Page 217 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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Dynamic and Time-Dependent Fracture 197

Theoretical fracture mechanics analyses that incorporate viscoelastic material response are
relatively new, and practical applications of viscoelastic fracture mechanics are rare, as of this
writing. Most current applications to polymers utilize conventional, time-independent fracture
mechanics methodology (see Chapter 6 and Chapter 8). Approaches that incorporate time depen-
dence should become more widespread, however, as the methodology is developed further and is
validated experimentally.
This section introduces viscoelastic fracture mechanics and outlines a number of recent
advances in this area. The work of Schapery [56–61] is emphasized, because he has formulated
the most complete theoretical framework, and his approach is related to the J and C* integrals,
which were introduced earlier in this text.

4.3.1 LINEAR VISCOELASTICITY
Viscoelasticity is perhaps the most general (and complex) type of time-dependent material response.
From a continuum mechanics viewpoint, viscoplastic creep in metals is actually a special case of
viscoelastic material behavior. While creep in metals is generally considered permanent deformation,
the strains can recover with time in viscoelastic materials. In the case of polymers, time-dependent
deformation and recovery is a direct result of their molecular structure, as discussed in Chapter 6.
Let us introduce the subject by considering linear viscoelastic material behavior. In this case,
linear implies that the material meets two conditions: superposition and proportionality. The first
condition requires that stresses and strains at time t be additive. For example, consider two uniaxial
strains ε and ε , at time t, and the corresponding stresses σ(ε ) and σ(ε ). Superposition implies
2
2
1
1
t
t
σε ( )] + σε ( )] = [ t σε ( )+ [ t ε ( )] (4.54)
[
1 2 1 2
If each stress is multiplied by a constant, the proportionality condition gives
λσ ε ( )] + [ t λ σ ε ( )] = [ t σ λ ε ( ) + [ t λ ε ( )] (4.55)
t
1 1 2 2 1 1 2 2
If a uniaxial constant stress creep test is performed on a linear viscoelastic material, such that
σ = 0 for t < 0 and σ = σ for t > 0, the strain increases with time according to
o
ε D = () σ() t o (4.56)
t

where D(t) is the creep compliance. The loading in this case can be represented more compactly
as σ H(t), where H(t) is the Heaviside step function, defined as
o
0 for t < 0
Ht() ≡ 
 1 for t > 0

In the case of a constant uniaxial strain, i.e., ε = ε H(t), the stress is given by
o
t
σ E = () ε() t o (4.57)

where E(t) is the relaxation modulus. When ε is positive, the stress relaxes with time. Figure 4.19
o
schematically illustrates creep at a constant stress, and stress relaxation at a fixed strain.
When stress and strain both vary, the entire deformation history must be taken into
account. The strain at time t is obtained by summing the strain increments from earlier times.