Introduction to Elasticity   362-1-3111   (Fall 2011)

Prof. Zohar Yosibash
 

Syllabus (pdf)

Homework:

Relations between elastic constants in isotropic materials (tiff format)

Kinematic relations and equilibrium equations in cylindrical coordinates (jpg format)

Kinematic relations and equilibrium equations in spherical coordinates (jpg format)

A nice tutorial on 2-D Problems in Elasticity from the book Advanced Strength and Applied Elasticity by A. Ugural & S. Fenster. Prentice Hall , 2003, including the Compatibility equations in polar coordinates (p116)     

 

Example of a final exam given in 2004 (pdf file)

Lecture schedule:

·         Week 1: The stress tensor, equilibrium eqs. derivation, trasformation of coordinates

of a 2nd order tensor, principal stresses and stress tensor invariants.

·         Week 2: Kinematic relations between strains and displacements. Derivation of the Green and Almansi

strain tensors (finite strains). The small strain tensonr and the compatibility equations. The physical interpretation of shear strain and normal strain.

·         Week 3: Constitutive law – Hooke’s law for a general anisotropic material, and Isotropic materials.

                  The Lamme constants, Young modulus and

Poisson’s ratio – Incompressible materials.

·         Week 4: The Navier-Lamme equation, Saint-Venant  principle and the superposition principle.

                  Two dimensional problems – Generalized plane-strain, and the Navier-Lamme equations

 For pl. strain.

·         Week 5: The Airy stress function for 2-D isotropic material under pl. stain: derivation based on

compatibility equation, and equilibrium equation. Examples of Airy S.F. in a rectangle.

·         Week 6: Plane stress – the Navier-Lame equations, the Airy stress function. Example of the Airy

Stress function for solving the cantilever beam with an end shear force.

·         Week 7: Stress and strains in a Cylindrical coordinate system. Derivation of the equilibrium

                  equations in cylindrical coordinates. Plane strain and plane stress problems of

                  2-D bodies of circular shapes, independent of \theta.

 

·         Week 8: Solution of circular bodies problems in cylindrical coordinates: cylinder and disc under

                  internal and external pressure (plane stress/strain).

 

·         Week 9: Cylinder/disc rotating with constant angular velocity.

The Airy stress function in cylindrical coordinates.

 

·         Week 10: Equations of elasticity in axi-symmetric domains.

Introduction to thermo-elasticity.

 

·         Week 11: Formulation of the uncoupled Thermo-Elastic problem in 3-D domains.

                  Plane-stress and plane-strain situations of a thermo-elastic problem.

                  Solution of a plane-stress thermo-elastic disk under constant temperature field.

 

·         Week 12: Solution to HW problems and solving one of the previous year's test.

                 

·         Week 13: Elastic solution in the neighborhood of singular points in 2-D elasticity.

                  Eigen-pairs and the Airy stress function for singular solutions, and the asymptotic series.

                  Following are 3 good references to the subject:

1.      Williams, M. L., "Stress singularities resulting from various boundary conditions in angular corners of plates in extension", ASME Journal of Applied Mechanics, 19, (1952), pp. 526-528

2.      Williams, M. L., "On the stress distribution at the base of a stationary crack ", ASME Journal of Applied Mechanics, 24, (1957), pp. 109-114

3.      Karp, S.N. and Karal, F.C. Jr., "The elastic field behavior in the neighborhood of a crack of arbitrary angle", Communications on Pure & Applied Mathematics, 15, (1962), pp. 413-421

 

                                                           

Last Updated:  Nov, 3, 2011