Page 278 - Mechanical Behavior of Materials

Section 7.3 Maximum Normal Stress Fracture Criterion 279

We will now proceed to discuss various specific failure criteria, some of which are appropriate
for yielding, and others for fracture. In doing so, unless otherwise noted, the subscripts for principal
stresses σ 1 , σ 2 , and σ 3 will not be assumed to be assigned in any particular order relative to their
magnitudes.

7.3 MAXIMUM NORMAL STRESS FRACTURE CRITERION

Perhaps the simplest failure criterion is that failure is expected when the largest principal normal
stress reaches the uniaxial strength of the material. This approach is reasonably successful in
predicting fracture of brittle materials under tension-dominated loading.
To simplify the discussion, let us assume for the present that we have a material which fractures
if an ultimate strength σ u is exceeded in either tension or compression. That is, we are temporarily
assuming that σ ut =|σ uc |= σ u , where σ ut is the ultimate strength in tension and |σ uc | is the ultimate
strength in compression, expressed as a positive value.
For such a material, a maximum normal stress fracture criterion would be specified by a
function f as follows:

σ u = MAX(|σ 1 | , |σ 2 | , |σ 3 |) (at fracture) (7.6)

where the notation MAX indicates that the largest of the values separated by commas is chosen.
Absolute values are used so that compressive principal stresses can be considered. A particular set
of applied stresses can then be characterized by the effective stress

¯ σ N = MAX(|σ 1 | , |σ 2 | , |σ 3 |) (7.7)

where the subscript specifies the maximum normal stress criterion. Hence, fracture is expected when
¯ σ N is equal to σ u , but not when it is less, and the safety factor against fracture is
σ u
X = (7.8)
¯ σ N
7.3.1 Graphical Representation of the Normal Stress Criterion

For plane stress, such as σ 3 = 0, this fracture criterion can be graphically represented by a square
onaplotof σ 1 versus σ 2 , as shown in Fig. 7.2(a). Any combination of σ 1 and σ 2 that plots within the
square box is safe, and any on its perimeter corresponds to fracture. Note that the box is the region
that satisfies

(7.9)
MAX(|σ 1 | , |σ 2 |) ≤ σ u
Equations for the four straight lines that form the borders of this safe region are obtained as
shown in Fig. 7.2(b):

σ 1 = σ u , σ 1 =−σ u , σ 2 = σ u , σ 2 =−σ u (7.10)