Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

GTBL042-07 GTBL042-Callister-v2 August 9, 2007 13:52

7.15 Viscoelastic Deformation • 219

behavior.Stress relaxationmeasurements represent one possibility. With these tests,

a specimen is initially strained rapidly in tension to a predetermined and relatively

low strain level. The stress necessary to maintain this strain is measured as a function

of time, while temperature is held constant. Stress is found to decrease with time

due to molecular relaxation processes that take place within the polymer. We may

relaxation modulus define arelaxation modulusEr(t), a time-dependent elastic modulus for viscoelastic

polymers, as

Er(t)=

σ(t)

 0

(7.23)

Relaxation

modulus—ratio of

time-dependent

stress and constant

strain value

whereσ(t) is the measured time-dependent stress and 0 is the strain level, which is

maintained constant.

Furthermore, the magnitude of the relaxation modulus is a function of temper-

ature; to characterize more fully the viscoelastic behavior of a polymer, isothermal

stress relaxation measurements must be conducted over a range of temperatures.

Figure 7.27 is a schematic logEr(t)-versus-log time plot for a polymer that exhibits

viscoelastic behavior. Curves generated at a variety of temperatures are included.

Key features of this plot are that (1) the magnitude ofEr(t) decreases with time (cor-

responding to the decay of stress, Equation 7.23), and (2) the curves are displaced to

lowerEr(t) levels with increasing temperature.

To represent the influence of temperature, data points are taken at a specific

time from the logEr(t)-versus-log time plot—for example,t 1 in Figure 7.27—and

then cross-plotted as logEr(t 1 ) versus temperature. Figure 7.28 is such a plot for

an amorphous (atactic) polystyrene; in this case,t 1 was arbitrarily taken 10 s after

Log relaxation modulus,

Er

(t)

Log time, t

t 1

T 1

T T^2

3

T 6

T 4

T 5

T 7

T 7 > T 6 >... > T 1

Figure 7.27 Schematic plot of logarithm of

relaxation modulus versus logarithm of time

for a viscoelastic polymer; isothermal curves

are generated at temperaturesT 1 throughT 7.

The temperature dependence of the relaxation

modulus is represented as logEr(t 1 ) versus

temperature.