Failure theories

I have been working in the industry for more than a decade now and I wish to share some things with the reader that I have observed.

Material failure theories are an important part of the knowledge of a mechanical Design and analysis engineer and is a frequently discussed topic in interviews. The basic concept of stress is well known (I will assume the reader already knows it). We material failure occurs when stress exceeds the critical value. Why then is it necessary to know the failure theories? It is because there are many different types of “stresses” and you should know when to use which one. They come under the category of failure criteria. Now let us see how it works with examples.

Example1

Figure 1

Consider a rod being pulled by a tensile force and a normal stress of 200 MPa is induced. The yield strength of the material is 150 MPa and the ultimate strength is 280 MPa. Will the rod fail? Since the induced stress is greater than yield, the rod will yield but since the induced stress is less than the ultimate strength, it will not rupture. This one was a sitter. In this case, normal stress, principal stress and von Mises stress are the equal . This stress state occurs in the mechanical tensile specimen material test.

Example 2

Figure 2

Consider the system shown in fig.2: a brittle block of ultimate strength 250 MPa being compressed by pressure of 200 MPa on two of faces sides. Let us assume the block is thick enough in the plane perpendicular to the screen to avoid buckling and also that the contact between the walls and the block is frictionless (we have decided to ignore friction). The stress state in 2D would be as shown in fig.3where stresses normal to x, Sx and stresses normal to y, Sy are both 200 MPa.

Figure 3

Now, we have to decide whether the block is safe from rupture or not. The dilemma for a person not knowing the failure theories might be: 200 MPa is less than the ultimate stress of 250 MPa but there are two such stress, Sx and Sy. Singularly they may not cause failure but collectively will they cause failure (after all 200+200= 400 and if you have a good imagination: even MPa)? Using failure theories we can answer such questions.

The answer is: no. They will not cause rupture. The principal stress state is shown in fig.4.

Figure 4

According to the maximum principal stress theory which is most popular for rupture of brittle materials, the magnitude of maximum or the minimum principal stresses must be greater than the ultimate strength of the material. Since the principal stress is less than the ultimate strength of the material, it will not rupture…..

Example 3

This is the same as example 2 except that the brittle block is replaced by a block of ductile material of yield strength 220 MPa and ultimate strength 350 MPa. The system is shown in fig.5

Figure 5

Since the mechanics of the problem are not changing, the stress state is the same as the previous example but the material is different and ductile. Will it yield? Will it rupture?

The most commonly used failure theory for yielding of ductile isotropic materials is the von Mises theory. As per this theory, failure occurs when the induced von Mises stress exceeds a critical value. When the third principal stress = 0,

von Mises stress, Sv =

S1 = 200 MPa, S2 = 200 MPa

e Sv = 200 MPa

Since the von Mises stress is less than the yield stress, the block is safe from yielding.

Example 4

Consider the same problem with the block being made of material with yield point 120 MPa and ultimate stress = 210 MPa.

As in the previous example, the von Mises stress is 200 MPa. So yielding will occur.

But sometimes we are okay with yielding but not rupture, then we have to evaluate whether rupture would occur. This is where things are not very clear. The failure criteria for rupture are not as clear as failure criteria for yielding.

Some researchers use strain based methods (the strain multiplied by a function of the Davis triaxilality factor must exceed the rupture strain for rupture to take place) but most engineers I have seen in the industry use Principal stress obtained from a nonlinear analysis as the failure criterion for rupture with the understanding that rupture occurs when the largest principal stress exceeds the rupture strength. Alas! I remember one telephonic interview where the interviewer was not pleased because when asked for rupture criterion for ductile materials, I mentioned a strain based criterion.