Stationary straight cracks in quasicrystals

Stationary straight cracks in quasicrystals in linear elastic setting are under scrutiny. The analysis is developed by using Stroh formalism which is modified to account for a totally degenerate eigenvalue problem: in fact, the fundamental matrix of the governing equations of motion admits a repeated eigenvalue corresponding to a single eigenvector. Cases of a semi-infinite rectilinear crack loaded along its margins and a crack of finite length under remote loading conditions are considered. Standard and phason stresses display square-root singularities at crack tip. The latter stresses represent peculiar microstructural inner actions occurring in quasicrystals and are determined by rearrangements assuring quasi-periodicity of the atomic tiling—modes described by a vector field, called phason field, collecting the local degrees of freedom exploited by the atoms within the material elements. Energy release rate increases with the coupling parameter between displacement and phason fields.     

Phason dynamics in nonlinear photonic quasicrystals

We study the dynamics of phasons in a nonlinear photonic quasicrystal. The photonic quasicrystal is formed by optical induction, and its dynamics is initiated by allowing the light waves inducing the quasicrystal to nonlinearly interact with one another. We show quantitatively that, when phason strain is introduced in a controlled manner, it relaxes through the nonlinear interactions within the photonic quasicrystal. We establish experimentally that the relaxation rate of phason strain in the quasicrystal is substantially lower than the relaxation rate of phonon strain, as predicted for atomic quasicrystals. Finally, we monitor and identify individual 'atomic-scale' phason flips occurring in the photonic quasicrystal as its phason strain relaxes, as well as noise-induced phason fluctuations.     

Inverse fracture problems in piezoelectric solids by local integral equation method

The meshless local Petrov–Galerkin (MLPG) method is used to solve the inverse fracture problems in two-dimensional (2D) piezoelectric body. Electrical boundary conditions on the crack surfaces are not specified due to unknown dielectric permittivity of the medium inside the crack. Both stationary and transient dynamic boundary conditions are considered here. The analyzed domain is covered by small circular subdomains surrounding nodes spread randomly over the analyzed domain. A unit step function is chosen as test function in deriving the local integral equations (LIE) on the boundaries of the chosen subdomains. The Laplace-transform technique is applied to eliminate the time variation in the governing equation. The local integral equations are nonsingular and take a very simple form. The spatial variation of the Laplace transforms of displacements and electrical potential are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares (MLS) method. The singular value decomposition (SVD) is applied to solve the ill-conditioned linear system of algebraic equations obtained from the LIE after MLS approximation. The Stehfest algorithm is applied for the numerical Laplace inversion to retrieve the time-dependent solutions.     

Crack–surface displacements for cracks emanating from a circular hole under various loading conditions

The purpose of this paper is to calculate and develop equations for crack–surface displacements for two‐symmetric cracks emanating from a circular hole in an infinite plate for use in strip‐yield crack‐closure models. In particular, the displacements were determined under two loading conditions: (1) remote applied stress and (2) uniform stress applied to a segment of the crack surface (partially loaded crack). The displacements were calculated by an integral‐equation method based on accurate stress–intensity factor equations for concentrated forces applied to the crack surfaces and those for remote applied stress or for a partially loaded crack surface. A boundary‐element code was also used to calculate crack–surface displacements for some selected cases. Comparisons made with crack–surface displacement equations previously developed for the same crack configuration and loading showed significant differences near the location where the crack intersected the hole surface. However, the previous equations were fairly accurate near the crack‐tip location. Herein an improved crack–surface displacement equation was developed for the case of remote applied stress. For the partially loaded crack case, only numerical comparisons were made between the previous equations and numerical integration. A rapid algorithm, based on the integral‐equation method, was developed to calculate these displacements. Because cracks emanating from a hole are quite common in the aerospace industry, accurate displacement solutions are crucial for improving life‐prediction methods based on the strip‐yield crack‐closure models.     

