scholarly journals Variational Principles for Buckling of Microtubules Modeled as Nonlocal Orthotropic Shells

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Sarp Adali
Keyword(s):  
Boundary Conditions ◽  
Variational Formulation ◽  
Variational Principles ◽  
Orthotropic Shell ◽  
Rayleigh Quotient ◽  
Buckling Load ◽  
Mechanical Coupling ◽  
Elastic Filament ◽  
Orthotropic Shells ◽  
Filament Network

A variational principle for microtubules subject to a buckling load is derived by semi-inverse method. The microtubule is modeled as an orthotropic shell with the constitutive equations based on nonlocal elastic theory and the effect of filament network taken into account as an elastic surrounding. Microtubules can carry large compressive forces by virtue of the mechanical coupling between the microtubules and the surrounding elastic filament network. The equations governing the buckling of the microtubule are given by a system of three partial differential equations. The problem studied in the present work involves the derivation of the variational formulation for microtubule buckling. The Rayleigh quotient for the buckling load as well as the natural and geometric boundary conditions of the problem is obtained from this variational formulation. It is observed that the boundary conditions are coupled as a result of nonlocal formulation. It is noted that the analytic solution of the buckling problem for microtubules is usually a difficult task. The variational formulation of the problem provides the basis for a number of approximate and numerical methods of solutions and furthermore variational principles can provide physical insight into the problem.

Variational Formulation of Vibrations of Multi-Walled Carbon Nanotubes With Geometric and Physical Nonlinearities

2009 ◽  
Author(s):  
Sarp Adali
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Variational Formulation ◽  
Variational Principles ◽  
Beam Model ◽  
Nonlinear Force ◽  
Multi Walled Carbon Nanotubes ◽  
Geometric Boundary ◽  
Walled Carbon Nanotubes ◽  
Euler Bernoulli Beam

Variational principles are derived for multi-walled carbon nanotubes (CNT) undergoing nonlinear vibrations. Two sources of nonlinearity are considered in the continuum modeling of CNTs with the Euler-Bernoulli beam model describing the dynamics of the CNTs. One source is the geometric nonlinearity which may arise as a result of large deflections. The second source is due to van der Waals forces between the nanotubes which can be modeled as a nonlinear force to improve the accuracy of the physical model. After deriving the applicable variational principle, Hamilton’s principle is given. Natural and geometric boundary conditions are derived using the variational formulation of the problem. Several approximate and computational methods of solution such as Rayleigh-Ritz and finite elements employ the variational formulation of the problem and as such these principles are instrumental in obtaining the solutions of vibration problems under complicated boundary conditions.

scholarly journals A Paneitz–Branson type equation with Neumann boundary conditions

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Denis Bonheure ◽  
Hussein Cheikh Ali ◽  
Robson Nascimento
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Type Equation ◽  
Neumann Boundary ◽  
Rayleigh Quotient ◽  
Neumann Boundary Conditions ◽  
Asymptotic Estimates ◽  
Best Constant ◽  
Order Elliptic Problem ◽  
Nonconstant Solution

AbstractWe consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain threshold, namely, we prove that the best constant is achieved by a nonconstant solution of the associated fourth order elliptic problem under Neumann boundary conditions. Our arguments rely on asymptotic estimates of the Rayleigh quotient. We also show rigidity below another threshold.

Buckling analysis of functionally graded annular sector plates resting on elastic foundations

2010 ◽  
Vol 225 (2) ◽  
pp. 312-325 ◽  
Author(s):  
A Naderi ◽  
A R Saidi
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Buckling Load ◽  
Critical Buckling Load ◽  
Elastic Foundations ◽  
Annular Sector ◽  
Sector Plate ◽  
Annular Sector Plate

In this study, an analytical solution for the buckling of a functionally graded annular sector plate resting on an elastic foundation is presented. The buckling analysis of the functionally graded annular sector plate is investigated for two typical, Winkler and Pasternak, elastic foundations. The equilibrium and stability equations are derived according to the Kirchhoff's plate theory using the energy method. In order to decouple the highly coupled stability equations, two new functions are introduced. The decoupled equations are solved analytically for a plate having simply supported boundary conditions on two radial edges. Satisfying the boundary conditions on the circular edges of the plate yields an eigenvalue problem for finding the critical buckling load. Extensive results pertaining to critical buckling load are presented and the effects of boundary conditions, volume fraction, annularity, plate thickness, and elastic foundation are studied.

Study of Stokes dynamical system in a thin domain with Fourier and Tresca boundary conditions

2019 ◽  
pp. 2150007 ◽  
Author(s):  
Y. Letoufa ◽  
H. Benseridi ◽  
M. Dilmi
Keyword(s):  
Boundary Conditions ◽  
Variational Formulation ◽  
Reynolds Equation ◽  
Mixed Boundary ◽  
Three Dimensional ◽  
Mixed Boundary Conditions ◽  
Dynamic Regime ◽  
Thin Domain ◽  
Problem Statement ◽  
Stokes Fluid

Asymptotic analysis of an incompressible Stokes fluid in a dynamic regime in a three-dimensional thin domain [Formula: see text] with mixed boundary conditions and Tresca friction law is studied in this paper. The problem statement and variational formulation of the problem are reformulated in a fixed domain. In which case, the estimates on velocity and pressure are proved. These estimates will be useful in order to give a specific Reynolds equation associated with variational inequalities and prove the uniqueness.

