $$(1+\backslash nu)\backslash sigma\_+\backslash sigma\_=0.$$

A necessary, but insufficient, condition for compatibility under this situation is

\boldsymbol{\nabla} ^{4\boldsymbol{\sigma}} = \boldsymbol{ 0 } or \sigma _{ij,kk\ell\ell} = 0 .

These constraints, along with the equilibrium equation (or equation of motion for elastodynamics) allow the calculation of the stress tensor field. Once the stress field has been calculated from these equations, the strains can be obtained from the constitutive equations, and the displacement field from the strain-displacement equations.

An alternative solution technique is to express the stress tensor in terms of stress functions which automatically yield a solution to the equilibrium equation. The stress functions then obey a single differential equation which corresponds to the compatibility equations.

Solutions for elastostatic cases

Thomson's solution - point force in an infinite isotropic medium

The most important solution of the Navier–Cauchy or elastostatic equation is for that of a force acting at a point in an infinite isotropic medium. This solution was found by William Thomson (later Lord Kelvin) in 1848 (Thomson 1848). This solution is the analog of Coulomb's law in electrostatics. A derivation is given in Landau & Lifshitz.^[7] Defining$$a\; =\; 1-2\backslash nu$$$$b\; =\; 2(1-\backslash nu)\; =\; a+1$$where

\nu is Poisson's ratio, the solution may be expressed as $$u\_i\; =\; G\_\; F\_k$$ where F _{k} is the force vector being applied at the point, and G _{ik} is a tensor Green's function which may be written in Cartesian coordinates as:$$G\_\; =\; \backslash frac\; \backslash left[\backslash left(1\; -\; \backslash frac\{1\}\{2b\}\backslash right)\; \backslash delta\_\{ik\}\; +\; \backslash frac\{1\}\{2b\}\; \backslash frac\{x\_i\; x\_k\}\{r^2\}\; \backslash right]$$

It may be also compactly written as:$$G\_\; =\; \backslash frac\; \backslash left[\backslash frac\{\backslash delta\_\{ik\}\}\{r\}\; -\; \backslash frac\{1\}\{2b\}\; \backslash frac\{\backslash partial^2\; r\}\{\backslash partial\; x\_i\; \backslash partial\; x\_k\}\backslash right]$$and it may be explicitly written as:$$G\_=\backslash frac\; \backslash begin$$

1-\frac+\frac\frac & \frac\frac & \frac\frac \\

\frac\frac &1-\frac+\frac\frac & \frac\frac \\

\frac\frac & \frac\frac &1-\frac+\frac\frac \end

In cylindrical coordinates (

\rho,\phi,z ) it may be written as:where is total distance to point.

It is particularly helpful to write the displacement in cylindrical coordinates for a point force

F _{z} directed along the z-axis. Defining \hat{\boldsymbol{\rho}} and \hat{ z } as unit vectors in the \rho and z directions respectively yields:$$\backslash mathbf\; =\; \backslash frac\; \backslash left[\backslash frac\{1\}\{4(1-\backslash nu)\}\; \backslash ,\; \backslash frac\{\backslash rho\; z\}\{r^2\}\; \backslash hat\{\backslash boldsymbol\{\backslash rho\}\}\; +\; \backslash left(1-\backslash frac\{1\}\{4(1-\backslash nu)\}\backslash ,\backslash frac\{\backslash rho^2\}\{r^2\}\backslash right)\backslash hat\{\backslash mathbf\{z\}\}\backslash right]$$

It can be seen that there is a component of the displacement in the direction of the force, which diminishes, as is the case for the potential in electrostatics, as 1/r for large r. There is also an additional ρ-directed component.

Boussinesq–Cerruti solution - point force at the origin of an infinite isotropic half-space

Another useful solution is that of a point force acting on the surface of an infinite half-space. It was derived by Boussinesq^[8] for the normal force and Cerruti for the tangential force and a derivation is given in Landau & Lifshitz. In this case, the solution is again written as a Green's tensor which goes to zero at infinity, and the component of the stress tensor normal to the surface vanishes. This solution may be written in Cartesian coordinates as [recall: <math>a=(1-2\nu)</math> and <math>b=2(1-\nu)</math>, <math>\nu</math> = Poisson's ratio]:

$$G\_\; =\; \backslash frac\; \backslash begin$$

\frac+\frac-\frac-\frac &\frac-\frac&\frac-\frac\\

\frac -\frac&\frac+\frac-\frac-\frac &\frac -\frac\\

\frac-\frac&\frac-\frac&\frac+\frac\end

Other solutions

• Point force inside an infinite isotropic half-space.^[9]
• Point force on a surface of an isotropic half-space.
• Contact of two elastic bodies: the Hertz solution (see Matlab code).^[10] See also the page on Contact mechanics.

Elastodynamics in terms of displacements

Elastodynamics is the study of elastic waves and involves linear elasticity with variation in time. An elastic wave is a type of mechanical wave that propagates in elastic or viscoelastic materials. The elasticity of the material provides the restoring force of the wave. When they occur in the Earth as the result of an earthquake or other disturbance, elastic waves are usually called seismic waves.

