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# Geometry, topology, and solid mechanics

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Differential geometry in simple words is a generalization of calculus on some curved spaces called manifolds. An n-manifold is a space that locally looks like R^n but globally can be very different. The first significant application of differential geometry happened to be in Einstein’s theory of general relativity. The following quote from Kondo in 1954 is interesting: “Although the remarkable success of the general relativity theory impressed the importance of tensor calculus and Riemannian geometry on public opinion, it was unfortunate that it gave a metaphorical appearance to Riemannian expression, banishing it for a time from the attention of engineers.” “…it is strange that the first practical field of application was not the theory of elasticity, especially of residual strains.”. Topology is a branch of mathematics that studies classification of sets and their equivalence classes up to homoemorphisms (continuous bijections with continuous inverse). In other words if you can continuously deform a set to another one they are the same set topologically. These two branches of mathematics have important applications in solid mechanics that have been overlooked by most mechanicians in the last few decades. In my opinion, unfortunately, many of the existing works in geometric elasticity have been purely formal with no real contributions and this has perhaps been one reason that geometric methods have not been that popular in mechanics. The role of geometry should not be just reinterpreting the existing results; geometry has much more to offer and I’d like to discuss a few of the important applications in the following. I welcome your comments if I’m missing something here or if you like to add some more potential applications.

**1) Structure of the Governing Equations of Continuum Mechanics**

It has been known for a long time that the configuration spaces of physical theories are manifolds and that differential geometry is a natural framework for any field theory, and in particular, continuum mechanics. When formulated in the language of differential geometry the governing equations of continuum mechanics are written using some natural operators of differential geometry, e.g. Lie derivatives, covariant derivative, etc. [1]. As an example, in the geometric framework it is clear how the governing equations transform under changes of coordinates in both the reference and the current configurations [2] and whether they are invariant under isometries of the ambient space. In the geometric language some old controversies like superiority of one objective stress rate over another one can be easily resolved [1]. One recent application has been the solution to the old paradox of the lack of objectivity of linear elasticity [3,4]; if formulated properly linear elasticity would be covariant like any other physical theory. Another advantage of a geometric formulation is that one can formulate the theory of residually-stressed bodies very similarly to that of elastic bodies.

**2) Mechanics of Defects in Solids**

The first important application of differential geometry was in the mechanics of defects. Defects and their evolution control many of the mechanical properties of solids. A few decades before anyone knew anything about defects in solids, Vito Volterra [5] mathematically predicted defects and classified them into six types. Half of them are translational (called dislocations by Love) and the other half are rotational (called disclinations by Frank). Volterra’s motivation was elasticity of “multiply-connected” bodies and the possibility of (residual) stresses in the absence of external forces. Since the work of Volterra only very simple non-simply connected bodies have been considered in elasticity. Interestingly, Volterra’s friend Henri Poincare started creating what is now called Algebraic Topology (he called it Analysis Situs) a decade before Volterra’s work in elasticity. The goal of algebraic topology is to understand and classify spaces with “holes” using algebraic methods. For a recent discussion of applications of algebraic topology in nonlinear elasticity see [6].

The deep connection between mechanics of distributed defects and non-Riemannian geometries was independently discovered in the 1950s by Kondo [7,8] in Japan and Bilby and his co-workers [9] in the UK. The main focus of these seminal contributions was kinematics of defective solids in terms of Riemann’s curvature and Cartan’s torsion (Eli Cartan – the father of modern differential geometry – was influenced by the work of Cosserat brothers. In this sense solid mechanics has had an influence on the development of modern differential geometry). The main idea here is that in the presence of defects the natural (stress-free) configuration of the body is a non-Riemannian manifold with a geometry explicitly depending on the density of defects. There have been very few exact solutions for stress field of single and distributed defects in nonlinear solids. For dislocations we should mention [10-14] and for disclinations [12,15,16]. Interestingly, since the work of Love [17] more than ninety years ago on the stress field of a single point defect in linear elastic solids and the work of Eshelby in the 1950s [18] on distributed point defects in linear elastic solids there was no single stress calculation for either isolated or distributed point defects in nonlinear solids before our recent works [19]. This was accomplished using non-metricity, a geometric object that was introduced by Weyl in 1918 with the motivation of unifying electromagnetism and relativity (without success). The geometric approach can also be used in finding exact solutions for combination of defects (what we call *discombinations*) in nonlinear solids for which superposition cannot be used [20]. In my opinion, future work in this direction should focus on stability of the nonlinear solutions and dynamics of defects.

