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Linear elastic fracture mechanics

Crack Growth

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. 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

As cracks grow, energy is transmitted to the crack tip at an energy release rate $G$, which is a function of the applied load, the crack length (or area), and the geometry of the body. In addition, all solid materials have an intrinsic energy release rate $G_{C}$, where $G_{C}$ is referred to as the "fracture energy" or "fracture toughness" of the material. A crack will grow if the following condition is met

$G\geq G_{C}$

$G_{C}$ 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.

Crack growth stability

R-curves for a brittle material and a ductile material.

In addition, as cracks grow in a body of material, the material's resistance to fracture increases (or remains constant). The resistance a material has to fracture can be captured by the energy release rate required to propagate a crack, $R\left(a\right)$, which is a function of crack length $a$. $R\left(a\right)$ is dependent on material geometry and microstructure. The plot of $R\left(a\right)$ vs $a$ is called the resistance curve, or R-curve.

For brittle materials, $R$ is a constant value equal to $R_{0}=G_{C}$. For other materials, $R$ increases with increasing $a$, and it may or may not reach a steady state value.

The following condition must be met in order for a crack with length $a_{0}$ to advance an infinitesimally small crack length increment $\delta a$ :

$G(a_{0})=R(a_{0}+\delta a)$

Then, the condition for stable crack growth is:

${\frac {\delta G(a_{0})}{\delta a}}<{\frac {\delta R(a_{0})}{\delta a}}$

Conversely, the condition for unstable crack growth is:

${\frac {\delta G(a_{0})}{\delta a}}\geq {\frac {\delta R(a_{0})}{\delta a}}$

Predicting Crack Path

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. Several theories have been proposed to explain crack kinking and crack propagation in mixed-mode loading, and two are highlighted below.

Application of uniform tension $\sigma ~$on a crack of length $2a~$in an infinite planar body to induce mixed Mode-I and Mode-II loading. The solids lines extending out from the crack tips are the crack kinks.

Maximum hoop stress theory

Consider a crack of length $2a$ housed in an infinite planar body subjected to mixed Mode-I and Mode-II loadings via uniform tension $\sigma$, where $\beta$ is the angle between the original crack plane and the direction of applied tension and $\theta ^{\star }$ 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 $K_{I}=\sigma {\sqrt {\pi a}}\sin ^{2}(\beta )$ and $K_{II}=\sigma {\sqrt {\pi a}}\sin(\beta )\cos(\beta )$. Additionally, Erdogan and Sih postulated the following for this system:

1. Crack extension begins at the crack tip
2. Crack extension initiates in the plane perpendicular to the direction of greatest tension
3. The "maximum stress criterion" is satisfied, i.e., ${\sqrt {2\pi r}}\sigma _{\theta }(r,\theta ^{*})\geq K_{IC}$, where $K_{IC}$ is the critical stress intensity factor (and is dependent on fracture toughness $G_{C}$)

This postulation implies that the crack begins to extend from its tip in the direction $\theta ^{\star }$ along which the hoop stress $\sigma _{\theta }$ is maximum. In other words, the crack begins to extend from its tip in the direction $\theta ^{\star }$ that satisfies the following conditions:

${\partial \sigma _{\theta } \over \partial \theta }=0$ and ${\partial ^{2}\sigma _{\theta } \over \partial \theta ^{2}}<0$.

The hoop stress is written as

$\sigma _{\theta }={\frac {1}{\sqrt {2\pi r}}}\cos \left({\frac {\theta }{2}}\right)\left[K_{I}\cos ^{2}\left({\frac {\theta }{2}}\right)-{\frac {3}{2}}K_{II}\sin(\theta )\right]$

where $r$ and $\theta$ are taken with respect to a polar coordinate system oriented at the original crack tip. The direction of crack extension $\theta ^{\star }$ and the envelope of failure (plot of $K_{II}{\text{ vs }}K_{I}$) are determined by satisfying the postulated criteria. For pure Mode-II loading, $\theta ^{\star }$ is calculated to be $70.5^{\circ }$.

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.

Maximum energy release rate criterion

Negative crack kinking angle $-\theta ^{*}~$vs load application angle $\beta ~$as predicted by the maximum hoop stress theory. Angles are given in degrees.

