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# Future of fracture mechanics

Wed, 2007-06-06 19:06 - BoJing Zhu

If possible,share your opinion (idea) about the future trend of fracture mechanics. Thanks!

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## Fracture mechanics，F. Erdogan

Fracture mechanics，

F. Erdogan.International Journal of Solids and Structures.Volume 37, Issues 1-2, January 2000, Pages 171-183doi:10.1016/S0020-7683(99)00086-4

## The past of Fracture

The past of Fracture Mechanics is not less interesting than its future. Particularly, I would like to share some nightmare thoughts on the cornerstone contribution to Fracture Mechanics made by Griffith (1920). My doubts are about both experiment and theory of Griffith. To make sure, however, that we understand the Griffith contribution correctly I quote Erdogan (doi:10.1016/S0020-7683(99)00086-4):

"The starting point of Griffith's studies was the then current knowledge based on ample observations in glass and metal wires, rods, and plates that there is an approximately two orders of magnitude difference between theoretical strength and bulk strength of solids, and his conclusion, based again on observations, that various forms of imperfections, defects and scratches are primarily responsible for this discrepancy. The obvious approach would then be to calculate the correct values of the maximum stresses around these defects and compare them with the theoretical strength of the material. This Griffith did by simulating the defects with an elliptical hole, the solution for which was previously given by Inglis. The results showed that the calculated maximum stress is independent of the absolute size of the flaw and depends only on the ratio of the semiaxes of the ellipse. These findings were in apparent conflict with the test results and led Griffith to conclude that ‘maximum stress' may not be an appropriate strength criterion and an alternative theory was needed. The basic concept underlying Griffith's new theory was that, similar to liquids, solids possess surface energy and, in order to propagate a crack (or increase its surface area), the corresponding surface energy must be compensated through the externally added or internally released energy. For a linear elastic solid, this input energy which is needed to extend the crack may be calculated from the solution of the corresponding crack problem. Using Inglis' solution for a uniformly loaded plate with an elliptical hole, Griffith calculated the increase in strain energy and, from the energy balance, obtained the stress corresponding to fracture… Thus, regarding the fracture of brittle solids, Griffith's major contributions were that he was able … to show that the fracture stress is dependent on the flaw size through the expression σ=m/√a, where m is a material constant. He also verified this expression by performing some carefully designed experiments on pressurized glass tubes and spherical bulbs containing cracks of various sizes."

My experimental objection to Griffith is concerned with the fact that Griffith considered big and not small cracks in his report "The phenomena of rupture and flow in solids" (1920). The smallness of the cracks is the very heart of the problem because the linear elasticity solution for stresses/strains at the edge of the crack is size-independent for small cracks only. If a crack is big then the elasticity solution 'feels' it. The central question concerning the Griffith experiments is whether his cracks are small? I am giving the data from tables II and III of the Griffith report. In the case of the cracked thin spherical bulb, four cases were considered with the ratio of the crack length to the bulb diameter 0.15/1.49; 0.27/1.53; 0.54/1.60; 0.89/2.00. It seems that only the first case can hopefully be called the small crack, or, better say, the upper bound for small cracks. All other cases present big cracks. The situation is even worse in the six cases of the cracked thin cylindrical tube where the ratios of the crack length to the tube diameter are 0.25/0.59; 0.32/0.71; 0.38/0.74; 0.28/0.61; 0.26/0.62; 0.30/0.61. You can see that the average crack length is approximately equal to the tube radius. How could Griffith regard them as small cracks? His cracks are big and cannot be considered as a basis for his theoretical developments...

My theoretical objections to Griffith are the following:

1. The fracture occurs in the fracture zone at the tip of the crack and it is controlled by the bulk stresses/strains in the zone. These stresses/strains are crack-size-independent for small cracks and the fracture should be crack-size-independent too.

2. If the surface energy introduced by Griffith is important then it should be included in stress analysis. Griffith, however, does not include the surface energy in stress analysis. Instead, he uses the surface energy in a failure calculation which is separated from stress analysis in a tricky way.

3. Griffith prediction of the unlimited increase of the critical failure stress with the decrease of the crack length is unphysical.

## thanks. I learn a lot from

thanks. I learn a lot from your comment.

## Hutchinson and Evans: Top-down approaches to fracture

This review article might be of interest.

J.W. Hutchinson and A.G. Evans, Mechanics of materials: top-down approaches to fracture. Acta Materialia 48, 125-135 (2000).