Abstract
The aim of this contribution is to provide a brief overview of three-dimensional linear-elastic fracture mechanics (3D LEFM) as well as the latest advances in this area. The primary focus of this review is on the situations where the classical LEFM, which largely relies on plane stress or plane strain simplifications, provides peculiar or misleading results. As no exact analytical solutions are currently available for real cracks, which are inherently three-dimensional (3D), there are many controversial views in the literature and lack of understanding of the effects associated with 3D geometries. Fundamental results and general conclusions in 3D LEFM are largely based on dimensionless and energy considerations as well as on generalisations of outcomes of 3D numerical studies and application of asymptotic techniques. It is believed that 3D considerations alone cannot explain complex and diverse brittle fracture and fatigue phenomena, but these considerations can contribute into the further understanding of these phenomena.
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Abbreviations
- \(\mathrm {a}\) :
-
Radius of cylindrical hole
- \(\mathrm {A}\) :
-
Function of \(\mathrm {z}\)-coordinate
- \(\mathrm {b}\) :
-
Small number
- \(\mathrm {C}_{\mathrm {1}}...\mathrm {C}_{\mathrm {4}}\) :
-
Constants
- \(\mathrm {E}\) :
-
Young’s modulus
- \(\mathrm {h}\) :
-
Half plate thickness
- \(\mathrm {F}_{\mathrm {i}}\) :
-
Applied forces
- \(\mathrm {F}_{\mathrm {S}}\), \(\mathrm {F}_{\mathrm {A}}\) :
-
Functions of angular position \(\left( {\upvartheta ,\upvarphi }\right) \)
- \(\mathrm {f}_{\mathrm {I}}\), \(\mathrm {f}_{\mathrm {II}}\), \(\mathrm {f}_{\mathrm {III}}\) :
-
Functions
- \(\mathrm {G}\) :
-
Energy release rate
- \(\mathrm {G}_{\mathrm {I}}\) :
-
Energy release rate in mode I
- \(\mathrm {K}_{\mathrm {t}}\) :
-
Stress concentration factor
- \(\mathrm {K}_{\mathrm {c}}\) :
-
Apparent fracture toughness
- \(\mathrm {K}_{\mathrm {IC}}\) :
-
Fracture toughness in mode I
- \(\mathrm {K}_{\mathrm {I}}\), \(\mathrm {K}_{\mathrm {II}}\), \(\mathrm {K}_{\mathrm {III}}\) :
-
Stress intensity factors in mode I, II and III, respectively
- \(\mathrm {K}_{\mathrm {I}}^{{\infty }}\), \(\mathrm {K}_{\mathrm {II}}^{{\infty }}\) :
-
Remote stress intensity factors in mode I and II, respectively
- \(\mathrm {K}_{\mathrm {S}}\), \(\mathrm {K}_{\mathrm {A}}\) :
-
Stress intensity factors of symmetric and antisymmetric vertex singularities, respectively
- \(\mathrm {K}_{\mathrm {O}}\) :
-
Stress intensity factor of the out-of-plane mode
- \(\mathrm {K}_{\mathrm {vertex}}\) :
-
Stress intensity factor of vertex singularity, \(\mathrm {K}_{\mathrm {vertex}}=\left( \mathrm {K}_{\mathrm {S}},\mathrm {K}_{\mathrm {A}} \right) \)
- J:
-
J-integral
- \(\mathrm {M}\) :
-
Maximum modulus of the prescribed edge tractions
- \(\mathrm {n}\) :
-
Power exponent
- \(\mathrm {R}\) :
-
Distance from the edge of hole or vertex point
- \(\mathrm {r},\, {\upphi }\) :
-
Polar coordinates
- \(\mathrm {u}_{\mathrm {i}}\) :
-
Displacement components, \(\mathrm {i=}\left( \mathrm {x,y,z} \right) \)
- \(\mathrm {x, y,z}\) :
-
Coordinates
- \({\upalpha }\) :
-
Positive constant
- \({\upbeta }_{\mathrm {c}}\) :
-
Critical angle
- \({\uplambda }_{\mathrm {S}}\), \({\uplambda }_{\mathrm {A}}\) :
-
Strengths of symmetric and antisymmetric vertex singularities, respectively
- \({\uplambda }_{\mathrm {vertex}}, {\uplambda }_{\mathrm {line}}\) :
-
Strengths of vertex and line singularities, respectively
- \({\upsigma }\) or \({\upsigma }_{\mathrm {ij}}\) :
-
Stress components, \(\mathrm {i=}\left( \mathrm {x,y,z} \right) \) and \(\mathrm {j=}\left( \mathrm {x,y,z} \right) \)
- \({\upvarepsilon }\) or \(\mathrm {\varepsilon }_{\mathrm {ij}}\) :
-
Strain components
- \({\xi }\) :
-
Minimum distance to the observation point from the edge of plate or free surface
- \({\upeta }\) :
-
Coordinate associated with crack front
- \({\uplambda }\), \({\upmu }\) :
-
Lame’s constants
- \({\upnu }\) :
-
Poisson’s ratio
- \({\mathrm {R}}, \upvartheta ,\upvarphi \) :
-
Angular coordinates of a point
- \({\upbeta , \uptheta }\) :
-
Angles which identify the local geometry of the vertex point
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Acknowledgements
This research was sponsored by FEDER funds through the program COMPETE (Programa Operacional Factores de Competitividade) and by national funds through FCT (Fundação para a Ciência e a Tecnologia) under the project UIDB/00285/202.
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Zakavi, B., Kotousov, A. & Branco, R. Overview of three-dimensional linear-elastic fracture mechanics. Int J Fract 234, 5–20 (2022). https://doi.org/10.1007/s10704-021-00528-9
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DOI: https://doi.org/10.1007/s10704-021-00528-9