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_{ijkl} = K \, \delta_{ij}\, \delta_{kl} + \mu\, (\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}- \tfrac{2}{3}\, \delta_{ij}\,\delta_{kl}) 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: \sigma_{ij} = K \delta_{ij} \varepsilon_{kk} + 2\mu \left(\varepsilon_{ij} - \tfrac{1}{3} \delta_{ij} \varepsilon_{kk}\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_{ij} = \lambda \delta_{ij} \varepsilon_{kk}+2\mu\varepsilon_{ij} 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_{ij} = \frac{1}{9K} \delta_{ij} \sigma_{kk} + \frac{1}{2\mu} \left(\sigma_{ij} - \tfrac{1}{3} \delta_{ij} \sigma_{kk}\right) which is again, a scalar part on the left and a traceless shear part on the right. More simply: \varepsilon_{ij} = \frac{1}{2\mu}\sigma_{ij} - \frac{\nu}{E} \delta_{ij}\sigma_{kk} = \frac{1}{E} [(1+\nu) \sigma_{ij}-\nu\delta_{ij}\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 \sigma_{ji,j} + F_i = 0. In engineering notation (with tau as
shear stress), • \frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + F_x = 0 • \frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_y}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} + F_y = 0 • \frac{\partial \tau_{xz}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y} + \frac{\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: \sigma_{ij} = \lambda \delta_{ij} \varepsilon_{kk}+2\mu\varepsilon_{ij} = \lambda\delta_{ij}u_{k,k}+\mu\left(u_{i,j}+u_{j,i}\right). Differentiating (assuming \lambda and \mu are spatially uniform) yields: \sigma_{ij,j} = \lambda u_{k,ki}+\mu\left(u_{i,jj}+u_{j,ij}\right). Substituting into the equilibrium equation yields: \lambda u_{k,ki}+\mu\left(u_{i,jj} + u_{j,ij}\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_{i,jj} + (\mu+\lambda) u_{j,ji} + F_i = 0 where \lambda 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. {{math proof \sigma_x = 2 \mu \varepsilon_x + \lambda(\varepsilon_x + \varepsilon_y +\varepsilon_z) = 2 \mu \frac{\partial u_x}{\partial x} + \lambda \left(\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z}\right) \tau_{xy} = \mu\gamma_{xy} = \mu\left(\frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x}\right) \tau_{xz} = \mu\gamma_{zx} = \mu\left(\frac{\partial u_z}{\partial x} + \frac{\partial u_x}{\partial z}\right) Then substituting these equations into the equilibrium equation in the x\,\!-direction we have \frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + F_x = 0 \frac{\partial}{\partial x}\left( 2\mu\frac{\partial u_x}{\partial x}+ \lambda \left(\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y}+ \frac{\partial u_z}{\partial z}\right)\right) + \mu\frac{\partial}{\partial y} \left(\frac{\partial u_x}{\partial y}+ \frac{\partial u_y}{\partial x}\right)+ \mu\frac{\partial}{\partial z} \left(\frac{\partial u_z}{\partial x} + \frac{\partial u_x}{\partial z}\right) +F_x=0 Using the assumption that \mu and \lambda are constant we can rearrange and get: \left(\lambda+\mu\right)\frac{\partial}{\partial x} \left(\frac{\partial u_x}{\partial x} +\frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z}\right)+\mu \left(\frac{\partial^2 u_x}{\partial x^2} + \frac{\partial^2 u_x}{\partial y^2}+ \frac{\partial^2 u_x}{\partial z^2}\right) + F_x= 0 Following the same procedure for the y\,\!-direction and z\,\!-direction we have \left(\lambda + \mu\right) \frac{\partial}{\partial y} \left(\frac{\partial u_x}{\partial x} +\frac{\partial u_y}{\partial y} +\frac{\partial u_z}{\partial z}\right)+\mu\left(\frac{\partial^2 u_y}{\partial x^2} + \frac{\partial^2 u_y}{\partial y^2} + \frac{\partial^2 u_y}{\partial z^2}\right) + F_y = 0 \left(\lambda+\mu\right) \frac{\partial}{\partial z} \left(\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z}\right)+ \mu \left(\frac{\partial^2 u_z}{\partial x^2} + \frac{\partial^2 u_z}{\partial y^2} + \frac{\partial^2 u_z}{\partial z^2}\right) + F_z=0 These last 3 equations are the steady Navier–Cauchy equations, which can be also expressed in vector notation as (\lambda+\mu) \nabla(\nabla \cdot \mathbf{u}) + \mu \nabla^2\mathbf{u} + \mathbf{F} = \boldsymbol{0} }} 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_{j,ij} + \beta^2 u_{i,mm} = -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 (\alpha^2-\beta^2) u_{j,iij} + \beta^2u_{i,imm} = 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_{j,iij} = 0 from which we conclude that: u_{j,iij} = 0. Taking the
