In the study of the mechanics of deformable bodies we frequently refer to "failure". The term failure can mean many things. Failure usually is interpreted by the layman to mean the physical separation of a member, i.e. fracture. Certainly fracture is one valid definition, but failure may also refer to initiation of inelastic action, excessive inelastic deformations, or even excessive elastic deformations.
In the study of materials, theories of failure usually refer to either that stress condition which initiates yielding or causes fracture. For ductile materials such as those we have been testing, the initiation of yielding is usually of paramount interest.
The basic problem is that of taking data from a relatively simple test, e.g. a uniaxial tensile test, and extrapolating the results to more complex stress situations. For example, after measuring the tensile yield point from a simple tensile test, how can we predict initiation of yielding in a pressure vessel?
Several theories of failure have been proposed, each of which gives good results for some materials under some stress states. Unfortunately, none of the theories gives uniformly good results when applied to a large variety of materials and loading conditions.
A brief summary of three of the common theories used to predict yielding of ductile materials follows:
- Maximum Shear Stress Theory of Failure
Independent of the complexity of the stress state, yielding is assumed to occur when the maximum shearing stress in the material reaches a value equal to the maximum shearing stress for the material as determined from a tensile test at yield:
For a plane stress state where the two in-plane principal stresses are of opposite sign the maximum shear stress is given by:
If the in-plane principal stresses are of the same sign then we must consider the third principal stress. The third principal stress may be the maximum, minimum or intermediate principal stress. In a thin-walled pressure vessel for example, the in-plane principal stresses are both positive and the minimum normal stress acts normal to the surface of the pressure vessel.
This theory is appealing since for some ductile materials (e.g. hot-rolled carbon steel) we can observe slip occurring at orientations which appear to agree with the maximum shear planes. Recall the orientation of the slip planes from your tensile and torsion tests of hot rolled carbon steel.
This theory is quite simple to apply and gives reasonable results when applied to many ductile materials subjected to fairly simple loading states.
Examination of experimental results shows that the shearing stress at yielding as determined from a torsion test is slightly higher than that determined from a tensile test.
- Maximum Octahedral Shear Stress Theory of Failure
Independent of the complexity of the stress state, yielding is assumed to occur when the octahedral shearing stress in the material reaches a value equal to the octahedral shearing stress for the material as determined from a tensile test at yielding. The octahedral planes make equal angles with the three principal axes. The Octahedral shearing stress for a plane stress state can be shown to be:
In a uniaxial tensile test this reduces to:
- Maximum Distortion Energy Failure Theory
Strain energy can be separated into energy which is associated with volume change and energy which causes distortion of the element. The maximum distortion energy failure theory predicts yielding when the distortion energy reaches a critical value.
This theory of failure can be shown to be equivalent to the maximum octahedral shear stress theory of failure.
Summary
Below is a summary of the first two theories of failure applied to a simple uniaxial stress state and to a pure shear stress state.
Failure Criteria
Theory \ Loading Uniaxial Pure Shear Relationship Max. Shear Stress Theory Oct. Shear Stress Theory where σyp is the tensile (or compressive) yield point determined for uniaxial loading and τyp is the shearing yield point as determined from a pure shear (e.g. torsion) test.
Plastic Action (materials with a well defined yield point)
- Uniaxial Tension
The yield point is easily determined since the stress vs strain curve reaches a plateau just beyond the end of the linear elastic range. This test produces a constant state of stress on the cross section, thus the entire cross section starts yielding simultaneously.
- Torsion
The initiation of yielding in a torsion test on a solid circular member is somewhat difficult to determine directly. The difficulty arises from the fact that the transition from linear elastic action to plastic action is a very gradual transition, thus a sudden yielding of the entire cross section is not attained. For materials which exhibit a well defined yield point we can utilize the fully plastic phenomenon to predict the shearing yield point stress as follows:
where TFP is the fully plastic torque (refer to the lesson on inelastic torsion).
- Bending
Initiation of yielding in a bending test is analogous to the torsion test. The transition from linear elastic action to fully plastic action is a gradual one which makes it difficult to directly determine where yielding begins. However, we can determine the yield point stress from the fully plastic bending moment (Figure 1). The fully plastic bending moment for a solid member of circular cross section is related to the tensile (or compressive) yield point by the relationship:
Note the similarity between fully plastic bending theory and fully plastic torsion theory.
- Obtain two specimens of a hot-rolled low carbon steel e.g. A36. One specimen will be used for a tensile and torsion test and the second specimen will be used for plastic bending.
- Conduct a tensile test to determine the tensile yield point. It is not necessary to obtain a large amount of load vs deformation data, simply obtain sufficient data to determine the tensile yield point load. Stop the test as soon as it is clear that the tensile yield point has been reached. Do not severely cold work the specimen. From the tensile yield point load determine the tensile yield point stress.
- Using the specimen from your tensile test
conduct a torsion test to determine the fully plastic
torque. The fact that the specimen was tested to yielding
should not materially affect the torsion test results as
long as the specimen was not subjected to large inelastic
deformations in the tensile test. Again, it is not
necessary to take a large amount of torque vs angle of
twist data, simply take sufficient data to establish the
fully plastic torque.
From the fully plastic torque, determine the shearing yield point stress. - Conduct a simple bending test to determine
the fully plastic bending moment. This is most easily
accomplished by measuring center deflection as a function
of applied load. When the deflection increases with
constant load (or essentially constant load) you have
reached the fully plastic bending moment.
From the fully plastic load (and thus bending moment) calculate the tensile (and compressive) yield point stress.
- Tabulate the yield point stresses (normal or shear) for the three tests.
- In your table compare the results obtained with the results predicted by a) the maximum shear stress theory of failure, and b) the octahedral shear stress theory of failure.
- Write a brief discussion of your results.
Expt. Value τmax τoct Tensile Test σyp = Torsion Test τyp = Bending Test σyp =
Expt. Ratio Maximum Shear Stress Theory Octahedral Shear Theory 0.500 0.577 0.500 0.577
Mechanics of Materials, by Higdon, et al., 4th Ed, Wiley, 1985. "Theories of Failure" pp 487-495.
Mechanics of Materials, by Hibbeler, Macmillan, 1991, pp 486-491.
Mechanics of Materials, by Hibbeler, Macmillan, 2nd Ed, 1994, pp 536-544.