iMechanica - Comments for "Geometry of Non-Linear Continuum Mechanics "
https://imechanica.org/node/11545
Comments for "Geometry of Non-Linear Continuum Mechanics "enIam interesting too.. maybe
https://imechanica.org/comment/28417#comment-28417
<a id="comment-28417"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/17848#comment-17848">On the definition of time-derivative</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>Iam interesting too.. maybe our can teach me about it??</p>
<p>_____________________________________________</p>
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</ul>Tue, 28 Jun 2016 07:08:27 +0000galaxyeleven3comment 28417 at https://imechanica.orgDifferential geometry and fluid mechanics
https://imechanica.org/comment/27912#comment-27912
<a id="comment-27912"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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 have a few general questions. Is differential geometry being applied to the study of fluid mechanics at all? Are there people and or universities conducting research in this area? Is it at all effective to apply differential geometry to fluid mechanics?</p>
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</ul>Fri, 04 Dec 2015 21:06:32 +0000John A Aherncomment 27912 at https://imechanica.orgRe: On the role of a linear connection in Dynamics
https://imechanica.org/comment/17894#comment-17894
<a id="comment-17894"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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 Giavanni:</p>
<p>Thank you for your detailed and informative explanation. I agree that Euler-Lagrange equations coming from an action principle are all written in terms of partial derivatives. I think one thing one should consider carefully is that the Lagrangian density in elasticity explicitly depends on the metrics. This then leads to writing the EL equations in the equivalent and well-known form using the Levi-Civita connection (divergence of stress...). The following reference is relevant.</p>
<p>J.E. Marsden, et al., Variational methods, multisymplectic geometry and continuum mechanics, Journal of Geometry and Physics 38(3-4), pp. 253-284 , 2001.</p>
<p>Let me also add one more point. Having a volume form divergence is defined using the Lie derivative. Having a connection one can define the compatible volume form (in the obvious way). It turns out that the standard Riemannian volume form is the compatible volume form of the Levi-Civita connection.</p>
<p>Happy Holidays!</p>
<p>Regards,<br />
Arash</p>
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</ul>Sat, 24 Dec 2011 03:11:00 +0000Arash_Yavaricomment 17894 at https://imechanica.orgOn the role of a linear connection in Dynamics
https://imechanica.org/comment/17880#comment-17880
<a id="comment-17880"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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>
<br />
Dear Arash,
</p>
<p>
I deeply appreciate your question which will lead us to discuss about a basic issue in Classical Continuum Dynamics (<strong>CCD</strong>).
</p>
<p>
I posed the very same question to me some time ago and, in giving a satisfactory answer, a hard effort of investigation was required to me to recover the essential informations from the literature and to extract the underlying geometrical concepts and results, together with new contributions.
</p>
<p>
I will try to summarize hereafter the basic ideas, focusing on the role of a linear connection in the ambient space and quoting some recent articles by me and co-workers, where the foundations of <strong>CCD</strong> are treated in detail.
</p>
<p>
The foundations of <strong>CCD</strong> are laid down in the most general way by means of a variational principle concerning the trajectory and the relevant evolution operator. The principle may be put in the standard geometrical form of an <strong>Action Principle</strong> in which the tests in the variational principle are made by displacing the trajectory in the container manifold. To this end the trajectory is considered as a submanifold of the state-space manifold defined to be the velocity-time (or the covelocity-time) manifold and an <strong>action one-form</strong> on the state-space is devised by a suitable lifting of the scalar lagrangian. Under usual regularity assumptions, the Action Principle may be localized to provide the differential <strong>Euler-Lagrange</strong> <strong>equation</strong> and the <strong>corner conditions</strong> at singular points.
</p>
<p>
No geometric connection in the state-space manifold enters into the theory until this point and hence it can be affirmed that <strong>CCD</strong> may be founded in a natural way in terms of the evolution operator and of the lagrangian, without any additional assumptions, as illustrated in
</p>
<p>
<br /><span>G. Romano, R. Barretta and M. Diaco, On Continuum Dynamics,</span>
</p>
<p>
J. Math. Phys. 50 (2009) 102903-1-26. DOI: <span><a href="http://dx.doi.org/10.1063/1.3215979">http://dx.doi.org/10.1063/1.3215979</a></span>
</p>
<p>
</p>
<p>
An equivalent principle can be formulated by imposing that the displacement of the trajectory leaves the energy functional invariant, to get a generalized form of <strong>Maupertuis Least Action Principle</strong> as illustrated in
</p>
<p>
G. Romano, R. Barretta and A. Barretta, On Maupertuis principle in dynamics,
</p>
<p class="MsoNormal">
Rep. Math. Phys. 63 (3) (2009) 331-346. DOI: <span><a href="http://dx.doi.org/10.1016/S0034-4877%2809%2900015-9">http://dx.doi.org/10.1016/S0034-4877(09)00015-9</a></span>
</p>
<p>
<br />
The introduction of a linear connection provides however a very valuable tool of investigation about the properties of the evolution along the trajectory fulfilling the Action Principle, the choice of a special connection being a question of convenience. For instance, a curvilinear coordinate system will induce an associated distant parallelism and a corresponding linear connection which has vanishing torsion and curvature forms. The adoption of a <strong>Levi-Civita</strong> connection will instead induce a torsion-free and metric connection with a non-vanishing curvature. In this respect we must observe that only the torsion of the linear connection enters in the equations of Dynamics. An important example is provided by <strong>Poincaré</strong>'s law of Dynamics which is the outcome of taking the distant parallel transport induced by a mobile reference system associated with curvilinear coordinates. In this case the torsion form is equal to the opposite of the <strong>Lie bracket</strong> and hence the components of the Lie brackets of the basis vector fields (named <em>structure coefficients</em>) appear into the equation of Dynamics.
