Strain rate viscoelastic response

Factorizability has also been found to apply to polymer solutions and melts in that both constant rate of shear and dynamic shear results can be analyzed in terms of the linear viscoelastic response and a strain function. The latter has been called a damping function (67,68). 

Most pigmented systems are considered viscoelastic. At low shear rates and slow deformation, these systems are largely viscous. As the rate of deformation or shear rate increases, however, the viscous response cannot keep up, and the elasticity of the material increases. There is a certain amount of emphasis on viscoelastic behavior in connection with pigment dispersion as well as ink transportation and transformation processes in high-speed printing machines (see below). Under periodic strain, a viscoelastic material will behave as an elastic solid if the time scale of the experiment approaches the time required for the system to respond, i.e., the relaxation time. Elastic response can be visualized as a failure of the material to flow quickly enough to keep up with extremely short and fast stress/strain periods. 

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

Fig. 3.14. The data is for a very broad range of times and temperatures. The superposition principle is based on the observation that time (rate of change of strain, or strain rate) is inversely proportional to the temperature effect in most polymers. That is, an equivalent viscoelastic response occurs at a high temperature and normal measurement times and at a lower temperature and longer times. The individual responses can be shifted using the WLF equation to produce a modulus-time master curve at a specified temperature, as shown in Fig. 3.15. The WLF equation is as shown by Eq. 3.31 for shifting the viscosity. The method works for semicrystalline polymers. It works for amorphous polymers at temperatures (T) greater than Tg + 100 °C. Shifting the stress relaxation modulus using the shift factor a, works in a similar manner.

A unified approach to the glass transition, viscoelastic response and yield behavior of crosslinking systems is presented by extending our statistical mechanical theory of physical aging. We have (1) explained the transition of a WLF dependence to an Arrhenius temperature dependence of the relaxation time in the vicinity of Tg, (2) derived the empirical Nielson equation for Tg, and (3) determined the Chasset and Thirion exponent (m) as a function of cross-link density instead of as a constant reported by others. In addition, the effect of crosslinks on yield stress is analyzed and compared with other kinetic effects — physical aging and strain rate. 

When Zotefoam HDPE materials of density 98 kg m" were subjected to a single major compressive impact (419), after recovery at 50 °C for 1 hour, the performance, defined as the energy density absorbed before the compressive stress reached 2.5 MPa was back to 75% of the initial value. Further severe impacts caused a further deterioration of the performance of the recovered foam. Peak compressive strains of 80 to 90% caused some permanent buckling of the cell walls of HDPE foams. The recovery is much slower than the 0.1 second impact time, so is not a conventional linear viscoelastic response. It must be driven by the compressed air in internal cells in the gas, with some contribution from viscoelasticity of the polymer. Recovery of dimensions had slowed to a very low rate after 10 seconds at 20 °C or after 10 seconds at 50 °C. 

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. 

According to the change of strain rate versus stress the response of the material can be categorized as linear, non-linear, or plastic. When linear response take place the material is categorized as a Newtonian. When the material is considered as Newtonian, the stress is linearly proportional to the strain rate. Then the material exhibits a non-linear response to the strain rate, it is categorized as Non Newtonian material. There is also an interesting case where the viscosity decreases as the shear/strain rate remains constant. This kind of materials are known as thixotropic deformation is observed when the stress is independent of the strain rate [2,3], In some cases viscoelastic materials behave as rubbers. In fact, in the case of many polymers specially those with crosslinking, rubber elasticity is observed. In these systems hysteresis, stress relaxation and creep take place. 

The theory of sqeezing flow rheometry assumes that the sample is nonelastic. Tests on viscoelastic samples should therefore be carried out at low strain rates, to minimize elastic response, and results should be reported as apparent elongational viscosity. 

From a more fundamental point of view, the selection of different inden-ter geometries and loading conditions offer the possibility of exploring the viscoelastic/viscoplastic response and brittle failure mechanisms over a wide range of strain and strain rates. The relationship between imposed contact strain and indenter geometry has been quite well established for normal indentation. In the case of a conical or pyramidal indenter, the mean contact strain is usually considered to depend on the contact slope, 0 (Fig. 2a). For metals, Tabor [32] has established that the mean strain is about 0.2 tanG, i.e. independent of the indentation depth. A similar relationship seems to hold for polymers although there is some indication that the proportionality could be lower than 0.2 for viscoelastic materials [33,34], In the case of a sphere, an... 

Analysis of these effects is difficult and time consuming. Much recent work has utilized two-dimensional, finite-difference computer codes which require as input extensive material properties, e.g., yield and failure criteria, and constitutive laws. These codes solve the equations of motion for boundary conditions corresponding to given impact geometry and velocities. They have been widely and successfully used to predict the response of metals to high rate impact (2), but extension of this technique to polymeric materials has not been totally successful, partly because of the necessity to incorporate rate effects into the material properties. In this work we examined the strain rate and temperature sensitivity of the yield and fracture behavior of a series of rubber-modified acrylic materials. These materials have commercial and military importance for impact protection since as much as a twofold improvement in high rate impact resistance can be achieved with the proper rubber content. The objective of the study was to develop rate-sensitive yield and failure criteria in a form which could be incorporated into the computer codes. Other material properties (such as the influence of a hydrostatic pressure component on yield and failure and the relaxation spectra necessary to define viscoelastic wave propagation) are necssary before the material description is complete, but these areas will be left for later papers. 

