On the Numerical Solution of One-Dimensional Continuum Problems Using the Theory of a Cosserat Point

1985 ◽  
Vol 52 (2) ◽  
pp. 373-378 ◽  
Author(s):  
M. B. Rubin
Keyword(s):  
Numerical Solution ◽  
Body Force ◽  
Closed Form Solution ◽  
Form Solution ◽  
Element Approximation ◽  
One Dimensional ◽  
Linear Elastic ◽  
Elastic Bar ◽  
Cosserat Point ◽  
Nonlinear Elastic

The theory of a Cosserat point is specialized to describe the motion of a one-dimensional continuum. Attention is focused on two problems of an elastic bar. Vibration of a linear-elastic bar is considered in the first problem and static deformation of a nonlinear-elastic bar subjected to a uniform body force is considered in the second problem. A closed-form solution is derived for each problem by dividing the bar into two elements, each of which is modeled as a Cosserat point. The predictions of the two-element approximation are shown to be very accurate.

Numerical Analysis for Acoustic Resonance of One-Dimensional Nonlinear Elastic Bar

2011 ◽  
Vol 50 (7) ◽  
pp. 07HB02 ◽  
Author(s):  
Ryuichi Tarumi ◽  
Tomohiro Matsuhisa ◽  
Yoji Shibutani
Keyword(s):  
Numerical Analysis ◽  
Acoustic Resonance ◽  
One Dimensional ◽  
Elastic Bar ◽  
Nonlinear Elastic

Free Vibration of a Multilayered One-Dimensional Quasi-Crystal Plate

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Natalie Waksmanski ◽  
Ernian Pan ◽  
Lian-Zhi Yang ◽  
Yang Gao
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Closed Form Solution ◽  
Form Solution ◽  
Crystal Plate ◽  
Mode Shapes ◽  
Sandwich Plates ◽  
One Dimensional ◽  
Propagator Matrix ◽  
Multilayered Plates

An exact closed-form solution of free vibration of a simply supported and multilayered one-dimensional (1D) quasi-crystal (QC) plate is derived using the pseudo-Stroh formulation and propagator matrix method. Natural frequencies and mode shapes are presented for a homogenous QC plate, a homogenous crystal plate, and two sandwich plates made of crystals and QCs. The natural frequencies and the corresponding mode shapes of the plates show the influence of stacking sequence on multilayered plates and the different roles phonon and phason modes play in dynamic analysis of QCs. This work could be employed to further expand the applications of QCs especially if used as composite materials.

scholarly journals The generalized viscoelastic spring

2020 ◽  
Vol 476 (2236) ◽  
pp. 20190881
Author(s):  
Kostas P. Soldatos
Keyword(s):  
Viscous Flow ◽  
A Priori ◽  
Closed Form Solution ◽  
Form Solution ◽  
Viscoelastic Deformation ◽  
One Dimensional ◽  
Flow Formation ◽  
Inelastic Material ◽  
Flow Potential ◽  
Well Posed

A spring/rod model is presented that describes one-dimensional behaviour of solids susceptible to large or small viscoelastic deformation. Derivation of its constitutive equation is underpinned by the fact that the internal energy, which the elastic part of deformation stores in the spring, changes in time with the observed strain as well as with some, unknown part of the strain-rate. The latter emerges through the action of a viscous flow potential and is the source of inelastic deformation. Thus, unlike its conventional viscoelasticity counterparts, the model does not postulate a priori a rule that relates strain with viscous flow formation. Instead, it considers that such a rule, as well as other important features of combined elastic and inelastic material response, should become known a posteriori through the solution of a relevant, well-posed boundary value problem. This paper begins with considerations compatible with large viscoelastic deformations and gradually progresses through simpler modelling situations. The latter also include the case of small viscoelastic strain that underpins formulation of classical, spring-dashpot viscoelastic models. In an example application, a relevant closed-form solution is obtained for a spring undergoing small viscoelastic deformation under the influence of a viscous flow potential which is quadratic in the stress.

Numerical Solution Procedures for Nonlinear Elastic Curved Rods Using the Theory of a Cosserat Point

2005 ◽  
Vol 10 (1) ◽  
pp. 89-126 ◽  
Author(s):  
M. B. Rubin
Keyword(s):  
Numerical Solution ◽  
Circular Ring ◽  
Axisymmetric Deformation ◽  
Shear Locking ◽  
Cosserat Theory ◽  
Pure Bending ◽  
External Pressures ◽  
Cosserat Point ◽  
Curved Rods ◽  
Nonlinear Elastic

The numerical solution of problems of curved rods can be formulated using rod elements developed within the context of the theory of a Cosserat point. Although the general theory is valid for curved rods, the constitutive coefficients have been determined by comparison with exact linear solutions only for straight beams. The objective of this paper is to explore the accuracy of the predictions of the Cosserat theory for curved rods by comparison with exact solutions. Specifically, these problems include: linearized axisymmetric deformation of a circular ring loaded with internal and external pressures; nonlinear axisymmetric inversion of a circular ring; and linearized pure bending of a section of a circular ring. In all cases, the Cosserat theory performs well with no modifications of the constitutive constants, even in the limit of reasonably thick rods. Also, it is shown that the Cosserat theory does not exhibit shear locking in the limit of thin rods.

