iMechanica - continuum mechanics - Comments

Notes added

Teng Li — Fri, 19 Sep 2008 12:31:00 -0400

Prof. Casey,

Thanks for posting the lecture notes. A link to Prof. Naghdi's notes on continuum mechanics has been added to the lecture notes of interest to mechanicians.

Voids

Arash_Yavari — Tue, 12 Aug 2008 16:12:10 -0400

Dear Rezwan:

Thank you for your interest.

Of course you can think of "damage" or any measure that describes "damage" as a microstructure field. However, the question should be how useful the formulation is. My motivation in this paper was to see why there are so many different possibilities for "balance laws" and if there is any way to understand this starting from first principles. In many works balance laws are simply postulated and it's not clear if what one sees is the personal choice of the author(s) or there is more into the given formulation.

For damage you may look at the following paper:

Fu, M.F., Saczuk, J., Stumpf, H. 1998 'On fiber bundle approach to damage analysis' Int J Engng Science 36, 1741-1762.

The presentation is geometric but somewhere in the middle they assume Euclidean ambient spaces. Again, the question you should always ask yourself is whether you can gain anything using geometry. For your damage evolution problem, you should first see why the formulation based on Euclidean ambient spaces is not satisfactory (is this really the case?) and then try to formulate the theory geometrically. I had a look at the paper you sent me (don't want to mention the author) but didn't see anything but some "nice" interpretations of what is already known.

I hope this helps.

Regards,

Arash

RE: Balance Laws in Continua with Microstructure

Rezwanur Rahman — Sun, 10 Aug 2008 15:23:22 -0400

Hi Dr. Arash

This paper is very interesting indeed. This paper can be helpful regarding considering the issue of damages/voids/defects in the continuum (with respect to the microstructure field). I would like to mention that the Article 5.1 named: A Geometric Theory of Elastic Solids with Distributed Voids is expressive about cosidering damages/voids. According to our previous discussion, this material void velocity can be considered as "Local Martingale". And some random nucleation of new void(s) can cause this velocity to be a stochastic process containing sudden jumps. Besides that we can deal with a density function of void velocity instead of a single void. Please let me know about your opinion regarding this.

Thanks

Rezwan

many thanks

Dong Kong — Tue, 22 Jan 2008 23:53:28 -0500

Dear Prof. Chen and Prof. Eom,

Thanks a lot for your very useful comments.

Continuum description on protein molecules

Kilho Eom — Mon, 10 Sep 2007 03:32:13 -0400

As far as I know, there are very few works on continuum description on protein molecules. Very recently, as Prof. Chen commented, he and his collaborators have proposed the multi-scale model of protein, i.e. mechano-sensitive ion channel, based on combination of MD simulation and FE model (See here).

Except Prof. Chen's work, the continuum descriptions on single-molecule chain (e.g. DNA) have been proposed by Kratky and Porod, renowned as Worm-Like-Chain (WLC) model. WLC model has succeeded in depicting the nonlinear elastic behavior of single-stranded DNA based on pulling experiments by Bustamante group (See paper by Bustamante et al., Science, 265, p1599-1600, 1994). Such model has recently enabled Frey and coworkers to study the mechanical properties (bending properties) of microtubule based on real-time observation of microtubule tip connected to a bead with optical apparatus (for details, see here). Based on their experiments with comparison to WLC model, they showed that WLC model allows for depicting the bending behavior of microtubule as a function of contour length.

Such chain model have also allowed me and my advisors (Prof. Makarov and Prof. Rodin) to understand the mechanical properties of cross-linked single-molecule chain, which provides the insight into remarkable mechanical properties of some mechanical proteins such as titin Ig domain (See Refs: Ref at JPC and Ref at PRE).

