What is the spatial derivative of strain tensor?

Question

I'm looking for a metric that contains enough information of a 3D solid's deformation: stretch, bend and twist, etc. Stretch, bend and twist may be ambiguous for a 3D solid, since unlike a rod, it doesn't have a curve direction. Let's assume that we've found three directions, along which the deformations are independent and the combined deformation of the three directions is the total deformation of the solid. (Do the three directions even exist? I'm still looking for some references on this.)

Strain tensor only describes stretch at a material point, because it's the same order as the gradient of displacement. It needs the strain at neighbor positions to measure other deformation behavior. I'm thinking of taking the spatial derivative of the strain tensor and the resulting 3-order tensor may contains the information I want.

My question is: what does every entry of the "strain gradient" mean? Is it that some of them measure stretching, some measure bending and some others measure twisting? Does anyone know any books that may cover this topic?

Thanks!

Answer 1

What you are describing to me in the lead-up paragraph (as opposed to the discussion about the derivative of the strain tensor) reminds me strongly of Cosserat/micropolar elasticity theories. The classical theory of the continuum tries to capture only local displacements. This is the content of the strain tensor. It allows us to measure direction-dependent stretching and compressing of the material. The Cosserat/micropolar theory of elasticity adds to it a measure of local rotation. In three dimensions this means that every point has three additional degrees of freedom. (In differential geometric language, they add torsion to the connection.)

You can find more about these theories here and here.

One can generalise the idea even further. By considering at each point not only the point by also its tangent space (or rather, in the Cartan picture where we consider attached to each point three privileged "unit" vectors), we can consider the full generality of micromorphic continua, which admits 6 more degrees of freedom: infinitesimal volume stretching and infinitesimal shear of the attached frame. A modern reference for these ideas can be found in Eringen's Microcontinuum field theories.