Nonlinear viscoelastic response of carbon black reinforced rubber derived from moderately large deformations in torsion

J. Mech. Phys. Solids, Vol. 43, No. 2, pp. 295-318,1995

Pergamon 0022-5096(94)00070-0

Copyright 0 1995ElsevierScienceLtd Printed in Great Britain. All rights reserved 0022-5096/95$9.50+ 0.00

NONLINEAR VISCOELASTIC RESPONSE OF CARBON BLACK REINFORCED RUBBER DERIVED FROM MODERATELY LARGE DEFORMATIONS IN TORSION K. HAUSLER and M. B. SAYIR Swiss Federal Institute of Technology (ETH), Zurich, Switzerland

(Received 6 June 1994)

ABSTRACT The response of Butyl rubber loaded with carbon black is investigated experimentally, by subjecting it to a finite twist. The experimental set-up and the measurement procedure are described. The material shows a highly nonlinear behaviour for relatively small shear strains. Constitutive relations corresponding to the data are derived from a theory based on a nonlinear tensorial generalization of a three-parameter model. The sample stiffness and normal force for a cylindrical sample subjected to simple torsion are derived for the quasistatic case. To explain the marked drop in stiffness of the material beyond the linear range, a damage and rehealing model is introduced. It consists basically of an evolution equation for the model parameters. With the theory proposed here the measured response for torque and normal force corresponding to a given deformation history can be confirmed satisfactorily.

1.

INTRODUCTION

In this work we investigate the mechanical response of a Butyl rubber to small but finite strain histories in a range of deformation in which the response of the material changes from linear to a markedly nonlinear behaviour. Reviewing the literature on the subject of nonlinear viscoelasticity of rubbers (Lockett, 1973; Findley, 1976; Derman, 1978; McKenna, 1979; Dafalias, 1991) we found only a few references dealing with the type of behaviour observed in the experiments carried out in this work (Donnet and Voet, 1976 ; Rigbi, 1980 ; Shapery, 1985 ; Partom, 1983). This behaviour can be characterized briefly by the fact that the stiffness of the material drops sharply within a narrow range of strain beyond the linear range, by almost an order of magnitude. The experimental facts and considerations about the microstructure of rubberlike materials, especially of the particlereinforced type (Donnet and Voet, 1976 ; Partom, 1983), suggest that even at relatively low levels of stress or strain some mechanism of damage and rehealing or recovery of the material microstructure might be active. In the following a strain dependent formulation is chosen. The material is investigated experimentally in torsion. The time range of interest is from 0 to about 100 s, the shear strain is in the range from 0.0005 to 0.2 and the rate of shear strain from 0.1 to 0.85 s-l. In simple torsion, besides the torque, the normal force acting on the end plates must also be described by the constitutive equation. 295

296

K. HAUSLER and M. B. SAYIR end plates

(a)

03

Fig. 1. The sample. (a) M(I), N(t) are the torque and the normal force acting on the rigid end plates of the rubber sample. The rubber (Butyl, 80” A Shore, 30% carbon black) is vulcanized to the end plates. (b) The angular displacement C$is measured at the right end of the sample. y, is the corresponding shear angle at the lateral surface.

The rates of strain are sufficiently small, so that inertial effects can be neglected (quasistatic case). The material behaviour is described phenomenologically in a continuum mechanics approach based on three-dimensional tensorial constitutive relations with finite strain and strain rate tensors. The principal goal of this work is to simulate theoretically the measured torque and normal force versus time responses, as well as the isochrone torque and normal force characteristics, with the help of constitutive relations satisfying two requirements : (i) they should be sufficiently simple, so that the material parameters can be determined easily, and (ii) the numerical procedure needed for solving the equations should allow meaningful parameter studies. For these reasons constitutive relations of rate type are considered. All computations are performed with the program MATHEMATICAB (Wolfram, 1991). The material, a Butyl rubber reinforced with carbon black, is chosen for two reasons : (i) this type of rubber is widely used in technical applications ; (ii) compared to other rubber types it offers both experimental and theoretical challenges because of its pronounced nonlinear behaviour. In the following two sections the experiment and the observed behaviour will be described and discussed in some detail. In Section 4, a three-dimensional nonlinear theoretical model will be developed by generalizing simple linear viscoelastic behaviour. In Section 5, this model will be applied to the particular case of torsion realized in the experiment. In order to account for most of the nonlinear characteristics observed in the experimental study a damage and rehealing model is introduced in Section 6. In Section 7, the material constants of the constitutive relations will be determined from the experimental data. Further refinements of the theoretical model are made in Section 8. In the last section, the final results will be summarized and reviewed critically.

2.

THE EXPERIMENT

The sample used is shown schematically in Fig. 1. It consists of a short circular cylindrical rod whose length is equal to the radius (20 mm). The material is a Butyl

291

Response of reinforced rubber sample

spindlehead

1

*A J

I displacement transducer ---------_A

j ’

I I ,I

dissacement transducer

/ I

I I I I , I I I I ,

/ stepperr I

Fig. 2. The experimental set-up. Torsion of a rubber sample mounted on a stiff axial bearing. Measurement of torque M(t), normal force N(t) and angle of rotation $(I)

