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?