Asymptotic analysis of Bloch–Floquet waves in a thin bi-material strip with a periodic array of finite-length cracks

We present an asymptotic algorithm to solve a problem of wave propagation in a thin bi-material strip with an array of cracks situated at the interface between two materials. For small frequencies we construct an asymptotic solution which takes into account the singular behavior near the crack tips and the smooth nature of the oscillation far away from them. We construct the boundary layer solutions near the crack tips. The boundary layers are harmonic solutions in scaled domains. Dispersion equations are derived and solved within the frame of the asymptotic model.     

Non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane

The non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane was investigated using the generalized Almansi's theorem and the Schmidt method. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the jumps in displacements across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of crack length, the distance between two collinear cracks and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement or magnetic flux singularities are present at the crack tips in a magnetoelectroelastic composite material plane. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.     

Shielding effect and emission criterion of a screw dislocation near an interfacial blunt crack

Shielding effect and emission criterion of a screw dislocation near an interfacial blunt crack are dealt with in this paper. Utilizing the conformal mapping technique, the closed-form solutions are derived for complex potentials and stress fields due to a screw dislocation located near the interfacial blunt crack. The stress intensity factor on the crack tips and the critical stress intensity factor for dislocation emission are also calculated. The influence of the orientation of the dislocation and the morphology of the blunt crack as well as the material elastic dissimilarity on the shielding effect and the emission criterion is discussed in detail. The results show that positive screw dislocations can reduce the stress intensity factor of the interfacial blunt crack tip (shielding effect). The shielding effect increases with the increase of the shear modulus of the lower half-plane, but it decreases with the increase of the dislocation azimuth angle. The critical loads at infinity for dislocation emission increases with the increase of emission angle and curvature radius of blunt crack tip, and the most probable angle for screw dislocation emission is zero. The present solutions contain previous results as special cases.     

一类奇摄动薄板弯曲问题的匹配渐近解

研究了一类薄板弯曲问题. 对四阶奇摄动边值问题, 引入伸长变量, 构造边界附近的内层解, 然后与外部解匹配. 最后用合成展开式理论, 得到了原问题的的渐近解. 关键词： 薄板弯曲 挠度 渐近解     

Influence of nanotwin generation near crack twins on the fracture toughness of nanomaterials

A theoretical model of microscopic mechanisms of the nucleation and development of deformation twins in nanocrystalline materials has been developed. Within the model, we have studied the generation of deformation twins near crack tips, which occurs through multiple nanoscopic shears that represent nanoscopic regions of an ideal plastic shear. It has been shown that the nucleation of such nanotwins near crack tips reduces the high local stresses that arise near these tips. Thus, the generation and development of nanotwins near crack tips increases the fracture toughness of brittle nanocrystalline materials and serves as an efficient mechanism of improving the crack resistance of deformed nanocrystalline materials.     

Regularization of moving boundaries in a laplacian field by a mixed Dirichlet-Neumann boundary condition: exact results

The dynamics of ionization fronts that generate a conducting body are in the simplest approximation equivalent to viscous fingering without regularization. Going beyond this approximation, we suggest that ionization fronts can be modeled by a mixed Dirichlet-Neumann boundary condition. We derive exact uniformly propagating solutions of this problem in 2D and construct a single partial differential equation governing small perturbations of these solutions. For some parameter value, this equation can be solved analytically, which shows rigorously that the uniformly propagating solution is linearly convectively stable and that the asymptotic relaxation is universal and exponential in time.     

Some solutions of the 3D Laplace equation in a layer with oscillating boundary describing an array of nanotubes and an application to cold field emission. I. Regular array

The aim of this paper is to construct solutions of the Dirichlet problem for the 3D Laplace equation in a layer with highly oscillating boundary. The boundary simulates the surface of a nanotube array, and the solutions are applied to compute the cold field electron emission. We suggest a family of exact solutions that solve the problem for a boundary with appropriate geometry. These solutions, along with the Fowler-Nordheim formula, allow one to present explicit asymptotic formulas for the electric field and the emission current. In this part of the paper, we consider the main mathematical aspects, restricting ourselves to the analysis of properties of the potential created by a single tube and a regular array of tubes. In the next part, we shall consider some cases corresponding to nonregular arrays of tubes and concrete physical examples.     

