Failure theories help engineers predict when a material or component will fail.
Predicting failure is vital in the world of structural engineering and material science.
When applying loads to an object, predicting when static failure will occur becomes crucial.
Failure is defined as the onset of plastic deformation for ductile materials and fracture for brittle materials.
However, predicting failure in a complex case of tri-axial stress is much less straightforward.
There is no universally applicable method, and instead, we have to predict failure by selecting the most suitable one of a range of different theories of failure, each of which works relatively well under certain circumstances based on experimentation.
Introduction
Most failure theories are defined as a function of the principal stresses and the material strength.
The simplest failure theory is that failure occurs when the maximum or minimum principal stresses reach the yield or ultimate strengths of the material.
This is called the Maximum Principal Stress theory or Rankine theory. However, this theory is not an excellent failure theory, particularly for ductile materials, for reasons explained later.
The Tresca and von Mises failure criteria are two failure theories consistent with the observation that hydrostatic stresses do not cause yielding in ductile materials.
These are the two most commonly used failure theories for ductile materials.
Definition of Failure
Failure of an object occurs when the material can no longer withstand the applied load.
The definition of failure varies depending on the material type being assessed. For ductile materials, failure is generally considered to occur at the onset of plastic deformation.
For example, imagine folding up one end of a paper clip. The paper clip will lose its original shape if you use enough force. Its inability to return to its original form means it has plastically deformed.
Failure for brittle materials is when they fracture. A typical example would be a window shattering.
Complexity of Tri-Axial Stress
In uniaxial stress states, such as in a tensile test, failure occurs when the normal stress in the object reaches the yield strength of the material for ductile materials and the ultimate strength for brittle materials.
However, predicting failure in more complex cases of tri-axial stress is much more challenging.
This is where failure theories come into play.
These theories compare the stress state in the object being assessed with material properties such as the yield or ultimate strengths of the material, which can be obtained through uniaxial testing.
Most failure theories are defined as a function of the principal stresses and the material strength.
Failure theories are used to predict a material’s failure by comparing the object’s stress state with material properties that are easy to determine, such as the yield or ultimate strengths of the material.
The stress state at a point can be described using the three principal stresses, so most failure theories are defined as a function of the principal stresses and the material strength.
Failure theories that apply to ductile materials are usually not applicable to brittle materials and vice versa.
Maximum Principal Stress Theory
The most straightforward failure theory is that failure occurs when the maximum or minimum principal stresses reach the yield stress, or ultimate strength, of the material.
Where:
- σ1 denotes the maximum principal stress
- σut refers to the ultimate tensile strength of the material
This equation is comparing maximum stress to the tensile strength of the material. It can also be used for compressive strength.
This is known as the Maximum Principal Stress theory or Rankine theory. Although it’s simple to apply, it’s not a great failure theory, particularly for ductile materials.
It doesn’t consider the effect of hydrostatic stresses.
Understanding Hydrostatic and Deviatoric Stresses
Hydrostatic stresses cause a volume change, while deviatoric stresses cause shape distortion.
Hydrostatic stress
Hydrostatic stress is a scenario in which a material or object is experiencing equal stress in all directions. An example would be a pressure vessel or an object submerged underwater.
The stresses are uniform across the object or material. It is often defined as the three principal stresses divided by 3:
Where:
- σx denotes the normal stress in the x-direction
- σy denotes the normal stress in the y-direction
- σz denotes the normal stress in the z-direction
Hydrostatic stresses do not contribute to yielding in ductile materials.
Deviatoric stress
Deviatoric stress is the type of stress that changes the shape of an object but doesn’t make it bigger or smaller.
Imagine squeezing a rubber ball: the part of the stress that makes the ball change shape (flattening it, for instance) but doesn’t change its overall size is the deviatoric stress.
This differs from hydrostatic stress, which would push equally on the ball from all sides without changing its shape.
So, deviatoric stress focuses on shape-changing, while hydrostatic stress is all about size-changing.
Mohr’s circle
Mohr’s circle is a graphical representation of the stress state for a particular element or material. It’s a helpful way to get a handle on principal stresses and shear when considering failure theories.
