DISTORTION ENERGY THEORY OF FAILURE
A THEORY OF FAILURE APPLICABLE TO DUCTILE MATERIALS
Statement of the theory
When Yielding occurs in any material, the distortion strain energy per unit volume at the
point of failure equals or exceeds the distortion strain energy per unit volume when
yielding occurs in the tension test specimen.
The theory applies to ductile materials only, because it is based on yielding.
The three dimensional (triaxial) stress situation.
In the three dimensional stress situation, the state of stress at a particular location is fully
defined by three principal stress  1 ,  2 ,  3 .
Strain energy at a location of the element
The strain energy at a particular location of the element can be segregated into three
categories, namely:
(a) Total strain energy per unit volume of the stressed element, arising from the principal
stresses  1 ,  2 ,  3 .
(b) Strain energy per unit volume arising from the change of volume caused by a
hydrostatic stress, which is uniform in all three directions
(c) Strain energy per unit volume arising from distortion of the element, and which can be
considered as being the difference between (a) and (b).
ELASTIC STRESS-STRAIN RELATIONS
Uni-Axial stress
This is the case of a single principal stress  1 .
Principal strains are then given by the expressions
1 
1
E
,
 2  
1
E
,
 3  
1
E
Where,
1  Principal strain in the direction of the principal stress
  Poisson’s ratio for the material
E  Modulus of elasticity for the material
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Bi-Axial stress situation
In this case the stress situation consists of two principal stresses  1 ,  2 , and the strains1 are
given by:
1 
1
 1   2  ,
E
2 
1
 2  1  , and  3   1   1   2 
E
E
Tri-Axial stress situation
This is the case of three principal stresses  1 ,  2 ,  3 , and the strains in the directions of the
principal stresses are then given by:
1
 1    2   3 
E
1
 2   2    1   3 
E
1
 3   3    1   2 
E
1 
(1)
(2)
(3)
ENERGY PER UNIT VOLUME AT STRESS LOCATION
Total strain energy U
The total strain energy is the strain energy caused by the three principal stresses  1 ,  2 ,  3 .
It is given by
1
1
1
 11 +  2 2 +  3 3
2
2
2
U 
(4)
Substituting the three strains 1 ,  2 , and  3 in equations (1),(2) and (3) into equation (4)
yields:
U 


1
 12   2 2   32  2  1 2   2 3   1 3 
2E
(5)
Strain Energy due to Change of Volume (Hydrostatic stress) only
The stress that causes change of volume only (hydrostatic stress) may be considered as the
average of the three principal stresses  av , and derived from the expression:
 av 
1   2   3
(6)
3
Substituting for the hydrostatic stress  av , into equation (5) yields:
Uv 
1


1
3 av 2  2 3 av 2
2E

(7)
Mechanical Engineering Design; Shigley, Joseph, pg 124, McGraw Hill, Seventh Edition, 2004
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3 av 2
1  2  = 31  2  av2
2E
2E
Uv 
(8)
Substituting the value of  av from equation (6) into equation (8) yields:
Uv 
Uv 
Uv 
31  2    1   2   3 


2E 
3

1  2  
6E
1
  2   3 2 =
2
=
31  2 
 1   2   3 2
9 * 2E
1  2  
6E
2
1

  2 2   32  2 1 2   2 3   1 3 


1  2 2
 1   2 2   3 2  2 1 2   2 3   1 3 
6E
(9)
This U v is the strain energy per unit volume caused by the uniform (hydrostatic) stress,
which is part of the three principal stresses  1 ,  2 ,  3 .
Distortion Energy at the location of principal stresses  1 ,  2 ,  3
The distortion energy can then be obtained as the difference between the total strain energy at
the location of principal stresses, and the strain energy due to the hydrostatic portion of the
stresses at the same location. The distortion energy is then derived from the expression:
U d  U -Uv
Where,
U d  Distortion energy in the element at the location of principal stresses  1 ,  2 ,  3


1
 12   2 2   32  2  1 2   2 3   1 3 
2E
1  2 2
Uv 
 1   2 2   3 2  2 1 2   2 3   1 3 
6E
U 

