## Topics

- Deformation Gradients
- Polar Decompositions
- Rotation Matrices
- Finite Element Mapping
- Small Scale Strains
- Green & Almansi Strains
- Principal Strains & Invariants
- Hydrostatic & Deviatoric Strains
- Velocity Gradients
- True Strain
- Material Derivative
- Special Topics

## Summary

This section gets to the heart of what Continuum Mechanics is all about - dealing with large displacements
and

deformations of objects.
The ultimate goal is often the determination of the

stress,
strength, fatigue, and fracture properties of an object or material. However, all these objectives begin
with the same first step - quantifying the object's displacements and deformations.

Displacements are not usually the focus of attention. In fact, the term

*rigid body displacement*
implies this because it refers to the situation where the object moves, but does not stretch
or deform in any way. Such behavior does not generate

stress. It can, however, seriously
complicate the more important objective of determining

deformations.
The term

*deformation* refers to the much more interesting and complex situation
of material bending, twisting, stretching, etc. All of these deformation modes generate

stress. The challenge is to separate the displacements from the

deformations, and to quantify each.

It turns out that

deformations
are closely related to the gradient of the displacement field.
Since gradients quantify

*rate of change w.r.t. position*, it makes sense that if the
displacements at all points on an object are the same, then it is undergoing rigid body
displacement and there is no change in displacements and therefore no

strains, stresses, fatigue, etc.

The main complication to the above effort is...

rotations.
*Rigid body rotations* are a
subset of

*rigid body displacements* that complicate the whole process and
can appear (incorrectly) as

strains if they are not treated properly.