Plastics Engineering 3E Episode 4 doc

Mechanical Behaviour of Plastics 91 Solution The spring element constant, 61, for the Maxwell model may be obtained from the instantaneous strain, ~ 1.. Mechanical Behaviour of Plastics

Mechanical Behaviour of Plastics 89

Fig 2.37 Response of KelvinNoigt model

If the stress is removed, then equation (2.38) becomes

0 = C E + r)€

Solving this differential equation with the initial condition E = E’ at the time

of stress removal, then

i!

This represents an exponential recovery of strain which is a reversal of the predicted creep

(c) More Complex Models

It may be seen that the simple Kelvin model gives an acceptable first approx- imation to creep and recovery behaviour but does not account for relaxation The Maxwell model can account for relaxation but was poor in relation to creep

and recovery It is clear therefore that some compromise may be achieved by combining the two models Such a set-up is shown in Fig 2.38 In this case

the stress-strain relations are again given by equations (2.27) and (2.28) The

geometry of deformation yields

Total strain, E = e1 + €2 + &k (2.41)

5 2

Fig 2.38 Maxwell and Kelvin models in series

where &k is the strain response of the Kelvin Model From equations (2.27), (2.28) and (2.41)

From this the strain rate may be obtained as

(2.42)

(2.43)

The response of this model to creep, relaxation and recovery situations is the sum of the effects described for the previous two models and is illustrated in Fig 2.39 It can be seen that although the exponential responses predicted in

these models are not a true representation of the complex viscoelastic response

of polymeric materials, the overall picture is, for many purposes, an acceptable approximation to the actual behaviour As more and more elements are added

to the model then the simulation becomes better but the mathematics become complex

Example 2.12 An acrylic moulding material is to have its creep behaviour

simulated by a four element model of the type shown in Fig 2.38 If the creep

curve for the acrylic at 14 MN/m2 is as shown in Fig 2.40, determine the

values of the four constants in the model

Mechanical Behaviour of Plastics 91

Solution The spring element constant, 61, for the Maxwell model may be

obtained from the instantaneous strain, ~ 1 Thus

' - e1 0.005 The dashpot constant, V I , for the Maxwell element is obtained from the slope

of the creep curve in the steady state region (see equation (2.32))

The spring constant, 62, for the Kelvin-Voigt element is obtained from the maximum retarded strain, ~ 2 , in Fig 2.40

The dashpot constant, q2, for the Kelvin-Voigt element may be determined

by selecting a time and corresponding strain from the creep curve in a region where the retarded elasticity dominates (i.e the knee of the curve

in Fig 2.40) and substituting into equation (2.42) If this is done then q2 = 3.7 x lo8 MN.s/m2

Having thus determined the constants for the model the strain may be predicted for any selected time or stress level assuming of course these are within the region where the model is applicable

(d) Standard Linear Solid

Another model consisting of elements in series and parallel is that attributed to Zener It is known as the Standard Linear Solid and is illustrated in Fig 2.41 The governing equation may be derived as follows

In a similar manner to the previous models, equilibrium of forces yields

(TI = a3

Mechanical Behaviour of Plastics 93 Geometry of Deformation Equation

In this case the total deformation, E , is given by

This is the governing equation for this model

The behaviour of this model can be examined as before

(9 creep

If a constant stress, a is applied then the governing equation becomes

&{q3(6l + t2)1 + 6162e - 6100 = 0 The solution of this differential equation may be obtained using the boundary condition E = a,/(ij1 + 62) at t = 0 So

(2.50)

It may be seen in Fig 2.42 that this predicts the initial strain when the stress

is first applied as well as an exponential increase in strain subsequently

(ii) Relaxation

If the strain is held constant at E', then the governing equation becomes

v3a + 610 - 6162.s' = 0 This differential equation may be solved with the boundary condition that

a = a, = ~ ' ( 6 1 + 62) when the strain is first kept constant

Stress, a(t) = ~

This predicts an exponential decay of stress as shown in Fig 2.42

If the stress is at a value of 0’ and then completely removed, the governing

1)3(e1 + e2)& + e1e2& = 0

E’ = a’/(e1 + (2)

(2.52)

This predicts an instantaneous recovery of strain followed by an exponential decay

It may be observed that the governing equation of the standard linear solid

has the form

+ U,O = b l i + bo&

Mechanical Behaviour of Plastics 95

where a l , a,, bl and bo are all material constants In the more modem theories

of viscoelasticity this type of equation or the more general form given in equation (2.53) is favoured

