Engineering Rock Mechanics

Time-dependency 2 1 7

which upon differentiation gives

dEs-dEE I de v.

dt dt dt

Differentiating the fundamental constitutive relation for an elastic
element gives dE/dt = (l/E)do/dt. Substituting this, and the relation for the
viscous element, into the above relation gives

1do 1

dt E dt F

This is the differential equation governing the behaviour of a Maxwell
material. By considering two loading cases (constant stress and constant
strain), it is possible to demonstrate its behaviour more clearly. For example,
if we assume that from t = 0 to t = tl, a constant stress is applied, oo, and
then from t = tl constant strain is maintained,

de,- - +-o.

1 do CF

Id& = -I-dt + -g dt + C

E dt F

which after integration becomes, as the stress is constant,

oo

E=-+-t+C.

EF

At t = 0, the material behaves instantaneously as an elastic material, with
E = oo/E. Hence, C = 0. Consequently, under the action of constant stress,
the behaviour of a Maxwell material is

oo

E = - + - t linear creep.

EF

The strain that has accumulated at t = tl is thus

However, for constant strain, deldt = 0, so the basic differential equation
becomes

1do 1

0 = -~ + -0.

E dt F

Rearranging and integrating gives

E

F

log, o = -- t + c.

Now, at t = tl, o = 00 with the result that C = log,o + (E/F)tl, and hence