A Dynamic Non-Linear Viscoelastic Model

(1)

A Dynamic Non-Linear Viscoelastic Model

KANWALJEET KAUR

Faculty of Applied Sciences, BMSCE, Muktsar-152026, India

RAJNEESH KAKAR*

Principal, DIPS Polytechnic College, Hoshiarpur-146001, India

K. C. GUPTA

Faculty of Applied Sciences, BMSCE, Muktsar-152026, India *E-mail of Corresponding Author: [email protected]

Abstract:

The aim of this paper is to present a dynamic non-linear viscoelastic model which is useful to explain the essential characteristics of the dynamic soils behavior. The strain dependency and damping capacity of the soil is useful for a reliable prediction of the seismic response. With the help of this model the damping functions in terms of both strain and frequency are obtained. The model validity is checked for constitutive dynamic functions from experimental data.

Keywords: Kelvin model, Non-linear analysis, Fourier transforms

1. Introduction

The vibrations in earthquakes are due to differences in dynamic characteristics therefore the cyclic stress-strain behaviour of soils play a vital role for reliable prediction of the seismic response. Many researchers studied structural pounding during earthquakes.

Jankowski [1] discussed the linear viscoelastic model and the nonlinear viscoelastic model. Anagnostopoulos [2] studied the linear viscoelastic model of collision to simulate structural pounding. Jankowski et al. [3] studied the pounding of superstructure segments in bridges with the help of linear viscoelastic model. Muthukumar and DesRoches [4] made a comparison study using two single degree of freedom (SDOF) systems for capturing pounding. Westermo suggested linking buildings with beams which can transmit the forces between them eliminating dynamic contacts [5].

In this paper an analysis is made on the dynamic soils behaviour by using the damping capacity and strain dependency. It has been observed that during seismic load conditions there is a very small impact of frequency greater than 1 Hz on damping and modulus function; however the major weight is on the strain level on modulus and damping values. But this frequency has impact on the dynamic structural. The resonant column specimen along with the vibration device can be treated as a system which is made up of a single mass supported by a spring and having one degree of freedom.

2. Basic equations for non-linear viscoelastic soils

Let us consider the integral form of linear viscoelasticity law (Malvern 1969), the relation between stress and time is given by,

 





 

0

d

t

kl

t

G

klmn mn

G

klmn

t

mn











_



 











(1)

Where, G is relaxation modulus function,  is strain tensor,(t) is stress at present time,





 



t

and λ depends on G.

The function G can be written for isotropic materials as

 

2

 

1

 

1

 

ln



1

3

2

klmn kl mn km kn lm

G

t



_



G t



G t



_

 



G t

 



 

(2)

Where,

  

  



j i if 0

j i if 1

ij known as Kroneker's symbol,

G

1and

G

2 are independent functions.

The spherical tensors



ii and



ii are given as

(2)





3

kl ii kl

kl

e







 

(4)

Put Eq. (3), Eq. (4) in Eq. (1), we get

 

2



  

0

d

t

ii

t

G t

ii









  









(5)

 



  

kl 1

0

S

d

t

kl

t





G t







e









(6)

The stress and strain relations are given as



1 2 3



1

3

ii

3 σ



σ



σ



σ



σ

(7)



1 2 3



1

3

ii

3 ε



ε



ε

 

ε

(8)

The octahedral stress and strain invariants are given as



 

2

 

2



2

1 2 2 3 3 1

1

3 τ



σ



σ



σ



σ



σ



σ

(9)



 

2

 

2



2

1 2 2 3 3 1

2

3 γ



ε



ε



ε



ε



ε



ε

(10)

Eq. (5) and Eq. (6) are the constitutive law for volume changes and constitutive law for the shape changes respectively, therefore put

G

₂



K

and

G

₁



G

in Eq. (5) and Eq. (6), we get

 



  

0

d

t

K t









  









(11)

 



  

0

t

d

t

G t









  









(12)

The Eq. (11) and Eq. (12) can be reduced to linear elastic laws if time is fixed i.e. t  ct. In that case, the relaxation function reduces to

