Problems for Chapter 3

Constitutive Models: Relations between Stress and Strain

3.6.  Viscoelasticity

3.6.1.      The uniaxial tensile stress-strain behavior of a viscoelastic material is idealized using the spring-damper systems illustrated in the figure, as discussed in Section 3.6.3.

3.6.1.1.            Derive the differential equations relating stress to strain for each system.

3.6.1.2.            Calculate expressions for the relaxation modulus for the Maxwell material and the 3 parameter model.

3.6.1.3.            Calculate expressions for the creep compliance of all three materials

3.6.1.4.            Calculate expressions for the complex modulus for all three materials.

3.6.1.5.            Calculate expressions for the complex compliance for all three materials.

3.6.2.      The shear modulus of a viscoelastic material can be approximated by a Prony series given by $G\left(t\right)={G}_{\infty }+{G}_{1}{e}^{-t/{t}_{1}}$.

3.6.2.1.            Find the creep shear compliance of the material

3.6.2.2.            Find the complex shear modulus of the material

3.6.2.3.            Find the complex shear compliance of the material

3.6.3.      A uniaxial tensile specimen is made from a viscoelastic material with time independent bulk modulus K, and has a shear modulus that can be approximated by the Prony series $G\left(t\right)={G}_{\infty }+{G}_{1}{e}^{-t/{t}_{1}}$.  The specimen is subjected to step increase in uniaxial stress, so that ${\sigma }_{11}={\sigma }_{0}$ $t>0$ with all other stress components zero.  Find an expression for the history of strain the specimen.  It is easiest to solve this problem by first calculating the creep shear compliance for the material.

3.6.4.      A floor is covered with a pad with thickness h of viscoelastic material, as shown in the figure.  The pad is perfectly bonded to the floor, so that ${\epsilon }_{11}={\epsilon }_{33}=0$.   The pad can be idealized as a viscoelastic solid with time independent bulk modulus K, and has a shear modulus that can be approximated by the Prony series $G\left(t\right)={G}_{\infty }+{G}_{1}{e}^{-t/{t}_{1}}$.  The surface of the pad is subjected to a history of displacement $u=u\left(t\right){e}_{2}$.

3.6.4.1.            Calculate the history of stress induced in the pad by   $u\left(t\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t<0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(t\right)={u}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0$

3.6.4.2.            Calculate the history of stress induced in the pad by $u\left(t\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t<0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\left(t\right)={u}_{0}\mathrm{sin}\omega t\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0\text{\hspace{0.17em}}\text{\hspace{0.17em}}$

3.6.4.3.            Assume that the pad is subjected to a displacement $u\left(t\right)={u}_{0}\mathrm{sin}\omega t\text{\hspace{0.17em}}$ for long enough for the cycles of stress and strain to settle to steady state. Calculate the total energy dissipated per unit area of the pad during a cycle of loading.

 27 0C 35 0C 35 0C 55 0C 65 0C Time (Sec) Modulus (GPa) Time (Sec) Modulus (GPa) Time (Sec) Modulus (GPa) Time (Sec) Modulus GPa Time (Sec) Modulus GPa 9 23.95 13 18.89 17 8.81 15 5.67 17 4.92 17 23.91 23 18.46 31 8.61 27 5.16 30 4.86 29 23.84 41 18.04 55 8.34 48 5.00 55 4.76 52 23.72 73 17.36 98 7.91 85 4.97 97 4.59 93 23.72 131 16.27 174 7.27 152 4.95 174 4.32 166 23.51 233 14.67 310 6.46 271 4.92 309 3.92 295 23.16 415 12.63 552 5.67 482 4.86 550 3.41 525 22.61 738 10.61 982 5.16 857 4.76 979 2.92 933 21.79 1312 9.29 1746 5.00 1524 4.59 1742 2.60 1660 20.73 2334 8.81 3106 4.97 2711 4.32 3097 2.50 2952 19.66 4151 8.61 5523 4.95 4822 3.92 5508 2.50 5250 18.89 7381 8.34 9822 4.92 8574 3.41 8000 2.50 9337 18.46 13126 7.91 17467 4.86 15248 2.92 17500 2.50 16600 18.04

3.6.5.      The table above lists measured relaxation (shear) modulus for a (fictitious) polymer at various temperatures.  The polymer has a glass transition temperature of 30 0C.

3.6.5.1.            Plot a graph of the modulus as a function of time for each temperature, using a log scale for both axes

3.6.5.2.            Hence, show that the data for various temperatures can be collapsed onto a single master-curve by scaling the times in each experiment by a temperature dependent factor $A\left(T,{T}_{1}\right)$, as described in Section 3.6.1.  Plot the master-curve corresponding to relaxation at 27 0C.

3.6.5.3.            Plot a graph of $\mathrm{log}\left(A\left(T,{T}_{1}\right)\right)$ as a function of temperature, and show that the data can be fit by a function of the form $\mathrm{log}\left[A\left(T;{T}_{1}\right)\right]=-{C}_{1}\left(T-{T}_{1}\right)/\left\{{C}_{2}+\left(T-{T}_{1}\right)\right\}$.  Determine the values of ${C}_{1},{C}_{2}$ that best fit the data

3.6.5.4.            Hence, determine the constants ${C}_{1}^{g},{C}_{2}^{g}$ for the material, as discussed in Section 3.6.1

3.6.5.5.            Hence, scale the times in the experimental data to plot the relaxation modulus at the glass transition temperature $G\left(t,{T}_{g}\right)$.

3.6.5.6.            Find a Prony series fit to $G\left(t,{T}_{g}\right)$.  Use four terms in the series, together with an appropriate value for ${G}_{\infty }$

3.6.6.      An instrument with mass m=10kg is mounted a set of rubber pads with combined cross sectional area A=5 $c{m}^{2}$ and height h=3cm as shown in the figure.  The pads are made from polyisobutylene, with properties listed in Section 3.6.6.  The base vibrates harmonically with amplitude ${Y}_{0}$ and (angular) frequency $\omega$, causing the instrument to vibrate (also harmonically) with amplitude ${X}_{0}$.

3.6.6.1.            Find an expression relating ${X}_{0}/{Y}_{0}$ to the harmonic modulus $G\left(\omega \right)$ of the material and m, A, h and $\omega$.  Assume that the pads are all subjected to a uniaxial state of stress.

3.6.6.2.            Find an expression for the harmonic modulus $G\left(\omega \right)$ in terms of the material properties ${G}_{\infty }$ and ${G}_{i},{t}_{i}$.

3.6.6.3.            Hence, plot a graph showing the variation of ${X}_{0}/{Y}_{0}$ as a function of frequency, for temperatures ${0}^{0}C,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{25}^{0}C,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{40}^{0}C$