Temperature-dependent stress-strain curves

Figure 11. Temperature-dependent stress—strain curves for [0] composite laminates. (Reproduced from reference 12.)...

Great importance is attached to the determination of required materials data, since simulations can only be as reliable as the underlying input data. Here all caloric materials data were taken from literature or databases. Only mechanical data, in particular temperature depending stress-strain curves, are rarely available. Therefore one-dimensional tensile tests were performed at different temperatures by using the Gleeble3500 equipment. 

Kawashima, K. Ito, T. Sakuragi, M. Strain rate and temperature dependent stress-strain curves of Sn-Pb eutectic solder. J. Mater. Sci. 1992, 27, 6387-6390. 

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... 

Depending on the material and deformation conditions (strain rate, temperature) other stress-strain curve shapes can be observed (Fig. 2b and c). In Fig. 2b, the plastic flow occurs at the same stress level as that required for the yielding so the strain softening does not exist. In the case shown in Fig. 2c, the strain hardening happens very close to yielding, suppressing both strain softening and plastic flow behaviour. 

As a pipeline is heated, strains of such a magnitude are iaduced iato it as to accommodate the thermal expansion of the pipe caused by temperature. In the elastic range, these strains are proportional to the stresses. Above the yield stress, the internal strains stiU absorb the thermal expansions, but the stress, g computed from strain 2 by elastic theory, is a fictitious stress. The actual stress is and it depends on the shape of the stress-strain curve. Failure, however, does not occur until is reached which corresponds to a fictitious stress of many times the yield stress. 

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. 

The transition obtained under stress can be in some cases reversible, as found, for instance, for PBT. In that case, careful studies of the stress and strain dependence of the molar fractions of the two forms have been reported [83]. The observed stress-strain curves (Fig. 16) have been interpreted as due to the elastic deformation of the a form, followed by a plateau region corresponding to the a toward [t transition and then followed by the elastic deformation of the P form. On the basis of the changes with the temperature of the critical stresses (associated to the plateau region) also the enthalpy and the entropy of the transition have been evaluated [83]. 

Figure 18.17 shows that the characteristics of the stress-strain curve depend mainly on the value of n the smaller the n value, the more rapid the upturn. Anyway, this non-Gaussian treatment indicates that if the rubber has the idealized molecular network strucmre in the system, the stress-strain relation will show the inverse S shape. However, the real mbber vulcanizate (SBR) that does not crystallize under extension at room temperature and other mbbers (NR, IR, and BR at high temperature) do not show the stress upturn at all, and as a result, their tensile strength and strain at break are all 2-3 MPa and 400%-500%. It means that the stress-strain relation of the real (noncrystallizing) rubber vulcanizate obeys the Gaussian rather than the non-Gaussian theory. 

The compression stress-strain curves obtained over quite a broad temperature range are shown in Fig. 15 [33]. It appears that the strain softening, weak and slightly temperature-dependent in the temperature range from 40 to 100 °C, increases when lower temperatures are considered. 

Fig. 15 Temperature dependence of nominal stress-strain curves under uni-axial compression for PMMA at a strain rate of 2 x 10-3 s-1 (From [33])...

The effect of strain rate, sy at - 50 °C and 50 °C is shown in Fig. 16. Qualitatively, increasing the strain rate is analogous to decreasing temperature (the temperature and strain rate dependencies of the stress-strain curve shapes are summarised in Fig. 17). However, it is worth noting that an equivalence temperature-strain rate does not apply over the whole stress-strain curve. 

Fig. 17 Temperature and strain rate dependencies of the stress-strain curve shapes of PMMA small (less than 10 MPa) and larger strain softening (From [33])...

At this stage, the difference in temperature and strain rate dependencies of cry and Opf explains why a strain rate and temperature equivalence cannot be achieved over the whole stress-strain curve. 

As above described, depending on temperature and strain rate, the stress-strain curves present a strain softening of variable amplitude. 

Note that, since the stress-strain curves are dependent on the applied strain rate and the specimen temperature, both PED and AT], are functions of the strain, strain rate, and temperature. 

Polymeric materials show a wide range of stress-strain characteristics. One characteristic of polymers that is markedly different from metals and ceramics is that their mechanical properties are highly time- and temperature-dependent. An elastomer or a rubbery polymer shows a stress-strain curve that is nonlinear. 

In some cases, an extrudable and injectable paste may consist of 65% vol. ceramic powder and 35% vol. polymeric binder. In others, an extrudable paste may consist of a highly loaded aqueous suspension of clay particles such that its rheology is plastic. Hie low shear (i.e., <100 sec ) viscosity of such a paste is between 2000 and 5000 poise at ambient temperature. Highly nonlinear stress strain curves are typical of ceramic pastes, as well as time dependent thixotropy. In many cases, pastes behave like visco-elastic fluids. This complex rheological behavior of ceramic pastes has made theoretical approadies to these problems difficult. For this reason, the discussion in this chapter is limited to Newtonian fluids where analytical solutions are possible, with obvious consequences as to accuracy of these equations for non-Newtonian ceramic pastes. 

The most common type of stress-strain tests is that in which the response (strain) of a sample subjected to a force that increases with time, at constant rate, is measured. The shape of the stress-strain curves is used to define ductile and brittle behavior. Since the mechanical properties of polymers depend on both temperature and observation time, the shape of the stress-strain curves changes with the strain rate and temperature. Figure 14.1 illustrates different types of stress-strain curves. The curves for hard and brittle polymers (Fig. 14.1a) show that the stress increases more or less linearly with the strain. This behavior is characteristic of amorphous poly-... 

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