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Linear elastic fracture mechanics
[edit]Crack Growth
[edit]In general, the initiation and continuation of crack growth is dependent on several factors, such as bulk material properties, body geometry, crack geometry, loading distribution, loading rate, load magnitude, environmental conditions, time effects (such as viscoelasticity or viscoplasticity), and microstructure.[1] In this section, let's consider cracks that grow straight-ahead from the application of a load resulting in a single mode of fracture.
Crack path initiation
[edit]As cracks grow, energy is transmitted to the crack tip at an energy release rate , which is a function of the applied load, the crack length (or area), and the geometry of the body.[2] In addition, all solid materials have an intrinsic energy release rate , where is referred to as the "fracture energy" or "fracture toughness" of the material.[2] A crack will grow if the following condition is met
depends on a myriad of factors, such as temperature (in a directly proportional manner, i.e., the colder the material, the lower the fracture toughness, and vice versa), the existence of a plane strain or a plane stress loading state, surface energy characteristics, loading rate, microstructure, impurities (especially voids), heat treatment history, and the direction of crack growth.[2]
Crack growth stability
[edit]![](http://up.wiki.x.io/wikipedia/commons/thumb/7/7f/Fig1RCurveRevisedAgain.jpg/300px-Fig1RCurveRevisedAgain.jpg)
In addition, as cracks grow in a body of material, the material's resistance to fracture increases (or remains constant).[2] The resistance a material has to fracture can be captured by the energy release rate required to propagate a crack, , which is a function of crack length . is dependent on material geometry and microstructure.[2] The plot of vs is called the resistance curve, or R-curve.
For brittle materials, is a constant value equal to . For other materials, increases with increasing , and it may or may not reach a steady state value.[2]
The following condition must be met in order for a crack with length to advance an infinitesimally small crack length increment :
Then, the condition for stable crack growth is:
Conversely, the condition for unstable crack growth is:
Predicting Crack Path
[edit]In the prior section, only straight-ahead crack growth from the application of load resulting in a single mode of fracture was considered. However, this is clearly an idealization; in real-world systems, mixed-mode loading (some combination of Mode-I, Mode-II, and Mode-III loading) is applied. In mixed-mode loading, cracks will generally not advance straight ahead.[2] Several theories have been proposed to explain crack kinking and crack propagation in mixed-mode loading, and two are highlighted below.
![](http://up.wiki.x.io/wikipedia/commons/thumb/b/bf/Fig2MaxHoopStressLoadingSchematic.jpg/300px-Fig2MaxHoopStressLoadingSchematic.jpg)
Maximum hoop stress theory
[edit]Consider a crack of length housed in an infinite planar body subjected to mixed Mode-I and Mode-II loadings via uniform tension , where is the angle between the original crack plane and the direction of applied tension and is the angle between the original crack plane and the direction of kinking crack growth. Sih, Paris, and Erdogan showed that the stress intensity factors far from the crack tips in this planar loading geometry are simply and .[3] Additionally, Erdogan and Sih[4] postulated the following for this system:
- Crack extension begins at the crack tip
- Crack extension initiates in the plane perpendicular to the direction of greatest tension
- The "maximum stress criterion" is satisfied, i.e., , where is the critical stress intensity factor (and is dependent on fracture toughness )
This postulation implies that the crack begins to extend from its tip in the direction along which the hoop stress is maximum.[4] In other words, the crack begins to extend from its tip in the direction that satisfies the following conditions:
and .
The hoop stress is written as
where and are taken with respect to a polar coordinate system oriented at the original crack tip.[4] The direction of crack extension and the envelope of failure (plot of ) are determined by satisfying the postulated criteria. For pure Mode-II loading, is calculated to be .[4]
Maximum hoop stress theory predicts the angle of crack extension in experimental results quite accurately and provides a lower bound to the envelope of failure.[2]
Maximum energy release rate criterion
[edit]![](http://up.wiki.x.io/wikipedia/commons/thumb/0/02/Fig3NegativeThetaStarVsBeta.png/420px-Fig3NegativeThetaStarVsBeta.png)
Consider a crack of length housed in an infinite planar body subjected to a state of constant Mode-I and Mode-II stress applied infinitely far away. Under this loading, the crack will kink with a kink length at an angle with respect to the original crack. Wu[5] postulated that the crack kinks will propagate at a critical angle that maximizes energy release rate defined below. Wu defines and to be the strain energies stored in the specimens containing the straight crack and the kinked crack (or Z-shaped crack), respectively.[5] The energy release rate that is generated when the tips of the straight crack begin to kink is defined as
Thus, the crack will kink and propagate at a critical angle that satisfies the following maximum energy release rate criterion:
is unable to be expressed as a closed-form function, but it can be well approximated though numerical simulation.[5]
For crack in pure Mode-II loading, is calculated to be , which compares well with the maximum hoop stress theory.[5]
Anisotropy
[edit]Other factors can also influence the direction of crack growth, such as far-field material deformation (e.g., necking), the presence of micro-separations from defects, the application of compression, the presence of an interface between two heterogeneous materials or material phases, and material anisotropy, to name a few.[1]
In anisotropic materials, the fracture toughness changes as orientation within the material changes. The fracture toughness of an anisotropic material can be defined as , where is some measure of orientation.[2] Therefore, a crack will grow at an orientation angle when the following conditions are satisfied
