USE OF QR KINEMATICS IN RHEOMETRY

Abstract

Traditionally solutions of rheometry problems were obtained using convected, curvilinear coordinate system. It was shown that the use of QR kinematics in rheometry provides an alternate, yet simpler, approach to the rheometry problem by eliminating the use of convected coordinate and replacing it with orthonormal coordinate system. This orthonormal coordinate system arises from Gram-Schmidt factorization of the deformation gradient which offers significant computational advantages. Furthermore, this kinematic approach facilitates the development of a constitutive model which relates normal stress differences with shear stress using the components of Laplace stretch matrix. This solution approach was applied to a parallel plate rheometer to obtain rheological properties of a material. QR kinematics can be extensively used in rheology due to its ease of application from an experimenter’s standpoint. However, since different experimental setups are more amenable to different co-ordinate systems, it is important to derive and adopt transformation rules that help us navigate from one co-ordinate system to another with ease. Here the quotient rules for transforming vectors and tensors between different co-ordinate systems obtained from a QR decomposition of the deformation gradient has been derived. The use of these quotient rules in comparing results obtained from two different rheometry methods, viz., a cone and plate and a parallel plate rheometer has also been demonstrated.