Behaviour of polymers

Demonstration 5.3:Use a sample of viscoelastic "memory foam", squeeze it and let go.

It will recoverits shape slowly.

^Demonstration 5.4:Attach together a N.

metal spring and a cupboard doordamper push them in together and let go. The elastic spring will return back instantly but the damperonly slowly.

/Demonstration 5.5:To show the effect of speed, use a blob of⁴^ "Smartputty"(a). Roll it into a ball between palms(b). Dropthe ball gently on the floorit will hardly bounce, buthit the floorwith it and see it bounce rigourously and repeatedly. Mind the bounce!!!

On stressing a viscoelastic material, three deformation responses may be observed - an initial instant elastic response, then a time-dependent delayed elasticity (fully recoverable) and, finally, a viscous, non-recoverable flow. Experimental evidence for viscoelasticity is creep, stress relaxation and mechanical damping. Experimentally, thus, viscoelasticityis characterised by creep(creep compliance), stress relaxation(stress relaxation modulus) and by dynamic mechanical response(the storage and loss moduli).

Mathematical equationsto describe the stress-strain behaviourof the linear viscoelastic materials maybe derived by using simple mechanical modelsconsisting of springsand dashpots.The spring represents an elastic solid (obeying Hookes law in its mechanical behaviour) and the dashpot, containing oil that behaves as Newtonian fluid, represents viscous liquid.

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Behaviour of polymers

elastic solid: a as

viscousliquid: aade/dt

ds/dt

The spring and dashpot can be arranged simply in series or in parallel to illustrate the linear-viscoelastic behaviour. The assumptions for linear-viscoelastic behaviour:

1) Elastic strain and rate of viscous flow are directly proportional to stress.

2) Total deformation and stress are the arithmetic sum of viscous and elastic contributions, which may be treated
independently.

5.2.1 Voigt (Kelvin) Model

Voigt model (Figure 5.3) consists of a parallel combination of a spring and a dashpot. The model at best describes the creep behaviour of a real material.

Figure 5.3Voigt model, subjected to stress (a) and strain (e)

When loaded externally the model is assumed to undergo a uniform strain (iso-strain), i.e., the model and its components experience the same strain.

The stress (a) - strain (e) equations: for the spring, a = Es

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for the dashpot, a₂ = r\( de/dt)
.". for the Voigt model, a= a_l + a₂ = Ее + ц (de/dt) (Equation 5.1)

Note that: in creep testing ais constant, and e is time dependent and therefore expressed as a function of time, i.e., e(t). Material compliance (inverse of modulus) can be determined with creep testing.

Rearranging Equation 5.1:

De Eg

— + 8—= —

dt г) г)

Л

In terms of a time parameter (retardation time), T= —

E ds s _ с

dt I r|

The differential equation (as expressed in Equation 5.1) can be solved as below to determine £(t): a = Es + Г| (ds / dt) or a - Es = ц (ds / dt)

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Rearranging gives:

(dt / r|) = ds / (a - eE); multiplying across by (-E):

-E(dt / r|) = -E[ds / (a - sE)]; integrating:

-E(t / r|) = ln(cr - eE) + С

Where С is the constant of integration and it can be determined by substituting initial conditions: at t = 0, e(t) = 0, therefore С = - lncr.

Substitutingback forC, and also for n/ E = x: -t/x = ln[(a- eE) / a], or [(a- eE) / a]= e^(t/4 .-. e(t)= (a/E) [l-e⁽"^t/T)].

At t= x, e(t) = 0.63 (a/E)i.e., at retardation time x, an approximately 63% of the final deformation (viz. ст/E) occurs, x is a characteristic material response time.

The Voigt mechanical analogue of viscoelastic behaviour is used in many engineering applications such as the shock absorbers on a mountain bike (Figure 5.4).

Figure 5.4Shock absorber on a mountain bike frame (courtesy of www.chainreactioncvcles.coml

5.2.2 Maxwell Model

In the Maxwell model (Figure 5.5) a Hookean spring and a Newtonian dashpot are linked together in series. The model at best gives only a simple description of the stress-relaxation of a viscoelastic material.

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Figure 5.5Maxwell model, subjected to stress (a) and strain (e)

Stress-relaxation behaviour of a material is similar to its creep behaviour, however, for stress relaxation measurement the material is applied a given (fixed) strain, e, and stress, a, to maintain this strain is measured as a function of time, and a relaxation modulus is determined.

The Maxwell model is assumed to undergo a uniform stress (iso-stress), i.e., the model and its components experience the same stress.

The stress (a) - strain (e) equations:

for the spring, e_l = a IE or сЦ/ dt= (1/E) (da/dt)

for the dashpot, (de₂/dt) = a/ ц

■'■ for the Maxwell model, de/dt = (de/dt) + (de₂/dt) = (1/E) (da/dt) + a / rj

(Equation 5.2)

The conditionfor the stress-relaxationexperiment is that e is kept constant, therefore, de/dt = 0.

Substituting de/dt = 0 in Equation 5.2 for the model gives:

(1/E) (da/ dt)+ a/r| = 0, rearranging

(da/a) = -(E/r|) dtand integrating gives:

lna = -(E/ri) t+ с

where с is the constant of integration, or:

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When t = 0, a = a(0), i.e., the initial stress. Substituting in the above equation gives С = а (0).

Substituting back for C, and also for r)/E = x:

cr(t) = ct(0) e ^(tft) or as a modulus E(t) = [ст(0)/е] е ^(tft)

where, т (the relaxation time)= r|/E.

At t = x, a (t) = (1/e) a(0) « (1/2.7) a (0), i.e., stress decays (or relaxes) down to approximately 37% of the initial stress value after a period of time known as the relaxation time.

5.2.3 Shortcomings of the simple mechanical models

The Maxwell model describes the stress-relaxation behaviour by the equation

a (t) = a (0) e""\ The graphical representation of this prediction is shown in Figure 5.6.

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Figure 5.6Stress (a) vs. time (t) for a Maxwell model

According to the model the stress relaxes down to zero after a long time (t = oo). In real materials, this is not necessarily so. Therefore, in this respect, a modified Maxwell model(Figure 5.7) maybe a more appropriate model for actual polymers.

Figure 5.7 Modified Maxwell model Furthermore, the Maxwell model is not realistic under creep conditions:

Under creep loading, a = constant, therefore, da/dt = 0. Substituting this in the a-e relationship of the Maxwell model, de /dt = (1/E) (da/dt) + a/ц, gives (de/dt) = a /r\. The implication being that polymers behave as a viscous liquid under creep, which is not the case.

The Voigtmodel provides a very basic prediction for the creep behaviour of polymers by the equation:

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The graphical representation of this equation is shown in Figure 5.

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Figure 5.8Strain (e) vs. time (t) for Voigt model

The model implies that there is no elastic contribution to strain. In comparison an experimental curveindicates certain differences as shown in Figure 5.9.

Figure 5.9Features of a strain vs. time plot for a real specimen

The Kelvin/Voigt model needs to be modifiedto describe the creep behaviour more successfully. A comparison of the Figures 5.8 and 5.9 shows that a more accurate model needs to include elements to account for the initial elastic response of the material as well as the possibility of a permanent deformation, e.g., in the form flow of neighbouring molecular chains with respect to each other in thermoplastics. The four-element model (Figure 5.10) incorporates a spring and a dashpot in series with the Voigt unit to accommodate these two forms of deformations.

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Figure 5.10Illustration of the four-element model

Therefore the equation of the four-element model becomes:

El E₂

Date: 2015-12-11; view: 1223