Experiments on the Buckling of Simply-Supported Rectangular Plates

1969 ◽  
Vol 73 (703) ◽  
pp. 607-608 ◽  
Author(s):  
A. C. Mills
Keyword(s):  
Experimental Investigation ◽  
Theoretical Analysis ◽  
Boundary Conditions ◽  
Buckling Load ◽  
Theoretical Solution ◽  
Rectangular Plates ◽  
Uniform Load ◽  
Simply Supported

In ref. (1) Pope presents a theoretical analysis of the buckling of rectangular plates tapered in thickness under uniform load in the direction of taper. An experimental investigation into the end load buckling problem for a plate having simply-supported edges with the sides prevented from moving normally in the plane of the plate is described in ref. (2). For these boundary conditions the theoretical solution is exact. However, the compatability equation is not satisfied exactly when the sides are free to move in the plane of the plate. This experimental investigation demonstrates that the buckling load is nevertheless adequately predicted by the analysis in these circumstances.

scholarly journals Nonlinear Deformation of Flexible Orthotropic Shells of Variable Thickness in an Unsteady Magnetic Field

2019 ◽  
Vol 97 ◽  
pp. 02015 ◽  
Author(s):  
Zafar Abdullaev ◽  
Sayibdjan Mirzaev ◽  
Sobir Mavlanov
Keyword(s):  
Magnetic Field ◽  
Stress State ◽  
Electric Current ◽  
Variable Thickness ◽  
Nonlinear Deformation ◽  
Strain State ◽  
Orthotropic Shell ◽  
Mechanical Force ◽  
Time Varying ◽  
Orthotropic Shells

The analysis of the stress state of a flexible orthotropic shell under the influence of a time-varying mechanical force and a time-varying external electric current is performed, taking into account the mechanical and electromagnetic orthotropy. The effect of thickness on the stress-strain state of the orthotropic shell is investigated. The results obtained indicate the influence of thickness on the deformation of the shell and the need to take this factor into account in the design schemes.

scholarly journals Nonlocal strain gradient torsion of elastic beams: variational formulation and constitutive boundary conditions

2019 ◽  
Vol 90 (4) ◽  
pp. 691-706 ◽  
Author(s):  
R. Barretta ◽  
S. Ali Faghidian ◽  
Francesco Marotti de Sciarra ◽  
M. S. Vaccaro
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Variational Formulation ◽  
Elastic Beams ◽  
Nonlocal Strain Gradient

scholarly journals Study on Buckling Behavior of Tapered Friction Piles in Soft Soils with Linear Shaft Friction

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Junxiu Liu ◽  
Xianfeng Shao ◽  
Baoquan Cheng ◽  
Guangyong Cao ◽  
Kai Li
Keyword(s):  
Boundary Conditions ◽  
Soft Soil ◽  
Buckling Load ◽  
Lateral Stiffness ◽  
Equilibrium Equations ◽  
Tree Roots ◽  
Soft Soils ◽  
Buckling Behavior ◽  
Friction Pile ◽  
Shaft Friction

The buckling instability of long slender piles in soft soils is a key consideration in geoengineering design. By considering both the linear shaft friction and linear lateral stiffness of the soft soil, the buckling behaviors of a tapered friction pile embedded in heterogeneous soil are extensively studied. This study establishes and validates an analytical model to formulate the equilibrium equations and boundary conditions and then numerically solves the boundary value problem to obtain the critical buckling load and buckling shape by using software Matlab. The effects of boundary conditions, tapered ratio, stiffness ratio, friction ratio, lateral stiffness, and shaft friction on the buckling behavior of the friction pile are extensively explored. This study demonstrates that the buckling load decreases with the increase of friction ratio of the linear shaft friction. There exists an optimal tapered ratio corresponding to the maximum dimensionless buckling load in the tapered friction pile with linear shaft friction. The result means that the linear shaft friction should be considered in designing the tapered friction piles in heterogeneous soils. The results also have potential applications in the fields of growing of tree roots in soils, moving of slender rods in viscous fluids, penetrating of fine rods in soft elastomers, etc.

scholarly journals Variational Principles for Multiwalled Carbon Nanotubes Undergoing Vibrations Based on Nonlocal Timoshenko Beam Theory

2010 ◽  
Vol 2010 ◽  
pp. 1-7 ◽  
Author(s):  
Ismail Kucuk ◽  
Ibrahim S. Sadek ◽  
Sarp Adali
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Multiwalled Carbon Nanotubes ◽  
Timoshenko Beam ◽  
Variational Principles ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Small Scale ◽  
Multiwalled Carbon ◽  
Nonlocal Timoshenko Beam Theory

Variational principles are derived for multiwalled carbon nanotubes undergoing linear vibrations using the semi-inverse method with the governing equations based on nonlocal Timoshenko beam theory which takes small scale effects and shear deformation into account. Physical models based on the nonlocal theory approximate the nanoscale phenomenon more accurately than the local theories by taking small scale phenomenon into account. Variational formulation is used to derive the natural and geometric boundary conditions which give a set of coupled boundary conditions in the case of free boundaries which become uncoupled in the case of the local theory. Hamilton's principle applicable to this case is also given.

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