The surface correction to the quadrupole source term of the Ffowcs Williams and Hawkings integral in the frequency domain suffers from the computation of high-order derivatives of Green’s function.The far-field appro...The surface correction to the quadrupole source term of the Ffowcs Williams and Hawkings integral in the frequency domain suffers from the computation of high-order derivatives of Green’s function.The far-field approximations to the derivatives of Green’s function have been used without derivation and verification in previous work.In this work,we provide the detailed derivations of the far-field approximations to the derivatives of Green’s function.The binomial expansions for the derivatives of Green’s function and the far-field condition are employed during the derivations to circumvent the difficulties in computing the high-order derivatives.The approximations to the derivatives of Green’s function are systemically verified by using the benchmarks two-dimensional convecting vortex and the co-rotating vortex pair.In addition,we provide the derivations of the approximations to the multiple integrals of Green’s function by using the far-field approximations to the derivatives.展开更多
Calculating interacting stress intensity factors(SIFs)of multiple ellipticalholes and cracks is very important for safety assessment,stop-hole optimization design and resource exploitation production in underground ro...Calculating interacting stress intensity factors(SIFs)of multiple ellipticalholes and cracks is very important for safety assessment,stop-hole optimization design and resource exploitation production in underground rock engineering,e.g.,buried tunnels,deep mining,geothermal and shale oil/gas exploitation by hydraulic fracturing technology,where both geo-stresses and surface stresses are applied on buried tunnels,horizontal wells and natural cracks.However,current literatures are focused mainly on study of interacting SIFs of multiple elliptical-holes(or circularholes)and cracks only under far-field stresses without consideration of arbitrary surface stresses.Recently,our group has proposed a new integral method to calculate interacting SIFs of multiple circular-holes and cracks subjected to far-filed and surface stresses.This new method will be developed to study the problem of multiple elliptical-hole and cracks subjected to both far-field and surface stresses.In this study,based on Cauchy integral theorem,the exact fundamental stress solutions of single elliptical-hole under arbitrarily concentrated surface normal and shear forces are derived to establish new integral equation formulations for calculating interacting SIFs of multiple elliptical-holes and cracks under both far-field and arbitrary surface stresses.The new method is proved to be valid by comparing our results of interacting SIFs with those obtained by Green’s function method,displacement discontinuity method,singular integral equation method,pseudo-dislocations method and finite element method.Computational examples of one elliptical-hole and one crack in an infinite elastic body are given to analyze influence of loads and geometries on interacting SIFs.Research results show that whenσ_(xx)^(∞)≥σ^(yy′)^(∞),there appears a neutral crack orientation angle b0(without elliptical-hole’s effect).Increasing s¥xx/s¥yy and b/a(close to circularhole)usually decreases b0 of KI and benefits to the layout of stop-holes.The surface compressive stresses applied onto elliptical-hole(n)and crack(p)have significant influence on interacting SIFs but almost no on b0.Increasing n and p usually results in increase of interacting SIFs and facilitates crack propagation and fracture networks.The elliptical-hole orientation angle(a)and holed-cracked distance(t)have great influence on the interacting SIFs while have little effect on b0.The present method is not only simple(without any singular parts),high-accurate(due to exact fundamental stress solutions)and wider applicable(under far-field stresses and arbitrarily distributed surface stress)than the common methods,but also has the potential for the anisotropic problem involving multiple holes and cracks.展开更多
基金National Numerical Windtunnel project,and the National Natural Science Foundation of China(Nos.11922214,91752118).
文摘The surface correction to the quadrupole source term of the Ffowcs Williams and Hawkings integral in the frequency domain suffers from the computation of high-order derivatives of Green’s function.The far-field approximations to the derivatives of Green’s function have been used without derivation and verification in previous work.In this work,we provide the detailed derivations of the far-field approximations to the derivatives of Green’s function.The binomial expansions for the derivatives of Green’s function and the far-field condition are employed during the derivations to circumvent the difficulties in computing the high-order derivatives.The approximations to the derivatives of Green’s function are systemically verified by using the benchmarks two-dimensional convecting vortex and the co-rotating vortex pair.In addition,we provide the derivations of the approximations to the multiple integrals of Green’s function by using the far-field approximations to the derivatives.
基金supports by National Natural Science Foundation of China(Nos.51874351,51474251 and 12072309)Excellent Postdoctoral Innovative Talents Project of Hunan Province(No.2020RC2001).
文摘Calculating interacting stress intensity factors(SIFs)of multiple ellipticalholes and cracks is very important for safety assessment,stop-hole optimization design and resource exploitation production in underground rock engineering,e.g.,buried tunnels,deep mining,geothermal and shale oil/gas exploitation by hydraulic fracturing technology,where both geo-stresses and surface stresses are applied on buried tunnels,horizontal wells and natural cracks.However,current literatures are focused mainly on study of interacting SIFs of multiple elliptical-holes(or circularholes)and cracks only under far-field stresses without consideration of arbitrary surface stresses.Recently,our group has proposed a new integral method to calculate interacting SIFs of multiple circular-holes and cracks subjected to far-filed and surface stresses.This new method will be developed to study the problem of multiple elliptical-hole and cracks subjected to both far-field and surface stresses.In this study,based on Cauchy integral theorem,the exact fundamental stress solutions of single elliptical-hole under arbitrarily concentrated surface normal and shear forces are derived to establish new integral equation formulations for calculating interacting SIFs of multiple elliptical-holes and cracks under both far-field and arbitrary surface stresses.The new method is proved to be valid by comparing our results of interacting SIFs with those obtained by Green’s function method,displacement discontinuity method,singular integral equation method,pseudo-dislocations method and finite element method.Computational examples of one elliptical-hole and one crack in an infinite elastic body are given to analyze influence of loads and geometries on interacting SIFs.Research results show that whenσ_(xx)^(∞)≥σ^(yy′)^(∞),there appears a neutral crack orientation angle b0(without elliptical-hole’s effect).Increasing s¥xx/s¥yy and b/a(close to circularhole)usually decreases b0 of KI and benefits to the layout of stop-holes.The surface compressive stresses applied onto elliptical-hole(n)and crack(p)have significant influence on interacting SIFs but almost no on b0.Increasing n and p usually results in increase of interacting SIFs and facilitates crack propagation and fracture networks.The elliptical-hole orientation angle(a)and holed-cracked distance(t)have great influence on the interacting SIFs while have little effect on b0.The present method is not only simple(without any singular parts),high-accurate(due to exact fundamental stress solutions)and wider applicable(under far-field stresses and arbitrarily distributed surface stress)than the common methods,but also has the potential for the anisotropic problem involving multiple holes and cracks.
