Continuum Theory

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Continuum theory for liquid crystals

Continuum theory for liquid crystals

The continuum theories for liquid crystals that are in use today largely stem from the original work of Oseen [88, 89] and Zocher [115, 116], and the general classifications of liquid crystal phases described by Friedel [44], in the 1920s and 1930s. Friedel proposed a classification scheme for liquid crystals that consists of three broad categories called nematic, cholesteric and smectic. This classification has been widely adopted and is now in general usage. In 1958, Frank [41] gave a direct formulation of the energy function needed for the continuum theory and this rejuvenated interest in the subject in the 1960s when Ericksen [34] developed a static theory for nematic liquid crystals which consequently led to balance laws for dynamical behaviour [33]. This work encouraged Leslie [60, 61] to formulate constitutive equations for dynamics and thereby complete what has turned out to be an extremely successful and comprehensive dynamic theory for nematic liquid crystals. The continuum theory, and crucial continuum descriptions, will be reviewed below. More recently, there has been a renewal of interest in biaxial nematics and smectic liquid crystals and continuum theories that extend the notions developed by Ericksen and Leslie to these phases of liquid crystals will also be mentioned in this Chapter.
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The continuum theory of shear localization in two dimensional foam

The continuum theory of shear localization in two dimensional foam

Abstract. We review some recent advances in the rheology of two-dimensional liquid foams, which should have implications for three-dimensional foams, as well as other mechanical systems that have a yield stress. We focus primarily on shear localization under steady shear, an effect first highlighted in an experiment by Debr´egeas et al. A continuum theory which incorporates wall drag has reproduced the effect. Its further refinements are successful in matching results of more extensive observations and make interesting predictions regarding experiments for low strain rates, and non-steady shear. Despite these successes, puzzles remain, particularly in relation to quasistatic simulations. The continuum model is semi- empirical: the meaning of its parameters may be sought in comparison with more detailed simulations and other experiments. The question of the origin of the Herschel-Bulkley relation is particularly interesting.
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New Discovery on Planck Units and Physical Dimension in Cosmic Continuum Theory

New Discovery on Planck Units and Physical Dimension in Cosmic Continuum Theory

Conversion is one of the core ideas of Cosmic Continuum Theory. The mutual transformation of mass, energy, and dark mass makes the universe colorful. However, these transformations are not arbitrary, but are determined by the ex- istence boundaries of particles, quantum, and dark particles. With the mutual conversion between particles, quantum, and dark particles, space, time, and dark space also follow.

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Continuum theory of biaxial nematic liquid crystals

Continuum theory of biaxial nematic liquid crystals

The first attempt at a continuum theory was by Saupe [9], with other formulations following by Liu [10], Kini [11], Govers and Vertogen [12–14], Chaur´ e [15] and Leslie, Laverty and Carlsson [16]. These authors essentially obtained the same continuum the- ory from differing approaches and viewpoints, as mentioned by Leslie and Carlsson [17] when they examined theoretically flow alignment in a biaxial phase. The formulation by Leslie et al. [16] is slightly more general and therefore it is this theory that will be adopted here. There has been a resurgent interest in the continuum modelling of biaxial nematic liquid crystals, largely due to emerging experimental results that have appeared in the literature, especially those motivated by more recent confirmation of biaxial nematic phases [18–22]. A series of comments on some of these results has been made by Luckhurst [23].
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Cosmic Continuum Theory: A New Idea on Hilbert’s Sixth Problem

Cosmic Continuum Theory: A New Idea on Hilbert’s Sixth Problem

Based on quantum theory, the universe is quantized, that is, the mass body is composed of “particles”, the energy body by “quanta” and the dark mass body by “dark particles”. Further, such particles, quanta and dark particles are not infi- nitely divisible, but limited within boundaries. In other words, there exists ele- mentary particles, elementary quanta and elementary dark particles. Thus, from a mathematical perspective, all particles, quanta and dark particles in the un- iverse can build up to a denumerable set.