More recently, the single-molecule experiments on spider silk fiber and spider silk protein by Hansma and coworkers have revealed that the mechanical behaviors of both spider silk fiber and spider silk protein are well delineated by scaling law, indicating that fiber consists of single-molecule chain in a hierarchical manner (See here). Such experimental results have resulted in the emergence of hierarchical model of spider silk fiber based on single-molecule chain (e.g., See here). Such hierarchical model based on single-molecule chain such as WLC has enabled the quantitative descriptions on the nonlinear elastic behavior of biological gels such as actin, collagen, neurofilament, etc. (for details, see here). In the issue of last week (Aug. 30, 2007) at PRL, the cover article is about the model of a fiber made of WLC chains with chirality (see here).

As shown above, the continuum-like model for a fiber made of single-molecules in a hierarchical manner has been one of hot issues in recent modeling researches in biomolecules. Such issue may be intriguing mechanicist, physicist, and chemists whose backgrounds are based on both theory and experiments.

We have some preliminary works in this area

Xi Chen — Fri, 07 Sep 2007 08:54:38 -0400

See http://www.imechanica.org/node/92 which is a top-down approach. More publications along such direction are forthcoming.

x + h off the shell surface

Amit Acharya — Tue, 15 May 2007 21:04:23 -0400

Nachiket,

Yes, you are correct.

does the point leave the surface?

Nachiket Gokhale — Tue, 15 May 2007 20:51:55 -0400

You wrote:

For geometric understanding, consider a scalar function φ defined on a
two dimensional shell surface in 3-d space. Let h be a tangent vector
to the shell at x and let the shell not be flat at x. Then φ(x + h) is
not strictly defined for every non-zero tangent vector h, however small.

I think this is because the point (x+h) leaves the surface of the shell. Am I correct?

-Nachiket

Differentiation

Amit Acharya — Mon, 14 May 2007 23:36:12 -0400

Biswajit,

Consider the invertible tensors I and -I. Their sum is not invertible. Hence. not a vector space.

Differentiation of a function at x requires the value of the function to be defined at x + h for h sufficiently small. If the domain of the function involved is a vector space this is not a problem, and it is not a problem even if it is not the entire space but only an open set of a vector space.

For geometric understanding, consider a scalar function φ defined on a two dimensional shell surface in 3-d space. Let h be a tangent vector to the shell at x and let the shell not be flat at x. Then φ(x + h) is not strictly defined for every non-zero tangent vector h, however small. For the same reason the definition of the directional derivative also changes to

d/ds φ(f(s))|s=0, where f is a curve on the manifold with f(0) = x.

If (y^i) is a local parametrization of the shell at x, and identifying the derivative at x with the gradient vector

grad φ = φ_,i e^i where (e^i) is the dual basis corresponding to the parametrization. Thus the gradient vector is tangent to the shell whereas if φ was defined everywhere then the gradient vector would have a component normal to the shell.

Practically, this comes up all the time when dealing with finite rotations in FE implementations of shells, while deriving Jacobians.

Hope this helps.

Re: Tensor derivatives

Biswajit Banerjee — Mon, 14 May 2007 14:56:39 -0400

Amit,

Thanks for the pointer. I'll try to write out the derivation when I get the time and post it here.

Could you elaborate on why "the set of all invertible second order tensors is not a vector space" and how that affects differentiation? An example of a situation where an excursion leaves the manifold will really be helpful.

Biswajit

derivative of the inverse

Amit Acharya — Mon, 14 May 2007 07:39:02 -0400

Of course, the easiest way for the deriv. of the inverse is to work from SS^-1 = I.

a slightly easier way

Amit Acharya — Sun, 13 May 2007 23:09:29 -0400

Biswajit,

Indeed, algebra and calculus of the type that never loses its charm.

Another way: The derivatives of the first and second invariants are easily done by realizing that the trace of a second order tensor is its inner product with the identity. For the second invariant, couple this observation with the chain rule.

The third invariant, as well as the derivative of the inverse of a (invertible) tensor is more interesting. I learnt the following from my advisor Don Carlson.

Once you know how to do the derivatives of the first and second invariants as above, and then realize that a tensor satisfies its own characteristic equation (Cayley-Hamilton theorem), then taking a derivative of the tensor characteristic equation gives the derivative of the third invariant in terms of the derivatives of the first two invariants, and the derivatives of the tensor itself and its square.