**3) Mechanics of Biological Systems with Bulk Growth**

In recent years many researchers have been interested in formulating continuum theories for bodies with bulk growth [21-32]. Most of these formulations are based on a multiplicative decomposition of deformation gradient into elastic and growth parts, which is an idea borrowed from finite plasticity [32-34] and thermoelasticity [35]. There are also works on finite thermoelasticity and growth mechanics using a Riemannian material manifold [36,37]. One open problem here is the form of the evolution equation for the growth tensor (or material metric) as a function of stress. The existing formulations are heuristic and more research in this direction is needed. In the case of growing plates and shells there have been recent works among which we should mention [38]. The assumed constitutive equations are linear but this restriction can be released if needed. We should also mention the work of Guven and his co-workers [39] on bio membranes using variational methods. Speaking of a truly geometric theory of shells one should mention [40,41] in which everything is written in terms of the first and second fundamental forms of the surface. The first fundamental form is the induced metric and quantifies the in-plane strains (membrane strains) while the second fundamental form tells us how a submanifold is embedded in the ambient space and quantifies the bending strains. As future work, developing a theory of growing shells in terms of only the first and the second fundamental forms would be interesting. I find the recent book [42] on the geometry of submanifolds quite interesting and easy to read.

**4) Computational Mechanics**

There are many different numerical schemes the literature and perhaps the most successful one in solid mechanics has been the finite element method. There have been difficulties in using finite element method, e.g. in problems with internal constraints like incompressibility (volume locking and checkerboarding of pressure). The solution to this problem has been a mixed formulation in which displacement and pressure fields are discretized independently. In any numerical scheme one has to worry about two things: 1) convergence and 2) stability. The first stable discretization scheme for linear elasticity was introduced by Arnold and his co-workers [43-45] based on the idea of differential complexes. A differential complex is a sequence of linear spaces with some linear operators between them such that the successive application of any two is null. Kroner [34] introduced a differential complex for linear elasticity with the motivation of understanding the kinematics of linear dislocation mechanics. Two years later Calabi [46] introduced a complex for Riemannian manifolds with constant curvature (flat Euclidean space is a special case). Eastwood [47] realized the connection between the two complexes (obviously Eastwood was not aware of Kroner’s work and didn’t cite it). Arnold and his co-workers successfully used the ideas of Eastwood in linear elasticity and derived a Hilbert complex for linear elasticity and then discretized it. This led to the first stable discretization of linear elasticity. Recently, we have introduced some differential complexes in nonlinear elasticity [48]. These complexes are potentially useful for discretization of nonlinear elasticity.

It has been observed in the literature that preserving some structure of a continuum system at the discrete level may lead to “better” discretization schemes [49,50]. One example is Arnold, et al.’s work in which the differential complex structure of linear elasticity is preserved at the discrete level. Another example is numerical dissipation in time discretizations of conservative systems. Following the ideas of Veselov [51], instead of discretizing the Euler-Lagrange equations one can directly discretize the action integral to get an action sum and then apply Hamilton’s principle to it, e.g. Hamilton’s principle is preserved at the discrete level [52]. This idea has led to variational integrators with excellent conservation of energy and momenta [53, 54]. Structure preserving discretization of nonlinear elastostatics is an open problem. It seems that exploring the discretization of the differential complexes we introduced recently [48] may be a good research direction.

Let me end this discussion by acknowledging that the main obstacle for using geometric methods by mechanicians is the required background that is not part of any standard engineering curriculum. I do not have a good solution for this but from personal experience I think one should try to learn the mathematics that is needed for solving a particular problem. To learn geometry and topology I would suggest the excellent books [55-63].

In response to Zhigang's question I have added a new reference [64] on inclusions in nonlinear solids.

An applied introduction to differential geometry can be seen in the book [65]. For an introduction to Cartan's calculus and moving frames I highly recommend the recent excellent book [66] by Prof. Shlomo Sternberg.

