Large deformation in solid mechanics

[Chestermiller]

A lot of the confusion in this thread is due to the wording in the OP. Instead of "linear strain" and "nonlinear strain," you should say "small strain" and "large strain." The word "nonlinear" refers to the constitutive relation between stress and strain, and in this thread is specifically referring to 3D large strain nonlinear elasticity (i.e. the way rubber is often modeled in Finite Element Analysis).

3D large strain (can be nonlinear) elasticity is a.k.a. "hyperelasticity." There's lots of choices for hyperelastic constitutive relations, mostly because there's a variety of ways that materials can exhibit 3D nonlinear behavior. But I'll stick to the OP's questions, which I believe are 1) can we ever assume small strains for a large strain scenario and 2) what large strain tensor options are out there.

1)
At Chestermiller's link, you'll see E_{ij}=/frac{1}{2}\bigg(\frac{\partial u_i}{\partial X_j}+\frac{\partial u_j}{\partial X_i}+\frac{\partial u_k}{\partial X_i}\frac{\partial u_k}{\partial X_j}\bigg) near the top.
That's the equation for the Green (Lagrangian) strain tensor, which works for large strains. Remove the last term and you have the small strain version, which is used for small strain linear elasticity (I prefer to call this "Generalized Hooke's Law").
You can invent a structure (e.x. a cube) and write an equation for the node displacements "u" so that it undergoes a really simple large deformation (e.x. is stretched into a longer skinnier rectangle).
Then find the strain using the large strain version of the equation and the small strain version, in order to compare your answers.
If you haven't seen that equation before or don't know how to use it, then you'll just have to trust what everyone else has correctly said in this thread about the need for the large strain tensor.

2)
There are a lot of ways that the large strain tensor can be written.
When you read literature in this field of solid mechanics, sometimes a particular hyperelastic constitutive relation will be written in terms of the Almansi (Eulerian) strain and other times it will be written in terms of the Green (Lagrangian).. and many times it won't even be written in terms of strain, but rather "stretch."
These different choices mean that a particular author prefers a particular reference frame. Think "true stress" versus "engineering stress" (also relevant is the decision about whether to have your coordinate system rotate with a point on a rotating structure or remain in its original orientation).

Hope that helps -- epecially #2, because that is a common point of confusion.

Regarding item (1), I would add that the equation written for the strain applies also to large displacements but small strains.

Regarding item (2), I disagree with the implication that the different strain tensors are exclusively related to the frame of reference of the observer. They are defined using different mathematical contractions of the deformation gradient tensor with its transpose (or inverses of these tensors), all guaranteed to factor out any rotations of the frame of reference of the observer.

 

[afreiden]

Regarding item (2), I disagree with the implication that the different strain tensors are exclusively related to the frame of reference of the observer. They are defined using different mathematical contractions of the deformation gradient tensor with its transpose (or inverses of these tensors), all guaranteed to factor out any rotations of the frame of reference of the observer.

For a given deformation, the #'s in the 3x3 Almansi matrix would change under a rigid body rotation. OTOH, the #'s in the Green matrix would not change their values. I interpret that to mean that the Almansi matrix does not "factor out" rotations, but I think we're talking semantics because you are probably referring to a non-stationary "observer" in that circumstance.

 

[Chestermiller]

For a given deformation, the #'s in the 3x3 Almansi matrix would change under a rigid body rotation. OTOH, the #'s in the Green matrix would not change their values. I interpret that to mean that the Almansi matrix does not "factor out" rotations, but I think we're talking semantics because you are probably referring to a non-stationary "observer" in that circumstance.

I think the numbers in both matrices would change because these are just the components of the tensors. But, the tensors themselves would not change.

 

[afreiden]

I think the numbers in both matrices would change because these are just the components of the tensors. But, the tensors themselves would not change.

Strains are nice because we can physically see them!
Credit for example #1 in the figure goes to www.utsv.net. That example by itself is pretty convincing to me that the Green strain matrix is invariant to rigid body rotation and the Almansi strain is not. But to double-check, I created example #2.