Stress-strain curves linear viscoelasticity

When the magnitude of deformation is not too great, viscoelastic behavior of plastics is often observed to be linear, i.e., the elastic part of the response is Hookean and the viscous part is Newtonian. Hookean response relates to the modulus of elasticity where the ratio of normal stress to corresponding strain occurs below the proportional limit of the material where it follows Hooke s law. Newtonian response is where the stress-strain curve is a straight line. 

When an engineering plastic is used with the structural foam process, the material produced exhibits behavior that is easily predictable over a large range of temperatures. Its stress-strain curve shows a significantly linearly elastic region like other Hookean materials, up to its proportional limit. However, since thermoplastics are viscoelastic in nature, their properties are dependent on time, temperature, and the strain rate. The ratio of stress and strain is linear at low strain levels of 1 to 2%, and standard elastic design... 

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

The mechanical response of polypropylene foam was studied over a wide range of strain rates and the linear and non-linear viscoelastic behaviour was analysed. The material was tested in creep and dynamic mechanical experiments and a correlation between strain rate effects and viscoelastic properties of the foam was obtained using viscoelasticity theory and separating strain and time effects. A scheme for the prediction of the stress-strain curve at any strain rate was developed in which a strain rate-dependent scaling factor was introduced. An energy absorption diagram was constructed. 14 refs. 

The effect of gas compression on the uniaxial compression stress-strain curve of closed-cell polymer foams was analysed. The elastic contribution of cell faces to the compressive stress-strain curve is predicted quantitatively, and the effect on the initial Young s modulus is said to be large. The polymer contribution was analysed using a tetrakaidecahedral cell model. It is demonstrated that the cell faces contribute linearly to the Young s modulus, but compressive yielding involves non-linear viscoelastic deformation. 3 refs. 

PP bead foams of a range of densities were compressed using impact and creep loading in an Instron test machine. The stress-strain curves were analysed to determine the effective cell gas pressure as a function of time under load. Creep was controlled by the polymer linear viscoelastic response if the applied stress was low but, at stresses above the foam yield stress, the creep was more rapid until compressed cell gas took the majority of the load. Air was lost from the cells by diffusion through the cell faces, this creep mechanism being more rapid than in extruded foams, because of the small bead size and the open channels at the bead bonndaries. The foam permeability to air conld be related to the PP permeability and the foam density. 15 refs. 

It is necessary to state more precisely and to clarify the use of the term nonlinear dynamical behavior of filled rubbers. This property should not be confused with the fact that rubbers are highly non-linear elastic materials under static conditions as seen in the typical stress-strain curves. The use of linear viscoelastic parameters, G and G", to describe the behavior of dynamic amplitude dependent rubbers maybe considered paradoxical in itself, because storage and loss modulus are defined only in terms of linear behavior. 

Figure 1. Stress-strain curves in cyclic tensile straining (a) linear viscoelastic (b) nonlinear viscoelastic.

There is linear viscoelastic behaviour in the stress region where the isochronous stress-strain curve is linear (to within 5%). The creep compliance /( ), defined by Eq. (7.4), is independent of stress. However, above this stress region (stresses >1 MPa for the data in Fig. 7.7 for a time of 1 year) there is non-linear viscoelastic behaviour and the creep compliance becomes stress dependent... 

Figure 7.7 Isochronous stress-strain curve at a time of I year constructed from the creep data in Figure 7.6. The broken line represents linear viscoelastic behaviour.

Figure 7.8 shows a possible cross section for the beam. The 2 mm thick section was chosen so the injection moulding cycle time is short, while the I beam is efficient in bending (Chapter 13). From the linear portion of the isochronous stress-strain curve, the linear viscoelastic compliance is 7(1 year) = 3.3 x 10 m N . Substituting this and the deflection limit in... 

Ideal yielding behaviour is approached by many glassy polymers well below their glass-transition temperatures, but even for these polymers the stress-strain curve is not completely linear even below the yield stress and the compliance is relatively high, so that the deformation before yielding is not negligible. Further departures from ideality involve a strain-rate and temperature dependence of the yield stress. These two features of behaviour are, of course, characteristic of viscoelastic behaviour. 

A perfectly elastic material obeys Hooke s law, which gives a linear stress-strain curve where the stretch is proportional to the stress and the slope is the modulus of elasticity. Most textile fibers are not completely elastic. They may be linear or Hookean under low stresses, but when higher stresses are applied, they elongate out of proportion to the amount of stress applied. This causes the stress-strain curve to become nonlinear. Materials that show this behavior are called viscoelastic. Acetate and triacetate fibers are viscoelastic. 

