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I. BASICS OF TENSOR ALGEBRA AND
TRANSFORMATION
- Definitions of scalars, vectors and tensors
- Vector transformation
- Tensor transformation
- Fundamentals of tensor algebra and calculus
- Kronecker delta function, tensor
contraction and alternating tensor
- Gauss divergence theorem
CHOU ve PAGANO
II. ANALYSIS OF STRAIN
- Deformation vector
- Strain and rotation tensors
- Geometric construction of the strain and rotation tensors in a
Cartesian reference frame
- Normal strain in an arbitrary direction
- Strain quadric of Cauchy
- Transformation of the strain tensor
- The eigenvalue problem to determine the
principal strains, directions and strain invariants
- Compatibility
III. ANALYSIS OF STRESS
- Definition of the stress (traction) vector
- Stress state at a point
- Stress (traction) vector on an arbitrarily oriented plane
- Normal stress in an arbitrary direction
- Stress quadric of Cauchy
- Transformation of the stress tensor
- The eigenvalue problem to determine the
principal stresses, directions and stress invariants
- Mohr's circles
- Equilibrium equations
IV. CONSTITUTIVE RELATIONS
- General linear-elastic constitutive relationship
- Anisotropic, monoclinic, orthotropic, transversely isotropic,
tetragonal and cubic materials
- Isotropic materials
- Functionally graded materials (FGMs)
- Lamé's constants
- Engineering parameters
- Generalized Hooke's Law
V. FORMULATION OF ELASTICITY PROBLEMS
- Equations of elasticity
- Boundary conditions
- Stress-based formulation
- Beltrami-Mitchell compatibility equations
- Displacement-based formulation
- Navier's equilibrium equations
VI. SOLUTION APPROACHES IN PLANAR ELASTICITY
- Plane strain
- Generalized plane strain
- Plane stress
- Airy stress function and biharmonic
equation
- Solutions of various problems:
- Simply supported beam under pure moments
- Beam subjected to sinusoidal load
- A surface loaded by concentrated normal and tangential forces (Flamant's solution)
VII. SOLUTIONS USING POLAR COORDINATES
- Equilibrium equations in polar coordinates
- Geometric construction of the strain tensor in a polar coordinate
system
- Compatibility in polar coordinates
- Solutions of various problems:
- Circular hole in a strained plate
- Stresses in rotating disks and cylinders
VIII. OTHER APPLICATIONS: CONTACT
MECHANICS, THERMOELASTICITY
- Contact mechanics
- Extension of Flamant's solution to
contact mechanics analysis
- Frictional stamp problems
- Reduction to singular integral equations
- Evaluation of contact pressure and influence of the coefficient of
friction
- Thermoelasticity
- Two-dimensional formulation of thermoelasticity problems (plane stress and strain)
- Thermoelastic
solution for rectangular beams under arbitrary temperature distributions
- Thermal stresses in thin disks and long
cylinders
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