Page 221 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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Dynamic and Time-Dependent Fracture 201

e
e
and strains in the elastic body are σ and ε , respectively, while the corresponding quantities
ij
ij
in the viscoelastic body are σ and ε , the stresses, strains, and displacements are related as
ij
ij
follows [59]:
σ ij σ = e ij , ε ij Dd ε = { ij} e , u = { i Ddu } i e (4.71)
Equation (4.71) defines a correspondence principle, introduced by Schapery [59], which allows the
solution to a viscoelastic problem to be inferred from a reference elastic solution. This correspon-
dence principle stems from the fact that the stresses in both bodies must satisfy equilibrium, and
the strains must satisfy compatibility requirements in both cases. Also, the stresses are equal on
the boundaries by definition:

T i i j n = j σ e i j n = σ j

Schapery [59] gives a rigorous proof of Equation (4.71) for viscoelastic materials that satisfy
Equation (4.68).
Applications of correspondence principles in viscoelasticity, where the viscoelastic solution is
related to a corresponding elastic solution, usually involve performing a Laplace transform on a
hereditary integral in the form of Equation (4.62), which contains actual stresses and strains. The
introduction of pseudo-quantities makes the connection between viscoelastic and elastic solutions
more straightforward.

4.3.2.3 Generalized J Integral

The correspondence principle in Equation (4.71) makes it possible to define a generalized time-
dependent J integral by forming an analogy with the nonlinear elastic case:

J = v ∫ Γ    w dy − e σ ij j u ∂ x ∂ e i ds    (4.72)
n

e
where w is the pseudo-strain energy density:

w e ∫ ij d = σε e ij (4.73)

The stresses in Equation (4.72) are the actual values in the body, but the strains and displacements
are pseudo-elastic values. The actual strains and displacements are given by Equation (4.71).
Conversely, if ε and u are known, J can be determined by computing pseudo-values, which are
i
v
ij
inserted into Equation (4.73). The pseudo-strains and displacements are given by
e
ε ij e Ed } and u = { Edu} (4.74)
ε = {
i
i
ij
Consider a simple example, where the material exhibits steady-state creep at t > t . The hereditary
o
integrals for strain and displacement reduce to
ε e ij ε = ˙ ij and u e i u = ˙ i