Large deflections of cantilever beams of non-linear elastic material under a combined loading

Large deflection of cantilever beams made of Ludwick type material subjected to a combined loading consisting of a uniformly distributed load and one vertical concentrated load at the free end was investigated. Governing equation was derived by using the shearing force formulation instead of the bending moment formulation because in the case of large deflected member, the shearing force formulation possesses some computational advantages over the bending moment formulation. Since the problem involves both geometrical and material non-linearities, the governing equation is complicated non-linear differential equation, which would in general require numerical solutions to determine the large deflection for a given loading. Numerical solution was obtained by using Butcher's fifth order Runge-Kutta method and are presented in a tabulated form.     

Non-linear propagation of laser beam and focusing due to self-action in optical fiber: Non-paraxial approach

A somewhat more general analysis for solving spatial propagation characteristics of intense Gaussian beam is presented and applied to the laser beam propagation in step-index profile as well as parabolic profile dielectric fibers with Kerr non-linearity. Considering self-action due to saturating and non-saturating non-linearity in the refractive index, a general theory has been developed without any kind of power series expansion for the dielectric constant as is usually done in other theories that make use of paraxial approximation. Result of the steady state self-focusing analysis indicates that the Kerr non-linearity acts as a perturbation on the radial inhomogeneity due to fiber geometry. Analysis indicates that the paraxial rays and peripheral rays focus at different points, indicating aberration effect. Calculated critical power matches with the experimentally reported result.     

On the controllability for the Khokhlov–Zabolotskaya–Kuznetsov-like equation

Recalling the proprieties of the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation, we prove the controllability of moments result for the linear part of the KZK equation and its non-linear perturbation.     

Sharp transition between extinction and propagation of reaction

We consider the reaction-diffusion equation

on with and . In 1964 Kanel proved that if is an ignition non-linearity, then as when , and when L_1$">. We answer the open question of the relation of and by showing that . We also determine the large time limit of in the critical case , thus providing the phase portrait for the above PDE with respect to a 1-parameter family of initial data. Analogous results for combustion and bistable non-linearities are proved as well.

     

Characteristics and application of laser-generated acoustic shock waves in air

When a high-power laser beam is focused at a point, the air at the focal point is heated to temperatures of thousands of degrees within several nanoseconds and breaks down. This generates a spark that, in turn, is accompanied by an acoustic shock wave. The acoustic shock waves generated by focussing the beam from a pulsed laser with a 1064 nm wavelength and a power of 800 mJ per pulse have been measured using 1/4″ and 1/8″ B&K microphones. Nonlinear sound levels are observed up to 1.5 m from the laser-induced sparks. Beyond a certain region close to the source, levels are found to decrease in a manner consistent with spherical spreading plus nonlinear hydrodynamic losses. Analysis of the waveforms shows that the acoustic pulses associated with the laser-induced sparks are more repeatable and have higher intensity than those from an electrical spark source. Laser-generated acoustic shock waves are ideal for simulating a blast wave or a sonic boom in the laboratory and for studying the associated propagation effects. To illustrate this application, the propagation of the laser generated shock waves over a series of different hard, rough surfaces has been investigated. The results show the distinctive influences of ground roughness on the propagation of the shock wave.     

Fabrication of PbS Nanoparticles Embedded in Silica Gel by Reverse Micelles and Sol-Gel Routes

PbS doped-silica gels showing a visible absorption onset were prepared by the sol-gel method. PbS nanoparticles with strong quantum-confinement effect were obtained from sodium sulfide and lead nitrate by the reverse micelle method. Chemical parameters such as the water/surfactant and the [Pb²⁺]/[S²⁻] ratios play a very important role in the PbS particle size and in their absorption threshold. The PbS nanoparticles were dispersed in a hydrolyzed solution of TEOS and converted to homogeneous gels after heating. The absorption threshold of PbS doped-gel is blue shifted compared to the one of the as-prepared PbS particles. The non-linear optical properties of the PbS nanoparticle solution were measured by degenerate four-wave mixing and theX ⁽³⁾ value was estimated to be 1.95 10⁻¹¹ esu.     

