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Crack Bridging. Lecture 1
These notes belong to a course on fracture mechanics
Following Griffith (1921), we distinguish two processes: deformation in the body and separation of the body. Up to this point, the process of deformation has been described by field theories of various kinds, such as
- linear elastic theory (infinitesimal deformation, linear elastic material)
- nonlinear elastic theory (finite deformation, nonlinear elastic material)
- deformation theory of plasticity (infinitesimal deformation, fictitious nonlinear elastic material)
By contrast, the process of separation has been described, if it is described at all, by micrographs, cartoons, and words. A picture is worth a thousand words, but an equation is worth a thousand pictures. So far we have not used a single equation to describe any process of separation.
This negligence in describing the process of separation seems odd, particularly in a subject called fracture mechanics. Without specifying a process of separation, the artificial singularity would remain the black hole in our subject.
Barrenblatt (1959) modeled the process of separation by an array of nonlinear springs. By now this idea has permeated into fracture mechanics in many ways.
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Hi Zhigang, Sorry to
Hi Zhigang,
Sorry to miss two of the classes and the discussion (because of the MRS). One question about the argument of 1/2 singularity: doesn't the quadratic dependence of the energy release rate on the applied load already implicitly assume 1/2 singularity already? If it does, the reasoning is circulating.
Li Han
An argument for the square-root singularity
I'm glad you notice this argument for the square-root singularity. I have also developed a similar argument for the HRR singularity.
Now return to your quastion. Recall the definition of the energy release rate G. G is the reduction of the potential energy associated with advanment of the crack by unit area. For a linearly elastic material, the potential energy is quadratic in the applied stress, so that G is also quadratic in the applied stress.
Incidentally, this quadratic dependence was discussed in the lecture on the Griffith paper.
Excellent notes!
Dr. Suo,
I returned to iMechanica after a long hibernation, and am thrilled to see these fantastic notes - they are lucid, and address the very questions that always were on my mind during my doctoral work. Thanks so much, I will track these more closely going forward!
Dhruv