</p>
<p>
The Levi-Civita connection on the trajectory, with the Lagrangian given by the kinetic energy per unit mass, leads to generalized <strong>Euler</strong>'s and <strong>d'Alembert</strong>'s laws of Dynamics. The standard formulations are recovered in the <strong>Euclid</strong> space endowed with the parallel transport by translation. These issues are treated in detail in the cited references.
</p>
<p>
In conclusion the <strong>answer</strong> to the question is:
</p>
<p>
Balance of linear momentum (which is part of Euler's formulation of the law of Dynamics) is the outcome of choosing in the Euclid space the standard connection associated with the parallel transport by translation. It is a special form of a general expression which doesn't depend on the choice of a connection.
</p>
<p>
Best regards and wishes of Merry Christmas and Happy new Year
</p>
<p>
Giovanni Romano
</p>
<p>
</p>
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</ul>Wed, 21 Dec 2011 11:20:20 +0000Giovanni Romanocomment 17880 at https://imechanica.orgArash, I agree with Amit.
https://imechanica.org/comment/17872#comment-17872
<a id="comment-17872"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/17866#comment-17866">Re: Arash on geometrization of CM</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, I agree with Amit. This is a stimulating discussion. If you have time, please start a separate thread.
</p>
<p>
</p>
<p>
Alireza-
</p>
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</ul>Tue, 20 Dec 2011 16:18:05 +0000Alireza Sarvestanicomment 17872 at https://imechanica.orgRe: On the foundations of Continuum Mechanics
https://imechanica.org/comment/17868#comment-17868
<a id="comment-17868"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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 Giovanni:</p>
<p>I completely agree that one should be open minded and willing to question what is already accepted by most people. I also believe that in a scientific discussion one should be very specific and clear in asking/answering questions. Let me repeat one of my previous (unanswered) questions. Do you need a connection in writing the balance of linear momentum? If not, how would you write it? </p>
<p>Regards,<br />
Arash</p>
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</ul>Tue, 20 Dec 2011 04:46:30 +0000Arash_Yavaricomment 17868 at https://imechanica.orgRe: Arash on geometrization of CM
https://imechanica.org/comment/17866#comment-17866
<a id="comment-17866"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/17861#comment-17861">Re: eqns in terms of \dot F for analysis of bvp</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>
You ask an interesting philosophical question (geometry has been essential in many developments....) that I would enjoy discussing, as we find time. But perhaps you should first start a separate thread.
</p>
<p>
- Amit
</p>
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</ul>Mon, 19 Dec 2011 16:00:20 +0000Amit Acharyacomment 17866 at https://imechanica.orgOn the foundations of Continuum Mechanics
https://imechanica.org/comment/17865#comment-17865
<a id="comment-17865"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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 align="justify">
</p>
<p class="MsoNormal" align="justify">
Dear collegues,
</p>
<p class="MsoNormal" align="justify">
let me say some words about the<br />
hope that this blog could serve to provide a well-sustained discussion about<br />
the foundations of Continuum Mechanics (<strong>CM</strong>).<br />
This is the aim why I opened the blog and, in my opinion, the goal can be<br />
reached only if there will be no hurry to claim things just as anyone has<br />
learned in his training, but rather an attitude to think and ponder with<br />
patience and deepness will be experimented, to provide constructive criticisms.
</p>
<p class="MsoNormal" align="justify">
I have learned this lesson once<br />
more at the end of my academic career when I realized that, to overcome a deep<br />
feeling of unsatisfaction for most treatment of <strong>CM</strong>,
</p>
<p class="MsoNormal" align="justify">
especially concerning rate formulations and constitutive theory,<br />
and to find a rationale out of a highly confused situation, I had to learn fundamental<br />
notions and concepts of Differential Geometry which were not teached to me in<br />
my training as a student of engineering and were later considered in the<br />
scientific community as a mathematical sophisticated discipline of no concern<br />
in applied mechanics.
</p>
<p class="MsoNormal" align="justify">
There is a common old saying in<br />
Naples, Italy, whose translation could be:
</p>
<p class="MsoNormal" align="justify">
<strong>The best deaf is who doesn’t intend to listen.</strong>
</p>
<p class="MsoNormal" align="justify">
I understand that changing ideas<br />
on fundamentals is a very hard task for experts, but the extraordinary<br />
experience with <strong>Einstein</strong>’s<br />
relativity theory (<em>absit iniuria verbis</em>)<br />
should learn us to be open minded.
