Page 277 - Mechanical Behavior of Materials

278 Chapter 7 Yielding and Fracture under Combined Stresses

strengths in tension and compression, σ ut and σ uc , are needed. In tension tests on materials that
behave in a brittle manner, yielding is in most cases not a well-defined event, and the ultimate
strength and fracture events occur at the same point. Hence, using σ ut for brittle materials is the
same as using the engineering fracture strength, σ f .
Failure criteria for isotropic materials can be expressed in the mathematical form

f (σ 1 ,σ 2 ,σ 3 ) = σ c (at failure) (7.1)
where failure (yielding or fracture) is predicted to occur when a specific mathematical function
f of the principal normal stresses is equal to the failure strength of the material, σ c ,asfroma
uniaxial test. The failure strength is either the yield strength σ o , or the ultimate strength, σ ut or σ uc ,
depending on whether yielding or fracture is of interest.
A requirement for a valid failure criterion is that it must give the same result regardless of the
original choice of the coordinate system in a problem. This requirement is met if the criterion can be
expressed in terms of the principal stresses. It is also met by any criterion where f is a mathematical
function of one or more of the stress invariants given in the previous chapter as Eq. 6.29.
If any particular case of Eq. 7.1 is plotted in principal normal stress space (three-dimensional
coordinates of σ 1 , σ 2 , and σ 3 ), the function f forms a surface that is called the failure surface.
A failure surface can be either a yield surface or a fracture surface. In discussing failure criteria,
we will proceed by considering various specific mathematical functions f , hence various types of
failure surface.
Consider a point in an engineering component where the applied loads result in particular values
of the principal normal stresses, σ 1 , σ 2 , and σ 3 , and where the materials property σ c is known, and
also where a specific function f has been chosen. It is then useful to define an effective stress, ¯σ,
which is a single numerical value that characterizes the state of applied stress. In particular,

¯ σ = f (σ 1 ,σ 2 ,σ 3 ) (7.2)
where f is the same function as in Eq. 7.1. Thus, Eq. 7.1 states that failure occurs when

¯ σ = σ c (at failure) (7.3)
Failure is not expected if ¯σ is less than σ c :

¯ σ< σ c (no failure) (7.4)
Also, the safety factor against failure is

σ c
X = (7.5)
¯ σ
In other words, the applied stresses can be increased by a factor X before failure occurs. For
example, if X = 2, the applied stresses can be doubled before failure is expected. 1

1
Safety factors may also be expressed in terms of applied loads, according to Eq. 1.3. If loads and stresses are
proportional, as is frequently the case, then safety factors on stress are identical to those on load. But caution is needed
if such proportionality does not exist, as for problems of buckling and surface contact loading.