Fracture Mechanics Online Class

Crack Growth > Stability

Considering a cracked body with $G \geqslant G_C$, then the crack propagates. However the crack can stop or not.

As the stability depends on the evolutions of the energy release rate and of the fracture energy, a crack growth depends on the geometry, the loading and the material behavior.

Example: DCB specimen

Remember the case of the composite laminate delamination studied previously. In linear elasticity, the compliance and its derivative with respect to the crack surface $A=at$ read

\begin{equation} \begin{cases}C=\frac{u}{Q} = \frac{8 a^3}{Eth^3}\\ \partial_A C=\frac{1}{t}\partial_a \frac{8 a^3}{Eth^3} =\frac{24 a^2}{Et^2h^3}\end{cases}.\label{eq:CDCB}\end{equation}

Prescribed loading

Picture V.5: Double Cantilever Beam (DCB) model with prescribed loading.

On the one hand if loading is prescribed, see Picture V.5, the energy release rate reads:

\begin{equation} G = \frac{Q^2}{2}\partial_A C = \frac{12Q^2a^2}{Et^2h^3}\label{eq:CDCBQ} \end{equation}

As the crack is growing, $G$ increases. For a perfectly brittle material, $G_C$ is constant, so the crack keeps propagating. This is an unstable crack.

Prescribed displacement

Picture V.6: Double Cantilever Beam (DCB) model with prescribed displacement.

On the other hand, if the displacement is prescribed, see Picture V.6, the energy release rate reads:

\begin{equation} G = \frac{u^2}{2C^2}\partial_A C = \frac{3u^2Eh^3}{16a^4}\label{eq:CDCBu}. \end{equation}

As the crack is growing, $G$ decreases. For a perfectly brittle material, $G_C$ is constant, so at the some point the crack stops propagating. The crack is stable.

We have seen that for the DCB problem, a prescribed loading condition can lead to an unstable crack growth, while a prescribed displacement condition leads to a stable crack. But is it a general rule? What is happening for hyperstatic structures?

General loading conditions

Picture V.7: General loading model.

Practically structures are hyperstatic. This means that if a crack appears in a component, the compliance of this component increases (the stiffness decreases) and there is a new loading distribution in the structure. This means that the component with the crack is neither subject to a prescribed loading condition nor to a prescribed displacement condition, but to a mixed one. This can be modeled with the compliant machine, see Picture V.7, made of:

The displacements are defined as follows:

\begin{align} u_T & = Q(a) \left( C(a) + C_M \right) \\ u& = C(a) Q(a) = \frac{C(a)}{C(a) + C_M} u_T ,\end{align}

and the related internal energy becomes:

\begin{eqnarray} E_\text{int}&=&\frac{Qu}{2}+\frac{Q(u_T-u)}{2} \nonumber\\&=& \frac{C}{2(C+C_M)^2}u^2_T+\frac{C_M}{2(C+C_M)^2}u^2_T\nonumber\\&=&\frac{u_T^2}{2(C+C_M)}. \end{eqnarray}

From the previous relations, we can deduce the energy release rate:

\begin{equation} G = -\partial_A \left.E_\text{int}\right|_{u_T}= \frac{u_T^2}{2\left(C+C_M\right)^2} \partial_A C .\label{eq:complianceG}\end{equation}

The variation of this energy release rate with respect to the crack surface reads:

\begin{align} & \partial_A G = -\frac{u_T^2}{\left(C+C_M\right)^3} \left(\partial_A C\right)^2 + \frac{u_T^2}{2\left(C+C_M\right)^2} \partial^2_{AA} C \\ \Leftrightarrow & \partial_A G = - \frac{Q^2}{C+C_M} \left(\partial_A C\right)^2 + \frac{Q^2}{2} \partial^2_{AA} C . \label{eq:dAG}\end{align}

Dead load case

The dead load case (a constant load over time such as the structural weight itself) can be modeled with a spring of infinite compliance ($C_M \rightarrow \infty$). As the first term of (\ref{eq:dAG}) is negative, this value of $C_M \rightarrow \infty$ is the one for which $\partial_A G$ is maximum (as this cancels the negative first term), with

\begin{equation} \left.\partial_A G\right|_{\text{Dead load}}= \frac{Q^2}{2} \partial^2_{A^2} C.\label{eq:dAGDeadLoad} \end{equation}

Mixed case

As the spring stiffens toward a fixed grip, $C_M$ decreases and so does $\partial_A G$, which tends to stabilize the crack growth.