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Angular dependence of ultrasonic wave propagation in a stressed, orthorhombic continuum: Theory and application to the measurement of stress and texture

Angular dependence of ultrasonic wave propagation in a stressed, orthorhombic continuum: Theory and application to the measurement of stress and texture

continuum is presented. Since many of the material parameters which appear in the analysis are unknown, in particular the third‐order elastic constants of polycrystalline metals, emphasis is placed on the angular dependence of the velocities. An expansion to first order in stress‐induced anisotropy and to second order in textural anisotropy reveals terms with twofold, fourfold, and sixfold symmetry. Scenarios are proposed for using various properties of this symmetry to deduce the difference in magnitude and directions of the principal stresses independent of textural anisotropy and the textural anisotropy independent of the stresses. Experimental results are presented for the cases of aluminum, 304 stainless steel, and copper.
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On   a Continuum Theory for Crack Propagation in Brittle Materials

On a Continuum Theory for Crack Propagation in Brittle Materials

This paper is concerned with the development of a theory for crack propagation in brittle solids. In order to characterize the cohesion state of the material, an order parameter field, which is related to the population of defects, is introduced. The microforce system, which acts in response to changes in the population of defects, is introduced and it is presumed consistent with its own balance, the microforce balance. Kinetic equations are obtained from the microforce balance and appropriate constitutive equations. The theory, which is described within the framework of modern continuum mechanics, can be considered as the first step toward a unifying route to incorporate not only physical processes on the microscale but also non-local effects.
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Continuum and Computational Modeling of Flexoelectricity

Continuum and Computational Modeling of Flexoelectricity

In this chapter, we will introduce a general framework for finite element solutions of problems for an elastic dielectric with flexoelectricity and/or piezoelectricity. The general- ized gradient theory developed by Mindlin (1964) is used to model the gradient effect of elasticity. Piezoelectric as well as flexoelectric coupling are introduced into the formulation by adding polarization as a variable in the energy storage function. The energy storage function depends on the strain tensor, second gradient of displacement, and polarization. To avoid using C 1 finite elements in our numerical solution, a mixed formulation based on the work of Amanatidou & Aravas (2002) is developed. In this formulation, displacement and displacement gradient are treated as separate degrees of freedom and their relationship is enforced in the variational form. This framework is entirely consistent with the continuum theory of flexoelectricity and is capable of capturing fine structures due to gradient effects. The finite element code is validated against benchmark problems with known analytical solutions. Then it is employed to study three important classes of problems: plate with an elliptical hole, stationary crack, and periodic meta structures. In the static crack problem, for which an asymptotic solution has been developed in Chapter 4, the validity and region of dominance of the asymptotics is determined. The elliptical hole and periodic structure provide an alternative means of generating large strain gradients; the finite element results show how these large gradients influence classical observations and generate crucial insights that can lead to better measurement in experiments as well as improved functionality in applications.
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Atomistic to continuum coupling

Atomistic to continuum coupling

Peridynamics seeks to model atomistic systems with defects by a nonlocal continuum theory (Du, Gunzburger, Lehoucq and Zhou 2013, Silling and Lehoucq 2010). As a continuum theory it must be discretized for numerical implementation, thus leading to a nonlocal particle model in regions where the mesh is refined to length scales less than the ‘horizon’, which plays the role of a cut-off radius in a nonlocal atomistic model. To reach larger length scales (Seleson, Beneddine and Prudhomme 2013), the mesh must be coarsened beyond the length scale of the horizon, which leads to local ap- proximations when quadrature points are added to the interior of elements (this can be seen by adapting the derivation in Section 4.3 for a nonlocal atomistic model to a nonlocal peridynamic continuum model). The mathe- matical framework and results presented in this article for a/c coupling can thus potentially be applied to the analysis of numerical approximations of peridynamics models.
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Continuum mathematics at the nanoscale

