Elastic plates under bending solved by pseudocaustics

The method pf pseudocaustics was applied to the study of out-of-plane bending in elastic plates. It is shown that for bending problems where the loading mode is given, the method determines experimentally the complex potential function at selected points along the boundaries. A conformal mapping of the closed smooth curves of each boundary of the plate on to a unit circle allows the determination of the complex potential ϕ (ζ), expressed in the form of a Laurent series. This in turn yields the complete solution of the bent plate. In order to show the efficiency of the method it was applied to two typical examples of thin infinite plates in cylindrical bending, having either a circular central hole, or a square hole. The experimental results corroborate the theoretical results, thus proving that this combined theoretical and experimental method is suitable for solving elastic problems in applications with high accuracy, where other methods fail to yield satisfactory results.     

Experimental study of plane elastic contact problems by the pseudocaustics method

The optical method of pseudocaustics can be used for the experimental solution of plane elasticity, smooth contact problems for finite or infinite media in contact of arbitrary shape. This technique constitutes an alternative to the various numerical and experimental techniques for the approximate solution of plane elasticity contact problems. The success in the application of the method of pseudocaustics to plane elasticity contact problems is due to the possibility inherent in this method of the direct determination of the derivative Φ'(z) of the complex potential Φ(z) of N.I. Muskhelishvili along the boundaries of the media in contact. Then, the Muskhelishvili complex potentials, Φ(z) and Ψ(z), completely characterizing the state of stress and strain in a plane elastic medium, can easily be determined at any point of the media in contact after simple algebraic calculations. Two applications of the proposed method to contact problems of practical interest are also made.     

Efficient computation of order and mode of three-dimensional stress singularities in linear elasticity by the boundary finite element method

The Boundary Finite Element Method (BFEM), a novel semi-analytical boundary element procedure solely relying on standard finite element formulations, is employed for the investigation of the orders and modes of three-dimensional stress singularities which occur at notches and cracks in isotropic halfspaces as well as at free edges and free corners of layered plates. After a comprehensive literature review and a concise introduction to the standard three-dimensional BFEM formulation for the static analysis of general unbounded structures, we demonstrate the application of the BFEM for the computation of the orders and modes of two-dimensional and three-dimensional stress singularities for several classes of problems within the framework of linear elasticity. Special emphasis is placed upon the investigation of stress concentration phenomena as they occur at straight free edges and at free corners of arbitrary opening angles in composite laminates. In all cases, the BFEM computations agree excellently with available reference results. The required computational effort is found to be considerably lower compared to e.g. standard Finite Element Method (FEM) computations. In the case of free laminate corners, numerous new results on the occurring stress singularities are presented. It is found that free-corner problems generally seem to involve a more pronounced criticality than the corresponding free-edge situations.     

Solutions of the generaln-th order variable coefficients linear difference equation

In this paper.variable operator and its product with shifting operator are studied.The product of power series of shifting operator with variable coefficient is defined andits convergence is proved under Mikusinski’s sequence convergence.After turning ageneral variable coefficient linear difference equation of order n into a set of operatorequations.we can obtain the solutions of the general n-th order variable coefficientlinear difference equation.     

On the intensity of linear elastic high order singularities ahead of cracks and re-entrant corners

The paper deals with high order elastic singular terms at cracks and re-entrant corners (sharp V-notches), which are commonly omitted in linear elastic analyses by the argument that the strain energy and displacements in the near-tip region should be bounded. The present analysis proves that these terms are fully included in the elastic part of complete elastic–plastic stress and strain solutions.The intensities of high order singular terms are found to be linked to the linear elastic stress intensity factor and the extension of the plastic zone along the crack bisector line. The smaller the plastic radius, the smaller the intensities of high order singular terms are.A physical justification of the existence of high order singular terms is provided on the basis of the strain energy density distribution detected along the crack bisector line. Finally, the influence of the V-notch opening angle is made explicit, discussing also the relationship between the singularity orders and the solution of a Williams’ type sinusoidal eigen-equation.     

Design of analog variable fractional order differentiator and integrator

This paper deals with the design of analog variable fractional order differentiator s ^m and integrator s ^?m , for 0<m<1, for a given frequency band, a subject that has not been yet investigated. The main feature of this analog variable fractional order integrator or differentiator is that its frequency characteristics can be changed without redesigning a new one. First, analog rational function approximation of the fractional order differentiator s ^m and integrator s ^?m are derived with the new idea to keep all its poles to be independent of the fractional orders?m. Next, we have used the polynomial interpolation method to design the variable fractional order analog integrator and differentiator that can be implemented by an analog structure like the digital Farrow structure. Finally, some examples are presented to illustrate the efficiency and the effectiveness of the proposed design method.     

Evaluation of the second order elastic moduli

The expressions of the apparent linear elastic moduli and their first and second derivatives, with respect to hydrostatic pressure, are obtained according to the second order elasticity theory. As a particular case when the material is hyperelastic, formulae of the first derivatives of the linear elastic moduli reduce to those obtained by Seeger and Buck.     

Fractional integration and differentiation of variable order: an overview

We give an overview of a selection of studies on fractional operations of integration and differentiation of variable order, when this order may vary from point to point. We touch on both the Euclidean setting and also the general setting within the framework of quasimetric measure spaces.     

