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# Balance Laws in Continua with Microstructure

This paper revisits continua with microstructure from a geometric point of view. We model a structured continuum as a triplet of Riemannian manifolds: a material manifold, the ambient space manifold of material particles and a director field manifold. Green-Naghdi-Rivlin theorem and its extensions for structured continua are critically reviewed. We show that when the ambient space is Euclidean, postulating a single balance of energy and thinking of microstructure manifold as the tangent space of the ambient space manifold, postulating energy balance invariance under time-dependent isometries of the Euclidean ambient space, one obtains conservation of mass, balances of linear and angular momenta but not a separate balance of linear momentum. This will be associated with the rigid structure of Euclidean space. We develop a covariant elasticity theory for structured continua by postulating that energy balance is invariant under time-dependent spatial diffeomorphisms of the ambient space, which in this case is the product of two Riemannian manifolds. We then introduce two types of constrained continua in which microstructure manifold is linked to the reference and ambient space manifolds. We show that when at every material point the microstructure manifold is the tangent space of the ambient space manifold at the image of the material point, covariance gives us balances of linear and angular momenta with contributions from both forces and micro-forces and two Doyle-Ericksen formulas. We show that a generalized covariance can lead to two balances of linear momentum and a single coupled balance of angular momentum. We then covariantly obtain the balance laws for two specific examples, namely elastic solids with distributed voids and mixtures. Lagrangian field theory of structured elasticity is revisited and a connection is made between covariance and Noether's theorem.

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## Comments

## RE: Balance Laws in Continua with Microstructure

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 Voidsis 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

## Voids

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