Cracks at interfaces—Atomistic versus continuum models

Sharp cracks with tips at interfaces in a bimaterial medium show a much richer variety of singularities in the stress fields near the tips as predicted by continuum linear elastic models than do similar cracks in a homogeneous elastic medium. The real or complex magnitudes of these singularities depend upon the ratio of the elastic constants of the two media as well as upon the angle between the crack and the interface. Problems incurred in comparing results of atomistic simulations of such cracks with the relevant continuum solutions are discussed.     

Elastic interaction of a dislocation array with a phase boundary

The equilibrium configuration of an array of dislocations in parallel equidistant slip planes under an external shear stress near a welded boundary between two isotropic half-spaces having different elastic constants is computed. For large external stress, the dislocations are arranged into an arc concave when seen from the boundary. It is concluded that such an arc is formed at the tip of a twin or of a martensitic plate near a phase boundary. The tensile stress across the boundary due to an edge dislocation array is discussed in connection with the formation of an interfacial crack.     

Elastic analysis of a mode Ⅱ crack in an icosahedral quasicrystal

Based on the displacement potential functions, the elastic analysis of a mode II crack in an icosahedral quasicrystal is performed by using the Fourier transform and dual integral equation theory. By the solution, the analytic expressions for the displacement field and stress field are obtained. The asymptotic behaviours of the phonon and phason stress fields around the crack tip indicate that the stresses near the crack tip exhibit a square root singularity. The most important physical quantities of fracture theory, crack stress intensity factor and energy release rate, are evaluated in an explicit version.     

Actual light deflections in regions of crack tips and their influence on measurements in photomechanics

Crack-tip photomechanics procedures are based on certain simplifying assumptions that are seldom discussed. In a recent paper the theoretical bases of the shadow optical methods of caustics have been analysed and tested using the results obtained by three analytical-experimental procedures, namely classical strain gage techniques, isodynes, and strain-gradient index method. It has been concluded that the straing-radient index method appears to be a suitable tool for analysis of stress states near crack tips and notches and, in particular, for testing the predictive power of the pertinent singular solutions of the linear elastic fracture mechanics and the ranges of their applicability. In the present paper, a more detailed analysis of all results obtained in light deflection experiments allows to quantify the contribution of both involved effects and to determine the distortion of the faces of the investigated plates along their crack planes. The ability of the strain-gradient light bending method to analyse some features of the three-dimensional stress-state is reported. Finally, the presented experimental evidence allows to draw conclusions related to limits of applicability of certain photomechanical measurements near crack tips.^†     

Elastic analysis of a mode II crack in a decagonal quasi—crystal

The elastic analysis of a mode II Griffith crack penetrating through a decagonal quasi-crystal along the periodic axis is made within the context of the continuum theory. By using a general solution obtained previously, the problem in the case of uniform shear stress at infinity is solved, and the analytical expressions for the entire stress field disturbed by an internal crack are derived in an explicit form. The asymptotic fields of the displacement and stress around a crack tip in both phonon and phason fields indicate that the stresses near a crack tip exhibit the square-root singularity. The formula for evaluating the energy release rate is also given. If imposing that the phason field is absent, the well-known results of a mode II crack in a conventional material are recovered from the present results.     

The density profile for the Klein-Kramers equation near an absorbing wall

We derive asymptotic series for the expansion coefficients of a function in terms of the Pagani functions, which occur in the boundary layer solutions of the Klein-Kramers equation. The results enable us to determine the density profile in the stationary solution of this equation near an absorbing wall from the numerically determined velocity distribution at the wall, with an accuracy of about 2%. We also obtain information about the analytic behavior of the density profile: this profile increases near the wall with the square root of the distance to the wall. Finally, the asymptotic analysis leads to an understanding of the slow convergence of variational approximations to the solution of the absorbing-wall problem and of the exponents that occur when one studies the variational approximations to various quantities of interest as functions of the number of terms in the variational ansatz. This is used to obtain a better variational estimate for the density at the wall.     