- Marsden, J.E. and T.J.R. Hughes [1983], Mathematical Foundations of Elasticity, Dover, New York.
- Yavari, A., J.E. Marsden and M. Ortiz [2006], On the spatial and material covariant balance laws in elasticity, Journal of Mathematical Physics 47: 042903; 85 -112.
- D.J. Steigmann [2007], On the frame invariance of linear elasticity theory, ZAMP 58(1):121-136.
- Yavari, A. and A. Ozakin [2008], Covariance in linearized elasticity,
*ZAMP***59**(6):1081-1110 - Volterra, V. [1907], Sur l'equilibre des corps elastiques multiplement connexes. Annales Scientifiques de l'Ecole Normale Superieure, Paris 24 (3):401-518.
- Yavari, A. [2013], Compatibility equations of nonlinear elasticity for non-simply-connected bodies,
*Archive for Rational Mechanics and Analysis*,**209**(1):237-253. - Kondo, K. [1955], Geometry of elastic deformation and incompatibility, Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry, (K. Kondo, ed.), vol. 1, Division C, Gakujutsu Bunken Fukyo-Kai, 1955, pp. 5-17.
- Kondo, K. [1955], Non-Riemannian geometry of imperfect crystals from a macroscopic viewpoint, Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry, (K. Kondo, ed.), vol. 1, Division D-I, Gakujutsu Bunken Fukyo-Kai, 1955, pp. 6-17.
- Bilby, B. A., R., Bullough, and E., Smith [1955], Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry, Proceedings of the Royal Society of London A231(1185): 263-273.
- Gairola, B.K.D. [1979], Nonlinear elastic problems, in Dislocations in Solids, (F.R.N. Nabarro, ed.), vol. 1, Amsterdam, North-Holland.
- Rosakis, P. and Rosakis, A. J. [1988], The Screw dislocation problem in incompressible finite elastostatics - a discussion of nonlinear effects. Journal of Elasticity 20 (1): 3-40.
- Zubov, L.M. [1997], Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies, Springer, Berlin.
- Acharya, A. [2001], A model of crystal plasticity based on the theory of continuously distributed dislocations. Journal of the Mechanics and Physics of Solids 49:761-784.
- Yavari, A. and A. Goriely [2012], Riemann-Cartan geometry of nonlinear dislocation mechanics. Archive for Rational Mechanics and Analysis 205 (1):59–118.
- Derezin, S. V. and Zubov, L. M. [2011], Disclinations in Nonlinear Elasticity. ZAMM 91 (6):433-442.
- Yavari, A. and A. Goriely [2013], Riemann-Cartan geometry of nonlinear disclination mechanics. Mathematics and Mechanics of Solids 18 (1): 91-102.
- Love, A.H. [1927], Mathematical Theory of Elasticity. Cambridge University Press, Cambridge.
- Eshelby, J.D. [1954], Distortion of a crystal by point imperfections. Journal of Applied Physics 25 (2):255-261.
- Yavari, A. and A. Goriely [2012], Weyl geometry and the nonlinear mechanics of distributed point defects. Proceedings of the Royal Society A 468:3902-3922.
- Yavari, A. and A. Goriely, The geometry of discombinations and its applications to semi-inverse problems in anelasticity, Proceedings of the Royal Society A 470, 2014, 20140403.
- Ambrosi, D. and Guana, F. [2007], Stress-modulated growth. Mathematics and Mechanics of Solids 12 (3):319-342.
- Ben Amar, M. and Goriely, A. [2005], Growth and instability in elastic tissues. Journal of the Mechanics and Physics of Solids 53:2284-2319.
- Epstein, M. and Maugin, G. A. [2000], Thermomechanics of volumetric growth in uniform bodies. International Journal of Plasticity 16:951-978.
- Goriely, A., Robertson-Tessi, M., Tabor, M., and Vandiver, R. [2008], Elastic growth models. In Mathematical Modelling of Biosystems, Ed. R. Mondaini, Springer-Verlag.
- Klarbring, A. and Olsson, T. and Stalhand, J. [2007], Theory of residual stresses with application to an arterial geometry. Archives of Mechanics 59:341-364.
- Lubarda, V. A. and Hoger, A. [2002], On the mechanics of solids with a growing mass. International Journal of Solids and Structures 39:4627-4664.
- Rodriguez, E. K. and Hoger, A. and McCulloch, A. D. [1994], Stress-dependent finite growth in soft elastic tissues. Journal of Biomechanics 27:455-467.
- Skalak, R. and Dasgupta, G. and Moss, M. and Otten, E. and Dullemeijer, P. and Vilmann, H. [1982], Analytical description of growth. Journal of Theoretical Biology 94:555-577.
- Skalak, R. and Zargaryan, S. and Jain, R. K. and Netti, P. A. and Hoger, A. [1996], Compatibility and the genesis of residual stress by volumetric growth. Journal of Mathematical Biology 34:889-914.
- Takamizawa, K. and Matsuda, T. [1990], Kinematics for bodies undergoing residual stress and its applications to the left ventricle. Journal of Applied Mechanics 57:321-329.
