Page 389 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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Fracture Testing of Nonmetals 369

8.1.5 EXPERIMENTAL ESTIMATES OF TIME-DEPENDENT FRACTURE PARAMETERS

While J values and J-R curves may be indicative of a polymer’s relative toughness, the existence
Ic
of a unique correlation between J and the crack-growth rate is unlikely. Parameters such as J may
v
be more suitable for some viscoelastic materials. For polymers that experience large-scale yielding,
neither J nor J may characterize crack growth.
v
This section outlines a few suggestions for inferring crack-tip parameters that take into account
the time-dependent deformation of engineering plastics. Since most of these approaches have yet
to be validated experimentally, much of what follows contains an element of conjecture. These
proposed methods, however, are certainly no worse than conventional J integral approaches, and
may be considerably better for many engineering plastics.
The viscoelastic J integral J can be inferred by converting physical displacements to pseudo
v
e
displacements. For a constant rate test, Equation (8.11) gives the relationship between ∆ and ∆ .
The viscoelastic J integral is given by Equation (8.13); J can also be evaluated directly from the
v
e
area under the P-∆ curve:
η ∆ e
J = b ∫ 0 Pd∆ e (8.28)
v
for a specimen with unit thickness. If the load-pseudo displacement is a power law (Equation (8.18)),
Equation (8.28) becomes

η M()∆ eN+1
J = (8.29)
v
bN + ) 1
(
Comparing Equation (8.29) and Equation (8.15) leads to

b  M
∂ 
η =−   (8.30)
M  a ∆
∂  e
Since M does not depend on time, the dimensionless η factor is the same for both J and J .
v
Computing pseudo-elastic displacements might also remove hysteresis effects in unloading
e
compliance tests. If the unload-reload behavior is linear viscoelastic, the P-∆ unloading curves
would be linear, and the crack length could be correlated to the pseudo-elastic compliance.
Determining pseudo displacements from Equation (8.11) or the more general expression
(Equation (8.10)) requires a knowledge of E(t). A separate experiment to infer E(t) would not be
particularly difficult, but such data would not be relevant if the material experienced large-scale
yielding in a fracture test. An alternative approach to inferring crack-tip parameters that takes time
effects into account is outlined below.
Schapery [10] has suggested evaluating a J-like parameter from isochronous (fixed time) load-
displacement curves. Consider a series of fracture tests that are performed over a range of crosshead
rates (Figure 8.12(a)). If one selects a fixed time and determines the various combinations of load
and displacement that correspond to this time, the resulting locus of points forms an isochronous
load-displacement curve (Figure 8.12(b)). Since the viscoelastic and yield properties are time
dependent, the isochronous curve represents the load-displacement behavior for fixed material
properties, as if time stood still while the test was performed. A fixed-time J integral can be defined
as follows:

η  ∆ 
∆
J = b ∫ Pd  (8.31)

t

0
t=constant