A framework for nonlinear viscoelasticity on the basis of logarithmic strain and projected velocity gradient

Abstract

Most existing nonlinear viscoelastic models are founded on the Finger tensor and its evolution during deformation and flow. In this paper, a new framework for nonlinear viscoelasticity on the basis of a projected velocity gradient tensor is presented. The pairing of the logarithmic strain tensor with the projected velocity gradient tensor is considered the cornerstone of the proposed formulation. The resulting linear relation for strain evolution in affine deformation provides a facile passage to connect small deformation mechanics with large deformation mechanics. Accordingly, proven linear viscoelastic models may now be extended to large deformation. Another salient feature is that the deformation is decoupled into stretching and rotation and each of them has its own evolution equation. The relaxation process is considered coaxial, whereas rotational retardation is included in the formulation to tackle with shear-related rotational softening. In this way, the similarity and difference between rotational deformation and coaxial deformation are explicitly explored. Strain shape functions are added to the linear version of the model to render more realistic nonlinear effects. The overall framework uses three elements for model formulation, namely, a stretch evolution equation, a rotation evolution equation, and a stress law. Example models are provided to illustrate the proper combination of these elements for creation of useful models. A particular model with a fractional relaxation process is used for model testing against typical observations in simple shear. With five model parameters (one for fractionality, two for linear viscoelasticity, one for straining, and the last one for rotation) is able to fit startup shear viscosity of a polystyrene solution in high accuracy. With additional understanding of the role of entropic strain in the relaxation process, simple and unified constitutive equations for modeling general 3D viscoelastic deformation and flow may be developed.Most existing nonlinear viscoelastic models are founded on the Finger tensor and its evolution during deformation and flow. In this paper, a new framework for nonlinear viscoelasticity on the basis of a projected velocity gradient tensor is presented. The pairing of the logarithmic strain tensor with the projected velocity gradient tensor is considered the cornerstone of the proposed formulation. The resulting linear relation for strain evolution in affine deformation provides a facile passage to connect small deformation mechanics with large deformation mechanics. Accordingly, proven linear viscoelastic models may now be extended to large deformation. Another salient feature is that the deformation is decoupled into stretching and rotation and each of them has its own evolution equation. The relaxation process is considered coaxial, whereas rotational retardation is included in the formulation to tackle with shear-related rotational softening. In this way, the similarity and difference between rotational...