## Wednesday, 30 March 2011

### (5) Maximum Principal Strain theory

1.The maximum principal stress theory :
The theory associated with Rankine.

This theory is approximately correct for cast iron and brittle materials generally.

According to this theory failure will occur when the maximum principal stress in a system reaches the value of the maximum strength at elastic limit in simple tension.

\begin{align} \sigma_{max} &= \left(\frac{\sigma_x + \sigma_y}{2} \right )+\sqrt{(\frac{\sigma_x- \sigma_y}{2})^2+ {\tau_{xy}}^2} \end{align}
 Failure is predicted when either of the principal stresses, σ1 or σ2, equals or exceeds the yield strength, σyp, of the material.

σ1 = σyp   OR   σ2  = σyp

The design Factor of Safety 'n' for the two dimensional case
n =σyp/σ1
= Elastic Limit from tensile test / highest principal stress.

### 2.The Maximum Shear Stress Theory :

The theory associated with Tresca and Guest.This is very relevant to ductile metals. It is conservative and relatively easy to apply.
According to this theory, that failure occurs when a maximum shear stress attains a certain value.   This value being the value of shear strength at failure in the tensile test.
In this instance it is appropriate to choose the yield point as practical failure. If the yield point = σyand this is obtained from a tensile test and thus is the sole principal stress then,
the maximum shear stress tsy is easily identified as σy /2 .
tsy = σy /2
In the context of a complicated stress system the initial step would be to determine the principle stress i.e. σ1, σ1 and σ3-
then the maximum shear stress would be determined
The Maximum Shear Stress = τ max
will be Greatest of      ( σ1 - σ2 ) / 2   :    ( σ2 - σ3 ) / 2   :    ( σ1 - σ3 ) / 2

=     ( σ1 - σ3 ) / 2
The factor of safety selected would be
n = σy / ( 2 . τ max )    =   σy / ( σ1 - σ3 )

3.Strain Energy Theory :

The theory associated with Haigh.

This theory is based on the assumption that strains are recoverable up to the elastic limit, and the energy absorbed by the material at failure up to this point is a single valued function independent of the stress system causing it.

The strain energy per unit volume causing failure is equal to the strain energy at the elastic limit in simple tension.

The following relationship can be derived from this theory.
σy  - is the yield point in simple shear and
ν = poissons ratio.
Failure Occurs with this theory when,
12 + (σ22 + (σ3 ) 2 -  2v. (σ12 + σ23 + σ13 )     ≥   (σy2
4.Distortion Energy Theory :
Also called Shear Strain Energy Theory

This theory is also known as the Von Mises-Hencky theory. Detailed studies have indicated that yielding is related to the shear energy rather than the maximum shear stress..

Strain energy is energy stored in the material due to elastic deformation.   The energy of strain is similar to the energy stored in a spring.   Upon close examination, the strain energy is seen to be of two kinds : one part results from changes in mutually perpendicular dimensions , and hence in volume, with no change angular changes: the other arises from angular distortion without volume change.  The latter is termed as the shear strain energy , which has been shown to be a primary cause of elastic failure..

Failure is predicted when the distortional energy associated with the principal stresses, σ1,2,3 equals or exceeds the distortional energy corresponding to that for the yield strength, σy, of the material in uniaxial tension or compression.
1 - σ22 + (σ2 - σ32 + (σ1 - σ3 ) 2  = 2 σy2

Comparison of Theories:

Out of these four theories of failure, the maximum normal stress theory is only applicable for brittle materials, and the remaining three theories are applicable for ductile materials. Out of these three, the distortion energy theory provides most accurate results in majority of the stress conditions.The strain energy theory needs the value of Poisson’s ratio of the part material, which is often not readily available.The maximum shear stress theory is conservative. For simple unidirectional normal stresses all theories are equivalent, which means all theories will give the same result.