Mohr’s circle is a 2-D plot, where the x-axis is normal stress, and the y-axis is shear stress. However, Mohr’s circle can be created for two-dimensional or three-dimensional stress elements.
A simplified Mohr’s circle:
More in-depth information about Mohr’s circle is the topic for another post.
Von Mises Criterion
The von Mises criterion is a failure theory often used in engineering. It is for ductile materials and is often used for ductile metals.
It is also called the maximum distortion energy theory, or the von Mises yield criterion.
It states that yielding occurs when the maximum distortion energy in a material is equal to the distortion energy at yielding in a uniaxial tensile test.
Experimental data from a simple tension test provides important aspects of material behavior such as elastic limit.
The distortion energy is the portion of strain energy in a stressed element corresponding to the effect of the deviatoric stresses.
This theory also considers only the deviatoric stresses and is independent of the hydrostatic stress.
In a 3-D stress state, the von Mises equation is given as:
\(\Large\sigma_{\text{v}} = \sqrt{\frac{( \sigma_{xx} – \sigma_{yy} )^2 + ( \sigma_{yy} – \sigma_{zz} )^2 + ( \sigma_{zz} – \sigma_{xx} )^2 + 6( \tau_{xy}^2 + \tau_{yz}^2 + \tau_{zx}^2 )}{2}}\)Where:
- σxx denotes the normal stress in the x-direction
- σyy denotes the normal stress in the y-direction
- σzz denotes the normal stress in the z-direction
- τxy denotes the shear stress on the plane perpendicular to the x-axis, acting in the y-direction
- τyz denotes the shear stress on the plane perpendicular to the y-axis, acting in the z-direction
- τzx denotes the shear stress on the plane perpendicular to the z-axis, acting in the x-direction
- σv represents the Von Mises stress, a derived stress value that provides a single stress metric for evaluating yield failure
The equation takes on other, simpler forms in 2-D and 1-D scenarios.
Overall, the criterion helps determine when a ductile material will yield.
Tresca Criterion
The Tresca failure criterion often called the maximum shear stress theory, suggests that a material will start to deform when its highest shear stress matches the shear stress level where it begins to yield in a pulling test.
This idea aligns with the fact that stresses affecting only the volume of the material don’t influence when it starts to yield.
The Tresca theory can be expressed mathematically, in terms of maximum and minimum principal stresses, as:
Where:
- σmax denotes the maximum principal stress
- σmin denotes the minimum principal stress
- σy represents the yield stress of the material
Both the Tresca and von Mises failure criteria are consistent with the observation that hydrostatic stresses do not affect yield.
However, the von Mises theory is more widely used in modern engineering practice due to its better agreement with experimental observations.
Plotting Yield Surfaces for Both Theories
The yield surface for von Mises and Tresca are represented on a plot. This plot shows a failure envelope of the stress limits defined by both criteria.
The plot shown above is specifically for a 2-D stress state. The yield surfaces can be defined similarly for a 3-D stress state.
Stresses within the envelope will not cause failure. The outer perimeter, or edge, shows the stress values that would cause failure.
You can see from the plot that the Tresca yield limits are more conservative than the von Mises.
In other words, Tresca defines a failure point that otherwise would not indicate material failure in the von Mises regime.
However, von Mises is more often used in the engineering world. I can personally attest to this, as I have dealt with von Mises extensively in finite element modeling.
Key Takeaways
When designing structures, engineers rely on stress analysis to understand how materials will behave under different types of external forces, from tensile load to shear load.
A cornerstone in this is failure theories. They provide invaluable insights for safe design.
These theories take into account various parameters like principal stress space, uniaxial tension test results, and the shear strength of materials.
Experimental tests often validate these theories. The complexity grows when dealing with triaxial stress states or plane stress conditions, requiring more specific application of different failure theories.
Some theories focus on the equivalent stress under complex loading conditions, while others delve into the shear strain energy or the hydrostatic pressure affecting the material.
It’s also essential to understand the failure mechanisms, which can range from mechanical failure due to shear stresses to failure of ductile materials caused by reaching their ultimate tensile load.
By integrating a wide variety of parameters and theories, from stress analysis to fatigue failure, engineers can better predict how materials will behave under real-world conditions, ensuring the safety and longevity of structures.