Page 110 - Plastics Engineering

Mechanical Behaviour of Plastics 93

Geometry of Deformation Equation
In this case the total deformation, E, is given by
E = E2 = El + E3 (2.48)
From equation (2.48)
&=&1+&3
but from equation (2.47)
a1 =a - a2

~II~U~=CT-U~

Rearranging gives
(2.49)
This is the governing equation for this model.
The behaviour of this model can be examined as before

(9 creep
If a constant stress, a,, is applied then the governing equation becomes

&{q3(6l + t2)1 + 6162e - 6100 = 0
The solution of this differential equation may be obtained using the boundary
condition E = a,/(ij1 + 62) at t = 0. So

(2.50)

It may be seen in Fig. 2.42 that this predicts the initial strain when the stress
is first applied as well as an exponential increase in strain subsequently.

(ii) Relaxation
If the strain is held constant at E', then the governing equation becomes
v3a + 610 - 6162.s' = 0

This differential equation may be solved with the boundary condition that
a = a, = ~'(61 + 62) when the strain is first kept constant.

(2.51)
61 +h
Stress, a(t) = ~
This predicts an exponential decay of stress as shown in Fig. 2.42.