Large amplitude free vibration of shear deformable laminated composite annular sector plates by a sector p-element

A sector p-element is presented for the large amplitude free vibration analysis of laminated composite annular sector plates. The effects of out-of-plane shear deformations, rotary inertia, and geometric non-linearity are taken into account. The shape functions are derived from the shifted Legendre orthogonal polynomials. The element stiffness and mass matrices are integrated analytically with the aid of symbolic computing. The method consists of modeling the annular sector plate as one element. The accuracy of the solution is improved simply by increasing the polynomial order. The time-dependent coefficients are described by a truncated Fourier series. The equations of free motion are obtained using the harmonic balance method and solved by the linearized updated mode method. Results for the linear and non-linear frequencies of clamped laminated composite annular sector plates are obtained. The case of a clamped isotropic annular sector plate is also shown. The linear frequencies are found to converge rapidly downwards as the polynomial order is increased. Comparisons of the linear frequencies with published results show excellent agreement. The effects of sector angle, inner-to-outer radius ratio, thickness-to-outer radius ratio, moduli ratio, number of plies, and layup sequence on the backbone curves are also investigated. It is shown that the hardening behavior increases or decreases depending on geometric and lamination parameters.     

Nonlinear natural frequency of shallow conical shells with variable thickness

IntroductionTheplatesandtheshellswithvariablethicknessarewidelyusedinengineering .Theproblemaboutstaticshasbeenstudiedbymanyscholars;therearemanyRefs .[1 -4 ]inthisfield .Papersaboutnonlineardynamicsaremuchless[5 ,6 ].Inthispaper,selectingthemaximumamplitudeinthecenterofshallowconicalshellswithvariablethicknessasperturbationparameter,thenonlinearnaturalfrequencyofshallowconicalshellswithvariablethicknessisobtainedbymethodgiveninRef.[7] .Thenonlinearnaturalfrequencyisnotonlyconnectedwiththeva…     

Size-dependent dynamic pull-in analysis of geometric non-linear micro-plates based on the modified couple stress theory

This paper focuses on the size-dependent dynamic pull-in instability in rectangular micro-plates actuated by step-input DC voltage. The present model accounts for the effects of in-plane displacements and their non-classical higher-order boundary conditions, von Kármán geometric non-linearity, non-classical couple stress components and the inherent non-linearity of distributed electrostatic pressure on the micro-plate motion. The governing equations of motion, which are clearly derived using Hamilton's principle, are solved through a novel computationally very efficient Galerkin-based reduced order model (ROM) in which all higher-order non-classical boundary conditions are completely satisfied. The present findings are compared and successfully validated by available results in the literature as well as those obtained by three-dimensional finite element simulations carried out using COMSOL Multyphysics. A detailed parametric study is also conducted to illustrate the effects of in-plane displacements, plate aspect ratio, couple stress components and geometric non-linearity on the dynamic instability threshold of the system.     

Surface and non-local effects for non-linear analysis of Timoshenko beams

In this paper, we present a non-local non-linear finite element formulation for the Timoshenko beam theory. The proposed formulation also takes into consideration the surface stress effects. Eringen׳s non-local differential model has been used to rewrite the non-local stress resultants in terms of non-local displacements. Geometric non-linearities are taken into account by using the Green–Lagrange strain tensor. A C⁰ beam element with three degrees of freedom has been developed. Numerical solutions are obtained by performing a non-linear analysis for bending and free vibration cases. Simply supported and clamped boundary conditions have been considered in the numerical examples. A parametric study has been performed to understand the effect of non-local parameter and surface stresses on deflection and vibration characteristics of the beam. The solutions are compared with the analytical solutions available in the literature. It has been shown that non-local effect does not exist in the nano-cantilever beam (Euler–Bernoulli beam) subjected to concentrated load at the end. However, there is a significant effect of non-local parameter on deflections for other load cases such as uniformly distributed load and sinusoidally distributed load (Cheng et al. (2015) [10]). In this work it has been shown that for a cantilever beam with concentrated load at free end, there is definitely a dependency on non-local parameter when Timoshenko beam theory is used. Also the effect of local and non-local boundary conditions has been demonstrated in this example. The example has also been worked out for other loading cases such as uniformly distributed force and sinusoidally varying force. The effect of the local or non-local boundary conditions on the end deflection in all these cases has also been brought out.