- Takamizawa, K. [1991], Stress-free configuration of a thick-walled cylindrical model of the artery - An application of Riemann geometry to the biomechanics of soft tissues. Journal of Applied Mechanics 58:840-842.
- Bilby, B.A., Gardner, L.R.T., Stroh, A.N.: Continuous distribution of dislocations and the theory of plasticity. Proceedings of the Ninth International Congress of Applied Mechanics, Brussels, 1956, Universite de Bruxelles, pp. 35–44, 1957.
- Kroner, E. [1959], Allgemeine kontinuumstheorie der versetzungen und eigenspannungen, Archive for Rational Mechanics and Analysis 4:273-334.
- Lee, E.H., Liu, D.T. [1967], Finite-strain elastic-plastic theory with application to plane-wave analysis. Journal of Applied Physics 38:19–27.
- Stojanovic, R., Djuric, S., and L. Vujov sevic [1964], On finite thermal deformations. Archiwum Mechaniki Stosowanej 16: 103 -108.
- Ozakin, A. and A. Yavari [2010], A geometric theory of thermal stresses, Journal of Mathematical Physics 51, 032902.
- Yavari, A. [2010], A geometric theory of growth mechanics,
*Journal of Nonlinear Science*20(6):781-830. - Efrati, E. and Sharon, E. and Kupferman, R. [2009], Elastic theory of unconstrained non-Euclidean plates. Journal of the Mechanics and Physics of Solids 57 (4):762-775.
- Capovilla, R. and J Guven [2002], Stresses in lipid membranes,
*Journal of Physics A: Mathematical and General***35**6233. - Koiter,W.T. [1966], On the nonlinear theory of thin elastic shells. Proc. K. Ned. Akad. Wet. B 69, 1-54 (1966).
- Balaban, M. M., A.E. Green, and P.M. Naghdi [1967], Simple force multipoles in the theory of deformable surfaces, Journal of Mathematical Physics 8, 1026.
- Greenfield, P. [2013], Introduction to Tensor Analysis and the Calculus of Moving Surfaces, Springer.
- Arnold, D.N. and R. Winther [2002], Mixed finite elements for elasticity, Numer. Math. 92: 401-419.
- Arnold, D.N. and R. S. Falk and R. Winther [2006], Finite element exterior calculus, homological techniques, and applications, Acta Numerica 15:1-155.
- Arnold, D.N. and R. S. Falk and R. Winther [2010], Finite element exterior calculus: from Hodge theory to numerical stability, Bul. Am. Math. Soc 47: 281-354.
- Calabi, E. [1961], On compact Riemannian manifolds with constant curvature, Differential Geometry, 155-180, Proc. Symp. Pure Math. vol. III, Amer. Math. Soc.
- Eastwood, M.G. [1999], A complex from linear elasticity, Rend. Circ. Mat. Palermo, Serie II, Suppl. 63, 23-29.
- Angoshtari, A. and A. Yavari [2014], Differential complexes in continuum mechanics, arXiv:1307.1809v2.
- Yavari, A. [2008], On geometric discretization of elasticity
*, Journal of Mathematical Physics***49**, 022901, 1-36. - Angoshtari, A. and A. Yavari [2013], A geometric structure-preserving discretization scheme for incompressible linearized elasticity,
*Computer Methods in Applied Mechanics and Engineering***259**:130-153. - Veselov, A. [1988], Integrable discrete-time systems and difference operators, Functional Analysis and its Applications 22 (2):83–93.
- Marsden, J.E. and M. West [2001], Discrete mechanics and variational integrators, Acta Numerica 10:357 -514.
- Kane, C., J. E. Marsden,M. Ortiz, and M.West [2000],Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems, Int. J. Num. Math. Eng., 49, 1295–1325.
- Lew, A., J.E. Marsden, M. Ortiz, and M. West [2003], Asynchronous variational integrators, Archive for Rational Mechanics and Analysis 167:85-146.
- Munkres, J.R., Elementary Differential Topology, Annals of Mathematics Studies 54, Princeton University Press, 1966.
- Munkres, J.R., Elements of Algebric Topology, Springer, New York/Addison-Wesley, Menlo Park CA, 1984.
- Munkres, J.R., Topology, Prentice Hall, NJ, 2000.
- Nakahara, M., Geometry, Topology and Physics, Taylor & Francis, New York, 2003.
- Flanders, H., Differential Forms and Application to Physical Sciences, Dover, New York, 1990.
- Frankel, T., The Geometry of Physics: An Introduction,Cambridge University Press, Cambridge, 1997.
- do Carmo, M.P., Differential Geometry of Curves and Surfaces, 1976.
- do Carmo, M.P., Riemannian Geometry, 2013.
- Kreyszig, E., Differential Geometry, Dover Books on Mathematics, 1991.
- Yavari, A. and A. Gorley, Nonlinear elastic inclusions in isotropic solids, Proceedings of the Royal Society A 469, 2013, 20130415.
- Burke, W.L., Applied Differential Geometry, 1985.
- Sternberg, S., Curvature in Mathematics and Physics, 2012.
- A. Ozakin and A. Yavari, Affine development of closed curves in Weitzenbock manifolds and the Burgers vector of dislocation mechanics,
*Mathematics and Mechanics of Solids***19**, 2014, pp. 299-307.

