ME 502

Call No. 14464

S. F. Felszeghy Fall, 1996

THEORY OF ELASTICITY

Text:Elasticity in Engineering Mechanics, A. P. Boresi and K. P. Chong, Elsevier, 1987.

WEEK DATE TOPICS PROBLEMS 0 Sep. 26 Preface, 1-1 thru 1-6, 2-1 to 2-3 1 Oct. 1 2-3 thru 2-5 2-4:1 Oct. 3 2-6 to 2-10 2-6:1, 3; 2-8:1 2 Oct. 8 2-10 to 2-15 2-8:3; 2-11:3, 8* (*Find principal directions) Oct. 10 2-15, 2-16 2-13:4, 6; 2-15:1 3 Oct. 15 3-0 to 3-5 2-15:4; 2-16:2; 3-3:7 Oct. 17 3-5 thru 3-8 3-3:8, 10; 3-4:2 4 Oct. 22 MIDTERM Oct. 24 4-0 to 4-6 3-5:2; 3-6:5; 3-8:3 5 Oct. 29 4-6 thru 4-8, 4-11, 4-12, 4-14 4-4:1; 4-6:1, 4 Oct. 31 4-14 to 4-18 4-7:1; 4-14:3, 5 6 Nov. 5 4-18, 4-19 4-17:1; 4-18:6; 4-19:3 Nov. 7 5-0 thru 5-3 5-1:4; 5-2:4; 5-3:1 7 Nov. 12 5-4 to 5-7 5-4:5; 5-6:1 Nov. 14 MIDTERM 8 Nov. 19 5-7, 6-0 thru 6-3, 6-5, 6-6 5-7:6; 6-5:1 Nov. 21 6-6, 6-7, 6-10 6-8:5; 6-10:1 9 Nov. 26 6-10, 6-11, 7-0 thru 7-2 6-11:6, 9 10 Dec. 3 7-3 thru 7-6 Dec. 5 7-7, 7-8, 7-10

FINAL EXAM: Tuesday, Dec. 10, 7:30 - 10:00 p.m.

The continuum model of matter. Density.

A continuum occupying a region of space called a body.

Continuum mechanics. The foundations of the theory of elasticity.

Static response prediction of elastic bodies to applied loads involves solving boundary-value problems.

Analytical solutions to boundary-value problems are often difficult to find.

Need for numerical methods: Ritz, weighted residual, finite difference, finite-element methods.

Kinematics: material point (or particle), material (Lagrangian) coordinates, spatial (Eulerian) coordinates, the motion of a body, existence of inverse motion.

Material and spatial descriptions; material derivative.

Deformation: components of infinitesimal material vector. Summation convention. The deformation gradient.

Coordinate rotations. Kronecker delta. Second-order tensor transformation law.

Jacobian greater than zero.

Displacement;displacement gradient.

Extension of an infinitesimal material line element: the right Cauchy-Green tensor (symmetric). Stretch. Magnification factor. Unit extension. Lagrangian strain tensor (symmetric). Lagrangian strain-displacement relations.

Interpretation of the components of the Lagrangian strain tensor.

Extrema of magnification factor. Lagrange multiplier method. The eigenvalue problem. Characteristic equation. Principal strains. Invariants. Principal directions (axes).

Statement of polar decomposition of deformation gradient. Interpretation.

Reciprocal strain ellipsoid. Conjugate diameters are mapped into orthogonal line elements.

Infinitesimal deformation. Symmetric and skew-symmetric parts of displacement gradient.

The infinitesimal rotation tensor. The infinitesimal strain tensor. Normal strain. Shear strain.

Equations of compatibility. Derivation of equations. Existence of single-valued continuous displacement field when strain functions given. Line integral. Simply connected region. Six equations of compatibility.

Body forces, usually measured per unit mass. Contact (surface) forces. The stress principle of Euler and Cauchy. Stress vector.

Limit of summation of surface forces on infinitesimally small tetrahedron is zero as tetrahedron size approaches zero. Stress tensor. Classification of stress tensor components: normal stress and shearing stress.

Balance laws of linear momentum and angular momentum. Local forms: Cauchy's equation of motion, symmetry of stress tensor.

Normal and shearing stress on an olique plane expressed in terms of stress tensor components.

Principal stresses. Principal directions (axes). Stress invariants.

Maximum shearing stresses. Planes on which they act. Mohr's circle for stress.

Need for constitutive equations. No. of governing equations = 12. No. of dependent variables = 18. Need six more equations.

Limit ourselves to infinitesimal displacement gradients. No distinction between material and spatial coordinate in writing field equations.

The ideal elastic solid (or Hookean solid). Hooke's law for simple tension. Generalized Hooke's law with 81 constants. Elastically homogeneous material. Symmetries reduce 81 constants to 36.

Six-by-one matrix equation form of stress-strain relations. The strain energy-density function and its derivatives. Symmetry of six-by-six coefficient matrix of stress-strain relations reduces number of independent constants to 21. Consequences of material symmetries on number of independent constants. One plane of elastic symmetry: monoclinic symmetry; 13 independent constants. Three orthogonal planes of symmetry: orthotropic symmetry; nine independent constants. Material with no preferred directions: isotropy; two independent constants (Lamé constants). Special states of stress. Simple tension: modulus of elasticity (Young's modulus), Poisson's ratio. Pure shear: modulus of rigidity (shear modulus). Hydrostatic compression: volumetric strain, dilatation, bulk modulus (modulus of compression), incompressible material.

Thermoelasticity equations for isotropic media.

Compatibility equations in terms of stress. Beltrami-Michell compatibility equations.

The boundary-value problems in elastostatics. Navier's equations.

Uniqueness of solution.

Torsion of a circular shaft. Saint-Venant's Principle. Pure bending of a beam.

Plane strain.

Plane stress.

Generalized plane stress.

Plane strain and generalized plane stress problems are governed, except for a constant, by identical equations in two-dimensions.

Compatibility equations in terms of stress. For plane strain. For plane stress.

Airy stress function. Fundamental biharmonic boundary-value problem (FBBVP).

Representation of biharmonic function solution to FBBVP by means of two analytic functions of a complex variable.

Polynomial solutions to FBBVP in rectangular coordinates. Example: uniformly loaded beam; comparison to strength of materials solution.

Displacement vector. Strain-displacement equations. Stress vector. Stress tensor. Equations of equilibrium. Stress-strain relations.

Plane strain: displacements; strain-displacement equations; stress-strain equations; equilibrium equations; stress-compatibility equations; Airy stress function.

Axially symmetric problems. Airy stress function. Stresses. Strains. Displacements.

Examples: thick-walled cylinder; pure bending of a curved bar.

Stress concentration problem of circular hole in plate.

The warping function. The Neumann boundary-value problem in potential theory. Torsional rigidity.

Torsion problem formulated as Dirichlet problem.

Torsion problem formulated with Prandtl torsion function. Example: bar with elliptical cross section; lines of shearing stress; maximum shearing stress vector. Example: bar with elliptical cross section and concentric elliptical hole. Example: thin-walled tube.

Influence of location of origin of x-y axes. Torsion of rectangular bars.

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