(5)To derive the relationship between the SERR and crack length (G vs. a ) requires an analysis of the structure/joint geometry and loading of interest. For some simple joint geometries and loadings, an analytical expression can be derived based on compliance change considerations using Equation (4) or more generally finite element analysis (FEA) can be used.
To model a crack in a finite element model requires the creation of duplicate nodes along the plane of the crack growth. This can be done at the pre-processing stage or during the solution process when the likely crack path may not be known a priori. Coincident nodes can then be connected using multipoint constraints (MPCs) and the constraints released to create different crack lengths. There is no preferred size of element, but the smaller the elements, the more refined are the changes in G with crack length. This may be important for short cracks.
Calculation of G from FEA solutions can be done in post-processing. One method is to consider the global energy balance for two solutions with difference crack areas (Ai and AI+1 ), from which:
(5)
where U is the strain energy and A is the crack area for the ith solution. This required two separate FE solutions and gives the total SERR, Gtot , but contributions from separate modes cannot be derived. To calculate G and the separate modes, the virtual crack closure technique (VCCT) can be used. This is based on the work done close the crack tip. If the incremental change in G is small between compared with the overall crack length, G can be calculated from the nodal forces and displacements a single FE solution. For a two dimensional model with nodes defined as in Figure 4, VCCT gives:
(6)
(7)
where
FXF , FXG force in X direction in the respective node
FYF , FYG force in Y direction in the respective node
UB , UC , UD , UE X displacements in respective node
VB , VC , VD , VE Y displacements in respective node
|
X |
|
Y |
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Figure 4: Nodes used in VCCT
If the geometry of the structure or joint is large, then it may be necessary to perform the FEA in two stages following a global/local approach. In the first stage the complete geometry is included in a global FEA and the region for crack analysis is identified based on the global stress distributions. In the second stage the crack analysis is carried out on a local model of the critical region of the structure or joint, enabling a more refined mesh to be used in the critical region.
Boundary conditions may either be simplified or taken from the global model solutions. For the later, there will be a limit to the crack size for which G can be obtained accurately, since the boundary conditions for the global mode will not account for changes due to crack growth.
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