A finite deformation, finite strain nonclassical internal polar continuum theory for solids

ABSTRACT

A nonclassical internal polar continuum theory for finite deformation and finite strain for isotropic, homogeneous compressible and incompressible solids is presented in this paper. Since the Jacobian of deformation $\mathbf{J}$ is a fundamental measure of deformation in solid continua, $\mathbf{J}$ in its entirety must be incorporated in the thermodynamic framework. Polar decomposition of $\mathbf{J}$ into right stretch tensor ${\mathbf{S}}_{\mathbf{r}}$ and pure rotation tensor $\mathbf{R}$ shows that the entirety of $\mathbf{J}$ implies the entirety of ${\mathbf{S}}_{\mathbf{r}}$ and $\mathbf{R}$. The classical continuum theories for isotropic and homogeneous solid continua are based only on ${\mathbf{S}}_{\mathbf{r}}$. The influence of $\mathbf{R}$ on the thermodynamic framework is ignored altogether. The purpose of this research is to present a thermodynamic framework for finite deformation and finite strain of solids that incorporates complete deformation physics described by $\mathbf{J}$. This can be accomplished by incorporating the additional physics due to $\mathbf{R}$ in the current theories as these theories already contain the physics due to ${\mathbf{S}}_{\mathbf{r}}$. We note that the rotation tensor $\mathbf{R}$ results due to deformation of solid continua, hence arises in all deforming solids. Thus, this theory can be referred to as internal polar nonclassical theory for solid continua. The use of internal polar nonclassical is appropriate as the theory considers internal rotations. When the varying internal rotations and the rotation rates are resisted by the solid continua, conjugate internal moments, which together with rotations and rotation rates can result in additional energy storage, dissipation and memory. The objective of the nonclassical continuum theory presented here is to present a new thermodynamic framework for solid continua with finite deformation and finite strain that is consistent with the complete deformation physics in isotropic, homogeneous continua, which necessitates that $\mathbf{J}$ in its entirety must form the basis for derivation of conservation and balance laws and constitutive theories.

Keywords:

• Finite deformation
• internal polar continuum theory
• Jacobian of deformation
• non-classical continuum theory
• polar decomposition
• rotation tensor
• solid continua
• stretch tensor

Acknowledgments

The first and third authors are grateful for the support provided by their endowed professorships during the course of this research. The facilities provided by the Computational Mechanics Laboratory (CML) of the Mechanical Engineering department of the University of Kansas is gratefully acknowledged. The financial support provided to the second author by the Naval Air Warfare Center is greatly appreciated.