Continuum mathematics at the nanoscale

This paper provides a brief overview of three pertinent problems from the nano field. In the first problem we considered a model for fluid flow in carbon nanotubes. At present there is no theory for calculating the slip lengths for a liquid moving over a solid surface. We used an equivalent theory with a depletion layer over a solid surface to obtain an ex- pression for the slip length in terms of the depletion layer thickness and its viscosity. Next, we presented a mathematical model that successfully captured the ‘abrupt melting’ phe- nomenon of nanoparticles. In particular, the model demonstrated the sharp increase in melting rate as the size of the nanoparticles decreased. Finally, we considered the ther- mal conductivity of nanofluids, and our analysis led to an expression for solid-in-liquid suspensions, derived in a distinct way to the seminal Maxwell model. The model showed excellent agreement with experimental data for particle volume fractions greater than %. While this work is at the limit of continuum theory, in each case the continuum mod- els demonstrated good agreement with experimental data and provided valuable insight. Moreover, the results of the nanoscale modelling in some cases led to a deeper under- standing of the macroscale problems. At present there is great interest in nanotechnology with applications in a wide range of areas. We have presented results for just three spe- cific problems, however the success of the models indicates that there is great scope for applying continuum models at the nanoscale.
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The continuum disordered pinning model

The continuum disordered pinning model

α ∈ (0, 1), has a non-trivial scaling limit, known as the α-stable regenerative set. In this paper we consider Gibbs transformations of such renewal processes in an i.i.d. random environment, called disordered pinning models. We show that for α ∈ 1 2 , 1 these models have a universal scaling limit, which we call the continuum disordered pinning model (CDPM). This is a random closed subset of R in a white noise random environment, with subtle features:

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Health as a Space-Time Continuum

Health as a Space-Time Continuum

The present study investigates the properties the Space-Time Continuum (STC) of human health from the standpoint of health concept reliability of complex systems (RCS). It is known that human life characterized by different periods of activity morphogenesis, maturation and functioning of organs and body systems, speed evolutive and involutive processes [21]. There are many theories of aging the human body [4, 11, 12], anyway explanatory gradual involution physiological mechanisms for human adaptability to the environment. On the basis of these theories was put forward "recovery" concept [11, 12, 30], which, despite their consistency did not bring the expected progress in the extension of high-grade bio- social life throughout its entire length.
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Continuum descriptions of cytoskeletal dynamics

Continuum descriptions of cytoskeletal dynamics

This tutorial presents an introduction into continuum descriptions of cytoskeletal dynamics. In contrast to discrete models in which each molecule keeps its identity, such descriptions are given in terms of averaged quantities per unit volume like the number density of a certain molecule. Starting with a discrete description for the assembly dynamics of cytoskeletal filaments, we derive the continuity equation, which serves as the basis of many continuum theories. We illustrate the use of this approach with an investigation of spontaneous cytoskeletal polymerization waves. Such waves have by now been observed in various cell types and might help to orchestrate cytoskeletal dynamics during cell spreading and locomotion. Our analysis shows how processes at the scale of single molecules, namely, the nucleation of new filaments and filament treadmilling, can lead to the spontaneous appearance of coherent traveling waves on scales spanning many filament lengths. For readers less familiar with calculus, we include an informal introduction to the Taylor expansion.
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Precarious homes: The sharing continuum

Precarious homes: The sharing continuum

4 The theoretical framework for understanding sharing in relation to the home which is put forward in this chapter, explains sharing as a continuum. At different points along this continuum three different mode of sharing can be distinguished: involuntary (being forced into sharing by external circumstances); uninformed (sharing by choice, but without being aware of the implications); and intentional (a deliberate and positive choice to share, on the basis of full information). These are ideal types, with many possible variations and combinations along the continuum. Drawing on empirical research data from my study of shared residential space 16 and on a wide range of
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Continuum spectroscopy of light nuclei

