Papers by ALI REZA HADJESFANDIARI
arXiv (Cornell University), Apr 24, 2016
Several different versions of couple stress theory have appeared in the literature, including the... more 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 sizedependence 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 skewsymmetric couple stress theory, which inherently shows no size effect.
International Journal for Multiscale Computational Engineering, 2022
By investigating the benefits and shortcomings of the existing form of continuous defect theory (... more 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 forcestresses in classical continuum mechanics.
European Journal of Mechanics A-solids, Jul 1, 2023
arXiv (Cornell University), Dec 20, 2017
In this paper, a new size-dependent Timoshenko beam model is developed based on the consistent co... more 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 sizedependency 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.
arXiv (Cornell University), Oct 11, 2018
In this paper, we investigate the inherent physical and mathematical character of higher gradient... more 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.
Journal of Non-newtonian Fluid Mechanics, Sep 1, 2018
Consistent couple stress theory is extended to Boussinesq thermal convection problem Momentum... more Consistent couple stress theory is extended to Boussinesq thermal convection problem Momentum balance involves a new length scale parameter and fourth order derivatives Fluid element bending provides an additional energy dissipation mechanism Critical Rayleigh number for square cavity becomes size dependent at small scales Moment traction versus vorticity boundary conditions significantly affect stability
arXiv (Cornell University), Jan 10, 2022
In this paper, the concept of moment and couple in mechanics is examined from a fundamental persp... more In this paper, the concept of moment and couple in mechanics is examined from a fundamental perspective. It turns out that although representing a couple by its moment vector is very useful in rigid body mechanics and strength of materials, it has been very misleading in continuum mechanics. To specify the effect of a concentrated couple in continuum mechanics, not only the couple moment, but also the line of action of its constituent parallel opposite forces must be specified. However, in the governing equations of equilibrium or motion of a continuum, only moment of body couple, moment of couple-tractions, and moment of couple-stresses appear without specifying the line of action of any couple density forces. This results in non-uniqueness of the state of stresses and deformation in the continuum, which has shown itself in the indeterminacy of the couple-stress tensor. Nevertheless, the physical state of stress and deformation in the continuum is unique and determinate. Therefore, this imposes some restrictions on the form of body couple, couple-traction and couple-stresses. Here, the uniqueness of interactions in the continuum is used to establish that the continuum does not support a distribution of body couple, and a distribution of surface twisting couple-traction with normal moment. From this the mechanism of action of the couple-traction as a double layer of shear force-tractions, and the skew-symmetric character of the couple-stress moment tensor in continuum mechanics are also established.
Engineering Analysis With Boundary Elements, May 1, 2021
Summary In order to investigate size-dependent creeping plane microfluidic flow, a boundary eleme... more Summary In order to investigate size-dependent creeping plane microfluidic flow, a boundary element method is implemented that involves the calculation of interior quantities and multi-domain problems. The governing equations are formulated using the skew-symmetric character of the couple-stress tensor and its energy conjugate mean-curvature tensor, as established in consistent couple stress theory. This theoretical formulation includes one characteristic material length scale parameter l that represents the size-dependency of the problem. Here, we present the boundary integral representation and numerical implementation for two-dimensional size-dependent steady state creeping incompressible flow, in which velocities, angular velocities, force- and couple-tractions are the primary variables. Beyond the previously known singular fundamental solution kernels for point force and point couple in the velocity and angular velocity integral equations, here we present the integral equations for calculating internal mean curvatures and couple-stresses, along with their corresponding kernels. The formulation is then applied to solve some computational problems both to investigate the size effects on the response resulting from the theory, and to validate the strength of the numerical method. For this purpose, we study the important effect of different boundary conditions on the flow pattern and consider a multi-fluid domain Couette flow problem.
Engineering Analysis With Boundary Elements, Mar 1, 2022
Journal of Engineering Mechanics-asce, Feb 1, 2019
AbstractThe vibrations of ultrathin silicon cantilever microbeams are studied using consistent co... more AbstractThe vibrations of ultrathin silicon cantilever microbeams are studied using consistent couple-stress theory to investigate size-dependent effects. The corresponding Euler-Bernoulli beam the...
In this paper, we examine the mathematical and physical consistencies of the three primary couple... more In this paper, we examine the mathematical and physical consistencies of the three primary couple stress theories: original Mindlin-Tiersten-Koiter couple stress theory (MTK-CST), modified couple stress theory (M-CST) and consistent couple stress theory (C-CST). As has been known for many years, MTK-CST suffers from some fundamental inconsistencies, such as the indeterminacy of the couple-stress tensor. Therefore, despite the fact that MTK-CST has a fundamental position in the evolution of size-dependent continuum mechanics, it is not a reliable theory within continuum mechanics, for example, in developing new size-dependent multi-physics formulations. We also observe that M-CST not only inherits all inconsistences from the original MTK-CST, but also suffers from new additional inconsistencies, such as the introduction of a new non-physical governing equation. These inconsistencies refute the claim of those who state that the couple-stress tensor may be chosen symmetric. Therefore, the apparent success of MTK-CST and M-CST in describing a size-effect for some problems, such as two-dimensional plate and beam bending, is not enough to justify these theories as suitable for general cases. In fact, the symmetric couple-stresses in M-CST create torsional or anticlastic deformation, not bending. On the other hand, C-CST, with a skew-symmetric couple-stress tensor, is the consistent continuum mechanics suitable for solving different size-dependent solid, fluid and multi-physics problems.
