Conversion of work of deformation to heat in polymers

Conversion of work of deformation to heat in polymers P. Zoller and H. Bont Kunststofflabor, Neu-Technikum Buchs, CH-9470 Buchs, Switzerland (Received 30 July 1973; revised 3 December 1973)

The time dependence of the conversion of mechanical work of deformation to heat in polymers is investigated using viscoelastic models. The deformation chosen is a constant strain rate, 40, in tension for t<~to and a constant strain ~o=~0to for t>~to. For a simple Maxwell model with only one relaxation time, ~-, the ratio of heat produced to work done ('conversion fraction') is calculated as a function of time for different values of to/r. At t=to the conversion fraction is high only when ~-0.1to but extends with reasonable values into the region -r < 0"1to. A formula is given which allows the calculation of the conversion fraction at t=to from the distribution of relaxation times/-/(~-).

INTRODUCTION It is well known that polymers in states ranging from the glassy or crystalline states at low temperatures to polymer melts have mechanical properties intermediate between those of Hookean elastic solids and ideal viscous fluids. Manifestations of this more complicated behaviour of polymers are, e.g., the existence of creep, stress relaxation, melt elasticity etc. The phenomenological theory of linear viscoelasticity 1 is a suitable framework for describing the observed phenomena. One consequence of viscoelasticity is the fact that energy is dissipated (converted to heat) whenever a viscoelastic body is deformed. In a Hookean solid all work of deformation is stored as potential energy; in an ideal viscous fluid all work is converted to heat instantaneously. Here, too, the behaviour of a viscoelastic body is intermediate between the limiting cases: some work is stored as potential energy, some is dissipated, and the fraction which is dissipated can be an explicit function of time. The most familiar examples of this behaviour are the experiments involving periodic (sinusoidal) deformations of a polymer. In such experiments a loss angle is determined as the phase angle between stress and strain, and the tangent of the phase angle is proportional to the ratio of energy dissipated in one cycle to the maximum energy stored in one cycle. Under the usual conditions of the experiment the absolute magnitude of the energy dissipated is too small to affect the temperature of the sample appreciably, especially since the sample will also exchange heat with its surroundings. However, under conditions of ultrasonic welding (very high frequencies) the temperature rise is sufficient to melt and weld polymer samples. In a number of other experiments a temperature rise is more apparent, especially in situations involving large strains, such as stretching of films for orientation,

tensile testing etc. In such experiments the amount of heat generated can be a nuisance because it destroys the assumption of constant sample temperature. One can go one step further and propose to use the energy dissipated whenever a viscoelastic body is deformed to plasticize the material without any additional sources of heat. In this way one would hope, starting from 'cold' material, to reach temperatures sufficient for further processing (such as bottle blowing, injection moulding etc). Such a process would become interesting if it heated up a polymer either faster or more uniformly than conventional means of heating. Both these advantages are claimed for a process developed by Menges and coworkersL A slug of non-molten polymer is extruded with great force through a narrow die. The observed temperature rise is due to the conversion of mechanical work of extrusion to heat (plus some friction between polymer and extrusion barrel). The temperature uniformity claimed by Menges is questioned by Schenkel 3 on the basis of non-uniformity of stresses and deformations in the extrusion die. The other claim is that the polymer is heated 'within fractions of a second'. The purpose of this paper is to shed some light on the mechanism of this work to heat conversion process, especially the speed with which such a conversion can take place. We do this first on the basis of a simple Maxwell model (spring and dashpot in series) with a single relaxation time, ~-. We then extend the results to the discussion of a continuous Maxwell model with a distribution of relaxation times H(T). We realize, of course, that the process described by Menges 2 does not belong to the framework of linear viscoelasticity because of the large strains involved. Nevertheless, we believe that the discussion of these models will provide some guidance for the discussion of the work to heat conversion mechanisms in the real processes.

POLYMER, 1974, Vol 15, April

239

Conversion of work of deformation to heat in polymers: P. Zoller and H. Bont THEORY FOR THE SIMPLE MAXWELL MODEL Considering the simple Maxwell model of a viscoelastic body, the differential equation connecting stress, ~(t) and strain, ~(t) in tension is given by:

~1 + G 1o = ~

(1)

,/ and G are the two parameters of the model. ,//G has the dimension of a time and is called the relaxation time 7. By choice we have written the differential equation for stress and strain in tension. It could just as well have been written for a shear stress and a shear strain without changing the further course of the development. It is well known that a Maxwell viscoelastic body, characterized by equation (1), can be represented by an ideal spring and an ideal dashpot in series. Let Ei(t) denote the strain of the spring, c2(t) the strain of the dashpot. In the spring the stress a(t) is proportional to the strain ca(t): ~(t) = G el(t)

