Power Singularities Research Articles

Analysis of singular stress fields at multi‐material corners under thermal loading

AbstractThe scaled boundary finite‐element method is extended to the modelling of thermal stresses. The particular solution for the non‐homogeneous term caused by thermal loading is expressed as integrals in the radial direction, which are evaluated analytically for temperature changes varying as power functions of the radial coordinate. When applied to model a multi‐material corner, only the boundary of the problem domain is discretized. The boundary conditions on the straight material interfaces and the side‐faces forming the corner are satisfied analytically without discretization. The stress field is expressed semi‐analytically as a series solution. The stress distribution along the radial direction, including both the real and complex power singularity and the power‐logarithmic singularity, is represented analytically. The stress intensity factors are determined directly from their definitions in stresses. No knowledge on asymptotic expansions is required. Numerical examples are calculated to evaluate the accuracy of the scaled boundary finite‐element method. Copyright © 2005 John Wiley & Sons, Ltd.

Stress singularities in classical elasticity—II: Asymptotic identification

This review article (Part II) is a sequel to an earlier one (Part I) that dealt with means of removal and interpretation of stress singularities in elasticity, as well as their asymptotic and numerical analysis. It reviews contributions to the literature that have actually effected asymptotic identifications of possible stress singularities for specific configurations. For the most part, attention is focused on 2D elastostatic configurations with constituent materials being homogeneous and isotropic. For such configurations, the following types of stress singularity are identified: power singularities with both real and complex exponents, logarithmic intensification of power singularities with real exponents, pure logarithmic singularities, and log-squared singularities. These identifications are reviewed for the in-plane loading of angular elastic plates comprised of a single material in Section 2, and for such plates comprised of multiple materials in Section 3. In Section 4, singularity identifications are examined for the out-of-plane shear of elastic wedges comprised of single and multiple materials, and for the out-of-plane bending of elastic plates within the context of classical and higher-order theory. A review of stress singularities identified for other geometries is given in Section 5, axisymmetric and 3D configurations being considered. A limited examination of the stress singularities identified for other field equations is given as well in Section 5. The paper closes with an overview of the status of singularity identification within elasticity. This Part II of the review has 227 references.

On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations

We study the Dirichlet problem in half-space for the equation Δ u + g ( u ) | ∇ u | 2 = 0 , {\Delta u+g(u)|\nabla u|^2=0,} where g g is continuous or has a power singularity (in the latter case positive solutions are considered). The results presented give necessary and sufficient conditions for the existence of (pointwise or uniform) limit of the solution as y → ∞ , y\to \infty , where y y denotes the spatial variable, orthogonal to the hyperplane of boundary-value data. These conditions are given in terms of integral means of the boundary-value function.

On Singularities of the Physical Fields at the Edge of a Composite Wedge

We propose analytic procedures capable of evaluation of the orders of power singularities of magnetic fields at the edge of a composite ferromagnetic wedge bordering with a vacuum. The same procedure is used to study the singularities at the edge of a composite piezoceramic wedge. As a result, we present the plots characterizing the dependences of the order of singularities on the wedge angles of the wedges used to form the analyzed composite wedge and the combination of their physical parameters.

Stresses near Crack Tips on the Boundary of Two Media in the Presence of Plastic Strips

Under the conditions of plane deformation, we study stresses in a piecewise-homogeneous isotropic body near the tip of a mode I crack appearing at the angular point on the boundary of two media. It is assumed that plastic strips (modeled by plastic slip lines) are formed on the boundary of the media. To determine the stresses, we use the Mellin integral transformation and the Wiener–Hopf method. The angular point is a stress concentrator with power singularity and, therefore, immediately after the appearance of lateral plastic strips (zones), a new plastic zone begins to develop from this point. We study the dependence of the power of singularity of stresses on the angle made by the boundary and the elastic characteristics of the media.

PERTURBATION OF SOLUTIONS WITH MOVING SINGULARITIES

We study the Cauchy problem for a nonlinear second-order differential equation with a small parameter in the case where the exact solution has a power singularity depending on a small parameter. We propose an asymptotic method similar to the Krylov–Bogoliubov method for localizing the singularity up to the accuracy of any order and construct an asymptotic expansion of the solution in the domain of regular behavior.