The models described earlier are special cases of this equation

of the recovered strain which occurs during the rest periods of conversely the accumulated strain after N cycles of load changes

There are several approaches that can be used to tackle this problem and two of these will be considered now

2.12.1 Superposition Principle

The simplest theoretical model proposed to predict the strain response to a complex stress history is the Boltzmann Superposition Principle Basically this principle proposes that for a linear viscoelastic material, the strain response to

a complex loading history is simply the algebraic sum of the strains due to each step in load Implied in this principle is the idea that the behaviour of a plastic

is a function of its entire loading history There are two situations to consider

(a) Step Changes of Stress

t l , then the creep strain, &(t), at any subsequent time, t , may be expressed as

When a linear viscoelastic material is subjected to a constant stress, u1, at time,

(2.54)

where E ( t - t l ) is the time-dependent modulus for the elapsed time (t - t l )

Then suppose that instead of this stress q , another stress, a2 is applied at

some arbitraxy time, t2, then at any subsequent time, t , the stress will have been applied for a time (t - t2) so that the strain will be given by

1

- t2)

& ( t ) = .fJ1

Now consider the Situation in which the stress, 01, was applied at time, t l ,

and an additional stress, 02, applied at time, r2, then Boltzmanns’ Superposition Principle states that the total strain at time, t, is the algebraic sum of the two independent responses

where a is the step change of stress which occurs at time, t i

To illustrate the use of this expression, consider the following example

by a Maxwell model is to be subjected to the stress history shown in

Fig 2.43(a) If the spring and dashpot constants for this model are 20 GN/m2

and 1000 GNs/m2 respectively then predict the strains in the material after 150

seconds, 250 seconds, 350 seconds and 450 seconds

Solution From Section 2.1 1 for the Maxwell model, the strain up to 100s is given by

Mechanical Behaviour of Plastics 97

The predicted strain variation is shown in Fig 2.43(b) The constant strain

rates predicted in this diagram are a result of the Maxwell model used in this example to illustrate the use of the superposition principle Of course superposition is not restricted to this simple model It can be applied to any type of model or directly to the creep curves The method also lends itself to a

graphical solution as follows If a stress 01 is applied at zero time, then the creep curve will be the time dependent strain response predicted by equation (2.54) When a second stress, a2 is added then the new creep curve will be obtained

by adding the creep due to a2 to the anticipated creep if stress a1 had remained

alone This is illustrated in Fig 2.44(a) Then if all the stress is removed this

is equivalent to removing the creep strain due to 6 1 and 02 independently as

shown in Fig 2.44(b) The procedure is repeated in a similar way for any other stress changes

i

- 1 -

Fig 2.44(a) Stress history

Fig 2.44(b) Predicted strain response using Boltzmann’s superposition principle

(b) Continuous Changes of Stress

If the change in stress is continuous rather than a step function then equa-

tion (2.55) may be generalis4 further to take into account a continuous loading cycle So

t

(2.57)

where a(t) is the expression for the stress variation that begins at time, tl The lower limit is taken as minus infinity since it is a consequence of the

Mechanical Behaviour of Plastics 99 Superposition Principle that the entire stress history of the material contributes

to the subsequent response

It is worth noting that in exactly the same way, a material subjected to a continuous variation of strain may have its stress at any time predicted by

(2.58)

-m

To illustrate the use of equation (2.57) consider the following Example

Example 2.14 A plastic is subjected to the stress history shown in Fig 2.45

The behaviour of the material may be assumed to be described by the Maxwell model in which the elastic component 6 = 20 GN/m2 and the viscous compo- nent 9 = lo00 GNs/m2 Determine the strain in the material (a) after u1 seconds (b) after u2 seconds and (c) after u3 seconds

Stress (MN/m2)

Fig 2.45 Stress history to be analysed Solution As shown in the previous Example, the modulus for a Maxwell element may be expressed as

Substituting into equation (2.57)

It is interesting to note that if K1 was large (say K1 = 10 in which case

T = 1 second) then the strain predicted after application of the total stress (10 MN/m*) would be ~ ( 1 ) = 0.0505% This agrees with the result in the

previous Example in which the application of stress was regarded as a step function The reader may wish to check that if at time T = 1 second, the stress was held constant at 10 MN/m2 then after 100 seconds the predicted strain using the integral expression would be ~ ( 1 0 0 ) = 0.1495% which again agrees with the previous example