Laplacian of both sides of the elastostatic equation, and assuming in addition F_{i,kk}=0\,\!, we have (\alpha^2-\beta^2) u_{j,kkij} + \beta^2u_{i,kkmm} = 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_{i,kkmm} = 0 from which we conclude that: u_{i,kkmm} = 0 or, in coordinate free notation \nabla^4 \mathbf{u} = 0 which is just the
biharmonic equation in \mathbf{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: \varepsilon_{ij,km}+\varepsilon_{km,ij}-\varepsilon_{ik,jm}-\varepsilon_{jm,ik}=0. In engineering notation, they are: \begin{align} &\frac{\partial^2 \epsilon_x}{\partial y^2} + \frac{\partial^2 \epsilon_y}{\partial x^2} = 2 \frac{\partial^2 \epsilon_{xy}}{\partial x \partial y} \\ &\frac{\partial^2 \epsilon_y}{\partial z^2} + \frac{\partial^2 \epsilon_z}{\partial y^2} = 2 \frac{\partial^2 \epsilon_{yz}}{\partial y \partial z} \\ &\frac{\partial^2 \epsilon_x}{\partial z^2} + \frac{\partial^2 \epsilon_z}{\partial x^2} = 2 \frac{\partial^2 \epsilon_{zx}}{\partial z \partial x} \\ &\frac{\partial^2 \epsilon_x}{\partial y \partial z} = \frac{\partial}{\partial x} \left ( -\frac{\partial \epsilon_{yz}}{\partial x} + \frac{\partial \epsilon_{zx}}{\partial y} + \frac{\partial \epsilon_{xy}}{\partial z}\right) \\ &\frac{\partial^2 \epsilon_y}{\partial z \partial x} = \frac{\partial}{\partial y} \left ( \frac{\partial \epsilon_{yz}}{\partial x} - \frac{\partial \epsilon_{zx}}{\partial y} + \frac{\partial \epsilon_{xy}}{\partial z}\right) \\ &\frac{\partial^2 \epsilon_z}{\partial x \partial y} = \frac{\partial}{\partial z} \left ( \frac{\partial \epsilon_{yz}}{\partial x} + \frac{\partial \epsilon_{zx}}{\partial y} - \frac{\partial \epsilon_{xy}}{\partial z}\right) \end{align} 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_{ij,kk} + \frac{1}{1+\nu}\sigma_{kk,ij} + F_{i,j} + F_{j,i} + \frac{\nu}{1-\nu}\delta_{i,j} F_{k,k} = 0. In the special situation where the body force is homogeneous, the above equations reduce to (1+\nu)\sigma_{ij,kk}+\sigma_{kk,ij}=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. 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_{ik} 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_{ik} = \frac{1}{4\pi\mu r} \left[ \left(1 - \frac{1}{2b}\right) \delta_{ik} + \frac{1}{2b} \frac{x_i x_k}{r^2} \right] It may be also compactly written as: G_{ik} = \frac{1}{4\pi\mu} \left[\frac{\delta_{ik}}{r} - \frac{1}{2b} \frac{\partial^2 r}{\partial x_i \partial x_k}\right] and it may be explicitly written as: G_{ik}=\frac{1}{4\pi\mu r} \begin{bmatrix} 1-\frac{1}{2b}+\frac{1}{2b}\frac{x^2}{r^2} & \frac{1}{2b}\frac{xy} {r^2} & \frac{1}{2b}\frac{xz} {r^2} \\ \frac{1}{2b}\frac{yx} {r^2} & 1-\frac{1}{2b}+\frac{1}{2b}\frac{y^2}{r^2} & \frac{1}{2b}\frac{yz} {r^2} \\ \frac{1}{2b}\frac{zx} {r^2} & \frac{1}{2b}\frac{zy} {r^2} & 1-\frac{1}{2b}+\frac{1}{2b}\frac{z^2}{r^2} \end{bmatrix} In cylindrical coordinates (\rho,\phi,z\,\!) it may be written as: G_{ik} = \frac{1}{4\pi \mu r} \begin{bmatrix} 1 - \frac{1}{2b} \frac{z^2}{r^2} & 0 & \frac{1}{2b} \frac{\rho z}{r^2}\\ 0 & 1 - \frac{1}{2b} & 0\\ \frac{1}{2b} \frac{z \rho}{r^2}& 0 & 1 - \frac{1}{2b} \frac{\rho^2}{r^2} \end{bmatrix} 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{\mathbf{z}} as unit vectors in the \rho and z directions respectively yields: \mathbf{u} = \frac{F_z}{4\pi\mu r} \left[\frac{1}{4(1-\nu)} \, \frac{\rho z}{r^2} \hat{\boldsymbol{\rho}} + \left(1-\frac{1}{4(1-\nu)}\,\frac{\rho^2}{r^2}\right)\hat{\mathbf{z}}\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.