The linear momentum equation is simply the equilibrium equation with an additional inertial term:$$\backslash sigma\_+\; F\_i\; =\; \backslash rho\backslash ,\backslash ddot\_i\; =\; \backslash rho\; \backslash ,\; \backslash partial\_\; u\_i.$$

If the material is governed by anisotropic Hooke's law (with the stiffness tensor homogeneous throughout the material), one obtains the displacement equation of elastodynamics:$$\backslash left(C\_\; u\_,\_\backslash right),\_+F\_=\backslash rho\; \backslash ddot\_.$$

If the material is isotropic and homogeneous, one obtains the (general, or transient) Navier–Cauchy equation:$$\backslash mu\; u\_\; +\; (\backslash mu+\backslash lambda)u\_+F\_i=\backslash rho\backslash partial\_u\_i\backslash quad\; \backslash text\; \backslash quad\backslash mu\; \backslash nabla^2\backslash mathbf\; +\; (\backslash mu+\backslash lambda)\backslash nabla(\backslash nabla\backslash cdot\backslash mathbf)\; +\; \backslash mathbf=\backslash rho\backslash frac.$$

The elastodynamic wave equation can also be expressed as$$\backslash left(\backslash delta\_\; \backslash partial\_\; -\; A\_[\backslash nabla]\backslash right)\; u\_l\; =\; \backslash frac\; F\_k$$where$$A\_[\backslash nabla]=\backslash frac\; \backslash ,\; \backslash partial\_i\; \backslash ,\; C\_\; \backslash ,\; \backslash partial\_j$$is the acoustic differential operator, and

\delta _{kl} is Kronecker delta.

In isotropic media, the stiffness tensor has the form$$C\_=\; K\; \backslash ,\; \backslash delta\_\backslash ,\; \backslash delta\_+\; \backslash mu\backslash ,\; (\backslash delta\_\backslash delta\_\; +\; \backslash delta\_\; \backslash delta\_\; -\; \backslash frac\backslash ,\; \backslash delta\_\backslash ,\; \backslash delta\_)$$where

K is the bulk modulus (or incompressibility), and \mu is the shear modulus (or rigidity), two elastic moduli. If the material is homogeneous (i.e. the stiffness tensor is constant throughout the material), the acoustic operator becomes:$$A\_[\backslash nabla]\; =\; \backslash alpha^2\; \backslash partial\_i\; \backslash partial\_j\; +\; \backslash beta^2\; (\backslash partial\_m\; \backslash partial\_m\; \backslash delta\_\; -\; \backslash partial\_i\; \backslash partial\_j)$$

For plane waves, the above differential operator becomes the acoustic algebraic operator: $$A\_[\backslash mathbf\{k\}]\; =\; \backslash alpha^2\; k\_i\; k\_j\; +\; \backslash beta^2(k\_m\; k\_m\; \backslash delta\_-k\_i\; k\_j)$$where$$\backslash alpha^2\; =\; \backslash left(K+\backslash frac\backslash mu\backslash right)/\backslash rho\; \backslash qquad\; \backslash beta^2\; =\; \backslash mu\; /\; \backslash rho$$are the eigenvalues of

A[\hat{ k }] with eigenvectors \hat{ u } parallel and orthogonal to the propagation direction \hat{ k }\,\!, respectively. The associated waves are called longitudinal and shear elastic waves. In the seismological literature, the corresponding plane waves are called P-waves and S-waves (see Seismic wave).

Elastodynamics in terms of stresses

Elimination of displacements and strains from the governing equations leads to the Ignaczak equation of elastodynamics^[11] $$\backslash left(\backslash rho\; ^\; \backslash sigma\; \_,\_\backslash right),\_\; -\; S\_\; \backslash ddot\_\; +\; \backslash left(\backslash rho\; ^\; F\_\backslash right),\_\; =\; 0.$$

In the case of local isotropy, this reduces to$$\backslash left(\backslash rho\; ^\; \backslash sigma\; \_,\_\backslash right),\_\; -\; \backslash frac\; \backslash left(\backslash ddot\_\; -\; \backslash frac\backslash ddot\_\backslash delta\_\backslash right)\; +\backslash left(\backslash rho\; ^\; F\_\backslash right),\_\; =\; 0.$$

The principal characteristics of this formulation include: (1) avoids gradients of compliance but introduces gradients of mass density; (2) it is derivable from a variational principle; (3) it is advantageous for handling traction initial-boundary value problems, (4) allows a tensorial classification of elastic waves, (5) offers a range of applications in elastic wave propagation problems; (6) can be extended to dynamics of classical or micropolar solids with interacting fields of diverse types (thermoelastic, fluid-saturated porous, piezoelectro-elastic...) as well as nonlinear media.

Anisotropic homogeneous media

See main article: Hooke's law.

