Note on Energy bands on crack

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Energy bands on the crack

All brittle materials contain population of cracks and flows that have a variety of shapes, size and geometrics (orientation).

When magnitude of tensile stress at the tip of one of these crack or flow exceeds critical stress i.e.

$$\sigma_c=\sqrt\frac{2E\gamma_s}{\pi a}\dotsm (1)$$


\(\gamma_s\)=specific surface energy

E= Young’s modulus of elasticity

a=half length of internal flaw or full length of surface flaw

What is the Griffith criterion for the formation of cracks and its propagation to result in fracture in

a) Brittle material

b) ductile material

In ductile material the condition for condition of crack and its propagation is that the value of \(\gamma_s\) is replaced by \(\gamma_s+\gamma_p\) in equation (2) where,\(\gamma_p\) is energy for plastic deformation.

$$(\sigma_c)_{ductile}=\sqrt{\frac{2E(\gamma_s+\gamma_p)}{\pi a}}\dotsm(3)$$ and tensile stress must be greater than this value of critical stress.

Time dependent stress- fatigue

When fluctuating or cyclic stress is applied on material or metal the failure can occur at loads considerably lower than tensile or yield strength of the material under state load, this type of failure is known as fatigue. Almost 90% of causes of failure in metal structure such as bridge, aircraft and machine component is due to cyclic stress or fatigue. Fatigue failure is brittle like even in normally ductile materials thus it is sudden and catastrophic.

The cause of fatigue may axial stress that may be compression, tension. Flexural stress due to bending or torsional stress is due to twisting.

Fatigue failure proceed in three distinct stages

  1. Crack initiation in the areas of stress concentration
  2. Incremental crack propagation
  • Final catastrophic failure

Cyclical stresses (periodic stresses)

fig:cyclic stress
fig:cyclic stress(

Periodic stress is time independent and it is characterize by maximum stress(\(\sigma_{max.}\)), minimum stress (\(\sigma_{min.}\)), mean stress (\(\sigma_m\)), range of stress (\(\sigma_r\)), stress amplitude (\(\sigma_a\)) and stress ratio.


Mean stress=\(\sigma_m=\frac{\sigma_{max}+\sigma_{min.}}{2}\)

Range of stress=\(\sigma_r=\sigma_{max}-\sigma_{min}\)

Stress amplitude=\(\sigma_a=\frac{\sigma_r}{2}=\frac{\sigma_{max}-\sigma_{min}}{2}\)

Stress ratio =R=\(\frac{\sigma_{min}}{\sigma_{max}}\)

In the above diagram tensile stress are positive and compressive stress are negative.

S-N curve

S=periodic stress

N=number of cycle o failure

Fatigue properties of material are tested in laboratory by rotating bending test in fatigue test apparatus. The result of the experiment is commonly plotted as stress along Y-axis along number of cycle to failure along X-axis.

fig: S-N curve
fig: S-N curve(

Low cycle fatigue

In this low cycle fatigue, the amount of load required is high and material easily transform from plastic as well as elastic deformation.

High cycle fatigue

In high cycle fatigue, the number of cycle is high so low loads will results in elastic deformation.

What is endurance limit?

It is the maximum stress applied below which the material never fails no matter how large the number of cycle is. The S-N curve is different for different types of material. In most alloy S decreases continuously with N.

Fatigue strength

It is the stress at which fracture occurs after a specified number of cycle. In figure \(10^7\) is approximate value of fatigue strength.

It is the for the fatigue life. Number of cycle failure of metal at a specified stress level.


Crack initiation and propagation

The three different stages of fatigue failure are:

  1. Crack initiation in area of stress concentration
  2. Increment crack propagation
  3. Final rapid crack propagation after crack reaches critical size


The total number of cycle for the failure of metal or material is the sum of cycle in the first and second stages.


\(N_i\)= number of cycles for crack initiation

\(N_p\)=number of cycles for crack propagation

\(N_f\)= number of cycles to failure

High cycle fatigue

\(N_i\) is relatively high with increasing stress level, \(N_i\) decreases and \(N_p\) dominates.

Crack is always initiates at the side of stress concentration. Such as microvoids, scratches, indents, interior corner, dislocation etc.


Callister, W.D and D.G Rethwisch. Material Science and Engineering. 2nd. New Delhi: Wiley India, 2014.

Lindsay, S.M. Introduction of Nanoscience . New York : Oxford University Press, 2010.

Patton, W.J. Materials in industry . New Delhi : Prentice hall of India, 1975.

Poole, C.P. and F.J. Owens. Introduction To Nanotechnology. New Delhi: Wiley India , 2006.

Raghavan, V. Material Science and Engineering. 4th . New Delhi: Pretence-Hall of India, 2003.

Tiley, R.J.D. Understanding solids: The science of Materials. Engalnd : John wiley & Sons , 2004.


$$\sigma_c=\sqrt\frac{2E\gamma_s}{\pi a}$$

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