total strain. Theloss tangenttandis a measure of the nature of the material.
For very small tand, the material is solidlike (‘‘elastic’’ behavior); for large
tand, it is more liquidlike (more ‘‘viscous’’ behavior).
Note If the material consists of a solid matrix interspersed with a
continuous liquid—as is the case for most gels—deformation will
always lead to flow of the liquid with respect to the matrix. This
causes frictional energy dissipation, hence a finite value ofG^00 and a
finite tand. Nevertheless, the system may have a perfect memory,
i.e., a response like that in Figure 5.8b. In other words, such a
matrix may be called viscoelastic without showing any flow. To be
sure, many gels do show some flow upon applying a stress.
The values ofG^0 ;G^00 , and tandall tend to depend ono. The most
common situation is thatG^0 andG^00 increase in magnitude with increasing
frequency, though at different rates. Viscoelastic behavior depends on the
time scale of deformation, but the relations vary widely among materials.
This will be further discussed below.
Relaxation Time. Figure 5.10 illustrates what may happen in stress
relaxation experiments. A material is somehow deformed until a given strain
is obtained and then kept at that strain (a). In (b) the response of the stress is
given. For a Newtonian liquid, the stress will instantaneously go to zero.
For a purely elastic solid, the stress will remain constant. For a viscoelastic
material, the stress will gradually relax. The figure illustrates the simplest
case, where
s¼s 0 et=t ð 5 : 13 Þ*Heretis the relaxation time (defined as the time needed for the stress to
relax to 1=e& 0 :37 of its initial value). In most materials the stress
relaxation follows a different course, since there may be a number (a
distribution) of relaxation times. Nevertheless, the relaxation time, even if it
merely concerns an order of magnitude, is a useful parameter.
Actually, all materials have relaxation times, but these vary
tremendously in magnitude. The value is directly related to the proportion
of the bonds in the material that spontaneously break per unit time. If this
rate is large, tis small. In liquids, all bonds between molecules break
spontaneously, but they have a finite time scale, even if it is only about
10 ^12 s, as in water. This implies that at shorter time scales water would
behave as a solid. The bonds in elastic materials may be very long lasting.
Solid rocks can havet& 1014 s (a few million years), implying that a rock
(e.g., a mountain) will exhibit flow at such time scales. It thus depends on the