MATERIALS ENGINEERING CONCEPTS - Materials for Civil and Construction Engineers - page 17

In the Kelvin model [Figure 1.11(b)] the deformation of each of the ele-
ments must be equal at all times, due to the way the model is formulated.
Thus, the total deformation is equal to the deformation of each element
Since the elements are in parallel, they will share the force
such that the total force is equal to the sum of the force in each element. If
at time when a constant force F is applied, Equation 1.4 then re-
quires zero force in the spring. Hence, when the load is initially applied, be-
fore any deformation takes place, all of the force must be in the dashpot.
Under constant force, the deformation of the dashpot must increase, since
there is force on the element. However, this also requires deformation of the
spring, indicating that some of the force is carried by the spring. In fact, with
time, the amount of force in the dashpot decreases and the force in the
spring increases. The proportion is fixed by the fact that the sum on the
forces in the two elements must be equal to the total force. After a sufficient
amount of time, all of the force will be transferred to the spring and the
model will stop deforming. Thus, the maximum deformation of the Kelvin
model is Mathematically, the equation for the deformation in a
Kelvin model is derived as

1 d = d 1 = d 2 2 .

d = 0 t = 0

d = F/M.

Md + bd # F = F 1 + F 2 = (1.8)

Integrating Equation 1.8, using the limits that d = 0 at t = 0, and solv-

ing for the deformation d at time t results in

a F
M

e -Mt

> b

d = b 1 1 - 2 (1.9)

The Prandtl model [Figure 1.11(c)] consists of St. Venant and Hookean bodies
in series. The Prandtl model represents a material with an elastic-perfectly
plastic response. If a small load is applied, the material responds elasticly
until it reaches the yield point, after which the material exhibits plastic
deformation.
Neither the Maxwell nor Kelvin model adequately describes the behav-
ior of some common engineering materials, such as asphalt concrete. How-
ever, the Maxwell and the Kelvin models can be put together in series,
producing the Burgers model, which can be used to describe simplistically
the behavior of asphalt concrete. As shown in Figure 1.12, the Burgers
model is generally drawn as a spring in series with a Kelvin model in series

F

M 1 d 1

t

0

M 2 d 2 b 2 d 2

d

b 3 d 3

FIGURE 1.12 Burgers
model of viscoelastic materials.

t

F 0