Page 284 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C006.fm Page 264 Monday, May 23, 2005 5:50 PM

264 Fracture Mechanics: Fundamentals and Applications

where
Q = activation energy for viscous flow (which may depend on temperature)
T = absolute temperature
R = gas constant (= 8.314 J/(mole K))

In the Maxwell model, the stresses in the spring and dashpot are equal, and the strains are
additive. Therefore,

σ 1 d σ
˙ ε = + (6.9)
η E dt
For a stress-relaxation experiment (Figure 4.19(b)), the strain is fixed at ε , and ˙ ε = 0 . Inte-
o
grating stress with respect to time for this case leads to

σ σ() t = o e − / tt R (6.10)

where σ is the stress at t = 0, and t + η/E is the relaxation time.
R
o
When the spring and dashpot are in parallel (the Voigt model) the strains are equal and the
stresses are additive:
σ ε() t + η E = ε ˙ (6.11)

For a constant stress creep test, Equation (6.11) can be integrated to give

σ
ε() = t o ( − 1 e − / tt R ) (6.12)
E

Note that the limiting value of creep strain in this model is σ /E, which corresponds to zero stress
o
on the dashpot. If the stress is removed, the strain recovers with time:

ε ε() t = o e − / tt R (6.13)

where ε is the strain at t = 0, and zero time is defined at the moment the load is removed.
o
Neither model describes all types of viscoelastic response. For example, the Maxwell model
does not account for viscoelastic recovery, because the strain in the dashpot is not reversed when
the stress is removed. The Voigt model cannot be applied to the stress relaxation case, because
when strain is fixed in Equation (6.11), all of the stress is carried by the spring; the problem reduces
to simple static loading, where both stress and strain remain constant.
If we combine the two models, however, we obtain a more realistic and versatile model of
viscoelastic behavior. Figure 6.6(c) illustrates the combined Maxwell-Voigt model. In this case, the
strains in the Maxwell and Voigt contributions are additive, and the stress carried by the Maxwell
spring and dashpot is divided between the Voigt spring and dashpot. For a constant stress creep
test, combining Equation (6.9) and Equation (6.13) gives

σ σ σ t
ε() = t o + o ( − 1 e − / tt R () ) + o (6.14)
2
E E η
1 2 1
All three models are oversimplifications of actual polymer behavior, but are useful for approximat-
ing different types of viscoelastic response.