K t

 



K

and

G t

 



G

, where K is bulk modulus of elasticity and G is shear modulus of elasticity [10], hence







3 K



and





2G



(13) On applying the above reductions for soils, one gets a viscoelastic state into non-linear elastic state, therefore

K



K

 

t

and

G



G

 

t

must reduce to "variable moduli"

K



K

 



and

G



G

 



_, Apply these

conditions, the Eq. (11) and Eq. (12) become by changing integral arguments as

K



K

 



,

t

and

G



G

 



,

t

_, therefore non-linear viscoelastic constitutive equations for soils are

 



  

0

,

d

t

K

t





_





  









(14)

 



  

0

t

,

d

t

G

t





_





  









(15)

On the basis of triaxial tests, the “separated variables” are given as

 

,

0 1

   

2

(3)

Where,

 

0

0 1 2

0

,

t

,

e

c

,

e

t

g

G

t

g

a b



g

t

m

n



 



 

 



 

(a, b, c, m, n and  are the experimental

material parameters)

3. Dynamic behaviour of the viscoelastic model

Let Eq. (16) is under strain-history of the following form,

 

t

0

exp



i t





 







(17)

Where, 0 are the strain amplitude and  is the excitation frequency

From Eq. (17) and Eq. (15), we get

 



0

  

*

τ γ

,t



G

γ

,i

ω γ



t

(18)

Where, G* is the complex modulus function and given by



0





0





0



*

re im

G

γ

,i

ω



G

γ

,

ω



iG

γ

,

ω

(19)

The real and imaginary parts of G* are Gre (the storage modulus function) and Gim (the loss modulus function) respectively and are given as



0



 

0

 

0

sin

d

re

G

γ

,

ω

G

γ

ω

G

γ

,s

G

γ

ω

s















_



_



(20)



0



 

0

cos

d

im

G

γ

,

ω

G

γ

,s

G

γ

ω

s











_



_



(21)

Where,

G

_

 

γ

is the limit value,

 

0

 

t

G



γ



G

γ

,t

_ (22)

By using the Eq. (16), Eq. (20) and Eq. (21), the nonlinear relaxation function

G

 



,

t

is given as



0



0 1

 

0

 

re re

G

γ

,

ω



g



g

γ



g

ω

(23)



0



0 1

 

0

 

im im

G

γ

,

ω



g



g

γ



g

ω

(24)

Where,

 

2 2 2

re

ω

g

ω

m n

ω

β

 



(25)

 

2 2

im

ωβ

g

ω

n

ω

β





(26) At very slow or very high frequencies,





 









 





0 0 0 0

0 0 0

;

0 ;

0

re _ω t im ω

G

γ

,

ω

G

γ

,t

G

γ

,

ω

G

γ

,

ω

G

γ

,t

G

γ

,

ω

  

  



(27)

Hence, the constitutive equation (18) represents the nonlinear viscoelastic solid which responds to the nonlinear elastic manner at slow or fast processes. It tallies with corresponding classical results given by Christensen (1978) and Malvern (1969).

Case 1

For,  = 0 i.e. for static loads, the stresses are calculated by using the stabilized values of the nonlinear relaxation function.

Case 2

(4)

Also, the complex modulus functions G* Eq. (19) can be alternatively written as









 

0 0

i δ γ,ω

*

G

γ

,i

ω



G

γ

,

ω

e

(*) Where,





2





2





0 re 0 im 0

G

γ

,

ω



G

γ

,

ω



G

γ

,

ω

(28)



0





0





0



tan

δ γ

,

ω



G

_im

γ

,

ω

/ G

_re

γ

,

ω

(29) The damping properties of the material,

G





0

,





is known as dynamic modulus function and

tan







0

,





is known as loss tangent function.



is the phase difference between stresses and strains.

