Solutions of multiple crack problems of a circular region with free or fixed boundary condition in antiplane elasticity

Four cases of multiple crack problems in antiplane elasticity, circular region or an infinite region exterior to a circle with free or fixed boundary condition, are considered in this paper. For all these cases, the presented elementary solution is a particular solution of the circular region containing one crack. The solution consists of two parts and satisfies the following conditions: (a) The first part corresponds to a pair of longitudinal forces acting at a prescribed point on both edges of a single crack; (b) The second part corresponds to some distributed forces along both edges of the crack; (c) The elementary solution, i.e. the sum of the first and second parts, satisfies the above mentioned boundary value conditions along the circular boundary. As we have done in [1–3], the system of Fredholm integral equations for the undetermined densities of the elementary solutions can be easily estabished and the stress intensity factors at crack tips can also be evaluated.     

Solutions of the torsion problem for some kinds of bars with multiply connected section

This paper is a continuation of the senior author's previous papers[1, 2]. In the paper the multiply connected region dealt with is a rectangular section containing several rectangular holes (Figs. 2–4). The boundary value problem of such a multiply connected region in the case of torsion problem can be divided into two similar problems. Each one of these problems belongs to a Dirichlet problem of Laplace equation [3]. There is only one undetermined constant $\text{K}$, which is determined from the single value condition of the displacement [3]. When solving these boundary value problems, the original section is divided into several rectangular subregions (Figs. 2–4), and then the Duhem theorem (see Appendix I) is used. After two boundary value problems are solved, the undetermined constant $\text{K}$ can be found by using the above mentioned condition. Finally, the torsional rigidity can be easily deduced. Three sets of numerical results are shown in Tables 1–3.     

A note on the boundary value problem arising in free convection in porous media

By similarity transformation and governing equations of free convection on a heated vertical plate embedded in porous medium are reduced to coupled nonlinear equations. The equations are numerically integrated using the boundary conditions at the plate and at ‘infinity’. Assuming that the plate is subjected to a prescribed temperature [1–3] or to a prescribed heat flux [4, 5], the boundary value problems have been solved independently. These researchers seem to have not noted that the solutions for the two cases are dependent on each other. In the present note we consider yet another thermal boundary condition, namely, radiation boundary condition [6] at the plate and show that the solutions for the three cases are dependent and one can pass from one solution to the other easily.     

The structure of a migrating low-angle tilt grain boundary in strontium titanate

Weak beam transmission electron microscopy and stereomicroscopy have been used to characterize the three-dimensional structure of a severely deformed low-angle tilt grain boundary in a strontium titanate ceramic. Various interactions between crystal lattice dislocations and grain boundary dislocations in this boundary have been analysed. The deformed low-angle tilt grain boundary is a result of partial glide. The boundary is composed of dislocations with Burgers vectorsa [¯100] and the deformation of the boundary is interpreted as having occurred by the interactions of the boundary with crystal lattice dislocations during grain boundary migration. Observed dislocation network in the grain boundary are a result of the reactionb ₃=b₁+b₂=a [¯100]+a [00¯1]=a [¯10¯1], and the resultant dislocations are sessile. Also, many crystal lattice dislocations are pinned by the grain boundary and produce a complex dislocation structure for the boundary.M. Fujimoto is a Visiting Scientist at the Massachusetts Institute of Technology on leave from Taiyo Yuden Co., Ltd., Tokyo, Japan.     

First-principles calculation of grain boundary energy and grain boundary excess free volume in aluminum: role of grain boundary elastic energy

We examined the grain boundary energy (GBE) and grain boundary excess free volume (BFV) by applying the first-principles calculation for six [110] symmetric tilt grain boundaries in aluminum to clarify the origin of GBE. The GBE increased linearly as BFV increased. The elastic energy associated with BFV, namely the grain boundary elastic energy, was estimated as a function of BFV and the shear modulus. The grain boundary elastic energies were close in value to the GBEs. The charge density distributions indicated that the bonding in the grain boundary region is significantly different from the bonding in the bulk. The grain boundary elastic energies were 15–32% higher than the GBEs. This overestimation of the grain boundary elastic energy is caused by the characteristics of the electronic bonding at the grain boundary, which is different from bonding in the bulk. We have concluded that GBE results mainly from the grain boundary elastic energy.     

Diffusion boundary layer of a rotating disk electrode as a thin-layer spectroelectrochemical cell

A UV-visible rapid scan spectrophotometer (RSS) was coupled to a Au rotating disk electrode (RDE) for monitoring at near-normal incidence the reflection-absorption spectrum of the diffusion boundary layer in [Fe(CN)(6)](4)(-) aqueous solutions over a potential region in which [Fe(CN)(6)](4-) oxidizes, generating highly absorbing [Fe(CN)(6)](3-) (lambda(max) = 420 nm). Measurements were performed under steady-state conditions at rotation rates, omega, in the range 300 相似文献   

Onset of bouyancy-driven convection in superposed reacting fluid and porous layers

A linear stability analysis is applied to the onset of bouyancy-driven convection in a horizontal layer of reacting fluid overlying a porous region saturated with the same fluid. The fluid is assumed to be undergoing zero-order exothermic reactions in both regions. At the interface between fluid and porous layers the boundary conditions proposed by Nield [1] are employed; these include the empirical slip condition suggested by Beavers and Joseph [2]. Predictions for the onset of convection and critical wavenumbers are obtained from the analysis by the collocation method and solution of the resulting generalized eigenvalue problem. The effect of variable fluid/porous layer depth ratio, Frank–Kamenetskii number or thermal boundary conditions on the onset of fluid motions is studied, and ignition values of the Frank–Kamenetskii number for the system are calculated.     

