Large deformation in solid mechanics

[mertcan]

Hi initially I am aware that large deformation in solid mechanics requires non linear strain theory in the lieu of infinitesmall strain theory. But I wonder that if we can approximate large deformation of material using infinitesmall strain of small elements employing and summing linear strains of infinitesmall elements??

 

[sophiecentaur]

Non-linearity is non-linearity, however you care to model it.

 

[Dr.D]

The classical analysis of the elastica is a good example of linear theory giving good results for very large deformations.

 

[Chestermiller]

The classical analysis of the elastica is a good example of linear theory giving good results for very large deformations.

Please provide more information about this. It seems obviously incorrect to me.

 

[Dr.D]

Obviously incorrect, huh?

How about these:
Timoshenko, S.P. & Gere, J.M., Theory of Elastic Stability, McGraw-Hill, 1961, p.76 ff.
Ashwell, D.G., "Nonlinear Problems," in Handbook of Engineering Mechanics, S. Flugge, ed., McGraw-Hill, 1962, p. 45-3 ff.
Frisch-Fay, R, Flexible Bars, Butterworths, 1962 (the whole book)

In each of these, a linear constitutive model is used to develop a nonlinear bending model that produces good results.

 

[Andy Resnick]

The classical analysis of the elastica is a good example of linear theory giving good results for very large deformations.

The term 'linear theory' may need some clarification here. Certainly, use of linear or infinitesimal field theory can be extended to include a wide range of finite strain phenomena: winds and tides, sounds, structures and earthquakes, sailing and flying. However, there are simple phenomena that cannot be explained with a linear theory: paints and polymers will climb up a rotating rod. Similarly, there are materials such as hyper- or hypo-elastic materials, materials with memory, etc. that cannot be modeled with a linear constitutive relation.

 

[Andy Resnick]

I wonder that if we can approximate large deformation of material using infinitesmall strain of small elements employing and summing linear strains of infinitesmall elements??

The quick answer is "sometimes yes, sometimes no. It depends."

 

[Chestermiller]

Obviously incorrect, huh?

How about these:
Timoshenko, S.P. & Gere, J.M., Theory of Elastic Stability, McGraw-Hill, 1961, p.76 ff.
Ashwell, D.G., "Nonlinear Problems," in Handbook of Engineering Mechanics, S. Flugge, ed., McGraw-Hill, 1962, p. 45-3 ff.
Frisch-Fay, R, Flexible Bars, Butterworths, 1962 (the whole book)

In each of these, a linear constitutive model is used to develop a nonlinear bending model that produces good results.

So it is a geometric non-linearity and not a non-linearity of the constitutive model? Are there any linear large deformation constitutive models that you are aware of (i.e., that are properly invariant under change of observer)? If so, please write out the constitutive model in tensorial form. I am particularly interested in the strain tensor involved.

 

[sophiecentaur]

The classical analysis of the elastica is a good example of linear theory giving good results for very large deformations.

A rubber band and a polythene bag will both exhibit easily measurable departure from linear stress / strain. Dunnit many times in a School lab.

 

[Dr.D]

And the rubber band and the polyethylene bag say what about the elastica?

 

[Dr.D]

So it is a geometric non-linearity and not a non-linearity of the constitutive model? Are there any linear large deformation constitutive models that you are aware of (i.e., that are properly invariant under change of observer)? If so, please write out the constitutive model in tensorial form. I am particularly interested in the strain tensor involved.

My original (bold face is stuck on!) reply was to Sophiecentaur's comment that, "Non-linearity is non-linearity, however you care to model it." My point was that linear material models have worked well in certain particular cases with large deformations.

I really know very little about large deformation constitutive models, so I can't post anything useful about them.

 

[Nidum]

Elastica based methods can be used for geometrically large deflection problems where the elements of a structure are either relatively slender or made from material which is particularly flexible .

In this class of problems the components of the structure may collectively or individually deform by a large amount but the material of the components remains within the elastic limit .

It's a bit hard going but this explains the basics of the method reasonably well :

Link

Never very much used in practical problem solving and now almost completely forgotten with the advent of sophisticated non linear FEM .

 

[sophiecentaur]

And the rubber band and the polyethylene bag say what about the elastica?

Elastica are a band but not elastic or rubber.

