Fracture Mechanics Online Class

HRR theory > Outcomes of the HRR models

So far, we have derived the HRR asymptotic solution. At this point, we want to discuss the practical outcomes of the theory, and what we can actually learn from these models.

HRR solution can explain 3D effects

Picture VIII.30: Location of the free and mid surfaces of a 3D crack.

One of the advantages of the HRR solution is that it can explain 3D effects in crack propagation. Let us assume a specimen with an edge-crack. When applying the HRR theory, the solution is either

Both solutions are depicted in Picture VIII.30, where it can be seen that the shape of the process zone evolves from a "propeller-like" at mid-section to the more diffuse process zone at the specimen free faces.

Picture VIII.31: Example of shear lips.

Although the $J$-integral is formally not constant along the crack front through the thickness, it remains that

Picture VIII.32: Effect of thickness on the measured $K$ at fracture.

Assuming SSY, and performing a toughness test, the real value of $K$ changes along the crack front, so that the measured value of $K$ is an average one. As a results

Nevertheless, there actually exist complex 3D effects:

Picture VIII.33: triaxiality effect.

Rigorously, the effect of the specimen thickness results from triaxiality effect. Remember that under SSY assumption, the stress intensity factor characterizes the dominant term at crack tip, but at a larger distance, the other series terms are not negligible anymore, see Lecture 5. Effect of $T$-stress, see Picture VIII.33, should also be considered:

Effective crack length for SSY

Another outcome of the HRR theory was the prediction of the process zone size. If the SSY assumption holds true, then

Picture VIII.34: Redistribution of the stress for a perfectly plastic material.

In the case of perfectly plastic materials $\left( n\rightarrow \infty \right)$, Rice has developed a model to predict the process zone size, based on a redistribution of the stress at crack tip, see Picture VIII.34. When assuming linear elasticity, the Mode I asymptotic solution predicts an infinite stress at crack tip, following

\begin{equation} \cauchy_{yy}\left(r,\theta=0\right) = \frac{K_I}{\sqrt{2\pi r}}\,. \label{eq:LEFM}\end{equation}

blue line, which is not physical. At crack tip the stress is limited by the yield stress $\sigma_p^0$. However, there should be a stress redistribution so that the total load remains constant, leading to the green line for which the process zone extend to a length $r_p$. The model is then solved in two steps:

We thus have two equations with two unknowns $\eta$ and $r_p$. Using Eq. (\ref{eq:LEFM}) in (\ref{eq:rice_2}) yields

\begin{equation}\label{eq:rice_3} \sigma_p^0 r_p = \int_0^{\eta r_p} \frac{K_I}{\sqrt{2\pi r}} dr \,,\end{equation}

so that the plastic zone becomes

\begin{equation}\label{eq:rice} r_p = \frac{ 2K_I}{\sqrt{2\pi}\sigma_p^0}\sqrt{ \eta r_p} = \frac{K_I^2}{\pi} \frac{1}{\left(\sigma_p^0\right)^2}\,, \end{equation}

with $\eta = \frac{1}{2}$. So the behavior of perfectly plastic materials is as if the crack had an effective length of $a+\eta r_p = a + r_p/2$ on which the LEFM solution is applied.

Nevertheless, this solution was obtained for perfectly plastic materials. From numerical simulations, considering

and a more general estimation is

\begin{equation}\label{eq:etarP} \eta r_p=\frac{r_p}{2} = \left\{ \begin{array}{cc} \frac{n-1}{n+1}\frac{1}{2\pi}\left(\frac{K_I\left(a_\text{eff}\right)}{\sigma_p^0}\right)^2 & \text{ in plane-}\sigma\\ \frac{n-1}{n+1}\frac{1}{6\pi}\left(\frac{K_I\left(a_\text{eff}\right)}{\sigma_p^0}\right)^2 & \text{ in plane-}\varepsilon\text{.} \end{array}\right. \end{equation}

Therefore, if $\sigma_{\infty}$ remains lower than about $50\%$ of $\sigma_p^0$ (to avoid large scale yielding), then a second order SSY assumption holds true, and the LEFM methodology can be adapted following

So there is an iterative procedure to follow:

  1. Compute $K$ from $a$;
  2. Compute the effective crack size;
  3. Compute the new $K$ from $a_{eff}$ and go back to step 2. if needed.

This approach is a correction for linear fracture mechanics, but does not allow considering problems with large yielding.