# Homework Help: Continuum Mechanics deformation definitions

1. Feb 12, 2013

### EngSciNZ

1. The problem statement, all variables and given/known data

What do you understand by the following terms; (i) principal stretch (ii) an
anisotropic material (iii) a dilatant deformation, (iv) a Lagrangian description of a
deformation, and (v) a pure deformation.

2. Relevant equations

Am just trying to find descriptions for these, am trying to research my papers before I sit them this year, but there are no suggested texts and couldn't find good definitions for all of these from searching on the internet.

3. The attempt at a solution

I pressume principal stretch is going to be similar to principal strain? And hence the ammount of elongation/contraction in the principal strain direction...

Anisotropy is differing properties in along different axis.

Dilatancy is to do with increased shear stress causing increased viscosity and sometimes solidification? But couldn't work out what the deformation was.

Didn't have much luck with the last two as they seem to be quite general terms.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Feb 18, 2013

### afreiden

Principal stretch: The principal stretches are the three eigenvalues of either the "Right stretch tensor," typically denoted U, or the "Left stretch tensor," denoted V. These eigenvalues are the same for both tensors, though you could think of them as being "ordered" differently depending on which tensor you are interested in. Sometimes stress-strain relationships are given in terms of principal stress and principal stretch.

Dilatancy: Did you mean to ask about the definition of "dilatancy" or did you mean "dilatational"?
I'm not a fluid mechanics person, so I can't comment on dilatancy.

Pure Deformation: This is not really a common phrase. I would assume that it means "homogeneous" deformation and additionally that no rigid body rotations are present.

Lagrangian description: The tensor U is a Lagrangian tensor. The tensor V is an Eulerian tensor (another common Lagrangian strain tensor that I'm sure you've come across is E -- and its Eulerian counterpart e). Under rigid body rotation, U would be unchanged, whereas V would change. The Lagrangian description is often referred to as the "material" description.

Under deformation without rigid body rotation (pure deformation?), U and V are the same.

To see what the Lagrangian description means, physically, look at the example at the bottom of this page:
http://utsv.net/solid-mechanics/4-stress/alternative-measures-of-stress
Look closely at the axes in the figure at the bottom of that page. Note that E is a Lagrangian strain tensor, and observe that it is invariant to the rigid body rotation.