WHAT IS VON MISES STRESS?

 

What is Stress?

Every object is made up of multiple tiny particles that has a certain mass and occupies space. When a certain force is applied to the object as a whole, this force is transferred through the object and each individual particle experiences that force and subsequently transfers that force to the next particle. Stress is a physical quantity that expresses the internal forces that the neighboring particles of a continuous matter exert on each other.

Therefore it can be safely said that stress is a consequence of the external force. Another consequence of application of force is what is called strain. Strain is the measure of the amount of deformation of the material. How do we measure the deformation of the material? By comparing it with its initial state. And therefore, strain is rightly defined as the change in dimension to the original dimension of the object. The relation between stress and strain was well established by Robert Hook who defined the Hooke’s law in the 17^th century.

Why is stress important in Engineering Applications?

The understanding of stress, strain and strength of materials is what keeps us safe. It is what prevents the lift from collapsing or your chair from breaking. It is what keeps the wheels of your car stay attached to your car and prevent the pressure cooker from blowing off. But how?

Each and every material will have a particular strength limit before which it becomes useless, that is, it either breaks or deforms. In the case of a ductile material, this limit is the Yield Strength and in case of a brittle material, it is the Ultimate Tensile Strength. This limit is measured by a simple tensile test using the Universal Testing Machine and it varies from material to material.

Engineers calculate the maximum amount of force that is going to be applied on an object during its application and ensures that the material selected and the corresponding design of the object is such that it can endure the total stress developed within the object. This basically means that the total stress developed inside the object will be allowed to a certain multiple less than the maximum stress the object can withstand (YS or UTS). This multiple is what is called as the Factor of Safety (FOS). For this it is essential that the stresses developed by the individual particles due to the application of force are determined accurately.

What are the tools used to determine stress or stress distribution in the particles?

Stress is a tensor quantity. This means that stress has a magnitude and direction, but does not comply by the vector law of addition. This makes it very difficult to predict the direction of stress flow especially when there are multiple forces involved. In a simple object or part like a rod it is simple to understand the distribution of stress and deformation. Stress can be simply calculated as the force applied divided by the area of cross section of the part. In more complex structures, like a stapler or the knuckle of your car, this is not possible.

In such situations, the idea of nodes and elements come into play. Basically, the entire part or object is taken and subdivided into a finite number of smaller pieces called elements. Each and every element is studied based on its behavior (defined by shape functions) and the stress and strain are analysed. This is called Finite Element Analysis and is performed today using various softwares like Abacus, Ansys etc.

So what exactly is the Von Mises Stress?

Maximum Normal Stress Theory/ Maximum Principle Stress Theory

There are three principle stresses in a body. One each along the x,y and z axis. The Maximum Normal Stress Theory says that the object in consideration will fail when one of these three principle stresses exceeds the yield stress or the ultimate stress of the material. The important point here is to note that this theory conveniently leaves out the effect and understanding of the forces that are not uniaxial in nature.

Von Mises Criterion/ Distortion Energy Theory

The maximum distortion energy theory, however, considers the angular distortion that occurs in the object. It originated when it was observed that materials, especially ductile ones, behaved differently when a non-simple tension or non-uniaxial stress was applied, exhibiting stress values that are much larger than the ones observed along the principle axis.

According to distortion energy theory, it is the angular distortion or the shape change occurring due to the intermolecular slip that leads to the failure of the component. The energy required to cause the shape change is called distortion energy. Distortion energy is of two types: 1] Distortion energy in a simple tension test 2] Distortion energy developed in the actual object. When the distortion energy in a simple tension test is less than the distortion energy in actual object, the object fails. The equivalent stress caused this energy is called Von Mises stress.

This is how the Von Mises Criterion is safer and hence better preferred in engineering applications to analyze machine components especially with the ones that deal with ductile materials.