# Rheology

Rheology, as it is defined today, is a study of the deformation and flow of matter. A classic Newtonian fluid is described by Newton’s law, which relates the stress to the strain rate via a constant of proportionality, called viscosity. For Newtonian fluids, the viscosity is constant. Whenever the viscosity becomes a function of strain rate, we have what researchers refer to as a non-Newtonian fluid. Typical examples of non-Newtonian fluids are: polymer solutions, thermo\-plastics, drilling fluids, paints, fresh concrete, and biological fluids, to name a few. One of the most striking properties of non-Newtonian fluids is that they exhibit normal-stress differences. Non-Newtonian fluids display many effects that are not seen in Newtonian fluids, such as shear-thickening, shear-thinning, thixotropy, rheoplexy, extrudate swell, and normal-stress differences, among many others (Paul et al. 2021).

### Characterization of normal stress differences for viscoelastic solids

In standard rheometry, these normal-stress differences are measured using a parallel plate rheometer in conjunction with a cone and plate rheometer. When used together, these rheometers allow for the accurate measurement of normal-stress differences. The cone and plate setup for a rheometer is especially advantageous, because it produces a constant state of stress at the cone surface.  The need to use both rheometers becomes problematic whenever one has a soft material that is more of a viscoelastic solid as opposed to a viscoelastic liquid. This prohibits the use of a cone and plate setup, and thus, accurate measurement of the normal-stress differences is not feasible. In this paper, we introduce an alternative method for the calculation of both normal-stress differences by adopting a Gram-Schmidt or QR decomposition of the deformation gradient (Paul et al. 2021).

Relevant Publications

1. Paul, Sandipan, Alan D. Freed, and Chandler C. Benjamin. “Application of the Gram–Schmidt factorization of the deformation gradient to a cone and plate rheometer.” Physics of Fluids 33.1 (2021): 017113.

Publications by Collaborators

1. Paul, Sandipan, and Alan D. Freed. “A simple and practical representation of compatibility condition derived using a QR decomposition of the deformation gradient.” Acta Mechanica 231 (2020): 3289-3304.
2. Paul, Sandipan, and Alan D. Freed. “Characterizing geometrically necessary dislocations using an elastic–plastic decomposition of Laplace stretch.” Zeitschrift für angewandte Mathematik und Physik 71.6 (2020): 1-22.