rubber, with a hardness of 80” A Shore and reinforced with 30% carbon black. The end faces are vulcanized to the steel mounting plates. Other rubbers were also considered, but Butyl rubber showed the most pronounced nonlinear behaviour. The sample is twisted by rotating the mounting plates relative to each other through an angle 4. The length of the sample is held constant and the end faces remain practically rigid. The experimental set-up is shown schematically in Fig. 2. The torque M(t), the normal force N(t) and the angle 4(t) of twist are measured in the experiment. One end of the sample is clamped onto a dynamometer. The other end is clamped by the pliers of the spindlehead of a precision lathe. The spindle is driven by a stepper motor over a lever and excenter assembly. The stepper motor and the sampling of the data are controlled by a computer. This allows the choice of the stepper and sampling frequencies and of the number and direction of the motor steps. In order to measure the normal force with some accuracy, a high precision spindle head is used to rotate one end of the sample. This spindle head allows the axial play to be eliminated by slightly preloading the bearings axially. The frame of the machine has to be very stiff. The torque, the normal force and the angular displacement have to be measured simultaneously over time intervals of 0 to about 60 s. For the displacement-time functions of twist, the trapezoidal functions shown in Fig. 3 are used. The rise time of the displacement ramp t, is of the order of 0.024.4 s. The range of angular displacement C$used in the experiment is from 0 to 20”. The corresponding shear strain y, on the lateral surface (Fig. 1) is in the range of O-0.17. For small amplitudes (4 : O-6”) the angular displacement is measured by means of an inductive (eddy-current type) displacement transducer, with a linear range of 5 mm, a cut-off frequency of 100 kHz and with a resolution of better than 0.005”. For larger angles (4 up to 20”) a precision potentiometer with sufficiently high sweeping velocity and a resolution of 0.01” is used. The torque M and the normal force N are measured with a dynamometer. In this device a piezoelectric torque transducer and a piezoelectric force transducer are integrated. The linear range is -5 to 20 kN and - 100 to 100 Nm, the cut-off frequency is at 3 kHz. The resolution of the transducers is 0.02 N and 0.02 Nm. The use of a stepper motor has some disadvantages. Each motor step causes some

298

K. HAUSLER and M. B. SAYIR

t, t,=t*+t(

t, t, (4

(4

Fig. 3. The torsion experiment. (a) Typical signals for the angular displacement C#J, the torque M and the normal force N (schematic). (b) Isochrone plots for t = t,,, corresponding to (a).

vibrations which can disturb the torque and normal force measurement, since piezoelectric transducers are very sensitive to accelerations. Using rubber mounts for the motor, the vibrations can be reduced to a tolerable level. Piezoelectric transducers have the advantage of high speed, sensitivity and linearity but they can not be used for long term measurements. The inevitable drift due to finite electrical resistance limits the maximum measurement interval to less than 5 min. As mentioned above the whole experiment is computer-controlled. With the system computer-multiplexer used (HP 1000) a maximum rate of 1.5 kHz per channel could be achieved, resulting in a time delay of about 0.6 ms between the displacement, torque and normal force signals. This is more than two orders of magnitude smaller than the rise time of the displacement step and will be neglected. The equivalent of the electric hum and noise is 0.005” for the displacement, 0.2 N for the normal force and 0.05 Nm for the torque. Figure 3(a) shows schematically a series of measurements for three trapezoidal deformation histories with different amplitudes of the angular displacement 4, and Fig. 3(b) the corresponding isochrone plots: torque M(t) vs 4(t) and normal force N(t) vs f#~(t)for a given value of r = t,,. ti, designates a given time at which the angular displacement $, the torque M and the normal force N have the values c$(Q, M(t,,), N(t,), respectively. t, is the ramp rise time after which the shear remains constant until time t2. t3 is the time after which the shear remains zero. Since the slope d$/dt is the same for the different deformation histories, the magnitudes oft,, tZ, t, depend on the respective amplitudes. For a nonlinear viscoelastic material, the response to a deformation history may depend on the state of the material from which it is subjected to that deformation. It is known that rubbers after vulcanization are initially not in a stable mechanical state. It is only after subjecting the material to a few cycles of rather large deformation (called “breaking in”) that the microstructure of the material reaches a stable configuration corresponding to a “rest state”. This state, if it exists, is defined by the fact that the response to a given deformation (or stress) history which is applied repeatedly is always the same, provided that one waits long enough for the material to be able

Response

of reinforced

rubber

299

to return to its rest state. The existence of such a rest state means that no degradation or permanent damage of the material occurs. The experiments performed on the Butyl rubber show that for the range of deformations used, this assumption is valid and that this state is reached after leaving the samples for about 24 h at rest. The temperature is held constant at 22°C ( f 1’C) for all measurements. Repeated measurements are made on five samples of the same material. The scatter of the repeated measurements is 5% on the same sample and 12% between different samples.

3.

THE EXPERIMENTAL

RESULTS

In this section some typical measurements of the response of rubber samples used in the torsion experiment described in Section 2 are presented. Although the measured quantity is the angular displacement 4, in representing the results the shear angle ?/, on the lateral surface of the sample (see Fig. 1) is better suited for illustrating connections between forces and deformations. We set (3.1)

l?,(t) = r,of(t)

where y , ,, designates the maximum value of the shear angle, called “amplitude” in the following andf(t) is the (trapezoidal) time function. Obviously 0
t

[sl

-0.2

Fig. 4. Small amplitude response [torque M(t)] to a trapezoidal time function for the deformation a (twist per unit length) in the torsion experiment. Shear amplitude y ,,, = 0.00 I8 I. Rise time for step = 0.06 s.

-N(t),[Nl M(t),[N ml

(4

04

Fig. 5. Torque M(t) and normal force N(f) measured on a Butyl rubber sample subjected to a trapezoidal time function for the deformation a (twist per unit length) in the torsion experiment. Shear amplitude Y,0 = 0.102 and 0.172. Shear rate: 0.4 SK’.

K. HAUSLER

300

and M. B. SAYIR

W,)

M(L) 0.5 0.4 0.3 0.2

04

(a)

140

:

120 100 80 60 40 20

Y&&J 0

0.1

0.05

0.15

0.2

w

Fig. 6. Measured isochrone characteristics s (see Fig. 3). In (b) the small amplitude

of (a) torque vs shear and (c) normal force YS shear for t,,, = 2 range of diagram (a) is magnified : linear range of the material. Range of y, : 0.0005-0.17.

recorded in the experiment, but simply join the measured values characterized by dots. Figure 4 shows the small amplitude response, in this case the torque M(t), to a trapezoidal shear deformationtime function of amplitude y , ,, = 0.00 181 and a step rise time of 0.06 s. Figure 5 shows the large deformation response M(t) and the normal force N(t) for two different shear amplitudes. We note the fact that M, -M, < M, in Fig. 5(a). This is a nonlinear effect. Referring to Fig. 3(a) this means that the material at time tz is not in the same state as at time t = 0. This will be discussed in detail in Section 7. Figures 6(a), (c) show respectively the isochrone plots for M(t,,,) vs y,(l,,) and N(t,,,) vs ri(t,_). Figure 6(b) is the small amplitude extension of the plot of Fig. 6(a). The straight line (slope 2 c,/m, see Section 4) in Fig. 6(a) corresponds to the sample stiffness at small amplitudes as given by the plot in Fig. 6(b). The three curves in each of these figures refer to measurements on three different samples. Several experimental facts are of interest: (i) a linear range (y, < 0.002) for this material exists ; (ii) there is a range of small finite deformations (0.002 < y , < 0.17) in which this material is quasilinear as far as the isochrone amplitude dependence of the torque is concerned; (iii) the stiffness value corresponding to the linear range of deformations is about six times higher than the one for larger amplitudes ; (iv) the isochrone amplitude dependence of the normal force is almost linear with respect to

Response

of reinforced

301

rubber

the shear y, instead of being proportional to y: as predicted by classical theoretical models. In the next section a constitutive equation for the material in the range of time and deformation considered in this experiment is derived.