Using the Eq. (23), Eq. (24), Eq. (28) and Eq. (29), the real and imaginary functions

G

re





0

,





and





₀

,





im

G

become,



0

,



0 1

   

0 2

G

 



g



g





g



(30)

 

_

₂ ₂

_

₂

tan

n

m

n



 











(31) Where,

 

₂ 2 ₂ 2 ₂ ₂ 2

2 g _                      

 m n n

(32)

It is seen that the function

tan



 



is a linear damping, which is not useful for soils modelling because the damping in soils dependents on the strain level 0. Therefore, instead of

tan



 



we use the following

damping function

D





0

,





for soils.





₀

,







d

₀



d

₁

 



₀



d

₂

 



D

(33) The damping function

D





₀

,





and dynamic modulus function

G





₀

,





satisfy the resonant column test data.

4. Numerical results and discussion

The model is modified by including a new damping function

D





₀

,





and the experimental test can be obtained with the help of Drnevich resonant column apparatus. The dynamic response of soils can be measured by passing the steady-harmonic shear waves in a cylindrical soil specimen in the presence of resonant frequency conditions.

Therefore, for steady-harmonic shear waves

ωt

ω

ωt ; sin

sin 2

0

0   

 

 

The velocity of wave propagation

v

_s , the shear modulus G, the damping ratio D and the amplitude of the strain

invariant can be obtained by giving current, resonant frequency and acceleration

A







. [Bratosin 11]

4 2

0

6.59 10

2

;

s

;

0.249

r

;

2 r

r

A

I

A

G

v

D

A

 

I

 









  











The spatial diagrams are obtained for frequencies above 1 Hz. Similarly, the triaxial data is obtained by performing the same above procedure. The parameters required for plotting Fig.1 and Fig.2 is

1.

G



γ

₀

,

ω





g

₀



g

₁

   

γ

₀



g

₂

ω

Where,

 

0 1 0 1 89 2 1 97

0

0 586

0 502

136 MPa ;

0 414

;

1 1 460

.

1 881

.

g

γ

.

g

ω

γ

ω





 

(5)

2.

D



γ

₀

,

ω





d

₀



d

₁

   

γ

₀



d

₂

ω

Where,

 

0 1 0 1 01 2 1 162

0

0 864

0 062

0.22 ;

1 083

;

0 94

1 7 204

.

1 1 018

.

d

γ

.

d

ω

.

γ

.

ω









Above determinations are obtained at constant amplitude of excitation and a constant cell pressure. Other values of the above mechanical characteristics for nonlinear and dissipative behaviour of the real soils can also be obtained by changing the above conditions. Fig.1 and Fig.2 is obtained from the above data for

G



γ

0

,

ω



and





₀

,





D

.

Figure-1 Variations of modulus functions in terms of strain and frequency

Figure-2 Variations of damping functions in terms of strain and frequency

Fig.3 and Fig.4 are plotted for seismic load conditions. It is seen that the modulus and damping functions has no impact for frequency  > 1 Hz. But this frequency has impact on the dynamic structural.

Therefore, dependence of dynamic functions on frequency can be neglected, on imposing this condition on the complex modulus function given by Eq. (19), we get,

 

1   0

tan

0 0

i Dγ

*

G

γ

G

γ

e





(35)

And Eq. (28) become

 

0

 

0

 

0

*













im

re

i

G

i

G

(36)

Where,

 

   

 

0 0 0

0 ₂ im 0 ₂

0 0

, G

1

re

G

D

G

D









(37)

And Eq. (29) become

 

0  *

 

0 0

 G i (38) Eq. (38) is for the non-viscous type Kelvin model given by Levy S. and Wilkinson J.P.D (1976)

The real part of Eq. (38) is given as

 





 







(6)

Eq. (39) represents the relation between the stress and strain amplitudes and it is responsible for the curves. The parameters to plot Fig.3, Fig.4 and Fig.5 are

1.

G

 

γ



G G

₀



_n

 

γ

Where, ₀

 

3

 

_{1 89}

10 %

0 586

136 MPa ;

0 414

1 460

n .

.

G

γ

.

γ



_ 





2.

D

 

γ



D

₀



D

_n

 

γ

Where, ₀



,



_1%

0.22 ;

 

1 083

0 864

_1.01

1 7 204

n

.

D

γ

.

γ



 

_







Figure-3 Normalized form



₀



_.

