Today we will understand here the theories of failure, in strength of material, with the help of this post.

As we know very well that when a body or component or material will be subjected with an external load, there will be developed stresses and strains in the body or component.

As per hook’s law, stress will be directionally proportional to the strain within the elastic limit or we can say in simple words that if an external force is applied over the object, there will be some deformation or changes in the shape and size of the object. Body will secure its original shape and size after removal of external force.

Within the elastic limit, there will be no permanent deformation in the body i.e. deformation will be disappeared after removal of load.

If external load is applied beyond the elastic limit, there will be a permanent deformation in the body i.e. deformation will not be disappeared after removal of load. Component or material or body will be said to be failed, if there will be developed permanent deformation in the body due to external applied load.

Theories of failure help us in order to calculate the safe size and dimensions of a machine component when it will be subjected with combined stresses developed due to various loads acting on it during its functionality.

There are following theories as listed here for explaining the causes of failure of a component or body subjected with external loads.
5. The maximum shear strain energy theory

We will now understand here the maximum shear strain energy theory of failure with the help of this article.

The maximum shear strain energy theory of failure is also termed as maximum distortion energy theory or von mises criterion of failure. This theory is the best theory for failure of ductile material.

According to the maximum shear strain energy theory of failure, “The failure of a material or component will occur when the total shear strain energy per unit volume exceeds the limiting value of shear strain energy per unit volume i.e. value of shear strain energy per unit volume corresponding to the yield point of the material under tension test”.

Therefore in order to avoid the condition of failure of the component, total shear strain energy per unit volume must be below than the value of shear strain energy per unit volume corresponding to the yield point of the material under tension test.

Let us consider the condition of three dimensional stress systems. Where σ1, σ2 and σ3 are the principal stresses developed in the material under tension test.

Total shear strain energy per unit volume will be given as mentioned here

U = (1/12E) x [(σ1 – σ2) 2 + (σ2 – σ3) 2+ (σ3 – σ1) 2]

At the elastic limit under tension test, σt, 0 and 0 will be the principal stresses developed in the material.

Let us determine the value of shear strain energy per unit volume corresponding to the yield point of the material under tension test will be given by following equation as mentioned here

Ut = (1/2E) x [2 x σt2]
Where σt is the principle stress at elastic limit under tension test

### Condition of failure

Total shear strain energy per unit volume > Value of shear strain energy per unit volume corresponding to the yield point of the material under tension test

(1/12E) x [(σ1 – σ2) 2 + (σ2 – σ3) 2+ (σ3 – σ1) 2] > (1/2E) x [2 x σt2]
[(σ1 – σ2) 2 + (σ2 – σ3) 2+ (σ3 – σ1) 2] > [2 x σt2]

### Condition for safe design

Total shear strain energy per unit volume Permissible value of shear strain energy per unit volume

Permissible principle stress = Principle stress at elastic limit / F.O.S

Permissible principle stress = σt / F.O.S
σP = σt / F.O.S

For three dimensional stress systems
For tri-axial state of stress, we will have following equation
[(σ1 – σ2) 2 + (σ2 – σ3) 2+ (σ3 – σ1) 2] [2 x σt2]

For two dimensional stress systems
For two dimensional state of stress, σ3 will be zero and hence we will have following equation
[(σ1 – σ2) 2 + (σ2) 2+ (– σ1) 2] [2 x σt2]
[2σ1 2+ 2 σ2 2 – 2 σ2σ1] [2 x σt2]
σ1 2 + σ2 2 – σ2σ1 σt2

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We will now discuss the assumptions made in the theory of simple bending, in the category of strength of material, in our next post.

### Reference:

Strength of material, By R. K. Bansal