iMechanica - Comments for "Objective Stress Rates in Finite Strain of Inelastic Solid and Their Energy Consistency"
https://www.imechanica.org/node/12432
Comments for "Objective Stress Rates in Finite Strain of Inelastic Solid and Their Energy Consistency"enObjectivity and work conjugacy
https://www.imechanica.org/comment/19058#comment-19058
<a id="comment-19058"></a>
<p><em>In reply to <a href="https://www.imechanica.org/node/12432">Objective Stress Rates in Finite Strain of Inelastic Solid and Their Energy Consistency</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>[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]
</p><p>Dear All,</p>
<p>I just got the paper and I've started studying it. I'm a young and new researcher.</p>
<p>Kind Regards</p>
<p>Rotimi </p>
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</ul>Mon, 21 May 2012 09:51:32 +0000Rotimi Adeleyecomment 19058 at https://www.imechanica.orgRe: Objectivity
https://www.imechanica.org/comment/19051#comment-19051
<a id="comment-19051"></a>
<p><em>In reply to <a href="https://www.imechanica.org/comment/19049#comment-19049">Objectivity</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Mohsen:
</p>
<p>
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.</p>
<p>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.</p>
<p>Regards,<br />
Arash
</p>
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</ul>Fri, 18 May 2012 17:07:47 +0000arash_yavaricomment 19051 at https://www.imechanica.orgObjectivity
https://www.imechanica.org/comment/19049#comment-19049
<a id="comment-19049"></a>
<p><em>In reply to <a href="https://www.imechanica.org/comment/19044#comment-19044">Re: Objectivity</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Arash,
</p>
<p>
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.
</p>
<p>
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.
</p>
<p>
Regards
</p>
<p>
Mohsen
</p>
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</ul>Fri, 18 May 2012 07:25:19 +0000M. Jahanshahicomment 19049 at https://www.imechanica.orgRe: Objectivity
https://www.imechanica.org/comment/19044#comment-19044
<a id="comment-19044"></a>
<p><em>In reply to <a href="https://www.imechanica.org/comment/19043#comment-19043">Objectivity</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Dear Mohsen:</p>
<p>"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.</p>
<p>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.</p>
<p>Regards,<br />
Arash</p>
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</ul>Thu, 17 May 2012 17:21:13 +0000arash_yavaricomment 19044 at https://www.imechanica.orgObjectivity
https://www.imechanica.org/comment/19043#comment-19043
<a id="comment-19043"></a>
<p><em>In reply to <a href="https://www.imechanica.org/comment/19021#comment-19021">Re: Objective Stress Rates in Finite Strain of Inelastic Solid..</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Arash,
</p>
<p>
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:
</p>
<p>
σ_ij d_ij = S_ij ∂(E_ij)/∂t
</p>
<p>
And I would be grateful if you would give more description about your last question.
</p>
<p>
Mohsen
</p>
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</ul>Thu, 17 May 2012 09:15:16 +0000M. Jahanshahicomment 19043 at https://www.imechanica.orgRe: work conjugacy
https://www.imechanica.org/comment/19028#comment-19028
<a id="comment-19028"></a>
<p><em>In reply to <a href="https://www.imechanica.org/comment/19027#comment-19027">Re: work conjugacy</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
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.
</p>
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</ul>Mon, 14 May 2012 18:55:50 +0000Amit Acharyacomment 19028 at https://www.imechanica.orgRe: work conjugacy
https://www.imechanica.org/comment/19027#comment-19027
<a id="comment-19027"></a>
<p><em>In reply to <a href="https://www.imechanica.org/comment/19026#comment-19026">work conjugacy</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Hi Amit,</p>
<p>Yes, it is definitely relevant (has been cited in this paper).</p>
<p>Regards,<br />
Arash</p>
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</ul>Mon, 14 May 2012 18:33:21 +0000arash_yavaricomment 19027 at https://www.imechanica.orgwork conjugacy
https://www.imechanica.org/comment/19026#comment-19026
<a id="comment-19026"></a>
<p><em>In reply to <a href="https://www.imechanica.org/node/12432">Objective Stress Rates in Finite Strain of Inelastic Solid and Their Energy Consistency</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Arash,
</p>
<p>
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.
</p>
<p>
- Amit
</p>
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</ul>Mon, 14 May 2012 18:26:57 +0000Amit Acharyacomment 19026 at https://www.imechanica.orgRe: Objective Stress Rates in Finite Strain of Inelastic Solid..
https://www.imechanica.org/comment/19021#comment-19021
<a id="comment-19021"></a>
<p><em>In reply to <a href="https://www.imechanica.org/node/12432">Objective Stress Rates in Finite Strain of Inelastic Solid and Their Energy Consistency</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Dear Zdenek:</p>
<p>Thank you for sharing this. The practical implications are very interesting. A couple of quick questions/comments:</p>
<p>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?</p>
<p>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).</p>
<p>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.</p>
<p>Regards,<br />
Arash</p>
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</ul>Mon, 14 May 2012 04:06:55 +0000arash_yavaricomment 19021 at https://www.imechanica.org