Faculty of Civil and Environmental Engineering

Computational Engineering Continuum Mechanics Prof. Dr.-Ing. H. Steeb

Mechanical Modelling of Materials WS 2014/15

Exercise 5 (Rheology)

Task 5.1

Consider a specimen of polymeric material that behaves like a Maxwell-element with the

elastic modulus of spring 510 PaE = and with the viscosity of dashpot element 710 Pa sη = ⋅ (Figure 1). At 0t = a constant stress of 4

0 10 Paσ = is applied to the specimen and held constant during a period of 500 s . From 500 st = the strain is kept constant while the stress relaxation is recorded as a function of time.

a) What is the value of the strain 0ε immediately after applying the stress at 0t = ? b) What is the value of the strain ε at 500 st = ? c) What is the value of the stress σ at 800 st = ? d) Draw the curves of ( )tσ and ( )tε as functions of time between 0t = and 800 st = . e) To what extent will a real polymer deviate from this behavior?

Figure 1

ηE

Mechanical Modelling of Materials WS 2014/15

Task 5.2

Consider a polymeric material that behaves like the Kelvin-Voigt model (Figure 2a) with the elastic modulus of spring 82.5 10 PaE = ⋅ and the viscosity of dashpot element

108.5 10 Pa sη = ⋅ ⋅ . The applied loading history is shown in Figure 2b with 70 1.6 10 Paσ = ⋅

and 1 150s=t .

1. Compute the strain ε at time 1t . 2. Draw the curve ( )tε as a function of time between 0t and 1t . 3. Determine the time 2t such that 2 1( ) 2 ( )t tε ε= .

Figure 2a

Figure 2b

Task 5.3

The stress relaxation modulus of a certain polymer can be described approximately by

0( ) exp( )EG t G tη= − and has the values 92 10 Pa⋅ and 91 10 Pa⋅ at 0t = and 410 st = ,

respectively.

Calculate the creep compliance as the function of time and determine the strain 1000s

after the rapid application of a stress of 810 Pa .

η

E

Mechanical Modelling of Materials WS 2014/15

Task 5.4

A material can be modeled as a Standard Linear Solid (Figure 3) with an unrelaxed modulus 9

0 10 PaE = and a relaxed modulus 90.5 10 PaRE = ⋅ at 0t = and t →∞ , respectively. When a constant strain is applied, the stress in the material reduces from

72 10 Pa⋅ to 71.5 10 Pa⋅ over a period of 500 seconds.

Determine the dashpot viscosity.

Figure 3 Task 5.5 Consider the Maxwell model with the elastic modulus of spring 103.6 10 PaE = ⋅ and the viscosity of dashpot element 112.72 10 Pa sη = ⋅ ⋅ (Figure 1).

Using these values of material parameters, determine the variations of the storage modulus ( )E′ ω , the loss modulus ( )E′′ ω , and the loss factor tan δ for the angular

frequency in the range 10.001 100ω s−≤ ≤ .

η2E

1E