4.

THE CONSTITUTIVE

EQUATION

Mainly for reasons of computational simplicity a rate type constitutive relation will be chosen. As a starting point for this type of constitutive relation we choose the three-parameter model of Fig. 7, well-known from classical linear viscoelasticity. In its uniaxial elementary version the constitutive relation is ma’ + f3 = 2nd + 2ce.

(4.1)

cr is the uniaxial stress and E the corresponding infinitesimal strain, ( )’ is the time derivative d( )/dt. The coefficients m, n, c follow from the spring constants k,, kz and the damper viscosity yeaccording to m=&

2n=&

2c=&.

(4.2)

In the following we will generalize relation (4.1) in successive steps and try to adapt it to the requirements of the measured data. To generalize (4.1) to the case of three-dimensional finite deformations, the infinitesimal strain E and its time derivative have to be replaced respectively by the strain tensor G and the deformation rate tensor D, defined as follows : G = OS(B-I),

B = FFT,

D = 0.5(L+LT),

L = F’F-’

(4.3)

(F : deformation gradient, I : unit tensor, L : velocity gradient). The material is supposed to be homogeneous, isotropic and incompressible. The latter assumption means that the stress tensor o contains a hydrostatic pressure p which is not connected to the deformation by a constitutive relation and can only be determined after solving the boundary value problem. We write c = -pI+T,

(4.4)

where T is the so-called “extra stress” tensor. The tensorial constitutive relation generalizing (4.1) for finite deformation and formulated in terms of the extra stress tensor T is therefore mT’+T

= 2nD+2cG.

(4.5)

( )” is one of the many possible objective time derivatives. With the so-called upper

Fig. 7. The 3-parameter

model for solids.

302

K. HAUSLER

and M. B. SAYIR

convected derivative (Truesdell and Nell, 1965 ; Joseph, 1990) To = T’ - LT -TLT and (4.5) we get mT’ + T = m(LT + TL=) + 2nD + 2cG.

(4.6)

It can be shown that for example the use of the lower convected derivative To = T’-LTT-TL leads to a normal force which changes its sign and is thus in contradiction with the measured behaviour. For small but finite deformations the convective terms in the stress derivative and the last term involving G in (4.6) can be shown to introduce second order terms with respect to the magnitude of the deformation. Thus more terms are needed in (4.6) if we require consistency up to this second order. We therefore set mT’ + T = m(LT + TL=) + 2nD + 2cG + 4qG2 + 4r(DG + GD) + 4sD’.

(4.7)

As will be shown in Section 6, the inclusion of these higher order terms is indeed necessary to adjust the theoretical model to the experimental data given for example in Fig. 5 and Fig. 6. In addition we also have to allow the parameters c, m, n, q, r, s to depend if necessary on the invariants of G, for example on tr[G’] and, through an evolution equation, on time t. We note however that (4.7) represents a system of differential equations linear in the components of T.

5.

SIMPLE TORSION

In this section the constitutive relation (4.7) is used to derive the expression for the torque and the normal force acting on the sample. The mode of deformation is assumed to be simple torsion. Thus points of the body in the reference configuration with cylindrical coordinates R, 0, 2 are mapped into points with coordinates r, 4, z, by rotating each cross section perpendicular to the Z-axis through an angle a Z (Fig. 8). Thus the deformation is given by r=R,

cp=Q+crZ,

O
0<@<22n,

O
(5.1)

For simple torsion, the twist per unit length or(t) is independent of Z. The shear strain y is given by

dZ

Z Y

Fig. 8. Simple torsion.

a(f)

: twist/unit

length.

Response of reinforced rubber 10~0

= 4OR

140

with

= YVL 0 = Y,JW.

303 (5.2)

ylo is the maximum shear strain at the outer surface of the sample andf(t) deformation time function already introduced in Section 3. The fact that a is independent of Z is based on the assumption that the sample inertia is neglected (quasistatic case), as expressed by the equilibrium equation (6.8). In a frame of cylindrical coordinates the matrices of the deformation gradient F, the deformation tensor G and the deformation rate tensor D are given by

Introducing G and D into the constitutive equation (4.7) we obtain simple linear differential equations in time for the components of the extra stress : m(T,,)‘+

T,, = 2my’T,,+cy2+qy2

m(T,,)‘+T,,

+2ryy’+~(y’)~ +qy4,

= my’T,,+ny’+cy+qy3+ry2y’,

m(TzJ’ + T,, = qy2 + 2ryy’ + So, T,, = T,, = T,, = 0.

(5.4) (5.5) (5.6) (5.7)

The value of the largest shear deformation at the surface of the sample did not rise beyond y10= 0.17 in the experiment described in Section 3. Thus yyo < 0.03 and $,, < 0.005. Based on these evaluations we will neglect in the following terms of higher orders than y:,,. Thus (5.4H5.6) reduce to m(T,,)‘+

Trpp= 2my’T,,+cy2

+qy2 +2ryy’+W2,

m(T,,)’ + T,, = ny’ +

cy,

m(TA + Tz, = qy2 + 21-y~’+ s(#.

(5.8) (5.9) (5.10)

The torque M acting on the cylindrical sample is defined as

s s R,

2rrR2T,,dR.

Iv(t) =

(5.11)

0

The normal force is defined as

RI

N(t) =

hR(T,;

-p) dR.