If $\partial_A G$ becomes negative, the crack becomes stable for a perfectly brittle material ($G_c$ is a constant).

Fixed grip case

The fixed grip case (a constant displacement $u$ over time) can be modeled with a spring of zero-compliance ($C_M \rightarrow 0$). In that case (\ref{eq:dAG}) is minimum and reads

\begin{equation} \left.\partial_A G\right|_{\text{Fixed grip}}= - \frac{Q^2}{C} \left(\partial_A C\right)^2 + \frac{Q^2}{2} \partial^2_{A^2} C = - \frac{u^2}{C^3} \left(\partial_A C\right)^2 + \frac{u^2}{2C^2} \partial^2_{A^2} C .\label{eq:dAGFixedGrip}\end{equation}

Note that a fixed grip is always more stable than a dead load as $ \partial_A G$ is smaller.

Resistance curve

Picture V.8: Resistance of a non-perfectly brittle material.

Remember we have already introduced the resistance curve. For a perfectly brittle material the fracture energy $G_C$ is constant and equal to $2\gamma_s$. For other materials there will be a plastic flow at the crack tip as a stress cannot physically reach the infinity as predicted by the LEFM approach. This plastic flow arises in the active plastic zone as illustrated in Picture V.8. Evidently this active plastic zone moves with the crack tip in case of crack growth, meaning that the crack is opening in a zone affected by permanent plastic deformations: the plastic wake. This makes the crack more difficult to be opened, which corresponds to an increase of the apparent fracture energy $G_C$ with the crack propagation $\Delta a$. This is call the resistance and $G_c$ is replaced by $R_c(a)$. Some material can have a steady-state in the resistance curve as in Picture V.8, other not. Note that the curve does not only depend on the material, but also on the geometry as the thickness.

So for non-perfectly brittle materials, a crack can be stable even if $\partial_A G > 0$ as $R_c$ also depends on the crack surface. In 2D, $R_c (\Delta A) = R_c (t \Delta a)$

Example: delamination of a composite with initial crack $a_0$

On the one hand, for composites, as the crack propagates, fibers in the wake tend to close the crack tip so more energy is required for the crack to grow, and we have a curve $R_c (t \Delta a)$. On the other hand, we have previously evaluated the energy release rate of the DCB problem from the compliance variation (\ref{eq:CDCB}). We can now study the stability of the delamination.

Prescribed loading

Picture V.9: DCB stability for the dead load case with $G = \frac{Q^2}{2}\partial_A C = \frac{12Q^2a^2}{Et^2h^3}$.

We have previously computed the shape of the energy release rate (\ref{eq:CDCBQ}) as being $G = \frac{Q^2}{2}\partial_A C = \frac{12Q^2a^2}{Et^2h^3}$. As the crack is growing, $G$ also increases, see Picture V.9. Let us consider different loading values:

Prescribed displacement

Picture V.10: DCB stability for the fixed grip case with $G = \frac{u^2}{2C^2} \partial_A C= \frac{3u^2Eh^3}{16a^4}$.

We have previously computed the shape of the energy release rate (\ref{eq:CDCBu}) as being $G = \frac{Q^2}{2}\partial_A C = \frac{3u^2Eh^3}{16a^4}$. As the crack is growing, $G$ decreases, see Picture V.10, contrarily to the dead load case. In this case, whatever the value $u$ of the grip and whatever the material, $G$ always becomes smaller than $G_C$ (or $R_c$) for a given crack size, and the crack stops. This is a stable configuration.


In this part we have considered mode I failure. The case of mixed mode is considered in the next page.