Geometry of Non-Linear Continuum Mechanics

Hi Biswajit,

let me try to answer your interesting question with clear arguments.

To begin, let us consider the following issue taken from a comment in your blog:

The logarithmic spin Submitted by Andrew Norris on Mon, 2007-07-02 14:5,

where the well-known formula L = \dot{F}F^{-1} = d+W is quoted.

The point is that this well-known formula deals with the time derivative \dot{F}

whose definition is never explicitly given. If one tries to provide an explicit one,

he will soon be compelled to recognize that there is no suitable way to give the definition

of time-derivative of a time-dependent "deformation gradient" F(t).

In fact, the deformation map φ is a diffeomorphism between a source and a target body placement

and the "deformation gradient" F is the corresponding tangent map Tφ which is an isomorphsm

between the tangent bundles to the involved placements of the body.

Thus F is a material map which transforms, in a linear way, tangent vectors based at a point in the

source placement, into tangent vectors based at the corresponding displaced point in the target placement.

Assuming that the target placement is time-dependent, the sole manner to append a physical meaning

to the time-derivative of F is to provide for it a "natural" definition,

that is a definition which does not involve any arbitrary choice.

Moreover it is required that the time-derivative is still a linear transformation acting on

tangent vectors based at a point in the source placement, a property that makes it a material tensor

to appear in a constitutive relation.

The tool offered by differential geometry is then uniquely detected: the Lie time-derivative along the motion.

Unfortunately this time derivative vanishes identically because it is just the partial time-derivative,

at the fixed source point, of the pull-back of the push forward, which is the identity map.

The tool implicitly adopted in literature is then, with any evidence,

the parallel (or covariant) time-derivative along the motion,

according to a connection in the space manifold.

This derivative is however affected by two fatal bugs:

1) The choice of the connection is quite arbitrary, so that naturality is lost.

2) For lower dimensional bodies (such as wires and membranes in the 3D Euclid space) the parallel time-derivative

of F(t) (or rather of its spatial immersion) will have a range outside the tangent bundle to the source placement,

because parallel transport in space does not transform tangent vectors to the target placement into

tangent vectors to the source placement.

As a consequence, the parallel time-derivative of F(t) cannot appear in a constitutive relation.

Then, being the time derivative of F(t) not adoptable in a constitutive relation, what does it remain of

a theory of plasticity in which the kinematics is described by the chain decomposition

of a time-dependent deformation gradient ?

No mention will be made here of the well-discussed shortcomings of the multiplicative decomposition,

such as the inherent indeterminacy of the intermediate linear space

(which is a better name for the intermediate configuration).

A detailed presentation of the whole matter will appear in a

paper submitted for publication and I will let you be aware of it.

Sincerely

Giovanni Romano

Dear Arash,

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).

I would like to bring to your attention the following issues.

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.

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 not natural.

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 the velocity vector which may not be tangent to the submanifold.

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.

Best regards

Giovanni

Dear Arash,

let us put the discussion of a firm basis.

In the letter  Failure of the multiplicative decomposition , in replay to the request by Biswajit, I have exposed the motivations why I consider the definition of the time derivative \dot{F} geometrically untenable.

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{F}?

This will help us to come to the heart of the question.

Regards,

Giovanni

Let me explain in greater detail the concept of naturality and the geometric motivation why the seemengly appropriate covariant time-derivative in the ambient space is not apt to define the time rate of material tensors (the ones entering in a constitutive relation) which act on vectors tangent to the current body placement.

1) The convective time-derivative, depending only on the motion,
which is an essential ingredient of the theory, is a natural notion.
On the contrary, the choice of a connection in the trajectory manifold to perform
the parallel time-derivative of a material tensor makes
naturality of the notion of time-rate to be lost.

2) For lower dimensional bodies
(such as wires and membranes in the 3D Euclid space, sketched in fig. 1,2
the pull-back along the motion transforms material vectors into material vectors
(red arrows in fig. 1,2).
On the contrary, the parallel transport, of immersed material tangent vectors along a particle,
will in general yield a spatial vector which is not tangent to the body placement
(black arrows in fig. 1,2).

Then, for lower dimensional bodies, the definition of time-derivative of F 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.

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. 

As a last observation, I would point out that the theory ensuing from the adoption of the natural definition of time rates is definitely simpler and more suitable for computational applications.

Another point to think about is that Euler'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 F 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.

Best regards

Giovanni Romano

I see that a fish fryer came into the blog in a somewhat impolite way.

I don't want to get him carried away but rather to suggest him to learn the art of cooking before spitting his recipes. Wrongly fried fishes may have an awful taste and be dangerous for the health.

For other friends reading this blog, I notice that Fréchet derivatives are defined in linear spaces with a suitable topology. In the discussion opened with this blog the issue is rather how to reconduct a time-derivative along the non-linear trajectory manifold 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 Introduction to Continuum Mechanics.

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.

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.