(5.12)

0

To evaluate the right-hand sides of (5.11) and (5.12) more information is needed on the material parameters m, n, c, q, I and s. This will be discussed in the next section.

304

K. HAUSLER and M. B. SAYIR

6.

COMPARING THE EXPERIMENTAL DATA WITH THE THEORY: THE DAMAGE AND RECOVERY HYPOTHESIS

In the comments to the experimental data shown in Figs 4-6 a number of nonlinear effects were noted. In order to obtain these effects both qualitatively and quantitatively from the theoretical frame set by the constitutive relation (4.7) some important features must be included. In the following paragraphs such features will be introduced in successive steps. (a) Large amplitude isochrone response The expression (5.6) shows the necessity of the higher order terms D2, (DG + GD), G2 introduced in (4.7), since otherwise the normal force would be entirely given by the hydrostatic pressure integrated over the cross-section. This is in contradiction with experimental evidence. Furthermore, the torque A4 given by (5.9) (5.11) would be linear in CIif m, n and c were kept independent of the deformation [see (5.2)J. The experiments show however, as noted in Section 3, that the isochrone torque vs shear is highly nonlinear [Fig. 6(a)]. This can be accounted for theoretically by assuming that the coefficients n and c on the right-hand side of (5.9) depend on the invariants of G, or more simply, on some measure of the magnitude of the strain. In the present context we choose g : = tr (G’) to represent this magnitude. The coefficients m, q, r and s may still be treated as constants without contradicting the experimental results. For the sake of simplicity the strain dependence of n and c may be characterized by a single function (we recall that c is somehow connected with the “elastic stiffness” of the material and n with its viscous characteristics) h(g) by writing {n, c> = {n,, c,}h(g) ; g = tr (G2L (6.1) where n, and c0 are constant. h(g) is a monotonously decreasing function of g which starts from a value which may be chosen as 1 without loss of generality. Let its value for g + co be labelled h,. Obviously h, < 1. The role of h(g) is to render the material softer (decreasing c which represents the “elastic stiffness” for large times) and less viscous (Hausler, 1983) (decreasing n which is connected with the “inherent viscosity”) as the strain increases. The following simple expression fulfils the requirements mentioned so far :

1+gho

h(g)= l+g.

(6.2)

But the next step will require a more general expression which can be formulated by introducing a functionalf(g} which vanishes for g 3 0 (zero strain history) and increases monotonously with g. Hence we obtain for h more generally the functional expression

hb) =

1+fb>ho 1+fbl .

(6.3)

(b) Large amplitude time response : damage and recovery hypothesis The experiments show that the observed drop in stiffness is reversible. If after an experiment with a large amplitude step of say ylo = 0.17, we let the sample rest for

Response of reinforced rubber

305

one day (see discussion at end of Section 2) and then make a measurement in the linear range with say yI = 0.001, the sample shows the high value of the stiffness corresponding to the linear range of deformation starting from the rest state [see Fig. 6(b)]. But if the stiffness is measured a short time after a large amplitude deformation, then the shorter the delay the lower the stiffness value observed. One way to describe this phenomenon is to assume that the material undergoes some change at the microstructural level as the strain is increased. We may conjecture that some rupture, reversible in time, of certain bonds in the rubber networks (Donnet and Voet, 1976 ; Rigbi, 1980) is responsible for the observed severe drop in the stiffness. This kind of damage and rehealing mechanism can be represented by assuming that the material parameters n, c not only depend on the magnitude of the strain but also, through evolution equations, explicitly on time. Thus we consider that it takes a certain time for the damage due to increasing or sustained strain to develop. The same holds for the ensuing rehealing process as the strain is decreased. The corresponding time dependence has already been implied by consideringf{g} and hence h(g) in (6.3) to be functionals of the strain history g(t). Note that with increasing values off(g(t)) the resulting value of h(g(t)} decreases. Thus increasing S{ g( r)} means growing damage. Hence f{g(t)} will be called the “damage functional”. We introduce the following evolution equation for the damage functional :

r- ’ +f’ =

d, g + &g’ .

(6.4)

Here z, d, and d, are material constants with a well-defined physical meaning. The constant d, corresponds to the value of the damage functional for a unit step in the magnitude of the strain. The constant d, is the (asymptotic) value of the damage functional for a sustained level of constant strain corresponding to unity. Finally 7 is a time constant characterizing both the rate of change of damage and of rehealing (here obviously the same time constant is chosen for both damage and rehealing ; in more complicated cases one might wish to introduce different constants for these two processes). (c) Application to simple torsion

For simple torsion, neglecting y’: with respect to y:, one obtains for the strain magnitude g = 0.5 y:, or using (5.2), g = 0.5 R’ cx*.In this case the functionalfmay also be replaced by& = f/R*, so that (6.4) becomes

= OS~I,a~+d;~acc’. fMt)l = RZJ,b401; z(fT)'+,fr

(6.5)

With the help of the above damage and recovery model and using (5.11) we can now proceed to integrate (5.9). This leads to the following differential equation for the torque M: i?lM’+lM=

271R’(ny’+cy)dR.

Using (6.1) (6.3), (6.4) and (6.5) we obtain for the right side of (6.6)

(6.6)

306

K. HAUSLER and M. B. SAYIR

mM+M

= (noa’+c,a)Zo{a(t)}

Zo{a(t)} = 0.57&h,

1+2

;

1

1 -h, k&f&(~))

l-

In (1 + R:f,(a(t)}

JWTMO~

)I .

(6.7)

Thus the functional Z, with its dependence on the history of the twist per unit length x(t) plays for simple torsion the same role as h{g(t)} for the general constitutive equation (4.7). To derive the expression for the normal force N we have to evaluate the right-hand side of (5.12) which contains the hydrostatic pressure p. This quantity appears also in the equilibrium condition (quasistatic case)

which can also be written as -P,R

=

f

Trpq

by using (4.4), (5.7). To integrate (6.8) one needs (5.8) from which one obtains eventually -mR(p,R)‘-

Rp,R = 2mRa’T,,+

R2(c,,h{g(t)}az+qa2+2raa’+sa’*).

(6.10)

Now using (5.10), (5.12), integrating (6.10) and using the boundary condition o,,(&, t) = -_p(R,, t) = 0,

(6.11)

we obtain mN’+ N = -ma’M-0.5c,a2Zo+

71R: ~(qa*+2raa’+sa”).