Control of wave propagation response using quasi crystals: A formulation based on spectral finite element

This paper presents wave propagation studies in quasi crystal structures and quasi crystal reinforced aluminium structures. The analysis is performed using frequency domain spectral finite element formulation. The analysis considers different 2-D decagonal and 3-D icosahedral quasi crystals. First, wave propagation analysis of quasi crystal structure alone is performed and the propagation of phonon and phason modes for different quasi crystals are studied. The study includes the propagation of axial and transverse wave responses in these quasi crystals. The study has found that the amplitude of the phason modes is very small compared to the phonon modes and the increase of the phason mode content (through increase in R) increases the phason mode amplitude, without affecting the phonon mode amplitudes. It is shown that the dominant axial phonon mode is non-dispersive and the dominant flexural phonon mode is dispersive. In the next study, the aluminium beam structure is reinforced with different quasi crystals in different configurations and the wave propagation of axial and transverse responses are studied. For all the combinations of quasi crystal aluminium beam combination, there is substantial suppression of responses both for the axial and the bending responses. Unsymmetrical configuration produces substantial non-dominant phonon modes which propagate dispersively. It is found that for a symmetric bi-morph configuration, the response is reduced significantly, about 68% and 75% for axial loading and 80% and 78% for flexural loading, respectively, for the 2-D decagonal quasi crystal and the 3-D icosahedral quasi crystal.     

Modeling of porous piezoelectric structures by the meshless local Petrov-Galerkin method

A meshless method based on the local Petrov-Galerkin approach is proposed to solve initial-boundary value problems of porous piezoelectric solids. Constitutive equations for porous piezoelectric materials possess a coupling between mechanical displacements and electric intensity vectors in both solid and fluid phases. Stationary and transient 2-D and 3-D axisymmetric problems are considered in this article. Nodal points are spread on the problem domain, and each node is surrounded by a small circle for simplicity. The spatial variation of displacements and electric potentials for both phases is approximated by the moving least-squares scheme. After performing the spatial integration, one obtains a system of ordinary differential equations for certain nodal unknowns. The resulting system is solved numerically by the Houbolt finite-difference scheme as a time stepping method. The proposed method is applied to bending problems associated with a porous piezoelectric 2-D plate and 3-D axisymmetric cylinder under simply supported and clamped boundary conditions.     

Axisymmetric problem for a spherical crack on the interface of elastic media

A problem concerning a spherical interfacial crack is solved by the eigenfunction method. The problem is reduced to a coupled system of dual-series equations in terms of Legendre functions and then to a system of singular integral equations for two unknown functions. The behaviour of the solution near the edge of the spherical crack, and the stress-intensity factors and crack-opening displacements are studied. The case when the crack surfaces are under normal internal pressure of constant intensity is examined.     

Two bonded half planes with a crack going through the interface

The plane problem of two bonded elastic half planes containing a finite crack perpendicular to and going through the interface is considered. The problem is formulated as a system of singular integral equations with generalized Cauchy kernels. Even though the system has three irregular points, it is shown that the unknown functions are algebraically related at the irregular point on the interface and the integral equations can be solved by a method developed previously. The system of integral equations is shown to yield the same characteristic equation as that for two bonded quarter planes in the general case of the through crack, and the characteristic equation for a crack tip terminating at the interface in the special case. The numerical results given in the paper include the stress intensity factors at the crack tips, the normal and shear components of the stress intensity factors at the singular point on the interface, and the crack surface displacements.     

Meshless Local Petrov-Galerkin Method for Plane Piezoelectricity

Piezoelectric materials have wide range engineering applications in smart structures and devices. They have usually anisotropic properties. Except this complication electric and mechanical fields are coupled each other and the governing equations are much more complex than that in the classical elasticity. Thus, efficient computational methods to solve the boundary or the initial-boundary value problems for piezoelectric solids are required. In this paper, the Meshless local Petrov-Galerkin (MLPG) method with a Heaviside step function as the test functions is applied to solve two-dimensional (2-D) piezoelectric problems. The mechanical fields are described by the equations of motion with an inertial term. To eliminate the time-dependence in the governing partial differential equations the Laplace-transform technique is applied to the governing equations, which are satisfied in the Laplace-transformed domain in a weak-form on small subdomains. Nodal points are spread on the analyzed domain and each node is surrounded by a small circle for simplicity. The spatial variation of the displacements and the electric potential are approximated by the Moving Least-Squares (MLS) scheme. After performing the spatial integrations, one obtains a system of linear algebraic equations for unknown nodal values. The boundary conditions on the global boundary are satisfied by the collocation of the MLS-approximation expressions for the displacements and the electric potential at the boundary nodal points. The Stehfest's inversion method is applied to obtain the final time-dependent solutions.     

