arXiv subjects

Gary F. Dargush

Publications and source records attributed to Gary F. Dargush.

At least 19 recordsLinked to original sources

arXiv2024

A Real-Valued Description of Quantum Mechanics with Schrodinger's 4th-order Matter-Wave Equation

Using a variational formulation, we show that Schrodinger's 4th-order, real-valued matter-wave equation which involves the spatial derivatives of the potential V(r), produces the precise eigenvalues of Schrodinger's 2nd-order, complex-valued matter-wave equation together with an equal number of negative, mirror eigenvalues. Accordingly, the paper concludes that there is a real-valued description of non-relativistic quantum mechanics in association with the existence of negative (repelling) energy levels. Schrodinger's classical 2nd-order, complex-valued matter-wave equation which was constructed upon factoring the 4th-order, real-valued differential operator and retaining only one of the two conjugate complex operators is a simpler description of the matter-wave, since it does not involve the derivatives of the potential V(r), at the expense of missing the negative (repelling) energy levels.

quant-ph↗

arXiv2021

Variational principles and finite element Bloch analysis in couple stress elastodynamics

We address the numerical simulation of periodic solids (phononic crystals) within the framework of couple stress elasticity. The additional terms in the elastic potential energy lead to dispersive behavior in shear waves, even in the absence of material periodicity. To study the bulk waves in these materials, we establish an action principle in the frequency domain and present a finite element formulation for the wave propagation problem related to couple stress theory subject to an extended set of Bloch-periodic boundary conditions. A major difference from the traditional finite element formulation for phononic crystals is the appearance of higher-order derivatives. We solve this problem with the use of a Lagrange-multiplier approach. After presenting the variational principle and general finite element treatment, we particularize it to the problem of finding dispersion relations in elastic bodies with periodic material properties. The resulting implementation is used to determine the dispersion curves for homogeneous and porous couple stress solids, in which the latter is found to exhibit an interesting bandgap structure.

cond-mat.mtrl-sci↗

arXiv2021

MCBTE: A variance-reduced Monte Carlo solution of the linearized Boltzmann transport equation for phonons

MCBTE solves the linearized Boltzmann transport equation for phonons in three dimensions using a variance-reduced Monte Carlo solution approach. The algorithm is suited for both transient and steady-state analysis of thermal transport in structured materials with size features in the nanometer to hundreds of microns range. The code is portable and integrated with both first-principles density functional theory calculations and empirical relations for the input of phonon frequency, group velocity, and mean free path required for calculating the thermal properties. The program outputs space- and time-resolved temperature and heat flux for the transient study. For the steady-state simulations, the frequency-resolved contribution of phonons to temperature and heat flux is written to the output files, thus allowing the study of cumulative thermal conductivity as a function of phonon frequency or mean free path. We provide several illustrative examples, including ballistic and quasi-ballistic thermal transport, the thermal conductivity of thin films and periodic nanostructures, to demonstrate the functionality and to benchmark our code against available theoretical/analytical/computational results from the literature. Moreover, we parallelize the code using the Matlab Distributed Computing Server, providing near-linear scaling with the number of processors.

cond-mat.mtrl-sci↗

arXiv2019

Finite element modeling of micropolar-based phononic crystals

The performance of a Cosserat/micropolar solid as a numerical vehicle to represent dispersive media is explored. The study is conducted using the finite element method with emphasis on Hermiticity, positive definiteness, principle of virtual work and Bloch-Floquet boundary conditions. The periodic boundary conditions are given for both translational and rotational degrees of freedom and for the associated force- and couple-traction vectors. Results in terms of band structures for different material cells and mechanical parameters are provided.

physics.comp-ph↗

arXiv2018

An assessment of higher gradient theories from a continuum mechanics perspective

In this paper, we investigate the inherent physical and mathematical character of higher gradient theories, in which the strain or distortion gradients are considered as the fundamental measures of deformation. Contrary to common belief, the first or higher strain or distortion gradients are not proper measures of deformation. Consequently, their corresponding energetically conjugate stresses are non-physical and cannot represent the state of internal stresses in the continuum. Furthermore, the governing equations in these theories do not describe the motion of infinitesimal elements of matter consistently. For example, in first strain gradient theory, there are nine governing equations of motion for infinitesimal elements of matter at each point; three force equations, and six unsubstantiated artificial moment equations that violate Newton's third law of action and reaction and the angular momentum theorem. This shows that the first strain gradient theory (F-SGT) is not an extension of rigid body mechanics, which then is not recovered in the absence of deformation. The inconsistencies of F-SGT and other higher gradient theories also manifest themselves in the appearance of strains, distortions or their gradients as boundary conditions and the requirement for many material coefficients in the constitutive relations.

