The behavior of two collinear anti-plane shear cracks in a piezoelectric layer bonded to two half spaces is investigated by the Schmidt method. The cracks are vertically to the interfaces of the piezoelectric layer. B...The behavior of two collinear anti-plane shear cracks in a piezoelectric layer bonded to two half spaces is investigated by the Schmidt method. The cracks are vertically to the interfaces 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 using the Schmidt method. This process is quite different from that adopted previously. Numerical examples are provided to show the effect of the geometry of the interacting cracks and the piezoelectric constants of the material upon the stress intensity factor of the cracks.展开更多
In this paper, the interaction between two collinear cracks inpiezoelectric materials under anti-plane shear loading wasinvestigated for the impermeable crack face conditions. By using theFourier transform, the proble...In this paper, the interaction between two collinear cracks inpiezoelectric materials under anti-plane shear loading wasinvestigated for the impermeable crack face conditions. By using theFourier transform, the problem can be solved with two pairs of tripleintegral equations. These equations are solved using Schmidt'smethod. This process is quite different from that adopted previously.This makes it possible to understand the two collinear cracksinteraction in piezoelectric materials.展开更多
Field equations of the non-local elasticity are solved to determine the state of stress in a plate with a Griffith crack subject to uniform tension. Then a set of dual-integral equations is solved using a new method, ...Field equations of the non-local elasticity are solved to determine the state of stress in a plate with a Griffith crack subject to uniform tension. Then a set of dual-integral equations is solved using a new method, namely Schmidt's method. This method is more exact and more reasonable than Eringen's a one Sor solving this kind of problem. Contrary to the solution of classical elasticity, it is found that no stress singularity is present ar the crack tip. The significance of this result is that the fracture criteria are unified at both the macroscopic and the microscopic scales. The finite hoop stress at the crack tip depends on the crack length.展开更多
Crack problems are often reduced to dual integral equations,which can be solved by expanding the displacement integral equation as a series in the form of Chebyshev-like or Jacobi polynomials.Schmidt’s multiplying-fa...Crack problems are often reduced to dual integral equations,which can be solved by expanding the displacement integral equation as a series in the form of Chebyshev-like or Jacobi polynomials.Schmidt’s multiplying-factor integration method has been one of the most favorable techniques for determining the expansion coefficients by constructing a well-posed system of linear algebraic equations.However,Schmidt’s method is less efficient for numerical computation because the matrix elements of the linear equations are evaluated from dual integrals.In this study,we propose a modified method to construct linear equations to efficiently determine the expansion coefficients.The modified technique is developed upon the application of certain multiplying factors to the traction integral equation and then integrating the resulting equation over“source”regions.Such manipulations simplify the matrix elements as single integrals.By carrying out numerical examples,we demonstrate that the technique is not only accurate but also very efficient.In particular,the method only needs approximately 1/5 of the computation time of Schmidt’s method.Therefore,this method can be used to replace Schmidt’s method and is expected to be very useful in solving crack problems.展开更多
This paper proves that the weighting method via modified Gram-Schmidt（MGS） for solving the equality constrained least squares problem in the limit is equivalent to the direct elimination method via MGS（MGS-eliminat...This paper proves that the weighting method via modified Gram-Schmidt（MGS） for solving the equality constrained least squares problem in the limit is equivalent to the direct elimination method via MGS（MGS-elimination method）. By virtue of this equivalence, the backward and forward roundoff error analysis of the MGS-elimination method is proved. Numerical experiments are provided to verify the results.展开更多
基金supported by the Post Doctoral Science Foundation of Heilongjiang Province, the Natural Science Foundation of Heilongjiang Province，哈尔滨工业大学校科研和教改项目
文摘The behavior of two collinear anti-plane shear cracks in a piezoelectric layer bonded to two half spaces is investigated by the Schmidt method. The cracks are vertically to the interfaces 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 using the Schmidt method. This process is quite different from that adopted previously. Numerical examples are provided to show the effect of the geometry of the interacting cracks and the piezoelectric constants of the material upon the stress intensity factor of the cracks.
基金the Post-Doctoral Science Foundationthe Natural Science Foundation of Heilongjiang Province
文摘In this paper, the interaction between two collinear cracks inpiezoelectric materials under anti-plane shear loading wasinvestigated for the impermeable crack face conditions. By using theFourier transform, the problem can be solved with two pairs of tripleintegral equations. These equations are solved using Schmidt'smethod. This process is quite different from that adopted previously.This makes it possible to understand the two collinear cracksinteraction in piezoelectric materials.
文摘Field equations of the non-local elasticity are solved to determine the state of stress in a plate with a Griffith crack subject to uniform tension. Then a set of dual-integral equations is solved using a new method, namely Schmidt's method. This method is more exact and more reasonable than Eringen's a one Sor solving this kind of problem. Contrary to the solution of classical elasticity, it is found that no stress singularity is present ar the crack tip. The significance of this result is that the fracture criteria are unified at both the macroscopic and the microscopic scales. The finite hoop stress at the crack tip depends on the crack length.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.11802074,42074057,11972132,and 11734017)China National Postdoctoral Program for Innovative Talents(Grant No.BX201700066)+1 种基金China Postdoctoral Science Foundation(Grant No.2018M630345)the Fundamental Research Funds for the Central Universities(Grant No.HIT.NSRIF.2020016)。
文摘Crack problems are often reduced to dual integral equations,which can be solved by expanding the displacement integral equation as a series in the form of Chebyshev-like or Jacobi polynomials.Schmidt’s multiplying-factor integration method has been one of the most favorable techniques for determining the expansion coefficients by constructing a well-posed system of linear algebraic equations.However,Schmidt’s method is less efficient for numerical computation because the matrix elements of the linear equations are evaluated from dual integrals.In this study,we propose a modified method to construct linear equations to efficiently determine the expansion coefficients.The modified technique is developed upon the application of certain multiplying factors to the traction integral equation and then integrating the resulting equation over“source”regions.Such manipulations simplify the matrix elements as single integrals.By carrying out numerical examples,we demonstrate that the technique is not only accurate but also very efficient.In particular,the method only needs approximately 1/5 of the computation time of Schmidt’s method.Therefore,this method can be used to replace Schmidt’s method and is expected to be very useful in solving crack problems.
基金supported by the Shanghai Leading Academic Discipline Project (Grant No.J50101)
文摘This paper proves that the weighting method via modified Gram-Schmidt（MGS） for solving the equality constrained least squares problem in the limit is equivalent to the direct elimination method via MGS（MGS-elimination method）. By virtue of this equivalence, the backward and forward roundoff error analysis of the MGS-elimination method is proved. Numerical experiments are provided to verify the results.