Internal oxidation: does the classical model contradict local equilibrium conditions?

Abstract

Previously, Morral and Li have published an anomalous behaviour of the classical model of internal oxidation with respect to their local-equilibrium conditions/rules. The local-equilibrium conditions were interpreted from the local-equilibrium expression derived from the flux balance at the inter-region boundary – the boundary between a two-phase region and a single-phase region – under the local-equilibrium assumption. These local-equilibrium conditions predict a continuous average composition profile for the system where the two-phase region is growing at the expense of the single-phase region. In contrast, the classical model of internal oxidation shows a composition jump at the boundary of the growing two-phase region and the shrinking single-phase region.

Here, a complete interpretation of the local-equilibrium expression is delineated including the case of zero solubility-product. The relative behaviour of two of the composition-vectors in the local-equilibrium expression undergoes an abrupt change at zero solubility-product. This change in behaviour predicts a composition jump at the inter-region boundary for the growing two-phase region. Consequently, the classical model of internal oxidation is shown to follow the local equilibrium on all counts and thereby eliminating all contradictions.     

Concerning closed-streamline flows with discontinuous boundary conditions

High-Reynolds-number (Re) flow containing closed streamlines (Prandtl-Batchelor flows), within a region enclosed by a smooth boundary at which the boundary conditions are discontinuous, is considered. In spite of the need for local analysis to account fully for flow at points of discontinuity, asymptotic analysis for Re 1 indicates that the resulting mathematical problem for determining the uniform vorticity ₀) in these situations, requiring the solution of periodic boundary-layer equations, is in essence the same as that for a flow with continuous boundary data. Extensions are proposed to earlier work [3] to enable ₀ to be computed numerically; these require coordinate transformations for the boundary-layer variables at singularities, as well as a two-zone numerical integration scheme. The ideas are demonstrated numerically for the classical circular sleeve.     

Partial-differential-equation-constrained amplitude-based shape detection in inverse acoustic scattering

In this article we discuss a formal framework for casting the inverse problem of detecting the location and shape of an insonified scatterer embedded within a two-dimensional homogeneous acoustic host, in terms of a partial-differential-equation-constrained optimization approach. We seek to satisfy the ensuing Karush–Kuhn–Tucker first-order optimality conditions using boundary integral equations. The treatment of evolving boundary shapes, which arise naturally during the search for the true shape, resides on the use of total derivatives, borrowing from recent work by Bonnet and Guzina [1–4] in elastodynamics. We consider incomplete information collected at stations sparsely spaced at the assumed obstacle’s backscattered region. To improve on the ability of the optimizer to arrive at the global optimum we: (a) favor an amplitude-based misfit functional; and (b) iterate over both the frequency- and wave-direction spaces through a sequence of problems. We report numerical results for sound-hard objects with shapes ranging from circles, to penny- and kite-shaped, including obstacles with arbitrarily shaped non-convex boundaries. Partial support for this work was provided by the US National Science Foundation under grant award CMS-0348484.     

An effective hybrid displacement function element method for solving the edge effect of Mindlin–Reissner plate

In bending problems of Mindlin–Reissner plate, the resultant forces often vary dramatically within a narrow range near free and soft simply‐supported (SS1) boundaries. This is so‐called the edge effect or the boundary layer effect, a challenging problem for conventional finite element method. In this paper, an effective finite element method for analysis of such edge effect is developed. The construction procedure is based on the hybrid displacement function (HDF) element method [1], a simple hybrid‐Trefftz stress element method proposed recently. What is different is that an additional displacement function f related to the edge effect is considered, and its analytical solutions are employed as the additional trial functions for the first time. Furthermore, the free and the SS1 boundary conditions are also applied to modify the element assumed resultant fields. Then, two new special elements, HDF‐P4‐Free and HDF‐P4‐SS1, are successfully constructed. These new elements are allocated along the corresponding boundaries of the plate, while the other region is modeled by the usual HDF plate element HDF‐P4‐11 β [1]. Numerical tests demonstrate that the present method can effectively capture the edge effects and exactly satisfy the corresponding boundary conditions by only using relatively coarse meshes. Copyright © 2015 John Wiley & Sons, Ltd.     

