Correspondence strain-time

The dashpot constant, rj2, for the Kelvin-Voigt element may be determined by selecting a time and corresponding strain from the creep curve in a region where the retarded elasticity dominates (i.e. the knee of the curve in Fig. 2.40) and substituting into equation (2.42). If this is done then r)2 = 3.7 X 10 MN.s/m ... 

This mathematical condition can be replaced by the following physical argument. If only one normal stress is applied at a time, the corresponding strain is determined by the diagonal elements of the compliance matrix. Thus, those elements must be positive, that is,... 

The brittle behaviour of the silica aerogel remains upon the introduction of TMS-modihed PNPs. However, even a low polymer content promotes a clear improvement of the mechanical properties of the material Young s modulus increases and the maximum compression strength and the corresponding strain become three to five times higher. Furthermore, the improvement in mechanical behaviour is noticeable... 

In normal motion, the load exerted on the hip joint is 2.5 times body weight, (a) Estimate the corresponding stress (in MPa) on an artificial hip implant with a cross-sectional area of 5.64 cm in an average-sized patient, and (b) calcnlate the corresponding strain if the implant is made of Ti-6A1-4V alloy. Clearly state yonr assnmptions and sonrces of data. 

In bodies showing retarded elasticity, the deformation is a function of time as well as stress. Such a stress-strain curve is shown in Figure 8-10. The upward part of the curve represents increasing values of stress when the stress is reduced, the corresponding strains are greater on the downward part of the curve. When the stress reaches 0, the strain has a finite value, which will slowly return to zero. There is no permanent deformation. The corresponding relaxation (stress-time) and creep (strain-time) curves... 

The viscoelastic stress-strain equation, Equation (4) can be expressed in finite element formulation which relates the stress tensor a.. at time index n and cell centre (ij) to the corresponding strain tensor arising from the movement of the adjoining cell corners. Using backward differences for the time step, at time index n. 

The influence of temperature on the stress-strain behavior of polymers is generally opposite to that of straining rates. This is not surprising in view of the correspondence of time and temperature in the linear viscoelastic region (Section I l.5.2.iii). The curves in Fig. 11-23 are representative of the behavior of a partially crystalline plastic. 

Applying stress to any material results in a corresponding strain. For the common building materials such as steel and concrete the strain occurs virtually instantaneously with stress application. In contrast, fine-grained soils generally exhibit a measurable time lag between stress application and the resulting strain. This phenomenon is called consolidation. 

This solution illustrates an important point for a linear viscoelastic material, it is possible to convert between the viscoelastic response functions, without Imowledge of the stress/strain/time differential equations to which th correspond. 

Calculate an equivalent strain rate corresponding to each yield stress by dividing yield strain time to yield. 

The strain-time creep curves of alumina based fibers exhibit a steady state (or secondary) e-t linear domain, the corresponding strain rate, e, depending on the applied stress, a the grain size, a and the test temperature, T, according to the Dorn equation. 

Fig. 7 Computerized measurement and the creep rate vs compressive strain plot obtained for polycarbonate near the yield point y = 6.5% [12]. Each point corresponds to a deformation increment of 300 nm (0.005%). The variance of the creep rate reflects the jump-like development of deformation (see Sect. 5). IrneV. stationary creep in the strain-time coordinates, at 70 MPa and room temperature...

The physical meaning of the relaxation modulus C(t) is apparent in terms of this simple experiment. In general, the ratio of a stress to the corresponding strain is called a modulus, and for a perfectly elastic solid the equilibrium shear modulus G is defined as a/y, G(t) is its time-dependent analog as measured in an experiment with this particular time pattern. This experiment can be performed on both viscoelastic liquids and viscoelastic solids. 

Figure 7.2 shows that the strains produced by a shear stress are different depending on whether the material is a solid or a fluid. For a soUd, the strain is finite the rectangle will quasi-instantaneously turn into a slanted parallelepiped, and come to rest in that state of strata Consequently, to atty stress state there is a corresponding strain state. The fluid, on the other hand, will flow. This means that the parallelepiped leans over even more as time goes on To every stress state there is not a corresponding strain state, but a rate at which the parallelepiped leans over. This is referred to as a strain rate. 

Usually, creep models for engineering materials are based on the use of the strain-time response of the material. Other quantities, such as deflection and creep modulus, can be obtained using the corresponding relationships. Figure 4.4 shows the general strain versus time response of rPET polymer concrete subjected to constant stress and temperature. 

Consider a stress programme starting at time t = 0 in which the stress decreases exactly as the relaxation function G(t). In this case, the corresponding strain must remain constant as in a typical stress relaxation experiment. Thus, if... 

Consequently, only the components a, a y of the stress tensor survive. On taking time Fourier transforms, these are given by 2fi(co) multiplied by the corresponding strains, as follows from (1.9.18). The dynamical equation takes the form... 

Fig. 19 Onset of steady shear flow for unfilled LDPE melt at different shear rates represented as a a time dependence and b a corresponding strain dependence. T = 240 °C...

The method preferred by most design engineers is the use of isochronous (equal time) stress-strain curves. The stress versus corresponding strain is plotted at a specihc time of loading pertinent to particular application. 

A phenomenological explanation has been given for equations 6.27-6.29 by this author [12] without reference to Schapery s theory. It is a rational analysis of the relationship between macroscopic deformation and the deformation in the segmental level of polymer chains. An increase of free volume resulting from deformation in the dynamic state is the explanation for the strain-time correspondence. 

In order to differentiate between gel types, large deformation measurements must be performed and the strain-time correspondence mnst be used for linearisation. The elongation measurements performed at a constant temperature and various elongation rates are expressed... 

The data at different elongation rates do not obey strain-time correspondence but form different curves, which have an upturn indicating limited extensibility [22]. 

Some gum rubbers obey strain-time correspondence. Other rubbers do not obey the principle, because of strain-hardening. There is yet another type which does not obey strain-time correspondence, because it exhibits linear behaviour even at rather large deformations. In general this type of rubber is soft, and it may be difficult to obtain accurate elongational data. By using a rotational rheometer operated at very low speeds in order to avoid slip, it is possible to obtain accurate data at large shear strains [24]. 

For non-linear behaviour which obeys strain-time correspondence, viscosities Tj and It] I are equal at equal values of ro and 1/ott. Instead of equation (6.35), viscosity has the form... 

The points in the figures represent tj and the lines It] . Figure 6.10a shows that the linear form, equation (6.35), holds, and Figure 6.10b shows that the non-linear, strain-time correspondence, equation (6.37), does not hold. 

Figure 6.10 (a) Complex viscosity of ethylene-propylene rubber (EPR) at 30 °C. The line represents the observed data and the markings are the data calculated from the stress rise at a constant deformation rate. The calculation is based on linear viscoelasticity. The stress rise measurements at e = 0.025 s were performed in triplicate. The figure shows agreement between calculated and observed data, (b) The same data as in (a), except that the calculation is based on the non-linear relation, the strain-time correspondence. 

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