Energy release rate (fracture mechanics)

This gives The energy release rate is directly related to the stress intensity factor associated with a given two-dimensional loading mode (Mode-I, Mode-II, or Mode-III) when the crack grows straight ahead.

depending on whether the material is under plane stress or plane strain: For Mode-II, the energy release rate is similarly written as For Mode-III (antiplane shear), the energy release rate now is a function of the shear modulus

, For an arbitrary combination of all loading modes, these linear elastic solutions may be superposed as Which can be seen to be equivalent to the previous representation through the relationship between Young's modulus and the shear modulus: Crack growth is initiated when the energy release rate overcomes a critical value

Some are dependent on certain criteria being satisfied, such as the material being entirely elastic or even linearly-elastic, and/or that the crack must grow straight ahead.

for arbitrary conditions is to calculate the total potential energy and differentiate it with respect to the crack surface area.

(area under the curve) is equal to [3] Using the compliance method, one can show that the energy release rate for both cases of prescribed load and displacement come out to be [3] Consider a double cantilever beam (DCB) specimen as shown in the right figure.

for where one computes the simpler integral to be leaving the energy release rate as the expected relation.

This allows one to indirectly retrieve the stress intensity factor for this problem as In certain situations, the energy release rate

This section will elaborate on some relatively simple methods for fracture analysis utilizing numerical simulations.

If the crack is growing straight, the energy release rate can be decomposed as a sum of 3 terms

The energy release rate is calculated at the nodes of the finite element mesh for the crack at an initial length and extended by a small distance

In this case, a new FEA simulation is performed (for the next time step) where the node at the crack tip is released.

For a bounded substrate, we may simply stop enforcing fixed Dirichlet boundary conditions at the crack tip node of the previous time step (i.e. displacements are no longer restrained).

represents the direction corresponding to the Cartesian basis vectors with origin at the crack tip, and

MCCI is more computationally efficient than the nodal release method because it only requires one analysis for each increment of crack growth.

Additionally, this method requires sufficient discretization such that over the length of one element stress fields are self-similar.

Similar to the nodal release method, if the crack were to propagate one element length along the line of symmetry (parallel to the

Once again, making the assumption of self-similar straight crack growth the energy release rate can be calculated utilizing the following equations:

Like with the nodal release method the accuracy of MCCI is highly dependent on the level of discretization along the crack tip, i.e.

The energy release rate can then be calculated over the area bounded by the contour using an updated formulation:

The formula above may be applied to any annular area surrounding the crack tip (in particular, a set of neighboring elements can be used).

The above-mentioned methods for calculating energy release rate asymptotically approach the actual solution with increased discretization but fail to fully capture the crack tip singularity.

[10] These elements have a built-in singularity which more accurately produces stress fields around the crack tip.

The advantage of the quarter-point method is that it allows for coarser finite element meshes and greatly reduces computational cost.

For the purposes of this section elastic materials will be examined, although this method can be extended to elastic-plastic fracture mechanics.

Calculating the normal strain involves using the chain rule to take the derivative of displacement with respect to

This impedes the ability to capture the angular dependence of the stress fields which is critical in determining the crack path.

The collapsed rectangle can more easily surround the crack tip but requires that the element edges be straight or the accuracy of calculating the stress intensity factor will be reduced.

The element's geometry allows for the crack tip to be easily surrounded and meshing is simplified.

This can greatly reduce computation when compared to other 3-dimensional methods but can lead to errors if that crack tip propagates with a large degree of curvature.

Plot of Load vs. Displacement
Double cantilever beam (DCB) specimen under tensile load.
A Double Cantilever Beam Specimen with Thickness B (not shown)
Graphical Illustration of G under Fixed Displacement and Fixed Load Conditions
Double cantilever beam (DCB) specimen under tensile load.
Double cantilever beam (DCB) specimen under tensile load.
J-integral path for the DCB specimen under tensile load.
Figure 1: Crack debounding between two consecutive time steps using nodal release (NR)
Figure 2: Schematic of the MCCI process for 4-node linear rectangular elements.
Figure 3: Schematic of the MCCI process for 9-node quadratic rectangular elements.
Figure 4: Contour domain integral for the J-integral
Figure 5: Mapped and Parent Element of the 8-node isoparametric finite element. Note the quarter point location of the nodes in the mapped element.
Figure 6: Triangular Element mapped from 8-node rectangular element.
Figure 7: Natural Triangle Element