</p>
<p class="MsoNormal" align="justify">
The fortune of that theory was<br />
that people like <strong>Hilbert</strong>, <strong>Klein</strong>, <strong>Poincaré</strong> and <strong>Minkowski</strong>,<br />
very knowledgeable and authoritative experts, were intrigued by the approach by Einstein and<br />
shared his childish enthusiasm.
</p>
<p class="MsoNormal" align="justify">
There are some significant<br />
similarities between that situation and the present one concerning Continuum<br />
Mechanics.
</p>
<p class="MsoNormal" align="justify">
In both cases one is faced with a<br />
well-consolidated theory rich of implications and interpretations of<br />
experimental facts, but with some unexplicable difficulties and paradoxes.
</p>
<p class="MsoNormal" align="justify">
In both cases the right idea<br />
comes by collecting hints and partial answers by older proposals made by other<br />
valuable researchers.
</p>
<p class="MsoNormal" align="justify">
In the field of our concern,<br />
Continuum Mechanics, the leading new ideas are the following.
</p>
<p class="MsoListParagraphCxSpFirst" align="justify">
<span><span>1)<span> <br /></span></span></span>A careful geometric definition and distinction<br />
between spatial and material tensor fields.
</p>
<p class="MsoListParagraphCxSpMiddle" align="justify">
<span><span>2)<span> <br /></span></span></span>The statement of the Covariance Paradigm (<strong>CP</strong>) expressing that the rule for the comparison<br />
between material tensors is the push according to the relevant diffeomorphic<br />
transformation.
</p>
<p class="MsoListParagraphCxSpMiddle" align="justify">
The first item is basic but simple and evident, although<br />
never explicitly stated in a proper geometric form. The second item is made<br />
more subtle by a dimensional coincidence between the body and the ambient<br />
space, but is self-proposing for lower dimensional bodies.
</p>
<p class="MsoListParagraphCxSpMiddle" align="justify">
To grasp the motivation it should be observed that<br />
comparison between material tensors means that two placements of a body along a<br />
trajectory (whether real or virtual) are considered, as related by the<br />
evolution diffeomorphism between them. The <strong>natural<br /></strong>way to make the comparison is just to resort to the evolution<br />
diffeomorphism itself, and to its tangent map, to perform the push-pull<br />
transformations. This idea goes back as far as to <strong>Euler </strong>and to his<strong> </strong>celebrated<strong> </strong>formula for the<strong> </strong>stretching.
</p>
<p class="MsoListParagraphCxSpMiddle" align="justify">
The conceptual clarity of this approach is not questionable<br />
and the extraordinary effectiveness of its adoption becomes evident as soon as<br />
it is applied to formulate constitutive relations, to discuss basic issues such<br />
as time independence, time invariance, frame invariance and integrability<br />
conditions.
</p>
<p class="MsoListParagraphCxSpMiddle" align="justify">
Fictitious difficulties faced with in <strong>CM</strong> are swept away by the <strong>CP</strong><br />
which leads to formulate rate constitutive relations for<br />
elasto-visco-plasticity (and similar models of material behavior) in a direct<br />
and definite way and resolves the long lasting debate about rates of material<br />
tensors by giving a unique, simple and well-defined answer.
</p>
<p class="MsoListParagraphCxSpMiddle" align="justify">
A first exposure of this new approach has been contributed<br />
in the paper<br />
by G. Romano and R. Barretta: Covariant hypo-elasticity, 2011, <a id="ddJrnl" href="http://www.sciencedirect.com/science/journal/09977538"><strong>European Journal of Mechanics - A/Solids</strong></a>. <a href="http://dx.doi.org/10.1016/j.euromechsol.2011.05.005" target="_blank"><img src="http://www.sciencedirect.com/scidirimg/clear.gif" border="0" alt="" width="1" height="10" />doi:10.1016/j.euromechsol.2011.05.005</a>
</p>
<p class="MsoListParagraphCxSpMiddle" align="justify">
A more comprehensive treatment is <em>in itinere</em>.
