Isolated crack in three-dimensional piezoelectric solid. Part II: Stress intensity factors for circular crack

This is part II of the work concerned with finding the stress intensity factors for a circular crack in a solid with piezoelectric behavior. The method of solution involves reducing the problem to a system of hypersingular integral equations by application of the unit concentrated displacement discontinuity and the unit concentrated electric potential discontinuity derived in part I [1]. The near crack border elastic displacement, electric potential, stress and electric displacement are obtained. Stress and electric displacement intensity factors can be expressed in terms of the displacement and the potential discontinuity on the crack surface. Analogy is established between the boundary integral equations for arbitrary shaped cracks in a piezoelectric and elastic medium such that once the stress intensity factors in the piezoelectric medium can be determined directly from that of the elastic medium. Results for the penny-shaped crack are obtained as an example.     

The method of analysis of crack problem in three-dimensional non-local elasticity

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The analysis of crack problems with non-local elasticity

IntroductionTheclassicalconhnuummechanicshasbeenusedtosolvemanyproblemsinmacrofracturemechanics,butencountersdifficulheswhentheeffectofITilcrocharacteristicdimensionshouldbetakenintoaccount.Thestressfieldverynearthecracktipisstillnotclear.Somephenomenaofshortcrackscannotbeexplained["']andsomemechanismoffracturehasnotbeensolvedyet.Thenon-localelashcitytheoryseemsattractivetotheseproblems.Thetheoryofnon-localelasticity,establishedanddevelopedbyEringenetal[3),connectstheclassicalcontinuummechan…     

Smoothing the discontinuities of an electrical charge in electrodynamics as a result of diffusion processes

Electrohydrodynamic flows in which there are zones of abrupt changes in the electric charge (while remaining bounded, by assumption) are investigated. In a diffusionless approximation such flows are characterized by a discontinuity in the electric charge q. Examples of such motions are nonstationary flows with moving electrical charge fronts [1], stationary flows in which the electrical charge is lumped in just part of the hydrodynamic stream [2, 3], flows with discontinuity in q [4–7], boundary layers near an electrode grid mounted perpendicularly to the electrohydrodynamic stream. Diffusion effects of charged particles should cause smoothing of the electrical charge discontinuities. The diffusion structure of such discontinuities is studied for high electrical Peclet numbers. The distribution of q in gasdynamic jumps is analyzed taking account of the viscous and diffusion structure of the discontinuities in the small parameter approximation of the electrogasdynamic interaction. Three problems about flows with charged particle diffusion are examined: the problem of scattering of a finite electric charge in a medium at rest, initially concentrated at a point on a line of unit length; the boundary layer on an electrode grid perpendicular to the direction of the charged fluid stream; electrogasdynamic flows with an abrupt change in velocity not accompanied by the appearance of a surface charge.     

The nature of the triple point singularity in the case of stationary reflection of weak shock waves

The structure of flow in the vicinity of a triple point in the problem of stationary irregular reflection of weak shock waves is numerically investigated within the framework of the Euler model, including the von Neumann paradox range. To improve the accuracy of the solution near singular points a new technology including a grid contracted toward the triple point and the discontinuity fitting is applied. It is shown that in the four-wave flow pattern the curvatures of the tangential discontinuity and the Mach front at the triple point are finite. The singularity is concentrated only in a sector between the reflected wave front and the expansion fan. When the three-wave flow pattern is realized, the curvatures of the tangential discontinuity and both wave fronts at the triple point are infinite. On the range of weak and moderate shock waves the logarithmic singularity in subsonic sectors near the triple point conserves up to transition to the regular reflection.     

