The different failure theories help in describing the mechanical components when it starts showing inelastic behaviour. Different types of stress are put in components during its functioning period. This stress can be very high and could be subjected to failure of functioning of the component. Therefore, Failure theories help in understanding and bordering the safer dimensions of functioning of these mechanical components. It is believed that any mechanical component is bound to face a failure if the elastic limit is succeeded by the stress put in the component. This action will create a permanent deformation of the mechanical component. 

Let us look at the modes of failure concerning the material.

Mode of Failure and others

The failure in materials is bound to happen in two modes- Ductile Failure and Brittle Failure. These failures depend on loading over the mechanical component. 

• Brittle Failure- Brittle Failure points out that the material failed because of a sudden fracture. The material does not deform, in this case. Instead, it breaks down suddenly. Examples of Brittle material may include glasses, types of polymers, ceramics, etc. 
• Ductile Failure- It is one of the most intriguing and crucial concepts in material engineering. Ductile Failure includes the deformation of the material, as a whole. This process includes huge quality of energy and slow propagation before fracturing. The synonym of Ductile Failure can be Ductile Fracture.

The principal failing factor of material is- Strain, Energy, and Stress.

Before moving to the failure theories, let us discuss the uniaxial, biaxial, and triaxial loadings briefly. 

• Uniaxial Loading is the stress that works only in one direction or with one axis. Various metals are subjected to Uniaxial stress. 
• Biaxial Loading in the stress that works via two axes or in two directions.
• Axial Loading is the stress that is put directly along the axes of a structure.
• Triaxial Loading is the stress that works in three directions or with three axes.

Now that we are aware of the modes of failure, let us discuss the failure theories briefly. 

Failure theories

As we already know,  failure theories help in describing the mechanical components when it starts showing inelastic behaviour. These theories help in predicting the capacity of a material to stand against the unlimited combinations of non-standard loads. Failure theories are used to assess the safety of mechanical components in the engineering field. 

Different theories are induced and proposed by scientists and physicists for various modes of failure and types of materials. The failure theories include-

• Rankin’s Theory- Rankin’s theory is commonly known as the Maximum Principal Stress Theory.
• Tresca and Guest’s theory- Tresca and Guest’s theory is widely known as the Maximum shear stress Theory.
• Saint Venant’s theory- Saint Venant’s theory is widely known as the Maximum Principal Strain theory.
• Haigh’s theory- Haigh’s theory is widely known as the Total Strain Energy theory. 
• Huber Von Mises and Hencky’s theory- Huber Von Mises and Hencky’s theory is widely known as the Maximum Shear Strain Energy theory. It is also known as the Maximum Distortion Energy theory.

Let us discuss the Maximum shear stress theory or Tresca’s Theory of failure. 

Tresca’s Theory of failure

Maximum Shear Stress theory is one of the failure theories. It is also known as Tresca’s theory of Failure. Just like all other failure theories, the Maximum Shear Stress theory also is helpful in the safe designing of the mechanical components. However, this theory is suitable only for Ductile materials. 

The Maximum Shear Stress theory states that the failure is bound to happen when the shear stress at uniaxial loading succeeds the maximum shear stress. 

That is, 

Maximum shear stress ≤ Uniaxial stress

= t max ≤  σn

In the shear stress maximum in biaxial load, any one of the main stress will be zero

Therefore, the relation will be-

σa – σb ≤ σn

σb – 0  ≤ σn = σb ≤ σn

σa – 0 ≤ σn = σa  ≤ σn

Here, we have considered σc = 0. 

Therefore, the shear stress maximum in triaxial load will be given by-

σa – σb ≤ σn

σb – σc ≤ σn

σa – σc ≤ σn

In the above relation, σa, σb, and σc are the main stresses that are put.

Conclusion

Failure theories help in describing the mechanical components when it starts showing inelastic behaviour. Different modes of stress are put in components. This stress can be very high and could be subjected to failure of functioning of the component. The theories which help in predicting the capacity of a material to stand against the unlimited combinations of non-standard loads are widely known as Failure theories. Maximum Shear Stress theory is one of the failure theories. Maximum Shear Stress theory also is helpful in the safe designing of the mechanical components. However, this theory is suitable only for Ductile materials.