# What is a coordinate-free formulation of deformation theory?

For example how are stress, strain and shear tensors described invariantly, without any coordinates, purely in a geometric manner?

A formulation that avoids indices coordinates and matrices, even in practical calculations.

• Have you looked at the brief description of the tensor form of the linear elasticity equations on Wikipedia? Jun 5, 2020 at 12:40
• Yea but it's too brief, and it doesn't show any concrete examples, without Cartesian or other coordinates. Jun 5, 2020 at 13:03
• You cannot avoid coordinates and indices in practical (physics) calculations.
– user21299
Jun 5, 2020 at 13:37
• I'm not looking for opinions here Jun 5, 2020 at 13:41
• @Ezio It's "too brief" because that's all there is to it, and there aren't any examples because for almost all practical purposes you need a definite coordinate system. How would you propose to define the stress-strain relationship for a general anisotropic material, unless you have some way to talk about the orientation of the material at different points in space, for example - and that is going to involve defining some basis vectors, which is the same thing as a coordinate system. Jun 5, 2020 at 14:27

I would recommend looking at works of W. Noll, C. Truesdell and collaborators. They have been working on the mathematical foundations for continuous mechanics since 1950-s producing several textbooks & monographs with most notable being The non-linear field theories of mechanics by C. Truesdell & W. Noll.

For more modern exposition see the paper:

From the introduction:

This paper is intended to serve as a model for the first few chapters of future textbooks on continuum mechanics and continuum thermomechanics. It may be considered an update of the paper Lectures on the Foundations of Continuum Mechanics and Thermodynamics [N2] by one of us (W.N.), published in 1973,and an elaboration of topics treated in Part 3, entitled Updating the Non-Linear Field Theories of Mechanics, of the booklet [FC] by W.N.$${}^1$$.

The present paper differs from most existing textbooks on the subject in several important respects:

1. It uses the mathematical infrastructure based on sets, mappings, and families, rather than the infrastructure based on variables, constants, and parameters. (For a detailed explanation, see The Conceptual Infrastructure of Mathematics by W.N. [N1].)
2. It is completely coordinate-free and $$\mathbb{R}^n$$-free when dealing with basic concepts.
3. It does not use a fixed physical space. Rather, it employs an infinite variety of frames of reference, each of which is a Euclidean space. The motivation for avoiding physical space can be found in Part 1, entitled On the Illusion of Physical Space, of the booklet [FC]. Here, the basic laws are formulated without the use of a physical space or any external frame of reference.
4. It considers inertia as only one of many external forces and does not confine itself to using only inertial frames of reference. Hence kinetic energy, which is a potential for inertial forces, does not appear separately in the energy balance equation. In particle mechanics, inertia plays a fundamental role and the subject would collapse if it is neglected. Not so in continuum mechanics, where it is often appropriate to neglect inertia, for example when analyzing the motion of toothpaste when it is extruded slowly from a tube.

See also PhD thesis Frame-Free Thermomechanics (2010) by Seguin and other papers (including those referenced in the above quote) at Noll's webpage.

• Nolls stuff is really interesting I like his approach. The link you put in seems only quasi coordinate free. It says it's c free only for basic objects. Jun 6, 2020 at 14:44
• The first link that is Jun 6, 2020 at 14:46
• The 2-nd list item of quote should be read as It is (completely coordinate-free) and ($\mathbb{R}^n$-free when dealing with basic concepts) : the basic concepts bit refers to $\mathbb{R}^n$ independence, at some point they do assume Euclidean spaces, theory remains coordinate-free even after that. Jun 6, 2020 at 15:13
• Euclidean space is not the same thing as Rn. I do appreciate the abstraction, but I would like to see how this formalism applies to some concrete examples. Otherwise I can't see the point of all this structure. One of the main reasons I asked for coord free is for more simplicity. For example by a tensor I simply mean some linear function of some sort of vectors.. In theory these papers seem nice,but I don't see how this apparent loss of simplicity is justified practically. Jun 6, 2020 at 15:57
• My impression is that higher level of abstraction could be used to simplify description of complicated materials: nematic liquid crystals, materials with memory etc. Jun 6, 2020 at 16:29

The infinitessimal strain tensor is defined by $$\textstyle{\frac 12} {\mathcal L}_{\boldsymbol \eta} {\bf g}$$ where $${\bf g}$$ is the usual metric of our 3-d euclidean world. Here $${\mathcal L}_{\boldsymbol \eta}$$ is the Lie derivative with respect to the displacement vector field $${\boldsymbol \eta}$$. For large displacements that take a point $${\bf r}$$ to $$\phi({\bf r})$$ we define the finite strain as $$\textstyle{\frac 12}( \phi^*({\bf g})-{\bf g})$$. Here $$\phi^*{\bf g}({\bf x},{\bf y})= {\bf g}(\phi_*({\bf x}),\phi_*({\bf y}))$$. In other words take two small displacements $${\bf x}$$, $${ \bf y}$$ in the undeformed material and take their inner product. Now deform the material so that the displacement vectors get moved (possibly a long way) and stretched and rotated to (still small) displacements $$\phi_*({\bf x})$$ and $$\phi_*({\bf y})$$. Take their new inner product (in our ambiant 3-space). The difference between the original inner product and the one of the deformed vectors defines the finite strain tensor $${\bf e}$$ evaluated on $${\bf x},{\bf y}$$ . None of the these concepts need a coordinate system.

• OK this makes sense. Something along the lines of what I had in mind. So let's say than for example cauch stress tensor. It would be some linear function that would take in vectors(functions, displacements). What are some practical or real world examples of this. Without having to resort to coordinates? Jun 6, 2020 at 14:43