A viscoelastic polymer that can be assumed to obey the Boltzmann superposition principle is subjected to the following deformation cycle. At a time, t = 0, a tensile stress of 20 MPa is applied instantaneously and maintained for 100 s. The stress is then removed at a rate of 0.2 MPa s-1 until the polymer is unloaded. If the creep compliance of the material is given by

The question is incomplete. The complete question is:

A viscoelastic polymer that can be assumed to obey the Boltzmann superposition principle is subjected to the following deformation cycle. At a time, t = 0, a tensile stress of 20 MPa is applied instantaneously and maintained for 100 s. The stress is then removed at a rate of 0.2 MPa s-1 until the polymer is unloaded. If the creep compliance of the material is given by:

J (t) = Jo (1 - exp (-t/to))

Where,

Jo = 3m^2 / GPA

to = 200s

Determine

a) the strain after 100's (before stress is reversed)

b) the residual strain when stress falls to zero.

Answer:

a) - 60GPA

b) 0

Explanation:

Given t = 0,

σ = 20Mpa

Change in σ = 0.2Mpas^-1

For creep compliance material,

J (t) = Jo (1 - exp (-t/to))

J (t) = 3 (1 - exp (-0/100)) = 3m^2/Gpa

a) t = 100s

E (t) = ΔσJ (t - Jo)

= 0.2 * 3 (100 - 200)

= 0.6 (-100)

= - 60 GPA

Residual strain, σ = 0

E (t) = Jσ (Jo) ∫t (t - Jo) dt

3 * 0 * 200 ∫t (t - Jo) dt

E (t) = 0