physics.gen-ph↗

arXiv2018

Fundamental governing equations of motion in consistent continuum mechanics

We investigate the consistency of the fundamental governing equations of motion in continuum mechanics. In the first step, we examine the governing equations for a system of particles, which can be considered as the discrete analog of the continuum. Based on Newton's third law of action and reaction, there are two vectorial governing equations of motion for a system of particles, the force and moment equations. As is well known, these equations provide the governing equations of motion for infinitesimal elements of matter at each point, consisting of three force equations for translation, and three moment equations for rotation. We also examine the character of other first and second moment equations, which result in non-physical governing equations violating Newton's third law of action and reaction. Finally, we derive the consistent governing equations of motion in continuum mechanics within the framework of couple stress theory. For completeness, the original couple stress theory and its evolution toward consistent couple stress theory are presented in true tensorial forms.

physics.gen-ph↗

arXiv2018

Consistent continuous defect theory

By investigating the benefits and shortcomings of the existing form of continuous defect theory (CDT) and using recent advancements in size-dependent continuum mechanics, we develop a fully coherent theoretical framework, denoted as Consistent Continuous Defect Theory (C-CDT). Among several important potential applications, C-CDT may provide a proper foundation to study the continuum theory of crystal plasticity. The development presented here includes an examination of the character of the bend-twist tensor, Weingarten's theorem, Burgers and Frank vectors, continuous dislocation and disclination density tensors, and the dualism between geometry and statics of CDT based on couple stress theory (CST). Then, by using Consistent Couple Stress Theory (C-CST), the new C-CDT is derived in a totally systematic manner. In this development, the geometry of C-CDT is dual to the statics formulation in C-CST. Previously, the fundamental step in the creation of C-CST was recognizing the skew-symmetric character of the couple-stress tensor, which requires the skew-symmetrical part of the bend-twist tensor to be the additional measure of deformation in size-dependent continuum mechanics. Via Weingarten's theorem and arguments from C-CST, we establish that in defect theory the dislocation density tensor must be skew-symmetric and thus can be represented by an equivalent dislocation density vector. In addition, we investigate the character of a classical version of C-CDT with unexpected consequences. For full consistency, there can be no continuous dislocation density tensor within classical continuum mechanics, and the continuous disclination density tensor becomes symmetric. This clearly is analogous to the absence of couple-stresses and the symmetry of force-stresses in classical continuum mechanics.

physics.gen-ph↗

arXiv2017

Size-dependent couple stress Timoshenko beam theory

In this paper, a new size-dependent Timoshenko beam model is developed based on the consistent couple stress theory. In the present formulation, the governing equations and corresponding boundary conditions are obtained. Afterwards, this formulation is used to investigate size-dependency for several elementary beam problems. Analytical solutions are obtained for pure bending of a beam and for a cantilever beam with partially and fully clamped boundary conditions. These analytical results are then compared to the numerical results from a two-dimensional finite element formulation for the corresponding couple stress continuum problem.

physics.gen-ph↗

arXiv2017

Integral representation for three-dimensional steady state size-dependent thermoelasticity

Boundary element methods provide powerful techniques for the analysis of problems involving coupled multi-physical response, especially in the linear case for which boundary-only formulations are possible. This paper presents the integral equation formulation for size-dependent linear thermoelastic response of solids under steady state conditions. The formulation is based upon consistent couple stress theory, which features a skew-symmetric couple-stress pseudo-tensor. For general anisotropic thermoelastic material, there is not only thermal strain deformation, but also thermal mean curvature deformation. Interestingly, in this size-dependent multi-physics model, the thermal governing equation is independent of the deformation. However, the mechanical governing equations depend on the temperature field. First, thermal and mechanical weak forms and reciprocal theorems are developed for this general size-dependent thermoelastic theory. Then, an integral equation formulation for the three-dimensional isotropic case is derived, along with the corresponding singular infinite space fundamental solutions or kernel functions. Remarkably, for isotropic materials within this theory, there is no thermal mean curvature deformation, and the thermoelastic effect is solely the result of thermal strain deformation. As a result, the size-dependent behavior is specified entirely by a single characteristic length scale parameter, while the thermal coupling is defined in terms of the thermal expansion coefficient, as in the classical theory of steady state isotropic thermoelasticity. This simplification permits the development of the required kernel functions from previously defined fundamental solutions for isotropic media.

