This article focuses its discussion on how to calculate Von Mises stress from principal stresses. Von Mises theory is one of the five theories of failure.
We will discuss about theories of failure first and then read in detail about Von Mises theory of failure (also known as distortion theory of failure).
Theories of failure
It is very important to know the failure point of any material or specimen that is being used for specific applications. For different types of materials, one single way of finding the failure point won’t suffice. Which is why there are five theories of failure for different material types (ductile or brittle) and different failure modes (shear, tension, strain etc)
There are total five theories of failure. They are given below-
- Maximum principal stress theory or Rankine’s theory– This theory states that failure will occur when maximum principal stress reaches the maximum value of stress at the elastic limit of the material.
- Maximum shear stress theory or Guest and Tresca’s theory– This theory states that material will fail when the maximum shear stress reaches the value of maximum shear stress in tension at elastic limit.
- Maximum principal strain theory or St. Venant’s theory– According to this theory, failure occurs when principal strain reaches the strain value at elastic limit.
- Maximum strain energy theory or Beltrami and Haigh’s theory– The yield criterion for this theory states that failure occurs in a material when the total strain energy reaches the strain energy at elastic limit in simple tension.
- Maximum shear strain theory or distortion theory– This theory is also known as Von Mises theory, according to this theory failure occurs when the strain energy per unit volume reaches a value of shear strain energy per unit volume at elastic limit in tension.
From above mentioned theories, only maximum principal stress theory is suitable for brittle materials rest all are suitable for ductile materials such as metals. The maximum principal stress theory is not suitable for ductile materials.
Von Mises theory
As discussed in above section, this theory states that the material fails when the strain energy per unit volume reaches a value of strain energy per unit volume at elastic limit of the material. This theory is valid for ductile materials and works best for materials undergoing shear.
This theory was proposed by Von Mises-Henky. Mathematically, this theory can be written as-
The sigma values represent principal stress values.
Von Mises stress
The value of stress which determines whether a given material will yield or fracture os the Von Mises Stress. The criterion for yield states that the material will yield or fracture under a given load when the Von Mises stress of the material exceeds the Von Mises stress of that material at elastic limit under simple tension.
What are principal stresses?
Before discussing about principal stresses lets discuss about principal planes. Principal planes are the planes within the material such that resultant stresses are normal stresses or planes in which shearing does not occur.
Principal stresses are the stresses which are acting on the principal planes. The planes which carries maximum principal stress and minimum principal stress are called as major principal axis and minor principal axis respectively.
How to calculate Von Mises stress from principal stresses?
We can calculate Von Mises stress from the principal stresses by using the formula given below-
The above formula is for 3D system of forces. Subscripts 1, 2 and 3 represent 1st, 2nd and 3rd principal stress respectively.
Von Mises stress vs principal stress
Von Mises stress and principal stress hold different meanings and different significance. Von Mises stress is calculated using the values of different principal stresses.
Von Mises stress is the stress which determines the distortion energy density whereas principal stress is the real stress. Von Mises stress gives the information about yield or fracture limit of a material (mostly ductile) whereas principal stresses are the actual stresses acting on the respective principal planes.
Von Mises stress example
We have discussed the formula for calculating Von Mises stress. The Von Mises stress gives us an idea whether the material fail under the given set of stresses or not.
Let us assume the following data for a 3D system of forces-
First principal stress:110MPa
Second principal stress:90MPa
Third principal stress:100 MPa
Calculate the Von Mises stress form the above data. If the yield stress is 20MPa then tell whether the material will yield or not.
Substituting the values in formula discussed in above sections, we get Von Mises stress as 17.32 MPa. As the yield stress is 20 MPa, the material will not fail as Von Mises stress is lesser than the Von Mises stress at yield point.