Fracture Mechanics Online Class

Energetic Approach > The energy release rate

In the previous section we have deduced the energy released by the system when the crack surface increases and we have found

\begin{equation} G = \begin{cases} & - \partial_A \left( E_\text{int} - Q u\right) &\text { for a prescribed loading} \\ &- \partial_A E_\text{int} & \text { for a prescribed displacement}\end{cases}.\label{eq:GPrescribedCases}\end{equation}

Note that these equations hold for any elastic materials, linear or not. The only requirement is the existence of a material internal potential $U$. Nevertheless the expression of $G$ depends on the loading conditions, and one would like to define it for a general loading condition.

General Loading

Let us define the potential energy of the specimen as

\begin{equation}\Pi_T = E_\text{int}-W_{\text{ext}}.\label{eq:potEnergy}\end{equation}

The energy release rate is thus defined as the energy, per unit crack surface, released by the system assuming a crack grows

\begin{equation} G = -\partial_A \left(E_\text{int}-W_\text{ext}\right) = - \partial_A \Pi_T .\label{eq:G}\end{equation}

Note that in the case of prescribed loading or displacement, (\ref{eq:G}) simplifies into (\ref{eq:GPrescribedCases}).

Crack growth

What has to be understood from the definition of the energy release rate (\ref{eq:G}) is that the crack growth is purely virtual: We evaluate the energy released by the system if a crack would grow. But what is required for a crack to grow?

Let us now consider the total energy of the system

\begin{equation}E=\Pi_T + \Gamma,\label{eq:sysEnergy}\end{equation}

where $\Gamma$ is the energy required to create a crack surface $A$ in the body. Two kinds of materials can be considered

For both cases we define the fracture energy as the energy required to form a unit surface in the material

\begin{equation}G_C = \partial_A \Gamma.\label{eq:Gc}\end{equation}

As the derivation of (\ref{eq:sysEnergy}) implies

\begin{equation} \partial_A E= \partial_A \Pi_T +\partial_A \Gamma=G_C-G,\label{eq:dsysEnergy}\end{equation}

we have now a crack propagation criterion:

Thus the crack propagation criterion simply reads

\begin{equation} G \begin{cases} & < G_C &\text { no propagation} \\ &\geq G_C & \text { propagation}\end{cases}.\label{eq:Gcriterion}\end{equation}

In this relation $G$ depends on the sample geometry (including crack length) and boundary conditions, while $G_C$ depends on the material (for ductile materials it can also depend on the crack advance).

We have now introduced the energy release rate concept and have related it to a fracture criterion. These considerations are valid for elastic materials (or materials that can be considered as it) as we have assumed a material internal potential $U$. We did not make assumption on the linearity or not of the elastic material. However, things become easier for linear elasticity.

Linear case and compliance

In particular, in linear elasticity the expression (\ref{eq:GPrescribedCases}) of $G$ for different loading conditions can be unified in terms of the structure compliance $C$. Indeed as the material is linear the compliance of the specimen, for a given crack surface $A$, is defined as

\begin{equation} C\left(A\right)=\frac{u}{Q}\label{eq:C}.\end{equation}

The system energies can directly be computed from this compliance

\begin{equation} \begin{cases} E_\text{int} &=& \frac{1}{2}Qu=\frac{u^2}{2C} = \frac{Q^2C}{2}\\ W_\text{ext} &=& Qu = \frac{u^2}{C}=Q^2C\end{cases}\label{eq:ELinear}.\end{equation}

We can now analyze the two cases of interest.

Prescribed displacements

In that case, (\ref{eq:GPrescribedCases}) simplifies into

\begin{equation} G =- \left.\partial_A E_\text{int}\right|_u = -\left.\partial_A \left(\frac{u^2}{2C}\right)\right|_u = \frac{u^2}{2C^2}\partial_AC = \frac{Q^2}{2}\partial_A C .\label{eq:GLinearPrescribedDisp}\end{equation}

The physical interpretations of this relation are

Prescribed loading

In that case, (\ref{eq:GPrescribedCases}) simplifies into

\begin{equation} G =- \left.\partial_A \left(E_\text{int}-Q u\right)\right|_Q = -\left.\partial_A \left(\frac{Q^2C}{2}-Q^2C\right)\right|_Q = \frac{Q^2}{2}\partial_AC.\label{eq:GLinearPrescribedLoading}\end{equation}

This expression is similar as (\ref{eq:GLinearPrescribedDisp}) obtained for a prescribed displacement. However the implication is quite different. Indeed, starting again from (\ref{eq:GPrescribedCases}) and using (\ref{eq:GLinearPrescribedLoading}), we have
\begin{equation} \begin{cases} G =- \left.\partial_A \left(E_\text{int}-Q u\right)\right|_Q = -\left.\partial_A E_\text{int}\right|_Q + \left. \partial_A\left(Q
u\right)\right|_Q \\ \iff G + \left.\partial_A E_\text{int}\right|_Q = Q \partial_A u =
Q^2\partial_AC = 2G\\\iff G =\left.\partial_A E_\text{int}\right|_Q\end{cases} .\label{eq:GLinearPrescribedLoading1}\end{equation}

The physical interpretations of this relation are

Application of the compliance method: Delamination of a composite laminate

Picture III.10: Double Cantilever Beam (DCB) model.

Delamination is a common failure mode in composite structures. The composite laminate is made of a stacking sequence of composite plies and two plies can unglue from each other due to the shear force, leading to delamination. This effect can be modeled by considering two clamped beams as in Picture III.10. This model is valid when the crack is long compared to the ply thickness: $a>>>h$. In that case we can consider two identical (semi-)cantilever beams of length $a$. The question is how to determine the energy release rate of this structure.

The resolution method is as follows:

Picture III.11: Experiment on a DCB.

This model can be used to measure the delamination energy of composite plies following a rigorous standard (ISO 15024). This process is summarized by