(6.12)

The torque M(t) and the normal force N(t) are given by the solution of the differential equations (6.5), (6.6) and (6.12). The differential equations were solved with MATHEMATICAB with different sets of material constants, the solutions for the torque M(t) and the normal force N(t) were then compared with the experimental data presented in Section 3. Details of this comparison will be presented in the next section.

7.

DETERMINATION

OF THE MATERIAL EXPERIMENT

CONSTANTS

FROM THE

The material constants are successively determined as follows (a) m, no, co are determined from the data for the small amplitude response. (b) ho, z, d, and d, are determined respectively from (i) the drop in stiffness beyond the linear range ; (ii) the slope of the torque M(C) at t = 0 ; (iii) the isochrone amplitude

Response of reinforced rubber

307

dependence of the torque in the region of transition between small and large amplitudes; and (iv) the rate of recovery of the material. (c) q, r, s are determined from the time response and isochrone amplitude dependence of the normal force N. (a) Determination of m, n,, c,, In a first step the constants m, n,, c0 are determined from the data for the small amplitude response Fig. 4. for y10= 0.0018. From Fig. 6(b) we can see that the isochrone amplitude dependence of the torque M is nearly linear up to y, = 0.002. In this range of shear strains the torque M(t) is given by (6.7) with Z, = OSxRf. The normal force N(t) x 0. Referring to Fig. 3(a), Fig. 4 and (6.7) it can be shown that -the constant m is proportional to the slope of M(t) at t = t: (t, : end of the ramp of the time function for the shear yl) ; -the maximum value of the torque, M(t ,) is proportional to m/no ; -the value of the torque A4 for large times is proportional to cO. Taking these values from the experimental data (Fig. 4) as initial values, a least square fit yields the following values : m = 0.65 s, no = 2.6 x IO’ N m-*, c,, = 2.5 x 10’ N rn-*. (b) Determination of ho, r, d, and di For a + co, according to (6.7), the value of Z,, tends to (0.5 TcR:) h,,, hence to the product of Z,{a -+ 0} with the material constant h,. Thus, considering the ratio of the slopes of the isochrone torque [see Fig. 6(a)] for large shear strains (y, > 0.03) and for small values (y, < 0.002) one obtains directly ho = 0.164. The assumption that the experiment can indeed be considered to be quasistatic can now be justified. It has to be shown that the time of propagation of a disturbance across the length L = 0.05 m of the sample is much smaller than the rise time of the deformation ramp t, (Fig. 3). The large amplitude shear modulus is [(6. l), (6.3)] c = c0 h,, = 0.41 x 10’ N mP2. The shear wave propagation speed is c, = (c/p)“” = 54 m SC’. The propagation time across the sample is therefore L/es = 0.37 ms which is indeed much smaller than t, = 0.4 s. The physical significance of the constants z, d, and d, was already discussed briefly in the comment to (6.4). Figures 9(a), (b) shows I0 taken from (6.7) for trapezoidal shear strain histories of different amplitudes with different time scales. Hereby the values for r, aI, and d, as determined by the procedure described below were used. Figure 9(c) shows an example of a trapezoidal history. The time for the material to recover, i.e. for ZJ(O.5 TcR:) to return to its initial value 1, increases with increasing deformation amplitude. This also explains the following experimental observation (see also the discussion at the end of Section 2). Assume that we measure the torque M(t) of a sample subjected to a small deformation step in the linear range, (i) from a state of rest of the material and, (ii) after a time delay after subjecting the sample to a large pulse. The experiment shows that for a delay of the order of several minutes the response to the small step is reduced by about 30% by the effect of the previous large pulse. It takes about 24 h

308

K. HAUSLER and M. B. SAYIR I,

(ty(0.5n Ft)

I,

(1)/(0.5 z RI)

