What is a coordinate-free formulation of deformation theory?

Question

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.

Answer 1

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:

• Noll, W., & Seguin, B. (2010). Basic concepts of thermomechanics. Journal of Elasticity, 101(2), 121-151, doi:10.1007/s10659-010-9253-x, free pdf.

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.

Answer 2

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.