Differential-algebraic approach to large deformation analysis of frame structures subjected to dynamic loads

A nonlinear mathematical model for the analysis of large deformation of frame structures with discontinuity conditions and initial displacements, subject to dynamic loads is formulated with arc-coordinates. The differential quadrature element method （DQEM） is then applied to discretize the nonlinear mathematical model in the spatial domain, An effective method is presented to deal with discontinuity conditions of multivariables in the application of DQEM. A set of DQEM discretization equations are obtained, which are a set of nonlinear differential-algebraic equations with singularity in the time domain. This paper also presents a method to solve nonlinear differential-algebra equations. As application, static and dynamical analyses of large deformation of frames and combined frame structures, subjected to concentrated and distributed forces, are presented. The obtained results are compared with those in the literatures. Numerical results show that the proposed method is general, and effective in dealing with disconti- nuity conditions of multi-variables and solving differential-algebraic equations. It requires only a small number of nodes and has low computation complexity with high precision and a good convergence property.     

Electrochemical and optical studies of a stable to unstable transition in a polymer solution flowing close to a rotating disc

Summary A transition has been displayed in a concentrated solution of a linear chain polymer flowing close to a rotating disk. For a critical angular speed of rotation, a discontinuity of the mass transfer rate by diffusion and convection, as well as the existence of flow birefringence are observed. This transition is interpreted as an effect of conformational change (coiled to stretched shape) of the polymer under the local strain in the fluid.With 4 figures     

Anti-plane shear cracks in ideally plastic crystals

Cracks in ductile single crystals are analyzed here for geometries and orientations such that two-dimensional states of anti-plane shear constitute possible deformation fields. The crystals are modelled as ideally plastic and yield according to critical resolved shear stresses on their slip systems. Restrictions on the asymptotic forms of stress and deformation fields at crack tips are established for anti-plane loading of stationary and quasistatically growing cracks, and solutions are presented for several specific orientations in f.c.c. and b.c.c. crystals. The asymptotic solutions are complemented by complete elastic-plastic solutions for stationary and growing cracks under small scale yielding, based on previous work by Rice (1967, 1984) and Freund (1979). Remarkably, the plastic zone at a stationary crack tip collapses into discrete planes of displacement and stress discontinuity emanating from the tip; plastic flow consists of concentrated shear on the displacement discontinuities. For the growing crack these same planes, if not coincident with the crack plane, constitute collapsed plastic zones in which velocity and plastic strain discontinuities occur, but across which the stresses and anti-plane displacement are fully continuous. The planes of discontinuity are in several cases coincident with crystal slip planes but it is shown that this need not be the case, e.g., for orientations in which anti-plane yielding occurs by multi-slip, or for special orientations in which the crack tip and the discontinuity planes are perpendicular to the activated slip plane.     

Analytical model for a rigid, linear-hardening plastic tube subjected to impulsive torsional moment

An analytical model of a rigid, linear-hardening plastic tube subjected to impulsive torsion at its free end is suggested. It is shown that an impulsive torsional moment at the free end of the tube gives rise to a torsional disturbance of strong discontinuity that propagate at a constant speed and the plastic deformation is concentrated at the front of the disturbances. Two examples are given to illustrate the characterization of the structure under the intense dynamic pure torsional loading condition.     

Singular integral equations in elastic dielectrics

Galerkin representations for the displacement vector, polarization vector and the potential field are obtained by elementary matrix inversions of the equations of equilibrium. Matrices of fundamental solutions of an infinite elastic dielectric continuum subjected to a concentrated body force, an electric force, and a charge density, are constructed. Theorems are proved on the discontinuity of double layer potentials and R, M, M operators of single layer potentials. By means of these theorems, the solution of the two basic boundary value problems has been reduced to the solution of a system of seven singular integral equations.     

Shock waves in diatomic chains—II. nonlinear analysis

The effectsof nonlinear terms in the force-displacement relations of a general diatomic chain are studied by methods closely analogous to the analysis of a monatomic chain by J. Tasi (1972, 1973). Two-time parameter perturbation solutions of the equations of motion are obtained for the nonlinear growth of the shock response at the head of the wave. These solutions are similar in form to the asymptotic results obtained by M.J.P. Musgrave and J. Tasi (1976) but depend upon a dispersive wave coordinate modified by the nonlinear parameters. Partial differential equations are derived for the far-field velocity response of each typical particle in a unit cell. In the ease of a general diatomic chain, these equations are not identical and discontinuity in strain within the unit cell occurs. Graphical plots of solutions are presented.     

