1. Fracture of Solids Theoretical tensile strength of a solid U(r) a r

2. Fracture of Solids Work of fracture F(r) a 2g r
Work of fracture for a defect-free solid

3. Stress Concentrations
Airy stress function a

4. Stress Concentrations
Elliptical flaw 2a 2b
The radius of curvature at a is, so
for
Def: Stress Concentration factor, Kt
For the circular flaw

5. Stress field of an Elliptical flawy ao bo x
ao is the ellipse defined by the flaw. 2a
The Cartesian coordinates, x, y are
connected to the elliptical coordinates
a, b by a b

6. Stress field of an Elliptical flaw
r is the distance along y = 0. The relative values of r/a and b/a compared to
1 determine the behavior of and define regions of interest.
In the region r/a < 1 the leading term (dominant) including the bluntness
contribution of b/a is,
r < a
As b/a  0 (a sharp “crack”) the stress field decays as (a/r)1/2.
In the region, r/a >1
and the stress field scales as (a/r)2

7. Sharp Cracks 2a
Central crack of length 2a in an infinite plate under uniform tension. The
leading terms for r <<a,
Def: Stress Intensity Factor, K

8. Sharp Cracks
The stress intensity factor defines the strength of the crack in much the same way as the Burgers vector defines the strength of a dislocation.

9. Crack Loading Modes Mode III: Out-of plane or longitudinal shear
Opening
Mode II:
In-plane shear

10. Units of K
Y = numerical factor depending on geometry and loading
MPa

12. Sharp Cracks Mode I stress field Cartesian coordinates
Cylindrical polar coordinates

13. Sharp Cracks Mode II stress field Cartesian coordinates
Mode III stress field
Cartesian coordinates

14. Griffith theory of fracture
Consider a crack of length 2a
in a 2D plate of infinite extent
under external boundary tractions. 2a da
2(a+da)
The total energy, UT of the
system is composed of 3 terms.
Here U load is the work done
by the applied loads Ti on the
system.
We will subsequently show that

15. Griffith theory of fracture
For a thin plate under load the excess elastic energy in the system owing to
the presence of the crack is,
The surface energy of the crack system is
The total energy may now be written as U a a*

16. Griffith theory of fracture
We can find the position of the energy for fracture, corresponding to the
maximum in UT,
For the case of plain strain

17. Griffith theory of fracture
Def: Crack extension force
G is the (negative) change of potential
energy per unit crack extension per unit
width.
Energy/area = Force/length

18. Griffith theory of fracture
Mechanical work during crack extension 2a da
2(a+da)
Apply tractions loading the crack system
During crack extension the work done is

19. Griffith theory of fracture
Mechanical work during crack extension
The elastic energy at location 3 is just
The change in elastic energy is just
Then

20. Griffith theory of fracture
Plane Strain
We can rearrange this to read
What is a “typical value” of K at which fracture of a brittle solid is predicted
E ~ 1011 Pa, g ~ 1Jm-2
Many materials exhibit critical K values times larger then this. Why?

21. Crack extension force a+da a P D Def: Crack extension force mg
Crack growth under fixed load

22. Crack extension force a+da a P D Def: Crack extension force
Crack growth under fixed displacement

23. Crack extension force Crack growth under fixed load
Crack growth under fixed displacement

24. 1st estimate of the plastic zone radius
At edge of plastic zone

25. Reminder: Mode I Stress Intensity Factor
y (x2) r q
x (x1)
z (x3)

26. Von Mises criterion (It helps to obtain principal stresses, first)

27. Identify elastic-plastic boundary, r(q)
Principal stresses
Identify elastic-plastic boundary, r(q)
Substitute these stresses into Von Mises criterion & solve for r as function of q
where

28. SIZE & SHAPE OF THE PLASTIC ZONE
1.0 x y
SIZE & SHAPE OF THE PLASTIC ZONE
PLANE STRESS r(q)/ry
PLANE STRAIN r(q)/ry