Page 374 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C008.fm Page 354 Monday, May 23, 2005 5:59 PM

354 Fracture Mechanics: Fundamentals and Applications

Let us examine the basis for applying K and J to viscoelastic materials, as well as the limitations
on these parameters.
8.1.1.1 K-Controlled Fracture

In linear viscoelastic materials, remote loads and local stresses obey the same relationships as in
the linear elastic case. Consequently, the stresses near the crack tip exhibit a 1/ r singularity:

K
σ = I f θ () (8.1)
ij
2 π r ij

and K is related to remote loads and geometry through the conventional linear elastic fracture
I
mechanics (LEFM) equations introduced in Chapter 2. The strains and displacements depend on
the viscoelastic properties, however. Therefore, the critical stress-intensity factor for a viscoelastic
material can be rate dependent; a K value from a laboratory specimen is transferable to a structure
Ic
only if the local crack-tip strain histories of the two configurations are similar. Equation (8.1)
applies only when yielding and nonlinear viscoelasticity are confined to a small region surrounding
the crack tip.
Under plane strain linear viscoelastic conditions, K is related to the viscoelastic J integral J ,
v
I
as follows [1]:
2
K 1( −ν 2 )
J = I E R (8.2)
v

where E is a reference modulus, which is sometimes defined as the short-time relaxation modulus.
R
2
Figure 8.1 illustrates a growing crack at times t and t + t . Linear viscoelastic material surrounds
o
o
ρ
a Dugdale strip-yield zone, which is small compared to specimen dimensions. Consider a point A,
which is at the leading edge of the yield zone at t and is at the trailing edge at t + t . The size of
o
o
ρ
the yield zone and the crack-tip-opening displacement (CTOD) can be approximated as follows
(see Chapter 2 and Chapter 3):
π  K  2
ρ =  Ic  (8.3)
c
8  σ 
cr
and
K 2
δ ≈ Ic (8.4)
c
σ Et ()
ρ
cr
where σ is the crazing stress. Assume that crack extension occurs at a constant CTOD. The time
cr
interval t is given by
ρ
ρ
t = c (8.5)
ρ
a ˙

2 This derivation, which was adapted from Marshall et al. [8], is only heuristic and approximate. Schapery [9] performed
a more rigorous analysis that led to a result that differs slightly from Equation (8.9).