Theory of Elasticity

1. Theory of Elasticity • Theory of elasticity governs response • Symmetric stress & strain components • Governing equations • Equilibrium equations (3) • Strain-displacement equations (6) • Constitutive equations (6) • Unknowns • Stress (6) • Strain (6) • Displacement (3)

2. Boundary & Initial Conditions • Linear elastic material • Three partial differential equations in displacements • Second order in each coordinate • Second order in time • On a free surface in each direction • Specify stress or displacement but not both • Initial conditions for each direction specify • Displacement and velocity

3. Surface Forces • Specify pressure (szz), shears (szx, szy) or • Specify displacements (u,v,w)

4. Rigid Body Motion – 2D

5. Rigid Body Displacements – 2D • Strains vanish • Integrating normal strains • Integrating shear strain • Hence (U, V are constants) • Displacement solution

6. Reactions with Excessive Constraints Extra constraint in horizontal direction will add excessive stress Vertical constraints and loads produce point load infinite stresses

7. Rigid Body Displacements – 3D • All strains vanish • In terms of displacements • Integrating yields displacements • Where • And

8. Self-Equilibrating Forces • Examples include: • Uniform pressure (submarine or bathysphere) • Thermal expansion • BCs remove rigid body translations & rotations • Constrain six degrees of freedom (3 dofs at one point, 2 dofs at a second and 1 dof at a third)

9. Plate & Beam Dofs at Each Node • Beam – 3 translations 3 rotations • Plate – 3 translations 2 rotations

10. Nastran FE Code – Plate Elements • All nodes for all elements types have six dofs • 3 for translation • 3 for rotation • Flat plate models need dofs perpendicular to plane of model constrained (set to zero) • Shells made of plate elements do not • Solid elements need all three rotations at each node set to zero

11. Simply Supported Beam Example Fix six dofs – 5 translation and 1 rotation

12. Simply Supported Beam Example Fix six dofs – 5 translation and 1 rotation

13. Cantilever Beam Fix six dofs – 3 translation and 3 rotation

14. Internal Surfaces & Cracks • Cracks • Internal Surfaces

15. Hertz Contact - Gaps & Friction • Hertz Contact • Gap & Friction Elements

16. Transformations

17. Use of Symmetry • Makes a large problem smaller • Axisymmetry reduces a 3D problem to 2D • Recall stress & strain symmetric • Examples:

18. Periodic Boundary Conditions • Stress & Strain are periodic • Mean displacements can vary linearly with coordinates due to expansion and rotation • For

19. Multi-Point Constraints - Tying or where xi are specified degrees of freedom, cij dj are known constants.

20. Distant Boundary Conditions • Build a sufficiently large model • At least 20 times length of largest dimension of interest • Substructure a large coarse model • Use output from large model as input to a refined local model • Use super-elements or substructuring • Use infinite elements (when available)