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## Comments

## Re: Geometry, topology, and solid mechanics

Dear Arash: Thank you so much for this intriguing entry. Over the years I have been fascinated by your description of the links bettwen geometry, topology and mechanics. Perhaps this time I should take the plunge and do some serious reading. Is it possible for you to describe one particular paper that highlights this approach? I'll then try to read the paper, and come back to you. Reading one particular paper will never achieve the effect of reading the 63 works, but it is an improvement over reading zero. Thank you for your consideration.

## Re: Re: Geometry, topology, and solid mechanics

Dear Zhigang:

Thanks for your kind message. I think reading for the purpose of understanding a particular problem would be the fastest route. I added one more reference that perhaps can help. It’s a recent work on eigenstrains and inclusions in nonlinear solids (reference [64]). The problem is: Given a finite ball made of an arbitrary isotropic and incompressible solid with a distribution of radial and circumferential eigenstrains that respect the spherical symmetry what is the resulting residual stress field? As a particular case, you can look at an inclusion, which means eigenstrains are supported on a subset of the ball. The main idea here is to rewrite this problem as if it’s a standard nonlinear elasticity problem. This we do by constructing the material manifold, which is a space in which the body is stress free. For this example, this space is the ball with a metric that explicitly depends on the distribution of eigenstrains. The ball with this metric is a Riemannian manifold that you can construct easily (the solid cylinder is very similar). The rest would be classical nonlinear elasticity. I hope this can help motivate you to read this paper and hopefully the rest later on.

Regards,

Arash

## nonlinear elastic inclusion

Dear Arash: Thank you so much. Ref. 64 helps me focus. I have two questions for you

Thank you very much for guiding me through this.

## Re: nonlinear elastic inclusion

Dear Zhigang:

1) An eigenstrain may have different sources. Growth is one. It can be due to temperature changes (and the body may be thermally anisotropic), etc. So, for the purpose of this discussion we can assume eigenstrains come from growth.

2) The approach of using a multiplicative decomposition of deformation gradient is equivalent to what you see in [64]. Actually, if you look at [22] they solve similar problems using F=AG. Some people may argue that this decomposition is vague (you need to specify the rotation) and this is true but for these simple geometries they are really equivalent. As I discuss in [37] the material metric (I call it “G”) is equal to G^TG in the F=AG approach. For inclusions as we discuss in [64] in the reference (Diani JL, Parks DM. 2000 Problem of an inclusion in an infinite body, approach in large deformation. Mech. Mater. 32, 43–55.) a similar decomposition was used in their numerical simulations.