Coupled Electroelastic Fields Near the Vertex of a Compound Piezoceramic Wedge

The plane and antiplane deformations of a compound piezoceramic wedge are considered on the assumption that its components are in ideal mechanical and electric contact, while its outside faces are either free from forces and in contact with vacuum or fixed and covered with grounded electrodes. Mechanical and electric quantities are represented in a complex form to analyze the order of the power singularity at the wedge vertex. Calculated data are presented for various compositions

Asymptotic fields for dynamic crack growth in pressure-sensitive elastic–plastic materials

Asymptotic analyses for dynamic propagation of mode I planar cracks in pressure-sensitive elastic–plastic materials have been carried out. The material model adopted is based on the Drucker–Prager yield surface obeying the associate flow rule with linear isotropic hardening. The asymptotic solution is assumed to be of the variable-separable form with a power singularity in the radial coordinate from the crack tip. Attention is focused on the inertia effect on some features of the asymptotic solutions. It is found that for plane-strain cases, the range of pressure sensitivity can be expanded by increasing the crack speed due to a delay in the occurrence of a hydrostatic stress state ahead of the crack tip. An increase in crack speed produces a corresponding change in the characteristics of the governing equations because of a tendency to produce strain and stress ‘jumps’. Material and loading mode-dependent speed limits have also been studied under plane-strain and plane-stress conditions. In addition, numerical results are presented for the strength of singularity, the angular distributions of stresses and velocities, and the crack-tip constraint.

Postcritical regimes in the nonlinear problem of vortex motion under the free surface of a weighable fluid

An improved Levi-Civita method in which the singularities of the desired function are taken into account by introducing terms containing power singularities is proposed. Results of numerical analysis of the nonlinear problem of a vortex in a bounded flow of an ideal weighable fluid (Fr>1) are given. The following limiting flow regimes are studied: the Stokes waves with one and two crests, emergence of a critical point on the surface, and the detachment of a vortex from a soliton and a uniform flow. It is shown that nonperiodic waves can form in a local zone in the vicinity of the critical point.

Electrode-Ceramic Interfacial Cracks in Piezoelectric Multilayer Materials

A thin electrode layer embedded at the interface of two piezoelectric materials represents a common feature of many electroceramic multilayer devices. The analysis of interface cracks between the embedded electrode layer and piezoelectric ceramic leads to a nonstandard mixed boundary value problem which likely prevents a general analytical solution. The present work shows that the associated mixed boundary value problem does indeed admit an exact elementary solution for a special case of major practical interest in which the two piezoelectric half-planes are poled in opposite directions perpendicular to the electrode layer. In this case, it is found that oscillatory singularity disappears, in spite of the unsymmetric characters of the problem, and electroelastic fields exhibit power singularities. Particular emphasis is placed on the near-tip singular stresses along the bonded interface. The results show that tensile stress exhibits the square root singularity along the interface whereas shear stress exhibits the dominant-order nonsquare root singularity. In addition, the present model indicates that a pure electric-field loading could induce the dominant-order singular shear stress directly ahead of the interface crack tip. [S0021-8936(00)00602-4]

Graviton exchange and complete four-point functions in the AdS/CFT correspondence

The graviton exchange diagram for the correlation function of arbitrary scalar operators is evaluated in anti-de Sitter space, AdSd+1. This enables us to complete the computation of the four-point amplitudes of dilaton and axion fields in IIB supergravity on AdS5×S5. By the AdS/CFT correspondence, we obtain the four-point functions of the marginal operators Tr(F2+…) and Tr(FF̃+…) in N=4, d=4 SU(N) SYM at large N, large gYM2N. The short distance asymptotics of the amplitudes are studied. We find that in the direct channel the leading power singularity agrees with the expected contribution of the stress-energy tensor in a double OPE expansion. Logarithmic singularities occur in the complete four-point functions at subleading orders.

Thermal stresses in an anisotropic plate with angular inclusions

We reduce the plane problem of anisotropic thermoelasticity for bodies with angular inclusions to a Fredholm system of integral equations. We construct potentials with power singularities at the angular points in the explicit form and present an example of application of the obtained results to the investigation of the dependence of local stresses in a plate with elastic stringer on its physicomechanical characteristics.

Stress singularity near the end of a cylindrical interface and the linear elasticity of push-in tests

Investigated is the push-in test used to determine the interfacial strength of fiber-reinforced composites via the method of singular integral equation. Singularity analysis shows that stresses near the end for an intact interface have a power singularity, and the singularity index is only dependent on the material constants of the fiber and the matrix. In order to describe the interfacial strength at the initial debonding of the push-in test, critical interfacial shear stress intensity factor and the related singularity index are adopted. This interfacial strength characteristic parameter is applied to analysis of the push-in test results of carbon fiber reinforced epoxy resin composites.

