NIST-03 (Linear Elasticity)¶

This problem is a coupled system of two equations with a mixed derivative in the coupling term (Lame equations); the context of the problem comes from the subject of linear elasticity.

Model problem¶

Equation solved: Coupled system of two equations

$-E \frac{1-\nu^2}{1-2\nu} \frac{\partial^{2} u}{\partial x^{2}} - E\frac{1-\nu^2}{2-2\nu} \frac{\partial^{2} u}{\partial y^{2}} -E \frac{1-\nu^2}{(1-2\nu)(2-2\nu)} \frac{\partial^{2} v}{\partial x \partial y} - F_{x} = 0.$

$-E \frac{1-\nu^2}{2-2\nu} \frac{\partial^{2} v}{\partial x^{2}} - E\frac{1-\nu^2}{1-2\nu} \frac{\partial^{2} v}{\partial y^{2}} -E \frac{1-\nu^2}{(1-2\nu)(2-2\nu)} \frac{\partial^{2} u}{\partial x \partial y} - F_{y} = 0.$

where $F_{x} = F_{y} = 0$ , $u$ and $v$ are the $x$ and $y$ displacements, $E$ is Young’s Modulus, and $\nu$ is Poisson’s ratio.

Domain of interest: $(-1, 1)^2$ with a slit from $(0, 0)$ to $(1, 0)$ .

Boundary conditions: Dirichlet, given by exact solution.

Exact solution¶

Known exact solution for mode 1:

$u(x, y) = \frac{1}{2G} r^{\lambda}[(k - Q(\lambda + 1))cos(\lambda \theta) - \lambda cos((\lambda - 2) \theta)].$

$v(x, y) = \frac{1}{2G} r^{\lambda}[(k + Q(\lambda + 1))sin(\lambda \theta) + \lambda sin((\lambda - 2) \theta)].$

here lambda = 0.5444837367825, and Q = 0.5430755788367.

Known exact solution for mode 2:

$u(x, y) = \frac{1}{2G} r^{\lambda}[(k - Q(\lambda + 1))sin(\lambda \theta) - \lambda sin((\lambda - 2) \theta)].$

$v(x, y) = -\frac{1}{2G} r^{\lambda}[(k + Q(\lambda + 1))cos(\lambda \theta) + \lambda cos((\lambda - 2) \theta)].$

here lambda = 0.9085291898461, and Q = -0.2189232362488. Both in mode 1 and mode 2, $k = 3 - 4 \nu$ , and $G = E / (2(1 + \nu))$ .

Sample solution¶

Solution for mode 1:

Solution-u.

Solution-v.

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