Consider a crack of length $2a$ 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 $\epsilon$ at an angle $\alpha$ with respect to the original crack. Wu postulated that the crack kinks will propagate at a critical angle $\alpha =\alpha _{C}$ that maximizes energy release rate defined below. Wu defines $U\left({\boldsymbol {\sigma }}\right)$ and $U_{z}\left({\boldsymbol {\sigma }},\alpha ,\epsilon \right)$ to be the strain energies stored in the specimens containing the straight crack and the kinked crack (or Z-shaped crack), respectively. The energy release rate that is generated when the tips of the straight crack begin to kink is defined as

$G\left({\boldsymbol {\sigma }},\alpha \right)={\frac {1}{2}}\lim _{\epsilon \to 0}{\frac {U_{z}-U}{\epsilon }}$

Thus, the crack will kink and propagate at a critical angle $\alpha =\alpha _{C}$ that satisfies the following maximum energy release rate criterion:

${\partial G \over \partial \alpha }{\bigg |}_{\alpha =\alpha _{C}}=0$

${\partial ^{2}G \over \partial \alpha ^{2}}{\bigg |}_{\alpha =\alpha _{C}}\leq 0$

$G\left({\boldsymbol {\sigma }},\alpha \right)\geq G_{C}$

$G\left({\boldsymbol {\sigma }},\alpha \right)$ is unable to be expressed as a closed-form function, but it can be well approximated though numerical simulation.

For crack in pure Mode-II loading, $\alpha _{C}$ is calculated to be $75.6^{\circ }$, which compares well with the maximum hoop stress theory.

Anisotropy

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.

In anisotropic materials, the fracture toughness changes as orientation within the material changes. The fracture toughness of an anisotropic material can be defined as $G_{C}\left(\theta \right)$, where $\theta$ is some measure of orientation. Therefore, a crack will grow at an orientation angle $\theta =\theta ^{*}$ when the following conditions are satisfied

$G\left(\theta ^{*}\right)\geq G_{C}\left(\theta ^{*}\right)$ and ${\partial G\left(\theta ^{*}\right) \over \partial \theta }={\partial G_{C}\left(\theta ^{*}\right) \over \partial \theta }$

The above can be considered as a statement of the maximum energy release rate criterion for anisotropic materials.

Crack Path Stability

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 $K_{II}=0$ 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.

If a crack path proceeds with a discontinuously sharp change in direction, then $K_{II}=0$ 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 $K_{II}=0$ is satisfied.

Consider a semi-infinite crack in an asymmetric state of loading. A kink propagates from the end of this crack to a point $\left(l,\lambda (l)\right)$ where the $(x,y=\lambda (x))$ coordinate system is aligned with the pre-extended crack tip. Cotterell and Rice applied the $K_{II}=0$ 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.

Cotterell and Rice: First-Order Form of the Stress Intensity Factors for the Kinked Crack Tip and First-Order Form of the Kinked Crack Path