Formation fronts of a nonlinear elastic medium from a medium without shear stresses

The fronts of phase transition of a medium without shear stresses to a nonlinear incompressible anisotropic elastic medium are considered. The mass flux through unit area of a front is assumed to be known. The variation of the tangential components of the medium’s velocity and the variation of the arising shear stresses are studied. An explicit form of boundary conditions is found using the existence condition of a discontinuity front structure. The Kelvin–Voight viscoelastic model is adopted for this structure.     

应变局部化分析的嵌入强间断多尺度有限元法

固体材料的应变局部化行为是导致结构破坏失效的重要因素之一,开展相关数值模拟分析对于结构安全性评估具有重要意义.然而由于材料的非均质和多尺度特性,采用传统数值方法进行求解时通常需要从最小特征尺度离散求解的结构,这将大幅度增加计算规模和成本.针对这一问题,本文提出了一种基于嵌入强间断模型的多尺度有限元方法.该方法从粗细两个尺度离散求解模型,首先在细尺度单元上引入嵌入强间断模型来描述单元间断特性,所附加的跳跃位移自由度则通过凝聚技术进行消除,从而保持细尺度单元刚度阵维度不变.其次,提出了一种增强多节点粗单元技术,其可根据局部化带与粗单元边界相交情况自适应动态地增加粗节点,新构造的增强数值基函数可以捕捉细尺度间断特性,完成物理信息从细单元到粗单元的准确传递以及宏观响应的快速分析;再次,在细尺度解的计算中,将细尺度解分解为降尺度解与单胞局部摄动解,从而消除弹塑性分析时单胞内部的不平衡力.最后,通过两个典型算例分析,并与完全采用细单元的嵌入有限元结果进行对比,验证了所提出算法的正确性与有效性.     

Stability of surfaces of strong discontinuity in gases

The existence of solutions with surfaces of strong discontinuity is one of the principal features of the continua whose motions are described by systems of differential equations of hyperbolic type. Shock waves in gas dynamics, magnetohydrodynamics and in solids, detonation waves and combustion fronts, contact discontinuities, etc. are well-known examples of these surfaces. The discontinuities are usually investigated in accordance with the following scheme: 1) derivation of the boundary conditions on the discontinuity from the input system of differential equations in integral form; 2) verification of the fulfilment of the evolution conditions; 3) solution of the problem of the discontinuity structure and, when the occasion requires, obtaining supplementary boundary conditions; 4) investigation of the stability of the discontinuity. Only after obtaining positive results in all fours stages can we assert that the existence of the discontinuity is theoretically justified and that it can be used for constructing the solutions of particular boundary value problems. In the present paper attention will be concentrated on the problem of the stability of discontinuities, all the material, with the exception of the general results of Sec.1, being concerned with gas media and relating to discontinuities on whose surface the normal mass flow is nonzero. Having no way of exploring all the aspects of the problem of the stability of discontinuities in the same detail within the limited context of this paper, the authors hope to demonstrate the most general ideas and approaches which could subsequently be used to investigate the stability of discontinuities in various particular models of continua.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–22, March–April, 1996.     

Three-dimensional interfacial fracture analysis of a one-dimensional hexagonal quasicrystal coating

In this paper, the three-dimensional (3D) interfacial fracture is analyzed in a one-dimensional (1D) hexagonal quasicrystal (QC) coating structure under mechanical loading. A planar interface crack with arbitrary shape is studied by a displacement discontinuity method. Fundamental solutions of interfacial concentrated displacement discontinuities are obtained by the Hankel transform technique, and the corresponding boundary integral-differential equations are constructed with the superposition principle. Green’s functions of constant interfacial displacement discontinuities within a rectangular element are derived, and a boundary element method is proposed for numerical simulation. The singularity of stresses near the crack front is investigated, and the stress intensity factors (SIFs) as well as energy release rates (ERRs) are determined. Finally, relevant influencing factors on the fracture behavior are discussed.     

