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200 Fracture Mechanics: Fundamentals and Applications

Equation (4.66) or Equation (4.67), provided both configurations are subject to the same applied
loads. This is a special case of a correspondence principle, which is discussed in more detail below;
note the similarity to Hoff’s analogy for elastic and viscous materials (Section 4.2).

4.3.2 THE VISCOELASTIC J INTEGRAL
4.3.2.1 Constitutive Equations

Schapery [59] developed a generalized J integral that is applicable to a wide range of viscoelastic
materials. He began by assuming a nonlinear viscoelastic constitutive equation in the form of a
hereditary integral:
e
ij
ε ij E () = R ∫ t D ( − t τ t t , ) ∂ ετ() τ d (4.68)
0 τ ∂
−
where the lower integration limit is taken as 0 . The pseudo-elastic strain ε e ij is related to stress
through a linear or nonlinear elastic constitutive law. The similarity between Equation (4.66) and
Equation (4.68) is obvious, but the latter relationship also applies to certain types of nonlinear
viscoelastic behavior. The creep compliance D(t) has a somewhat different interpretation for the
nonlinear case.
The pseudo-strain tensor and reference modulus in Equation (4.68) are analogous to the linear
case. In the previous section, these quantities were introduced to relate a linear viscoelastic problem
to a reference elastic problem. This idea is generalized in the present case, where the nonlinear
viscoelastic behavior is related to a reference nonlinear elastic problem through a correspondence
principle, as discussed below.
The inverse of Equation (4.68) is given by

ij
ε ij e E () = R −1 ∫ t E ( − t τ t t , ) ∂ ετ() τ d (4.69)
0 ∂τ
Since hereditary integrals of the form of Equation (4.68) and Equation (4.69) are used extensively
in the remainder of this discussion, it is convenient to introduce an abbreviated notation:

{Ddf } E R ∫ t D (t − τ , ) f ∂ dτ (4.70a)
≡
t
0 ∂τ
and

≡
t
{Edf } E R −1 ∫ t ( E t − τ , ) f ∂ dτ (4.70b)
0 ∂τ
−
where f is a function of time. In each case, it is assumed that integration begins at 0 . Thus
Equation (4.68) and Equation (4.69) become, respectively,

ε ij D () = { d ε t ij} e and ε ij e Ed }
ε = {
ij
4.3.2.2 Correspondence Principle

Consider two bodies with the same instantaneous geometry, where one material is elastic and
the other is viscoelastic and is described by Equation (4.68). Assume that at time t, a surface
traction T = σ n is applied to both configurations along the outer boundaries. If the stresses
ij j
i