Fracture Mechanics Online Class

J-integral > Fracture toughness testing for elasto-plastic materials

In the previous section, we have seen how to evaluate the $J$-integral of a cracked body. In this section we will apply these methods to assess the maximum $J$-integral that an elasto-plastic material can sustain.

Normalized procedure

Picture IX.22: Single Edge Notch Bend (SENB) specimen with crack length $a$ measured from support line.
Picture IX.23: Compact Tension Specimen (CTS) with crack length $a$ measured from loading support line.

The purpose of fracture toughness testing in the case of elasto-plastic materials is to evaluate limit values of the $J$-integral in order to further assess the behavior of a cracked body. In particular we are interested in three values

Plane-$\varepsilon$ conditions are usually considered for conservatism reasons as it will be justified later.

The testing process should strictly follow the norm, e.g. the ASTM E1820 norm, and is usually performed on either the Single Edge Notch Bend (SENB) specimen, see Picture IX.22, or on the Compact Tension Specimen (CTS), see Picture IX.23. During the sample loading, the evolution of the crack mouth opening $v$ is measured since it is more accurate than the measurement of the deflection $u$. The testing process will be summarized in the following section.

R-curve method

First step: Fatigue crack growth

A cracked sample cannot be manufactured in representative way. Therefore, the samples are manufactured with a notch, and a crack is initaited and grown by fatigue. To this end a cyclic loading reaching the maximum load $Q_\text{fat}$ is applied to the notched sample. In order to avoid plastic flow during the unloading stage, the last three cycles are in between $\frac{Q_\text{fat}}{2}$ and $Q_\text{fat}$.

Before conducting the toughness test, one needs to know what the initial size $a_0$ of the crack originating from the fatigue loading is. An accurate measurement is not possible, so we estimate the compliance of the sample and use formula that have been obtained by finite element analyzes of the normalized sample with different crack lengths $a$. Since we are not propagating the crack at this stage, the formula are the ones of linear elasticity, and imply to apply a reduced loading $Q$. As previously stated, the measurement of the crack mouth opening $v$ is more accurate than that of the deflection $u$, and the compliance is thus defined in terms of $\frac{v}{Q}$.

For the SENB specimen, the following relation of the crack length in terms of the normalized compliance $U$ has been derived:

\begin{equation}\begin{cases} \frac{a}{W} &= 0.9997-3.95 U +2.982 U^2-3.214 U^3 + 51.52 U^4 - 113.0 U^5;\text{ with}\\ U&= \frac{1}{1+\sqrt{\frac{4E't v W}{LQ}}}.\end{cases}\label{eq:SENBcrack} \end{equation}

These formula can be inverted in order to define the elastic compliance $C_e$ for a known crack length, leading to, for the SENB specimen

\begin{equation} C_e=\frac{v}{Q} = 6 \frac{L}{E'tW}\frac{a}{W}\left[0.76-2.28\frac{a}{W}+3.87\left(\frac{a}{W}\right)^2-2.04\left(\frac{a}{W}\right)^3+\frac{0.66}{\left(1.-\frac{a}{W}\right)^2}\right].\label{eq:SENBvCompliance} \end{equation}

We finally note that similar formula exist when considering the elastic compliance $C_{LLe}$ in terms of the deflection $u$, with for the SENB specimen

\begin{equation} C_{LLe}=\frac{u}{Q} = \frac{1}{E't}\left(\frac{L}{W-a}\right)^2 \left[1.193-1.98\frac{a}{W}+4.478\left(\frac{a}{W}\right)^2-4.443\left(\frac{a}{W}\right)^3+1.739\left(\frac{a}{W}\right)^4\right]. \label{eq:SENBuCompliance}\end{equation}

Second step: Sample loading

Picture IX.24: Loading and partial unloading during an elasto-plastic toughness test.

Once the initial crack length $a_0$ has been determined, the toughness test can proceed. To this end the load $Q$ is increased and the crack mouth opening $v$ recorded. Once the crack starts propagating, several cycles of partial unloading/reloading are applied, see Picture IX.24. During the loading, the crack size increases and the partial unloading steps are necessary to determine its evolution using the compliance method, e.g. using Eqs. (\ref{eq:SENBcrack}) for the SENB specimen. The unloading stages are partial, i.e. the load is not decreased down to zero in order to avoid reverse plasticity at crack tip.

Once enough, i.e. at least 8, pairs $(a_i,\,Q_i)$ of crack length and loading have been obtained, the sample is totally unloaded and broken (after cooling down if needed to reduce the required load), and the final crack is marked.

Third step: Data reduction

Picture IX.25: Evaluation of the $J$-integral at different crack increments $\Delta a$.

We can now evaluate the evolution of the $J$-integral during the crack propagation using the $\eta$-approach. To this end, for each pair $(a_i,\,Q_i)$, one has to

Using these values $J_i$ evaluated for the different crack lengths $a_i$ allows drawing the $J$-$\Delta a$ curve, see Picture IX.25, with $\Delta a=a-a_0$ computed from the initial crack length $a_0$.

Fourth step: Analysis

Picture IX.26: Extrapolation of the resistance curve.

We can now analyze the data $(J_i,\,a_i)$ of Picture IX.25 following the approach illustrated in Picture IX.26 in order to determine the values of interest. To do so, we proceed as follows

Typical resistance curves

Picture IX.27: Effect of the sample thickness on the resistance curve.
Picture IX.28: Effect of the crack length on the resistance curve.

Picture IX.27 and Picture IX.28 illustrate typical resistance curves of a ductile material as can be found in "Andrews WR and Shih CF (1979), Thickness and side -groove effects on J- and d- resistance curves for A533-B steel at 93$^o$C, ASTM STP 668 , 426-450" for steel A533-B, values for indication purpose only. Picture IX.27 shows that the resistance curves, but also the values of $J_{IC}$, are not unique for a given material, but depend on the thickness of the sample. Actually, when the thickness increases, the stress-triaxiality tends to that of a plane-$\varepsilon$ state and the sample is less resistant to crack propagation. This behavior has been discussed in the HRR theory finding. This means that in order to ensure conservatism, the sample should be thick enough, justifying the condition (\ref{eq:checkToughness}). Another solution is to groove, i.e. dig half-cylindrical shapes in the alignment of the crack, both sides of the sample. In that way the plane-$\sigma$ state of the sample sides is cancelled, lowering the resistance curves. The values also depend, because of the change of triaxiality, on the other geometrical parameters such as the initial crack length in Picture IX.28, but to a lower extent.