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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 Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mec ...
. The fundamental assumptions of linear elasticity are infinitesimal strains — meaning, "small" deformations — and linear relationships between the components of stress and strain — hence the "linear" in its name. Linear elasticity is valid only for stress states that do not produce yielding. Its assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in
structural analysis Structural analysis is a branch of solid mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on physical structures and their c ...
and engineering design, often with the aid of
finite element analysis Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the traditional fields of structural ...
.


Mathematical formulation

Equations governing a linear elastic
boundary value problem In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
are based on three
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s for the balance of linear momentum and six infinitesimal strain-
displacement Displacement may refer to: Physical sciences Mathematics and physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
relations. The system of differential equations is completed by a set of
linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
algebraic constitutive relations.


Direct tensor form

In direct
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
form that is independent of the choice of coordinate system, these governing equations are: * Cauchy momentum equation, which is an expression of
Newton's second law Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...
. In convective form it is written as: \boldsymbol \cdot \boldsymbol + \mathbf = \rho \ddot * Strain-displacement equations: \boldsymbol = \tfrac \left boldsymbol\mathbf + (\boldsymbol\mathbf)^\mathrm\right/math> *
Constitutive equations In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance o ...
. For elastic materials,
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
represents the material behavior and relates the unknown stresses and strains. The general equation for Hooke's law is \boldsymbol = \mathsf:\boldsymbol, where \boldsymbol is the
Cauchy stress tensor In continuum mechanics, the Cauchy stress tensor (symbol \boldsymbol\sigma, named after Augustin-Louis Cauchy), also called true stress tensor or simply stress tensor, completely defines the state of stress at a point inside a material in the d ...
, \boldsymbol is the infinitesimal strain tensor, \mathbf is the
displacement vector In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along ...
, \mathsf is the fourth-order stiffness tensor, \mathbf is the body force per unit volume, \rho is the mass density, \boldsymbol represents the
nabla operator Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denot ...
, (\bullet)^\mathrm represents a
transpose In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other ...
, \ddot represents the second
material derivative In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material de ...
with respect to time, and \mathsf:\mathsf = A_B_ 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 In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
system, the governing equations of linear elasticity are: *
Equation of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathem ...
: \sigma_ + F_i = \rho \partial_ u_i where the _ subscript is a shorthand for \partial / \partial x_j and \partial_ indicates \partial^2 / \partial t^2, \sigma_ = \sigma_ 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 Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...
equations with 6 independent unknowns (stresses). In engineering notation, they are: \begin \frac + \frac + \frac + F_x = \rho \frac \\ \frac + \frac + \frac + F_y = \rho \frac \\ \frac + \frac + \frac + F_z = \rho \frac \end * Strain-displacement equations: \varepsilon_ =\frac (u_ + u_) where \varepsilon_=\varepsilon_\,\! is the strain. These are 6 independent equations relating strains and displacements with 9 independent unknowns (strains and displacements). In engineering notation, they are: \begin \epsilon_x=\frac \\ \epsilon_y=\frac \\ \epsilon_z=\frac \end \qquad \begin \gamma_=\frac+\frac \\ \gamma_=\frac+\frac \\ \gamma_=\frac+\frac \end *
Constitutive equations In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance o ...
. The equation for Hooke's law is: \sigma_ = C_ \, \varepsilon_ where C_ 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 C_ = C_ = C_ = C_. 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). By specifying the boundary conditions, the boundary value problem is fully 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 are \begin & \frac + \frac\frac + \frac + \cfrac(\sigma_-\sigma_) + F_r = \rho~\frac \\ & \frac + \frac \frac + \frac + \frac\sigma_ + F_\theta = \rho~\frac \\ & \frac + \frac\frac + \frac + \frac \sigma_ + F_z = \rho~\frac \end The strain-displacement relations are \begin \varepsilon_ & = \frac ~;~~ \varepsilon_ = \frac \left(\cfrac + u_r\right) ~;~~ \varepsilon_ = \frac \\ \varepsilon_ & = \frac \left(\cfrac\cfrac + \cfrac- \cfrac\right) ~;~~ \varepsilon_ = \cfrac \left(\cfrac + \cfrac\cfrac\right) ~;~~ \varepsilon_ = \cfrac \left(\cfrac + \cfrac\right) \end and 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 are \begin & \frac + \cfrac\frac + \cfrac\frac + \cfrac (2\sigma_-\sigma_-\sigma_+\sigma_\cot\theta) + F_r = \rho~\frac \\ & \frac + \cfrac\frac + \cfrac\frac + \cfrac \sigma_-\sigma_)\cot\theta + 3\sigma_+ F_\theta = \rho~\frac \\ & \frac + \cfrac\frac + \cfrac\frac + \cfrac(2\sigma_\cot\theta+3\sigma_) + F_\phi = \rho~\frac \end The strain tensor in spherical coordinates is \begin \varepsilon_ & = \frac\\ \varepsilon_& = \frac \left(\frac + u_r\right)\\ \varepsilon_ & = \frac \left(\frac + u_r\sin\theta + u_\theta\cos\theta\right)\\ \varepsilon_ & = \frac \left(\frac \frac + \frac - \frac\right) \\ \varepsilon_ & = \frac \left frac\frac +\left(\frac - u_\phi \cot\theta\right)\right\ \varepsilon_ & = \frac \left(\frac \frac + \frac - \frac\right). \end


