3-D elasticity

3-D elasticity is one of three methods of structural analysis. This method is used for analyzing structures that behave in a linearly elastic fashion. There are 15 partial differential equations that must be simultaneously solved to get the state of stress at any point in an arbitrary structure. These are categorized into equilibruim, strain-displacement (or kinematic) relations, and the constitutive equations (or the general form of Hooke's law). Over the whole field the compatibility condition must be satisfied, which simply states that the displacement field is continuous and single-valued.

Equilibrium

The 3-D equilibrium equations are as follows:

${\frac {\partial \sigma _{x}}{\partial x}}+{\frac {\partial \tau _{xy}}{\partial y}}+{\frac {\partial \tau _{xz}}{\partial z}}+F_{x}=0$

${\frac {\partial \tau _{xy}}{\partial x}}+{\frac {\partial \sigma _{y}}{\partial y}}+{\frac {\partial \tau _{yz}}{\partial z}}+F_{y}=0$

${\frac {\partial \tau _{xz}}{\partial x}}+{\frac {\partial \tau _{yz}}{\partial y}}+{\frac {\partial \sigma _{z}}{\partial z}}+F_{z}=0$

Where $\sigma _{i}$ is the normal stress in the i direction, $\tau _{ij}$ is the shear stress on the i face in the j direction, and $F_{i}$ is the body force in the i direction.

For a body in equilibrium, the body forces are often sufficiently small to neglect. So, because $\tau _{ij}=\tau _{ji}$, these equations give us only 6 unknown quantities. With only 3 independent equations, they cannot be solved at this time.

Strain-Displacement Equations

The 3-D strain-displacement equations are as follows:

$\epsilon _{x}={\frac {\partial u}{\partial x}}$

$\epsilon _{y}={\frac {\partial v}{\partial y}}$

$\epsilon _{z}={\frac {\partial w}{\partial z}}$

$\gamma _{xy}={\frac {1}{2}}\left({\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}}\right)$

$\gamma _{yz}={\frac {1}{2}}\left({\frac {\partial v}{\partial z}}+{\frac {\partial w}{\partial y}}\right)$

$\gamma _{zx}={\frac {1}{2}}\left({\frac {\partial u}{\partial z}}+{\frac {\partial w}{\partial x}}\right)$

Where $\epsilon _{i}$ is the normal strain in the i direction, $\gamma _{ij}$ is the shear strain in the ij plane, and u, v, and w are the respective displacements in the x, y, and z directions.

These equations have 9 more unknown quantities, and only 6 more equations. With equilibrium there are a total of 15 unknowns and 9 equations. They cannot be solved yet.

Constitutive

The constitutive equation (or Generalized 3-D Hooke's law) is as follows:

$\epsilon _{x}={\frac {1}{E}}\left(\sigma _{x}-\nu (\sigma _{y}+\sigma _{z})\right)$

$\epsilon _{y}={\frac {1}{E}}\left(\sigma _{y}-\nu (\sigma _{x}+\sigma _{z})\right)$

$\epsilon _{z}={\frac {1}{E}}\left(\sigma _{z}-\nu (\sigma _{x}+\sigma _{y})\right)$

$\gamma _{xy}={\frac {\tau _{xy}}{2G}}$

$\gamma _{yz}={\frac {\tau _{yz}}{2G}}$

$\gamma _{xz}={\frac {\tau _{xz}}{2G}}$

Where E is the modulus of elasticity, ν is Poisson's ratio, G is the shear modulus, and all other variables are as defined above.

Since no new unknowns were introduced while 6 more independent equations now exist, it is possible to solve for the state of stress at an arbitrary point. Once this is done for every point in a body, compatibility must be satisfied for the displacement field to be physically possible.

For non-isotropic materials, this may be written in matrix form as:

${\begin{bmatrix}\epsilon _{1}\\\epsilon _{2}\\\epsilon _{3}\\\gamma _{23}\\\gamma _{13}\\\gamma _{12}\end{bmatrix}}={\begin{bmatrix}s_{11}^{E}&s_{12}^{E}&s_{13}^{E}&0&0&0\\s_{12}^{E}&s_{11}^{E}&s_{13}^{E}&0&0&0\\s_{13}^{E}&s_{13}^{E}&s_{33}^{E}&0&0&0\\0&0&0&s_{44}^{E}&0&0\\0&0&0&0&s_{55}^{E}&0\\0&0&0&0&0&s_{66}^{E}\end{bmatrix}}{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\tau _{23}\\\tau _{13}\\\tau _{12}\end{bmatrix}}$

with s₁₁^E=1/Y₁₁, s₂₂^E=1/Y₂₂, s₃₃^E=1/Y₃₃, s₁₂^E=$-\nu _{12}/Y_{11}$, etc.

Compatibility

The 3-D equations of compatibility may be derived directly from the strain-displacement equations. They are as follows:

${\frac {\partial ^{2}\epsilon _{x}}{\partial y^{2}}}+{\frac {\partial ^{2}\epsilon _{y}}{\partial x^{2}}}={\frac {\partial ^{2}\gamma _{xy}}{\partial x\partial y}}$

${\frac {\partial ^{2}\epsilon _{y}}{\partial z^{2}}}+{\frac {\partial ^{2}\epsilon _{z}}{\partial y^{2}}}={\frac {\partial ^{2}\gamma _{yz}}{\partial y\partial z}}$

${\frac {\partial ^{2}\epsilon _{x}}{\partial z^{2}}}+{\frac {\partial ^{2}\epsilon _{z}}{\partial x^{2}}}={\frac {\partial ^{2}\gamma _{xz}}{\partial x\partial z}}$

$2{\frac {\partial ^{2}\epsilon _{x}}{\partial y\partial z}}={\frac {\partial }{\partial x}}\left(-{\frac {\partial \gamma _{yz}}{\partial x}}+{\frac {\partial \gamma _{xz}}{\partial y}}+{\frac {\partial \gamma _{xy}}{\partial z}}\right)$

$2{\frac {\partial ^{2}\epsilon _{y}}{\partial x\partial z}}={\frac {\partial }{\partial y}}\left({\frac {\partial \gamma _{yz}}{\partial x}}-{\frac {\partial \gamma _{xz}}{\partial y}}+{\frac {\partial \gamma _{xy}}{\partial z}}\right)$

$2{\frac {\partial ^{2}\epsilon _{z}}{\partial x\partial y}}={\frac {\partial }{\partial z}}\left({\frac {\partial \gamma _{yz}}{\partial x}}+{\frac {\partial \gamma _{xz}}{\partial y}}-{\frac {\partial \gamma _{xy}}{\partial z}}\right)$

References

• A.C. Ugural, S.K. Fenster, Advanced Strength and Applied Elasticity, 4th ed.