physics.class-ph↗

arXiv2016

Couple stress theories: Theoretical underpinnings and practical aspects from a new energy perspective

In this paper, we examine theoretical and practical aspects of several versions of couple stress theory. This includes indeterminate Mindlin-Tiersten-Koiter couple stress theory (MTK-CST), indeterminate symmetric modified couple stress theory (M-CST) and determinate skew-symmetric consistent couple stress theory (C-CST). We observe that MTK-CST and M-CST not only suffer from inconsistencies, these theories also cannot describe properly several elementary deformations, such as pure torsion of a circular bar and pure bending of a plate. By using an energy method, we also demonstrate another aspect of the inconsistency of the indeterminate MTK-CST and M-CST for elastic solids. This is achieved by deriving the governing equilibrium equations for elastic bodies in MTK-CST, M-CST and C-CST. This development shows that the direct minimization of the total potential energy for MTK-CST and M-CST violates the divergence free compatibility condition of the rotation vector field. On the other hand, the direct minimization of the total potential energy for C-CST satisfies this compatibility condition automatically. This result demonstrates another aspect of the inner consistency of C-CST.

physics.gen-ph↗

arXiv2016

Pure plate bending in couple stress theories

In this paper, we examine the pure bending of plates within the framework of modified couple stress theory (M-CST) and consistent couple stress theory (C-CST). In this development, it is demonstrated that M-CST does not describe pure bending of a plate properly. Particularly, M-CST predicts no couple-stresses and no size effect for the pure bending of the plate into a spherical shell. This contradicts our expectation that couple stress theory should predict some size effect for such a deformation pattern. Therefore, this result clearly demonstrates another inconsistency of indeterminate symmetric modified couple stress theory (M-CST), which is based on considering the symmetric torsion tensor as the curvature tensor. On the other hand, the fully determinate skew-symmetric consistent couple stress theory (C-CST) predicts results for pure plate bending that tend to agree with mechanics intuition and experimental evidence. Particularly, C-CST predicts couple-stresses and size effects for the pure bending of the plate into a spherical shell, which represents an additional illustration of its consistency.

physics.gen-ph↗

arXiv2016

Comparison of theoretical elastic couple stress predictions with physical experiments for pure torsion

Several different versions of couple stress theory have appeared in the literature, including the indeterminate Mindlin-Tiersten-Koiter couple stress theory (MTK-CST), indeterminate symmetric modified couple stress theory (M-CST) and determinate skew-symmetric consistent couple stress theory (C-CST). First, the solutions within each of these theories for pure torsion of cylindrical bars composed of isotropic elastic material are presented and found to provide a remarkable basis for comparison with observed physical response. In particular, recent novel physical experiments to characterize torsion of micro-diameter copper wires in quasi-static tests show no significant size effect in the elastic range. This result agrees with the prediction of the skew-symmetric C-CST that there is no size effect for torsion of an elastic circular bar in quasi-static loading, because the mean curvature tensor vanishes in a pure twist deformation. On the other hand, solutions within the other two theories exhibit size-dependent torsional response, which depends upon either one or two additional material parameters, respectively, for the indeterminate symmetric M-CST or indeterminate MTK-CST. Results are presented to illustrate the magnitude of the expected size-dependence within these two theories in torsion. Interestingly, if the material length scales for copper in these two theories with size-dependent torsion is on the order of microns or larger, then the recent physical experiments in torsion would align only with the self-consistent skew-symmetric couple stress theory, which inherently shows no size effect.

physics.gen-ph↗

arXiv2015

Cubic Single Crystal Representations in Classical and Size-dependent Couple Stress Elasticity

Beginning with Cosserat theory in the early 20th century, there have been several different formulations for size-dependent elastic response. In this paper, we concentrate on the application of classical Cauchy theory and the recent parsimonious consistent couple stress theory to model a homogeneous linear elastic solid, exemplified by a pure single crystal with cubic structure. The focus is on an examination of elastodynamic response based upon wave velocities from ultrasonic excitation and phonon dispersion curves, along with adiabatic bulk moduli measurements. In particular, we consider in detail elastic parameter estimation within classical elasticity and consistent couple stress theory for four different cubic single crystals (NaCl, KCl, Cu, CuZn). The classical theory requires the estimation of three independent material parameters, while only one additional parameter relating skew-symmetric mean curvature to skew-symmetric couple-stress is needed for the size-dependent consistent couple stress theory. This additional parameter can be defined for cubic crystals in terms of a material length scale, which is found to be on the order of tens of microns for the four materials studied here. Furthermore, a detailed statistical investigation provides strong to very strong evidence that couple stress theory is superior to classical Cauchy elasticity for representing the wave velocities and adiabatic bulk moduli for all four single crystals.