Normal mode solutions of linear dynamic field theories using Green's extended identity

Normal mode solutions of certain classes of linear, spatially time-invariant, self-adjoint and nonself-adjoint differential operators, with inhomogéneous boundary conditions in a finite region of arbitrary shape, are obtained by the use of Green's extended identity [1] in conjunction with the eigenvalue problems associated with the differential operators. Thus, continuum field theories belonging to these classes of operators, which encompass arbitrary (a) material and geometrical parameters, (b) spatial and time-dependent boundary conditions, (c) initial conditions can be solved by this technique. This is illustrated by the determination of the transient response of an axisymmetric, finite, thick transversely isotropic elastic hollow cylinder under inhomogeneous boundary conditions of all types (both pure and mixed). It is also shown that the displacement (separation-of-variables, integral transforms) and acceleration (Mindlin-Goodman, Williams) methods currently used for solutions of dynamic problems can both be derived from Green's extended identity.     

Free‐surface effects on an oblique towing flat plate

Abstract

This paper applies a finite difference method in calculating the free surface profiles, the lateral hydrodynamic forces as well as the moments acting on a plate during the oblique towing of a flat plate on the free surface. The 3‐D fluid motion in the infinite region is approximated by the unsteady 2‐D flow in the finite region with an artificial open boundary on which a more relaxed Sommerfeld's radiation condition is imposed. The free surface profiles, nondimensional side forces and yawing moments are calculated by the present method and are compared satisfactorily with the experimental results [15], Chapman's results [4], and Yamasaki's computational results [16].     

Morphotropic phase boundary in the system Pb(ZrxTi1−x)O3

The room temperature phase boundary dividing the rhombohedral and tetragonal in the solid solution system lead zirconate titanate Pb(Zr_xTi_1?x)O₃ is examined carefully using an ultrahomogeneous synthesis technique. There is no evidence for a broad transition region of co-existing phases as understood earlier but a sharp boundary located at x = 0.510 ± 0.002.     

Fatigue damage in nickel and copper single crystals at nanoscale

Nanoscale fatigue damage simulations using molecular dynamics were performed in nickel and copper single crystals. Cyclic stress–strain curves and fatigue crack growth were investigated using a middle-tension (MT) specimen with the lateral sides allowing periodic boundary conditions to simulate a small region of material as a part of a larger component. The specimen dimensions were in the range of nanometers, and the fatigue loading was strain controlled under constant and variable amplitude. Four crystal orientations, [111], [100], [110] and [101] were analyzed, and the results indicated that the plastic deformation and fatigue crack growth rates vary widely from one orientation to another. Under increasing strain amplitude loading, nickel nanocrystals experienced a large amount of plastic deformation causing at least in one orientation, [101], out-of-plane crack deviation in a mixed mode I+ II growth. Under constant amplitude loading, the fatigue cracks were a planar mode I type. Double slip is observed for some orientations, while for others, many more slip systems were activated causing a more evenly distributed plastic region around the crack tip. A comparative analysis revealed that small cracks grow more rapidly in copper than in nickel single crystals.     

Solution of the problem of a crack in a finite plane region using an indirect boundary-integral method

In a previous paper [1], an indirect boundary-integral approach was developed for the treatment of a finite, plane, linear-elastic region weakened by a hole of arbitary shape. It was suggested there that this method would yield excellent results on and near the hole boundary.

In this presentation, the method is applied to the problem of a finite, plane, linear-elastic region containing a sharp crack. Numerical results are obtained for a simple geometry and the crack-opening-displacement and crack tip stresses are compared to the known solution. It is shown that the boundary-integral solution is quite accurate.     

Mechanism for the dependence of the photoemission threshold of a photocathode on its irradiance

Secondary electron emission is shown to occur at the boundary of the working region of a photocathode when current flows through this region, causing a deviation from the known classical laws of photoemission. It is established that the sensitivity and photoemission threshold of the photocathode depend on the strength of the photocurrent flowing through it. Pis’ma Zh. Tekh. Fiz. 24, 1–6 (September 26, 1998)     

Interface structure in the growth of semiconductor crystals using the Bridgman–Stockbarger method

In this paper we assume, like in Balint et al. [Mater. Sci. Semiconductor Process. V3/3 (2000) 115–121], that the melt/solid interface in the case of vertically stabilized Bridgman–Stockbarger semiconductor growth system is a thin layer, masking the crystal, where a weak form of the periodical structure of the crystal exists. In these conditions, using a new diffusion coefficient in the equation of the dopant transport in the interface region, we compute the axial and radial variation of the dopant field in this region for crystal and melt with thermophysical properties similar to the gallium-doped germanium. We compare the results to those obtained in Mater. Sci. Semiconductor Process. V3/3 (2000) 115–121 [1], where we have changed the diffusion coefficient only in the boundary condition for the dopant concentration at the melt/solid interface.     

两种材料界面裂纹的应力强度因子

选用复变函数的幂级数作为应力函数,分析在两种材料界面上存在裂纹的情况。利用裂纹自由表面力的边界条件及材料界面上的连续条件求得特征值及待定系数之间的关系式,从而求得应力、位移及应力强度因子表达式。本文修改了Lin[1]和Mar[2]的应力函数表达式,使所得应力、位移表达式与Williams[6]及Sih—Rice[3,4]结果一致。基于上述解析解建立了裂纹尖端的奇异元。并以含有界面中心裂纹平板拉伸情况作为计算实例。