A mapping finite difference model for infinite elastic media

A simple mapping finite difference model is presented for the solution of boundary-value problems in the theory of time-harmonic elastic vibrations. The finite problem domain is condensed by mapping into a smaller finite domain using a suitable coordinate transformation. The field equations and the boundary conditions are also appropriately transformed. The radiation condition at infinity is satisfied through a change of the dependent variable. Finite difference forms of the transformed equations are then solved in the mapped domain, subject to the transformed boundary conditions.     

Snap-through buckling of a shallow arch resting on a two-parameter elastic foundation

In this paper, we study the static and dynamic snap-through of a shallow arch resting on a two-parameter elastic foundation under a point load moving at a constant speed. The deformation of the arch is expressed in Fourier series. We extend the previous works when the arch resting on a certain model of two-parameter elastic foundation. Each model of foundation has a different definition for the second foundation parameter, where the model is discussed in this paper is Pasternak model. For quasi-static analysis, it is noted that the first four modes in the expansion are sufficient to predict the response of the arch. Similarly, when the point load moves with a significant speed, we integrate the equation of motion numerically using the first modes in the expansion.     

Mathematical modelling of unsteady flow of a Sisko fluid through an anisotropically tapered elastic arteries with time-variant overlapping stenosis

In the present investigation, we have studied the influence of heat and chemical reactions on blood flow through anisotropically tapered elastic artery with time-variant overlapping stenosis. The nature of blood in small arteries are analyzed mathematically by considering it as a Sisko fluid. The analysis is carried out for an artery with a mild stenosis. Analytical expressions for the axial velocity, the stream function, the temperature distribution, the concentration of fluid, the pressure gradient, the resistance impedance and the wall shear stress distribution have been computed numerically and the results were studied for various values of the physical parameters, such as the Sisko parameter b^∗, the power index n, the taper angle ?  , the maximum height of stenosis δ^∗${\delta }^{\ast }$, the Soret number S_r, the Brickmann number B_r, the total mass of the vessel and the surrounding tissues M and the longitudinal contributions of the viscous and elastic constraints to the total tethering C and K respectively. The obtained results show that the magnitude of the axial velocity is higher for a Newtonian fluid than that for a Sisko fluid and it decreases by increasing of the power index n also the transmission of axial velocity curves through a tethered tube is substantially higher than that through the free tube. The wall shear stress distribution and resistance impedance profiles with the time have an oscillation form through the tapered overlapping stenosed arteries and this oscillation decaying as the time increases. The temperature profile increase by increasing the Sisko parameter b^∗ and the power index n   but the concentration profile has an opposite behavior as compared to the temperature profile. For a fixed flux, the magnitude of the pressure drop for a shear-thinning fluid (n<1)$(n<1)$ is much larger than that through a shear-thickening (n>1)$(n>1)$. The stream lines separate and the trapping bolus appear by increasing the maximum height of the stenosis δ^∗${\delta }^{\ast }$. The trapping bolus increase in size toward the line center of the tube as the power index n increases and it appear gradually by increasing the Sisko parameter b^∗. Finally the size of trapped bolus for the stream lines in the free isotropic tube is smaller than those in the tethered tube.     

Vibration characteristics of single-walled carbon nanotubes based on an anisotropic elastic shell model including chirality effect

Single-walled carbon nanotubes (SWCNTs) exhibit remarkable chirality-dependent mechanical phenomena. In present paper, an anisotropic elastic shell model is developed to study the vibration characteristics of chiral SWCNTs. Analytical solution is presented by using the Flügge shell theory and complex method. The suggested model is justified by a good agreement between the present results and some experimental and numerical available data in literature. Furthermore, the model is used to elucidate the effect of tube chirality on the frequencies of SWCNTs. Finally, the influences of the externally applied mid-face axial force and torque on longitudinal, radial and torsional frequencies of SWCNTs are investigated.     

一类弹性梁的能量衰减估计

研究一类弹性梁的耗散性.首先应用非线性算子半群理论证明了系统的适定性;进而运用能量方法结合乘子技巧得到弹性梁系统的能量衰减估计.     

