FRACTURE MECHANICS

PRESENTATION ON
GRIFFITH AND IRWIN THEORY
 Introduction
Fracture mechanics is the field of mechanics concerned with the study of the propagation of
cracks in materials.

There are three ways of applying a force to enable a crack to

propagate:

a) Mode I fracture – Opening mode (a tensile stress

normal to the plane of the crack),
b) Mode II fracture – Sliding mode (a shear stress acting
parallel to the plane of the
crack and perpendicular to the crack front), &
c) Mode III fracture – Tearing mode (a shear stress acting
parallel to the plane of the crack m
and parallel to the crack front).
Source:Fracture Toughness Determinations by Means of
Indentation Fracture (Book) By Enrique Rocha-Rangel
 Fracture Modes
• Classification is based on the ability of material to experience plastic deformation.

• Ductile fracture
Accompanied by significant plastic deformation

• Brittle fracture
Little or no plastic deformation
Sudden, Catastrophic
 Griffith Criterion
• Explains the failure of brittle materials.

Motivation
 The stress needed to fracture bulk glass is around 100 MPa.

 The theoretical stress needed for breaking atomic bonds of glass is approximately 10,000 Mpa.

Theory
 Experiments on glass fibers suggested that the fracture stress increases

as the fibre diameter decreases due to the presence of microscopic flaws

in the bulk materials.

Source : DTD Handbook Section 2.2.6.1.1.

Experiment

 Griffith introduced an artificial flaw in his experimental glass specimen.

 Experiment showed that the product of the square root of the flaw length (a) and the stress at fracture (σf) was
nearly constant, which is expressed by the equation:

Contradiction

 Linear elasticity theory predicts that stress (and hence the strain) at the tip of a sharp flaw in a
linear elastic material is infinite.
 Griffith Energy Balance
• The energy approach states that crack extension (i.e., fracture) occurs when the energy available for crack growth is
sufficient to overcome the resistance of the material.

• The material resistance may include the surface energy, plastic work, or other types of energy dissipation associated
with a propagating crack.

According to the first law of thermodynamics, when a system goes from a non-equilibrium state to equilibrium, there is a
net decrease in energy-
• A crack can form (or an existing crack can grow) only if the process causes the total energy to decrease or remain
constant.
• Thus the critical conditions for fracture can be defined as the point where crack growth occurs under equilibrium
conditions, with no net change in total energy.
• Consider a plate subjected to a constant stress σ which contains a
crack 2a long .
Assumption:
a) Plate width>>2a and
b) Plane stress conditions.

For the crack to increase-

Sufficient potential energy must be available in the plate to overcome
the surface energy of the material.

Fig.1. Cracked specimen

 The Griffith energy balance for an incremental increase in the crack area dA, under equilibrium conditions, can be
expressed in the following way:

Where
E = total energy
Π= potential energy supplied by the internal strain energy
and external force
Ws= work required to create new surfaces

For the cracked plate illustrated in Fig.1, Griffith used the stress analysis of Inglis [3] to show that

Πo= potential energy of an uncracked plate and

B = plate thickness.
Since the formation of a crack requires the creation of two surfaces, Ws is given by

γs=Surface energy of the material contd.

Thus,

(1)

and
(2)

Equating eqn.(1) & (2), & solving for fracture stress, we get,

Which shows ,

Hence,
 Irwin Theory or Modified Griffith theory
Motivation
For ductile materials such as steel although the relation still holds, the surface energy (γ) predicted by
Griffith's theory is usually unrealistically high.

Theory

 In ductile materials (and even in materials that appear to be brittle), a plastic zone develops at the tip of the crack.

 As the applied load increases, the plastic zone increases in size until the crack grows and the elastically strained
material behind the crack tip unloads.

 The plastic loading and unloading cycle near the crack tip leads to the dissipation of energy as heat. Hence, a
dissipative term has to be added to the energy balance relation devised by Griffith for brittle materials.
Irwin’s strategy was to partition the energy into two parts:
a) The stored elastic strain energy which is released as a crack grows. This is the thermodynamic driving force for
fracture.
b) The dissipated energy which includes plastic dissipation and the surface energy (and any other dissipative forces
that may be at work). The dissipated energy provides the thermodynamic resistance to fracture. Then the total
energy is

γ = surface energy and

Gp = plastic dissipation (and dissipation from other sources)
per unit area of crack growth.

The modified version of Griffith's energy criterion can then be written as

Source: Wikiwand Fracture mechanics

 References
1. Griffith, A. A. (1921), "The phenomena of rupture and flow in solids" , Philosophical Transactions of the
Royal Society of London, A, 221: 163–198.

2. Irwin G (1957), Analysis of stresses and strains near the end of a crack traversing a plate, Journal of
Applied Mechanics 24, 361–364.

3. Inglis, C.E., “Stresses in a Plate Due to the Presence of Cracks and Sharp Corners.” Transactions of the
Institute of Naval Architects, Vol. 55, 1913, pp. 219–241.

4. T.L. Anderson (1995). Fracture Mechanics: Fundamentals and Applications. CR Press. ISBN 970849316562.
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