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## 3-D elasticity
## Additional recommended knowledge
## Equilibrium equationsThe equilibrium equations of 3-D elasticity are formulated through the principle of conservation of linear momentum (i.e., Newton's second law of motion) as applied to a continuum. The equations may be expressed in Cartesian coordinates as Where σ
The solution of the equilibrium equations fully defines the nine stress components throughout a structure. The conservation of angular momentum principle, however, reveals that not all nine of the stress components are independent. This principle shows that in the absence of body couples the stress tensor is symmetric (i.e., τ In general the equations of equilibrium cannot be solved without introducing additional equations. In other words, 3-D elastic structures are statically indeterminate. ## Strain-displacement equationsThe 3-D strain-displacement equations are as follows: where ε These equations have 9 more unknown quantities, and only six more equations. With equilibrium there are a total of 15 unknowns and 9 equations. They cannot be solved yet. ## Constitutive equationsThe constitutive equation (or generalized 3-D Hooke's law) is as follows: where 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 with where In isotropic and orthotropic materials symmetry reduces the equations to: ## Compatibility equationsThe 3-D equations of compatibility may be derived directly from the strain-displacement equations. They are as follows: ## See also- Elastic
- Solid mechanics
- Linear elasticity
- Young's modulus
- Stress (physics)
- Strain (materials science)
- Stiffness
- Poisson's ratio
- Yield (engineering)
- Yield surface
## References- A.C. Ugural, S.K. Fenster,
*Advanced Strength and Applied Elasticity*, 4th ed.
Categories: Continuum mechanics | Materials science |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "3-D_elasticity". A list of authors is available in Wikipedia. |