ME 5438: Plasticity and Viscoelasticity - Continuum Mechanics Review

Introduces the full tensor form of Hooke's Law to describe a linear elastic material.

Develops equations of motion and equilibrium in 1D and in 3D.

Defines hydrostatic (spherical) and deviatoric stresses. Shows that the principal directions are the same for deviatoric and full stress tensor. Develops expression for J2 deviatoric stress invariant.

Develops the concept of stress invariants by derivation and expansion of the characteristic equation.

Develops the eigenvalue/eigenvector problem leading to principal stresses as extreme values of normal stresses.

Develops the Cauchy stress formula and definition of traction vector.

Develops a measure for shape change (shear) from deformation gradient tensor and links with off-diagonal terms in the small strain tensor.

Develops the Lagrange finite strain tensor and the small strain tensor in terms of displacement gradients.

Continuation of a discussion of the Lagrange finite strain tensor. Introduces stretch and shows how both stretch and nominal (engineering) strain are related to the Lagrange finite strain tensor.

Develops the Lagrange finite strain tensor from a basic measure of length change and the deformation gradient tensor.

Introduces Eulerian versus Lagrangian reference frames and discusses motion (rigid body and deformation) in the Lagrangian reference frame.

Develops coordinate transformation relationships for vectors and tensors

Overview of tensor representation, Kronecker delta, and permutation tensor (alternating tensor)

Overview of my vector notation (direct, symbolic, index) and Einstein summation convention