Equivalent Von Mises Stress: What, How, Examples, Exhaustive Facts

Equivalent Von Mises Stress are generally applied to determine yielding of ductile materials.

Equivalent Von Mises Stress is used to forecast yielding of material when multi axial loads are applied on the body with the help of the results obtained from simple uni-axial tensile tests.

In the above, “equivalent” refers to the combined 3 x 3 stress and it is equivalent to the state of simple tension in such a manner that the magnitude of the stress system can be derived using the data obtained from material’s tensile test.

Von Mises is one of the most commonly referred Equivalent stresses used in material science to estimate and materials strength and durability.

The Von Mises relationship is most widely used by engineers and it has a great importance in data treatment and continuum calculations.

Von Mises stress can be calculated from both individual stress components as well as from the principal stress. The depiction of Von Mises stress through principal stress is much more easier to visualise and hence more commonly used for stress calculation. In terms of principal stress, the Von Mises stress is characterised by equal difference between individual components.

The Von mises stress can be best depicted by the stresses a cube experiences when thrown into a deep sea.

Equivalent Von Mises stress
Equivalent Von Mises stress for a sinking cube, Image Credit: courses.ansys.com

In the above example, the principal stresses that the cube experiences are the buoyancy forces which are equal on all sides with minimal variation based on the difference of the area of each surface. The value of stress would depend only upon the hydrostatic head which increases with the depth of the sea as the cube continues to sink. Since the individual stress components on all sides remain more or less same, the difference between them remains same.

What is Equivalent Von Mises Stress?

Equivalent stress allows one to view stress acting on a structure by one plot. Von Mises equivalent stress is one of the most widely used.

Equivalent Von Mises stress predicts the yielding of materials under a condition of multiaxial loading with the help of the results from simple uniaxial tensile tests. It is indicated by σv.

Von Mises stress is a most widely used equivalent stress can be represented as

In terms of stress components

In terms of principal stresses

Both the expression give the same equivalent stress value for same stress state. Von Mises stress is only dependent on the difference between the three principal stresses, therefore it is a good equivalent stress to represent the distortion of a material.

In the principal coordinate system we can plot the expression with principal components as below

The cylindrical surface is a perfect circle in octahedral plane; Image credit: courses.ansys.com
cylindrical surface in the principal coordinate system, Image Credit: courses.ansys.com

Equivalent Von Mises Stress Formula

To verify the point at which a mechanical design has started yielding, a metric of calculation known as Von Mises stress is applied.

The stresses determined at any point of a mechanical structure can be expressed mathematically in terms of a scalar quantity known as Von Mises stress which can be varified with experimentally determined yield points.

The Von Mises relationship for equivalent stress is represented by

The stress components in the above equation are principal stresses. Equivalent stress is obtained with these principal stresses. In case of biaxial stresses σ3= 0 and the equation is reduced to

Using rectangular coordinate system, we get the equations as below:

And

What is Von Mises Equivalent Strain?

In classical mechanics just like Equivalent Von Mises Stress result , Equivalent Von Mises strain is also available.

The equivalent Von Mises strain is given by

The strain elements in the above equation are principal strains and can be obtained from principal stress values.

In rectangular coordinate system,

How do you calculate Von Mises stress in 2D?

Von Mises stress is an evaluation of all the stresses acting on a mechanical structure taking into account normal stresses in both the directions(x and y)and the shear stress.

Von Mises stress in terms of principal stress can be represented by

The strain energy density experienced at a point of a material can be expressed in terms of principal stresses as given below:

The strain energy density experienced at a point in a material can be classified as bellow:

  • Dilatational strain energy density, Uh, associated with change in volume
  • Distortion strain energy density, Ud, associated with change in shape.

By substracting Uh i.e Dilatational Strain energy density from U0 i.e total energy density we will get Distortion strain energy(Ud) part.

Ud in terms of equivalent Von Mises stress σVM can be written as

In the above formula, Ud is the distortion energy density and when it reaches a critical value, the yielding of ductile material begins. This idea is given by Von Mises theory.

Since this is generally applicable for an uniaxial stress state, we can easily calculate this critical value of distortional energy from uniaxial test.

Here,

 σ1 = σY and σ2 = σ3 = 0.

Where,

σ1, σ2, σ3 are principal stresses, σY is the Yield Stress

The energy density associated with yielding is given by

Replacing Von Misses stress in equation A with principal stress, we obtain equation B. The energy density obtained in equation B is the critical value of the distortional energy density for the material.

As per the Von Mises’s failure criterion, When a material is subjected to multiaxial loading, yielding of the material takes place when the distortional energy = or > than the critical value for the material

Therefore, the yielding of a material starts when the Von Mises stresses acting on the material are greater than the yield stress experienced by the material in a uniaxial tensile test.

In terms of stress components Von Mises stress can be written as

For 2D plane stress state σ3=0, the Von Mises stress in terms of principal stress can be expressed as follows

In terms of general stress components,

Failure envelope of the distortion energy theory;Image Credit: mae.ufl.edu

How do you calculate Von Mises stress 3D?

Von Mises stress can be expressed in six stress components as follows:

Stress Components in 3D;Image Credit: www.researchgate.net
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