Page 484 - Introduction to Continuum Mechanics

468 Linear Maxwell Fluid

For a Maxwell fluid, consider the stress relaxation experiment with the displacement field
given by

where H(i) is the unit step function defined in Eq, (8.1.10). Neglect inertia effects,
(i) obtain the components of the rate of deformation tensor.
(ii) obtain r 12 at t = 0.

(iii) obtain the history of the shear stress r^.
Solution. Differentiate Eq. (i) with respect to time, we get

where 6(t) is the Dirac delta function defined in Eq. (8.1.11). The only non-zero rate of
e 0 d(t)
deformation component is D^i = —~—. Thus, from the constitutive equation for the linear

Maxwell fluid, Eq. (S.l.lb), we obtain

Integrating the above equation from J=0-e to f=0+e, we have

The integral on the right side of the above equation is equal to 1 [see Eq. (8.1.12)]. As e-^0,
the first integral on the left side of the above equation approaches zero whereas the second
integral becomes:

Thus, since ^(O-) = 0, from Eq. (iv), we have

s tnat
For t> 0, <5(0=0 ° Eq. (iii) becomes
» _

The solution of the above equation with the initial condition