A semi-infinite hydraulic fracture with leak-off driven by a power-law fluid

2017 ◽  
Vol 837 ◽  
pp. 210-229 ◽  
Author(s):  
E. V. Dontsov ◽  
O. Kresse
Keyword(s):  
Numerical Solution ◽  
Power Law ◽  
Hydraulic Fracture ◽  
Closed Form Solution ◽  
Analytic Solutions ◽  
Form Solution ◽  
Rapid Evaluation ◽  
Power Law Fluid ◽  
Parametric Space ◽  
Fluid Rheology

This study investigates the problem of a semi-infinite hydraulic fracture that propagates steadily in a permeable formation. The fracturing fluid rheology is assumed to follow a power-law behaviour, while the leak-off is modelled by Carter’s model. A non-singular formulation is employed to effectively analyse the problem and to construct a numerical solution. The problem under consideration features three limiting analytic solutions that are associated with dominance of either toughness, leak-off or viscosity. Transitions between all the limiting cases are analysed and the boundaries of applicability of all these limiting solutions are quantified. These bounds allow us to determine the regions in the parametric space, in which these limiting solutions can be used. The problem of a semi-infinite fracture, which is considered in this study, provides the solution for the tip region of a hydraulic fracture and can be used in hydraulic fracturing simulators to facilitate solving the moving fracture boundary problem. To cater for such applications, for which rapid evaluation of the solution is necessary, the last part of this paper constructs an approximate closed form solution for the problem and evaluates its accuracy against the numerical solution inside the parametric space.

scholarly journals MHD Nanofluids in a Permeable Channel with Porosity

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 378 ◽  
Author(s):  
Ilyas Khan ◽  
Aisha Alqahtani
Keyword(s):  
Magnetic Parameter ◽  
Flow Direction ◽  
Perturbation Technique ◽  
Closed Form Solution ◽  
Form Solution ◽  
Convection Flow ◽  
Temperature Profiles ◽  
Uniform Temperature ◽  
One Dimensional ◽  
Physical Conditions

This paper introduces a mathematical model of a convection flow of magnetohydrodynamic (MHD) nanofluid in a channel embedded in a porous medium. The flow along the walls, characterized by a non-uniform temperature, is under the effect of the uniform magnetic field acting transversely to the flow direction. The walls of the channel are permeable. The flow is due to convection combined with uniform suction/injection at the boundary. The model is formulated in terms of unsteady, one-dimensional partial differential equations (PDEs) with imposed physical conditions. The cluster effect of nanoparticles is demonstrated in the C 2 H 6 O 2 , and H 2 O base fluids. The perturbation technique is used to obtain a closed-form solution for the velocity and temperature distributions. Based on numerical experiments, it is concluded that both the velocity and temperature profiles are significantly affected by ϕ . Moreover, the magnetic parameter retards the nanofluid motion whereas porosity accelerates it. Each H 2 O -based and C 2 H 6 O 2 -based nanofluid in the suction case have a higher magnitude of velocity as compared to the injections case.

New exact results for the defective square lattice

1976 ◽  
Vol 80 (2) ◽  
pp. 365-381 ◽  
Author(s):  
G. Ronca
Keyword(s):  
Closed Form Solution ◽  
Exact Result ◽  
Form Solution ◽  
Square Lattice ◽  
Zero Temperature ◽  
Real Crystal ◽  
One Dimensional ◽  
Resonance Modes ◽  
The One ◽  
Dimensional Crystal

Since the publication of the fundamental papers by Lifshitz (1, 2) and Montroll and Potts (3, 4) many authors have investigated the effect of an isotopic impurity on the lattice vibrations of a harmonic crystal at zero temperature. A fairly broad knowledge is now available on scattering amplitudes, localized modes and resonance modes (6, 7). Nevertheless, as pointed out by Maradudin and Montroll (see (7), p. 430), a closed form solution to the problem has been found only for the one-dimensional crystal, the work done on two and three-dimensional crystals being predominantly numerical. Unfortunately the one-dimensional crystal, as an approximation for a real crystal is an oversimplified model, incapable as it is of exhibiting resonance modes. To the author's knowledge the most significant exact result concerning the classical behaviour at zero temperature of crystals having a dimensionality higher than one is the connexion, calculated by Mahanty et al. (5) between localized mode frequency and impurity mass for the case of a square lattice undergoing planar vibrations.

Structure design and kinematics of a robot manipulator

Robotica ◽  
1988 ◽  
Vol 6 (4) ◽  
pp. 299-309 ◽  
Author(s):  
Kesheng Wang ◽  
Terje K. Lien
Keyword(s):  
Numerical Solution ◽  
Closed Form ◽  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Robot Manipulator ◽  
Closed Form Solution ◽  
Form Solution ◽  
Structure Design ◽  
General Algorithm ◽  
6 Degrees Of Freedom

SUMMARYIn this paper we show that a robot manipulator with 6 degrees of freedom can be separated into two parts: arm with the first three joints for major positioning and wrist with the last three joints for major orienting. We propose 5 arms and 2 wrists as basic construction for commercially robot manipulators. This kind of simplification can lead to a general algorithm of inverse kinematics for the corresponding configuration of different combinations of arm and wrist. The approaches for numerical solution and closed form solution presented in this paper are very efficient and easy for calculating the inverse kinematics of robot manipulator.

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