First, Cotterell and Rice showed that the stress intensity factors for the extended kinked crack tip are to first order ${\begin{Bmatrix}K_{I}\\K_{II}\end{Bmatrix}}=\left({\frac {2}{\pi }}\right)^{1/2}\int _{-\infty }^{l}\left[{\begin{Bmatrix}T_{y}-\lambda ^{\prime }(l)T_{x}\\T_{x}+\lambda ^{\prime }(l)T_{y}\end{Bmatrix}}{\frac {1}{(l-x)^{1/2}}}+{\begin{Bmatrix}T_{x}\\T_{y}\end{Bmatrix}}{\frac {\left[\lambda (l)-\lambda (x)-\lambda ^{\prime }(l)(l-x)\right]}{2(l-x)^{3/2}}}\right]$ where $T_{x}$ and $T_{y}$ are the tractions on the extended kinked crack from the origin to $\left(l,\lambda (l)\right)$. Using the stress field on the $x$-axis from the Williams solution, the tractions $T_{x}$ and $T_{y}$ can be written to first order as ${\begin{Bmatrix}T_{x}\\T_{y}\end{Bmatrix}}={\frac {1}{\sqrt {2\pi x}}}\left[k_{I}{\begin{Bmatrix}{\frac {\lambda (x)}{2x}}-\lambda ^{\prime }(x)\\1\end{Bmatrix}}+k_{II}{\begin{Bmatrix}1\\{\frac {\lambda (x)}{2x}}-\lambda ^{\prime }(x)\end{Bmatrix}}\right]-{\begin{Bmatrix}\lambda ^{\prime }(x)T\\0\end{Bmatrix}}$ where $k_{I}$ and $k_{II}$ are the stress intensity factors for the pre-extended crack tip, $\lambda ^{\prime }(x)={\operatorname {d} \!\lambda (x) \over \operatorname {d} \!x}$, and $T$ corresponds to the value of the local stress acting parallel to the pre-extended crack tip, called the $T$ stress. For instance, $T=-\sigma$ for a straight crack under uniaxial normal stress $\sigma$. Substituting the tractions into the stress intensity factors and then imposing the $K_{II}=0$ criterion of local symmetry at the tip of the extending kinking crack leads to the following integral equation of the crack path $y=\lambda (x)$ $\theta _{0}=\lambda ^{\prime }(l)-{\frac {\beta }{\pi ^{1/2}}}\int _{0}^{l}{\frac {\lambda ^{\prime }(x)}{(l-x)^{1/2}}}$ where $\beta ={\frac {2{\sqrt {2}}T}{k_{I}}}$ can be considered as a normalized $T$ stress and $\theta _{0}={\frac {-2k_{II}}{k_{I}}}$ 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 $\lambda (x)$ is $\lambda (x)={\frac {\theta _{0}}{\beta }}\left[\exp(\beta ^{2}x){\text{erfc}}(-\beta x^{1/2})-1-2\beta \left({\frac {x}{\pi }}\right)^{1/2}\right]$

The solution for the crack path $\lambda (x)$ is

Kinked crack paths (dashed/dotted lines) extending from a semi-infinite crack (solid line) in an asymmetric state of loading with $\theta _{0}=0.2~$radians. Note the directionally unstable crack growth for positive $\beta ~$(i.e., for positive $T~$stress), the neutrally stable crack growth for zero $\beta ~$(i.e., for $T~$stress), and the directionally stable crack growth for negative $\beta ~$(i.e., for negative $T~$stress).

$\lambda (x)={\frac {\theta _{0}}{\beta }}\left[\exp(\beta ^{2}x){\text{erfc}}(-\beta x^{1/2})-1-2\beta \left({\frac {x}{\pi }}\right)^{1/2}\right]$

For small values of $\beta ^{2}x$, the solution for the crack path $\lambda (x)$ reduces to the following series expansion

$\lambda (x)=\theta _{0}x\left[1+{\frac {4T}{3k_{I}}}\left({\frac {2x}{\pi }}\right)^{1/2}+4{\frac {T^{2}x}{k_{I}^{2}}}+\dots \right]$

Crack Path $\lambda (x)$Parameters
$\theta _{0}={\frac {-2k_{II}}{k_{I}}}$ where $k_{I}$ and $k_{II}$ are the stress intensity factors for the pre-extended crack tip
$\beta ={\frac {2{\sqrt {2}}T}{k_{I}}}$ where $T$ corresponds to the value of the local stress acting parallel to the pre-extended crack tip, called the $T$ stress
${\text{erfc}}()$ is the complementary error function

When $T>0$, 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. When $T=0$, the crack path continuously extends its initial path. This is considered as neutrally stable kinked crack growth. When $T<0$, 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.

These theoretical results agree well (for $\theta _{0}\leq 15^{\circ }$) with the crack paths observed experimentally by Radon, Leevers, and Culver in experiments on PMMA sheets biaxially loaded with stress $\sigma$ normal to the crack and $R\sigma$ parallel to the crack. In this work, the $T$ stress is calculated as $T=(R-1)\sigma$.

Since the work by Cotterell and Rice was published, it has been found that positive $T$ 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 $T$ stress in a periodic (regularly-spaced) array of cracks. 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 $\tau _{xy}=\tau _{xy}^{\infty }$.