</p>
<p class="MsoListParagraphCxSpLast" align="justify">
Sincerely
</p>
<p align="justify">
Giovanni Romano
</p>
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</ul>Mon, 19 Dec 2011 11:19:14 +0000Giovanni Romanocomment 17865 at https://imechanica.orgRe: eqns in terms of \dot F for analysis of bvp
https://imechanica.org/comment/17861#comment-17861
<a id="comment-17861"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/17859#comment-17859">eqns in terms of \dot F for analysis of bvp</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 Amit:</p>
<p>People (even the most distinguished) have been confused for a long time with objective time derivatives. I think the main reason has been the lack of familiarity with the geometric techniques. For some reason workers in continuum mechanics have isolated themselves (not all but most) thinking that they are working with a very special theory not realizing that this is just one of the many field theories of physics. Geometry has been essential in many developments in physics and we should learn from that experience. I think you would agree with me? </p>
<p>Deformation gradient has a fixed leg and a leg that moves with the deformable body (two-point tensor). For this reason one cannot take a (naive) time derivative unless everything is Euclidean. Most likely this is what Gurtin does too? </p>
<p>I believe geometric notions are always important when you are dealing with bodies with residual stresses regardless of dimensionality.</p>
<p>Simo was among the very few who both understood geometry and properly used it in computational mechanics. Unfortunately, he died too young.</p>
<p>Let me repeat that I strongly believe solving specific (nonlinear) problems is the right direction in convincing others of usefulness of geometric techniques.</p>
<p>Regards,<br />
Arash</p>
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</ul>Sun, 18 Dec 2011 20:04:46 +0000Arash_Yavaricomment 17861 at https://imechanica.orgRe: On the definition of time-derivative
https://imechanica.org/comment/17860#comment-17860
<a id="comment-17860"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/17847#comment-17847">On the definition of time-derivative</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 Giovanni:</p>
<p>I understand that you like to work with the pulled-back metric and its time derivative. This is perfectly fine. You can equivalently work with the Lie derivative of the spatial metric. I assume you agree that continuum mechanics can be consistently formulated using both spatial and material descriptions? I also understand that you would like to work with a minimal geometric structure. Having the motion Lie derivative can be defined without even any need for a metric. Now my point is: this argument would not be enough for dismissal of <strong>F</strong> as a deformation measure. Let me emphasize that I am not an advocate for the multiplicative decomposition of deformation gradient but at the same time don't think it is inconsistent. What is problematic is to work with an intermediate "configuration", which is defined only locally and as you mentioned earlier it is not even a configuration.</p>
<p>I also assume that you agree that metric is an essential ingredient and we must have a way of measuring distances to calculate the elastic energy? Then the natural connection in this Riemannian manifold is the Levi-Civita connection (in the classical framework. If one is interested in relativistic effects perhaps more complicated spacetimes can be used to account for cosmological defects) and using the covariant time derivative of <strong>F</strong> should lead to an acceptable theory. One may still argue what is more natural or fundamental. What I think is an invariant is the time derivative of energy density (a scalar) and this scalar can be written in terms of the time derivative of the pulled-back metric and the second Piola-Kirchhoff stress or the covariant time derivative of <strong>F</strong> and the first Piola-Kirchhoff stress. What do you think?</p>
<p>Let us also remember that for writing equilibrium equations we do need to use a covariant derivative, i.e. using the Levi-Civita connection is inevitable; divergence of Cauchy stress requires the use of the Levi-Civita connection. I would like to know your opinion on this point too.</p>
<p>Regards,<br />
Arash</p>
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</ul>Sun, 18 Dec 2011 19:50:28 +0000Arash_Yavaricomment 17860 at https://imechanica.orgeqns in terms of \dot F for analysis of bvp
https://imechanica.org/comment/17859#comment-17859
<a id="comment-17859"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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>
A look at Hill's Invariance in Solid Mechanics (1978) article for when it is convenient to look at objective rate constitutive equations transformed to be in terms of deformation gradient rate and the nominal stress (rate) may be useful here. This turns out to be very useful in the analysis of bvp and bifurcations in the rate-independent setting. This is also all in the context of mechanics of 3-d bodies. Of course, Hill was not a proponent of the multiplicative decomposition and I agree with that viewpoint. I think that what is most important is to understand the transformations between different equivalent representations.
</p>
<p>
In that article, Hill also explains convected derivatives (in what amounts to a slightly more general setting than with respect to a flow of a manifold - e.g. the Jaumann rate (for a deformable body) can only be understood in that context), that may also be useful for readers of this blog. In my opinion, he does a good job of explaining the Lie derivative idea of pull-back-time derivative-push forward, for a first exposure to the idea for people who do not have any exposure to DG. Once one understands that, then it is perhaps easy to adapt it to the situation when tangent spaces of the same material point along a flow cannot be declared as identical (as for shells, membrane, rods) - of course, it is also in this context that one can appreciate why this is a useful concept.
</p>
<p>
Notions of DG are of course important in CM for lower-dimensional bodies - people who have done 3-d CM as well as played with shell theories (and implemented them numerically) know and appreciate this. But one of the main questions in this blog had to do with the failure of the multiplicative decomposition, and whether Norris's statement made sense. These developments were made in the context of 3-d bodies in 3-d space - there they make perfect precise sense (at least to me).
</p>
<p>
There is also a past history of DGizing in CM starting from the works of Simo. It is probably also an open-minded endeavor for readers of this blog (especially young researchers) to take a look at that body of work and come to their own conclusions as to what was learned from that exercise. May be Prof. Romano's paper already covers this point.
</p>
<p>
</p>
<p>
</p>
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</ul>Sun, 18 Dec 2011 15:02:51 +0000Amit Acharyacomment 17859 at https://imechanica.orgSilly fishes
https://imechanica.org/comment/17857#comment-17857
<a id="comment-17857"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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 see that a <strong>fish fryer</strong> came into the blog in a somewhat impolite way.
</p>
<p>
I don't want to get him carried away but rather to suggest him to <strong>learn the art of cooking</strong> before spitting his recipes. Wrongly fried fishes may have an awful taste and be dangerous for the health.