Inspection of this equation shows that it models reasonably well, on a very superficial level, a stress-strain curve of the type shown in Fig. 1(b), curve (4). In other words it raises the question as to whether the deviations from linear stress-strain relationships observed in constant strain-rate tests might not be merely resulting from the intrinsic time-dependence of the linear viscoelasticity, which can be more clearly studied in creep or stress-relaxation and not due to some new process starting at high stresses. It does not take long to show that at the strain-levels of 3-5% experienced at yield, the response of most polymers is highly non-linear (r(t)/ is a function of strain-rate S as well as t, and so eqn. (14) cannot adequately describe the behaviour. However, it is also clear that at... 

The problems of exact design for a viscoelastic polymer with non-linear properties are severe. For example, in Figure 8.1 a) the stress-strain curve is linear only at the smallest strains (below 0.2%). Most plastic parts are designed to operate at strains well above 0.2%, and in this case exact stress analysis is impossible. In practice, a safe approximate procedure known as the pseudo-elastic design method is used. The salient features of the method, which is veiy straightforward to apply, are as follows ... 

The molecular theory of extensional viscosity of polymer melts is again based oti the standard tube model. It considers the linear viscoelastic factors such as reptation, tube length fluctuations, and thermal constraint release, as well as the nonlinear viscoelastic factors such as segment orientations, elastic contractimi along the tube, and convective constraint release (Marrucci and lannirubertok 2004). Thus, it predicts the extensional stress-strain curve of monodispersed linear polymers, as illustrated in Fig. 7.12. At the first stage, the extensional viscosity of polymer melts exhibits the Newtonian-fluid behavior, following Trouton s ratio... 

For linear materials, the stress is proportional to the strain. For linear elastic materials, Young s modulus ( ), equal to the slope of the stress-strain curve, is constant. For viscoelastic materials, E is dependent on the deformation rate. A material is called linear viscoelastic if the stress is proportional to the strain, despite the time-dependence. In this case, the Young s modulus is a function of the deformation rate only. In Fig. 2.13, the stress-strain behavior of a hnear viscoelastic material is shown schematically for different deformation rates, assuming that the material is exposed to a constant strain. 

If the viscoelastic behavior is nonlinear, stress-strain curves at constant rate of loading or deformation will be so a fortiori, since they can depart from linearity even without this complication. Calculations by Van Holde show that the nonlinearity of tensile creep in nitrocellulose implies a stress-strain curve at constant rate of loading with a sharp change in slope at strains of about 5% which resembles the apparent yield points observed in such experiments on many textile fi-bers. 5 6... 

If the linear viscoelasticity theory can be applied, the initial forward stress-strain curve is given by equation 56 of Chapter 3 written for extension ... 

Fillers of various dimensions are added to polymers to alter its processability, properties and uses. Such micro and nano composites obtained may have tremendous possibilities in industries and information on their viscoelasticity is very necessary as far as their processing and applicability are concerned. The dynamic properties of filled elastomers have been a subject of active research since they affect the performance of tyres such as skid, traction, and rolling resistance. Elastomer nanocomposites are most important materials characterized by excellent elasticity and flexibility, and are widely used in various applications such as cables, tyres, tubing, dielectric materials and sensors [1-5]. The non linear features observed in filled elastomers upon a simple shear are as follows. The dynamic storage and loss moduli of the composites are only dependent on the dynamic strains and not on the static strain. In the same way the stress strain curves also do not depend on static strain. Moreover the initial modulus under constant strain rate is highly rate dependent whereas the terminal modulus is independent of strain rate. This initial to terminal modulus ratio in the stress-strain curves is the same as the ratio of the dynamic storage moduli obtained at low and high strains. 

Typical examples of tensile (isochronous) linear and nonlinear stress-strain diagrams for elastic and viscoelastic materials are shown in Fig, 10.1. For elastic materials, the response is time independent, so there is a single curve for multiple times and the nonlinearity is apparent as a deviation of the stress-strain response from linear. For linear viscoelastic materials, the isochronous response is linear, but the effective modulus decreases with time so that the stress-strain curves at different times are separated from one another. When a viscoelastic material behaves nonlinearly, the isochronous stress-strain curves begin to deviate from linearity at a certain stress level. Fig. 10.2 shows creep compliance data for an epoxy adhesive as a function of stress level for various time intervals after initial loading. 

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