Numerical evaluation of the high accuracy of an integral with power singularities

Numerical evaluation of the high accuracy of an integral with power singularities

Application of Adaptive Quadrature to Axi-symmetric Vortex Sheet Motion

Studies of the formation of fine structures on free surfaces in liquids, such as curvature singularities or interface pinching, demand that the motion of the interface must be computed very accurately. Boundary integral techniques are a popular choice in such studies because they reduce the dimension of the problem by one. On the other hand, the boundary integrals are singular, and their accurate evaluation can prove quite challenging. In two-dimensional motion, the interface is just a curve. When this curve is closed or periodic, the singularity in the integrand may be removed and the trapezoidal rule may be applied with spectral accuracy. Unfortunately, the nature of the singularity in the integrand for three-dimensional motion is much more difficult to treat. In this paper, we present an accurate adaptive quadrature to compute the motion of a vortex sheet in axi-symmetric flow. The technique is based on a vector-potential formulation which offers some computational advantages over other methods based on the Biot–Savart integral. Direct numerical computations show that our technique is much more accurate and efficient than existing techniques. We apply our technique to study the evolution of an initially spherical vortex sheet. We present evidence of the formation of a 3/2 power singularity in the curvature of the vortex sheet.

Identification of Weakly Singular Memory Kernels in Viscoelasticity

Inverse problems for the identification of memory kernels in the linear theory of viscoelasticity with constitutive stress-strain-relation of Boltzmann type are dealt with in the case of weakly singular kernels in the space L p , and of continuous kernels with power singularity at zero. The problems are reduced to nonlinear Volterra integral equations of convolution type for which by the method of contraction with weighted norms global existence, uniqueness, and stability of solutions are proved.

On the singularity of the electron-electron pair potential in quantum wires

We exactly obtain the momentum distribution for the electrons in a one-dimensional model of a quantum wire viewed as a Luttinger liquid that contains impurities which cause the scattering of plasmons. From our results and experimental data we conclude that the electron-electron pair interaction cannot have inverse power singularities at zero momentum transfer.

Asymptotic fields for dynamic crack growth at a ductile-brittle interface

In this work, the asymptotic fields near a crack tip propagating dynamically under plane strain conditions at a ductile-brittle interface are derived. The ductile material is taken to obey the J 2 flow theory of plasticity with linear isotropic strain hardening. The asymptotic solution is assumed to be of the variable-separable form with a power singularity in the radial coordinate from the crack tip. Results have been generated for different bi-material combinations over a range of crack speeds. At a given crack speed, two solutions resembling the Mode I and Mode II asymptotic fields in a homogeneous material are obtained for all strain hardening between a lower and upper bound. The effect of mismatch in elastic stiffness and density, as well as strain hardening of the ductile phase on the singularity strength, near-tip mixities and the angular stress and velocity functions are examined. Attention is focused on the influence of crack speed on the above features of the asymptotic solution.

Identification of Weakly Singular Memory Kernels in Heat Conduction

AbstractInverse problems of identification of memory kernels in linear heat conduction are dealt with in case of weakly singular kernels in the space Lp and of continuous kernels with power singularity. The problems are reduced to nonlinear Volterra integral equations of convolution type for which by the method of contraction with weighted norms global existence and stability of solutions are proved.

Imagining democracy: political culture and democratisation in Buganda

Most of the recent literature on democratisation in Africa has paid insufficient attention to popular understandings of democracy and the local reception of democratic practices. This article examines the articulation of the concept of democracy with existing socio-political conceptions in contemporary Buganda. The standard translation of the word ‘democracy’ into Luganda tends to assimilate it to a local political cosmology which emphasises the values of justice, civility and open communication between rulers and subjects, and involves a conception of sociopolitical hierarchy modelled on the system of clans and kingship. Key ideological features of this conception include its construction from the bottom up, the singularity of power, regulated competition and nested solidarities. Such liberal democratic practices and institutions as elections, political parties and representation are not part of the local definition of democracy. In fact, political parties are widely condemned as antithetical to democratic governance. At a more pragmatic level, however, some of the democratisation initiatives of the current Ugandan government have given rise to a new popular allegiance to democratic elections. These reforms are unusual in that they resonate significantly with local political values and conceptions. The article suggests that more attention should be devoted to the coherence of democratisation initiatives with local socio-political conceptions.