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 \sqrt{-1}q_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_{ij}(q) = \frac{1}{\mu}\bigg[\frac{\delta_{ij}}{|q|^2} -\frac{1}{b}\frac{q_iq_j}{|q|^4}\bigg] The stress to strain Green's function \Gamma is \Gamma_{khij} = \frac{1}{4\mu |q|^2}(\delta_{ki}q_hq_j+\delta_{hi}q_kq_j+\delta_{kj}q_hq_i+\delta_{hj}q_kq_i) -\frac{\lambda+\mu}{\mu(\lambda+2\mu)}\frac{q_iq_jq_kq_h}{|q|^4} where \epsilon_{kh} = \Gamma_{khij}\sigma_{ij}
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. for the normal force and Cerruti for the tangential force and a derivation is given in Landau & Lifshitz. • Contact of two elastic bodies: the Hertz solution (see Matlab code). 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: \sigma_{ji,j}+ F_i = \rho\,\ddot{u}_i = \rho \, \partial_{tt} 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_{ijkl} u_{(k},_{l)}\right) ,_{j}+F_{i}=\rho \ddot{u}_{i}. If the material is isotropic and homogeneous, one obtains the (general, or transient)
Navier–Cauchy equation: \mu u_{i,jj} + (\mu+\lambda)u_{j,ij}+F_i=\rho\partial_{tt}u_i \quad \text{or} \quad \mu \nabla^2\mathbf{u} + (\mu+\lambda)\nabla(\nabla\cdot\mathbf{u}) + \mathbf{F}=\rho\frac{\partial^2\mathbf{u}}{\partial t^2}. The elastodynamic wave equation can also be expressed as \left(\delta_{kl} \partial_{tt} - A_{kl}[\nabla]\right) u_l = \frac{1}{\rho} F_k where A_{kl}[\nabla]=\frac{1}{\rho} \, \partial_i \, C_{iklj} \, \partial_j is the
acoustic differential operator, and \delta_{kl} is
Kronecker delta. In
isotropic media, the stiffness tensor has the form C_{ijkl} = K \, \delta_{ij}\, \delta_{kl} + \mu\, (\delta_{ik}\delta_{jl} + \delta_{il} \delta_{jk} - \frac{2}{3}\, \delta_{ij}\, \delta_{kl}) 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_{ij}[\nabla] = \alpha^2 \partial_i \partial_j + \beta^2 (\partial_m \partial_m \delta_{ij} - \partial_i \partial_j) For
plane waves, the above differential operator becomes the
acoustic algebraic operator: A_{ij}[\mathbf{k}] = \alpha^2 k_i k_j + \beta^2(k_m k_m \delta_{ij}-k_i k_j) where \alpha^2 = \left(K+\frac{4}{3}\mu\right)/\rho \qquad \beta^2 = \mu / \rho are the
eigenvalues of A[\hat{\mathbf{k}}] with
eigenvectors \hat{\mathbf{u}} parallel and orthogonal to the propagation direction \hat{\mathbf{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 \left( \rho ^{-1} \sigma _{(ik},_{k}\right) ,_{j)} - S_{ijkl} \ddot{\sigma}_{kl} + \left( \rho ^{-1} F_{(i}\right) ,_{j)} = 0. In the case of local isotropy, this reduces to \left( \rho ^{-1} \sigma _{(ik},_{k}\right) ,_{j)} - \frac{1}{2\mu } \left( \ddot{\sigma}_{ij} - \frac{\lambda }{3 \lambda +2\mu }\ddot{\sigma}_{kk}\delta _{ij}\right) +\left( \rho ^{-1} F_{(i}\right) ,_{j)} = 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 ==