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(t)= G + G 1 e t/ t 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacMcacqGH9a qpcaWGhbWaaSbaaSqaaiabg6HiLcqabaGccqGHRaWkcaWGhbWaaSba aSqaaiaaigdaaeqaaOGaamyzamaaCaaaleqabaGaeyOeI0IaamiDai aac+cacaWG0bWaaSbaaWqaaiaaigdaaeqaaaaaaaa@3F4F@ .

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(t)= G + G 1 e t/ t 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacMcacqGH9a qpcaWGhbWaaSbaaSqaaiabg6HiLcqabaGccqGHRaWkcaWGhbWaaSba aSqaaiaaigdaaeqaaOGaamyzamaaCaaaleqabaGaeyOeI0IaamiDai aac+cacaWG0bWaaSbaaWqaaiaaigdaaeqaaaaaaaa@3F4F@ .  The specimen is subjected to step increase in uniaxial stress, so that σ 11 = σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaGimaaqabaaaaa@37A6@   t>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshacqGH+aGpcaaIWaaaaa@3343@  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 ε 11 = ε 33 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaH1oqzdaWgaaWcbaGaaG4maiaaiodaaeqaaOGa eyypa0JaaGimaaaa@39F8@ .   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(t)= G + G 1 e t/ t 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacMcacqGH9a qpcaWGhbWaaSbaaSqaaiabg6HiLcqabaGccqGHRaWkcaWGhbWaaSba aSqaaiaaigdaaeqaaOGaamyzamaaCaaaleqabaGaeyOeI0IaamiDai aac+cacaWG0bWaaSbaaWqaaiaaigdaaeqaaaaaaaa@3F4F@ .  The surface of the pad is subjected to a history of displacement u=u(t) e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwhacqGH9aqpcaWG1bGaaiikaiaads hacaGGPaGaaCyzamaaBaaaleaacaaIYaaabeaaaaa@37AE@ .

3.6.4.1.            Calculate the history of stress induced in the pad by   u(t)=0t<0u(t)= u 0 t>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhacaGGOaGaamiDaiaacMcacqGH9a qpcaaIWaGaaGPaVlaaykW7caaMc8UaamiDaiabgYda8iaaicdacaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG1bGaaiikaiaads hacaGGPaGaeyypa0JaamyDamaaBaaaleaacaaIWaaabeaakiaaykW7 caaMc8UaaGPaVlaadshacqGH+aGpcaaIWaaaaa@53C6@

3.6.4.2.            Calculate the history of stress induced in the pad by u(t)=0t<0u(t)= u 0 sinωtt>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhacaGGOaGaamiDaiaacMcacqGH9a qpcaaIWaGaaGPaVlaaykW7caaMc8UaamiDaiabgYda8iaaicdacaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG1bGaaiikaiaads hacaGGPaGaeyypa0JaamyDamaaBaaaleaacaaIWaaabeaakiGacoha caGGPbGaaiOBaiabeM8a3jaadshacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaamiDaiabg6da+iaaicdacaaMc8UaaGPa Vdaa@62A6@

3.6.4.3.            Assume that the pad is subjected to a displacement u(t)= u 0 sinωt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhacaGGOaGaamiDaiaacMcacqGH9a qpcaWG1bWaaSbaaSqaaiaaicdaaeqaaOGaci4CaiaacMgacaGGUbGa eqyYdCNaamiDaiaaykW7aaa@3DED@  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(T, T 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeacaGGOaGaamivaiaacYcacaWGub WaaSbaaSqaaiaaigdaaeqaaOGaaiykaaaa@35FA@ , 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 log(A(T, T 1 )) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacYgacaGGVbGaai4zaiaacIcacaWGbb GaaiikaiaadsfacaGGSaGaamivamaaBaaaleaacaaIXaaabeaakiaa cMcacaGGPaaaaa@3A23@  as a function of temperature, and show that the data can be fit by a function of the form log[A(T; T 1 )]= C 1 (T T 1 )/{ C 2 +(T T 1 ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaciiBaiaac+gacaGGNbGaai4waiaadg eacaGGOaGaamivaiaacUdacaWGubWaaSbaaSqaaiaaigdaaeqaaOGa aiykaiaac2facqGH9aqpcqGHsislcaWGdbWaaSbaaSqaaiaaigdaae qaaOGaaiikaiaadsfacqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqa aOGaaiykaiaac+cadaGadaqaaiaadoeadaWgaaWcbaGaaGOmaaqaba GccqGHRaWkcaGGOaGaamivaiabgkHiTiaadsfadaWgaaWcbaGaaGym aaqabaGccaGGPaaacaGL7bGaayzFaaaaaa@4DDE@ .  Determine the values of C 1 , C 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaaGymaaqabaGcca GGSaGaam4qamaaBaaaleaacaaIYaaabeaaaaa@34A1@  that best fit the data

3.6.5.4.            Hence, determine the constants C 1 g , C 2 g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaqhaaWcbaGaaGymaaqaaiaadE gaaaGccaGGSaGaam4qamaaDaaaleaacaaIYaaabaGaam4zaaaaaaa@367B@  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(t, T g ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacYcacaWGub WaaSbaaSqaaiaadEgaaeqaaOGaaiykaaaa@3651@ .

3.6.5.6.            Find a Prony series fit to G(t, T g ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaamiDaiaacYcacaWGub WaaSbaaSqaaiaadEgaaeqaaOGaaiykaaaa@3651@ .  Use four terms in the series, together with an appropriate value for G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeadaWgaaWcbaGaeyOhIukabeaaaa a@32F1@

 

 

 

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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogacaWGTbWaaWbaaSqabeaacaaIYa aaaaaa@334B@  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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMfadaWgaaWcbaGaaGimaaqabaaaaa@324C@  and (angular) frequency ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3baa@3255@ , causing the instrument to vibrate (also harmonically) with amplitude X 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIfadaWgaaWcbaGaaGimaaqabaaaaa@324B@ .

3.6.6.1.            Find an expression relating X 0 / Y 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIfadaWgaaWcbaGaaGimaaqabaGcca GGVaGaamywamaaBaaaleaacaaIWaaabeaaaaa@34CC@  to the harmonic modulus G(ω) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaeqyYdCNaaiykaaaa@347A@  of the material and m, A, h and ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3baa@3255@ .  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(ω) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeacaGGOaGaeqyYdCNaaiykaaaa@347A@  in terms of the material properties G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeadaWgaaWcbaGaeyOhIukabeaaaa a@32F1@  and G i , t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEeadaWgaaWcbaGaamyAaaqabaGcca GGSaGaamiDamaaBaaaleaacaWGPbaabeaaaaa@353B@ .

3.6.6.3.            Hence, plot a graph showing the variation of X 0 / Y 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIfadaWgaaWcbaGaaGimaaqabaGcca GGVaGaamywamaaBaaaleaacaaIWaaabeaaaaa@34CC@  as a function of frequency, for temperatures 0 0 C, 25 0 C, 40 0 C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaicdadaahaaWcbeqaaiaaicdaaaGcca WGdbGaaiilaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGOmaiaa iwdadaahaaWcbeqaaiaaicdaaaGccaWGdbGaaiilaiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaisda caaIWaWaaWbaaSqabeaacaaIWaaaaOGaam4qaaaa@505A@