(2)

In the dashpot the stress o(t) is proportional to the time rate of change of the strain c2(t): o(t)=,/~2(t)

(3)

The stress is the same in both elements because the elements are in series, and the total strain c(t) of the Maxwell element is c(t)=cl(t)+~2(t). Using this to combine equations (2) and (3) yields equation (1). This model is subjected to the strain history given in Figure 1. Starting from t = 0 the sample is strained with a constant strain rate ~0 until a total strain c0 is reached. This occurs at t =to=co/~o. For t >~t0 the strain is kept constant. This process was chosen because of its analogy with tensile testing, an experiment on which tests for the theoretical ideas developed here can most conveniently be made. The differential equation (1) can be solved explicitly for the strain history of Figure 1, giving the stress as a function of time: t ~
(4)

t >_.to: ~(t)=~o[exp(--{t--to)/7)--exp(--t/T)]

(5)

When the Maxwell body is deformed, mechanical work is done on the spring and on the dashpot. The work done on the spring is stored as potential energy, the work done on the dashpot is dissipated as heat. We will proceed to calculate the work WM(t) done in deforming the whole Maxwell model and the work WD(t) done

z/'

Eo

w

d

Wz2=f;i,,(t')~(t')dt'

(6)

This expression is applied to the Maxwell model as a whole, by substituting the strain history of Figure 1 and the stress of equations (4) and (5):

t <~to: WM(t)=~l~[t + r e x p ( - t / 7 ) - 7 ]

(7)

t >_-to: WM(t)= WM(t0)=constant

(8)

Applied to the dashpot alone equation (6) yields: 1 Wn(t)=fto,,(t')~z(t')dt'=~ fto~2(t')dt'

(9)

(where we have made use of equation 3). Carrying out the integration with the stress of equations (4) and (5) yields: t ~
WD(t) = ~7g~[t + 7{2-- 0"5exp(-- t/~))exp(-- t/r)-- 1"57] (10) t ~>t0:

WD(t ) = •/ ~02[t0+ 7exp(- to~r) - 7 - 0.5 7exp{- 2(t - t0)/~'}+ 7exp( - (2t - to)/'r} -0.57exp(- 2t/T)]

(1 1)

The work to heat conversion percentages C~(t)= IOOWD(t)/WM(t) thus become: t ~
C~(t)=lO0[1

0.5exp(-2t/7)-exp(-t/7)+0.5]~_:l

j

(12)

t ~>to: {0"5 + 0"5exp(2t0/T)--

C,(t) = 100 1 --

]

exp(t0/7)}exp(-- 2t/r) I --j

to/7+exp(--to/r)--I

(13)

Note that in equations (12) and (13) the parameters ,/ and G of the model enter only through their ratio 7=~/G, and the process parameters c0 and ~0 only through their ratio c0/~0= to. Moreover, the times t and to enter only through their ratios to the relaxation time r: t/7 and to/7. Accordingly we have evaluated C~(t) as a function of the reduced time t/7 with to/r as a parameter. The resulting family of curves is given in Figure 2.

DISCUSSION OF THE SIMPLE MAXWELL MODEL

to

Figure I

in deforming the dashpot alone from time zero up to time t. The ratio WD(t)/WM(t) is (as a function of time t) the fraction of the total work done up to time t which has been converted to heat at time t. The general expression for work done (per unit volume) when a sample is stretched according to the strain function E(t) from time tl to time t2 is given by:

Strain, ~ as a f u n c t i o n of t i m e

240 POLYMER, 1974, Vol 15, April

After a sufficiently long time all work done in stretching the Maxwell model is converted to heat, irrespective of the relaxation time, r. This is intuitively clear: even after one has stopped stretching the Maxwell model as a whole the strain in the dashpot is increasing at the expense of the strain in the spring, keeping the total

Conversion of work of deformation to heat in polymers: P. Zoller and H. Bont I00

C u"

o c

o.

50

c O > ¢O

t.)

