THE EXTENSION OF GRIFFITH’S ANALYSIS

1.   INTRODUCTION

 

In my earlier article on: “Energy release rate in the context of Fracture Mechanics”, https://www.dhirubhai.net/pulse/energy-release-rate-context-fracture-mechanics-ajay-taneja?trk=pulse_spock-articles, I had attempted to explain the process of crack growth in brittle materials – as elucidated through Griffith’s theory.

In the section 3 of the above article, it was made clear that crack growth begins when an energy equal to the resistance “R” – the surface energy of the material can be delivered to form two new crack surfaces as shown in the figure below;

Figure: Energy contribution to crack growth in constant displacement case

I had tried to bring out through the above article that the source of energy to crack propagation will be a mix of:

1. constant load case
2. constant displacement case

In a constant load case, the source of energy to crack propagation is through the work done on the system and in a constant displacement case, the source of energy to crack propagation is though the release of strain energy stored in the system.

I had pointed out that in case of brittle materials, the resistance “R” remains constant and the process of crack growth is spontaneous and failure catastrophic.

Figure: Crack resistance ‘R’ in brittle materials

 In this article, I’ve attempted to explain the about the extension of Griffith’s analysis to ductile materials as postulated by Irwin and Orawan. Section 2 of this article provides some details on the questionable aspects of Griffith’s analysis as raised and by Godier in 1968 and finally concluded by Kannimen later in 1985.

Section 3 of this article involves the extension of the concept of surface energy in brittle materials (which was well explained by Griffith) to ductile materials – as postulated by Irwin and Orawan.

Section 4 and 5 conceptually explain the “R” curve in brittle and ductile materials and the subtle differences in cases involving plane strain and plane stress cases.

Finally, to conclude, the discussion of sections 2 through 5, leads to the necessary and sufficient conditions of fracture in brittle and ductile materials. This forms the section 6 of this article.

2.   SOME QUESTIONABLE ASPECTS ON GRIFFITH’S APPROACH

 

Background of Griffith’s theory:

Griffith was able to explain about crack growth/propagation through the concept of surface energy in solids.

That is: similar to the surface tension in liquids, all solids are associated with surface energies or free energies. The energies are developed because atoms close to the surface of the solid behave differently from the atoms in the interior of the solid. The interior atoms are attracted or repulsed by the neighbouring atoms, more or less uniformly in all directions. On the contrary, the atoms on the free surface have neighbouring atoms on one side of the surface, thus, resulting in a different equilibrium.

Thus, there exist additional strains/stresses around the free surface of the solid and the energy associated with these additional strains/stresses is the “free surface energy”.

Griffith realised that a crack will not grow / propagate [assuming there is already an initial crack / flaw in the solid] until there was sufficient energy availability equal to the surface energy [or, the resistance ‘R’] of the material that would cause the formation of 2 new surfaces.

 

Griffith’s expression for fracture strength in a fixed-fixed plate:

It may be recalled, that, in my article on: “History of the development of Fracture Mechanics - https://www.dhirubhai.net/pulse/history-development-fracture-mechanics-ajay-taneja?trk=prof-post, the fracture strength (based on Griffith’s approach) was derived as;

Where;

σ = fracture strength of the material

E = modulus of elasticity

γ = surface energy of the material

a = critical crack length

Goodier / Kanninen’s questions, research and conclusions:

Goodier in 1968 had pointed out that Griffith had neglected the stresses due to surface tension whilst calculating the strain energy in the fixed plate problem.

It should be underscored that the strain energy expression used in deriving the above relation did not consider the normal traction due to surface tension. Kanninen in 1985 had in fact extended Griffith’s work to include the normal traction term in the computation of fracture strength and found that the fracture strength with improved definition of the boundary value problem was found to be;

This was comparable to the expression obtained through Griffith’s approach neglecting the stresses due to surface tension the definition of the boundary value problem. Thus, it could be concluded that it was justifiable to neglect the normal traction term in determining the fracture strength.

 

3.   SURFACE ENERGY IN BRITTLE AND DUCTILE MATERIALS

 

Griffith’s most work was confined to brittle materials wherein he observed (through his research / experiments on glass) that once the energy released was equal to surface energy of the material, the crack advancement was spontaneous and the failure catastrophic.

For most of the engineering materials (such as metals), a lot more energy was required in addition to the surface energy in order to grow a crack. Therefore, besides surface energy of solids some other mechanisms must be operative i.e. some additional energy must be required in order to grow a crack. The concepts involving the growth of crack in ductile materials are (to a large extent) the result of the research work of Irwin and Orawan.