Edge resonance in an elastic semi-infinite cylinder

We study the three-dimensional elasticity operator in a semi-infinite circular cylinder subject to free boundary conditions, in the case of zero Poisson ratio. We prove, adapting the method from [15], i.e., by first finding an invariant subspace for the elasticity operator such that the essential spectrum has a strictly positive lower bound and then finding a test function in this space for which the variational quotient takes a value below the bottom of the essential spectrum, that there is an eigenvalue embedded in the continuous spectrum. Physically, an eigenvalue corresponds to a "trapped mode", that is, a harmonic oscillation localized near the edge. This effect, known in mechanics as the "edge resonance" has been extensively studied numerically and experimentally. Our paper extends the mathematical justification of such phenomena provided by [15] to a three-dimensional setting     

Mathematical modelling of spark-ignition engines

The flow field, scavenging efficiency, power output, heat transfer losses, and unburned hydrocarbon emissions have been numerically studied by means of a two-equation model of turbulence in a four-stroke, homogeneous-charge, spark-ignition engine. The engine is equipped with an intake valve, an exhaust valve, and a constant rate heat source which simulates the spark plug. Combustion has been modelled by means of a one-step irreversible chemical reaction whose rate is controlled by an Arrhenius-type expression. The numerical results indicate that the intake stroke is characterized by the formation of two eddies which persist in the compression stroke. Turbulence is generated at the shear layers of the air jet drawn into the cylinder, but its level decreases in the compression stroke. Due to the heat released by the spark plug and the chemical reaction, a spherical flame kernel is formed. This kernel evolves into a cylindrical flame when the flame front reaches the piston. Fuel remains unburnt at the corner between the cylinder head and the cylinder wall due to heat transfer losses. The numerical results also indicate that despite uncertainties about the turbulence and heat transfer models, an engine model such as the one studied here can be used to understand the flow field, heat transfer losses, scavenging efficiency, and power output in conventional spark-ignition engines. Such capabilities are very helpful in the development and optimization stages of engines. For example, here the engine model thermal and scavenging efficiencies are 15.69% and 94%, respectively. The peak pressure is 33 atm and occurs at 6° ATDC. The unburnt hydrocarbon emissions are 7.41% of the total fuel admitted into the cylinder.     

Finite element modelling of surface waves in a channel

A variational formulation of the vertically-integrated differential equations for free surface wave motion is presented. A finite element model is derived for solving this nonlinear system of hydrodynamic equations. The time integration scheme employed is discussed and the results obtained demonstrate its good stability and accuracy.Several applications of the model are considered: the first problem is an open channel of uniform depth and the second an open channel of linearly varying depth. The ‘inflow’ boundary condition is prescribed in terms of the velocity which represents a wavemaker and/or a flow source, while the ‘outflow’ boundary condition is specified in terms of the water elevation. The outflow condition is adjusted for two cases, a reflecting boundary (finite channel) and a non-reflecting boundary (open-ended channel). The latter boundary condition is examined in some detail and the results obtained show that the numerical model can produce the non-reflecting boundary that is similar to the analytical radiation condition for waves. Computational results for a third problem, involving wave reflection from a submerged cylinder, are also presented and compared with both experimental data and analytical predictions.The simplicity and the performance of the computational model suggest that free surface waves can be simulated without excessively complicated numerical schemes. The ability of the model to simulate outflow boundary conditions properly is of special importance since these conditions present serious problems for many numerical algorithms.     

The Reissner-Sagoci type problem for a non-homogeneous elastic cylinder embedded in an elastic non-homogeneous half-space

The problem considered is that of the torsion of a non-homogeneouselastic cylinder, which is embedded in a non-homogeneous elastichalf-space (matrix) of different rigidity modulus. A rigid discis bonded to the flat surface of the cylinder and torque isapplied to the cylinder through a rigid disc. It is assumedthat there is perfect bonding at the common cylindrical surface.Using integral transformation techniques the solution of theproblem is reduced to dual integral equations. Later on thesolution of the dual integral equations is transformed intothe solution of a Fredholm integral equation of the second kind.Solving the Fredholm integral equation numerically the numericalresults for torque and shear stress inside the cylinder areobtained and displayed graphically to demonstrate the effectof non-homogeneity of the elastic material on the torque andshear stress.     

Some studies on mathematical models for general elastic multi-structures

The aim of this paper is to study the static problem about a general elastic multi-structure composed of an arbitrary number of elastic bodies, plates and rods. The mathematical model is derived by the variational principle and the principle of virtual work in a vector way. The unique solvability of the resulting problem is proved by the Lax-Milgram lemma after the presentation of a generalized Korn's inequality on general elastic multi-structures. The equilibrium equations are obtained rigorously by only assuming some reasonable regularity of the solution. An important identity is also given which is essential in the finite element analysis for the problem.     

