- 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
This section gets to the heart of what Continuum Mechanics is all about - dealing with large displacements
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
The challenge is to separate the displacements from the
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
The main complication to the above effort is... rotations.
Rigid body rotations
subset of rigid body displacements
that complicate the whole process and
can appear (incorrectly) as strains
if they are not treated properly.