TITLE:
Finite Element Processes Based on GM/WF in Non-Classical Solid Mechanics
AUTHORS:
K. S. Surana, R. Shanbhag, J. N. Reddy
KEYWORDS:
Non-Classical Continua, Polar Continua, Lagrangian Description, Internal Rotations, Galerkin Method with Weak Form
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.7 No.3,
September
5,
2017
ABSTRACT: In non-classical thermoelastic solids incorporating
internal rotation and conjugate Cauchy moment tensor the mechanical deformation
is reversible. This suggests that within the realm of linear mathematical
models that only consider small strains and small deformation the mechanical
deformation is reversible. Hence, it is possible to recast the conservation and
balance laws along with constitutive theories in a form that adjoint A* of the differential operator A in
mathematical model is same
as the differential operator A. This
holds regardless of whether we consider an initial value problem (IVP) (when
the integrals over open boundary are neglected) or boundary value problem
(BVP). Thus, in such cases Galerkin method with weak form (GM/WF) for BVPs and
space-time Galerkin method with weak form (STGM/WF) for IVPs are highly meritorious
due to the fact that: 1) the integral form for BVPs is variationally consistent (VC) and 2) the space-time
integral forms for IVP are space time variationally consistent (STVC). The
consequence of VC and STVC integral forms is that the resulting coefficient
matrices are symmetric and positive definite ensuring unconditionally stable
computational processes for both BVPs and IVPs. Other benefits of GM/WF and
space-time GM/WF are simplicity of specifying boundary conditions and initial
conditions, especially traction boundary conditions and initial conditions on
curved boundaries due to self-equilibrating nature of the sum of secondary
variables that only exist in GM/WF due to concomitant. In fact, zero traction
conditions are automatically satisfied in GM/WF, hence need not be specified at
all. While VC and STVC feature also exists in least squares process (LSP) and
space-time least squares finite element processes (STLSP) for BVPs and IVPs,
the ease of specifying traction boundary conditions feature in GM/WF and
STGM/WF is highly meritorious compared to LSP and STLSP in which zero traction
conditions need to be explicitly specified. A disadvantage of GM/WF and STGM/ WF is that the mathematical models (momentum
equations) needed in the desired form contain higher order derivatives of
displacements (upto fourth order), hence necessitate use of higher order spaces
in their solution. As well known, this problem can be easily overcome in LSP
and STLSP by introduction of auxiliary equations and auxiliary variables, thus
keeping the highest orders of the derivatives of the dependent variables to one
or any other desired order. A serious disadvantage of this approach in LSP is
the significant increase in the number of dependent variables, hence poor
computational efficiency. In this paper we consider non-classical continuum
models for internally polar linear elastic solids in which internal rotations
due to displacement gradient tensor (hence internal polar physics) are
considered in the conservation and the balance laws and the constitutive theories.
For simplicity, we only consider isothermal case; hence energy equation is not
part of mathematical model. When using mathematical models derived in
displacements in GM/WF and LSP in constructing integral forms, we note that in
GM/WF the number of dependent variables is reduced drastically (only three in R3), whereas in case of first order systems used in
LSP and STLSP we may have as many as 22 dependent variables for isothermal
case. Thus, GM/WF results in dramatic improvement in computational efficiency
as well as accuracy when minimally conforming spaces are used for
approximations. In this paper we only consider mathematical model in R2 for BVPs (for
simplicity). Mathematical models for IVP and BVP in R3 will be considered in
subsequent paper. The integral form is derived in R2 using GM/WF. Numerical
examples are presented using GM/WF and LSP to
demonstrate advantages of finite element process derived using integral
form based on GM/WF for non-classical linear theories for solids incorporating
internal rotations due to displacement gradient tensor.