Now what is the advantage of the approach of [64]? If it is equivalent to what you have used (which is the case) there is no real advantage. However, the geometric approach is more general and you can use it in more complicated problems. Here, we are able to directly write the growth tensor or the material metric. This cannot be done that easily, in general. For defects you’re given the distribution of Burger’s vectors, Frank’s vectors or some volume density of point defects. The material metric can be very systematically calculated in the geometric approach (of course, there have been some exact solutions for defects using other approaches as well). One can also deal with combinations of defects as well as we discuss in the recent work [20].

I don’t know if the following answers your question or not. But I’m wondering why after so many years since the work of Eshelby on inclusions there hasn’t been any 3D analytic solutions for inclusions in nonlinear solids? I don’t know why. However, when you look at this problem in the geometric framework it’s so clear how it should be formulated. I think geometry is just a tool that can be very useful sometimes.

Regards,

Arash

## Thank you, Arash. You have

Thank you, Arash. You have answered my questions. It is often significant to have multiple methods for the same problem, because they might lead to different insights and extensions.

Now I should read more to learn about these different insights. Your formulation in ref 64 assumes that the reader knows about reimann manifold. I don't. I can guess what it might be. But even Wikipedia page is convoluted. Can you write down basic definitions with familiar words? Make them self contained. Like many mechanicians, I have a working knowledge of calculus and linear algebra, but do not know their modern developments.

Thank you very much.

## Defects, continuum mechanics, differential geometry

Apropos "different insights and extensions," an approach to the subject of connections between continuum mechanics and differential geometry in the specific context of statics and dynamics of line and wall defects can be found in the paper at the following link. An express aim of the paper was to be accessible to an audience with background in multi-variable calculus and linear algebra required for a thorough grasp of continuum kinematics.

http://imechanica.org/node/15585

In the context of the current discussion,

Section 2 - Notation

Section 3 - Motivation for a fundamental kinematic decomposition

Section 7 - Contact with the differential geometric point of view

may be particularly relevant, as well as

Section 4.7 - Contact with the classical view of modeling defects: A Weingarten theorem for g.disclinations and associated dislocations

We have developed finite elelement algorithms for computing static solutions for energy/stress fields of defect distributions at small and finite strains, e.g. five twin boundaries meeting at almost a point, and Chiqun Zhang, my student, will be presenting the work at the MMM2014 conference at Berkeley.

## Re: Beginners' differential geometry

I had the opportunity to sit in on Andrejs Treiberg's lectures on differential geometry from Cartan's point of view. Some of his expositions at http://www.math.utah.edu/~treiberg/Lect.html are enlightening. Another document I like is from TAMU by Ivey and Landsberg, "Cartan for begineers" at http://www.math.tamu.edu/~jml/EDStmp.pdf. However, it's never easy reading and one has to work through the content systematically to gain some understanding.

I admire Arash's attempts at making differential geometry understandable to engineers.

-- Biswajit

## Re: Re: Beginners' differential geometry

Dear Biswajit:

Thank you for your kind message and the references. The book “Cartan for Beginners” is very nice but it’s not really written for beginners in differential geometry! A better book to start with is the excellent book by Prof. Shlomo Sternberg of Harvard U. It’s written beautifully. In simple words he starts explaining what geometry is and then presents a vivid exposition of Cartan’s calculus. Actually I learned Cartan's calculations from a presentation from him that I found after a google search. I added this book to the list of references (reference [66]). I also added another reference [65] that is a good book for beginners.

Regards,

Arash

## Re: Riemannian manifold

Dear Zhigang:

First let me explain what a manifold is. An n-dimensional manifold is a space that locally looks like R^n. Let’s start with 2-manifolds. A sphere is an example of a 2-manifold. Locally it looks like a plane (this is perhaps why for a long time people thought the surface of Earth was flat). You can describe the whole sphere by some “nice” mappings that map parts of the sphere to planes. A cylinder is another example of a 2-manifold and so is the plane itself. Each of these “nice” mappings is a coordinate chart. For the sphere it turns out that you need at least two coordinate charts to cover the whole space.