Slip effects on the flow of a third order fluid with variable suction

The effect of slip condition on the flow of third order fluid past a porous plate with variable suction is investigated. Perturbation solution of the resulting problem is derived. Several limiting solutions have been deduced. Graphs are plotted and discussed.     

反平面V形切口塑性应力奇异性分析

论文提出了用插值矩阵法计算幂硬化塑性材料反平面V形切口和裂纹尖端区域的应力奇异性.首先在切口和裂纹尖端区域采用自尖端径向度量的渐近位移场假设,将其代入塑性全量理论的基本微分方程后,推导出包含应力奇异性特征指数和特征角函数的非线性常微分方程特征值问题.然后采用插值矩阵法迭代求解导出的控制方程,得到一般的塑性材料反平面V形切口和裂纹的前若干阶应力奇异阶和相应的特征角函数,该法的重要优点是以上求解的特征角函数和它们各阶导函数具有同阶精度,并且一次性地求出前若干阶特征对.同时,插值矩阵法计算量小,易于和其他方法联合使用,这些优点在后续求解尖端区域完全应力场非常优越.论文方法的计算结果与现有结果对照,发现吻合良好,表明了论文方法的有效性.     

Approximation and application of the Riesz-Caputo fractional derivative of variable order with fixed memory

Meccanica - In this paper, the Riesz-Caputo fractional derivative of variable order with fixed memory is considered. The studied non-integer differential operator is approximated by means of...     

A single cell high order scheme for the convection-diffusion equation with variable coefficients

A new finite difference scheme for the convection-diffusion equation with variable coefficients is proposed. The difference scheme is defined on a single square cell of size 2h over a 9-point stencil and has a truncation error of order h⁴. The resulting system of equations can be solved by iterative methods. Numerical results of some test problems are given.     

Evaluation of fracture mechanics parameters for free edges in multi-layered structures with weak singularities

This study focuses on the stress intensity factors for free edges in multi-layered structural components. The effects of elastic constants of various material combinations on the weak singularity at free edges are analyzed. Using the H-integral approach, the effects of elastic mismatch parameters, the bond area and the thickness of the thin metal layer on the stress intensity factor are quantified and the results are compared with detailed finite element solutions. A good agreement between numerical predictions obtained from the H-integral and the detailed FE results is achieved, showing the applicability of this approach. Similar to a crack problem, the singular elastic field dominates in an annular region adjacent to the notch tip. The relationship between the valid range of the K-dominated field and the thin-film thickness is then demonstrated. Furthermore, the competition of crack initiation between the free edge interface (180° opening angle) and a 90° notch interface in a generic specimen is investigated, in order to find out which is the prevailing failure mode. Comparison between isotropic Si and anisotropic Si substrate is also illustrated. Anisotropy of the Si substrate has a significant influence on the stress intensity factor when combined with an Au or Al metal layer but not with a Cu layer. Additionally, standardized numerical formulae of the dimensionless stress intensity factor have been derived to guide the engineering application.     

The occurrence of singularities in solutions of homogeneous systems of two first order quasilinear hyperbolic equations with smooth initial data

An unfolding procedure for an arbitrary initial curve in the hodograph plane is used to determine the influence of initial values on the breakdown of the solution to quasi-linear hyperbolic systems involving two first order homogeneous equations in one space dimension and time. The notion of additively separable Riemann invariants is introduced and a general expression for the critical time at which breakdown occurs in systems with this property is obtained and then applied to several physical examples. Some important properties of a system that is both reducible and strictly exceptional are also derived establishing its connection with the linear homogeneous wave equation.     

Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation

We investigate the dynamics and control of a nonlinear oscillator that is described mathematically by a Variable Order Differential Equation (VODE). The dynamic problem in question arises from the dynamical analysis of a variable viscoelasticity oscillator. The dynamics of the model and the behavior of the variable order differintegrals are shown in variable phase space for different parameters. Two different controllers are developed for the VODEs under study in order to track an arbitrary reference function. A generalization of the van der Pol equation using the VODE formulation is analyzed under the light of the methods introduced in this work.     

On the pressure and stress singularities induced by steady flows of incompressible viscous fluids

Design for structural integrity requires an appreciation of where stress singularities can occur in structural configurations. While there is a rich literature devoted to the identification of such singular behavior in solid mechanics, to date there has been relatively little explicit identification of stress singularities caused by fluid flows. In this study, stress and pressure singularities induced by steady flows of viscous incompressible fluids are asymptotically identified. This is done by taking advantage of an earlier result that the Navier-Stokes equations are locally governed by Stokes flow in angular corners. Findings for power singularities are confirmed by developing and using an analogy with solid mechanics. This analogy also facilitates the identification of flow-induced log singularities. Both types of singularity are further confirmed for two global configurations by applying convergence-divergence checks to numerical results. Even though these flow-induced stress singularities are analogous to singularities in solid mechanics, they nonetheless render a number of structural configurations singular that were not previously appreciated as such from identifications within solid mechanics alone.     

The stability of zero solution of second order system of variable coefficients with four non-linear terms

This paper discusses the following non-linear systems of second order coefficients: This paper is the generalization of works [1] and [2].