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# Maximum Distortion Energy Theory: Definition, Derivation, Examples, and Comparison with Other Failure Theories

## Introduction

Maximum distortion energy theory, also known as von Mises yield criterion or von Mises stress theory, is a part of plasticity theory that mostly applies to ductile materials, such as some metals. It was developed by M.T. Huber in 1904 and R. von Mises in 1913, based on the general conditions described by J.C. Maxwell in 1865.

This theory states that yielding of a material begins when the second invariant of deviatoric stress reaches a critical value. Deviatoric stress is the part of stress that causes change in shape or distortion of the material without changing its volume. The second invariant of deviatoric stress is also known as von Mises stress or equivalent tensile stress, which is a scalar value that can be computed from the Cauchy stress tensor. The critical value of von Mises stress is equal to the yield strength of the material in simple tension.

Maximum distortion energy theory is used to predict yielding of materials under complex loading from the results of uniaxial tensile tests. It satisfies the property where two stress states with equal distortion energy have an equal von Mises stress. Because this theory is independent of the hydrostatic component of the stress tensor, it is suitable for analyzing plastic deformation for ductile materials, as onset of yield for these materials does not depend on the hydrostatic pressure.

Maximum distortion energy theory is one of the most popular failure theories because it has many advantages over other theories, such as Tresca's maximum shear stress theory or Mohr's Coulomb theory. However, it also has some limitations and assumptions that need to be considered when applying it to real-world problems. We will discuss these aspects later in this article.

## Definition and Derivation of Maximum Distortion Energy Theory

In this section, we will define and derive the maximum distortion energy criterion from the concept of strain energy. Strain energy is the energy stored in a material due to deformation. It can be divided into two parts: the strain energy that causes change in volume (dilatational strain energy) and the strain energy that causes change in shape (distortional strain energy). Maximum distortion energy theory is based on the idea that failure occurs when the distortional strain energy per unit volume reaches a limit that is equal to the distortional strain energy at yielding in a simple tension test.

### What is von Mises stress and how is it related to distortion energy?

Von Mises stress, denoted by , is a scalar value of stress that represents the intensity of distortion in a material. It can be computed from the Cauchy stress tensor, denoted by , as follows:

Alternatively, von Mises stress can be expressed in terms of the principal stresses, denoted by , , and , as follows:

The relation between von Mises stress and distortion energy can be derived from the definition of strain energy density, denoted by , which is the strain energy per unit volume. For a linear elastic material, the strain energy density can be written as:

Where is the Young's modulus and is the Poisson's ratio of the material.

The strain energy density can be decomposed into two parts: the dilatational strain energy density, denoted by , and the distortional strain energy density, denoted by , as follows:

Where

And

Are the dilatational and distortional strain energy densities, respectively. Here, is the bulk modulus and is the shear modulus of the material, and are the principal strains.

in terms of the principal stresses as follows:

Substituting these expressions into the equations for and , we get:

And

Using the relation between , , and , we can simplify the expression for as follows:

Since , we have:

Now, we can define von Mises stress as the equivalent tensile stress that would produce the same distortional strain energy density as the actual stress state. In other words, we have:

Where is the equivalent tensile strain. Solving for , we get:

Using Hooke's law again, we can express the equivalent tensile stress as:

This is the relation between von Mises stress and distortion energy density. We can see that von Mises stress is proportional to the square root of distortion energy density, and it depends on the material properties , , and .

### How to derive the maximum distortion energy criterion from strain energy?

The maximum distortion energy criterion states that yielding of a material occurs when the distortional strain energy density reaches a critical value that is equal to the distortional strain energy density at yielding in a simple tension test. To derive this criterion, we need to find the expression for the critical value of distortional strain energy density, denoted by .

We can start by considering a simple tension test, where a material is subjected to a uniaxial tensile stress , as shown in the figure below. The material will yield when reaches the yield strength of the material, denoted by .

In this case, the principal stresses are , , and . The principal strains are , , and . Using Hooke's law, we can write:

Using these values, we can calculate the distortional strain energy density at yielding as follows:

## Advantages and Limitations of Maximum Distortion Energy Theory

Maximum distortion energy theory is one of the most widely used failure theories in engineering because it has many advantages over other theories, such as:

• It is applicable to ductile materials that undergo plastic deformation before failure, such as metals.

• It is independent of the hydrostatic component of the stress tensor, which means it can account for the effect of pressure on yielding.

• It is consistent with experimental observations and empirical data for various materials and loading conditions.

• It can predict yielding of materials under complex loading from the results of simple uniaxial tensile tests.

• It can be easily implemented in numerical methods and software tools for stress analysis and design.

However, maximum distortion energy theory also has some limitations and assumptions that need to be considered when applying it to real-world problems, such as:

• It is not applicable to brittle materials that fail by fracture without significant plastic deformation, such as ceramics or glass.

• It assumes that the material is isotropic and homogeneous, which means it has the same properties in all directions and at all points.

• It does not account for the effect of strain rate, temperature, or microstructure on yielding.

• It does not consider the possibility of localized failure modes, such as buckling, shear banding, or necking.

• It does not provide information about the post-yield behavior or ultimate strength of the material.

Therefore, maximum distortion energy theory should be used with caution and validation when dealing with materials or situations that do not meet its assumptions or criteria. In some cases, it may be necessary to use other failure theories or modify this theory for more accurate results.

## Conclusion

Maximum distortion energy theory is a part of plasticity theory that mostly applies to ductile materials, such as metals. It states that yielding of a material occurs when the second invariant of deviatoric stress reaches a critical value that is equal to the yield strength of the material in simple tension. This theory is based on the idea that failure occurs when the distortional strain energy density reaches a limit that is equal to the distortional strain energy density at yielding in a simple tension test.

This theory is useful for predicting yielding of materials under complex loading from the results of uniaxial tensile tests. It satisfies the property where two stress states with equal distortion energy have an equal von Mises stress. It is also independent of the hydrostatic component of the stress tensor, which means it can account for the effect of pressure on yielding.

This theory has many advantages over other failure theories, such as consistency with experimental data, applicability to ductile materials, and ease of implementation. However, it also has some limitations and assumptions that need to be considered when applying it to real-world problems, such as inapplicability to brittle materials, isotropy and homogeneity assumption, and ignorance of strain rate, temperature, microstructure, localized failure modes, and post-yield behavior.

## FAQs

Here are some common questions and answers related to maximum distortion energy theory:

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