Note that we don’t have to think of sphere as a surface living in a larger space (here R^3); sphere is defined as a 2-manifold that doesn’t have to be embedded in another bigger space. In other words sphere is defined intrinsically.

So far in this definition we don’t have a notion of distance. If you need to make sense of distance between two points in a manifold you need to introduce what is called “metric”. In Euclidean space given two points with Cartesian coordinates (X,Y,Z) and (X+a,Y+b,Z+c) the distance between them is

dS^2 = a^2 + b^2 + c^2

In this case we say that metric of the Euclidean space is represented by the identity matrix diagonal(1,1,1). Note that if I choose a cylindrical coordinate system (R,\Theta,Z) then the same thing is written as

dS^2=dR^2+R^2 d\Thata^2+dZ^2,

which means metric has the matrix representation G=diagonal (1, R^2, 1). If you use the spherical coordinates you would have a different diagonal matrix representation for the same metric. These are the same metric in the sense that they all give you the same distance between two given points A and B in the manifold.

The metric that we use in measuring distances between cities, etc. is the metric that is induced from the standard metric of R^3. For example, the distance between Boston and Atlanta is the length of the (great) circular arc between the two cities. This arc is called a geodesic (shortest distance curve). The shortest distance curve may not be unique depending on the manifold. For a sphere, there are infinitely many shortest distance curves between the north and south poles, for example.

Given an n-manifold you have a tangent space at each point. You can see this more clearly for a 2-manifold. At each point there is a tangent space (plane), which is 2-dimensional and is a linear space.

A metric at a point defines an inner product in the tangent space at that point. In the case of a 2-manifold at each point metric is represented by a symmetric 2 by 2 matrix. It may not be a diagonal matrix, in general. Note that inner product in a linear space is standard with the classical properties. For example only inner product of the zero vector by itself is zero, there is linearity, etc.

Now moving from a point to another point metric (or inner product of tangent vectors) may be represented by a different matrix. Or in other words, metric may vary from a point to another point. If metric varies smoothly from point to point your 2-manifold with this metric is called a Riemannian manifold. You can have something similar for an n-manifold.

So, a Riemannian manifold is a pair (B,G), where B is an n-manifold, which is endowed with a metric G (you need to have more to call it a Riemannian manifold but for this discussion this is good enough). Note that given a manifold you can endow it with different metrics.

If I give you a ball with an inclusion with pure dilatational eigenstrains the rest configuration is a Riemannian manifold, which outside the inclusion has the Euclidean metric while inside the inclusion its metric is some positive number times the Euclidean metric.

I hope this helps. Please let me know if something is unclear.

Regards,

Arash

## Dear Arash:

Dear Arash:

Coincidentally, I am a beginner of mechanics of defects in solids and here is my opinion. The classical Eshelby approach is identical to Green's function method, and eigenstrains as well as inhomogeneities are regarded as sources of internal stress. Eigenstrains were generalized in Toshio Mura's work (Micromechanics of defects in solids, 1987). I think nonlinear elastic inclusions can also be treated as general inhomogeneities which possess varying elastic properties other than those of matrix. With these consideration, the classical Eshelby approach, i.e. Green's function method, is still applicable. For the problems of finite domains, the corresponding Green's function should be choosen properly to satisfy the actual boundary conditions. A paper of Zhuping Huang written in Chinese discussed the applicability of Eshelby's equavalent inclusion method in nonlinear continumm mechanics and drew a negative conclusion. I think the reason for the discrepancy between two solutions lies in the absence of considering the elastoplastic region outside of the inclusion as an additional eigenstrain, which fades out till the elasticity-plasticity interface.

Your comment is appreciated.

Best regards.

liebealt

## Dear Liebealt:

Dear Liebealt:

Thanks for your comments and question. Please note that in [64] an inhomogeneity with eigenstrains is analyzed, i.e. the energy function of the inclusion can be different from that of the matrix. There have been recent works on extending the Green’s functions to finite domains. I am aware of the recent work of Prof. Shaofan Li on deriving Green’s functions for a finite spherical ball. I am not familiar with the paper of Zhuping Huang and would like to see an English translation if there is one.

Regards,

Arash

## To Dear Arash

Thanks for your comment and sharing of a vast amount of information. Reference [64] is then bound to be studied. I have spend a couple of hours in searching the English translation of Zhuping Huang's work, yet the result is disappointing. Please feel free to contact me to share it should you find Chinese edition is OK. Looking forward to having more discussion with you in the future.

