Chapter 8: Fracture of Cracked Members
8.1 - Introduction Pre-existing cracks, or flaws that can be
treated like cracks, exist in many materials. These flaws could
consist of voids in the material, voids in welds, gouges, foreign
material (impurity), delamination of layered materials, separation
of reinforcing bar from concrete, etc. Thermal stresses may lead to
a crack, even with no load on the material.
8.2.1 Cracks as Stress Raisers
Cracks can be considered infinitely sharp elliptical holes that
give rise to infinite stress concentration factors, or stress
raisers, as they are often called.
(see Section 2.18. Stress Concentrations, p. 107-8, Mechanics of
Materials, 4th ed., Beer, Johnston, and DeWolf, McGraw-Hill, 2006,
your deforms book)
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y = S 1 + 2 = S 1 + 2 d Kt =
c
c
(8.1)
yS
=1+ 2
c c =1+ 2 d
As d approaches zero or approaches zero the ellipse is slit-like
or crack-like and the stress concentration factor Kt approaches
infinity.
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8.2.2 Behavior at Crack Tips in Real Materials Define = crack
tip opening displacement (CTOD)
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8.2.3 Effect of Cracks on Strength To account for the effect of
a crack on the strength of a material, the stress intensity factor
K is defined. The quantity K characterizes the severity of the
crack and is a function of crack size, geometry of the cracked
structural member, and loading type (i.e., tension, torsion, ).
Example: Consider a center-cracked plate loaded in tension. The
stress intensity at the tip of crack is defined asK = S a
a is the half crack length S is the gross stress, a force/area
calculation If K < Kc, then no failure. Kc = critical stress
intensity factor, which is a material property. The values for Kc
for various materials can be found in Table 8.1, 8.2 on p. 318-19
in the text. The relation K = S a is a result of sophisticated
analysis from the theory of linear elasticity. The study of
fracture based on this approach is called Linear Elastic Fracture
Mechanics (LEFM). Rearranging the equation, the critical level of
the gross stress, Sc is given bySc = Kc a
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(Repeating formula for convenience)Sc = Kc a
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8.2.4 Effects of Cracks on Brittle versus Ductile Behavior
Consider the crack length a = at (t = transition) where the
critical level of gross stress S equals the yield stress of the
material, o .
Kc 1K Sc = 0 = at = c 0 at
2
(eq. 8.4)
at is material property since Kc and 0 are material
properties
For a > at Failure by fracture For a < at Little strength
reduction due to the crack since the material yields anyway If low
0 and high Kc , then at large. Component will fail by yielding. If
high 0 and low Kc , then at small. Component will fail by
fracture.
Figure 8.6 Transition crack length at for a low-strength,
hightoughness material (a), and for high-strength, low-toughness
material (b). If (b) contains internal flaws ai, its strength in
tension ut is controlled by brittle fracture.
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8.2.5 Internally Flawed Materials
If ai = represents inherent (i) half flaw size, then the
ultimate strength of the material is actually given by
ut = K c ai
(8.5)
even though u from a material property table has a different
value of u determined from testing the material with no flaws,
e.g., Table 4.2 on p. 121. Material is insensitive to cracks longer
than 2ai since the flaws inherent in the material are worse than
the crack itself. Pretty bad quality control of the material! Or,
for some materials, it is the character of the material. Other
Loading Modes
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8.3.1 Strain Energy Release Rate, G
G=
1 dU , strain energy release rate t daNm Nm i.e., U per crack
length = mm m2
(8.6)
Units of G:
Gc = critical strain energy release rate, a material property If
G = Gc, then the crack grows Using a numerical stress analysis,
e.g., the finite element method, for a particular structure, crack
length a, loading, and material, the crack can be allowed to grow
virtually an amount a and G computed and compared to Gc.
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8.3.2 Stress Intensity Factor K (More Formal View for Mode I
Loading)
x = y = xy = yz = 0
KI 3 cos 1 sin sin + ... 2 2 2 2 r KI 3 cos 1 + sin sin + ... 2
2 2 2 r KI 3 cos sin cos + ... 2 2 2 2 r
(8.7)
xz = 0
z = 0 for plane stress, z = (x + y) for plain strain (z =
0)FormallyK I limr 0, 0
(
y
2 r
)
All stresses are proportional to KI and
1 (square-root singularity). r
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Generalization of stress intensity equation For the cracked
plate of Fig. 8.5 the stress intensity factor was given byK = S
a
Though not stated explicitly, to derive this formula it was
assumed that a t for #1 to occur
(8.32)
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