Visualization of Material Rotation in Simple Shear: An Operator-Based Approach in Continuum Mechanics Implemented With Computer Graphics

Abstract

In this paper, we develop a physically meaningful and visually grounded framework for understanding mechanics of deformable media by combining the operator-based formulation of tensor calculus with computer graphics. The resulting images presented can be used as a powerful educational tool for teaching nonlinear continuum mechanics and, additionally, these images are able to accurately depict complex geometric transformations induced by finite deformations. Being generated by specialized programs, they cannot be reproduced manually.  We believe that a proper understanding of the physical meaning of tensors requires moving beyond their definition as simple “matrices of coefficients”, because matrices are only the components of a tensor that arise when it is expressed in terms of the dyadic products of basis vectors in a chosen coordinate system. Matrices are mainly used in computing. However, the true physical meaning of tensors becomes clear when they are interpreted as linear operators acting on vectors in three-dimensional Euclidean space. The operator-based approach lies at the core of the present work. As a representative example, we consider the deformation of a material under simple shear. The testing apparatus is fixed, and the rigid plates used to apply shear deformation move translationally, without any rotation. Nevertheless, each infinitesimal material element undergoes a rigid-body-like rotation. Although this behavior is well known in the context of finite deformation theory, it appears paradoxical to researchers accustomed to small strains. The physical interpretation of tensors in combination with computer-generated visualizations helps resolve this apparent contradiction. The paper clearly demonstrates the pedagogical advantages of the operator-based approach for teaching and understanding nonlinear theories of finite deformations.