(An)isotropic (in)homogeneous media

In
isotropic In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...
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 \, \delta_\, \delta_ + \mu\, (\delta_\delta_+\delta_\delta_- \tfrac\, \delta_\,\delta_) where \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
, ''K'' is the
bulk modulus The bulk modulus (K or B or k) of a substance is a measure of the resistance of a substance to bulk compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume. Other mo ...
(or incompressibility), and \mu is the
shear modulus In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the Elasticity (physics), elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear s ...
(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 Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
, then the elastic moduli will be independent of the position in the medium. The constitutive equation may now be written as: \sigma_ = K \delta_ \varepsilon_ + 2\mu \left(\varepsilon_ - \tfrac \delta_ \varepsilon_\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: \sigma_ = \lambda \delta_ \varepsilon_+2\mu\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: \varepsilon_ = \frac \delta_ \sigma_ + \frac \left(\sigma_ - \tfrac \delta_ \sigma_\right) which is again, a scalar part on the left and a traceless shear part on the right. More simply: \varepsilon_ = \frac\sigma_ - \frac \delta_\sigma_ = \frac 1+\nu) \sigma_-\nu\delta_\sigma_/math> where \nu is
Poisson's ratio In materials science and solid mechanics, Poisson's ratio (symbol: ( nu)) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value ...
and E is
Young's modulus Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Youn ...
.


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 \sigma_ + F_i = 0. In engineering notation (with tau as
shear stress Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
), * \frac + \frac + \frac + F_x = 0 *\frac + \frac + \frac + F_y = 0 *\frac + \frac + \frac + 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: \sigma_ = \lambda \delta_ \varepsilon_+2\mu\varepsilon_ = \lambda\delta_u_+\mu\left(u_+u_\right). Differentiating (assuming \lambda and \mu are spatially uniform) yields: \sigma_ = \lambda u_+\mu\left(u_+u_\right). Substituting into the equilibrium equation yields: \lambda u_+\mu\left(u_ + u_\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) \mu u_ + (\mu+\lambda) u_ + F_i = 0 where \lambda and \mu are
Lamé parameters In continuum mechanics, Lamé parameters (also called the Lamé coefficients, Lamé constants or Lamé moduli) are two material-dependent quantities denoted by ''λ'' and ''μ'' that arise in strain- stress relationships. In general, ''λ'' an ...
. 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: (\alpha^2-\beta^2) u_ + \beta^2 u_ = -F_i. Taking the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
of both sides of the elastostatic equation and assuming the body forces has zero divergence (homogeneous in domain) (F_=0\,\!) we have (\alpha^2-\beta^2) u_ + \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: \alpha^2 u_ = 0 from which we conclude that: u_ = 0. Taking the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
of both sides of the elastostatic equation, and assuming in addition F_=0\,\!, we have (\alpha^2-\beta^2) u_ + \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: \beta^2 u_ = 0 from which we conclude that: u_ = 0 or, in coordinate free notation \nabla^4 \mathbf = 0 which is just the
biharmonic equation In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. Specifically, it is used in the modeling of t ...
in \mathbf\,\!.


