Numerical Methods
This chapter introduces some existing numerical methods dedicated to the simulation of crack propagation:
 It mainly focuses on Linear Fracture Elastic Mechanics (LFEM) and considers the Finite Element Method (FEM), Cohesive Zone Methods (CZM), and the eXtended Finite Element Method (XFEM);
 The key ideas of Continuum Damage Models (CDM) for ductile materials are also given;
 Finally, multiscale methods are briefly introduced.
Numerical Methods > Reminder of LEFM
Definition of elastic fracture
Up to now we have mainly considered elastic fracture mechanics. Strictly speaking, elastic fracture means that the only
changes at the material level during the failure are atomic separations.
As this way of thinking is too restrictive for real life applications, a pragmatic definition
should be: The process zone, which is the region where the inelastic
deformations happen, is a small region compared to the specimen sizes (including crack size), and is
located at the crack tip. The inelastic deformations may include, among others, plastic flow, microfractures,
or void growth.
Summary of the previous lectures
The Stress Intensity Factor
For linear elastic materials, the linear elastic stress analysis and the asymptotic solution therefore describe the fracture process
with accuracy. In particular for the three modes of fracture (I for opening, II for
inplane sliding and III for outofplane shearing) the asymptotic solution is expressed in terms of the SIFs following
\begin{equation}\begin{cases} \mathbf{\sigma}^\text{mode i} = \frac{K_i}{\sqrt{2\pi r}} \mathbf{f}^\text{mode i}(\theta) \\ \mathbf{u}^\text{mode i} = K_i\sqrt{\frac{r}{2\pi}} \mathbf{g}^\text{mode i}(\theta) \end{cases},\label{eq:fandg}\end{equation}
The crack closure integral
The crack closure integral represents the energy required to close the crack
on an infinitesimal length $da$.

If an internal (material) potential exists (linear materials or not), then the energy release rate reads
\begin{equation} G
= \partial_A \left(E_{\text{int}}  W_{\text{ext}}\right) = \partial_A \Pi_T, \label{eq:GfromPiT}\end{equation}
where $A$ is the crack surface, and where the crack closure integral reads
\begin{equation} \Delta \Pi_T = \int_{\Delta A} \int_{[\![\mathbf{u}]\!]}^{[\![\mathbf{u}+\Delta
\mathbf{u}]\!]} \mathbf{t}([\![\mathbf{u}]\!]) \cdot [\![d\mathbf{u}]\!] d(\Delta A) .\label{eq:PiT}\end{equation}
In this expression $\mathbf{t}$ is the traction between the forming crack lips and $[\![d\mathbf{u}]\!]$ is the opening between the crack lips.

In linear elasticity (thus an internal potential always exists), (\ref{eq:GfromPiT}) simplifies into
\begin{equation} G = \lim_{\Delta A \rightarrow 0} \frac{1}{2
\Delta A} \int_{\Delta A} \mathbf{t} \cdot [\![\Delta \mathbf{u}]\!] d \partial A \label{eq:GfromPiTLinear},\end{equation}
where $\mathbf{t}$ is the surface traction in the material before the crack propagates, and where $[\![\Delta \mathbf{u}]\!]$ is the opening between the crack lips after the crack has propagated.

In linear elasticity (thus an internal potential always exists) and if the crack grows straight ahead we have a relation between the energy release rate and the SIFs
\begin{equation} G = \frac{K_I^2}{E'} + \frac{K_{II}^2}{E'} + \frac{K_{III}^2}{2\mu}.\label{eq:GfromSIF} \end{equation}
The Jintegral concept

The Jintegral is the energy that flows towards the crack tip
\begin{equation} J = \int_\Gamma (U(\varepsilon) \mathbf{n}_x  \partial_x \mathbf{u} \cdot
\mathbf{T}) dl .\end{equation}

If an internal potential exists

The Jintegral is path independent if
the contour $\Gamma$ embeds a straight crack tip. However, it does not assume
on a subsequent growth direction;

If the crack grows straight
ahead then $G = J$;

In linear elasticity (no assumption on the crack growth direction)
\begin{equation} J = \frac{K_I^2}{E'} + \frac{K_{II}^2}{E'} + \frac{K_{III}^2}{2\mu}.\label{eq:JfromSIF} \end{equation}

This concept can be extended to plasticity if there is no
unloading, as in this case we can define an internal potential (see later).
Crack Growth
As
previously detailed,
the growth can be analyzed as follows:
 Crack growth criterion: A crack will grow when the energy release rate is greater than the critical rate of the material. This condition can be written as: \begin{equation} G \geq G_c. \end{equation} In this relation G depends on the sample geometry (including crack length) and boundary conditions, while G C depends on the material:
 Stability of crack growth: A crack is stable if, during the crack propagation, the resistance rate growth is larger than the energy release rate growth: \begin{equation} \partial_a G \leq \partial_a R_c, \end{equation} and unstable otherwise.
 Crack growth direction: The crack growth direction for mixed loadings can be determined by the maximum hoop stress method:
 The crack propagation criterion reads:
\begin{equation} \left( \sqrt{2\pi r} \sigma_{\theta\theta}(r,\theta^*) \right) \geq K_c, \end{equation}
with $ \left.\partial_\theta \sigma_{\theta\theta}\right_{\theta^*} = 0$ & $\left.\partial_\theta^2 \sigma_{\theta\theta}\right_{\theta^*} = 0$ ;
 The mixed loading coefficient is defined aunder the form: $\cot \beta^* = \dfrac{K_{II}}{K_{I}}$;
 The maximum hoop stress criterion thus becomes:
\begin{equation} \begin{cases}
K_C = \left. \sqrt{2\pi r} \sigma_{\theta\theta}(r,\theta^*) \right_{\theta^*} = K_I \cos^3\dfrac{\theta^*}{2}  \dfrac{3K_{II}}{2} \sin \theta^* cos\dfrac{\theta^*}{2}, \\
2 \tan \dfrac{\theta^*}{2}  \cot \dfrac{\theta^*}{2} = \tan \beta^* = \dfrac{K_I}{K_{II}}
.\end{cases}\end{equation}