# Why are there lots of definitions for strain?

Why do we need Green or Almansi strains and what is True strain? I'm so confused about terminology.

## 1 Answer

A physical structure doesn't care what stress and strain measures you use to model it. It just does what it does.

However to make a useful mathematical model, the model has to be simple enough so you can actually work with it. That results in different stress and strain measures for different situations. The thing that needs to stay simple is actually the stress-strain relationship $$\sigma_{ij} = C_{ijkl}\epsilon_{kl}$$ where $$C$$ is a fourth-order tensor, with 21 independent components for a general material, and in general all those 21 components can be nonlinear functions of stress, strain, temperature, time, etc, etc ...

Life gets much simpler if you can make the approximation that $$C$$ is constant, and one way to do that is to get creative about how to define $$\epsilon$$ and $$\sigma$$.

The simplest situation is where the deformations can be assumed to be infinitesimally small. In that case, the only significant terms are the first derivatives of the deformation (i.e. the nine partial derivatives $$\partial u_i/\partial x_j$$), and products of two derivatives are negligible and can be ignored. Those assumptions give "engineering strain," and assuming $$C$$ is constant then gives "engineering stress."

Another situation is where the translational deformations can be large, but there is no significant rigid body rotation of the structure. In that case, the nine partial derivatives can be large (e.g. strains of order 1 or higher) but the absence of rotations means that products of the derivatives can still be ignored.

If you want to combine large strain increments, things work out better if you take the logarithm of the derivatives (for example if you stretch something by 50% of its original length and then stretch it by 50% of its new length, its final length is 2.25 times the original length, not 2.0 times). Those assumptions lead to "logarithmic strain" or "true strain" and "true stress".

A third combination is the "opposite" of the above: the strains are small, but there may be arbitrarily large rigid body rotations. Those assumptions lead to Cauchy-Green strains, and similar strain measures.

Of course the final situation is where everything is large - and in that case, it's sometimes not very clear whether an engineering model is "really" solid mechanics, or fluid dynamics of a non-Newtonian fluid!

The basic difference between Green strain and Almansi strain is that Green strain is based on the initial configuration of the material, and Almansi strain on its final configuration. To be honest I've never seen any use of Almansi strain at all, but no doubt there is some special application where it is the "best" strain measure to use.