Edge dislocations in nanoquasicrystalline materials

A theoretical model is proposed which describes energetic characteristics and behavioral features of edge dislocations in coarse-grained and nanostructured quasicrystals. The key point of the model is representation of dislocation-induced phason imperfections as centers of isotropic dilatation (flexes). The special attention in this paper is paid to the influence of the nanoscale structure on both energetic characteristics and behavioral features of edge dislocations in nanostructured quasicrystals (nanoquasicrystalline materials). It is shown, in particular, that the basic grain-size-sensitive contribution of dislocation-induced phason imperfections to the total energy of an edge dislocation varies directly with grain size, in which case the formation of edge dislocations is not suppressed in nanoquasicrystalline materials (in contrast to coarse-grained quasicrystalline materials.     

位错和均匀分布载荷作用下的二维十次对称准晶的弹性分析

复变函数方法是经典弹性理论中求解平面弹性与缺陷问题非常有效的方法。该文推广了经典弹性理论的复变函数方法,研究了位错和均匀分布载荷作用下的二维十次对称准晶的弹性场。根据偏微分方程理论,用四个解析函数给出了应力和位移的复表达式。在此基础上,结合边界条件,获得了声子场和相位子场应力及位错应变能的解析表达式。讨论了声子场与相位子场耦合弹性常数对应变能的影响。     

Longitudinal Shear of an Elastic Wedge with Cracks and Notches

We study the problem of longitudinal shear of an infinite wedge with cracks and notches. The integral representations of the complex stress potential are constructed in terms of the jumps of displacements and stresses on curvilinear contours identically satisfying the boundary conditions imposed on the faces of the wedge (stresses or displacements are equal to zero). By using these representations, we deduce singular integral equations of the analyzed problem for a wedge weakened by a system of cracks and holes of any shape. In some cases (a crack along the bisectrix of the wedge, a crack along a circular arc whose center is located at the edge of the wedge, and a circular notch near the edge of the wedge), we obtain exact closed solutions.     

Dynamic crack interactions in magnetoelectroelastic composite materials

In this paper, the dynamic interactions among cracks embedded in a two-dimensional (2-D) piezoelectric-piezomagnetic composite material are analyzed by means of a hypersingular formulation of the boundary element method. In the numerical solution procedure, extended crack opening displacements and extended traction jumps across the crack are considered as basic unknowns, so that only the traction boundary integral equations are needed on the crack surfaces. Quadratic discontinuous boundary elements are implemented together with discontinuous quarter-point elements placed next to the crack tips to ensure a proper representation of the square root asymptotic behavior. Several impermeable cracks configurations subjected to time-harmonic incident L-waves are analyzed in order to characterize the effects of the magnetoelectromechanical coupling on the dynamic crack interactions and to illustrate the dependence on such coupling of the fracture parameters: stress intensity factors, electric displacement intensity factor and magnetic induction intensity factor.     

A hypersingular time‐domain BEM for 2D dynamic crack analysis in anisotropic solids

A hypersingular time‐domain boundary element method (BEM) for transient elastodynamic crack analysis in two‐dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack‐faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time‐stepping scheme is obtained to compute the unknown boundary data including the crack‐opening‐displacements (CODs). Special crack‐tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time‐domain BEM. Copyright © 2008 John Wiley & Sons, Ltd.     

A time-domain collocation-Galerkin BEM for transient dynamic crack analysis in anisotropic solids

This paper presents a time-domain boundary element method (BEM) for transient elastodynamic crack analysis in homogeneous and linear elastic solids of general anisotropy. A finite crack subjected to a transient loading is investigated. Two-dimensional (2D) generalized plane-strain or plane-stress condition is considered. The initial-boundary value problem is described by a set of hypersingular time-dependent traction boundary integral equations (BIEs), in which the crack-opening displacements (CODs) are unknown quantities. The hypersingular time-domain BIEs are first regularized to weakly singular ones by using spatial Galerkin method, which transfers the derivatives of the fundamental solutions to the unknown CODs and the weight functions. To solve the time-domain BIEs numerically, a time-stepping scheme is developed. The scheme applies the collocation method for temporal discretization of the time-domain BIEs. As spatial shape-functions, two different functions are implemented. For elements away from crack-tips, linear spatial shape-function is used, while for elements near the crack-tips a special ‘crack-tip shape-function’ is applied to describe the local ‘square-root’ behavior of the CODs at the crack-tips properly. Special attention of the analysis is devoted to the numerical computation of the transient elastodynamic stress intensity factors for cracks in general anisotropic and linear elastic solids. Numerical examples are presented to verify the accuracy of the present time-domain BEM.     