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: \varepsilon_+\varepsilon_-\varepsilon_-\varepsilon_=0. In engineering notation, they are: \begin &\frac + \frac = 2 \frac \\ &\frac + \frac = 2 \frac \\ &\frac + \frac = 2 \frac \\ &\frac = \frac \left ( -\frac + \frac + \frac\right) \\ &\frac = \frac \left ( \frac - \frac + \frac\right) \\ &\frac = \frac \left ( \frac + \frac - \frac\right) \end 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: \sigma_ + \frac\sigma_ + F_ + F_ + \frac\delta_ F_ = 0. In the special situation where the body force is homogeneous, the above equations reduce to (1+\nu)\sigma_+\sigma_=0. A necessary, but insufficient, condition for compatibility under this situation is \boldsymbol^4\boldsymbol = \boldsymbol or \sigma_ = 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 In linear elasticity, the equations describing the deformation of an elastic body subject only to surface forces (or body forces that could be expressed as potentials) on the boundary are (using index notation) the equilibrium equation: :\sigma_=0\ ...
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

= Thomson's solution or Kelvin's solution is 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 Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric for ...
in
electrostatics Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric e ...
. A derivation is given in Landau & Lifshitz. Defining a = 1-2\nu b = 2(1-\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_ is a tensor
Green's function In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if L is a linear dif ...
which may be written in
Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
as: G_ = \frac \left \left(1 - \frac\right) \delta_ + \frac \frac \right/math> It may be also compactly written as: G_ = \frac \left frac - \frac \frac\right/math> and it may be explicitly written as: G_=\frac \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: G_ = \frac \begin 1 - \frac \frac & 0 & \frac \frac\\ 0 & 1 - \frac & 0\\ \frac \frac& 0 & 1 - \frac \frac \end 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 and \hat as unit vectors in the \rho and z directions respectively yields: \mathbf = \frac \left frac \, \frac \hat + \left(1-\frac\,\frac\right)\hat\right/math> 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.


Frequency domain Green's function

Rewrite the Navier-Cauchy equations in component form (\lambda + \mu)\partial_i \partial_j u_j +\mu\partial_j\partial_j u_i =-F_i Convert this to frequency domain, where derivative \partial_i maps to \sqrtq_i, where q is the wave vector (\lambda + \mu)q_i q_j u_j +\mu, q, ^2u_i =F_i Spatial frequency domain force to displacement Green's function is the inverse of the above G_(q) = \frac\bigg frac -\frac\frac\bigg/math> The stress to strain Green's function \Gamma is \Gamma_ = \frac(\delta_q_hq_j+\delta_q_kq_j+\delta_q_hq_i+\delta_q_kq_i) -\frac\frac where \epsilon_ = \Gamma_\sigma_


= 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 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 ecall: a=(1-2\nu) and b=2(1-\nu), \nu = Poisson's ratio G_ = \frac \begin \frac + \frac & \frac & \frac - \frac \\ \frac & \frac + \frac & \frac - \frac \\ \frac + \frac & \frac + \frac & b + \frac \end


= Other solutions

= * Point force inside an infinite isotropic half-space. * Contact of two elastic bodies: the Hertz solution (se
Matlab code
. See also the page on
Contact mechanics Contact mechanics is the study of the Deformation (mechanics), deformation of solids that touch each other at one or more points. A central distinction in contact mechanics is between Stress (mechanics), stresses acting perpendicular to the cont ...
.