O L-=~'`-"'-

O-OI

I

I

I

OI

I

IO

IOO

t/x Figure 2 Plot of the conversion percentage Cr (t) according to equations (12) and (13) as a function of reduced time t/r with to/7 as a parameter Table 1 C 7 (to) and t (90%) as a function of to/r

to/.

c, (to)

t (9o%)/to

0-01 0.1 0-5 1 2 5 10

0.79/o 6-4% 27.3% 45" 6 % 67.1% 87.7% 94"4%

116 12.5 3"2 1 •85 1-35 1- 02 0"6

Maxwell model, consisting of an infinite series of simple Maxwell models with different relaxation times in parallel to one another a. The model is usually completed by a spring in parallel to all the component models. This spring, however, needs to be introduced only for crosslinked materials, such as elastomers and thermosetting resins. It is therefore disregarded here. The infinite series of Maxwell models is described by a distribution of relaxation times H(z) a. This characteristic function (with the dimension of a modulus) can be determined for each material (as time consuming as this might be) from viscoelastic data such as creep and stress relaxation experiments, forced vibrations, torsional oscillations etc. The various approximations used for the determination of H(z) are discussed in the literature 5. In stretching the continuous Maxwell model all component models (being in parallel) are subjected to the same strain programme as the total model (Figure 1). For the component models with relaxation times in the interval (z, z + d z ) equations (12) and (13) for the conversion percentages are therefore still valid. To obtain a conversion factor for the model as a whole one also needs to know how the work of stretching is distributed among the components of the continuous Maxwell model. This information is contained in H(z). We will proceed to calculate the conversion percentage for the continuous Maxwell model at t = to. Let dA(r) denote the work required to stretch the elements with relaxation times z in the interval (z, z + dz) from t = 0 to t = to according to the strain programme of Figure 1. This work is given by equation (7) evaluated at t = to when the constant ~ is replaced by H(z)dz: dA(r) = H(z) ~02[t0+ z e x p ( - to/r)- r]dz

(14)

The work to heat conversion percentage at t =to for the continuous Maxwell model is then given by:

f c,(to)dA(z) strain constant. A changing strain in the dashpot means energy dissipation. Eventually the stress will decay to zero (spring unstretched), i.e. all the energy stored in the extended spring has been used to stretch the dashpot and is dissipated to heat in the process. Of special interest is the energy conversion percentage C,(to) at the termination of stretching. This time to is, after all, the time at which one would like to process the sample further. C,(to) is a strong function of the ratio to/z of the stretching time to to the relaxation time ~-. If t0< z only a small fraction of the work of stretching has been converted to heat at t =to, however, if, say, to > 10z the conversion is essentially complete at t = to. Table 1 gives some values of C,(to) as a functior of to[z. After the completion of stretching, the con version of work to heat continues, with the detailed time dependence to be taken out of Figure 2. Table 1 also gives values for t (90%) (expressed in units of stretching time to), which is the time at which 90% of the work of stretching has been converted to heat EXTENSION TO A CONTINUOUS MAXWELL MODEL The simple Maxwell model with a single relaxation time does not describe adequately real polymeric materials. One therefore introduces a continuous

R

K(to) =

fdA(z)

(15)

J

in which the integrations are extended over all relaxation times z from 0 to 0% and where C,(to)is the conversion percentage at t =to of the simple Maxwell model with relaxation time z (given by equation 12). Evaluating K(to) yields [changing the integration variable from z to log(z/t0)/:

L -f+_~n(z)f2(z/to)dlog(z/to)l

(16)

with"

f~(z/to) = (z/t0)[0.5exp(- 2t0/r) -

e x p ( - to/z) + 0.5]

fz(z/to) = [1 + (z/t0)exp(- to/z)- (z/t0)] For any material with a known relaxation time spectrum H(z) these integrals may be evaluated numerically and the conversion factor obtained. To help with this evaluation we have also calculated values of f l and f~ over a wide range of values of log(r/t0) in Table 2. In many cases a detailed integration may not be necessary. By inspection of Table 2 one finds that f l and f2 have comparable values only for z/to >0.1. For ~'/to
P O L Y M E R , 1974, V o l 15, A p r i l

241

Conversion of work of deformation to heat in polymers: P. Zoller and H. Bont Table 2 fl and /2 defined in equation (16) as a function of r/to

• /to

fi

f3

104 10-8

0 . 5 x 10-4 0 . 5 x 10-s

1 0.999

10-2 5 x 1 0 -~ 10-1

0 . 5 x 10-3 0"025 0"05

0"99 0"95 0"90

0.5

0-19

0"57

1 5

0"20 0" 082

0"37 0" 093

4.52x 10- 2 4.95x 10-3 5 x 10-4 5 x 10-5

4.83x 10-3 4.98x 10-3 5 x 104 5 x 10-5

101 103 103 104

is not peaked very strongly in the region r>t0/10, but continues with reasonably large values into the region ~'
242

POLYMER,

1974, V o l 15, A p r i l

require a large H ( , ) for T<0.01sec. Such relaxation time distributions are found e.g. for polyethylene and isotactic polystyrene for temperatures around 25°C 6. H(~-) for these materials changes by less than a factor of five for 10-asec