For anisotropic media, the stiffness tensor

C _{ijkl} is more complicated. The symmetry of the stress tensor \sigma _{ij} means that there are at most 6 different elements of stress. Similarly, there are at most 6 different elements of the strain tensor \varepsilon _{ij} . Hence the fourth-order stiffness tensor C _{ijkl} may be written as a matrix C _{\alpha} (a tensor of second order). Voigt notation is the standard mapping for tensor indices,

\begin11 & 22 & 33 & 23,32 & 13,31 & 12,21 \\\Downarrow & \Downarrow & \Downarrow & \Downarrow & \Downarrow & \Downarrow & \\1 &2 & 3 & 4 & 5 & 6\end

With this notation, one can write the elasticity matrix for any linearly elastic medium as:

As shown, the matrix

C _{\alpha} is symmetric, this is a result of the existence of a strain energy density function which satisfies \sigma _{ij} = (\partial W)/(\partial\varepsilon _{ij}) . Hence, there are at most 21 different elements of C _{\alpha} .

The isotropic special case has 2 independent elements:

The simplest anisotropic case, that of cubic symmetry has 3 independent elements:

The case of transverse isotropy, also called polar anisotropy, (with a single axis (the 3-axis) of symmetry) has 5 independent elements:

When the transverse isotropy is weak (i.e. close to isotropy), an alternative parametrization utilizing Thomsen parameters, is convenient for the formulas for wave speeds.

The case of orthotropy (the symmetry of a brick) has 9 independent elements:

Elastodynamics

The elastodynamic wave equation for anisotropic media can be expressed as$$(\backslash delta\_\; \backslash partial\_\; -\; A\_[\backslash nabla])\backslash ,\; u\_l\; =\; \backslash frac\; F\_k$$where$$A\_[\backslash nabla]=\backslash frac\; \backslash ,\; \backslash partial\_i\; \backslash ,\; C\_\; \backslash ,\; \backslash partial\_j$$is the acoustic differential operator, and

\delta _{kl} is Kronecker delta.

Plane waves and Christoffel equation

A plane wave has the form$$\backslash mathbf[\backslash mathbf\{x\},\; \backslash ,\; t]\; =\; U[\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{x\}\; -\; \backslash omega\; \backslash ,\; t]\; \backslash ,\; \backslash hat$$with

\hat{ u }\,\! of unit length.It is a solution of the wave equation with zero forcing, if and only if \omega ^{2} and \hat{ u } constitute an eigenvalue/eigenvector pair of the acoustic algebraic operator$$A\_[\backslash mathbf\{k\}]=\backslash frac\; \backslash ,\; k\_i\; \backslash ,\; C\_\; \backslash ,\; k\_j.$$This propagation condition (also known as the Christoffel equation) may be written as$$A[\backslash hat\{\backslash mathbf\{k\}\}]\; \backslash ,\; \backslash hat\; =\; c^2\; \backslash ,\; \backslash hat$$where\hat{ k } = \mathbf / \sqrtdenotes propagation direction and c = \omega / \sqrt{ k ⋅ k } is phase velocity.

See also

• Castigliano's method
• Cauchy momentum equation
• Clapeyron's theorem
• Contact mechanics
• Deformation
• Elasticity (physics)
• GRADELA
• Hooke's law
• Infinitesimal strain theory
• Michell solution
• Plasticity (physics)
• Signorini problem
• Spring system
• Stress (mechanics)
• Stress functions

Notes and References

1. Book: Slaughter, William S. . The Linearized Theory of Elasticity . 2002 . Birkhäuser Boston . 978-1-4612-6608-2 . Boston, MA . en . 10.1007/978-1-4612-0093-2.
2. Belen'kii . Salaev. 1988. Deformation effects in layer crystals. Uspekhi Fizicheskikh Nauk. 155. 5. 89–127. 10.3367/UFNr.0155.198805c.0089. free.
3. Book: Aki, Keiiti . Keiiti Aki . Quantitative seismology . Richards . Paul G. . Paul G. richards . University Science Books . 2002 . 978-1-891389-63-4 . 2 . Mill Valley, California.
4. Continuum Mechanics for Engineers 2001 Mase, Eq. 5.12-2
5. Book: Sommerfeld, Arnold . Mechanics of Deformable Bodies . Arnold Sommerfeld. 1964 . Academic Press . New York.
6. News: Elastic Deformation. tribonet. 2017-02-16 . Tribology . 2017-02-16 . en-US.
7. Book: Landau, L.D. . Theory of Elasticity . 3rd. Lev Landau . Lifshitz, E. M. . Evgeny Lifshitz . 1986 . Butterworth Heinemann . Oxford, England . 0-7506-2633-X.
8. Book: Boussinesq, Joseph . Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques . Joseph Boussinesq . 1885 . Gauthier-Villars . Paris, France .
9. Mindlin . R. D.. Raymond D. Mindlin . 1936. Force at a point in the interior of a semi-infinite solid . Physics . 7. 5. 195–202 . https://web.archive.org/web/20170923074956/http://www.dtic.mil/get-tr-doc/pdf?AD=AD0012375. dead. September 23, 2017. 10.1063/1.1745385 . 1936Physi...7..195M.
10. Hertz . Heinrich. Heinrich Hertz . 1882 . Contact between solid elastic bodies . Journal für die reine und angewandte Mathematik. 92.
11. [Ostoja-Starzewski, M.]