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 In physics, a mechanical wave is a wave that is an oscillation of matter, and therefore transfers energy through a material medium.Giancoli, D. C. (2009) Physics for scientists & engineers with modern physics (4th ed.). Upper Saddle River, N.J. ...
that propagates in elastic or
viscoelastic In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both Viscosity, viscous and Elasticity (physics), elastic characteristics when undergoing deformation (engineering), deformation. Viscous mate ...
materials. The elasticity of the material provides the restoring
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
of the wave. When they occur in the
Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
as the result of an
earthquake An earthquakealso called a quake, tremor, or tembloris the shaking of the Earth's surface resulting from a sudden release of energy in the lithosphere that creates seismic waves. Earthquakes can range in intensity, from those so weak they ...
or other disturbance, elastic waves are usually called
seismic wave A seismic wave is a mechanical wave of acoustic energy that travels through the Earth or another planetary body. It can result from an earthquake (or generally, a quake), volcanic eruption, magma movement, a large landslide and a large ma ...
s. The linear momentum equation is simply the equilibrium equation with an additional inertial term: \sigma_+ F_i = \rho\,\ddot_i = \rho \, \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: \left( C_ u_,_\right) ,_+F_=\rho \ddot_. If the material is isotropic and homogeneous, one obtains the (general, or transient) Navier–Cauchy equation: \mu u_ + (\mu+\lambda)u_+F_i=\rho\partial_u_i \quad \text \quad \mu \nabla^2\mathbf + (\mu+\lambda)\nabla(\nabla\cdot\mathbf) + \mathbf=\rho\frac. The elastodynamic wave equation can also be expressed as \left(\delta_ \partial_ - A_ nablaright) u_l = \frac F_k where A_ nabla\frac \, \partial_i \, C_ \, \partial_j is the ''acoustic differential operator'', and \delta_ is
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
. In
isotropic In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...
media, the stiffness tensor has the form C_ = K \, \delta_\, \delta_ + \mu\, (\delta_\delta_ + \delta_ \delta_ - \frac\, \delta_\, \delta_) where K is the
bulk modulus The bulk modulus (K or B or k) of a substance is a measure of the resistance of a substance to bulk compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume. Other mo ...
(or incompressibility), and \mu is the
shear modulus In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the Elasticity (physics), elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear s ...
(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_ nabla= \alpha^2 \partial_i \partial_j + \beta^2 (\partial_m \partial_m \delta_ - \partial_i \partial_j) For
plane waves In physics, a plane wave is a special case of a wave or field: a physical quantity whose value, at any given moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any tim ...
, the above differential operator becomes the ''acoustic algebraic operator'': A_ mathbf= \alpha^2 k_i k_j + \beta^2(k_m k_m \delta_-k_i k_j) where \alpha^2 = \left(K+\frac\mu\right)/\rho \qquad \beta^2 = \mu / \rho are the
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s of A
hat A hat is a Headgear, head covering which is worn for various reasons, including protection against weather conditions, ceremonial reasons such as university graduation, religious reasons, safety, or as a fashion accessory. Hats which incorpor ...
/math> with
eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...
s \hat parallel and orthogonal to the propagation direction \hat\,\!, 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 A seismic wave is a mechanical wave of acoustic energy that travels through the Earth or another planetary body. It can result from an earthquake (or generally, a quake), volcanic eruption, magma movement, a large landslide and a large ma ...
).


Elastodynamics in terms of stresses

Elimination of displacements and strains from the governing equations leads to the Ignaczak equation of elastodynamics Ostoja-Starzewski, M., (2018), ''Ignaczak equation of elastodynamics'', Mathematics and Mechanics of Solids. \left( \rho ^ \sigma _,_\right) ,_ - S_ \ddot_ + \left( \rho ^ F_\right) ,_ = 0. In the case of local isotropy, this reduces to \left( \rho ^ \sigma _,_\right) ,_ - \frac \left( \ddot_ - \frac\ddot_\delta _\right) +\left( \rho ^ F_\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

