Inverse problem of fracture mechanics for a disk fitted onto a rotating shaft

A plane problem of fracture mechanics for a circular disk fitted onto a rotating shaft is considered. The disk is assumed to be fitted tightly onto the shaft, and there are N randomly located straight-line cracks of length 2l_k (k = 1, 2, ..., N) near the inner surface of the disk. The interference between the disk and the rotating shaft, providing minimization of fracture parameters (stress intensity factor) of the disk, is theoretically studied on the basis of the minimax criterion. A closed system of algebraic equations is constructed, which allows the problem of optimal design to be solved. A simplified method of minimization of disk fracture parameters is considered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 201–209, July–August, 2009.     

Finite-part integral and boundary element method for interaction between two parallel planar cracks in three-dimensional finite bodies

By using the Somigliana representation and the concepts of finite-part integrals, a set of hypersingular integral equations of the interaction between two parallel planar cracks in a three-dimensional finite body subjected to arbitrary loads is derived, and then its numerical method is proposed by the finite-part integral method combined with the boundary element method. According to the analytic theory of hypersingular integral equations, the square root models of displacement discontinuities in the elements near the crack front are applied, and thus the computational precision is raised. Based on this, the stress intensity factors can be directly calculated. Finally, the stress intensity factors of several typical interaction problems are calculated.     

关于界面下裂纹对界面裂纹的屏蔽参数

讨论了界面下裂纹对界面裂干涉的合理屏蔽参数问题。对于一条在远场载荷作用下、受到附近界面下裂纹干涉的界面裂纹,采用伪力法计算不同长度单位时界面裂纹的G／G0、k1/k10和k11/k110以及K1／K10和K11／K110（G是能量释放率、k1 ik110是通常定义的应力强度因子、K10＋iK110是含标定长度的应力强度因子;G0、k10 ik110、K10＋iK110对应单一界面裂纹情况）。结果表明,G／G0、K1／K10和K11／K110是较为合理的描述界面下裂纹对界面裂纹屏蔽的参数。     

Basic solution of two parallel non-symmetric permeable cracks in piezoelectric materials

The behavior of two parallel non-symmetric cracks in piezoelectric materials subjected to the anti-plane shear loading was studied by the Schmidt method for the permeable crack electric boundary conditions. Through the Fourier transform, the present problem can be solved with two pairs of dual integral equations ip which the unknown variables are the jumps of displacements across crack surfaces. To solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials. Finally, the relations between electric displacement intensity factors and stress intensity factors at crack tips can be obtained. Numerical examples are provided to show the effect of the distance between two cracks upon stress and electric displacement intensity factors at crack tips. Contrary to the impermeable crack surface condition solution, it is found that electric displacement intensity factors for the permeable crack surface conditions are much smaller than those for the impermeable crack surface conditions. At the same time, it can be found that the crack shielding effect is also present in the piezoelectric materials.     

采用新方法研究加层压电材料中平行界面共线双裂纹的断裂问题

利用新的方法(Schmidt方法)研究加层压电材料中含共线并与材料界面平行的双裂纹在稳态弹性波作用下的动态问题,经富立叶变换使问题的求解转换为求解两对三重对偶积分方程.这些方程可以采用Schmidt方法来求解,这个方法不同与以前求解所利用的方法.结果表明应力强度因子不仅与裂纹的几何尺寸、入射波频率、加层厚度有关,而且与材料性质有关.     