At this point, one must recall, that once a ductile material reaches its yield point, it starts to deform plastically. This deformation is inelastic / irreversible i.e. once the structure is unloaded (after, it has deformed plastically), it will not regain its original shape. Such plastic deformation commences in regions of high stresses (stress being greater than the yield stress of the material), most commonly being in front of the crack tip. This plastic zone provides additional energy that adds to the resistance to crack growth.

It has been suggested by some authors – to clarify the concepts involving energy requirement for crack growth that it is useful to express surface energy as the sum of natural surface energy γn which the structure possesses even if it not subjected to plastic deformation (this was accounted by Griffith in his work for brittle materials) and the energy γp resulting from plastic deformation near the crack tip where the stresses are high;

That is;

For most ductile materials, the energy resulting from plastic deformation γis of several orders higher than the natural surface energy γ(see table below).

Table: Representative value of γγfor some structural materials:

Thus; it can be concluded, at this stage that the resistance “R” in ductile materials is many fold more than that of brittle materials. The resistance “R” in ductile materials is a function of plastic deformation and increases as the material deforms plastically. The plastic deformation provides “additional resistance”, thus, “additional energy requirements” to form 2 new free surfaces.

 

4.   THE ‘R’ CURVE IN BRITTLE MATERIALS

 

For a crack to grow, the crack resistance “R” is the energy required by the crack per unit increase in its area. For most engineering materials which are ductile in nature, the crack resistance increases with the crack length.

 

Crack resistance depends upon the plastic zone size. In a crack with large plastic zone size, higher energy is required to grow the crack as more material is subjected to plastic deformation.

 

In case of a brittle material, the size of the plastic zone in the vicinity of the crack tip is negligible and he growth of the plastic zone is negligible with the advancement of the crack. Consequently, the resistance “R” in brittle materials remains constant. As soon as the stress is large enough to make “G” overcome “R” the crack starts advancing. Crack growth is spontaneous and failure catastrophic.

Figure: R-curve of brittle materials

 

5.   THE ‘R’ CURVE IN DUCTILE MATERIALS: PLANE STRAIN AND PLANE STRESS CASES

 

In high strength ductile materials, the resistance to crack growth does not remain constant and increases as the crack grows.

The increase in resistance is smaller in case of thick plates (which emulate a plane strain condition closely) than in case of thin plates (which emulates a plane stress condition closely).

In case of thin plates (plane stress condition) , the value of the shear stress is very large which results in the generation of large size of the plastic zone whereas in thick plates the shear stress is considerably lower than in that of thin plates which does not allow the attainment of the plastic zone size as large as that in thin plates.

Thus, the increase in resistance in case of thin plates (plane stress case) is much steeper than in that of thick plates (plane strain case) as shown in the figure below;

Figure: ‘R’ curve in case of thick plates (plane strain case)

Figure: ‘R’ curve in case of thin plates (plane stress case)

Interpretation of the R curve in plane stress case:

Following can be interpreted by the “R” curve in case of plane stress cases;

1. Increase in the resistance is very steep (as the material continues to plasticize).
2. Fracture initiates at σ, G2. At this point, energy release rate G = R
3. If the material was brittle, at this point the further to this point the crack growth would have been very spontaneous and failure catastrophic.
4. The moment one comes to a plane stress case, the “R” curve is much steeper. If the stress is increased beyond σ, the energy release rate remains equal to “R”, but the fracture remains “stable”.
5. When, we say “stable”, it means that if the load is removed, the crack will have grown and stopped – it will not progress beyond that.
6. Whereas; if the load is continued to be increased, the crack will also continue to grow but will not lead to a catastrophic failure until a stage is reached where; σ = σ. At this point;

6.   CONCLUSIONS: NECESSARY AND SUFFICIENT CONDITIONS FOR FRACTURE

 

To conclude this article/blog, I find it worthwhile to do so, by examining the necessary and sufficient conditions for fracture.

The equation;

i.e. the energy release rate “G” equals the resistance “R” – the surface energy of the material is the necessary as well as the sufficient condition for fracture in case of brittle materials.

But, this is not the case when we extend Griffith’s analysis to ductile materials. As I’ve explained in the section 5 of this article, there arises another condition;

i.e. the slope of “G” and “R” should also match for fracture to become unstable - thus leading to failure.

Thus, in this article, I’ve attempted to explain the extension of Griffith’s analysis – which includes the concept of surface energy related to plastic deformation, effect of thickness on the fracture strength, necessary and sufficient conditions for fracture.