Principal stress is the maximum or minimum normal stress which may be developed on a loaded body. It is classified as major principal stress and minor principal stress. On the plane of principal stress shear stress value is termed as zero.
When principal stress gets on the major principal plane, it is called major principal stress and when it is found on the minor principal plane, it is known as minor principal stress. These are denoted as σ_{1 }and σ_{2}, respectively.
Shear stress is a sloping applied force that causes deformation due to lateral load. In any stress block surface, there is applied shear stress over the plane, but to stabilize the body there, we need to apply complementary shear stress. This is known as complementary shear stress.
The principle of complementary shear stress is when the same intensity but opposite direction shear stress is applied over the surface of the body, it creates a couple and this couple stabilizes the body. This balancing couple mechanism is known as the principle of complementary shear stress.
Principal stress is calculated on the principal plane as maximum stress is called major principal stress, and minimum stress on the principal plane is called minor principal stress.
Here;
Where
Maximum shear stress (τ_{max}) = (Maximum principal stress Minor principal stress)/2 =R
Here maximum shear stress is the magnitude of that point with an equal radius in a mohr circle, and principal stress is the end point of the circle’s diameter.
Here we take only twodimensional theory in which we consider σ_{1} ( the maximum principal stress ) and σ_{2} (the minimum principal stress). But in the case of threedimensional theory, we take σ_{1}, σ_{2 }and σ_{3}. In this case, we decide on the major or minor principal stress by using principal stress theory or Rankine, lames or maximum principal stress theory.
When the applied load needed to be calculated under the design criteria, the total applied load should be less than the ultimate yield capacity of material divided by the factor of safety. This theory is also a part of principal stress theories and its also known as Rankine’s theory, lame’s theory and reciprocal theory.
According to the maximum principal stress theory
σ_{1} + σ_{2} + σ_{3} ≤ yield strength/Factor of safety
Principal stress is related to the existing principal plane available stresses as when stress is line on maximum principal planes as positive and negative stress value higher in magnitudes called principal stress. Stress is a tensor because it follows the transformation equation.
In principal strain the maximum and minimum normal strains are obtained by differentiating x, and y coordinates and the orientation of the planes of the strains is determined there are two roots, p_{1} and p_{2}, for 1 and 2 directions. Principal strains are the nominal strains with higher magnitude.
Various load types or combinations act over the components while under operating conditions. So, the theories of failure help to give a safe dimension to machine components. Before learning about Theories of Failure, let us briefly learn about a machine’s failure.
The meaning failure of the machine is the nonability of the component or a machine to perform its function as per requirement. Failure of a machine or any mechanical components does not mean a total breakdown or not operating. It can operate but not as per our requirements.
The diagram shown below shows a graphical representation of various Theories of Failure for ductile material:
For example, A mechanical shaft transmits the power, but due to increased load, it gets bent slightly. Still, it can transmit power but not that efficiently. In this case, we can say that failure of the shaft happens.
Based on the loading condition (Tensile load, compressive load, and shear load) and material behavior (Ductile, plastic, and brittle), there are five main theories of failure:
Maximum Principal Stress Theory or Rankine’s Theory
In this case of complex loading, when the maximum principal stress value reaches the ultimate stress or yield stress of the material, the component will get failed.
As brittle material is weak in tension, this is most suitable for the brittle material
In simple tension, σ_{1}= σ_{yt}
In compression, σ_{2}= σ_{yc}
Where,
Maximum Principal Strain Theory or St. Venant’s Theory
According to this failure theory, components are safe if the maximum principal strain at a critical point in the loaded member will not exceed the strain under the yielding condition when subjected to uniaxial loading.
It is the most suitable for all types of material, like ductile material, brittle material, and material under hydrostatic conditions.
ε_{1}≤σ_{yt}/E
σ_{1}−[ μ ( σ_{2}+ σ_{3})]≤σ_{yt}/E
Where,
Maximum Shear Stress Theory or Guest and Tresca’s Theory
According to this theory, the material will be failed in the complex loading system at any place in components, and the magnitude of shear stress induced in the material is greater than or equal to the maximum shear stress of the material. Therefore, it is the most suitable theory of failure to experiment on ductile material, especially in the case of a shaft subjected to torque.
The limitation of maximum shear stress theory is that it will not give proper results for a pure shear loading as well as triaxial tensile or compressive loading having approximately equal magnitude (for example body submerge in water)
Maximum Strain Energy Theory or Haigh’s Theory
Per the maximum strain energy theory, failure occurs when the total strain energy in the fundamental component exceeds strain energy under yield conditions. According to this theory, strain can be fully recoverable up to the elastic limit. However, this theory will not give results in the case of triaxial loading having almost equal magnitude.
The Strain energy is developed and stored in the material after the elastic deformation. This energy will also be the work done during the elastic transformation. This theory says the developed strain energy will be responsible for the failure of the strain energy under yield conditions.
Maximum Shear Strain Energy Theory or VonMises and Henckey Theory
It is also known as the maximum distortion energy theory. As it was introduced by the scientists VonMises and Hickey, it is also known as von MisesHencky’s theory. When loads are applied to the component, it will deform, due to which the size and shape of the components will change. The energy responsible for changes in shape and volume is known as distortion energy.
Maximum shear strain theory state that distortion energy associated with the unit volume of the member exceeds the distortion energy at uniaxial loading per unit volume.
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1. What is the significance of principal stressstrain in material behavior? 
2. How do we calculate the principal stresses from a stressstrain curve? 
3. What are the theories of failure in material science? 
4. How does the Maximum Shear Stress Theory determine failure in materials? 
5. What is the relationship between principal stressstrain and the theories of failure? 

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