Nonlinear Disturbances and Weak Discontinuities in a Supersonic Boundary Layer

The processes of wave disturbance propagation in a supersonic boundary layer with self-induced pressure [1–4] are analyzed. The application of a new mathematical apparatus, namely, the theory of characteristics for systems of differential equations with operator coefficients [5–8], makes it possible to obtain generalized characteristics of the discrete and continuous spectra of the governing system of equations. It is shown that the discontinuities in the derivatives of the solution of the boundary layer equations are concentrated on the generalized characteristics. It is established that in the process of flow evolution the amplitude of the weak discontinuity in the derivatives may increase without bound, which indicates the possibility of breaking of nonlinear waves traveling in the boundary layer.     

Transient phenomena in thixotropic systems

A nonlinear rheological model which accounts for the time-dependent elastic, viscous and yielding phenomena is developed in order to describe the flow behavior of thixotropic materials which exhibit yield stress. A key feature of the formulation is a smooth transition from an ‘elastically’ dominated response to a ‘viscous’ response without a discontinuity in the stress–strain curve. The model is phenomenological and is based on the kinetic processes responsible for structural changes within the thixotropic material. As such, it can predict thixotropic effects, such as stress overshoot during start-up of a steady shear flow and stress relaxation after cessation of flow. Thus this model extends a previously proposed viscoplastic model [J. Rheol. 34 (1991) 647] to include thixotropy.An analysis and comparison to experimental data involving oscillatory shear flow are provided to evaluate the accuracy of the model and to estimate the model parameters in a prototype concentrated suspension. The experiments were conducted using a series of concentrated suspensions of silicon particles and silicon carbide whiskers in polyethylene. The data obtained with this experimental system indicated much better agreement between the theory and experiments that obtained in earlier work.     

Characterization of bifurcating non-linear normal modes in piecewise linear mechanical systems

The non-linear modal properties of a vibrating 2-DOF system with non-smooth (piecewise linear) characteristics are investigated; this oscillator can suitably model beams with a breathing crack or systems colliding with an elastic obstacle. The system having two discontinuity boundaries is non-linearizable and exhibits the peculiar feature of a number of non-linear normal modes (NNMs) that are greater than the degrees of freedom. Since the non-linearities are concentrated at the origin, its non-linear frequencies are independent of the energy level and uniquely depend on the damage parameter. An analysis of the NNMs has been performed for a wide range of damage parameter by employing numerical procedures and Poincaré maps. The influence of damage on the non-linear frequencies has been investigated and bifurcations characterized by the onset of superabundant modes in internal resonance, with a significantly different shape than that of modes on fundamental branch, have been revealed.     

Propagation of first-order electromagnetic discontinuities in an isotropic medium

The propagation of a first-order electromagnetic discontinuity is discussed. Expressions are obtained for the possible velocities of propagation as functions of the field strengths ahead of the surface of discontinuity. Expressions are also obtained for the growth in the magnitude of the discontinuity as the wave progresses.     

Analysis of the State for a Coal Massif Enclosing in-Seam Working and a Geological Discontinuity

A model of the geomechanical state for a rock massif with strength anisotropy has been developed. Themassif encloses an in-seamworking near the geological discontinuity. The developed models implemented on the basis of pseudo-load method and displacement discontinuity method. Based on the results obtained, stress diagrams and discontinuity zones of the massif in the vicinity of the mine working are constructed. The regularities of changing the state of the massif when its parameters are varied and the relative positions of the mine working and geological discontinuity are revealed.