 

[Chestermiller]

Elastica based methods can be used for geometrically large deflection problems where the elements of a structure are either relatively slender or made from material which is particularly flexible .

In this class of problems the components of the structure may collectively or individually deform by a large amount but the material of the components remains within the elastic limit .

It's a bit hard going but this explains the basics of the method reasonably well :

Link

Never very much used in practical problem solving and now almost completely forgotten with the advent of sophisticated non linear FEM .

As best I can tell from the link, this kind of methodology is limited to non-linear materials experiencing very simple deformational kinematics (typically 1D) so that the constitutive equation describing any and all deformations of the material does not have to be determined. The material behavior only needs to be characterized for the specific class of deformations occurring in the application of interest. I have personally used this approach in analyzing the inflation behavior of a balloon, in which the elastic sheet comprising the balloon membrane is experiencing an equal-biaxial deformation. In this situation, the in-plane tensile stress within the material $\sigma$ only needs to be expressed as a (typically) nonlinear function $\sigma(\lambda)$ of the stretch ratio $\lambda$ (as characterized by the ratio of the initial and present balloon radii). The required non-linear function can be measured in the laboratory using a biaxial stretching device. However, this kind of approach can only be applied if one is lucky enough to be working with a system that experiences a very simple type of deformation.

Another example of this kind approach, this time in fluid mechanics, is in the description of steady viscometric flows of viscoelastic fluids, in which the stresses and deformation of the fluid are fully characterized by the shear viscosity, first normal-stress difference, and second normal stress difference, each expressed as a non-linear function of the local shear rate. These three non-linear functions can be measured for a given fluid in laboratory viscometers, and then the results of these experiments can be applied to viscometric fluid flows in much more complex situations encountered in practice.

 

[mertcan]

Thanks for your responses, besides I would like to ask if there is another nonlinear strain-displacement relation formula except Almansi- Green strain ?

 

[Chestermiller]

Thanks for your responses, besides I would like to ask if there is another nonlinear strain-displacement relation formula except Almansi- Green strain ?

Yes. For large deformations, the strain tensor is not unique. There is also the Cauchy-Green strain tensor (aka the Green strain tensor). Here is a link that discusses both these large deformation strain tensors: http://www.continuummechanics.org/greenstrain.html

 

[mertcan]

Ok thanks for responses, so can we say that Almansi and cauchy green strain tensors always work for large deformation of all kind of material? ?

 

[Chestermiller]

Ok thanks for responses, so can we say that Almansi and cauchy green strain tensors always work for large deformation of all kind of material? ?

I don't understand what you mean by "work for." The constitutive equation for a material can be expressed in terms of either strain measure (and, in the case of liquids, in terms of their first and higher derivatives, or, in terms of "memory integrals" of the strain measures). For a purely elastic solid, it is not too difficult to determine the constitutive equation. In the case of viscoelastic liquids, rheologists are still working on how to express and measure the constitutive equation for real viscoelastic liquids. There are plenty of viscoelastic liquid mathematical models around, but none of them describe the behavior of a real viscoelastic liquid quantitatively for any and all deformations. And, if temperature is changing, the issue becomes even more complex.

 

[mertcan]

I don't understand what you mean by "work for." The constitutive equation for a material can be expressed in terms of either strain measure (and, in the case of liquids, in terms of their first and higher derivatives, or, in terms of "memory integrals" of the strain measures). For a purely elastic solid, it is not too difficult to determine the constitutive equation. In the case of viscoelastic liquids, rheologists are still working on how to express and measure the constitutive equation for real viscoelastic liquids. There are plenty of viscoelastic liquid mathematical models around, but none of them describe the behavior of a real viscoelastic liquid quantitatively for any and all deformations. And, if temperature is changing, the issue becomes even more complex.

I mean :

I don't understand what you mean by "work for." The constitutive equation for a material can be expressed in terms of either strain measure (and, in the case of liquids, in terms of their first and higher derivatives, or, in terms of "memory integrals" of the strain measures). For a purely elastic solid, it is not too difficult to determine the constitutive equation. In the case of viscoelastic liquids, rheologists are still working on how to express and measure the constitutive equation for real viscoelastic liquids. There are plenty of viscoelastic liquid mathematical models around, but none of them describe the behavior of a real viscoelastic liquid quantitatively for any and all deformations. And, if temperature is changing, the issue becomes even more complex.

you say that strain tensor is not unique there is also a cauchy green strain tensor besides the almansi strain in regard to nonlinear strain-displacement relation in post 16. So I would like to ask if there are just 2 (cauchy+almansi) strain definition for nonlinear strain- displacement relation?

