Page 224 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 204 Thursday, April 21, 2005 5:38 PM

204 Fracture Mechanics: Fundamentals and Applications

4.3.3 TRANSITION FROM LINEAR TO NONLINEAR BEHAVIOR

Typical polymers are linear viscoelastic at low stresses and nonlinear at high stresses. A specimen
that contains a crack may have a zone of nonlinearity at the crack tip, analogous to a plastic
zone, which is surrounded by linear viscoelastic material. The approach described in the previous
section applies only when one type of behavior (linear or nonlinear) dominates.
Schapery [61] has modified the J concept to cover the transition from small-stress to large-
v
stress behavior. He introduced a modified constitutive equation, where the strain is given by the
sum of two hereditary integrals: one corresponding to linear viscoelastic strains and the other
describing nonlinear strains. For the latter term, he assumed power-law viscoelasticity. For the case
of uniaxial constant tensile stress σ the creep strain in this modified model is given by
o
 σ  n
t
t
ε() = E R D ()  o D + L  () σ o (4.85)
t
 σ ref 
where D and D are the nonlinear and linear creep compliance, respectively, and σ is a reference
L
ref
stress.
At low stresses and short times, the second term in Equation (4.85) dominates, while the nonlinear
term dominates at high stresses or long times. In the case of a viscoelastic body with a stationary
crack at a fixed load, the nonlinear zone is initially small but normally increases with time, until the
behavior is predominantly nonlinear. Thus there is a direct analogy between the present case and the
transition from elastic to viscous behavior described in Section 4.2.
Close to the crack tip, but outside of the failure zone, the stresses are related to a pseudo-strain
through a power law:

 σ  n
e
ε =  o  (4.86)
 σ ref 

In the region dominated by Equation (4.86), the stresses are characterized by J , regardless of whether
v
the global behavior is linear or nonlinear:

1
 J n  +1
σ ij σ = ref  v  σ ˜ ij θ n (, ) (4.87)
 σ Ir
ref n 

If the global behavior is linear, there is a second singularity further away from the crack tip:
K
σ = I f θ () (4.88)
ij
2 πr ij

Let us define a pseudo-strain tensor that, when inserted into the path-independent integral of
Equation (4.72), yields a value J . Also suppose that this pseudo-strain tensor is related to the stress
L
tensor by means of linear and power-law pseudo-complementary strain energy density functions
(w and w , respectively):
cn
cl
∂
ε ij eL = ∂ σ ij ( fw + w ) (4.89)
cn
cl