Therefore,

(5)

(9)



Ud =
1
1  2 2
 12   2 2   32  2  1 2   2 3   1 3   1   2 2   3 2  2 1 2   2 3   1 3 
2E
6E
Ud =
1
3  12   2 2   3 2  6  1 2   2 3   1 3  6E







1
 12   2 2   32  2 1 2   2 3   1 3   2 *  12   2 2   32  2 * 2 1 2   2 3   1 3 
6E






Ud =
2
2
2
2
2
2

1 3  1   2   3  6  1 2   2 3   1 3    1   2   3


6 E  2          2 *  2   2   2  2 * 2        
1
2
2
3
1
3
1
2
3
1
2
2
3
1
3


Ud =
2
2
2
1 2  1   2   3  2  1 2   2 3   1 3  

6 E  2          2 *  2   2   2
1 2
2 3
1 3
1
2
3

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
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

Ud =
1
 12   2 2   32 2  2    1 2   2 3   1 3 2  2
6E
Ud =
12  2 
 12   2 2   3 2   1 2   2 3   1 3 
6E
Ud =

1    
3E
2
1
But

2
1




  2 2   3 2   1 2   2 3   1 3 

  2 2   3 2   1 2   2 3   1 3  =
Therefore
Ud =
Ud =
(10)
 1   2 2   2   3 2   1   3 2
2
1       2     2     2
1
2
2
3
1
3
2 * 3E


(11)
1    

(12)
6E
1
  2 2   2   3 2   1   3 2
THE CASE OF SIMPLE TENSION TEST WHEN YIELDING OCCURS
For the simple tension test specimen, the three principal stresses when yielding occurs are:
1 = S y ,
 2 =0,
 3 =0
Substituting for the principal stresses in equation (12) yields
Ud =
1    S
y
Ud =
1    2S
y
6E
6E


 
 0 2  0  02  S y  0 2
2

(13)
THE CASE OF THREE DIMENSIONAL STRESS WHEN YIELDING OCCURS
The distortion energy theory of failure states:
When Yielding occurs in any material, the distortion strain energy per unit volume at the
point of failure equals or exceeds the distortion strain energy per unit volume when
yielding occurs in the tension test specimen.
This can be restated that when yielding occurs in any situation:
Ud =
1    
6E
1
  2 2   2   3 2   1   3 2
EQUALS
Ud =
1    2S
6E
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y
2


(12)
(13)
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 1   2 2   2   3 2   1   3 2 = 2S y 2
    2     2     2 
2
2
3
1
3
 1

2


= Sy
(14)
EQUIVALENT (Von-Mises) STRESS
The expression on the left hand side of equation (14) is therefore considered as the
equivalent stress  e , which causes failure by yielding. The equivalent stress is then given
by:
    2     2     2 
1
2
2
3
1
3

2


e= 
(15)
The equivalent stress  e is also referred to as Von Mises stress.
DESIGN EQUATION BASED ON THE DISTORTION ENERGY THEORY
This is derived by adjusting the yield strength of the material in simple tension with an
appropriate factor of safety f .s. The design equation then becomes:
    2     2     2 
1
2
2
3
1
3

2


e= 
=
Sy
f .s.
(16)
APPLICATION OF THE DESIGN EQUATION
The principal stresses  1 ,  2 ,  3 are first determined by stress analysis. Such analysis
describes the principal stresses as a function of the load carried, and the geometry and
dimensions of the machine or structural element.
The equivalent stress in the design equation is then expressed in terms of the dimensions of
the machine or structural element, while the right hand side is the tensile yield strength of
the material.
The factor of safety is simply a number chosen by the designer. The factor of safety together
with the strength of the material, gives the working2 (design, allowable) stress expected in the
machine part. The solution to the design equation then gives the minimum dimensions
required to avoid failure of the element by yielding.
Working Stress, page 527,Handbook, Metals Engineering –Design, American Society of Mechanical
Engineers (ASME)
2
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