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  dual integral
    By Hankel integral transform, the problem is reduced to a set of dual integral equations which are transformed into a set of singular integral equations.
    文中采用Hankel积分变换,将散射问题转化为求解对偶积分方程,进而变换为奇异积分方程组.
短句来源
    by using the Fourier integral transform and the boundary conditions, the problem is reduced to a dual integral equations. The dynamic stress intensity factors at the crack tip are obtained by the using Copson methods and the numerical integral technique. As an example, the eects of the parameter and the frequency of SH wave on norm dynamic stress intensity factors are discussed.
    研究了正交各向异性功能梯度材料中直裂纹对SH波的散射问题,材料两个方向的剪切模量和密度假定为指数模型,通过积分变换-积分方程方法,建立数学模型,化为对偶积分方程,用Copson方法求解对偶积分方程,最后得到动应力强度因子,并且给出了数值算例,讨论了在SH波作用下,裂纹尖端的动应力强度因子与入射波的频率,入射角的关系.
短句来源
    Fourier transform is employed to reduce to this mixed boundary value problem to three pairs of dual integral equations; also the new additional boundary conditions are discussed.
    利用Fourier 变换技术将力电复合边值问题转化为三组对偶积分方程,并讨论了所需要新增的边界条件。
短句来源
    By integral transform method, the scattering of elastic wave by two cracks is reduced to a set of dual integral equations which are further transformed to a set of singular integral equations of the first kind. By means of Chebyshev polynomials of the first kind, the solutions of the singular integral equations have been obtained, and the formulae of the dynamic stress intensity factors are presented.
    文中采用积分变换法将双裂纹的弹性波散射化为对偶积分方程,进而将其转化为第一类奇异积分方程组,借助于第一类Chebyshev多项式,给出了奇异积分方程组的解答,并得到了双裂纹的动态应力强度因子的计算公式
短句来源
    The influence function of the axisymmetrical problem is given,and dual integral equations of the problem are derived.
    给出了轴对称问题的影响函数,导出了圆盘状Ⅰ型裂纹非局部理论解的对偶积分方程。
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In this paper, the three-dimensional elastic solid with internal rectangular crack is considered. Let the crack surfaces be subjected to equal and opposite normal tractions p0. This problem is reduced, by means of Fourier transforms, to the standard set of dual integral equations with two variables. Then the fomulas of analytic solution of the displacements on the crack surfaces and of the stress-intensity factors of crack border are obtained.

本文分析了含有矩形片状裂纹(裂纹上下表面受均匀压力作用)的三维弹性体.借助Fourier积分变换.将问题化归为二个变数的对偶积分方程,并获得裂纹面位移和裂纹前缘应力强度因子的解析表达式.

In this paper, the transient response of a pair of radial cracks which are of equal length and notched symmetrically from a borehole, hole-line-shaped (HLS) crack, is discussed in case of normal tractions being suddenly applied to the wall pf the borehole and the surface of the crack, which comes from the approximatioe of the initiation of the HLS crack under dynamic loading, that is, the first phasn of fracture control.Due to the complexity of the boundary and the existance of the inertia term in the dynamic...

