1. LINEAR ELASTIC FRACTURE MECHANICS
BDC40403
DR. AL EMRAN ISMAIL

2. PRINCIPAL OF FRACTURE MECHANCIS
Fracture mechanics is the study of mechanical behavior of cracked materials subjected to an applied load.
Irwin developed the field of fracture mechanics using the early work of Inglis, Griffith, and Westergaard.
Essentially, fracture mechanics deals with the irreversible process of rupture due to nucleation and growth of cracks.

3. PRINCIPAL OF FRACTURE MECHANCIS
The formation of cracks may be a complex fracture process, which strongly depends on the microstructure of a particular crystalline or amorphous solid, applied loading, and environment.
The microstructure plays a very important role in a fracture process due to dislocation motion, precipitates, inclusions, grain size, and type of phases making up the microstructure.
All these microstructural features are imperfections and can act as fracture nuclei under unfavorable conditions.

4. PRINCIPAL OF FRACTURE MECHANCIS
For instance, Brittle Fracture is a low-energy process (low energy dissipation), which may lead to catastrophic failure without warning since the crack velocity is normally high.
Therefore, little or no plastic deformation may be involved before separation of the solid.
On the other hand, Ductile Fracture is a high- energy process in which a large amount of energy dissipation is associated with a large plastic deformation before crack instability occurs.
Consequently, slow crack growth occurs due to strain hardening at the crack tip region.

5. STRESS CONCENTRATION FACTORS
Consider an infinite plate containing an elliptical hole with major axis 2a and minor axis 2b as shown in below, where the elliptical and Cartesian coordinates are (,) and (x, y) respectively.

6. STRESS CONCENTRATION FACTORS
The equation of an ellipse, the Cartesian coordinates, and the radius of curvature are given below, respectively.
(x2/a2)+(y2/b2) = 1
where, x = c cosh cos and x = c sinh sin
The radius of the ellipse is defined in terms of semi-axes by,  = b2/a.

7. STRESS CONCENTRATION FACTORS
It is desirable to derive the maximum elastic stress component at an elliptical crack tip along the major axis when the minor axis.
Inglis who derived the elastic stress distribution in an infinite plate studied this problem.
When a remote stress is applied perpendicular to the major axis 2a of the thin plate, the stress distribution expression for tan elliptical hole shown in figure above.

8. STRESS CONCENTRATION FACTORS
Where,

9. STRESS CONCENTRATION FACTORS
The resultant maximum axial stress at the edge of the ellipse is
Here, Kt is the stress-concentration factor and  is the nominal stress or the driving force. If a = b then Kt = 3 and max = 3 for a hole.
On the other hand, if b0 then max is singular and it is meaningless, and a very sharp crack is formed since 0.

10. STRESS CONCENTRATION FACTORS
In addition, Kt is used to analyze the a stress at a point in the vicinity of a notch having a radius  >> 0.
However, if a crack is formed having   0, the stress field at the crack tip is defined in terms of the stress-intensity factor, KI instead of the stress-concentration factor, Kt.

11. GRIFFITH CRACK THEORY
Griffith noted in 1921 that when a stressed plate of an elastic material containing a crack, the potential energy decreased and the surface energy increased.
Potential energy is related to the release of stored energy and the work done by the external loads.
The “surface energy” results from the presence of a crack as shown in next figure.
This energy arises from a non-equilibrium configuration of the nearest neighbor atoms at any surface in a solid.

12. GRIFFITH CRACK THEORY

13. U = Uo – (a22B)/E + 2(2aBs)
GRIFFITH CRACK THEORY
Thus, the total potential energy of the system is given by:
U = Uo – Ua +U
U = Uo – (a22B)/E + 2(2aBs)
where,
U = Potential energy of the cracked body
U0 = Potential energy of uncracked body
Ua = Elastic energy due to the presence of the crack
U = Elastic surface energy due to the formation of crack surfaces
s = Specific surface energy
 = 1 (plane stress) & 1-2 (plane strain)

14. GRIFFITH CRACK THEORY
Take the first order partial derivative of the equation above to have the equilibrium condition with respect to crack length, a:
dU/da = 0
2s = (a2)/E
a = (2sE/)
KI = a
KI is called stress intensity factor
KC is called critical stress intensity factor or fracture toughness
KIC is called plain strain fracture toughness and it is a mechanical property

15. GRIFFITH CRACK THEORY Combine  = b2/a and max = (1+2a/b) to give:
For sharp crack, a >>  and (a/) >> 1, yield the following:
max = (2(a/))
The theoretical Kt:
Kt = max /  = (2(a/)
Kt   as   0.
Therefore, KI is the most useful approach to analyze structural component containing sharp cracks.

16. Strain-energy release rate
It is well-known that plastic deformation occurs in engineering metal, alloys and some polymers.

17. Strain-energy release rate
Due to this fact, Irwin and Orowan modified Griffith’s elastic surface energy expression 2s = (a2)/E, by adding a plastic deformation energy or plastic strain work in the fracture process.
For tension loading, the total elastic-plastic strain-energy is known as the strain energy release rate which is the energy per unit crack surface area available for infinitesimal crack extension. Thus:

18. Strain-energy release rate
2s = (a2)/E 2(s + p) = (a2)/E GI = (a2)/E GI = (a2)/E’  = (E’GI/a) GI = K2I/E’ GI = The material’s resistance (R) to crack extension or crack driving force GIC = Critical strain energy release rate or fracture toughness

19. Strain-energy release rate

20. Modes of loading

21. Modes of loading

22. Specimen geometries (THROUGH-THE-THICKNESS CENTER CRACK)

23. Specimen geometries (THROUGH-THE-THICKNESS CENTER CRACK)

24. Specimen geometries (THROUGH-THE-THICKNESS CENTER CRACK)

25. Specimen geometries (THROUGH-THE-THICKNESS CENTER CRACK)

26. Specimen geometries (THROUGH-THE-THICKNESS CENTER CRACK)

27. PRINCIPLE OF SUPERPOSITION
It is very common in machine parts to have a combination of stress loadings that generate a combination of stresses on the same part.
Each case has the same stress-field distribution described by the local stresses such as x, y and x.
Consequently, the principle of superposition requires that the total stress intensity factor, KI be the sum of each stress intensity factor components.

28. PRINCIPLE OF SUPERPOSITION

29. ELLIPTICAL CRACKS
According to Irwin’s analysis on an infinite plate containing an embedded elliptical crack (A figure below) loaded in tension, the stress intensity factor is defined by:

30. ELLIPTICAL CRACKS

31. ELLIPTICAL CRACKS  can be expended as follows:
Neglecting higher order of , then:
Inserting  into KI in page 29, then:

32. ELLIPTICAL CRACKS

33. PART-THROUGH THUMBNAIL SURFACE FLAW
The KI for a plate of finite width being subjected to a uniform and remote tensile stress is further corrected as below:
M = Magnification correction factor
Mk = Front face correction factor
 = Applied hoop or design stresses
Q = Shape factor for a surface flaw
P = Internal pressure (MPa)

34. PART-THROUGH THUMBNAIL SURFACE FLAW

35. PART-THROUGH THUMBNAIL SURFACE FLAW
Combining and  into Q, therefore:
Manipulating the equation above, then:
Plotting the equation above to yield:

36. PART-THROUGH THUMBNAIL SURFACE FLAW

37. PART-THROUGH THUMBNAIL SURFACE FLAW
The magnification correction factor takes the form: