Page 87 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 67 Thursday, April 14, 2005 6:28 PM

Linear Elastic Fracture Mechanics 67

where σ is the effective stress, and σ , σ , and σ are the three principal normal stresses. According
e
3
2
1
to the von Mises criterion, yielding occurs when σ = σ , the uniaxial yield strength. For plane
YS
e
stress or plane strain conditions, the principal stresses can be computed from the two-dimensional
Mohr’s circle relationship:
/
σ σ +  σ  σ −  2  12
σσ = xx yy ±   xx yy  + τ 2  (2.83)
,
1 2   xy
2   2  
For plane stress σ = 0, and σ 3 σ = 1 + σ v( 2 ) for plane strain. Substituting the Mode I stress fields
3
into Equation (2.83) gives
θ  
σ = I cos  θ K 1+ sin    (2.84a)
1 πr  2    2  
2  
θ  
σ = I cos  θ K 1 − sin    (2.84b)
2 πr  2    2  
2  
σ = 0 (plane stress)
3
2vK  
θ
= I cos (plane strain) (2.84c)
2 r π  
2
By substituting Equation (2.84) into Equation (2.82), setting σ = σ , and solving for r, we obtain
YS
e
estimates of the Mode I plastic zone radius as a function of θ:
2
I
r θ () = 1   K     + θ 1 cos 3 sin 2 θ +  (2.85a)
y

4 πσ YS   2  
for plane stress, and
2
I
θ
r ( ) = 1   K     ( − 12 v) ( + 2 1 cos ) + θ 3 sin θ 2  (2.85b)
y
4 π  σ YS   2  
for plane strain. Equation (2.85a) and Equation (2.85b), which are plotted in Figure 2.34(a), define
the approximate boundary between elastic and plastic behavior. The corresponding equations for
Modes II and III are plotted in Figure 2.34(b) and Figure 2.34(c), respectively.
Note the significant difference in the size and shape of the Mode I plastic zones for plane stress
and plane strain. The latter condition suppresses yielding, resulting in a smaller plastic zone for a
given K value.
I
Equation (2.85a) and Equation (2.85b) are not strictly correct because they are based on a
purely elastic analysis. Recall Figure 2.29, which schematically illustrates how crack-tip plasticity
causes stress redistribution, which is not taken into account in Figure 2.34. The Irwin plasticity
correction, which accounts for stress redistribution by means of an effective crack length, is also
simplistic and not totally correct.
Figure 2.35 compares the plane strain plastic zone shape predicted from Equation (2.85b) with
a detailed elastic-plastic crack-tip stress solution obtained from the finite element analysis. The
latter, which was published by Dodds et al. [27], assumed a material with the following uniaxial
stress-strain relationship:

ε = σ + α  σ  n
ε o σ o  σ   (2.86)
o 