Basic solution of four parallel non-symmetric permeable Mode-III cracks in a piezoelectric/piezomagnetic composite plane

The interaction of four parallel non-symmetric permeable cracks in a piezoelectric/piezomagnetic composite plane subjected to anti-plane shear stress loading was studied by the Schmidt method. The problem was formulated through a Fourier transform into four pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the relationships among the electric displacement, magnetic flux and stress fields near the crack tips were obtained. The results show that the stress, the electric displacement and the magnetic flux intensity factors at the crack tips depend on the lengths and spacing of cracks. It was also revealed that the crack shielding effect is present in piezoelectric/piezomagnetic composites.     

Boundary Singularity of Moments for the Linearized Boltzmann Equation

We study the boundary singularity for stationary solutions of the linearized Boltzmann equation with hard-sphere potential. An asymptotic estimate for the gradient of the moments is established, which shows the logarithmic singularity near the boundary. Our formula holds for the solutions of the Milne and Kramers problems obtained by Bardos-Caflish-Nicolaenko (Commun. Pure Appl. Math. 49:323–452, 1986). Our theorem requires the Hölder continuity of the boundary data. In particular, it applies to the complete condensation problem for half space.     

Solutions to two or four parallel Mode-I permeable cracks in magnetoelectroelastic composite materials

The solutions to two or four parallel Mode-I permeable cracks in magnetoelectroelastic composite materials are derived using the generalized Almansi's theorem under permeable electric and magnetic boundary conditions. The problem can be solved through the Fourier transform with the help of two pairs of dual integral equations, in which unknown variables were jumps of displacements across crack surfaces, not dislocation density functions. To solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials to obtain the relations among the electric displacement intensity factors, the magnetic flux intensity factors and the stress intensity factors at the crack tips. The paper presents the interactions of two or four parallel Mode-I cracks in magnetoelectroelastic composite materials and the crack-shielding effect in magnetoelectroelastic composite materials.     

Crack propagation as a free boundary problem

A sharp interface model of crack propagation as a phase transition process is discussed. We develop a multipole expansion technique to solve this free boundary problem numerically. We obtain steady state solutions with a self-consistently selected propagation velocity and shape of the crack, provided that elastodynamic effects are taken into account. Also, we find a saturation of the steady state crack velocity below the Rayleigh speed, tip blunting with increasing driving force, and a tip splitting instability above a critical driving force.     

Wave function of the Universe in the early stage of its evolution

In quantum cosmological models, constructed in the framework of Friedmann–Robertson–Walker metrics, a nucleation of the Universe with its further expansion is described as a tunneling transition through an effective barrier between regions with small and large values of the scale factor a at non-zero (or zero) energy. The approach for describing this tunneling consists of constructing a wave function satisfying an appropriate boundary condition. There are various ways for defining the boundary condition that lead to different estimates of the barrier penetrability and the tunneling time. In order to describe the escape from the tunneling region as accurately as possible and to construct the total wave function on the basis of its two partial solutions unambiguously, we use the tunneling boundary condition that the total wave function must represent only the outgoing wave at the point of escape from the barrier, where the following definition for the wave is introduced: the wave is represented by the wave function whose modulus changes minimally under a variation of the scale factor a. We construct a new method for a direct non-semiclassical calculation of the total stationary wave function of the Universe, analyze the behavior of this wave function in the tunneling region, near the escape point and in the asymptotic region, and estimate the barrier penetrability. We observe oscillations of the modulus of the wave function in the external region starting from the turning point which decrease with increasing of a and which are not shown in semiclassical calculations. The period of such an oscillation decreases uniformly with increasing a and can be used as a fully quantum dynamical characteristic of the expansion of the Universe.     

Solution of collinear cracks in an electrostrictive solid

A general treatment is presented of the two-dimensional problem of N collinear cracks in an infinite electrostrictive material subjected to remote electric loads based on the complex variable method combined with analytical extension of the complex variable functions. First, for the case of permeable cracks, general solutions for the electric potentials, Maxwell stresses, electroelastic stresses and stress intensity factors are derived. As specific examples, explicit and concise results are obtained for the cases of one crack and two collinear cracks. Then, these results are extended to the cases of impermeable and conducting collinear cracks, respectively. It is found that, in general, the total stresses always have the classical singularity of the r ^{- 1/2} type at the crack tips, whereas the Maxwell stresses have an r ^{- 1} singularity for the above three crack models. Finally, it is concluded that the applied electric field may either enhance or retard crack growth depending on the electric boundary conditions adopted on the crack faces, and the Maxwell stresses on the crack faces and at infinity.