Page 379 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C008.fm Page 359 Monday, May 23, 2005 5:59 PM

Fracture Testing of Nonmetals 359

Solving for J in terms of physical displacement (Equation (8.11)) gives
v
Et 
∆ N+1  () N+1  ∂ M 
J =− N +   E     a ∂   ∆ (8.16)
v
1
R
Let us now evaluate J from the same constant rate test:

∂  ∆ 
J =−  ∫ Pd∆  (8.17)
a ∂  0  ∆

The load can be expressed as a function of physical displacement by combining Equation (8.11)
and Equation (8.14):

Et 
P M = ∆ N  ()  N (8.18)


E 
R
Substituting Equation (8.18) into Equation (8.17) leads to

N  ∂ M  1 t  E( )τ  N N
J =−∆ +1   a ∂   ∆ t N+1 ∫ 0   E   τ dτ (8.19)
R
˙
since ∆ ∆ = t . Therefore
J v J = φ() (8.20)
t

where

tE
t
φ τ() t = [( )] N+1  ∫ t [( )] N N τ d  −1 (8.21)
E
τ


N
( +1 ) E R  0 
Thus J and J are related through a dimensionless function of time in the case of a constant rate
v
test. For a linear viscoelastic material in plane strain, the relationship between J and K is given by
I
2
K 1( −ν 2 )
J = I (8.22)
E R φ t ()
The conventional J integral uniquely characterizes the crack-tip conditions in a viscoelastic
material for a given time. A critical J value from a laboratory test is transferable to a structure,
provided the failure times in the two configurations are the same.
A constant rate J test apparently provides a rational measure of fracture toughness in polymers,
but applying such data to structural components may be problematic. Many structures are statically
loaded at either a fixed load or remote displacement. Thus a constant load creep test or a load
relaxation test on a cracked specimen might be more indicative of structural conditions than a
constant displacement rate test. It is unlikely that the J integral would uniquely characterize
viscoelastic crack-growth behavior under all loading conditions. For example, in the case of viscous