33,652 research outputs found
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.Comment: 46 pages, 1 tabl
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.Comment: 30 page
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.Comment: 47 pages, 4 figures, 2 tables, 35 reference
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.Comment: 25 pages, 3 table
Analysis of bi-material interface cracks with complex weighting functions and non-standard quadrature
A boundary element formulation is developed to determine the complex stress
intensity factors associated with cracks on the interface between dissimilar
materials. This represents an extension of the methodology developed previously
by the authors for determination of free-edge generalized stress intensity
factors on bi-material interfaces, which employs displacements and weighted
tractions as primary variables. However, in the present work, the
characteristic oscillating stress singularity is addressed through the
introduction of complex weighting functions for both displacements and
tractions, along with corresponding non-standard numerical quadrature formulas.
As a result, this boundary-only approach provides extremely accurate
mesh-independent solutions for a range of two-dimensional interface crack
problems. A number of computational examples are considered to assess the
performance of the method in comparison with analytical solutions and previous
work on the subject. As a final application, the method is applied to study the
scaling behavior of epoxy-metal butt joints
Extraction of topological features from communication network topological patterns using self-organizing feature maps
Different classes of communication network topologies and their
representation in the form of adjacency matrix and its eigenvalues are
presented. A self-organizing feature map neural network is used to map
different classes of communication network topological patterns. The neural
network simulation results are reported.Comment: 8 Pages, 5 figures, To be appeared in IEE Electronics Letter Journa
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.Comment: 27 pages, 3 table
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.Comment: 38 pages, 10 figure
Oversampled Adaptive Sensing with Random Projections: Analysis and Algorithmic Approaches
Oversampled adaptive sensing (OAS) is a recently proposed Bayesian framework
which sequentially adapts the sensing basis. In OAS, estimation quality is, in
each step, measured by conditional mean squared errors (MSEs), and the basis
for the next sensing step is adapted accordingly. For given average sensing
time, OAS reduces the MSE compared to non-adaptive schemes, when the signal is
sparse. This paper studies the asymptotic performance of Bayesian OAS, for
unitarily invariant random projections. For sparse signals, it is shown that
OAS with Bayesian recovery and hard adaptation significantly outperforms the
minimum MSE bound for non-adaptive sensing. To address implementational
aspects, two computationally tractable algorithms are proposed, and their
performances are compared against the state-of-the-art non-adaptive algorithms
via numerical simulations. Investigations depict that these low-complexity OAS
algorithms, despite their suboptimality, outperform well-known non-adaptive
schemes for sparse recovery, such as LASSO, with rather small oversampling
factors. This gain grows, as the compression rate increases.Comment: To be presented in the 18th IEEE ISSPIT in Louisville, Kentucky, USA,
December 2018. 6 pages, 3 figure
Semi periodic maps on complex manifolds
In this letter we proved this theorem: \emph{if be a holomorphic mapping
of to a mapping manifold such that for every compact subset
the mapping is uniformly continues on and
is a relatively compact subset of . If the restriction of
to some hyperplane is semi periodic, then is an
semi mapping of to .
- …