Constant strain-rate modulus

For deformation in simple extension at constant rate of (practical) strain e, for which <7(r) is replaced by E(t), Smith has pointed out that it is convenient to deflne a constant-strain-rate modulus F(t) = This is related to the relaxation... 

J7 In a tensile test on a plastic, the material is subjected to a constant strain rate of 10 s. If this material may have its behaviour modelled by a Maxwell element with the elastic component f = 20 GN/m and the viscous element t) = 1000 GNs/m, then derive an expression for the stress in the material at any instant. Plot the stress-strain curve which would be predicted by this equation for strains up to 0.1% and calculate the initial tangent modulus and 0.1% secant modulus from this graph. 

If the elastic modulus is obtained from the slope of the elastic stress-strain curve, then we can evaluate the first term on the right-hand side in Equation (8.3) from experimental data elastic stress-strain curves. The second term on the right-hand side in Equation (8.3) can be evaluated from the product of the strain rate, which is set in a constant strain-rate experiment, and the viscosity. As we discussed in Chapter 3, the viscosity of a macromolecule is related to the shape factor v, therefore we can evaluate the second term on the right-hand side of Equation (8.3) from the product of the shape factor and the strain rate. 

Each type of propellant has specific mechanical characteristics, but the influence of test parameters (temperature, strain rate, and pressure) is the same for all propellants (11). Tensile tests are widely used to analyze propellant behavior as well as examine the manufacturing controls of the propellants. Because their behavior is not linear-elastic, it is necessary to define several parameters that allow a better representation of the experimental tensile curve. The stylistic experimental stress-strain response at a constant strain rate from a uniaxial tensile test is shown in Figure 7, where E is the elastic modulus (initial slope), Sr P is the tensile strength (used later for a failure criterion), and eXj> is the strain at tensile strength. 

Stress-strain curves are often measured by monitoring the tensile stress as a sample, originally at rest, is subjected to a constant tensile strain rate starting at t = 0. Show that, at any subsequent time during the constant-strain-rate period, the slope of the stress-strain curve is the tensile stress relaxation modulus ... 

Fig. 4.32. (a) A constant strain rate leads to a non-linear stress (b) the definition of (r), the mean modulus and (c) increasing the strain rate (k) increases the slope of the stress-strain curve. 

Step-strain stress-relaxation measurements have been frequently used to determine Sr(X) for polymer melts > . Equation (6) shows that if separability of time and strain effects is possible for the melt under consideration, the stress after a step elongational strain can be factored into a time-dependent function, the linear shear relaxation modulus G(t), and a strain-dependent function, the nonlinear strain measure Sr(X). Also other types of experiment may be oerformed to obtain Sr(X), such as constant-strain-rate experiments "", creep under constant stress and constant-stretching-rate experiments but these methods require more involved analytical and/or numerical calculations. 

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. 

As it is known [2, 12], within the frameworks of cluster model the elasticity modulus E value is defined by stiffness of amorphous polymers structure both components local order domains (clusters) and loosely packed matrix. In Fig. 13.3, the dependences E(v J are adduced, obtained for tensile tests three types with constant strain rate, with strain discontinuous change and on stress relaxation. As one can see, the dependences E(yJ are approximated by three parallel straight lines, cutting on the axis E loosely packed matrix elasticity modulus E different values. The greatest value E is obtained in tensile tests with constant strain rate, the least one - at strain discontinuous change and in tests on stress relaxation E = 0 [1]. 

FIGURE 13.3 The dependences of elasticity modulus E on entanglements cluster network density n, in tests with constant strain rate (1), strain discontinuous change (2) and on stress relaxation (3) for PASF [1]. 

If tests are performed at different constant strain rates or temperatures, stress-strain response similar to that shown in Fig. 3.6 is obtained for many polymers. Notice that modulus and intrinsic yield point vary with both rate and temperature. Also, the stress-strain response appears to be nonlinear even at low stress levels. However, caution on the interpretation of the information obtained from such elementary tests is suggested, as it will be shown in a later section that linearity as well as other essential mechanical properties should be deduced from isochronous stress-strain diagrams. 

A constant strain rate test may be used to determine the relaxation modulus and a constant stress-rate test may be used to find the creep compliance. Steady state oscillation tests may also be used to determine the viscoelastic properties of polymers. These details and the interrelation between various test approaches are given in Chapter 5. 