Existence theory for linearly elastic shells

We review existence and uniqueness results, recently obtained for three of the most important linear two-dimensional shell models: Koiter’s model, the bending model and the membrane model. They rely on a crucial lemma of J L Lions, used in an essential way for establishing in each case a generalized Korn’s inequality, which is then combined with a generalized rigid displacement lemma of a geometrical nature. Dedicated to the memory of Professor K G Ramanathan     

Quasistatic evolution of damage in an elastic body with random inputs

Abstract

A model for the quasistatic evolution of the motion of an elastic body which is subject to material damage is presented and analyzed. The model takes the form of an elliptic system for the displacements coupled with a parabolic inclusion for the damage field. In both the system and the inclusion the coefficient and input functions that are present are assumed to be stochastic processes dependent on a random variable. The existence of a weak solution to the model is established using a sequence of approximate problems and passage to a limit. Moreover, the weak solution is shown to be product measurable.     

Coupling of numerical Green''s matrix and boundary integral equations for the elastodynamic analysis of inhomogeneous bodies on an elastic half-space

A coupled system of integral equations (of the domain and boundary types) is formulated for the elastodynamic response analysis of a locally inhomogeneous body on a homogeneous elastic half-space. The method uses the fundamental solution for homogeneous elastostatics in the inhomogeneous domain owing to the lack of a fundamental solution in inhomogeneous elastodynamics.

The integral representation of displacements in the inhomogeneous domain is formulated with the help of this elastostatic fundamental solution by considering the term induced by the inhomogeneity of materials and the acceleration term as the body force term. Then the Green's matrix is obtained numerically from this integral representation and combined with the ordinary boundary integral equations, which are valid in the exterior homogeneous half-space.

Some numerical examples show the efficiency and the versatility of this coupled method.     

Application of Smoothed Particle Hydrodynamics for modelling gated spillway flows

Computational models of spillways are important for evaluating and improving dam safety, optimising spillway design and updating operating conditions. Traditionally, scaled down physical models have been used for validation and to collect hydraulic data. Computational fluid dynamics (CFD) models however provide advantages in time, cost and resource reduction. CFD models also provide greater efficiency when evaluating a range of spillway designs or operating conditions. Within the present literature, most studies of computational spillway models utilise a mesh-based method. In this work we use the particle based method of Smoothed Particle Hydrodynamics (SPH) to model weir flow through a four bay, gated, spillway system. Advantages of SPH for such modelling include automatic representation of the free surface flow behaviour due to the Lagrangian nature of the method, and the ability to incorporate complex and dynamic boundary objects such as gate structures or debris. To validate the SPH model, the reservoir water depth simulated is compared with a related physical study. The effect of SPH resolution on the predicted water depth is evaluated. The change in reservoir water level with discharge rates for weir flow conditions is also investigated, with the difference in simulated and experimental water depths found to range from 0.16% to 11.48%. These results are the first quantitative validation of the SPH method to capture spillway flow in three dimensions. The agreement achieved demonstrates the capability of the SPH method for modelling spillway flows.     

The singing wineglass: an exercise in mathematical modelling

Lecturers in mathematical modelling courses are always on the lookout for new examples to illustrate the modelling process. A physical phenomenon, documented as early as the nineteenth century, was recalled: when a wineglass ‘sings’, waves are visible on the surface of the wine. These surface waves are used as an exercise in mathematical modelling. Based on assumptions about the wine in the glass and observations illustrated with photographs, a mathematical problem is set up. This problem includes a non-homogeneous Neumann boundary condition on the lateral side of the glass. The solution to the mathematical problem is animated using Mathematica?. The predictions of the model are tested by comparing them with the known facts. The predictions of the model agree with the actual observations.     

Software for interpolating and modelling unknown functions

Software has been developed for fitting a stochastic process model to multi-dimensional data. Applications include contouring, cross-section plotting and optimization. The behaviour of the variance and the gradient of the interpolating function in the near neighbourhood of closely adjacent data points has been investigated.     

Asymptotic analysis of contact problems between an elastic material and thin-rigid plates

We consider an elastic material in contact with a three-dimensional rigid plate of varying thickness. We suppose that a perfect adhesion occurs along thin zones disposed in a self-similar way on the interface between the two materials. We suppose that the elasticity coefficients in the plate depend on its thickness and tend to infinity as this thickness tends to zero. We derive the effective material properties using Γ-convergence methods.     

New mathematical modelling and simulation of an industrial accumulator for elastic webs

This paper concerns the modelling of an accumulator used in industrial elastic web processing plant such as paper mills, fabric, rolling mills etc. Accumulators are used to allow rewind or unwind core changes while the process continues at a constant web velocity. A new nonlinear model of a pneumatic actuated industrial accumulator including pneumatic jack model, static friction representation and web weight is first detailed which enables to deduce a linear model. These models are derived from physical laws that describe web tension and velocity dynamics in each web span. In a second part, the effects of time-varying mechanical parameters, such as web Young modulus, web length and rollers inertia on accumulator dynamics are presented. The performances of the modeled accumulator are illustrated by simulations in Matlab/Simulink software environment.