scholarly journals Addendum to ‘Sharp inequalities that generalize the divergence theorem: an extension of the notion of quasi-convexity’

2015 ◽  
Vol 471 (2176) ◽  
pp. 20140886 ◽  
Author(s):  
Graeme W. Milton
Keyword(s):  
Quadratic Form ◽  
Boundary Conditions ◽  
Lower Bounds ◽  
Wide Class ◽  
Quadratic Functions ◽  
General Boundary ◽  
Divergence Theorem ◽  
General Boundary Conditions ◽  
Lagrange Equations ◽  
Quasi Convexity

The paper ‘Sharp inequalities that generalize the divergence theorem: an extension of the notion of quasi-convexity’ published in Proc. R. Soc. A 2013, 469, 20130075 ( doi:10.1098/rspa.2013.0075 ) is clarified. Notably, much more general boundary conditions are given under which sharp lower bounds on the integrals of certain quadratic functions of the fields can be obtained. More precisely, if the quadratic form is Q *-convex then any solution of the Euler–Lagrange equations will necessarily minimize the integral. As a consequence, strict Q *-convexity is found to be an appropriate condition to ensure uniqueness of the solutions of a wide class of linear Euler–Lagrange equations in a given domain Ω with appropriate boundary conditions.

Effect of the multiple projectile on the low-velocity impact response of CNTs reinforced beam

2020 ◽  
Vol 17 (1) ◽  
pp. 1-17
Author(s):  
Ali Sadik Gafer Qanber ◽  
Raed Salman Saeed Alhusseini ◽  
Bashar Dheyaa Hussein Al-Kasob ◽  
Manar Hamid Jasim ◽  
Mehdi Ranjbar
Keyword(s):  
Boundary Conditions ◽  
Volume Fraction ◽  
Ritz Method ◽  
Low Velocity Impact ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Lagrange Equations ◽  
Low Velocity ◽  
Content Type ◽  
Velocity Impact

PurposeThe main objective of this article is to develop a theoretical formulation for predicting the response of CNTs reinforced beam under multiple impactors with general boundary conditions, using first-order shear deformation beam theory.Design/methodology/approachThe rule of mixtures is implemented to derive the material properties of the beam. The nonlinear Hertz contact law is applied for simulation between impactors and the surface of the beam. A combination of approaches includes energy method, Ritz method and generalized Lagrange equations are used to extract the matrix form of equations of motion. The time-domain solution is obtained using implementing the well-known Runge Kutta 4th order method.FindingsAfter examining the accuracy of the present method, the effects of the number of impactors include one impactor, and three impactors in various CNTs volume fraction are studied for CNTs reinforced beam with clamped-clamped, clamped-free and simply supported boundary conditions under the low-velocity impact. The most important finding of this article is that contact force and beam indentation at the middle of the beam in the case of one impactor are greater than those reported in the case of three impactors.Originality/valueThis article fulfills an identified need to study how CNTs reinforced beam behaviour with general boundary conditions under multiple low-velocity impacts can be enabled.

Flexural Vibration Response of Beams With General Boundary Conditions: A Transfer Matrix Approach

1991 ◽  
Author(s):  
T. Önsay
Keyword(s):  
Boundary Conditions ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Vibration Response ◽  
General Boundary ◽  
Phase Variable ◽  
General Boundary Conditions ◽  
Transfer Matrices ◽  
Analytical Results

Abstract The wave-mode representation is utilized to obtain a more efficient form to the conventional transfer matrix method for bending vibrations of beams. The proposed improvement is based on a phase-variable canonical state representation of the equation governing the time-harmonic flexural vibrations of a beam. Transfer matrices are obtained for external forces, step-change of beam properties, intermediate supports and for boundaries. The transfer matrices are utilized to obtain the vibration response of a point-excited single-span beam with general boundary conditions. The general characteristic equation and the transfer mobility of a single-span beam are determined. The application of the analytical results are demonstrated on physical structures with different boundary conditions. A hybrid model is developed to incorporate measured impedance of nonideal boundaries into the transfer matrix method. The analytical results are found to be in excellent agreement with experimental measurements.

Vibration Analysis of Laminated Composite Rectangular Plates With General Boundary Conditions

2018 ◽  
Author(s):  
Yu Fu ◽  
Jianjun Yao ◽  
Zhenshuai Wan ◽  
Gang Zhao
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Laminated Composite ◽  
Free Vibration Analysis ◽  
Geometric Parameters ◽  
General Boundary ◽  
Rectangular Plates ◽  
General Boundary Conditions ◽  
Benchmark Solutions

In this investigation, the free vibration analysis of laminated composite rectangular plates with general boundary conditions is performed with a modified Fourier series method. Vibration characteristics of the plates have been obtained via an energy function represented in the general coordinates, in which the displacement and rotation in each direction is described as an improved form of double Fourier cosine series and several closed-form auxiliary functions to eliminate any possible jumps and boundary discontinuities. All the expansion coefficients are then treated as the generalized coordinates and determined by Rayleigh-Ritz method. The convergence and reliability of the current method are verified by comparing with the results in the literature and those of Finite Element Analysis. The effects of boundary conditions and geometric parameters on the frequencies are discussed as well. Finally, numerous new results for laminated composite rectangular plates with different geometric parameters are presented for various boundary conditions, which may serve as benchmark solutions for future research.

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