</p>
<p>
For other friends reading this blog, I notice that <strong>Fréchet derivatives</strong> are defined <strong>in linear spaces</strong> with a suitable topology. In the discussion opened with this blog the issue is rather how to reconduct a time-derivative along the <strong>non-linear trajectory manifold</strong> to a (Fréchet) derivation on a linear space. What I sustain is that, among the two possibilities offered by differential geometry (DG), viz. parallel transport and push-pull transformation along the motion, only the second is natural and suitable for material tensor fields entering in constitutive relations. This point was not detected by the treatment by Morton Gurtin in his nice book on <em>Introduction to Continuum Mechanics.</em>
</p>
<p>
Leaving this discussion apart and looking at DG as a useless fashion, would be as trying to make derivatives with the sole tools of linear algebra.
</p>
<p>
I am really surprised that also knowledgeable researchers could express agreement with such a stubbornly blind viewpoint. I am not interested in making proselytes with any effort, but rather to discuss in a polite way about scientific issues with other open minded people.
</p>
<p>
Giovanni Romano
</p>
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</ul>Sun, 18 Dec 2011 12:41:29 +0000Giovanni Romanocomment 17857 at https://imechanica.orgOn the definition of time-derivative
https://imechanica.org/comment/17848#comment-17848
<a id="comment-17848"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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 don't understand diff geometry but I find this explanation and the implications quite interesting. Thank you.
</p>
<p>
Nachiket
</p>
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</ul>Sat, 17 Dec 2011 17:30:54 +0000Nachiket Gokhalecomment 17848 at https://imechanica.orgOn the definition of time-derivative
https://imechanica.org/comment/17847#comment-17847
<a id="comment-17847"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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>
Let me explain in greater detail the concept of <strong>naturality</strong> and the geometric motivation why the seemengly appropriate covariant time-derivative in the ambient space is not apt to define the time rate of <strong>material tensors</strong> (the ones entering in a constitutive relation) which act on vectors tangent to the current body placement.
</p>
<p>
1) The convective time-derivative, depending only on the motion,<br />
which is an essential ingredient of the theory, is a <strong>natural</strong> notion.<br />
On the contrary, the choice of a connection in the trajectory manifold to perform<br />
the parallel time-derivative of a material tensor makes <br /><strong>naturality</strong> of the notion of time-rate to be lost.</p>
<p>2) For lower dimensional bodies <br />
(such as wires and membranes in the 3D Euclid space, sketched in fig. 1,2<br />
the pull-back along the motion transforms material vectors into material vectors <br />
(red arrows in fig. 1,2).<br />
On the contrary, the parallel transport, of immersed material tangent vectors along a particle, <br />
will in general yield a spatial vector which is not tangent to the body placement<br />
(black arrows in fig. 1,2).
</p>
<p>
<img class="image img_assist_properties" src="http://imechanica.org/files/images/Wire_0.img_assist_properties.jpg" alt="motion of a wire" title="motion of a wire" width="200" height="109" /><img class="image img_assist_properties" src="http://imechanica.org/files/images/Membrane.img_assist_properties.jpg" alt="motion of a membrane" title="motion of a membrane" width="200" height="101" /></p>
<p>
Then, for lower dimensional bodies, the definition of time-derivative of <strong>F</strong> by means of a parallel transport is not acceptable if the result is to be adopted in a constitutive relation because only material vectors (tangent to the body placement) and not spatial vectors (tangent to the ambient space) are admissible.
</p>
<p>
I am assuming that notions and definition of Continuum Mechanics should be equally be applicable to bodies of any dimensionality, viz. cable, membranes and balls.
</p>
<p>
As a last observation, I would point out that the theory ensuing from the adoption of the <strong>natural</strong> definition of time rates is definitely simpler and more suitable for computational applications.
</p>
<p>
Another point to think about is that <strong>Euler</strong>'s formula for the stretching is an expression of the convective time-derivative of the metric tensor. In accord with duality, the time derivative of the stress is also a convective time-derivative. Why then conceive to take a time derivative by parallel transport when dealing with the deformation gradient? Just because the convective derivative vanishes, but this is an indication that an essntial difficulty is involved. In fact there is no need to consider the time derivative of <strong>F </strong>in a constitutive relation, the proper candidate to measure deformation being the metric tensor in the ambient space (or better its pull-back to the body placement, to include lower dimensional bodies.
