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# Objective Stress Rates in Finite Strain of Inelastic Solid and Their Energy Consistency

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In many practical problems of solid mechanics, it is sufficient to characterize material deformation by the small (or linearized) strain tensor. But there are also many problems where the finiteness of strain must be taken into account. These are of two kinds: 1) Large nonlinear elastic deformations possessing a potential energy (exhibited, e.g., by rubber), in which the stress components are obtained as the partial derivatives of potential energy; and 2) inelastic deformations possessing no potential, in which the stress-strain relation is defined incrementally and the stress increment or rate must be formulated objectively (i.e., independently of coordinate rotations). The present paper reviews the latter kind. The presentations is focused on energy consistency (or work-conjugacy of the stress rate and the finite strain increment), which requires a variational energy approach instead of the narrow classical viewpoint of tensorial coordinate transformations The objective stress rates corresponding to various choices of the finite strain measure are presented and it is shown that the energy consistency is violated by the rate used in most commercial structural analysis software (using finite element analysis). While in many applications this causes only a negligible error, it is pointed out that the error in forces or displacements can reach 30% to 100% in some practical situations. These include highly compressible materials (e.g., rigid forms, honeycomb, compressible soils, soft wood, osteoporotic bone, various biologic tissues) and highly orthotropic materials very soft in shear (e.g., some fiber composites, sandwich plates, solids weakened by unidirectional cracking, layered elastomeric bearings).

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12-05-22-FinStrain-ObjStrRate.pdf | 1.29 MB |

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## Comments

## Re: Objective Stress Rates in Finite Strain of Inelastic Solid..

Dear Zdenek:

Thank you for sharing this. The practical implications are very interesting. A couple of quick questions/comments:

On page 3, you mention: "The objectivity of \sigma_{ij}^{(m)} is ensured by independence of \sigma_{ij}^{(m)} from rigid body motions…". If you have a (time-independet) rigid translation then an objective stress rate would be independent of this translation. What about rotations? If a stress rate is objective, wouldn't it mean that under a rigid rotation it is transformed like a second-order tensor? If so, then it would depend on the rotation?

At the end of the same page, you mention "Lie derivative". Lie derivative of what components of Cauchy stress (with respect to spatial velocity)? Marsden and Hughes (Mathematical Foundations of Elasticity, 1983) show that all the well-studied objective stress rates are Lie derivatives of different components (covariant, contravariant and mixed) of Cauchy stress or some linear combinations of them. This is interesting but of course one should worry if a given objective stress rate corresponds to some measure of strain (and this is what you're addressing here).

The following paper may be relevant (haven't read it): F. Molenkamp, Limits to the Jaumann stress rate, International Journal for Numerical and Analytical Methods in Geomechanics 10(2), 1986, 151-176.

Regards,

Arash

## Objectivity

Dear Arash,

I think, in this report the objectivity of S_ij is implied with respect to work conjugacy. In other words the following relation always holds regardless of any rigid body rotation:

σ_ij d_ij = S_ij ∂(E_ij)/∂t

And I would be grateful if you would give more description about your last question.

Mohsen

## Re: Objectivity

Dear Mohsen:

"σ_ij d_ij" is independent of rotations (or any coordinate transformation) because it is a scalar. However, this does not mean that "σ_ij" and/or "d_ij" remain unchanged under a rotation; they transform such that "σ_ij d_ij" remains unchanged. Note also that you should write this like σ_ij d^ij to be tenurially consistent.

Regarding the other comment, please have a look at the book by Marsden and Hughes (their discussion on applications of Lie derivatives) and then we can discuss it if you have any questions.

Regards,

Arash

## Objectivity

Dear Arash,

Yes it is a scalar. In this report it is correct that no objective rate of stress is used. Therefor perhaps the objectivity is implied with respect to work conjugacy.

In plasticity (hyperelastic formulation of plasticity) the evolution of C^p or b^e is expressed in terms of their Lie derivatives (one can refer to the work of Simo). And here you mentioned that one should be careful about the Lie derivative of stress without regard to the way that the corresponding strain is defined. I wanted to know if this relevance can also influence the way that we define the evolution of C^p or b^e.

Regards

Mohsen

## Re: Objectivity

Dear Mohsen:

This paper is fine. As long as one uses an objective stress rate and is careful with work conjugacy things would be ok. However, the statement that objective stress rates are independent of rotations is not correct. An objective quantity transforms like a tensor. So, any objective Cauchy stress rate would transform like a second-order tensor.

If all you need is a stress rate then Lie derivative of Cauchy stress (and its different associated components) are fine. One can also use any covariant time derivative of stress as well. Perhaps that wouldn't be as natural as a Lie derivative because you would need a connection. If you're given some "potential" function then you don't need to worry about work conjugacy as it'll be there automatically. But if for any reason you need to work with a pair of "stress" and "strain" they must be work conjugate and this is what Prof. Bazant is emphasizing.

Regards,

Arash

## work conjugacy

Arash,

Speaking of references, pages 14 (make it 12) - 23 of Hill's Aspects of Invariance in Solid Mechanics paper seems to be quite relevant for the present discussion.

- Amit

## Re: work conjugacy

Hi Amit,

Yes, it is definitely relevant (has been cited in this paper).

Regards,

Arash

## Re: work conjugacy

I actually took a quick look through the paper when it appeared but could not find it discussed in the text - may be I just missed it.

## Objectivity and work conjugacy

[img_assist|nid=12435|title=Thermoviscoelasticity|desc=Hi all, I'm trying to model high temperature viscoelasticity using hypoelastic constitutive equations. I'm not sure of how to include the rotation tensor in the generalized viscoelastic equation. kindly advise me. Thank you very much.|link=none|align=left|width=99|height=100]

Dear All,

I just got the paper and I've started studying it. I'm a young and new researcher.

Kind Regards

Rotimi