Continuum spectroscopy of light nuclei

Light nuclei near the drip lines have very few particle-bound levels, with most of their states being in the continuum. Of course those beyond the drip lines have all their levels in the continuum including their ground states. Thus the study of continuum spectroscopy is quite important for these light nuclei. There are many techniques to study these continuum states, however, in this work we will consider just the invariant-mass method and discuss re- sults obtained with the HiRA array [1] at the the National Superconducting Cyclotron Laboratory at Michigan State University and the Texas A&M University cyclotron.
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MoL 2014 10: 
  An Outline of Algebraic Set Theory with a View Towards Cohen's Model Falsifying the Continuum Hypothesis

MoL 2014 10: An Outline of Algebraic Set Theory with a View Towards Cohen's Model Falsifying the Continuum Hypothesis

Towards the end of the 19th century, Georg Cantor proved by his famous di- agonal argument that the real numbers have a strictly larger cardinality than the natural numbers. Cantor conjectured that all subsets of the reals are either countable or have the same cardinality as the reals themselves. The statement became known as the Continuum hypothesis (CH) and was considered as so im- portant that Hilbert placed it on top of his list of open mathematical problems to be solved in the 20th century. A partial answer to the conjecture was found in 1940 by Kurt Gödel who proved that CH is consistent with the other axioms of Zermelo-Fraenkel by constructing L, the universe of constructible sets [15]. A full answer to the conjecture was found in 1963 by Paul Cohen [11], who intro- duced the method of forcing to construct a model of set theory falsifying CH. The results of Gödel and Cohen established the independence of the Continuum hypothesis from the other axioms of Zermelo-Fraenkel set theory.
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OER adoption: a continuum for practice

OER adoption: a continuum for practice

doi http://doi.dx.org/10.7238/ rusc.v11i3.2102 Abstract Whilst Open Educational Resources OER offer opportunities for broadening participation in Higher Education, reducing course develo[r]

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Continuum robots and underactuated grasping

Continuum robots and underactuated grasping

In this paper, a novel idea to use a continuum manipula- tor for under-actuated continuum grasping has been proposed and demonstrated using the Octarm as a prototype model. Continuum manipulators can be approximated by the oper- ations of a multi-linked under-actuated chain, but the kine- matics, actuation and control strategies employed are di ff er- ent from the former. The Octarm lacks links and joints, but, having the air pressurized actuators arranged in a triangular pattern enables the arm to bend in an infinite number of di- rections. The grasp realized using a continuum arm is more qualified by its flexibility (compliance) and this compensates for the lack of accuracy when compared to rigid link robots in positioning the arm to grasp objects. The arm was manually operated and its potential to grasp was analyzed and quan- tified using empirical data. We are currently programming di ff erent trajectory motions for the Octarm to find a move- ment of the arm that increases the impact force at the contact point. Having this as the groundwork, in the short term, we intend to develop autonomous grasping algorithms to realize impulsive manipulation with continuum arms.
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Parton distribution functions on the lattice and in the continuum

Parton distribution functions on the lattice and in the continuum

The maximum nucleon momentum used in our calculation was 2.5 GeV. Inside this momentum range, the continuum dispersion relation for the nucleon was satisfied within the errors of the calcu- lation, indicating small lattice artifacts of O (ap). In figure 1(left) we plot the nucleon energy as a function of momentum along with the continuum dispersion relation.

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Linear vibrations of continuum with fractional derivatives

Linear vibrations of continuum with fractional derivatives

The general solution allows one to investigate the effects on a dynamic analysis of contin- uum whose damping term is modelled by a fractional derivative. An engineering problem which is a special application of the general model developed in this study was formerly considered in []. In our previous study [], the analysis of primary and parametric res- onance for the external excitation term having ε-order was performed. As the forced term is obtained in one-order, sum or difference type of resonance also appears in the present model. The method of multiple scales is used in the analysis. Thus, the amplitude and phase modulation equations are produced in terms of operators. In addition, the varia- tions of the curves with respect to the dimensionless parameters are presented. Finally, the effects of fractional damping on the linear vibrations of continuum are investigated in detail.
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