</p>
<p>
Best regards
</p>
<p>
Giovanni Romano
</p>
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</ul>Sat, 17 Dec 2011 08:56:53 +0000Giovanni Romanocomment 17847 at https://imechanica.orgRe: definition of \dot F
https://imechanica.org/comment/17846#comment-17846
<a id="comment-17846"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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 agree with Amit. My point was: Lie derivative of <strong>F</strong> vanishes ok but the covariant time derivative of <strong>F</strong> is a well-defined and valid time derivative that is also "objective".</p>
<p>In my opinion, we should avoid what happened to "rational mechanics" people (obsession with notation and funny things like the "principle of equipresence", etc.). If all differential geometry can offer is to make things look more complicated, then there is no real contribution and we should forget about it. Geometry can indeed lead to important contributions that are more than just simple reinterpretations of what is already known. What one should focus on is to solve problems that cannot be solved using the existing theories/approaches. </p>
<p>Regards,<br />
Arash</p>
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</ul>Sat, 17 Dec 2011 01:31:00 +0000Arash_Yavaricomment 17846 at https://imechanica.orgdefinition of \dot F
https://imechanica.org/comment/17845#comment-17845
<a id="comment-17845"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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>
come on gents, let's not get carried away and say that \dot F is not defined precsiely in continuum mechanics and a 'right' definition is required.
</p>
<p>
Just go take a look at Gurtin's Introduction to Continuum Mechanics. It is competely precise and self-consistent.
</p>
<p>
Now you might want to get all worked up and say I do not want to consider the tangent space of two points of a configuration of 3-d body immersed in 3-d Euclidean point space as the same - that is your choice, but that does not mean that the deformation gradient rate cannot be given precise mathematical (and physical) meaning within the context of differentiation in finite dimensional vector spaces. You recognize a point space and its associated translation space and learn a little about Frechet Derivatives and off you go.
</p>
<p>
Let's not make heavy weather of the things that we have some control over and know how to make precise sense of. There is bigger fish to fry in continuum mechanics - like solving some hard nonlinear PDE that arise in continuum mechanics - I can assure you no amount of differential geometrizing will ease that burden.
</p>
<p>
</p>
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</ul>Sat, 17 Dec 2011 01:14:58 +0000Amit Acharyacomment 17845 at https://imechanica.orgRe: Time derivative of the deformation gradient
https://imechanica.org/comment/17844#comment-17844
<a id="comment-17844"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/17843#comment-17843">Re: What the time derivative of deformation gradient is it?</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>
Thank you Arash and Givoanni for taking the time to explain your points of view. I don't understand much of it andmy understanding may be totally wrong.
</p>
<p>
Gioovanni's point appears to be that if there is no <strong>unique</strong> connection that applies to solid mechanics, then we are in the same situation as when nounique objective rates was the rage.Giovanni, could you explain your natural connection.
</p>
<p>
Arash says that general (local?) covariance is all we need; which is esentially objectivity in another guise. Arash, could you explain what covariance phyically means?
</p>
<p>
Also, since we are not dealing with relativistic mechanics, aren't things simpler in solid mechanics? Can we just take results from relativity and run with them?
</p>
<p>
How can these theories be guided and given a solid foundation by experiment? Is anything predicted that's unusual which can be used to provide us with a preference for the right definition of derivative?
</p>
<p>
-- Biswajit
</p>
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</ul>Sat, 17 Dec 2011 00:12:04 +0000Biswajit Banerjeecomment 17844 at https://imechanica.orgRe: What the time derivative of deformation gradient is it?
https://imechanica.org/comment/17843#comment-17843
<a id="comment-17843"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/17842#comment-17842">What the time derivative of deformation gradient is it?</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 Giovanni:</p>
<p>The ambient space is a Riemannian manifold. So, if you need a time derivative for <strong>F</strong>, covariant time derivative using the Levi-Civita connection is perfectly fine, in my opinion. If you think this is not a natural time derivative, I would like to know what your definition of a "natural time derivative" is.</p>
<p>Regards,<br />
Arash</p>
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</ul>Fri, 16 Dec 2011 19:07:08 +0000Arash_Yavaricomment 17843 at https://imechanica.orgWhat the time derivative of deformation gradient is it?
https://imechanica.org/comment/17842#comment-17842
<a id="comment-17842"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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>
let us put the discussion of a firm basis.
</p>
<p>
In the letter <a href="http://imechanica.org/node/11545#comment-17818" class="active">Failure of the multiplicative decomposition</a> , in replay to the request by Biswajit, I have exposed the motivations why I consider the definition of the time derivative \dot{<strong>F</strong>} geometrically untenable.
</p>
<p>
Since your opinion is not in agreement with my arguments against the multiplicative decomposition in plasticity, would you please provide a definition for the time derivative \dot{<strong>F</strong>}?
</p>
<p>
This will help us to come to the heart of the question.
</p>
<p>
</p>
<p>
Regards,
</p>
<p>
Giovanni
</p>
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</ul>Fri, 16 Dec 2011 17:16:07 +0000Giovanni Romanocomment 17842 at https://imechanica.orgRe: On the geometric approach to CM
https://imechanica.org/comment/17824#comment-17824
<a id="comment-17824"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/17820#comment-17820">On the geometric approach to CM</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 Giovanni:</p>
<p>Thank you for your kind words.</p>
<p>1) I agree that metric tensor is a fundamental ingredient of any geometric theory of continuum mechanics. In the case of solids, for the reference manifold one may lose the freedom in choosing an arbitrary metric if the reference configuration is required to be stress-free. </p>
<p>2) I agree with your statement regarding the Levi-Civita connection. Here, perhaps you should clarify what you precisely mean by "natural". The Levi-Civita connection is not always the natural connection. For example, given a solid with distributed dislocations the natural connection (in the reference configuration) is a Weitzenbock connection (metric-compatible, flat, but with torsion). This was first realized by Kondo and Bilby in the 1950s.</p>
<p>3) The induced connection acts in the tangent space of the submanifold. I don't think this is the main issue in this discussion though.</p>
<p>Regards,<br />
Arash</p>
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</ul>Thu, 15 Dec 2011 19:48:05 +0000Arash_Yavaricomment 17824 at https://imechanica.orgOn the geometric approach to CM
https://imechanica.org/comment/17820#comment-17820
<a id="comment-17820"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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 align="justify">
Dear Arash,
</p>
<p align="justify">
let me say that it is a pleasure to have an opportunity to discuss about fundamental issues with a scientist and researcher in Continuum Mechanics (CM) with a firm background in Differential Geometry (DG).