physics.gen-ph↗

arXiv2015

Foundations of consistent couple stress theory

In this paper, we examine the recently developed skew-symmetric couple stress theory and demonstrate its inner consistency, natural simplicity and fundamental connection to classical mechanics. This hopefully will help the scientific community to overcome any ambiguity and skepticism about this theory, especially the validity of the skew-symmetric character of the couple-stress tensor. We demonstrate that in a consistent continuum mechanics, the response of infinitesimal elements of matter at each point decomposes naturally into a rigid body portion, plus the relative translation and rotation of these elements at adjacent points of the continuum. This relative translation and rotation captures the deformation in terms of stretches and curvatures, respectively. As a result, the continuous displacement field and its corresponding rotation field are the primary variables, which remarkably is in complete alignment with rigid body mechanics, thus providing a unifying basis. For further clarification, we also examine the deviatoric symmetric couple stress theory that, in turn, provides more insight on the fundamental aspects of consistent continuum mechanics.

physics.gen-ph↗

arXiv2014

Evolution of generalized couple-stress continuum theories: a critical analysis

In this paper, we examine different generalized couple-stress continuum mechanics theories, including couple stress, strain gradient and micropolar theories. First, we investigate the fundamental requirements in any consistent size-dependent couple stress continuum mechanics, for which satisfying basic rules of mathematics and mechanics are crucial to establish a consistent theory. As a result, we show that continuum couple stress theory must be based on the displacement field and its corresponding macrorotation field as degrees of freedom, while an extraneous artificial microrotation cannot be a true continuum mechanical concept. Furthermore, the idea of generalized force and independent generalized degrees of freedom show that the normal component of the surface moment traction vector must vanish. Then, with these requirements in mind, various existing couple stress theories are examined critically, and we find that certain deviatoric curvature tensors create indeterminacy in the spherical part of the couple stress tensor. We also examine micropolar and micromorphic theories from this same perspective.

physics.gen-ph↗

arXiv2011

The mixed convolved action

A series of stationary principles are developed for dynamical systems by formulating the concept of mixed convolved action, which is written in terms of mixed variables, using temporal convolutions and fractional derivatives. Dynamical systems with discrete and continuous spatial representations are considered as initial applications. In each case, a single scalar functional provides the governing differential equations, along with all the pertinent initial and boundary conditions, as the Euler-Lagrange equations emanating from the stationarity of this mixed convolved action. Both conservative and non-conservative processes can be considered within a common framework, thus resolving a long-standing limitation of variational approaches for dynamical systems. Several results in fractional calculus also are developed.

math-ph↗

arXiv2011

Fundamental solutions for isotropic size-dependent couple stress elasticity

Fundamental solutions for two- and three-dimensional linear isotropic size-dependent couple stress elasticity are derived, based upon the decomposition of displacement fields into dilatational and solenoidal components. While several fundamental solutions have appeared previously in the literature, the present version is for the newly developed fully determinate couple stress theory. Within this theory, the couple stress tensor is skewsymmetrical and thus possesses vectorial character. The present derivation provides solutions for infinite domains of elastic materials under the influence of unit concentrated forces and couples. Unlike all previous work, unique solutions for displacements, rotations, force-stresses and couple-stresses are established, along with the corresponding force-tractions and couple-tractions. These fundamental solutions are central in analysis methods based on Green's functions for infinite domains and are required as kernels in the corresponding boundary integral formulations for size-dependent couple stress elastic materials.

math-ph↗

arXiv2010

Polar continuum mechanics

The existing polar continuum theory contains unresolved indeterminacies in the spherical part of the couple-stress tensor. This severely restricts its applicability in the study of micro and nano-scale solid and fluid mechanics and, perhaps more importantly, in the investigation of fluid turbulence phenomena, which involve a broad range of scales. In this paper, we rely on the energy equation, along with some kinematical considerations, to establish a consistent couple-stress theory for polar continua that resolves all indeterminacies. After presenting the general formulation and obtaining conservation laws, we concentrate exclusively on couple stress theory for polar fluid mechanics. We specialize the theory for linear viscous flow and consider several boundary value problems in couple-stress fluid mechanics. More generally, the resulting theory presented here may provide a basis for fundamental continuum-level studies at the finest scales.

physics.flu-dyn↗