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**

- 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__

- 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. - 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.