The observation that an increase in temperature or a decrease in rate both result in the same fracture response points toward a viscoelastic influence on thermoset fracture behavior, especially crack initiation. This characteristic behavior of epoxies has been explained qualitatively by consideration of the temperature and strain rate effects on the plasticity of the material at the crack tip . In effect, test conditions which promote the formation of a so-called crack tip plastic zone, or blunt the crack by a ductile process, promote unstable crack propagation. This aspect of unstable fracture is subsequently discussed in more detail. 

Strain rate of 5x10" s , while the strain rates within the epoxy surface layer are in the order of 10 s under fretting conditions. Accordingly, the values of the octahedral shear stress at the onset of yield are probably underestimated. In addition to the limited viscoelastic response of the epoxy material at the considered frequency and temperature (tan 8 = 0.005 at 25°C and 1 Hz, table I), this analysis supports the validity of a global elastic description of the contact stress environment. 

The particular response of a sample to applied stress or shear strain rate depends on the time scale of the experiment. If the shear strain rate is kept low, many materials appear to behave viscous, whereas at high shear strain rates, materials might behave rather elastically. The simultaneous existence of viscous and elastic properties in a material is called viscoelasticity, and one can assume that all real materials are viscoelastic in nature. 

Materials can show linear and nonlinear viscoelastic behavior. If the response of the sample (e.g., shear strain rate) is proportional to the strength of the defined signal (e.g., shear stress), i.e., if the superposition principle applies, then the measurements were undertaken in the linear viscoelastic range. For example, the increase in shear stress by a factor of two will double the shear strain rate. All differential equations (for example, Eq. (13)) are linear. The constants in these equations, such as viscosity or modulus of rigidity, will not change when the experimental parameters are varied. As a consequence, the range in which the experimental variables can be modified is usually quite small. It is important that the experimenter checks that the test variables indeed lie in the linear viscoelastic region. If this is achieved, the quality control of materials on the basis of viscoelastic properties is much more reproducible than the use of simple viscosity measurements. Non-linear viscoelasticity experiments are more difficult to model and hence rarely used compared to linear viscoelasticity models. 

Influence of the Strain Rate on Viscoelastic Response. In high-shear-strain experiments, we observe significant differences only with type 2 and type 3 blends (Figures 9b and 9c). The complex viscosity still depends on frequency, but it decreases as phase separation begins. Moreover, a crossover of the tan 8 curves appears at this moment. The differences reported in this land of experiment can be found at two levels ... 

In practice, interfaces are often subjected to a combination of the deformations mentioned. As in bulk rheology, there are some other variables. First, the response of a material to a force can be elastic or viscous. Elastic response means immediate deformation, where the strain (relative deformation, i.e., tan a. in shear and AA/A in dilatation) is related to the force on release of the force, the strain immediately becomes zero. In viscous deformation, the force causes flow or, more precisely, a strain rate (d tan a/dt or d In Ajdty, this occurs as long as the force lasts, and upon release of the force the strain achieved remains. For most systems, the behavior is viscoelastic. Second, deformation can be fast or slow, and time scales between a microsecond and more than a day may be of importance. Third, the relative deformation (strain) applied can be small—i.e., remain close to the equilibrium situation—or be large. 

Intermediate Response. Figure 6 is a double logarithmic plot of o/e vs. time in seconds at three different strain rates for the samples as a function of H O content. To extend the time scale and to correlate results at various , we have used the reduced-variables procedure shown to be applicable in describing the viscoelastic response of rubbery materials (8) as well as of several glassy polymers (6). (To compensate for the effect of different e we plot a/e vs. e/e the latter is simply the time, t.) Superposition over the entire time scale for 0% H2O (upper curve) is excellent except for times close to the fracture times of the materials tested e higher strain rates. For example, a deviat ipn occurs at 10 sec for the material at e = 3.3 x 10 sec... 

Fluoropolymers, as well as other thermoplastics, exhibit a complicated nonlinear response when subjected to loads. The behavior is characterized by initial linear viscoelasticity at small deformations, followed by distributed yielding, viscoplastic flow, and material stiffening at large deformations until ultimate failure occurs. The response is further complicated by a strong dependence on strain rate and temperature, as illustrated in Fig. 11.1. It is clear that higher deformation rates and lower temperatures increase the stiffness of the material. 

It is helpful to introduce mechanical elements as models of viscoelastic response, but neither the spring nor the dashpot alone accurately describes viscoelastic behavior. Some combination of both elements is more appropriate and even then validity is restricted to qualitative descriptions they provide valuable visual aids. In most polymers, mechanical elements do not provide responses beyond strains greater than about 1% and strain rates greater than 0.1 sr. ... 

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