(b) After the time T, the change in stress is given by

Mechanical Behaviour of Plastics 101

then for K2 = -0.2 MN/m2 s, T = 100 seconds and u2 = 125 seconds

Example 2.15 In the previous Example, what would be the strain after

125 seconds if (a) the stress remained constant at 10 MN/m2 after 100 seconds and (b) the stress was reduced to zero after 100 seconds

Solution

(a) If the stress was kept constant at 10 MN/m2 after 100 seconds as shown

in Fig 2.47 then the effective change in stress would be given by

Time (s)

Fig 2.47 Stress history: Example 2.15(a)

(b) If the stress was completely removed after 100 seconds as shown in Fig 2.48 then the effective change in stress would be given by

change in stress, a(t) = - K l ( t - T ) - Au

d a )

d t

Mechanical Behaviour of Plastics

of the predictions will reflect the accuracy with which the equation for

modulus (equation (2.33)) fits the experimental creep data for the material

In Examples (2.13) and (2.14) a simple equation for modulus was selected in

order to illustrate the method of solution More accurate predictions could have been made if the modulus equation for the combined MaxwellKelvin model

or the Standard Linear Solid had been used

2.12.2 Empirical Approach

As mentioned earlier, it is not feasible to generate test data for all possible combinations of load variations However, there have been a number of ex- perimental investigations of the problem and these have resulted in some very

useful design aids From the experimental point of view the most straightfor- ward situation to analyse and one that has considerable practical relevance is the loadno-load cycle In this case a constant load is applied for a period and

then completely removed for a period The background to this approach is as

follows

For a linear viscoelastic material in which the strain recovery may be

regarded as the reversal of creep then the material behaviour may be

represented by Fig 2.49 Thus the time-dependent residual strain, E,(?), may

be expressed as

where E, is the creep strain during the specified period denoted by (t) or (t - T) Since there can be an infinite number of combinations of creep and recovery periods it has been found convenient to express this behaviour in terms of two dimensionless variables The first is called the Fractional Recovery, defined as

Strain recovered E,(T) - E&)

(2.60)

-

Fractional recovery, F, = -

Max creep strain &,(TI

where E,(T) is the creep strain at the end of creep period and E,(?) is the residual strain at any selected time during the recovery period

The second dimensionless variable is called the Reduced Time, t ~ ,

defined as

(2.61)

Recovery time Creep time Reduced time, t~ =

Extensive tests have shown that if the final creep strain is not large then a graph of Fractional Recovery against Reduced Time is a master curve which

Mechanical Behaviour of Plastics 105

Reduced time, k

Fig 2.50 meal recovery behaviour of a plastic

describes recovery behaviour with acceptable accuracy (see Fig 2.50) The relationship between F, and f R may be derived in the following way

When creep curves are plotted on logarithmic strain and time scales they are

approximately straight lines so that the creep strain, E,(?) may be expressed as

to a complex problem For the purposes of engineering design the expression provides results which are sufficiently accurate for most purposes In addition,

equation (2.63) permits the problem of intermittent loading to be analysed in

a relatively straightforward manner thus avoiding uneconomical overdesign which would result if the recovery during the rest periods was ignored

From equation (2.63) and the definition of Fractional Recovery, F,, the residual strain is given by

& r ( t ) = & c ( T ) - Fr * &c(T)

If there have been N cycles of creep and recovery the accumulated residual

strain would be

x=N

&,(t) = EC(T) x=l [ (F) ?I - ( y - 1) '1

where t p is the period of each cycle and thus the time for which the total

accumulated strain is being calculated is t = t,N

Note also that the total accumulated strain after the load application for the

(N + 1)th time will be the creep strain for the load-on period ie c C ( T ) plus the

be estimated for any combinations of loading/unloading times

In many design calculations it is necessary to have the creep modulus in order to estimate deflections etc from standard formulae In the steady loading situation this is straightforward and the method is illustrated in the Exam-

ples (2.1)-(2.5) For the intermittent loading case the modulus of the material

is effectively increased due to the apparent stiffening of the material caused

by the recovery during the rest periods For example, if a constant stress of 17.5 MN/m2 was applied to acetal (see Fig 2.51) for 9600 hours then the total creep strain after this time would be 2% This would give a 2% secant modulus

of 17.5/0.02 = 875 MN/m2 If, however, this stress was applied intermittently for 6 hours on and 18 hours off, then the total creep strain after 400 cycles (equivalent to a total time of 9600 hours) would only be 1.4% This would be equivalent to a stress of 13 MN/m2 being applied continuously for 9600 hours and so the effective creep modulus would be 13/0.014 = 929 MN/m2