TY - JOUR
T1 - Surface singularity and crack propagation
AU - Bažant, Zdeněk P.
AU - Estenssoro, Luis F.
N1 - Funding Information:
Acknowledgements-Grateful acknowledgment is due to the U .S. Air Force Office of Scientitic Research for sponsoring the major part of this research under Grant No. AFOSR75-2859 to Northwestern University. Leon M. Keer, Professor at N orthwestern U niversity and co-director of the grant project, is thanked for his stimulating discussions and valuable suggestions during the progress of this work. The writers are also obliged to Jan P. Benthem, Professor at Delft University of Technology, for some very useful critical remarks and for clarifying the implications of his analytical solution for the p-value to be assumed, and to G. Sendeckyj, senior scientist at Air Force F1ight Dynamics Laboratory, Dayton, Ohio, for helpful suggestions.
PY - 1979
Y1 - 1979
N2 - The three-dimensional singular stress field near the terminal point 0 of the crack front edge at the surface of an elastic body is investigated, using spherical coordinates r, θ, φ and assuming all three displacements to be of the form rλppF(θ, λ) where p = distance from the singularity line (crack front edge or notch edge) and p = given constant. The variational principle governing the displacement distribution on a unit sphere about point 0, which has previously been obtained from the differential equations of equilibrium, is now derived more directly from potential energy. The previously developed finite element method on the unit sphere is used to reduce the problem to the form k(λ)X = 0 where X = column matrix of the nodal values of displacements on the unit sphere and k(λ) = square matrix, all coefficients of which are quadratic polynomials in λ. It is proven that the variational principle as well as matrix k must be nonsymmetric, which means that complex eigenvalues A are possible. The dependence of A upon Poisson's ratio v for Mode I cracks whose front edge is normal to the surface is solved numerically and it closely agrees with the analytical solution of Benthem. Previously unavailable solutions for Modes II and HI and for cracks (of all modes) with inclined front edge and inclined crack plane are also obtained. By energy flux argument, it is found that the front edge of a propagating crack must terminate at the surface obliquely, at a certain angle whose dependence upon the inclination of the crack plane is also solved. The angle is the same for Modes II and III, but different for Mode I. For this mode, the surface point trails behind the interior of the propagating crack, while for Modes II and III it moves ahead. Consequently, a combination of Mode I with Modes II and III is impossible at the surface terminal point of a propagating crack whose plane is orthogonal. When the plane is inclined, the three stress intensity factors can combine only in certain fixed ratios. The angle of crack edge is a function of the angle of crack plane. Some results with complex λ for two-material interfaces are also given.
AB - The three-dimensional singular stress field near the terminal point 0 of the crack front edge at the surface of an elastic body is investigated, using spherical coordinates r, θ, φ and assuming all three displacements to be of the form rλppF(θ, λ) where p = distance from the singularity line (crack front edge or notch edge) and p = given constant. The variational principle governing the displacement distribution on a unit sphere about point 0, which has previously been obtained from the differential equations of equilibrium, is now derived more directly from potential energy. The previously developed finite element method on the unit sphere is used to reduce the problem to the form k(λ)X = 0 where X = column matrix of the nodal values of displacements on the unit sphere and k(λ) = square matrix, all coefficients of which are quadratic polynomials in λ. It is proven that the variational principle as well as matrix k must be nonsymmetric, which means that complex eigenvalues A are possible. The dependence of A upon Poisson's ratio v for Mode I cracks whose front edge is normal to the surface is solved numerically and it closely agrees with the analytical solution of Benthem. Previously unavailable solutions for Modes II and HI and for cracks (of all modes) with inclined front edge and inclined crack plane are also obtained. By energy flux argument, it is found that the front edge of a propagating crack must terminate at the surface obliquely, at a certain angle whose dependence upon the inclination of the crack plane is also solved. The angle is the same for Modes II and III, but different for Mode I. For this mode, the surface point trails behind the interior of the propagating crack, while for Modes II and III it moves ahead. Consequently, a combination of Mode I with Modes II and III is impossible at the surface terminal point of a propagating crack whose plane is orthogonal. When the plane is inclined, the three stress intensity factors can combine only in certain fixed ratios. The angle of crack edge is a function of the angle of crack plane. Some results with complex λ for two-material interfaces are also given.
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U2 - 10.1016/0020-7683(79)90062-3
DO - 10.1016/0020-7683(79)90062-3
M3 - Article
AN - SCOPUS:0018294124
SN - 0020-7683
VL - 15
SP - 405
EP - 426
JO - International Journal of Solids and Structures
JF - International Journal of Solids and Structures
IS - 5
ER -