Failure Theories
How can we predict failure? This is the main question we ask when dealing with failure theories.
Ductile Materials (ex: Aluminum) → Failure occurs at the onset of plastic deformation, we figure this out by using the Yield Strength.
Brittle Materials (ex: Glass) → Failure occurs at fracture. We find these by using Ultimate Strength. In general, brittle materials have higher compressive strengths than tensile strengths.
These points are eastly to define for a uniaxial stress state.
When we come to triaxial stresses, predicting the failure is a lot harder to do.
This is where failure theories start to have a play.
Failure Theory: Allows us to predict failure of a material by comparing the stress state in the object we are assessing with the material properties that are easy to determine, like the yield or ultimate strength of materials (you can find these by using a uniaxial test).
Failure Theories
Rankine Theory or the Maximum Principal Stress Theory: Failure occurs when the maximum or minimum principal stresses reach the strength of the material.
THIS IS NOT A GREAT FAILURE THEORY. ESPECIALLY FOR DUCTILE MATERIALS.
Failure of Ductile Materials
→ Needs to be consistent with experimental observations we can make about how materials fail
→ This is very important in ductile materials
Stresses that cause a change in volume are known as hydrostatic stresses. The three principal stresses are equal and there is no shear stresses. Because of this, there is no yielding in ductile materials when it is in a hydrostatic stress state.
Deviatoric Stress is the component that is responsible for shape distortion and yielding.
THE REASON:
Triaxial Stress = Hydrostatic Stress Component (no shear stress) + shape distortion (has shear stress)
Shape distortion is the part that is responsible for yielding
Because ductile failure only takes into account the shape distortion, a good failure theory needs to show that yielding is not related to the hydrostatic stress.
MOHR’S CIRCLE
THEORIES
Rankine Theory
Failure occurs when the maximum or minimum principal stresses reach the strength of the material.
This is not a great failure theory for ductile materials because it does not take into account that hydrostatics stress does not cause yielding.
We have to use this method for the roll hoop.
Tresca Theory
Yielding occurs when the maximum shear stress is equal to the shear stress at yielding in a tensile test
Equation
This equation takes into account that hydrostatics stresses don't cause yielding.
It is more conservative than the Vin Mises theory
Von Mises Theory
Yielding occurs when the maximum distortion energy is equal to the distortion energy at yielding in an uniaxial test (ud = ud,y)