Tensorial Deformation Measures for Flexible Joints

Flexible joints, sometimes called bushing elements or force elements, are found in all multibody dynamics codes. In their simplest form, flexible joints simply consist of sets of three linear and three torsional springs placed between two nodes of a multibody system. For infinitesimal deformations, the selection of the lumped spring constants is an easy task, which can be based on a numerical simulation of the joint or on experimental measurements. If the joint undergoes finite deformations, the identification of its stiffness characteristics is not so simple, especially if the joint itself is a complex system. When finite deformations occur, the definition of deformation measures becomes a critical issue. Indeed, for finite deformation, the observed nonlinear behavior of materials is partly due to material characteristics and partly due to kinematics. This paper focuses on the determination of the proper finite deformation measures for elastic bodies of finite dimension. In contrast, classical strain measures, such as the Green–Lagrange or Almansi strains, among many others, characterize finite deformations of infinitesimal elements of a body. It is argued that proper finite deformation measures must be of a tensorial nature, i.e., must present specific invariance characteristics. This requirement is satisfied if and only if the deformation measures are parallel to the eigenvector of the motion tensor.