Variational formulation of the material failure process in solids by embedded discontinuities model

This work presents a variational formulation of the material failure process, idealized as strain or displacement discontinuities, by weak, strong, or discrete embedded discontinuities into a continuum. It is shown that the solution of the proposed variational formulation may be approximated by different types of finite elements with embedded discontinuities. The developed displacement approximation of a finite element split by the discontinuity leads to a symmetric stiffness matrix, which considers not only the continuity of tractions but also the rigid body relative motions of the portions in which the element is split. The variational formulation of a continuum with more than one discontinuity in its interior is developed. It is shown that this formulation may lead to finite elements with embedded discontinuities that can be classified as displacement, force, mixed, and hybrid models. To show the effectiveness of the proposed formulation, the classical example of a bar under tension is solved using one and 2D finite element approximations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Derivative discontinuities in density functional theory

Fifty years after the original formulation of density functional theory (DFT), subtle consequences of the mathematical mappings underlying its formalism continue to merit new views. In this article, we discuss the origin, the importance, and the challenges associated with finding the derivative discontinuity of the exchange-correlation (XC) energy of DFT at integer–electron numbers. We show how even the energy of a quantum electron gas with finite volume and number of electrons displays such derivative discontinuities, but continuous density functional approximations to the XC functional miss them entirely. We discuss some of the practical problems that arise due to this lack of derivative discontinuities in standard functionals, and explain new ways to recover them.     

Analysis of an interfacial crack in a piezoelectric bi-material via the extended Green’s functions and displacement discontinuity method

Based on the extended Stroh formalism, we first derive the extended Green’s functions for an extended dislocation and displacement discontinuity located at the interface of a piezoelectric bi-material. These include Green’s functions of the extended dislocation, displacement discontinuities within a finite interval and the concentrated displacement discontinuities, all on the interface. The Green’s functions are then applied to obtain the integro-differential equation governing the interfacial crack. To eliminate the oscillating singularities associated with the delta function in the Green’s functions, we represent the delta function in terms of the Gaussian distribution function. In so doing, the integro-differential equation is reduced to a standard integral equation for the interfacial crack problem in piezoelectric bi-material with the extended displacement discontinuities being the unknowns. A simple numerical approach is also proposed to solve the integral equation for the displacement discontinuities, along with the asymptotic expressions of the extended intensity factors and J-integral in terms of the discontinuities near the crack tip. In numerical examples, the effect of the Gaussian parameter on the numerical results is discussed, and the influence of different extended loadings on the interfacial crack behaviors is further investigated.     

Dispersion relation in the limit of high frequency for a hyperbolic system with multiple eigenvalues

The results of a previous paper (Muracchini et al., 1992) are generalized by considering a hyperbolic system in one space dimension with multiple eigenvalues. The dispersion relation for linear plane waves in the high-frequency limit is analyzed and the recurrence formulas for the phase velocity and the attenuation factor are derived in terms of the coefficients of a formal series expansion in powers of the reciprocal of frequency. In the case of multiple eigenvalues, it is also verified that linear stability implies λ$\lambda$-stability for the waves of weak discontinuity. Moreover, for the linearized system, the relationship between entropy and stability is studied. When the nonzero eigenvalue is simple, the results of the paper mentioned above are recovered. In order to illustrate the procedure, an example of the linear hyperbolic system is presented in which, depending on the values of parameters, the multiplicity of nonzero eigenvalues is either one or two. This example describes the dynamics of a mixture of two interacting phonon gases.     

A simple mathematical model for free vibration analysis of combined system consisting of framed tube,shear core,belt truss and outrigger system with geometrical discontinuities

Dynamic analysis of beam-like structure is significantly important in modeling real cases such as tall buildings, aircraft wings, spacecraft antennas and many other applications. This paper tries to determine the first natural frequency of tall buildings including framed tube, shear core, belt truss and outrigger system with multiple jumped discontinuities in the cross section of framed tube and shear core. In this regard, the entire length of the tall building is partitioned into uniform segments between each two successive discontinuity points. The effect of belt truss and outrigger system is modeled as a concentrated rotational spring applied at the belt truss and outrigger system location. Many cantilevered tall structures can be treated as cantilever bars with multiple jumped discontinuities in the cross section for the analysis of their free vibration. In this paper, the continuous approach was accepted and by using the Hamilton’s variational principle, the general form of governing equation for free vibration of tall building can be obtained. By applying the separation of variable method on time and space, the governing Partial Differential Equation (PDE) of motion is reduced to an Ordinary Differential Equation (ODE) with one variable coefficient while the other coefficients are constant based on the assumption that the transverse displacement is a harmonic vibration. To find exact solution of ODE, we must have exact distribution of EI(x), AG(x), N(x) and m(x) in the height of the structure. Some of these parameters such as EI(x), AG(x) and m(x), are constant throughout the height of each segment. These parameters can be expressed exactly by using of multi criteria function, while N(x) is variable in the height of each segment. Therefore, the ODE by using the method of variable separation and partitioned method can be expressed for each segment. We must apply the continuity conditions in conjunction with different segments for obtaining unique mode shape for mentioned system. Tall building characteristics matrix can be derived based on the boundary conditions and the continuity conditions applied at the partitioned points. This matrix is particularly used to find combined system first natural frequency and mode shape. Three numerical examples with different stepped discontinuities in their cross sections are studied to demonstrate the reliability of this method. The results of the proposed mathematical model give a good understanding of the structure’s dynamic characteristics; it is easy to use, yet reasonably accurate and suitable for quick evaluations during the preliminary design stages which require less time.     