I

tla~~~o

~~~ 5

10

15

(4

(4

20

25

0

20000

40000

(W

(4

Fig. 9. (a), (b) The value of the functional I0 for different trapezoidal shear histories vr(t) with amplitudes y10: 0.0018~. 17, according to (6.7) ; (b) shows the recovery of the material, i.e. the continuation in time of plot (a). It takes about 24 h for the material to recover from the largest step (v,,, = 0.17); (c) typical trapezoidal shear history corresponding to plot (a). For small values of a and fT the rate of change of damage (f,)’ is proportional to d, ; (d) parametric representation of I,,.

for the material to recover to its initial state from the effects of the large deformation, i.e. to get the response to the small step from the state of rest of the material. This effect can be explained by our theory, as shown in Figs 9(a) and (b). This observation is of significance when determining the linear viscoelastic properties: the material must initially be in a state of rest before any measurement is carried out. Even relatively small strains will disturb this state during a certain time during which the apparent sample stiffness will be considerably lowered. The constant r is the rate of recovery or rehealing. As mentioned in Section 2 the time of recovery from the effects of the largest shear amplitude (0.17) reached in the experiment is about 24 h [Fig. 9(b)]. F rom (6.5) one obtains then r = 1.3 x lo* s. Considering the limit [t + 0] in (6.5) one sees that d, is the value of the slope of& with respect to a2/2 at t = 0. Transferring this information to (6.7) one can show that d, is connected with the time response of the torque M(t) for times near t = 0. The value of d, corresponding to the measured value of the initial slope of M(t) [see Fig. 10(b) ; the dots are the measured values] is d, = 1.2 x 105. According to (6.5) d, corresponds to the value of the slope offr with respect to a2/2 for large times (constant a). Hence, if we refer to Fig. 6(a), its influence is strongly felt in the transition region between the small amplitude (linear) range, with negligible damage, and the large amplitude region, where the damage corresponding to a given amplitude is fully developed. Hence to determine ds we consider the measured

Response

of reinforced

rubber

309

6

?

4 2.5 0

b.."'

5

10

15

20

t

ISI 2 0-

t (61 0.2

0.4

0.6

0.6

1

1.2

1.4

-2.5

64

(W

Fig. 10. Measured (0) and computed values (curve) of (a), (b) : torque M(L) for trapezoidal displacement a(t) (twist/unit length) time function corresponding to strain amplitudes of y10 = 0.10 and 0.17.

isochrone torque M(t,,,) vs the shear strain amplitude 7,(&J depicted in Fig. 6(a) in the transition region where the values of the shear strain lie between 0.0017 and 0.015. Using a trial and error scheme we find d, = 2.2 x 10’. Using the values of the material constants m, n,, c,,, ho, T, d, and di determined so far we plot in Fig. 10 the torque M(t) as given by the solutions of the equation (6.7) as a function of time t. The dots are the measured values. This figure shows that the torque M(t) calculated matches the experimental values within lo%, except for t > t3 (see Fig. 3 for the definition of t,) when a(t) E 0. A modification to the theory to improve the time response of the torque M(t) and normal force N(t) is proposed in Section 8. Note the fact that in the experiment (see Fig. 5) M,( = 12.5 N m) > M, -M, (= 9.9 N m). This is a nonlinear coupling effect of the response to two consecutive ramps. In the linear theory the response to the second ramp should be identical to the response to the first ramp (performed from a state of rest). From Fig. 10 it can be seen that the theory proposed here also exhibits this nonlinear effect (M, = 13 N m, Ml--M3 = 10.5 N m). (c) Determination of q, r and s

From (6.7) and (6.12) it can be seen that q, Y, s affect the normal force N(t) but not the torque M(t). First an initial value for q was chosen for a shear amplitude y ,,, = 0.17, for the time interval where M’= 0 ; then Y and s were chosen to get a qualitatively correct time response for the normal force according to Fig. 5(b). These preliminary values were then used as starting values in a least square procedure which lead eventually to q = -0.23

NmP2,

r = -5.0x

106NsmP2,

s= -0.4x

10”Ns2mP2.

Using these values we plot in Fig. 11 the normal force N(t) as given by the solutions of the equation (6.12) as a function of time t. The dots are the measured values. We note that the response N(t) for the smaller amplitude y10 = 0.10 is almost 30% smaller than the measured values. Besides, the same remark made above for the torque M(t) for t > t3 also applies for the normal force N(t). Figure 12 shows the isochrone plots for the torque M(ti,,) vs yl(ti,J and the normal

K. HAUSLER and M. B. SAYIR

310 -N(t)

-N(t) 140 120 100 60 60 40 20 0

t 0.2

0.4

0.6

0.6

1

1.2

isI

1.4

04

(4

Fig. 11. Measured (0) and computed values (curve) of (a), (b) : normal force N(t) for trapezoidal displacement a(t) (twist/unit length) time function corresponding to strain amplitudes of y10= 0.10 and 0.17.

force N(t,,,) vs yl(S,) for tiso= 2 S. Again the curves represent the theoretical values and the dots (0) the experimental data. We note that the theoretical results related to the isochrone amplitude dependence of the torque correspond at least qualitatively to the experimental behaviour. In particular in both cases the stiffness drops already at relatively small amplitudes. The quantitative matching is not quite satisfactory. In the case of the isochrone amplitude dependence of the normal force N(tiso) vs yi(&) results are even worse. Obviously, the material constants determined from the time behaviour for one shear amplitude do not apply with sufficient accuracy to the others as noted above in connection with Fig. 10. W,)

M&o)

~~.~~~~--;i..) 0.05

0.1

0.15

0.2

0

o.w5

(a)

0.01

0.015

0.02

(b)

-W,)

60 60 40

-.

20 .. 100 ii/i 0

r,kJ 0.05

0.1

0.15

03

Fig. 12. Isochrone characteristics. (a), (b) : torque M(t,-) and (c) : normal force N(t,,) for ri, = 2 s. Here (b) shows the linear range of the material in terms of M(f,,) for small amplitudes up to yiO= 0.002.

Response of reinforced rubber

311

In order to improve the quantitative correspondence of the theoretical predictions with the measured data we will propose additional features in the next section.

8.

ADDITIONAL

FEATURES

TO THE CONSTITUTIVE

RELATIONS

Two additional features are proposed. The first serves to improve considerably the isochrone normal force dependence on the amplitude. The second will modify the time dependence of both the torque M(t) and the normal force N(t) and will lead to a better matching between theory and experiment over the whole range of times considered here. (a) Correction to normal force N The fact that the normal force N(tiso) depends on the amplitude in a quasilinear fashion as shown in Fig. 6(c) can be accounted for by introducing a strain dependence of the coefficient q = q(g) in the constitutive relation (4.7) as follows 4(g) = 40@ = 40(JWZP.

