3
0.1. Failure Theories
In the previous section, we introduced the concept of stress, strain and the relationship
between stresses and strains. We also discussed failure of materials under uniaxial state
of stress. Failure of engineering materials can be broadly classified into ductile and brittle
failure. Most metals are ductile and fail due to yielding. Hence, the yield strength
characterizes their failure. Ceramics and some polymers are brittle and rupture or fracture
when the stress exceeds certain maximum value. Their stress–strain behavior is linear up
to the point of failure and they fail abruptly.
The stress required to break the atomic bond and separate the atoms is called the
theoretical strength of the material. It can be shown that the theoretical strength is
approximately equal to
E
/3 where,
E
is Young’s modulus.
1
However, most materials fail
at a stress about one–hundredth or even one–thousandth of the theoretical strength. For
example, the theoretical strength of aluminum is about 22 GPa. However, the yield
strength of aluminum is in the order of 100 MPa, which is 1/220
th
of the theoretical
strength. This enormous discrepancy could be explained as follows.
In ductile material yielding occurs not due to separation of atoms but due to sliding of
atoms (movement of dislocations) as depicted in Figure 1.1. Thus, the stress or energy
required for yielding is much less than that required for separating the atomic planes.
Hence, in a ductile material the maximum shear stress causes yielding of the material.
In brittle materials, the failure or rupture still occurs due to separation of atomic
planes. However, the high value of stress required is provided locally by stress
concentration caused by small pre-existing cracks or flaws in the material. The stress
concentration factors can be in the order of 100 to 1,000. That is, the applied stress is
amplified by enormous amount due to the presence of cracks and it is sufficient to
separate the atoms. When this process becomes unstable, the material separates over a
large area causing brittle failure of the material.
Figure 1.1: Material failure due to relative sliding of atomic planes
Although research is underway not only to explain but also quantify the strength of
materials in terms of its atomic structure and properties, it is still not practical to design
machines
and
structures
based
on
such
atomistic
models.
Hence,
we
resort
to
phenomenological failure theories, which are based on observations and testing over a
period of time. The purpose of failure theories is to extend the strength values obtained
1
T.L. Anderson,
Fracture Mechanics – Fundamentals and Applications
, Third Edition, CRC
Press, Boca Raton, FL, 2006.