In this paper, the transient response of a pair of radial cracks which are of equal length and notched symmetrically from a borehole, hole-line-shaped (HLS) crack, is discussed in case of normal tractions being suddenly applied to the wall pf the borehole and the surface of the crack, which comes from the approximatioe of the initiation of the HLS crack under dynamic loading, that is, the first phasn of fracture control.Due to the complexity of the boundary and the existance of the inertia term in the dynamic equation, neither the integral transform for space veriable nor the conformal mapping can be applied to this kind of problems directly. But from Tweed's work for a crack notched from a circular hole in static case, we can get some inspiration. The problem can be solved by decomposition-superimposition-iteration (DSI)method in the following steps. First the initial-boundary value problem is changed into the boundary value problem by means of Laplace transformation. Next the solution region is decomposed into two simple regions, one is a circular hole, the other is a finite Line-shaped crack.The solutions of the decomposed problems can be obtained by using the separation of variables and the Fourier transform respectively. The displacement and the slress components are derived upon the super-imposition of the solutions. They have satisfied some conditions naturally. Then inserting them into the other conditions, the dual integral equations are obtained,where V(r, p) is a series. The coefficients in it can be determined from the borehole boundary conditions.The unknown function D(aaaaaaaaaaaaaaaaa, p) is included in these coefficients. To solve (1),V(r, p)is regarded as the free term temporarily, the predictor corrector method has been applied here- predicting V(r, p) in ( 1 ) to solve D(α, p), then making use of the borehole conditions to correct V(r, p), the corrected V(r, p) is inserted into ( 1 ) to solve D(α, p) again until the mstable V(t, p) is obtained. Thus ( 1 ) can be reduced to the standard Fredhol integral equation of the second kind.when the radius of the borehole approximates to zero, the equation is identical with Sih's solution for relevant problem of finite line-shaped crack in 1972. The solution of the equation ( 2 ) is got by iteration mentioned above and it leads to the Laplace transforms of the stress components, subsequently they are inverted by a combination of numerical means and an application of the Cagniard-DeHoop inversion technique. It has shown that the space distribution of the singular stress field of the crack discussed here is the same as Sih obtained, while the dynamic stress intensity factor k1,(t) reflects the affecting of the borehole. These are essential information in the application of the current theory of fracture mechnics, The numerical calculations demonstrate the feasibility of the DS1 method and the results provide the quantitative basis for us to simplify the practical problem, and make us fully understand the interaction of wave and the crack in this problem. The inferences from above results are in agreement .with others' experiments. Upon this the reasonable loading method is recommended. Moreover a number of theoretic extensions of the work and the ways to carry out the technique (fracture control) simply and economically in practice are proposed. Among these the most important one is that the DSI method can be extended to study the problem of the HLS crack moving at a constant velocity, which comes from the approximation of the propagation of the HLS crack under dynamic loading.

本文讨论由钻孔壁刻出的两个对称径向裂纹在孔壁及裂纹面上突加正压力作用下的瞬态响应,此问题来自于控制断裂。借助于叠加原理和积分交换方法,问题简化对偶积分方程,基于预测校正法的概念将孔壁边界条件对此方程的影响解耦,从而得到了第二类Fredholm积分方程,以迭代法和Cagniard-DeHoop反演技术求得解。结果表明在此裂端奇异场的空间分布与Sih 1972年就相应线状裂纹问题所得相同,而动态应力强度因子k_1(t)反映了孔的影响,当钻孔半径趋于零时解趋于Sih的解;并使得我们对本问题中波和裂纹的相互作用有了全面的了解,基于此提出了较为合理的加载方法,此外还提出了对本文工作的一系列理论推广以及简便和经济地应用控制断裂技术的方法。

In the present paper the general representations for the solution of the dynamical crack problems of the laminated media consist of different orthotropic layer are derived. Using conditions of the welding surfa.ce, we express all the quantities in terms of a single unknown function. Using method of integral transform, we formulate the impact loading problem as dual integral equations. Using the Copsons method, the solution is derived. The method in the paper is not difficult to generalize for the case of an...

In the present paper the general representations for the solution of the dynamical crack problems of the laminated media consist of different orthotropic layer are derived. Using conditions of the welding surfa.ce, we express all the quantities in terms of a single unknown function. Using method of integral transform, we formulate the impact loading problem as dual integral equations. Using the Copsons method, the solution is derived. The method in the paper is not difficult to generalize for the case of an arbiritrary loading applied to the crack.

本文对由不同正交异性材料组成的层状结构中受冲击载荷的Ⅲ型裂纹动力学问题给出了解的一般表示。文中利用结合面的边界条件,将问题中所有各量用单一未知函数表示,用积分变换方法将受冲击力作用问题化为对偶积分方程,并用Copson方法给出了解。本文方法也可以应用到裂纹表面受任意动载荷的情况。

 
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