and
The above can be considered as a statement of the maximum energy release rate criterion for anisotropic materials.[2]
Crack Path Stability
[edit]The above criteria for predicting crack path (namely the maximum hoop stress theory and the maximum energy release rate criterion) all have the implication that is satisfied at the crack tip as the crack extends with a continuously (or smoothly) turning path. This is often called the criterion of local symmetry.[6]
If a crack path proceeds with a discontinuously sharp change in direction, then may not necessarily coincide with the initial direction of the kinked crack path. But, after such a crack kink has been initiated, then the crack extends so that is satisfied.[6]
Consider a semi-infinite crack in an asymmetric state of loading. A kink propagates from the end of this crack to a point where the coordinate system is aligned with the pre-extended crack tip. Cotterell and Rice applied the criterion of local symmetry to deduce a first-order form of the stress intensity factors for the kinked crack tip and a first order form of the kinked crack path.[6]
Cotterell and Rice[6]: First-Order Form of the Stress Intensity Factors for the Kinked Crack Tip and First-Order Form of the Kinked Crack Path |
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First, Cotterell and Rice[6] showed that the stress intensity factors for the extended kinked crack tip are to first order
where and are the tractions on the extended kinked crack from the origin to . Using the stress field on the -axis from the Williams solution,[7] the tractions and can be written to first order as
where and are the stress intensity factors for the pre-extended crack tip, , and corresponds to the value of the local stress acting parallel to the pre-extended crack tip, called the stress. For instance, for a straight crack under uniaxial normal stress .[6] Substituting the tractions into the stress intensity factors and then imposing the criterion of local symmetry at the tip of the extending kinking crack leads to the following integral equation of the crack path
where can be considered as a normalized stress and can be considered as the initial angle of crack growth, which is necessarily small (so the small angle approximation can be applied). The solution for the crack path is
|
The solution for the crack path is
![](http://up.wiki.x.io/wikipedia/commons/thumb/c/cc/Fig4LambdaVsX.png/450px-Fig4LambdaVsX.png)
For small values of , the solution for the crack path reduces to the following series expansion
Crack Path Parameters |
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where and are the stress intensity factors for the pre-extended crack tip |
where corresponds to the value of the local stress acting parallel to the pre-extended crack tip, called the stress |
is the complementary error function |
When , the crack continuously turns further and further away from its initial path with increasing slope as it extends. This is considered as directionally unstable kinked crack growth.[6] When , the crack path continuously extends its initial path. This is considered as neutrally stable kinked crack growth.[6] When , the crack continuously turns away from its initial path with decreasing slope and tends to a steady path of zero slope as it extends. This is considered as directionally stable kinked crack growth.[6]
These theoretical results agree well (for ) with the crack paths observed experimentally by Radon, Leevers, and Culver in experiments on PMMA sheets biaxially loaded with stress normal to the crack and parallel to the crack.[8][9] In this work, the stress is calculated as .[6]
Since the work by Cotterell and Rice was published, it has been found that positive stress cannot be the only indicator for directional instability of kinked crack extension. Support for this claim comes from Melin, who showed that crack growth is directionally unstable for all values of stress in a periodic (regularly-spaced) array of cracks.[10] Furthermore, the kinked crack path and its directional stability cannot be correctly predicted by only considering local effects about the crack edge, as Melin showed through a critical analysis of the Cotterell and Rice solution towards predicting the full kinked crack path arising from a constant remote stress .[11]
References
[edit]- ^ a b Broberg, K. B. (1999). Cracks and fracture. San Diego: Academic Press. ISBN 0121341305. OCLC 41233349.
- ^ a b c d e f g h i j Zehnder, Alan T. (2012). "Fracture Mechanics". Lecture Notes in Applied and Computational Mechanics. doi:10.1007/978-94-007-2595-9. ISSN 1613-7736.
- ^ Sih, G. C.; Paris, P. C.; Erdogan, F. (1962). "Crack-Tip, Stress-Intensity Factors for Plane Extension and Plate Bending Problems". Journal of Applied Mechanics. 29 (2): 306. doi:10.1115/1.3640546.
- ^ a b c d Erdogan, F.; Sih, G. C. (1963). "On the Crack Extension in Plates Under Plane Loading and Transverse Shear". Journal of Basic Engineering. 85 (4): 519. doi:10.1115/1.3656897.
- ^ a b c d Wu, Chien H. (1978-07-01). "Maximum-energy-release-rate criterion applied to a tension-compression specimen with crack". Journal of Elasticity. 8 (3): 235–257. doi:10.1007/BF00130464. ISSN 1573-2681.
- ^ a b c d e f g h i j Cotterell, B.; Rice, J.R. (1980-04-01). "Slightly curved or kinked cracks". International Journal of Fracture. 16 (2): 155–169. doi:10.1007/BF00012619. ISSN 1573-2673.
- ^ Williams, M. L. (1957). "N/A". Journal of Applied Mechanics. 24: 109–114.
- ^ Radon, J.C., Leevers, P.S., and Culver, L.E. (1976). "Fracture trajectories in a biaxially stressed plate". J. Mech. Phys. Solids. 24: 381–395.
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: CS1 maint: multiple names: authors list (link) - ^ Radon, J.C., Leevers, P.S., Culver, L.E. "Fracture Toughness of PMMA Under Biaxial Stress". Fracture. 3: 1113–1118.
{{cite journal}}
: CS1 maint: multiple names: authors list (link) - ^ Melin, Solveig (1983-09-01). "Why do cracks avoid each other?". International Journal of Fracture. 23 (1): 37–45. doi:10.1007/BF00020156. ISSN 1573-2673.
- ^ Melin, S. (2002-04-01). "The influence of the T-stress on the directional stability of cracks". International Journal of Fracture. 114 (3): 259–265. doi:10.1023/A:1015521629898. ISSN 1573-2673.
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