Finite Element Processes Based on GM/WF in Non-Classical Solid Mechanics

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 How to cite this paper: Surana, K.S., Shanbhag, R. and Reddy, J.N. (2017) Finite Element Processes Based on GM/WF in Non-Classical Solid Mechanics. American Journal of Computational Mathematics, 7, 321-349. https://doi.org/10.4236/ajcm.2017.73024 Received: June 29, 2017 Accepted: August 31, 2017 Published: September 5, 2017 Copyright © 2017 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access


Introduction Literature Review, and Scope of Work
In Lagrangian description of deforming matter, the Jacobian of deformation is a fundamental quantity of the measure of deformation of the solid continua. In general, the Jacobian of deformation varies between material points, i.e. it varies between a material point and its neighbors. Polar decomposition of the Jacobian of deformation at material points into stretch (left of right) and pure rotation shows that if the Jacobian of deformation varies between a material point and its neighbors so do the rotations. We could also consider the decomposition of the displacement gradient tensor into symmetric and skew symmetric tensors. The skew symmetric tensor is a measure of pure rotations while the symmetric tensor is a measure of strains. Strain measures (such as Green's strain) are purely a function of stretch tensor or alternatively symmetric part of the displacement In non-polar continuum theories, only conjugate stress and strain tensors contribute to the stored energy in the deforming solid continua. Likewise, the dissipation mechanism is purely due to stress tensor and rates of conjugate strain tensor. In such theories, the influence of rotations and the influence of the rates of rotations on the mechanism of energy storage and dissipation is not The theory used here is a continuum theory in Lagrangian description for polar continuum and should not be confused with micropolar continuum theories [1]- [11] that are designed to accommodate effects at scales smaller than the continuum scale. Micropolar continuum theories require definitions of additional strain measures [6] related to micromechanics. The polar continuum theory used here incorporates standard measures of strains as used currently in non-polar continuum theories. In the polar continuum theory used here, the motivation is to account for the influence of varying rotations at neighboring material points that arise during evolution as these may result in additional energy storage in some solid continua. Polar decomposition of the Jacobian of deformation at neighboring material points clearly substantiates this. An important point to note is that the theory considered here can only account for local rotation effects due to deformation at material points; hence the theory used here is intrinsically a local polar continuum theory, thus cannot account for nonlocal effects.
In the following we present a brief literature review on micropolar theories, nonlocal theories and stress couple theories. A comprehensive treatment of micropolar theories can be found in the works by Eringen [1]- [9]. The concept of couple stresses is presented by Koiter [10]. Balance laws for micromorphic materials are presented in reference [11]. The micropolar theories consider micro deformation due to micro constituents in the continuum. In references [12] [13] [14] by Reddy et al. and reference [15] by Zang et al. nonlocal theories are presented for bending, buckling and vibration of beams, beams with nanocarbon tubes and bending of plates. The nonlocal effects are believed to be incorporated due to the work presented by Eringen [6] in which definition of a nonlocal stress tensor is introduced through integral relationship using the product of macroscopic stress tensor and a distance kernel representing the nonlocal effects. The polar continuum theory for solid continua presented in this paper is strictly local and non-micropolar. The concept of couple stresses was introduced by Voigt in 1881 by assuming a couple or moment per unit area on the oblique plane of the deformed tetrahedron in addition to the stress or force per unit area. Since the introduction of this concept many published works have appeared. We cite some recent works, most of which are related to micropolar stress couple theories. Authors in reference [16] report experimental study of micropolar and couple stress elasticity of compact bones in bending. Conservation integrals in couple stress elasticity are reported in reference [17]. A microstructuredependent Timoshenko beam model based on modified couple stress theories is reported by Ma et al. [18]. Further account of couple stress theories in conjunction with beams can be found in references [19] [20] [21]. Treatment of rotation gradient dependent strain energy and its specialization to Von Kármán plates and beams can be found in reference [22]. Other accounts of micropolar elasticity and Cosserat modeling of cellular solids can be found in references [23] [24] [25]. We remark that in references [16]- [25], Lagrangian description is used for solid matter, however the mathematical descriptions are purely derived using strain energy density functional and principle of virtual work. This approach works well for elastic solids in which mechanical deformation is reversible.
Extension of these works to thermoviscoelastic solids with and without memory is not possible. In such materials the thermal field and mechanical deformation are coupled due to the fact that the rate of work results in rate of entropy production. In references [26]- [37] various aspects of the kinematics of micropolar theories, stress couple theories, etc. are discussed and presented including some applications to plates and shells.
If the varying rotations and their rates result in energy storage and dissipation, then their energy conjugate moment (shown later in the paper) must exist in the deforming matter. This necessitates the existence of moment (per unit area) on the oblique plane of the deformed tetrahedron. Thus, at the onset, we consider average force per unit area and displacements, and average moment per unit area and the rotations on the oblique plane of the deformed tetrahedron. The work used here [38]- [44] follows a strictly thermodynamic approach using these i.e., for polar solid continua we consider: (i) Conservation of mass and present reasons for not deriving conservation of inertia (ii) Balance of linear momenta (iii) Balance of angular momenta (iv) Balance of moments of moments (or couples) (v) First law of thermodynamics and (vi) Second law of thermodynamics in Lagrangian description in which stress and strain, moment and rotations are energy conjugate pairs. The mathematical description for polar solid continua used here is applicable to polar thermoelastic solids for small deformation and small strain. In the present work the mathematical model derived by Surana, et al. [38]- [44] in Lagrangian description for thermoelastic polar solids incorporating internal