Best regards,

liebealt

## Re: Applicability of Eshelby's Equivalent Inclusion Method

Dear Liebealt:

I now have this short paper (6 pages). It’s in Chinese. If you or someone else can translate it, it would be great. If you email me (arash.yavari@ce.gatech.edu) I can email you the pdf file.

Regards,

Arash

## Geometry of Physics

Dear All,

I have come to toics in differential gemotry in the work of Simo and then the book by Hughes and Mardsen (Mathematical Foundations in Elasticity). The subject was so interesting that I began to study the book "Geometrical Methods in Mathematical Physics" by Bernard Shutz. I then intend to read the book suggested by Arash, namely the "Geometry of Physics" by Theodore Frankel. However the subject is too vast and wish to catch up as quickly as possible.

Regards

Mohsen

## Multiplicative decomposition

Dear Arash,

First of all, thanks for your nice posting! Although I don't know if my question is related to current forum topic, it's pleasure to hear your opinion on the multiplicative decomposition. My first question is the existence of stress-free intermediate configuration. It is referred that the intermediate configuration is a fictitious one and the deformation gradient of each elastic and plastic part is not corresponding the gradient of deformation. Can you elaborate on this? My second question is the use of the objectivity or principle of material-frame indifference. Some authors(e.g., Naghdi and Casey) used full restriction of both intermediate and deformation configuration imposed by superposed rigid body motion. But, others(e.g., Gurtin and Anand) opposed the use of restriction in the intermediate configuration. What do you think about this issue?

Best regards,

Sangyul

## Re: Multiplicative decomposition

Dear Sangyul:

The multiplicative decomposition of deformation gradient into an elastic and an inelastic part is certainly relevant to this discussion. This idea goes back to Bilby and Kroner in finite plasticity. It has been used in finite thermoelasticity and in more recent years in growth mechanics. We have had a few discussions on it in iMechanica in the past but I think it’s ok to discuss it again here.

In the literature there have been objections to the use of the multiplicative decomposition but most of these, in my opinion, are not constructive. This decomposition has been a useful approach and that should be acknowledged. First note that if F=AB is compatible, i,e, Curl F =0 (sufficient when the body is simply connected) you cannot conclude that A or B is compatible as well. So, the elastic and inelastic parts of F are not compatible, in general (they don’t correspond to “motions"). They would be compatible only in 1D and that you can see in some recent papers on growing rods. It is known that this decomposition has an SO(3) ambiguity (note that AB = AQ Q^T B for an arbitrary rotation Q) but this can be fixed in problems of interest. What is more troubling is what people call “intermediate configuration”. This is some "configuration” that is "defined only locally”, a statement that is mathematically vague and far from satisfying. It is ok to use this decomposition and, for example, say that elastic energy depends only on the elastic part of deformation gradient. But doing things like integration would not make sense in my opinion. In Ref [67] you can see a discussion on the definition of Burger’s vector and the issues regarding integration in the (undefined) intermediate configuration. In the geometric theory in Ref. [14] there is no intermediate configuration. There is a discussion there on the connection with the decomposition of deformation gradient. You could think of the “intermediate configuration” as the tangent space of the material manifold at a given point.

I haven’t read either the paper(s) by Naghdi and Casey or the one(s) by Gurtin and Anand. I don’t know what it means to even talk about a rigid translation in the “intermediate configuration”, for example.

Regards,

Arash

## Multiplicative Decomposition

Dear Arash,

Fixing the rotation Q and associating a physical interpretation to it is also an interesting topic. Moreover there exists publications in which the integration is suggested to be performed in intermediate configuration. I think the topic is so interesting that it deserves a totally separate discussion devoted to it.

Mohsen

## Regarding the multiplicative decomposition

Its value is pedagogical: it helps in creating an image, a geometry, or in discerning more easily relationships between variables.

Otherwise, what can/do we observe in reality ?

We observe changes, that is, rates.

I don't think that there's anyone on this planet who looks at a motion and sees... multiplicative (composed) motions.

If there are several phenomena taking place simultaneously (like elastic and inelastic), then we can certainly say something about

their velocities, and not, a priori, about their motions.