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Crack propagation in linear elastic solids
Problem description
Modelling fracture in linear elastic solids. Some examples below include contact stress analysis and solid-fluid interaction.
Physics and modelling
The mathematical model consists of two parts: the linear elastic solid model is used in the bulk of the domain and combined with a cohesive zone model to describe local fracture process along the prescribed trajectory of the crack. Cohesive zone model represents a predictive fracture model, whare crack initiation and propagation are a result of the analysis. Linear elastic stress analysis code, stressedFoam, is available as a part of the standard OpenFOAM release, whereas the fracture model was implemented by the research group.
OpenFOAM solver
contactStressFoam, customised version and purpose-written fracture codes written at the Strength of Materials Section, Dept. of Mechanical Engineering, Imperial College.
Images and animations
The images below present the results of three test cases on increasingly comples geometries, all including crack propagation. As is typical in these cases, fracture mechanics is only a part of the problem. In order to aid the visualization, in all cases the deformation of the geometry is exaggregated.

The first test represents the cracking of the cantilever beam specimen. Location of the crack tip can be recognized by the stress concentration in front of it. Although not immediately obvious, ths simulation also includes a contact problem at the left end of the specimen (the rigid part of the machinery was excluded from the analysis). Experimental study of the same problem shows the initial steady propagation of the crack followed by unstable crack propagation in the later stages.

Cantilever beam
Figure 17: Cracking of the cantilever beam specimen.

In the wedge peel test presented below, the specimen is shaped as a tuning fork with the structural adhesive applied between the two halves and forced by pushing a wedge through it. The computational model includes only one half of the geometry. The crack propagates along the x-axis in the negative direction. Two specimens with different wall thickness have been examined and show different crack propagation regimes, both well captured in the simulation. The thin specimen test exhibits a steaqdy-state crack growth whereas the thick speciemen shows transient or dynamic crack growth.

In this case, the importance of the contact problem is much more pronounced: any loss of accuracy in the contact region would significantly degrade the overall solution accuracy.

Wedge peel test
Figure 18: Crack propagation and stress state in the wedge peel specimen test. Courtesy of Imperial College.

Animation of the Impact Wedge Peel Test, thin specimen Size: 9.8MB. Courtesy of Imperial College.

Animation of the Impact Wedge Peel Test, thick specimen Size: 8.7MB. Courtesy of Imperial College.

The final test case consists of a pressurized pipe with an initial crack and represents a strongly coupled solid-fluid interaction problem. Here, the force driving the crack is the internal pressure in the pipe, while at the same time the cracked surface allows the fluid to escape into the environment. Thus, although the image only shows the solid part of the domain, both the solid and fluid side were simulated in a single coupled transient simulation with the pressure/uncovered surface data exchanged between the two materials in every time-step. For this purpose, a special OpenFOAM solver has been written to cover both the solid and the fluid side. The location of the crack at the pipe wall can be recognized by the stress concentration at its tip.

Cracking pipe
Figure 19: Solid-fluid interaction in fracture mechanics: depressurization and crack propagation in a pressurized pipe.

Simulation and fracture mechanics model courtesy of Dr. Vlado Trops  \special {t4ht=a and his supervisors, Dr. Alojz Ivankovi                          īc  \special {t4ht= and Prof. A.J. Kinloch, Imperial College of Science, Technology and Medicine, London.