Multiply the tensor characteristic equation by the inverse of the tensor and then take a derivative; then, if one knows the derivatives of the invariants, then the derivative of the inverse falls out.

The next interesting question is how to do all this if the domain of these functions was a nontrivial manifold, e.g. suppose your s was incompressible (i.e. det s = 1). Of course, even with the inverse there is a bit of an issue to wade through as the set of all invertible second order tensors is not a vector space, but one gets away because it is at least an open set and diferentiation works out as sufficiently small excursions from a point in the domain remains in the domain (a requirement for, say, your definition of the directional derivative to make sense).

best,

Amit

Mindlin Centennial

Henry Tan — Sun, 29 Apr 2007 22:07:46 -0400

Mindlin Centennial
http://imechanica.org/node/309

Micropolar material constants

Henry Tan — Sun, 29 Apr 2007 20:18:36 -0400

A link to discussions on Micropolar material constants
http://imechanica.org/node/1298

microstructure theory of elasticity

Henry Tan — Fri, 23 Mar 2007 19:05:11 -0400

for self-teaching (from Dr. Michail Kulesh (http://users.math.uni-potsdam.de/~mkulesh/rus/projectnons.html)):

In the framework of the Cosserat continuum theory the displacements of particles in the examined medium are described in terms of two variables - an ordinary displacement field and kinematically independent vector field, which is introduced to characterize small rotations of particles. Thus, in the couple-stress theory there are two independent kinematic unknown quantities, and the stress tensor and the couple-stress tensor are asymmetric. In the context of this theory the elastic behavior of isotropic linear medium is described by six elastic constants: two Lame constants and four new constants describing microstructure. In the case of quadratic-nonlinear medium the number of new constants increases to nine.

The history of microstructure theories goes back to works by W.Voigt, who was the first to introduce a model of the medium with rotational interaction of its particles for studying elastic properties of a crystal. An early effort to develop an elasticity theory with asymmetric stress tensor evidently belongs to E.Cosserat and F.Cosserat. According to the Cosserat brothers' conception, which takes into account rotational interactions of material particles the most effective approach to the problems of stress-strain state in deformable solids is to introduce in the problem formulation the couple-stresses (moment of force per unit of area) in addition to the ordinary stresses (force per unit of area).

There has been a number of works reported in the literature in which the asymmetric theory is extended to the case of thermoelasticity and large deformations. Few works presenting solutions to a number of dynamic problems are also available in the literature. This is, for example, a systematic development of the modern theory by V.I.Erofeev, who considers the problem of propagation and interaction of elastic waves in solids with microstructure. Moreover, the idea of allowing for the internal rotation vector is often used for modeling plastic deformation in materials. However, a detailed discussion of these problems is beyond the scope of this paper, which is restricted to a static state of plane bodies in the framework of the elastic Cosserat continuum theory.

The asymmetric theory of elasticity for the Cosserat continuum (especially for the pseudo-Cosserat continuum) was successfully used by many authors to construct exact analytical solutions. In the majority works the obtained solutions are analyzed and compared with the corresponding solutions of the classical elasticity theory. In this comparison, new physical constants specifying the contribution of the couple-stress components generally assume the values from the energetically admissible range. This can be explained by deficiency of information on the material constants of microstructure media, which is one of the main factors restricting further investigation of asymmetric media models.

In some works a comparison between the solutions of the asymmetric and classical theories is carried out based on the analysis of the stress concentration coefficient and its dependence on the characteristic dimension of the stress concentrator. The analysis clearly demonstrated that compared to the classical theory the coefficient of the stress concentration increases (or decreases) with characteristic dimension of the concentrator. Although this fact is of obvious interest, the use of the concentration coefficient as a measurable parameter seems to be rather problematic. Thus, for example, an attempt to measure variation of the concentration index by the photoelasticity method has failed, since the resolving power of this method is too low to apply strictly to the desired characteristic dimension of the concentrator.