Finite element analysis of viscoelastic structures using Rosenbrock-type methods

The consistent application of the space-time discretisation in the case of quasi-static structural problems based on constitutive equations of evolutionary type yields after the spatial discretisation by means of the finite element method a system of differential-algebraic equations. In this case the resulting system of differential-algebraic equations with the unknown nodal displacements and the evolution equations at all spatial quadrature points of the finite element discretisation are solved by means of a time-adaptive Rosenbrock-type methods leading to an iteration-less solution scheme in non-linear finite element analysis. The applicability of the method will be studied by means of a simple example of a viscoelastic structure.     

Circular cracks in tension and torsion

The concept of a dislocation continuum has been applied widely to the study of cracks in two, and is now employed in three dimensions. It is noted that surface displacements may be represented by a continuum of loops of dislocation conformal with the crack circumference or alternatively by an array of loops of dislocation each of infinitesimal area. In both cases the spatial dependence of crack surface displacement is formulated as an integral equation related to that of Abel. The distribution functions found from these equations are employed to determine Griffith crack propagation conditions for cases of potential interest.     

A SINGULAR INTEGRAL EQUATION SOLUTION FOR THE LINEAR ELASTIC CRACK OPENING DISPLACEMENT OF AN ARBITRARILY SHAPED PLANE CRACK: PART I FINITE-PART INTEGRAL SOLUTIONS

Abstract— The aim of the paper is to compute the local crack face displacements of a linear elastic body containing an arbitrarily shaped plane crack. From the crack face displacements the local stress intensity factors can be derived.
The boundary value problem for a plane crack of arbitrary shape, embedded in a linear elastic medium, has been treated by several authors by the singular integral equation (SIE) approach. Their computations lead to a set of hyper-singular integral equations for the Cartesian components of the unknown crack face displacements. To solve these equations the authors present a discretization procedure based on six-node triangular finite elements. A total set of 24 finite-part integrals defined over a triangular area can be developed. These 2D-finite-part integrals can be split into both a 1D-regular and a 1D-finite-part-integral by means of the polar coordinates so that they can be solved in closed form. Finally, the investigation of the SIEs is reduced to a discrete set of linear algebraic equations for the unknown nodal point values. The necessary steps will be demonstrated in detail. The derived closed-form solutions will be offered in the text and in the appendices.     

Elastodynamics of a crack on the bimaterial interface

The paper is an application of boundary integral equations to the problem of a crack located on the bimaterial interface under time-harmonic loading. A system of linear algebraic equations is derived for solving the problem numerically. The distributions of the displacements and tractions at the bimaterial interface are obtained and analysed for the case of a penny-shaped crack under normal tension-compression wave. The dynamic stress intensity factors (normal and shear modes) are also computed. The results are compared with those obtained for the static case.     

A magneto-electro-elastic material with a penny-shaped crack subjected to temperature loading

Summary The analysis of intensity factors for a penny-shaped crack under thermal, mechanical, electrical and magnetic boundary conditions becomes a very important topic in fracture mechanics. An exact solution is derived for the problem of a penny-shaped crack in a magneto-electro-thermo-elastic material in a temperature field. The problem is analyzed within the framework of the theory of linear magneto-electro-thermo-elasticity. The coupling features of transversely isotropic magneto-electro-thermo-elastic solids are governed by a system of partial differential equations with respect to the elastic displacements, the electric potential, the magnetic potential and the temperature field. The heat conduction equation and equilibrium equations for an infinite magneto-electro-thermo-elastic media are solved by means of the Hankel integral transform. The mathematical formulations for the crack conditions are derived as a set of dual integral equations, which, in turn, are reduced to Abel's integral equation. Solution of Abel's integral equation is applied to derive the elastic, electric and magnetic fields as well as field intensity factors. The intensity factors of thermal stress, electric displacement and magnetic induction are derived explicitly for approximate (impermeable or permeable) and exact (a notch of finite thickness crack) conditions. Due to its explicitness, the solution is remarkable and should be of great interest in the magneto-electro-thermo-elastic material analysis and design.