Viscoelastic Behaviour of Plastics

For most traditional materials, the objective of the design method is to determine stress values which will not cause fracture. However, for plastics it is more likely that excessive deformation will be the limiting factor in the selection of working stresses. Therefore this chapter looks specifically at the deformation behaviour of plastics and fracture will be treated separately in the next chapter. 

For a component subjected to a uniaxial force, the engineering stress, a, in the material is the applied force (tensile or compressive) divided by the original cross-sectional area. The engineering strain, e, in the material is the extension (or reduction in length) divided by the original length. In a perfectly elastic (Hookean) material the stress, a, is directly proportional to be strain, e, and the relationship may be written, for uniaxial stress and strain, as 

In a perfectly viscous (Newtonian) fluid the shear stress, t is directly proportional to the rate of strain (dy/dt or y) and the relationship may be written as 

Polymeric materials exhibit mechanical properties which come somewhere between these two ideal cases and hence they are termed viscoelastic. In a viscoelastic material the stress is a function of strain and time and so may be described by an equation of the form 

This type of response is referred to as non-linear viscoelastic but as it is not amenable to simple analysis it is often reduced to the form 

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Throughout this chapter the viscoelastic behaviour of plastics has been described and it has been shown that deformations are dependent on such factors as the time under load and the temperature. Therefore, when structural components are to be designed using plastics, it must be remembered that the classical equations which are available for the design of springs, beams, plates, cylinders, etc., have all been derived under the assumptions that... 

A Standard Model for the viscoelastic behaviour of plastics consists of a spring element in scries with a Voigt model as shown in Fig. 2.86. Derive the governing equation for this model and from this obtain the expression for creep strain. Show that the Unrelaxed Modulus for this model is and the Relaxed Modulus is fi 2/(fi + 2>. 

DeUa VaUe G, Buleon A, Carteau PJ, Lavioie PA, Vergnes B, (1998) Relationship between structure and viscoelastic behaviour of plasticized starch , J. Rheology, v42 n3 507-525. 

Over the years there have been many attempts to simulate the behaviour of viscoelastic materials. This has been aimed at (i) facilitating analysis of the behaviour of plastic products, (ii) assisting with extrapolation and interpolation of experimental data and (iii) reducing the need for extensive, time-consuming creep tests. The most successful of the mathematical models have been based on spring and dashpot elements to represent, respectively, the elastic and viscous responses of plastic materials. Although there are no discrete molecular structures which behave like the individual elements of the models, nevertheless... 

The viscoelastic behaviour of a certain plastic is to be represented by spring and dashpot elements having constants of 2 GN/m and 90 GNs/m respectively. If a stress of 12 MN/m is applied for 100 seconds and then completely removed, compare the values of strain predicted by the Maxwell and Kelvin-Voigt models after (a) 50 seconds (b) 150 seconds. 

Based on the stress-strain diagram the values tensile stress at yield cXy and tensile strength at maximum (7m as well as the associated normative yield strain and nominal strain 8tM or normative strain 8m at tensile strength as well as strain at break 8b can be calculated (Eqs. 4.6. 11). For completely recorded diagrams the nominal strain at break 8tB can be determined additionally (Eq. 4.12). Because of the dependence on software and test equipment, especially sampling rate, the tensile stress at break (Tb should not be used (Eq. 4.13). Due to the viscoelastic behaviour of the plastics modulus of elasticity in tension is determined as secant modulus between the strain limits of 0.05 % and 0.25 % (Eq. 4.14). If the transverse strain is recorded simultaneously using strain gauges Poisson ratio jl can be calculated (Eq. 4.15). 

Fig. 12. Diagram showing the time-dependent deformation behaviour of plastic> viscoelastic, and elastic substances before and after a deforming force is applied (modified according to Hessemer Dick, 1996)...

During loading time t), the bituminous mixture deforms the deformation (51) and, thus, the strain (e) increase rapidly at the beginning and then become quasi-constant. After removal of applied stress (rest period), part of the total deformation is recovered instantaneously (the elastic deformation), another part of deformation is recovered gradually and is time dependent (viscoelastic deformation) and another part of the deformation cannot be recovered (viscous or plastic or permanent deformation). Figure 7.15 explains the above, in terms of strain, which is known as creep behaviour. As it can be seen, the above behaviour is similar to the viscoelastic behaviour of the bitumen (see Section 4.21.1). 

Chazeau L, Cavaille JY, Canova G et al (1999a) Viscoelastic properties of plasticized PVC reinforced with cellulose whiskers. J Appl Polym Sci 71 1797-1808 Chazeau L, Cavaille JY, Terech P (1999b) Mechanical behaviour above Tg of a plasticised PVC reinforced with cellulose whiskers a SANS structural study. Polymer 40 5333-5344 Chazeau L, Paillet M, Cavaille JY (1999c) Plasticized PVC reinforced with cellulose whiskers. I. Linear viscoelastic behavior analyzed through the quasi-point defect theory. J Polym Sci Part B Polym Phys 37 2151-2164... 

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. 

To add to this picture it should be realised that so far only the viscous component of behaviour has been referred to. Since plastics are viscoelastic there will also be an elastic component which will influence the behaviour of the fluid. This means that there will be a shear modulus, G, and, if the channel section is not uniform, a tensile modulus, , to consider. If yr and er are the recoverable shear and tensile strains respectively then... 

For the investigation of the time and the temperature dependence of the fibre strength it is necessary to have a theoretical description of the viscoelastic tensile behaviour of polymer fibres. Baltussen has shown that the yielding phenomenon, the viscoelastic and the plastic creep of a polymer fibre, can be described by the Eyring reduced time (ERT) model [10]. The shear deformation of a domain brings about a mutual displacement of adjacent chains, the... 

A study was made of the impact and recovery behaviour of three HDPE closed-cell foams with varying densities. Impact stress-strain curves were measured using a falling striker impact rig and the recovery monitored from 10s after the impact. Cell deformation was observed during compression and recovery using SEM. Recovery was found to occur by the viscoelastic straightening of the buckled faces and to be incomplete due to plastic deformation in the structure. 6 refs. 

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