When dealing with the mechanics of materials, it is essential to understand the various measures of stress and strain that can affect materials under different loading conditions. Two of the most commonly used measures of stress are the principal stress and von Mises stress. While they both provide information about the stress state of a material, they are calculated differently and represent different aspects of the stress state. In this article, we will explore the difference between principal stress and von Mises stress, and their significance in the study of materials mechanics.
We will first discuss on the yielding of a ductile material based on the uni-axial tensile test. Then, we will explain the multi-axial stress state, principal stresses, and their representation as stress tensor. After that, we will develop von Mises stress–in terms of multi-axial stresses and principal stresses–which, also, helps us to find the relationships among these stresses. We use energy conservation principle to establish the relationship and derive the von Mises stress. Lastly, we will define the von Mises criteria, which gives us idea about the point at which the ductile material yields or fails.
Uni-axial tensile test of a ductile material
Uni-axial tensile test of a ductile material gives us the information about the point at which the material yields. In this test, the change in stress and strain is observed by applying uni-axial tensile force to the dog-legged specimen of the material. The point at which the material changes its behaviour from elastic to inelastic indicates the yielding of the material and is called yield stress or strength. When the stress goes beyond the yield point, the material is declared as failed or yielded. So, for ductile material, yield strength or stress is considered as the threshold for the material failure. Mathematically, failure criteria can be stated as S>=Sy(for failure to begin).
The stress-strain diagram in the uniaxial tensile test is shown below.
Multi-axial stress state
From the above discussion, we understood the concept to find the initiation of yielding through uni-axial tensile test–that is, apply the force and compare the induced stress and the yield stress. However, the approach cannot be applied directly to the structure when it is in the state of multiaxial stresses. Just by using our conscience, we can clearly declare that the comparison made, like in the uni-axial tensile test, is not the sufficient to dictate the yielding of material. It is in this scenario, von Mises stress is essential–to evaluate the yielding of a material under multi-axial stress state.
A 3 Dimensional stress state in a structure can be represented as shown in the picture below:
The diagonal components in the matrix are normal stresses, and the rest of the components represent shear stresses. Note that, strain can be represented in the similar form by applying constitutive or Hooke’s law in the linear elastic region.
In the crowd of such stresses, there are some planes with only normal stresses acting on them. These normal stresses are called principal stresses and the planes are called principal planes. Principal stresses are the extreme values of normal stress that can exist when a structure is under multiaxial stress state. Principal stresses are obtained from stresses existing in the structure.
Relationship between Principal Stresses and von-Mises stress
Von Mises stress is just a scalar value obtained from the existing stresses in the structure. As mentioned earlier, principal stresses are obtained from multiaxial stresses, and thereof, von Mises stress is calculated using the following expression.
As Principal stresses are obtained from the existing stress state, von-mises stress is also calculated directly from such stresses.
Derivation of von Mises Stress
When force is acted upon in any structure, the external work done in the structure is stored as strain energy. This equilibrium of energy exists by virtue of energy conservation–that is, energy can neither be created nor be destroyed. The stored energy will be consumed by the structure in two ways: volumetric distortion and shape distortion. When the energy stored in the system is defined in terms of unit volume, it is called energy density. Because of this, von Mises failure criteria is also called distortion energy density. Hence, by comparing the energy density at stress state with energy density at yielding, von Mises stress is obtained.
Von Mises Criteria
According to von Mises criteria, When the distortion energy density of the material at stress state is greater than or equal to the strain energy density at yield, the material starts to yield. Or, when the von Mises stress is higher than the yield strength of the material, the material yields.
Steps to check the criteria
Step 1: Calculate the principal stresses from the stresses acting on the structure
Step 2: Calculate the Principal Stresses
Step 3: Check the criteria
Note: There are several other expressions to define the criteria; however, the basic concept is same for all.
von Mises stress is a scalar quantity obtained from the stresses acting on any structure. This helps us to evaluate the yielding(or failure) of a ductile material. Principal stresses, on the other hand, are the components of stresses when the basis of other stress tensor are zero.