Numerical study of Griffith's principle

Nicholas Wheeler
Reed College Physics Department
4 November 2009

Hyperlink["Griffiths' Theory","http://en.wikipedia.org/wiki/Fracture_mechanics"]
Hyperlink["Fracture Kinetics","http://books.google.com/books?id=UWmvMMtDZtEC&pg=PA20&lpg=PA20&dq=Griffiths+crack+growth&source=bl&ots=W0aeQB7pNA&sig=YvoWhaHBxirEyHvqO6_KUCRISpw&hl=en&ei=fnzySom4GY3asgP71LQd&sa=X&oi=book_result&ct=result&resnum=10&ved=0CDUQ6AEwCQ#v=onepage&q=&f=false"]

I take a simple 100-node grid

and displace each of the nodes in the 10th column 1.9 units to the right. I then release pins, one after another, starting at the lower right hand corner and progressing upward.  I then produce the following succession of deformed grids, on each of which I have superimposed the preceding image of the unstressed grid. Here is the fully stressed grid (no pins yet removed)

and now I inaugurate "crack growth" up the right side of the grid:

In each case I computed the energy (Subscript[ℰ, n]) stored in the spring system after n nodes had been released, and found (Subscript[ℰ, n]) to be a monotonically decreasing function of n (i.e., of crack length):

Table[{n,(Subscript[ℰ, n])},{n,0,9}]//TableForm

I found, moreover, that the data conformed well to a quadratic function (as A. A. Griffith (1920) had speculated):

ℰ[n]

REMARK: Here the data point at 1 is the first data point, which is (Subscript[ℰ, 0]), not (Subscript[ℰ, 1]).

The spring energy released by crack-induced relaxation of the grid is

EnergyCredit[n_]:=ℰ[0]-ℰ[n]

EnergyCredit[n]

On the other hand, we expect the energy that must be invested to create a crack to depend linearly on crack length.

EnergyDebit[n_,strength_]:=-strength n

In the following figure (compare Gordon, page 101) I have set

strength=0.3;

Energy credit is green, energy debit is blue, their sum is black/red. Looking to the figure, we see that the growth of cracks of length n ≲ 2 is unlikely, because it would cost the system energy. Growth is more likely if 2 ≲ n < 3.5,  and is catastrophically inevitable if n ≳ 3.5.