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Welcome

1. Overview
   / 1 / 2 / 3 / 4 / 5 /
2. Asympthotic Solution
   / 1 / 2 / 3 / 4 / 5 / 6 /
3. Energetic Approach
   / 1 / 2 / 3 / 4 / 5 /
4. SIF Computation
   / 1 / 2 / 3 / 4 / 5 /
5. Crack Growth
   / 1 / 2 / 3 / 4 / 5 /
6. Numerical Methods
   / 1 / 2 / 3 / 4 /
7. Cohesive Zone
   / 1 / 2 / 3 / 4 /
8. Near-tip HRR Fields
   / 1 / 2 / 3 / 4 / 5 / app /
9. $J$-integral
   / 1 / 2 / 3 / 4 /
   

HRR theory > Other HRR solutions

In the previous section, we have derived the expressions of the near-tip stress and strain fields for a mode I crack in plane-$\varepsilon$ state. Moreover an uncompressible material has been assumed. In this section, a brief summary is given for other HRR solutions that were derived in the literature:

Mode I crack in plane-$\sigma$ state

Still assuming an uncompressible material, a similar analysis as in the plane-$\varepsilon$ state can be performed and yields different components tensors $\tilde{\cauchy} \left(\theta,n\right)$, $\tilde{\strain} \left(\theta,n\right)$, and $\tilde{\bf{u}} \left(\theta,n\right)$ fields.

Near-tip stress field

As we have derived in the previous section, the near-tip stress field reads

\begin{equation} \cauchy=\sigma_p^0\left(\frac{JE}{r\alpha\left(\sigma_p^0\right)^2I_n}\right)^{\frac{1}{n+1}}\tilde{\cauchy}\left(\theta,\,n\right) \, \text{.} \end{equation}.

Pictures VIII.17-VIII.20 compare the stress shape tensor $\tilde{\sigma}$ components between plane-$\sigma$ and plane-$\varepsilon$ conditions for two different hardening parameters. Moreover the integral $I_n$ is reported in Picture VIII.21 for the two different cases. These figures allow comparing the difference in the crack tip loading, for instance through the asymptotic hoop stress $\sigma_{\theta\theta}$, between a plane-$\varepsilon$ and a plane-$\sigma$ states. Indeed, for the same value of the $J$-integral, and at the same crack tip distance $r$, the ratio between the two hoop stresses for two different hardening parameters are

\begin{equation} \frac{\left.\cauchy_{\theta\theta}\left(0,\,r\right)\right|_{n=3,\,\text{plane }-\varepsilon}}{\left.\cauchy_{\theta\theta}\left(0,\,r\right)\right|_{n=3,\,\text{plane}-\sigma}} \simeq\frac{1.9}{1.1}\left(\frac{3.86}{5.81}\right)^\frac{1}{4} = 1.56 \, \text{,} \end{equation} \begin{equation} \frac{\left.\cauchy_{\theta\theta}\left(0,\,r\right)\right|_{n=13,\,\text{plane}-varepsilon}}{\left.\cauchy_{\theta\theta}\left(0,\,r\right)\right|_{n=13,\,\text{plane}-\sigma}} \simeq\frac{2.6}{1.2}\left(\frac{2.87}{3.4}\right)^\frac{1}{14} = 2.13\,. \end{equation}

It can be seen that because the material cannot expend laterally, the crack tip is more loading in plane-$\varepsilon$ state, for the same value of the $J$-integral.

Process zone shapes

The process zone was defined as the zone in which $\sigma_e > \sigma_p^0$, whose boundary corresponds to $\sigma_e=\sigma_p^0$, and was found to be

\begin{equation} \tilde{r}\left(\theta,\,n\right) = \frac{\left(\tilde{\sigma}_e\left(\theta,\,n\right)\right)^{n+1}}{I_n} \, \text{.}\label{eq:processZone} \end{equation}

in terms of a non-dimensional radius $\tilde{r}= \frac{r \alpha \left(\sigma_p^0\right)^2 }{JE}$. The process zones for $n=$ are illustrated in Picture VIII.24 for the plane-$\sigma$ and plane-$\varepsilon$ states. It can be seen that the process zone is more diffuse in plane-$\sigma$ state.

Crack Tip Opening Displacement

We have seen that there exists a direct relation between the crack tip opening displacement and the $J$-integral:

\begin{equation} \delta_t = d_n \frac{J}{\sigma^0_p}\,, \end{equation}

where $d_n$ depends on $n$ but also on the elastic strain of the considered material $\frac{\alpha\sigma^0_p}{E}$. The evolutions of $d_n$ in the cases of plane-$\varepsilon$ and plane-$\sigma$ states are respectively illustrated in Picture VIII.22 and Picture VIII.23.

Mode III crack loading

Assuming SSY, the process zone develops under mode III following a circular shape, as illustrated in Picture VIII.25 and Picture VIII.26 for respectively a perfectly plastic material and an elasto-plastic material with strain hardening.

Mode II and mixed mode crack loading

In SSY, the solution depends on the elastic mixity parameter given by

\begin{equation} M^e = \frac{2}{\pi}\arctan{\frac{K_I}{K_{II}}}\,, \end{equation}

which ranges from 0 (Mode I crack loading) to 1 (Mode II crack loading). Varying the mixity parameter the process zone changes orientation, from a"horizontal" shape in Mode II crack loading, as depicted in Picture VIII.27, rotating upward for a mixed mode crack loading as depicted in Picture VIII.28.

Compressibility effect