Mathematical Model
For non-classical elastic solid matter with internal rotation and conjugate moment physics undergoing small deformation and small strain, the mathematical model for BVPs has been presented by Surana et al. [38]- [44]. In present work we assume isothermal deformation process i.e. no entropy production due to mechanical work, hence the mathematical model in Lagrangian description consists of balance of linear momenta, balance of angular momenta, balance of moments of moments (as a balance law or its absence) [38]- [45] and the constitutive theories for: symmetric part of Cauchy stress tensor, symmetric part of Cauchy moment tensor and antisymmetric part of Cauchy moment tensor (if balance of moments of moments is not used as a balance law). We have the following dimensionless form of the mathematical model in 2  (neglecting body forces) assuming that balance of moments of moments is not a balance law [45]. Using the decomposition of the Cauchy moment tensor into symmetric and antisymmetric tensors The Cauchy stress tensor has also been decomposed into symmetric and antisymmetric tensors. In order to obtain the dimensionless Equations If we consider (1)-(9) are a system of eleven partial differential equations in eleven dependent Equations ((12) and (13)) form the basis for finite element formulation based on GM/WF.
Equations (38) imply that the element equations constructed from (38) Substituting from (39) We note that 11

Approximation Spaces and Some Remarks
1) Since the mathematical model ( (12) and (13) (12) and (13)  to the fact that the differential operator in (12) and (13) is linear in displacements u and v and the adjoint * A of the differential operator A is same as the operator A (when the mathematical model is expressed in displacements u and v) 6) In the study of the model problem we chose 0 β = (based on the material presented in [45]) i.e. we consider balance of moments of moments as a balance law, hence the Cauchy moment tensor is symmetric.

A Least Squares Formulation in 2  (Plane Stress) Based on Residual Functional
We consider the following mathematical model (obtained using (1)-(10)) in the dimensionless form (in the absence of balance of moments of moments as a balance law [45]) consisting of first order partial differential equations.

Model Problems
In this section we can consider three model problems in

Simply Supported and Fixed-Fixed Plate: Model Problems 1 and 2
We consider a thin plate of length ˆ2 0 l = inches with width ˆ0 .5 h = inches and thickness ˆ0 .1 t = inches. With 0 10 L = inches, the dimensionless plate is 2 0.05 0.01 L h t × × = × × . Figure 1(a) and Figure 1(b) show schematics of the plate, boundary conditions and loading for the formulation based on GM/WF for both simply supported and fixed-fixed boundary conditions. The load is applied over a length of 0.4 b = as forces at the nodes that corresponds to uniform stress in the y-direction (see Figure 1). Figure 2(a) and Figure 2 show same schematics with BCs and loading used in least squares formulation. In all numerical studies the plates are discretized using a 20 element uniform discretization (10 elements along the length and two elements width b) using a nine node p-version hierarchical higher order global differentiability finite elements. In all computations we choose Poisson's ratio of 0.3, C in x and y) with 5 p = . For this choice of k, integrals over the spatial discretization are Riemann.
2) For GM/WF we also consider 2 k = (solutions of class 1 C in x and y) with 7 p = . For this choice of k integrals over the spatial discretization are in Lebesgue sense.
3) Since the solution for LSP yields residual functional values of the order of ( ) 15 10 O − or lower, comparing the computed solutions from (1) and (2) we American Journal of Computational Mathematics confirm that when both solutions are almost indistinguishable from each other, the solution from GM/WF has good accuracy.

4) For Least squares formulation we consider solutions of class 0
C in x and y with p-level of nine [40] [45].

A Square Plate with a Circular Hole: Model Problem 3
We consider a 6" × 6" square plate of thickness 0.1" with a 0.48" diameter circular hole at the center. We use 0 1.  [57] [67]) with p-level of 7. For this choice of k, the integrals over the discretization are Lebesgue, but due to smoothness of the solution we can expect these solutions to converge to class 2 C in the weak sense.

Results and Discussion
The stresses s xx σ and s yy

General Remarks
In this section we make some remarks related to the two finite element formulations (GM/WF, LSP) in context with the numerical studies presented here.
1) It is obvious that for the model problems (in 2  ) the GM/WF has only two dependent variables u and v whereas LSP based on first order system of PDEs has nine dependent variables resulting in enormous computational inefficiency but permitting flexibility that permits 0 C local approximations.
2) In LSP there is no concept of secondary variables as in GM/WF, hence there are no self equilibrating quantities in LS finite element formulation. As a result, all dependent variables pertaining to the known physics, even those that American Journal of Computational Mathematics are zero, must be specified on the boundaries of the domain. For example in GM/WF stress free boundaries are automatically satisfied due to sum of secondary variables being zero at a node. Same is true for moment free boundaries. However, in LSP all boundary information must be defined in the problem data. Figure   1(a), Figure 1

Summary and Conclusions
In this paper the mathematical model consisting of conservation and balance laws in Lagrangian description for non-classical continuum theory for elastic solids (small strain small deformation physics without dissipation and memory) incorporating internal rotation physics due to displacement gradient tensor is considered (derived in reference [45]). In such solids the deformation physics due to mechanical work is reversible; hence the differential operator A in this mathematical model when expressed purely in terms of displacements is such that the adjoint * A of the differential operator A is same as A . Thus, in such mathematical models GM/WF is ideal for the finite element formulation of the corresponding BVPs. We make the following specific remarks and observations and draw some conclusions from the work presented in this paper.
1) GM/WF is ideal for reversible processes as in the present case. In such mathematical models * = A A holds.
2) LSP with first order system of PDEs is computationally non competitive with GM/WF. In the work presented here GM/WF has only two dependent variables whereas LSP has nine.
3) An important question is "could we have used LSP" for the mathematical model in displacements u and v derived for GM/WF. Of course we could but: (1) this would require solutions of class 4 C or of class 3 C for sure (2)