(8.1)

Introducing this into (6.10) and integrating we obtain for the normal force N 4

miV’+N

= -mct’M-0.5r,,tx210+F

n02+fi

qDc”-

2

RfSt12(‘+B)+ 2raor’+sa’2

1. (8.2)

For p = 0 we get back to (6.12). By trial and error we obtain : q,, = 0.045 N m-*, /l = -0.5 and proceeding as previously for r and s : r = - 1.l x lo6 N s rnm2, s = -0.8 x lo6 N s* m-‘. Comparing Fig. 13(c) with Fig. 12(c) shows that the amplitude dependence of the isochrone normal force is markedly improved by introducing the additional assumption (8.1). Comparing Figs 13(a), (b) and Figs 12(a), (b) shows that the isochrone torque M(tiso)~vs y 1(ti,,) is also improved by (8.1). (b) The 5-parameter model In order to improve the time dependence of torque M and normal force N, a Maxwell element can be added in parallel to the three-parameter element of Fig. 7 (see Fig. 14). In the one-dimensional linear case the stress aP added to g of (4.1) is given by the following differential equation mr,ap*+aP= 2n,E’.

(8.3)

Using the same constitutive relations (4.7), (6.1) and (6.3) we obtain the corresponding expressions for the additional torque IwJt) and Np(t) from (6.7) and (6.12) as (we simply set c0 = 0 and replace m and n, by mP and nPo, respectively) m,Mi+M,

= nPOa’IO,

(8.4)

K. HAUSLER and M. B. SAYIR

312

l----_~~td ~.~ y,(tJ

0.05

0

0.15

0.1

0.005

(4

0.01

0.015

0.02

04 -W d

(4 Fig. 13. Isochrone characteristics for extended theory. (a) Torque M(t,,,) and (b) normal force N(t,,,) for t150 = 2 s. Here (b) shows the linear range of the material in terms of M(r,,,) for small amplitudes up to ?,a = 0.002

Fig. 14. The five-parameter model.

4

m,Ni+N,,

= -m,a’M,+~(2rar’+s31’?).

(8.5)

The resulting total torque M,(t) and total normal force N,(t) are then given by (6.7), (6.12), (8.4) and (8.5) as M,(t) = M(t)+M,(t),

N,(t) = N(t)+Np(t).

(8.6)

Note that even more elements can be added in parallel to extend the validity of the theoretical structure to larger time ranges and various deformation histories. In the present case corresponding to the five-parameter model we leave the four material constants of damage and rehealing ho, z, d, and di unchanged. We also choose the same value for the exponent /I in the expression (8.1) for the parameter q introduced above. For the remaining eight material constants we proceed as above and start with the values from the three-parameter model and use again a least square scheme to obtain the following new values

Response

of reinforced

rubber

313

-NO) 140 120 12.5

loo

10

60

7.5

60

5

40

2.5

20

0 -2.5

(4

m = 0.25 s, nP0

(b)

(0) and computed values of (a) : torque M(r) and (b) : normal theory. Strain amplitudes of y10 = 0.10 and 0.17.

Fig. 15. Measured

no = 0.75 x

2.1 Nmm2,

=

q. =

107Nmm2,

c0 = 1.87NmP2,

-O.O45Nm-‘,

s = -0.8x

r = -

force N(t) for extended

mp = 2Os,

1.1 x 106NsmP2,

10”Ns2mP2.

Figure 15 shows that the addition of a Maxwell element to the original threeparameter model improves the time dependence of the theoretical torque M(t) and normal force N(t) with respect to the experimental values considerably, especially for times after t = t3. The results for the isochrone behaviour remain practically unmodified by the change from the three-parameter to the five-parameter representation so that Fig. 5 remains valid for the latter case. The extra effort of numerical evaluation associated with the introduction of the three additional constants m,, npo and p proposed in this section can be considered as negligible. The improvement of the quantitative correspondence between theoretical predictions and measured values is certainly worth this extra effort.

9.

CONCLUSION

A quantitative experimental study under simple torsion conditions and trapezoidal shear deformation histories lead to the following relatively simple tensorial constitutive relations which should describe the nonlinear viscoelastic behaviour of Butyl rubber: mT’ + T = m(LT + TLT) + 2nD + 2cG + 4qG’ + 4r(DG + GD) + 4sD2, m,(T,)’

+

T,, = m,(LT, + T,LT) + 2n,D + 4qG’ + 4r(DG + GD) + 4sD2, T, = TfT,.

(9.1)

Here T, is the total Cauchy stress tensor. The material parameters m, mp, r and s are constant. n, np and c are supposed to be functionals of the deformation history and behave according to the linear evolution equation (6.4) connected with (6.1) and (6.3).

314

K. HAUSLER and M. B. SAYIR

These simple relations describe apparently with sufficient accuracy the damage and rehealing phenomena observed in the experiment. A total of I3 material constants [m, no, co, mP, n,,, h,, T, d,, di, q,,,p , Y, s], is needed to characterize the full nonlinear viscoelastic response in a relatively wide time range. We note that five of these constants [m, no, co, mP, n,,] already intervene in a purely linear viscoelastic theory. The quantitative agreement of theory and experiment is such that the theoretical structure proposed here should be considered not only as a curve fitting exercise, but also as a valid system of constitutive modelling, since it leads to well-defined physical features observed in the experiment. The main qualitative features of the theory proposed that were confirmed by the experiment are (i) the fact of the marked drop in the sample stiffness at small deformations; and (ii) the nonlinear coupling effect in the time response as noted in the comment to Fig. 10. The case of simple torsion treated here is one of the few boundary value problems that can be solved exactly and experimentally realized with boundary conditions corresponding very accurately to the ones assumed in the theory. Since the torque A4 and the normal force N are coupled [(6.7), (6.12)], fitting both M(t) and hT(t) with the same set of material parameters represents a strong test of confirmation. Nevertheless, generally speaking, the single mode experiment in simple torsion reported here is of course not sufficient to establish with sufficient accuracy that the set of constitutive equations (6.1), (6.2) and (9.1) may be applied to more involved modes. Experiments with other deformation modes would be needed to confirm with more confidence these equations and the values of the corresponding parameters. The classical cases of simple shear and simple or biaxial extension, most frequently considered in the literature, are problematic as far as the experimental realization of the theoretically assumed boundary conditions is concerned. The assumption of quasistatic loading may also bring some difficulties if the samples are chosen to be long in orderZ to minimize the influence of boundary condition inaccuracies. In the Appendix to this paper we mention briefly the theoretical results obtained from the above set of constitutive equations for simple and biaxial extension. The behaviour corresponds at least qualitatively to what might realistically be expected. Experiments with more complicated modes (for example torsion and extension of cylindrical tubes and rods) and samples with different cross-sectional geometries are planned and would involve more sophisticated numerical analysis. The implementation of finite element schemes to solve more complicated three-dimensional problems should easily be possible since the proposed constitutive model has the advantage (compared to integral forms) to be relatively simple. ACKNOWLEDGEMENT The authors wish to express their thanks to Prof. S. R. Bodner, Technion, Haifa, Israel, for helpful discussions concerning the problems of damage and rehealing mechanisms.