THERMAL FRACTURE OF FUNCTIONALLY GRADED PLATE WITH PARALLEL SURFACE CRACKS

This work examines the fracture behavior of a functionally graded material （FGM） plate containing parallel surface cracks with alternating lengths subjected to a thermal shock. The thermal stress intensity factors （TSIFs） at the tips of long and short cracks are calculated using a singular integral equation technique. The critical thermal shock △Tc that causes crack initiation is calculated using a stress intensity factor criterion. Numerical examples of TSIFs and △Tc for an Al2O3/Si3N4 FGM plate are presented to illustrate the effects of thermal property gradation, crack spacing and crack length ratio on the TSIFs and △Tc. It is found that for a given crack length ratio, the TSIFs at the tips of both long and short cracks can be reduced significantly and △Tc can be enhanced by introducing appropriate material gradation. The TSIFs also decrease dramatically with a decrease in crack spacing. The TSIF at the tips of short cracks may be higher than that for the long cracks under certain crack geometry conditions. Hence, the short cracks instead of long cracks may first start to grow under the thermal shock loading.     

Dynamic stress intensity factors of two 3D rectangular cracks in a transversely isotropic elastic material under a time-harmonic elastic P-wave

The dynamic stress intensity factors (DSIFs) of two 3D rectangular cracks in a transversely isotropic elastic material under an incident harmonic stress wave are investigated by generalized Almansi’s theorem and the Schmidt method in the present paper. Using 2D Fourier transform and defining the jumps of displacement components across the crack surface as the unknown functions, three pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the geometric shape of the rectangular crack, the characteristics of the harmonic wave and the distance between two rectangular cracks on the DSIFs of the transversely isotropic elastic material.     

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采用有限元方法对表面含有两个不同大小半椭圆裂纹的有限厚矩形板在拉伸载荷作用下 进行参数化求解，得到两裂纹取不同尺寸和位置坐标时附属裂纹(尺寸较小裂纹)对主 导裂纹(尺寸较大裂纹)前沿参考点($\theta= 0, \pi/2, \pi $)处的应力强 度因子的影响系数$\beta$, 然后结合神经网络技术建立了相邻裂纹尺寸和位置参 数到主导裂纹前沿点处$\beta$的多变量非线性映射关系.     

Crack propagation in glass coatings under expanding spherical contact

The growth of transverse cracks under expanding spherical contact in a model system consisted of soda-lime glass bonded to a polycarbonate substrate is observed in situ from below or from the polished edge of the bilayer. Abrasion or chemical etching is employed on the coating surfaces to control the initial fracture. In the limit case of monoliths, the crack mouth becomes fully engulfed by the expanding contact, which results in a much steeper crack angle compared to the classical Hertzian cone case. As the coating thickness is reduced, flexure stresses are set in the coating which drive the cone crack to well away from the contact circle and initiate semi-elliptical-like radial cracks at the subsurface, right under the contact. Common to all three fracture modes is an initial unstable propagation phase following by a stable growth, with detrimental failure associated with severe damage to the top surface and/or delamination at the coating/substrate interface taking place at loads several times the fracture initiation loads.LEFM in conjunction with a large-strain FEM contact code is used to study the post-initiation fracture, with the crack path controlled by the principal stress trajectory or zero-mode II S.I.F. The analysis exposes the leading geometric and material parameters in each fracture mode, which may be useful in the design of bilayer structures for optimal mechanical performance. The well-known Auerbach law governing the initial fracture of monoliths is found to apply also to the bilayer crack systems within a certain range of the problem parameters. The numerical prediction for the crack profiles and the fracture envelopes generally collaborate well with the tests.     

Investigation of the dynamic behavior of two collinear anti-plane shear cracks in a piezoelectric layer bonded to two half spaces by a new method

The dynamic behavior of two collinear anti-plane shear cracks in a piezoelectric layer bonded to two half spaces subjected to the harmonic waves is investigated by a new method. The cracks are parallel to the interfaces in the mid-plane of the piezoelectric layer. By using the Fourier transform, the problem can be solved with two pairs of triple integral equations. These equations are solved by using Schmidt’s method. This process is quite different from that adopted previously. Numerical examples are provided to show the effect of the geometry of cracks, the frequency of the incident wave, the thickness of the piezoelectric layer and the constants of the materials upon the dynamic stress intensity factor of cracks.