 

[Chestermiller]

I mean :

you say that strain tensor is not unique there is also a cauchy green strain tensor besides the almansi strain in regard to nonlinear strain-displacement relation in post 16. So I would like to ask if there are just 2 (cauchy+almansi) strain definition for nonlinear strain- displacement relation?

I think there are probably others, but I'm not sure.

 

[Dr.D]

As best I can tell from the link, this kind of methodology is limited to non-linear materials experiencing very simple deformational kinematics (typically 1D) so that the constitutive equation describing any and all deformations of the material does not have to be determined.

The elastica is simply the extreme form of the classic column buckling problem. It is solvable in terms of elliptic integrals, but the material constitutive relation is the simple linear stress-strain model. There is no nonlinear material behavior, only gross deformation of the material. This is a very old solution, going back a long time (to Euler perhaps? I forget exactly who did it first).

I personally used this solution back in the mid-1960s in a MS thesis problem related to a spring based on large deformation column buckling.

 

[Dr.D]

Elastica are a band but not elastic or rubber.

The elastica is definitely elastic. That's where the name comes from. Rubber (and polyethylene) are examples of nonelastic materials.

 

[sophiecentaur]

The elastica is definitely elastic. That's where the name comes from. Rubber (and polyethylene) are examples of nonelastic materials.

I'm not too sure how the word 'elastic' should strictly be applied. I thought that it meant 'return to the same shape' - as opposed to 'plastic'. It doesn't imply linearity. From what I have read (recently, a bit) the elastica analysis assumes a uniform modulus. (?)

 

[Chestermiller]

The elastica is simply the extreme form of the classic column buckling problem. It is solvable in terms of elliptic integrals, but the material constitutive relation is the simple linear stress-strain model. There is no nonlinear material behavior, only gross deformation of the material. This is a very old solution, going back a long time (to Euler perhaps? I forget exactly who did it first).

I personally used this solution back in the mid-1960s in a MS thesis problem related to a spring based on large deformation column buckling.

In what you describe, the material behaves linearly elastically, but the geometrical effect is non-linear. On the other hand,the OP was asking about non-linear elastic material behavior.

 

[Chestermiller]

The elastica is definitely elastic. That's where the name comes from. Rubber (and polyethylene) are examples of nonelastic materials.

Then what are your thoughts on the well-established subject of Rubber Elasticity? Rubber is usually regarded as the "poster boy" for non-linear elasticity.

 

[Dr.D]

In what you describe, the material behaves linearly elastically, but the geometrical effect is non-linear. On the other hand,the OP was asking about non-linear elastic material behavior.

The OP spoke of large material deformation (IIRC), and I do not see that as necessarily implying nonlinear material behavior. That is why I mentioned the elastica.

Then what are your thoughts on the well-established subject of Rubber Elasticity? Rubber is usually regarded as the "poster boy" for non-linear elasticity.

I am not familiar with a well-estabished subject of rubber elasticity. All of my experience with rubber has shown it to be a lossy material, not fully returning to the original shape with the removal of load. I will certainly admit that this is not an area I follow very closely, however. Perhaps there is something new in this rubbery area.

 

[Chestermiller]

It was clear to me that the OP was referring to non-linear elastic behavior. There is a major difference between this and large-displacement-but-small-strain linear elastic behavior.

 

[Chestermiller]

Real rubber has a small amount of lossiness, but, in the ideal limit, rubber approaches non-linear elastic behavior (even at large deformations).

 

[sophiecentaur]

rubber has shown it to be a lossy material, not fully returning to the original shape with the removal of load

There are many applications in which rubber does return to its original shape after distortion. There is the famous example of the Rubber Cone suspension on the Morris Mini, which would last for tens of thousands of bumpy miles without lowering the level of the car body. The lossiness is a totally different issue. If steel is reckoned to be a typical elastic material, it isa fact that repeatedly operated springs lose their shape (air guns and motorcar valve springs are good examples where they need replacement when performance is important. So the distinction is not as clear as you suggest.

 

[afreiden]

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.

 

[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.