The Absolute Thermoelectric Power of Some Dilute Liquid Sodium Alloys

Abstract

The absolute thermoelectric powers of liquid sodium and of dilute liquid alloys of sodium with silver, cadmium, indium and tin have been measured within the temperature range 100–460°C. The thermopower of liquid sodium is decreased by the addition of cadmium, indium or tin, with cadmium having the smallest and indium the greatest effect, whereas it is increased slightly on the addition of 1 at. % of silver. Values of the dimensionless thermopower parameter, ?, derived from the experimental data, are compared with those predicted by the Faber-Ziman theory; the agreement is quite good for Na-Ag and Na-Cd but less satisfactory for Na-In and Na-Sn. The discrepancies are discussed in terms of the assumptions involved in the calculations, namely the use of only one structure factor and local dielectric screening of the ionic pseudopotentials.     

On the tangential stress anomaly in the displacement discontinuity method

It is shown that the anomaly associated with the incorrect evaluation of tangential stresses in the displacement discontinuity (DD) method, commonly used to solve crack problems, is related to the order of singularity of the fundamental solutions of linear elasticity. It is established here that a minimum of linear variation of the shear DD distribution is needed to obtain the correct tangential stress jump across a crack. Alternatively, a correction term (‘patch’) that improves tangential stresses is derived. It is also shown that need for higher functionality is a fundamental requirement rather than a convenient artifice for obtaining better accuracy.     

一种守恒型间断跟踪法中对任意多个间断的移动和相互作用的处理

对一种守恒型间断跟踪法设计了一种技巧来处理任意多个间断的移动和相互作用．由此技巧我们就可以建立一个“一般的强健的”间断跟踪法．由于采用了此技巧就会使得算法在某时刻在某网格上会存在非相容性并且会产生O(1)-强度的误差．但不管怎样,这些误差在后续的计算中会被算法的守恒性所消除．还给出了几个数值例子来显示这一技巧的有效性．     

Effects of the weak/micro-discontinuity of interface on the fracture behavior of a functionally graded coating with an inclined crack

The mechanical model was established for the anti-plane fracture problem of a functionally graded coating–substrate system with a coating crack inclined to the weak/micro-discontinuous interface. The Cauchy singular integral equation for the crack was derived using Fourier integral transform, and the Lobatto–Chebyshev collocation method put up by Erdogan and Gupta was used to get its numerical solution. Finally, the effects of the weak/micro-discontinuity of the interface on SIFs were analyzed, the “affected regions” corresponding to the two crack tips have been obtained and their engineering significance was discussed. It was indicated that, for the crack tip in the corresponding “affected region”, to reduce the weak-discontinuity of the interface and to make the interface micro-discontinuous are the two effective ways to reduce the SIF, and the latter way always has more remarkable SIF-reduction effect. For the crack tip outside the “affected region”, its SIF is mainly influenced by material stiffness, and to prevent such a tip from growing toward the interface “softer coating and stiffer substrate” is a more advantageous combination than “stiffer coating and softer substrate”.     

平面弹性裂纹分析的一种有效边界元方法

提出了一种简单而有效的平面弹性裂纹应力强度因子的边界元计算方法．该方法由Crouch与Starfield建立的常位移不连续单元和闫相桥最近提出的裂尖位移不连续单元构成A·D2在该边界元方法的实施过程中,左、右裂尖位移不连续单元分别置于裂纹的左、右裂尖处,而常位移不连续单元则分布于除了裂尖位移不连续单元占据的位置之外的整个裂纹面及其它边界．算例(如单向拉伸无限大板中心裂纹、单向拉伸无限大板中圆孔与裂纹的作用)说明平面弹性裂纹应力强度因子的边界元计算方法是非常有效的．此外,还对双轴载荷作用下有限大板中方孔分支裂纹进行了分析．这一数值结果说明平面弹性裂纹应力强度因子的边界元计算方法对有限体中复杂裂纹的有效性,可以揭示双轴载荷及裂纹体几何对应力强度因子的影响．