Giovanni Romano

Dear collegues,

let me say some words about the
hope that this blog could serve to provide a well-sustained discussion about
the foundations of Continuum Mechanics (CM).
This is the aim why I opened the blog and, in my opinion, the goal can be
reached only if there will be no hurry to claim things just as anyone has
learned in his training, but rather an attitude to think and ponder with
patience and deepness will be experimented, to provide constructive criticisms.

I have learned this lesson once
more at the end of my academic career when I realized that, to overcome a deep
feeling of unsatisfaction for most treatment of CM,

especially concerning rate formulations and constitutive theory,
and to find a rationale out of a highly confused situation, I had to learn fundamental
notions and concepts of Differential Geometry which were not teached to me in
my training as a student of engineering and were later considered in the
scientific community as a mathematical sophisticated discipline of no concern
in applied mechanics.

There is a common old saying in
Naples, Italy, whose translation could be:

The best deaf is who doesn’t intend to listen.

I understand that changing ideas
on fundamentals is a very hard task for experts, but the extraordinary
experience with Einstein’s
relativity theory (absit iniuria verbis)
should learn us to be open minded.

The fortune of that theory was
that people like Hilbert, Klein, Poincaré and Minkowski,
very knowledgeable and authoritative experts, were intrigued by the approach by Einstein and
shared his childish enthusiasm.

There are some significant
similarities between that situation and the present one concerning Continuum
Mechanics.

In both cases one is faced with a
well-consolidated theory rich of implications and interpretations of
experimental facts, but with some unexplicable difficulties and paradoxes.

In both cases the right idea
comes by collecting hints and partial answers by older proposals made by other
valuable researchers.

In the field of our concern,
Continuum Mechanics, the leading new ideas are the following.

1)   
A careful geometric definition and distinction
between spatial and material tensor fields.

2)   
The statement of the Covariance Paradigm (CP) expressing that the rule for the comparison
between material tensors is the push according to the relevant diffeomorphic
transformation.

The first item is basic but simple and evident, although
never explicitly stated in a proper geometric form. The second item is made
more subtle by a dimensional coincidence between the body and the ambient
space, but is self-proposing for lower dimensional bodies.

To grasp the motivation it should be observed that
comparison between material tensors means that two placements of a body along a
trajectory (whether real or virtual) are considered, as related by the
evolution diffeomorphism between them. The natural
way to make the comparison is just to resort to the evolution
diffeomorphism itself, and to its tangent map, to perform the push-pull
transformations. This idea goes back as far as to Euler and to his celebrated formula for the stretching.

The conceptual clarity of this approach is not questionable
and the extraordinary effectiveness of its adoption becomes evident as soon as
it is applied to formulate constitutive relations, to discuss basic issues such
as time independence, time invariance, frame invariance and integrability
conditions.

Fictitious difficulties faced with in CM are swept away by the CP
which leads to formulate rate constitutive relations for
elasto-visco-plasticity (and similar models of material behavior) in a direct
and definite way and resolves the long lasting debate about rates of material
tensors by giving a unique, simple and well-defined answer.

A first exposure of this new approach has been contributed
in the paper
by G. Romano and R. Barretta: Covariant hypo-elasticity, 2011, European Journal of Mechanics - A/Solids.  doi:10.1016/j.euromechsol.2011.05.005

A more comprehensive treatment is in itinere.

Sincerely

Giovanni Romano

 
Dear Arash,

I deeply appreciate your question which will lead us to discuss about a basic issue in Classical Continuum Dynamics (CCD).

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.

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 CCD are treated in detail.

The foundations of CCD 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 Action Principle 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 action one-form 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 Euler-Lagrange equation and the corner conditions at singular points.

No geometric connection in the state-space manifold enters into the theory until this point and hence it can be affirmed that CCD may be founded in a natural way in terms of the evolution operator and of the lagrangian, without any additional assumptions, as illustrated in

G. Romano, R. Barretta and M. Diaco, On Continuum Dynamics,

J. Math. Phys. 50 (2009) 102903-1-26. DOI: http://dx.doi.org/10.1063/1.3215979

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 Maupertuis Least Action Principle as illustrated in

G. Romano, R. Barretta and A. Barretta, On Maupertuis principle in dynamics,

Rep. Math. Phys. 63 (3) (2009) 331-346. DOI: http://dx.doi.org/10.1016/S0034-4877(09)00015-9

 
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 Levi-Civita 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 Poincaré'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 Lie bracket and hence the components of the Lie brackets of the basis vector fields (named structure coefficients) appear into the equation of Dynamics.

The Levi-Civita connection on the trajectory, with the Lagrangian given by the kinetic energy per unit mass, leads to generalized Euler's and d'Alembert's laws of Dynamics. The standard formulations are recovered in the Euclid space endowed with the parallel transport by translation. These issues are treated in detail in the cited references.

In conclusion the answer to the question is:

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.

Best regards and wishes of Merry Christmas and Happy new Year

Giovanni Romano