Page 346 - Mechanical Behavior of Materials

Section 8.3 Mathematical Concepts 347

These equations are derived on the basis of the theory of linear elasticity, as described in any
standard text on that subject, and they are said to describe the stress field near the crack tip. Higher
order terms that are not of significant magnitude near the crack tip are omitted. These equations
predict that the stresses rapidly increase near the crack tip. Confirmation of this characteristic of the
stress field is provided by a photograph of stress contours in a clear plastic specimen in Fig. 8.11.
If the cracked member is relatively thin in the z-direction, plane stress with σ z = 0 applies.
However, if it is relatively thick, a more reasonable assumption may be plane strain, ε z = 0, in
which case Hooke’s law, specifically Eq. 5.26(c), requires that σ z depend on the other stresses and
Poisson’s ratio, ν, according to Eq. 8.7(e).
The nonzero stress components in Eq. 8.7 are seen to all approach infinity as r approaches
zero—that is, upon approaching the crack tip. Note that this is specifically caused by these stresses
√
being proportional to the inverse of r. Thus, a mathematical singularity is said to exist at the crack
tip, and no value of stress at the crack tip can be given. Also, all of the nonzero stresses of Eq. 8.7
are proportional to the quantity K I , and the remaining factors merely give the variation with r and
θ. Hence, the magnitude of the stress field near the crack tip can be characterized by giving the

Figure 8.11 Contours of maximum in-plane shear stress around a crack tip. These were
formed by the photoelastic effect in a clear plastic material. The two thin white lines
entering from the left are the edges of the crack, and its tip is the point of convergence
of the contours. (Photo courtesy of C. W. Smith, Virginia Tech, Blacksburg, VA.)