For polymers, the torsion test is often the test of choice because, as discussed in Chapter 2, the time dependent (viscoelastic) behavior of polymers is principally due to the deviatoric (shear or shape change) stress components rather than the dilatoric (volume change) stress components. Typically, constant strain rate tests are often used for either tension, compression or torsion as discussed in Chapter 3. If the material is linear elastic, the stress rate is proportional to the strain rate as the modulus is time independent. That is. 

From this result it is apparent that the relaxation modulus can be found from a constant strain-rate test by dividing the slope of the stress output by the strain-rate. Similarly, the creep compliance can be found from an constant stress-rate test by dividing the strain output by the stress-rate,... 

To obtain the output for a Maxwell fluid in a constant strain rate test, the relaxation modulus, E(t) = must be inserted as... 

One way to obtain long-term information is through the use of the time-temperature-superposition principle detailed in Chapter 7. Indeed, J. Lohr, (1965) (the California wine maker) while at the NASA Ames Research Center conducted constant strain rate tests from 0.003 to 300 min and from 15° C above the glass transition temperature to 100° C below the glass transition temperature to produce yield stress master curves for poly(methyl methacrylate), polystyrene, polyvinyl chloride, and polyethylene terephthalate. It should not be surprising that time or rate dependent yield (rupture) stress master curves can be developed as yield (rupture) is a single point on a correctly determined isochronous stress-strain curve. Whether linear or nonlinear, the stress is related to the strain through a modulus function at the yield point (mpture) location. As a result, a time dependent master curve for yield, rupture, or other failure parameters should be possible in the same way that a master curve of modulus is possible as demonstrated in Chapter 7 and 10. 

Equation (3.26) gives a relaxation modulus that is independent of the applied strain magnitude, and will obey the homogeneity requirement of linearity for all scalars, with any arbitrary strain input. Equation (3.26) is not linear however as norms are not superposable except in the most trivial examples. The hypothetical material represented by Equation (3.26) is therefore non-linear, but for many types of tests used for material characterization it could not be distinguished from a linear viscoelastic material. In fact, the parameters n and P appearing in this constitutive equation can be adjusted so that the time derivative of the stress for a constant strain rate input is proportional to the relaxation modulus a commonly cited property of a linear viscoelastic material [12,37]. Careful examination of the stress output to various strain inputs confirms the non-linear nature of this equation and indicate it is within the range of this simple equation to describe the onedimensional response of solid propellants at small strains. To demonstrate this ability, the stress output for a variety of strain inputs have been determined for different values of n and p. 

In a fluid under stress, the ratio of the shear stress, r. to the rate of strain, y, is called the shear viscosity, rj, and is analogous to the modulus of a solid. In an ideal (Newtonian) fluid the viscosity is a material constant. However, for plastics the viscosity varies depending on the stress, strain rate, temperature etc. A typical relationship between shear stress and shear rate for a plastic is shown in Fig. 5.1. 

The elastic stress curve in figure perfectly follows elastic strain [2]. This constant is the elastic modulus of the material. In this idealized example, this would be equal to Young s modulus. Here at this point of maximum stretch, the viscous stress is not a maximum, it is zero. This state is called Newton s law of viscosity, which states that, viscous stress is proportional to strain rate. Rubber has some properties of a liquid. At the point when the elastic band is fully stretched and is about to return, its velocity or strain rate is zero, and therefore its viscous stress is also zero. 

Figure 47 shows taken from Equation 20 versus Vj. It shows that S. is quite sensitive to Vp and is therefore a good means to evaluate v, with the numerical values of Fig. 47. It can be estimated that the tensile modulus E of the bulk PMMA is not affected by the very low pressure toluene gas environment during the short duration of the experiment. The optical craze index in PMMA in air without load is known as n = 1.32, which corresponds to v = 0.6. From the optical interferometry, it is known that the craze just before breakage is twice as thick as unloaded, (v, = 0.3) and hence using Lorentz-Lorenz equation its optical index is n = 1.15. From Figs. 46 and 47 it can be concluded that the bulk modulus around the propagating crack is about 4400 MPa, which is a somewhat high value, in view of the strain rates at a propagating crack tip (10 to s" ). Using the scatter displayed in Fig. 46, it can be concluded from Fig. 47 that the fibril volume fraction is constant, v = 0.3, within a scatter band of 0.08, and is therefore not sensitive to the toluene gas.

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