REFERENCES Dafalias, Y. F. (1991) Constitutive model for large viscoelastic materials. Me& Res. Commun. 18(l).

deformations

of elastomeric

Response of reinforced rubber

315

Derman, D., Zaphir, Z. and Bodner, S. R. (1978) Nonlinear anelastic behaviour of a synthetic rubber at finite strains. J. Rheol. 22(3). Donnet, J. P. and Voet, A. (1976) Carbon Black. M. Dekker Inc., New York. Findley, W. N. et al. (1976) Creep and Relaxation of Nonlinear Viscoelastic Materials. NorthHolland, Amsterdam. Hausler, K. (1983) On the viscoelastic properties of prestrained rubber. J. Appl. Math. Phys. W). Joseph, J. J. (1990) Fluid Dynamics of Viscoelastic Liquids. Springer, New York. Locke& F. J. (1973) Nonlinear Viscoelastic Solids. Academic Press, New York. Mckenna, G. B. (1979) Nonlinear viscoelastic behaviour of PMM in torsion. J. Rheol. 23(2). Partom, Y. (1983) Modelling nonlinear viscoelastic response. Polym. Engng Sci. 15(23). Rigbi, Z. (1980) Reinforcement of rubber by carbon black. Advances in Polymer Science, Vol. 36 (ed. H. J. Cantow). Springer, New York. Shapery, R. A. (1985) A micromechanical model for nonlinear viscoelastic behaviour of particle-reinforced rubber with distributed damage. Mechanics of Damage and Fatigue, IUTAM Symposium (ed. S. R. Bodner and Z. Hashin). Pergamon Press, Oxford. Truesdell, C. and Nell, W. (1965) The Nonlinear Field Theories of Mechanics. Springer, New York. Wolfram, S. (1991) MATHEMATICA, 2nd edn. Addison-Wesley, New York.

APPENDIX

: SIMPLE

EXTENSION

AND BIAXIAL

EXTENSION

In the following the constitutive relations proposed in Sections 4, 6 and 8 are applied to the cases of simple extension and biaxial extension. The theoretical results for Ihe sample stiffness for these cases are then briefly discussed. Al. Simple extension We consider a rod of material as shown in Fig. Al. 1 subjected to the forces N: acting on the faces Z = 0, L,. The lateral faces of the rod are free of tractions. The deformation for simple extension-satisfying the condition of incompressibility-is x =

x/n, y = Y//i, z=12Z,

O
In a frame of Cartesian coordinates the matrices and the deformation rate tensor D are given by [F]=diag[l,l,L],

o< Y
of the deformation

2[G]=diag[E.-2-1,~-2-1,E?-1],

gradient

O
[D]=i.‘l-‘diag[-l,-1,2].

(Al.l) tensor G

(Al.2)

Since the problem is homogeneous the (quasistatic) equilibrium conditions are satisfied The boundary conditions for the stress-free lateral faces of the rod lead to N; = H;E.?(T,,As for the case of simple torsion

we consider

T,,).

only small finite deformations.

(A1.3) To this end we set (Al .4)

i.* = 1 +e. Solving

the set of differential

equations

obtained

from the constitutive

Fig. Al .l_ Simple extension.

relations

(9.1), (6.1), (6.3) by

K. HAUSLER and M. B. SAYIR

316

a(t)

a(t)

goE/io

25t[gl

0

;;

5

10

15

20

25

f=j----'*

-0.1 (4

64

150

0

loo

-50

50

-100

t

-150

0

-200

-50

-250

6) Fig. At.2

[sl

(4

Traction force N,(t) response to trapezoidal strains e(r) of different amplitudes in simple extension. Nz(t) is computed based on the constitutive relations (4.7), (6.1), (6.3), (6.4).

N,(t,)

-0.2

-0.15

-0.1 -0.05

0.05

0.1

0.15

' 0.2

e&J

Fig. Al .3. Isochrone characteristics for simple extension for deformation functions as shown in Fig. Al.2 for t,,, = 2 s.

using (Al ,2) for trapezoidal deformation functions e(t) [defined by (A1.4)] the force N:(t) acting on the sample can be computed. The same set of material parameters as evaluated from the simple torsion experiment are used for these calculations. Both the deformation function e(r) of simple extension and the force N,(t) are shown in Fig. Al.2 Figure Al .3 shows the isochrone characteristics for simple extension for the case of the deformation e(t) shown in Fig. A1.2, for ti, = 2 s. Both the time response N,(t) shown in Fig. Al.2 and the isochrone characteristic N,(t,,,) vs e(t,,,) as shown in Fig. Al.3 indicate that the proposed constitutive theory leads to reasonable results for simple extension. A2. Biaxial extension We consider a slab of material as shown in Fig. A2.1 subjected to uniformly distributed stresses with the resultant forces N,, N, acting respectively on the faces Z = 0, L1 and Y = 0, L,. The faces X = 0, L, of the slab are free of tractions. The deformation for biaxial extension is x = xi,,

y = Y&, z=I,Z,

OGX
o< YQL,,

O
(A2.1)

Response

of reinforced

317

rubber

Y Fig A2.1. Biaxial extension.

In a frame of Cartesian coordinates the matrices G and the deformation rate tensor D are given by [F]=diag[i,,,l,,1,],

of the deformation

2[G]=diag[l;‘-1,1;‘-l,i:-11,

gradient

F, the deformation

[D]=diag[i;i.,‘,I;l,‘,il,lj’].

(A2.2)

The equilibrium equations are trivially satisfied. The boundary condition of stress-free yields for the resultant forces Nz (on face A; : Z = 0, L,) and N, (on face A, : y = 0, L,)

Introducing

the condition

of incompressibility 1, = l+e,,

(A2.3)

i,, = l/(izi.,)

and the strains c,, c,, e; defined by

i, = I+?$,

Lz = I-te,,

(A2.4)

e,(t)

e,(t)

0

faces X = 0, L,

N._ = A,,i,&(T_;-T,,).

y, = A,,i.,13(T,,-T,,).

tensor

5

10

15

20

10

25

15,

20

25

(ii)

Cd N,(t) WI

t ISI

(b) Fig. A2.2. Biaxial extension.

(b) Calculated

quaistatic response N:(r), N,(r) to (a) trapezoidal e,(t), e,.(r).

strain functions

318

K. HAUSLER and M. B. SAYIR

into the constitutive relations (9.1), (6.1), (6.3) and then solving the corresponding set of differential equations, the following responses N,(r) and N,(r) to trapezoidal deformation functions e,(t) and e,(r), shown in Fig. A2.2, are obtained. Again the same set of material parameters as evaluated from the simple torsion experiment are used for these calculations. Note that the same time functions with different amplitudes were used for the trapezoidal deformation function e,,(t) and e=(r).The following cases were considered : (i) e,(t) = e,(f), (ii) e,(r) = -4-(f), (iii) e,(t) = 0 and (iv) simple extension : I, = l/A, or e, = (I/&) - 1,as special case of biaxial extension.