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Elastostatic Equation

$\mu\nabla^2\mathbf{u}+\frac{\mu}{1-2\nu}\nabla(\nabla\cdot\mathbf{u})=\mathbf{f} \,\,\,\,\,\,\,\,\mathrm{or}\,\,\,\,\,\,\,\, \mu\partial_j\partial_ju_i+\frac{\mu}{1-2\nu}\partial_i\partial_ju_j=f_i$

Constitutive equation, isotropic material ( $λ = 2μν / (1 - 2ν)$ )

$\sigma_{ij}=\lambda(\partial_k u_k)\delta_{ij}+\mu(\partial_i u_j+\partial_j u_i)\,$

1 Some particular solutions

[edit] Some particular solutions

[edit] Point force at the origin of an isotropic space (Thomson 1848)

(See Landau & Lifshitz § 8) Boundary Conditions: displacement is zero at infinity

$a=1-2\nu\,$

$b=2(1-\nu)=a+1\,$

$G_{ik}= \frac{1}{4\pi\mu r}\left[ \left(1-\frac{1}{2b}\right)\delta_{ik}+\frac{1}{2b}\frac{x_i x_k}{r^3} \right]$

$G_{ik}= \frac{1}{4\pi\mu}\left[\frac{\delta_{ik}}{r}-\frac{1}{2b}\frac{\partial r^2}{\partial x_i\partial x_k}\right]$

$G_{ik}=\frac{1}{4\pi\mu r}\begin{bmatrix} 1-\frac{1}{2b}+\frac{1}{2b}\frac{x^2}{r^2} & \frac{1}{2b}\frac{xy} {r^2} & \frac{1}{2b}\frac{xz} {r^2} \\ \frac{1}{2b}\frac{yx} {r^2} & 1-\frac{1}{2b}+\frac{1}{2b}\frac{y^2}{r^2} & \frac{1}{2b}\frac{yz} {r^2} \\ \frac{1}{2b}\frac{zx} {r^2} & \frac{1}{2b}\frac{zy} {r^2} & 1-\frac{1}{2b}+\frac{1}{2b}\frac{z^2}{r^2} \end{bmatrix}$

cylindrical coordinates ( $ρ,φ, z$ )

$G_{ik}=\frac{1}{4\pi \mu r}\begin{bmatrix} 1-\frac{1}{2b}\frac{z^2}{r^2}&0&\frac{1}{2b}\frac{\rho z}{r^2}\\ 0&1-\frac{1}{2b}&0\\ \frac{1}{2b}\frac{z \rho}{r^2}&0&1-\frac{1}{2b}\frac{\rho^2}{r^2} \end{bmatrix}$

spherical coordinates ( $r,θ,φ$ )

$G_{ik}=\frac{1}{4\pi \mu r}\begin{bmatrix} 1&0&0\\ 0&1-\frac{1}{2b}&0\\ 0&0&1-\frac{1}{2b} \end{bmatrix}$

z-force only (f=[0,0,1])

$\mathbf{u}=\frac{1}{4\pi \mu r} \left[\frac{1}{2b}\,\frac{z\mathbf{r}}{r^2}+\left(1-\frac{1}{2b}\right)\hat{\mathbf{z}}\right]$

Stress for f=[0,0,1]

$\sigma_{ik}=\begin{bmatrix} z(-ar^2+3x^2)&3xyz &x(ar^2+3z^2)\\ 3xyz &(-ar^2+3y^2)&y(ar^2+3z^2)\\ x(ar^2+3z^2) &y(ar^2+3z^2)&z(ar^2+3z^2) \end{bmatrix}$

z-stress on z=0 plane

$\sigma_z=\frac{-a}{2\pi b}\,\frac{\mathbf{r}}{r^3}$

[edit] Point force at the origin of an infinite isotropic half-space (Boussinesq 1885)

(See Landau & Lifshitz § 8)

Boundary Conditions: displacement is zero at infinity, and force is zero for z=0 (except at origin)

$a=1-2\nu\,$

$b=2(1-\nu)\,$

Cartesian coordinates:

$G_{ik}=\frac{1}{4\pi\mu}\begin{bmatrix} \frac{b}{r}+\frac{x^2}{r^3}-\frac{ax^2}{r(r+z)^2}+\frac{az}{r(r+z)} & \frac{xy}{r^3}-\frac{axy}{r(r+z)^2}& \frac{xz}{r^3}-\frac{ax}{r(r+z)}\\ \frac{yx}{r^3} -\frac{ayx}{r(r+z)^2}& \frac{b}{r}+\frac{y^2}{r^3}-\frac{ay^2}{r(r+z)^2}+\frac{az}{r(r+z)} & \frac{yz}{r^3} -\frac{ay}{r(r+z)}\\ \frac{zx}{r^3}+\frac{ax}{r(r+z)}& \frac{zy}{r^3}+\frac{ay}{r(r+z)}& \frac{b}{r}+\frac{z^2}{r^3} \end{bmatrix}$

at z=0:

$G_{ik}=\frac{1}{4\pi\mu}\begin{bmatrix} \frac{b}{r}+\frac{2\nu x^2}{r^3}& \frac{2\nu xy}{r^3}& -\frac{ax}{r^2}\\ \frac{2\nu yx}{r^3}& \frac{b}{r}+\frac{2\nu y^2}{r^3}& -\frac{ay}{r^2}\\ \frac{ax}{r^2}& \frac{ay}{r^2}& \frac{b}{r} \end{bmatrix}$

Cylindrical Coordinates ( $ρ,φ, z$ )

$G_{ik}=\frac{1}{4\pi\mu}\begin{bmatrix} \frac{b}{r}-\frac{a}{r+z}???+\frac{\rho^2}{r^3} & 0& -\frac{a\rho}{r(r+z)}+\frac{\rho z}{r^3}\\ 0& \frac{b}{r}-\frac{az}{r(r+z)}& 0\\ \frac{a\rho}{r(r+z)}+\frac{z \rho}{r^3}& 0& \frac{b}{r}+\frac{z^2}{r^3} \end{bmatrix}$

Spherical Coordinates ( $r,θ,φ$ )

$G_{ik}=\frac{1}{4\pi\mu}\begin{bmatrix} \frac{2}{r}+\frac{az}{r^2}& \frac{a\rho^3}{r^2(r+z)^2}& 0\\ \frac{-a\rho}{r^2}& \frac{b}{r}-\frac{az^2}{r^2(r+z)}& 0\\ 0& 0& \frac{b}{r}-\frac{az}{r(r+z)} \end{bmatrix}$

z-force only

$\mathbf{u}=\frac{1}{4\pi \mu} \left[\frac{z\mathbf{r}}{r^3}-\frac{a\mathbf{r}}{r(r+z)}+\left(\frac{b}{r}+\frac{az}{r(r+z)}\right)\hat{\mathbf{z}}\right]$

z-stress on z=0 plane is ZERO.

[edit] Nucleus of expansion at origin in infinite space

displacement (K is a constant)

$\mathbf{u}= K \frac{\mathbf{r}}{r^3}$

Stress

$\sigma_{ij}=K\mu\left(\frac{4\delta_{ij}}{r^3}-\frac{12x_ix_j}{r^5}\right)$

z-Stress on xy plane

$\sigma_{zj}=K\mu\left(\frac{4\hat{\mathbf{z}}}{r^3}-\frac{12z\mathbf{r}}{r^5}\right)$

[edit] Nucleus of expansion dipole in infinite space

dipole along z axis. Displacement

$\mathbf{u}=K\frac{z\mathbf{r}}{r^5}$

[edit] Nucleus of expansion at infinity in infinite space

displacement (K is a constant)

$\mathbf{u}= K \mathbf{r}$

Stress

$\sigma_{ij}=\frac{2K\mu b}{a}\,\delta_{ij}$

z-Stress on an xy plane, including z=0 plane

$\sigma_{zj}=\frac{2K\mu b}{a}\hat{\mathbf{z}}$

[edit] Semi-infinite line of dilatation in infinite space

Along negative z axis. Displacement

$\mathbf{u}= K\nabla\ln(r+z)= \frac{K}{r(r+z)} (\mathbf{r}+r\hat{\mathbf{z}})$

Stress

$\sigma_{ik}=\frac{4K\mu}{r^2}\begin{bmatrix} \frac{r}{r+z}-\frac{x^2(2r+z)}{r(r+z)^2}& -\frac{xy (2r+z)}{r(r+z)^2}& -\frac{x}{r}\\ -\frac{xy (2r+z)}{r(r+z)^2}& \frac{r}{r+z}-\frac{x^2(2r+z)}{r(r+z)^2}& -\frac{y}{r}\\ -\frac{x}{r}& -\frac{y}{r}& -\frac{z}{r}& \end{bmatrix}$