</p>
<p align="justify">
I would like to <span class="short_text"><span class="hps">bring</span> <span class="hps">to your attention</span></span> the following issues.
</p>
<p align="justify">
1) The metric tensor field in the space (and in the time) affine manifolds are basic ingredients of the theory of CM. Different choices lead to different theories.
</p>
<p align="justify">
2) The space-time manifold endowed with a metric tensor field is a Riemann manifold and the Levi-Civita connection is the only torsion-free (i.e. symmetric) and metric connection in that manifold, but different connections may and are in fact often considered, for instance the ones induced by general coordinate systems, which may well be non-symmetric. Connection dependent definitions of time-rates are then <strong>not natural</strong>.
</p>
<p align="justify">
3) Given a connection in the ambient space manifold, the induced connection in a placement of a lower dimensional body (a submanifold of the ambient space manifold) defines a covariant derivative which at each point is performed along vectors tangent to the submanifold. Covariant time derivatives are then outside its domain because they act along <strong>the velocity vector </strong>which<strong> </strong>may<strong> not </strong>be<strong> tangent to the submanifold</strong>.
</p>
<p align="justify">
To develop a general theory of CM the most suitable DG context is provided by the four-dimensional events manifold and by the trajectory submanifold.
</p>
<p align="justify">
Best regards
</p>
<p align="justify">
Giovanni
</p>
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</ul>Wed, 14 Dec 2011 21:41:52 +0000Giovanni Romanocomment 17820 at https://imechanica.orgRe: Failure of the multiplicative decomposition
https://imechanica.org/comment/17819#comment-17819
<a id="comment-17819"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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 Giovanni:</p>
<p>Thank you for the interesting discussion. I disagree with your reasoning regarding <strong>F</strong> not being an acceptable deformation measure.</p>
<p>I agree that the Lie derivative of <strong>F</strong> along the motion vanishes (I don't think this is "unfortunate"; it is simply a fact). It is true that you can choose different connections (in the ambient space here for a covariant time derivative). Let us not forget that we do need to choose a metric in the ambient space (and reference configuration) to be able to make a scalar out of <strong>F</strong> (a two-point tensor). This may seem "unnatural" or unpleasant but in elasticity we need to know how distances are measured in the ambient space. If an elastic sheet is attached to a curved 2-manifold, obviously, one must have the ambient space metric to calculate the stored energy. The point is: you must explicitly include the metrics of the ambient space and reference configuration explicitly in your stored energy (or Lagrangian density). Now having a metric the natural connection (in the ambient space) would be the Levi-Civita connection. However, you could choose any other connection and you would end up with a different measure of "time derivative" of <strong>F</strong>. This doesn't make the theory weak or inconsistent in any sense nor does it make <strong>F</strong> an unacceptable measure of deformation. Is it a disaster that one can have different measures of stress in continuum mechanics? No. As long as the appropriate deformation measure is chosen the theory is consistent.</p>
<p>Another point: if you choose <strong>F</strong> as a deformation measure then in calculating the Lie derivative of stored energy (assuming you're looking at nonlinear elasticity) Lie derivative of <strong>F</strong> vanishes but the Lie derivative of <strong>g</strong> (ambient space metric) does not and it would be your "rate of strain". If you choose to take a covariant time derivative you will have the covariant time derivative of <strong>F</strong> but the covariant time derivative of <strong>g</strong> vanishes (assuming the Levi-Civita connection is used).</p>
<p>Now regarding "lower dimensional bodies" the deformed body is a submanifold of the ambient space manifold and so you can use the induced connection to get the right covariant time derivative. </p>
<p>I agree that multiplicative decomposition of <strong>F</strong> has some issues but it is not a "failure". I also agree with you calling the "intermediate configuration", "intermediate linear space". </p>
<p>Regards,<br />
Arash</p>
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</ul>Wed, 14 Dec 2011 20:31:56 +0000Arash_Yavaricomment 17819 at https://imechanica.orgFailure of the multiplicative decomposition
https://imechanica.org/comment/17818#comment-17818
<a id="comment-17818"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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 align="justify">
Hi Biswajit,
</p>
<p align="justify">
let me try to answer your interesting question with clear arguments.
