Theory of Elasticity

Time and Location: Lecture, to be arranged

Concept of stress and strain; index notation, properties of tensors, generalized Hooke's law, the modulus and compliance tensors, field equations of elasticity; extension, torsion, and flexure of bars; two-dimensional and three dimensional elastostatic problems, stress concentrations, waves in elastic media.

Textbook: Mathematical Theory of Elasticity, I. S. Sokolnikoff. Krieger (1983).

Reference: Foundations of Solid Mechanics, Y. C. Fung. Prentice Hall (1968)

Coordinator: R. S. Lakes

Course Outline
Topic and Reading

Introduction.

Tensors. Rank of tensors. Coordinate transformations. Curvilinear coordinates.     Sokolnikoff Ch. 1, 2; Metric tensor, covariant differentiation.     Fung Ch. 2.

Tensor properties of materials. Elastic modulus as a tensor property. Strain. Principal strains. Compatibility. Infinitesimal and finite deformations.     Ch. 1.

Stress. Equilibrium of a continuous medium. Principal stresses.     Ch. 2.

Elastic behavior. Hooke's law. Anisotropic and isotropic media. Symmetry and material properties. Triclinic, monoclinic, orthorhombic, hexagonal, and cubic classes. Physical meaning of elastic constants.     Ch. 3.
Dynamics. Saint-Venant's principle.

Extension, torsion, and bending of beams. Beams of arbitrary cross section.     Ch. 4.
Three dimensional solutions. Warp of cross section of bars in torsion. Analysis of torsion of a bar of rectangular section. Grooved bars. Poisson's ratio and anticlastic curvature of bars in bending.

Two-dimensional problems. Derivation of stress concentration factors.     Ch.5.
Stress distribution around holes and notches. Airy stress function. Stress concentration.

Three-dimensional problems. Boussinesq's problem for a concentrated load.     Ch. 6.
Waves in elastic media. Shear waves and longitudinal waves. Dependence of wave speed on elastic modulus components. Potentials in the theory of elasticity.

Elastic materials with generalized constitutive relations. Stress, couple stress, displacement, and rotation. -
Anisotropic solids. Bending and torsion of anisotropic bars.
Elastic solids with microstructure. Dispersion of waves in heterogeneous media.

Variational formulation of elasticity theory. Reciprocity.     Ch. 7.