Viscoelasticity stress-strain curves

A typical stress—strain curve generated by a tensile tester is shown in Eigure 41. Creep and stress—relaxation results are essentially the same as those described above. Regarding stress—strain diagrams and from the standpoint of measuring viscoelastic properties, the early part of the curve, ie, the region... 

For a fiber immersed in water, the ratio of the slopes of the stress—strain curve in these three regions is about 100 1 10. Whereas the apparent modulus of the fiber in the preyield region is both time- and water-dependent, the equiUbrium modulus (1.4 GPa) is independent of water content and corresponds to the modulus of the crystalline phase (32). The time-, temperature-, and water-dependence can be attributed to the viscoelastic properties of the matrix phase. 

When a plastic material is subjected to an external force, a part of the work done is elastically stored and the rest is irreversibly (or viscously) dissipated hence a viscoelastic material exists. The relative magnitudes of such elastic and viscous responses depend, among other things, on how fast the body is being deformed. It can be seen via tensile stress-strain curves that the faster the material is deformed, the greater will be the stress developed since less of the work done can be dissipated in the shorter time. 

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... 

It has often been pointed out for a long time that the hysteresis energy given from the hysteresis loop under large extension is too big compared with the viscoelastic dissipation energy. For example, the hysteresis loop given from the stress relaxation is only 20%-30% of that from the stress-strain curve, when both measurements are performed at the same relaxation time and the... 

FIGURE 28.9 Idealized cyclic stress-strain curve, showing the fuU viscoelastic curve together with its elastic component. (Redrawn from Andrew, C., Introduction to Rubber Technology, Knovel e-book publishers, 1999.)... 

Figure 4 shows stress-strain curves measured at an extension rate of 94% per minute on the TIPA elastomer at 30°, —30°, and —40°C. With a decrease in temperature from 30° to -40°C, the ultimate elongation increases from 170% to 600%. The modulus Ecr(l), evaluated from a one-minute stress-strain isochrone, obtained from plots like shown in Figure 1, increases from 1.29 MPa at 30°C to only 1.95 MPa at —40°C. This small increase in the modulus and the large increase in the engineering stress and elongation at fracture results from viscoelastic processes. 

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. 

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. 

Williams, Landel, and Ferry equation (WLF) Used for predicting viscoelastic properties at temperatures above Tg when these properties are known for one specific temperature, yield point Point on a stress-strain curve below which there is reversible recovery. 

A viscoelastic material is characterized by at least three phenomena the presence of hysteresis, which is observed on stress-strain curves, stress relaxation which take place where step constant strain causes decreasing stress and creep occurs where step constant stress causes increasing strain. 

Solidified milk fat displays non-Newtonian behavior. It acts as a plastic material with a yield value (Sone, 1961 deMan and Beers, 1987). Throughout its wide melting range, milk fat, like butter, exhibits viscoelasticity, possessing both solid and liquid-like characteristics (Sone, 1961 Shama and Sherman, 1968 Jensen and Clark, 1988 Kleyn, 1992 Shukla and Rizvi, 1995). Several models to describe the complex rheological behavior of milk fat have been proposed. Figure 7.12 shows the corresponding stress-strain curves for the models discussed. 

Beside the consideration of the up-cycles in the stretching direction, the model can also describe the down-cycles in the backwards direction. This is depicted in Fig. 47a,b for the case of the S-SBR sample filled with 60 phr N 220. Figure 47a shows an adaptation of the stress-strain curves in the stretching direction with the log-normal cluster size distribution Eq. (55). The depicted down-cycles are simulations obtained by Eq. (49) with the fit parameters from the up-cycles. The difference between up- and down-cycles quantifies the dissipated energy per cycle due to the cyclic breakdown and re-aggregation of filler clusters. The obtained microscopic material parameters for the viscoelastic response of the samples in the quasi-static limit are summarized in Table 4. 

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. 

In spite of our reluctance to quote numerical values at this point, the effective modulus obtained from the initial portion of the tensile curve ranges from 1 to 5 X 10 dyn cm". Many individual PE crystals have moduli from 3 X 10 to 10 dyn cm" and fracture at forces of about 0.2 dyn. Orientation effects are expected to be present and are presently being investigated. There is no comparable experimental data with which we can directly relate these values. However, moduli are in the range found for bulk specimens but are considerably less than the value of 70 X 10 dyn cm reported by Perkins et al. (8) for ultra-drawn HDPE fibers. The x-ray measurements of the lattice moduli by Ito (9) using an x-ray technique for oriented sheet samples is perhaps the most relevant comparison. He found values of 240 X 10 dyn cm" in a direction parallel to the fiber axis and a value of 4 X 10 dyn cm for the perpendicular direction which would be the closest comparison with our orientation. We are not yet certain whether the initial portion of the stress-strain curve shows nonlinear viscoelastic effects such as found by Chen et al. (4) for springy polypropylene (PP) fibers. 

Figure 3. Stress-strain curves in nonlinear elastic and nonlinear viscoelastic responses...

When a viscoelastic material such as tire cord or rubber Is subjected to a small amplitude sinusoidal straining, the resulting stress-strain curve Is an ellipse and the material properties are characterized by the real and Imaginary moduli E and E" or the ratio E /E ( tan[Pg.372]

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

The behavior observed in the stress-strain curve corresponds to viscoelastic behavior that is typical of polymeric materials. The viscoelastic behavior is highly dependent on the temperature at which the test is performed and its relationship to the Tg of the sample. It is also dependent on the rate of deformation, as mentioned previously. In general, very rapid deformation does not allow time for molecular rearrangement to occur and results in behavior characteristic of a more brittle material. The effects of temperature and rate of testing on plastic materials are illustrated in Figs. 3.54 and 3.55, respectively. 

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... 

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