For anisotropic media, the stiffness tensor C_ is more complicated. The symmetry of the stress tensor \sigma_ means that there are at most 6 different elements of stress. Similarly, there are at most 6 different elements of the strain tensor \varepsilon_\,\!. Hence the fourth-order stiffness tensor C_ may be written as a matrix C_ (a tensor of second order). Voigt notation is the standard mapping for tensor indices, \begin ij & =\\ \Downarrow & \\ \alpha & = \end \begin 11 & 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: C_ \Rightarrow C_ = \begin C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \end. As shown, the matrix C_ is symmetric, this is a result of the existence of a strain energy density function which satisfies \sigma_ = \frac. Hence, there are at most 21 different elements of C_\,\!. The isotropic special case has 2 independent elements: C_ = \begin K+4 \mu\ /3 & K-2 \mu\ /3 & K-2 \mu\ /3 & 0 & 0 & 0 \\ K-2 \mu\ /3 & K+4 \mu\ /3 & K-2 \mu\ /3 & 0 & 0 & 0 \\ K-2 \mu\ /3 & K-2 \mu\ /3 & K+4 \mu\ /3 & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu\ & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu\ & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu\ \end. The simplest anisotropic case, that of cubic symmetry has 3 independent elements: C_ = \begin C_ & C_ & C_ & 0 & 0 & 0 \\ C_ & C_ & C_ & 0 & 0 & 0 \\ C_ & C_ & C_ & 0 & 0 & 0 \\ 0 & 0 & 0 & C_ & 0 & 0 \\ 0 & 0 & 0 & 0 & C_ & 0 \\ 0 & 0 & 0 & 0 & 0 & C_ \end. The case of transverse isotropy, also called polar anisotropy, (with a single axis (the 3-axis) of symmetry) has 5 independent elements: C_ = \begin C_ & C_-2C_ & C_ & 0 & 0 & 0 \\ C_-2C_ & C_ & C_ & 0 & 0 & 0 \\ C_ & C_ & C_ & 0 & 0 & 0 \\ 0 & 0 & 0 & C_ & 0 & 0 \\ 0 & 0 & 0 & 0 & C_ & 0 \\ 0 & 0 & 0 & 0 & 0 & C_ \end. 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: C_ = \begin C_ & C_ & C_ & 0 & 0 & 0 \\ C_ & C_ & C_ & 0 & 0 & 0 \\ C_ & C_ & C_ & 0 & 0 & 0 \\ 0 & 0 & 0 & C_ & 0 & 0 \\ 0 & 0 & 0 & 0 & C_ & 0 \\ 0 & 0 & 0 & 0 & 0 & C_ \end.


Elastodynamics

The elastodynamic wave equation for anisotropic media can be expressed as (\delta_ \partial_ - A_ nabla\, u_l = \frac F_k where A_ nabla\frac \, \partial_i \, C_ \, \partial_j is the ''acoustic differential operator'', and \delta_ is
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
.


Plane waves and Christoffel equation

A ''plane wave'' has the form \mathbf mathbf, \, t= U mathbf \cdot \mathbf - \omega \, t\, \hat with \hat\,\! of unit length. It is a solution of the wave equation with zero forcing, if and only if \omega^2 and \hat constitute an eigenvalue/eigenvector pair of the ''acoustic algebraic operator'' A_ mathbf\frac \, k_i \, C_ \, k_j. This ''propagation condition'' (also known as the Christoffel equation) may be written as A
hat A hat is a Headgear, head covering which is worn for various reasons, including protection against weather conditions, ceremonial reasons such as university graduation, religious reasons, safety, or as a fashion accessory. Hats which incorpor ...
\, \hat = c^2 \, \hat where \hat = \mathbf / \sqrt denotes propagation direction and c = \omega / \sqrt is phase velocity.


See also

* Castigliano's method * Cauchy momentum equation * Clapeyron's theorem *
Contact mechanics Contact mechanics is the study of the Deformation (mechanics), deformation of solids that touch each other at one or more points. A central distinction in contact mechanics is between Stress (mechanics), stresses acting perpendicular to the cont ...
* Deformation *
Elasticity (physics) In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are a ...
* GRADELA *
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
*
Infinitesimal strain theory In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
* Michell solution *
Plasticity (physics) In physics and materials science, plasticity (also known as plastic deformation) is the ability of a solid material to undergo permanent Deformation (engineering), deformation, a non-reversible change of shape in response to applied forces. For ...
* Signorini problem * Spring system *
Stress (mechanics) In continuum mechanics, stress is a physical quantity that describes forces present during deformation. For example, an object being pulled apart, such as a stretched elastic band, is subject to ''tensile'' stress and may undergo elongati ...
*
Stress functions In linear elasticity, the equations describing the deformation of an elastic body subject only to surface forces (or body forces that could be expressed as potentials) on the boundary are (using index notation) the equilibrium equation: :\sigma_=0\ ...


References

{{DEFAULTSORT:Linear Elasticity Elasticity (physics) Solid mechanics Sound