European Journal of Mechanics A-solids, Jul 1, 2021
Abstract Based upon the principle of minimum total potential energy for consistent couple stress ... more Abstract Based upon the principle of minimum total potential energy for consistent couple stress theory, a Ritz variational approach is developed using tensor product B-splines for both two- and three-dimensional elastostatic analysis. The underlying theory is size-dependent due to incorporation of the energy conjugate skew-symmetric couple-stress and mean curvature tensors, in addition to the force-stress and strain conjugate pair of classical theory. The use of B-splines as the basis functions can assure the required C 1 continuity of the displacement field, while also permitting higher order representations. Both displacements and rotations are essential boundary conditions in this theory. Displacements can be enforced in the usual way, but rotations require special treatment to maintain symmetry and positive definiteness of the stiffness matrix for well-posed elastostatic problems. Several computational examples are considered to validate the formulation, illustrate convergence characteristics, and investigate mechanical behavior under consistent couple stress theory. All previous numerical analysis of consistent couple stress theory has been limited to plane strain problems. Thus, the extension here to three-dimensions is of great importance, especially because the behavior in several cases shows significant deviation from the plane strain solutions. Consequently, the variational formulations and computational methodology presented in this paper can play a critical role in assessing the predictive capability of consistent couple stress theory and in understanding size-dependent elastic response.
Engineering Analysis With Boundary Elements, Sep 1, 2017
A boundary element method is developed to examine two-dimensional size-dependent thermoelastic re... more A boundary element method is developed to examine two-dimensional size-dependent thermoelastic response in isotropic solids. The formulation is founded on the recently established consistent couple stress theory, in which both the couple-stress tensor and its energy conjugate mean curvature tensor are skew-symmetric. For isotropic materials, there is no thermal mean curvature deformation, and the thermoelastic effect is solely the result of thermal strain deformation. As a result, size-dependency is quantified by one characteristic material length scale parameter l, while the thermal coupling is activated through the classical thermal expansion coefficient α. 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. Here, we develop the boundary integral representation and numerical implementation for this sizedependent thermoelastic boundary element method (BEM) for plane problems, which involves temperatures, displacements, rotations, normal heat fluxes, force-tractions and couple-tractions as primary variables within a boundary-only formulation. Then, we apply this new BEM formulation to several basic computational problems in an effort to validate the robustness of the numerical implementation and to examine size-dependent response.
A new boundary element formulation is developed to analyze two-dimensional size-dependent thermoe... more A new boundary element formulation is developed to analyze two-dimensional size-dependent thermoelastic response in linear isotropic couple stress materials. The model is based on the recently developed consistent couple stress theory, in which the couple-stress tensor is skew-symmetric. The size-dependency effect is specified by one characteristic parameter length scale l, while the thermal effect is quantified by the classical thermal expansion coefficient α and conductivity k. We discuss the boundary integral formulation and numerical implementation of this size-dependent thermoelasticity boundary element method (BEM). Then, we apply the resulting BEM formulation to a computational example to validate the numerical implementation and to explore thermoelastic couplings as the non-dimensional characteristic scale of the problem is varied. Interestingly, for a cantilever beam with a transverse temperature gradient, we find significantly reduced non-dimensional tip deflections as the beam depth h approaches the material characteristic length scale l. On the other hand, when l/h < 0.01, the classical size-independent deflections are recovered.
arXiv (Cornell University), Apr 6, 2017
Boundary element methods provide powerful techniques for the analysis of problems involving coupl... more Boundary element methods provide powerful techniques for the analysis of problems involving coupled multiphysical 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 couplestress 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 l , 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.
arXiv (Cornell University), Oct 1, 2018
We investigate the consistency of the fundamental governing equations of motion in continuum mech... more 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.
arXiv (Cornell University), Jun 1, 2016
In this paper, we examine the pure bending of plates within the framework of modified couple stre... more 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.
arXiv (Cornell University), Jul 29, 2015
In this paper, we examine the recently developed skew-symmetric couple stress theory and demonstr... more 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 couplestress 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.
arXiv (Cornell University), Jun 28, 2012
In this paper, a consistent theory is developed for size-dependent piezoelectricity in dielectric... more In this paper, a consistent theory is developed for size-dependent piezoelectricity in dielectric solids. This theory shows that electric polarization can be generated as the result of coupling to the mean curvature tensor, unlike previous flexoelectric theories that postulate such couplings with other forms of curvature and more general strain gradient terms ignoring the possible couple-stresses. The present formulation represents an extension of recent work that establishes a consistent size-dependent theory for solid mechanics. Here by including scale-dependent measures in the energy equation, the general expressions for force-and couple-stresses, as well as electric displacement, are obtained. Next, the constitutive relations, displacement formulations, the uniqueness theorem and the reciprocal theorem for the corresponding linear small deformation size-dependent piezoelectricity are developed. As with existing flexoelectric formulations, one finds that the piezoelectric effect can also exist in isotropic materials, although in the present theory the coupling is strictly through the skew-symmetric mean curvature tensor. In the last portion of the paper, this isotropic case is considered in detail by developing the corresponding boundary value problem for two dimensional analyses and obtaining a closed form solution for an isotropic dielectric cylinder.
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Papers by ALI REZA HADJESFANDIARI