# J controlled fracture#

## HRR#

Note

The HRR fields were published in 1968 by (J.R. Rice and G.F> Rosengren)[https://doi.org/10.1016/0022-5096(68)90013-6] and by (J.W. Hutchinson)[https://doi.org/10.1016/0022-5096(68)90014-8]

The **HRR** field, named after Hutchinson Rice and Rosengren is the solution for the mechanical fields at the tip of a stationary crack in a power law material.

The load is assumed to increase monotonically, thus allowing the use of the *deformation theory* of plasticity. As we mentioned previously, the *deformation theory* theory can be thought of as non-linear elasticity with the relation.

Writing down the \(J\) integral for \(r \to 0\) :

We can estimate that the integrand have the form of \(\frac{f(\theta)}{r}\)

This implies that

if we expand \(\sigma_{ij}\) into a series around \(r \to 0\) such that

we can obtain for \(r \to 0\) where the plastic strains are much greater than the elastic ones:

Since we required earlier that \(\sigma \cdot \epsilon \propto r^{-1}\) it implies that \(s+sn=-1\) and thus \(s\) which will lead to the strongest singularity is

Now, the **HRR** stress and strain fields (for \(r \to 0\)) are given by:

Note

For \(n \to 1\) we obtain a \(r^{-1/2}\) singularity

Using the **HRR** fields in the \(J\) integral will lead to

With \(k_n\) being:

and finally

**It is time to find the angular dependency of the stress and strain (but we wont)**

To do so, one need to introduce a stress function \(\phi\) whose derivatives define \(\sigma\).

## Crack growth resistance curves#

The value of the tearing modulus is calculated by applying:

on the polynomial fit to the data points between the exclusion lines.

Similar to the way we handeled \(G\) we can examine the stability of a growing crack by comparing the driving force for crack growth as :

with \(\color{red}{\Delta}\) being the applied displacement and \(\color{blue}{C_m,P}\) being the machine compliance and applied load.

We can easily obtain

When the machine is infinitely stiff (\(\color{blue}{C_m} \to \infty\) - load control) we are left with

A crack will grow in a stable manner when \(J=J_R\) and \(T_{app} \leq T_R\)

A crack will grow in an unstable manner when \(J=J_R\) and \(T_{app} \gt T_R\)