</p>
<p align="justify">
To begin, let us consider the following issue taken from a comment in your blog:
</p>
<p align="justify">
<a href="http://imechanica.org/node/1646#comment-3369" class="active">The logarithmic spin</a> Submitted by <a href="http://imechanica.org/user/17" title="View user profile.">Andrew Norris</a> on Mon, 2007-07-02 14:5,
</p>
<p align="justify">
where the well-known formula L = \dot{F}F^{-1} = d+W is quoted.
</p>
<p align="justify">
The point is that this <strong>well-known</strong> formula deals with the <strong>time derivative</strong> \dot{F}
</p>
<p align="justify">
whose definition is never explicitly given. If one tries to provide an explicit one,
</p>
<p align="justify">
he will soon be compelled to recognize that there is no suitable way to give the definition
</p>
<p align="justify">
of time-derivative of a time-dependent "deformation gradient" <strong>F</strong>(t).
</p>
<p align="justify">
In fact, the deformation map <strong>φ</strong> is a diffeomorphism between a source and a target body placement
</p>
<p align="justify">
and the "deformation gradient" <strong>F</strong> is the corresponding tangent map <strong>Tφ</strong> which is an isomorphsm
</p>
<p align="justify">
between the tangent bundles to the involved placements of the body.
</p>
<p align="justify">
Thus <strong>F</strong> is a material map which transforms, in a linear way, tangent vectors based at a point in the
</p>
<p align="justify">
source placement, into tangent vectors based at the corresponding displaced point in the target placement.
</p>
<p align="justify">
Assuming that the target placement is time-dependent, the sole manner to append a physical meaning
</p>
<p align="justify">
to the time-derivative of <strong>F </strong>is to provide for it a "natural" definition,
</p>
<p align="justify">
that is a definition which does not involve any arbitrary choice.
</p>
<p align="justify">
Moreover it is required that the time-derivative is still a linear transformation acting on
</p>
<p align="justify">
tangent vectors based at a point in the source placement, a property that makes it a material tensor
</p>
<p align="justify">
to appear in a constitutive relation.
</p>
<p align="justify">
The tool offered by differential geometry is then uniquely detected: the <strong>Lie</strong> time-derivative along the motion.
</p>
<p align="justify">
Unfortunately this time derivative <strong>vanishes identically</strong> because it is just the partial time-derivative,
</p>
<p align="justify">
at the fixed source point, of the pull-back of the push forward, which is the identity map.
</p>
<p align="justify">
The tool implicitly adopted in literature is then, with any evidence,
</p>
<p align="justify">
the parallel (or covariant) time-derivative along the motion,
</p>
<p align="justify">
according to a connection in the space manifold.
</p>
<p align="justify">
This derivative is however affected by two fatal bugs:
</p>
<p align="justify">
1) The choice of the connection is quite arbitrary, so that <strong>naturality is lost</strong>.
</p>
<p align="justify">
2) For lower dimensional bodies (such as wires and membranes in the 3D Euclid space) the parallel time-derivative
</p>
<p align="justify">
of <strong>F</strong>(t) (or rather of its spatial immersion) will have a range outside the tangent bundle to the source placement,
</p>
<p align="justify">
because parallel transport in space does not transform tangent vectors to the target placement into
</p>
<p align="justify">
tangent vectors to the source placement.
</p>
<p align="justify">
As a consequence, the parallel time-derivative of <strong>F</strong>(t) <strong>cannot appear in a constitutive relation</strong>.
</p>
<p align="justify">
Then, being the time derivative of <strong>F</strong>(t) not adoptable in a constitutive relation, what does it remain of
</p>
<p align="justify">
a theory of plasticity in which the kinematics is described by the chain decomposition
</p>
<p align="justify">
of a time-dependent deformation gradient ?
</p>
<p align="justify">
No mention will be made here of the well-discussed shortcomings of the multiplicative decomposition,
</p>
<p align="justify">
such as the inherent indeterminacy of the intermediate linear space
</p>
<p align="justify">
(which is a better name for the intermediate configuration).
</p>
<p align="justify">
A detailed presentation of the whole matter will appear in a
</p>
<p align="justify">
paper submitted for publication and I will let you be aware of it.
</p>
<p align="justify">
Sincerely
</p>
<p align="justify">
</p>
<p align="justify">
Giovanni Romano
</p>
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</ul>Wed, 14 Dec 2011 13:20:05 +0000Giovanni Romanocomment 17818 at https://imechanica.orgRe: Geometry of Non-Linear Continuum Mechanics
https://imechanica.org/comment/17805#comment-17805
<a id="comment-17805"></a>
<p><em>In reply to <a href="https://imechanica.org/node/11545">Geometry of Non-Linear Continuum Mechanics </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 Giovanni,
</p>
<p>
In one of your papers (or was it a presentation?) you mentioned that a multiplication decomposition of the deformation gradient is geometrically inconsistent. Could you elaborate a bit on that?
</p>
<p>
-- Biswajit
</p>
<p>
</p>
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</ul>Mon, 12 Dec 2011 21:16:58 +0000Biswajit Banerjeecomment 17805 at https://imechanica.org