0

, _ct

G   _ _ in terms of strain and frequency

Figure-4 Normalized form



₀



_.

0

, _ct

D  _ _ in terms of strain and frequency

(7)

Figure-6 Comparative Results

Figure-7 Analogic Model

The validity of the above proposed non-linear dynamic model can be checked by comparing the results of the resonant column data. Further, the resonant column specimen along with the vibration device can be treated as a system which is made up of a single mass supported by a spring and having one degree of freedom.

Hence, the system can be written for the torsional vibration case as

 

0 0

sin

J







c

 





k

 



M



t

(40)

Where,

c

 



and

k

 



are non-linear damping function and non-linear spring function respectively shown in fig.7.

Eq. (40) matches with the Voigt model equation i. e.

J

₀







c







k





M

₀

sin



t

where c = damper viscosity and k = spring stiffness.

Therefore, c and k can be assumed as a function of rotation, hence we have

 

[Nm]

[Nms]

2 0 0

   

    

G h I k

D J

c

p

Where, J0 = moment of inertia of the vibrator, 0  k/ J0 ,

32

4

/

I

_p





known as the polar moment of

the specimen, = diameter and h= height of the specimen.

Put







₀

t

and



 







 

t









/



₀



in Eq. (40), therefore, we get,

 

sin

C

K

(8)

Where,

 

   

 

_{ }

 

_{ }

 

2

0 0 0 0

0 0

2

0 0 0

2 , ,

0 0

, 0

n

st

c k k G

C C D K K G

J J k G

M M

J k

   

     

 



  

 

       

   

The equation (40) can be numerically solved for given normalized amplitude  and relative pulsation, and the solution is of the form

















;

,

(42) The amplitude of rotation 0 can be calculated by dropping the transitory part of the solution. Hence,

 

 

 

 



0 t (43)

Also, 0 can be obtained directly from resonant column and it is given by

2 0  _ 

a r

A

(44)

Where, A = accelerometer value, ra = 0.03175 m, thedistance from the axis of rotation to the accelerometer axis and  = 2f. On comparing the values of 0 which is obtained from both methods, a graph can be plotted which

validates the above discussion. The results of such testing are shown in fig.6.

5. Conclusions

I. Since the non-linear viscoelasticity depend on strain and damping therefore this theory can be applied for building a dynamic model for soils

II. The relationship between the modulus function

G





0

,





and the damping function

D





0

,





are given by the Fourier transforms.

III. It has been observed that during seismic load conditions there is a very small impact of frequency greater than 1 Hz on damping and modulus function; however the major weight is on the strain level on modulus and damping values. But this frequency has impact on the dynamic structural.

IV. The resonant column specimen along with the vibration device can be treated as a system which is made up of a single mass supported by a spring and having one degree of freedom.

Acknowledgments

The authors are thankful to the referees for their valuable comments.

References

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[3] Jankowski, R.; Wilde, K.; Fujino, Y. (1998): Pounding of Superstructure Segments In Isolated Elevated Bridge During Earthquakes. Earthquake Engineering and Structural Dynamics 27, pp 487-502.

[4] Muthukumar, S.; Desroches, R. (2006): A Hertz Contact Model with Nonlinear Damping for Pounding Simulation. Earthquake Engineering and Structural Dynamics 35, pp 811-828.

[5] Christensen R.M. (1971). Theory of Viscoelasticity. An Introduction, Academic Press, New York. [6] Malvern, L.E. (1969): Introduction to the Mechanics of a Continuous Medium, Prentice Hall, New Jersey.

[7] Westermo, Bd. (1989): The Dynamics of Interstructural Connection to Prevent Pounding. Earthquake Engineering Andstructural Dynamics; 18, pp 687–699.

[8] Lakes, R.S. (1998). Viscoelastic solids. CRC Press, New York,

[9] Findley, W.N.; Lai, J.S.; Onaran, K. (1976). Creep and relaxation of nonlinear viscoelastic materials. Dover, New York. [10] Oden, JT. (1972). Finite Elements for Nonlinear Continua. McGraw Hill.