Necessity of Additional Balance Law and the Importance of GM/WF in Non-Classical Continuum Theories for Solid Continua

Abstract

In non-classical continuum theories for solid continua incorporating the internal rota- tions due to displacement gradient tensor and conjugate moment tensor in the math- ematical description of the deformation physics, the fundamental question “are the conservation and balance laws derived for classical continuum theories sufficient to ensure equilibrium of the deforming solid matter” is addressed in the present work. It is shown that in the non-classical theories considered here for solid continua, an ad- ditional balance law, balance of moments of moments is required to ensure equilibrium of the deforming matter due to presence of additional physics of internal rotations and conjugate moment tensor that are absent in the classical continuum theories. Ther- moelastic non-classical solids with small deformation and small strain is used as an example to derive theoretical details as well as to present three model problems stud- ies in plane non-classical elasticity. The second part of the research presented here considers finite element processes based on GM/WF for non-classical solid continua with small strain and small deformation in which the deformation due to mechanical work is reversible. Using plane non-classical elasticity as an example, it is shown that when the conservation and balance laws are cast purely in terms of displacements in Lagrangian description, the balance of lin- ear momenta results in fourth order partial differential equations in displacements in which adjoint A ∗ of the differential operator A is same as A . A finite element formu- lation of the PDEs in displacements is constructed using GM/WF in which the inte- gral form is variationally consistent and is compared with least squares finite element formulation constructed for a first order system of PDEs in which the integral form is also variationally consistent. The meritorious features and advantages of GM/WF over least squares process are demonstrated. Model problem studies are also presented to illustrate these features.