Stress relaxation experiment

This is the fundamental differential equation for a shear stress relaxation experiment. The solution to this differential equation is an equation which gives a as a function of time in accord with experiment. 

The stress-relaxation behavior of a material is normally determined in either the tensile or the flexural mode. In these experiments, a material specimen is rapidly elongated or compressed to produce a specified strain level and the load exerted by the specimen on the test apparatus is measured as a function of time. Specimens of certain plastics may fail during tensile or flexural stress-relaxation experiments. 

Similar results were observed in the stress-relaxation experiments which are shown in Figure 2. The 5- and 10-day samples relax to the same stress level. The major difference in stress-relaxation behavior among the different samples occurs during the very beginning of the relaxation process. For that reason, and in order to better illustrate the first minutes of relaxation, the time scale is logarithmic. 

A similar superposition holds for stress-relaxation experiments in which the strain is changed during the course of the experiments. The Bolt/mann superposition principle for stress relaxation is... 

Moreover, in developing and testing the theory, biaxial stress-relaxation experiments were carried out. That is, square sheets were stretched in both directions but in unequal amounts. In all cases, the stress in the major stretch direction relaxed at the same relative rate as that in the minor... 

Time is the major (actor in determining the mechanical properties of a polymer. This is seen directly in creep and stress-relaxation experiments. These tests cover long periods of time, so that they are sensitive to the types of molecular motions that require long times. Tfrey give little direct information on the types of molecular motion that take place at short times. However, by using the time-temperature superposition principle and the WLF equations, access to these short times can be achieved even though they may not easily be attainable by direct experimentation. 

The spring is elastically storing energy. With time this energy is dissipated by flow within the dashpot. An experiment performed using the application of rapid stress in which the stress is monitored with time is called a stress relaxation experiment. For a single Maxwell model we require only two of the three model parameters to describe the decay of stress with time. These three parameters are the elastic modulus G, the viscosity r and the relaxation time rm. The exponential decay described in Equation (4.16) represents a linear response. As the strain is increased past a critical value this simple decay is lost. 

A stress relaxation experiment can be performed on a wide range of materials. If we perform such a test on a real material a number of deviations are normally observed from the behaviour of a single Maxwell model. Some of these deviations are associated with the application of the strain itself. For example it is very difficult to apply an instantaneous strain to a sample. This influences the measured response at short experimental times. It is often difficult to apply a strain small enough to provide a linear response. A Maxwell model is only applicable to linear responses. Even if you were to imagine an experiment where a strain is... 

The ideal stress relaxation experiment is one in which the stress is instantaneously applied. We have seen in Section 4.4.2 the exponential relaxation that characterises the response of a Maxwell model. We can consider this experiment in detail as an example of the application of the Boltzmann Superposition Principle. The practical application of an instantaneous strain is very difficult to achieve. In a laboratory experi-... 

Figure 4.13 The applied strain in a typical stress relaxation experiment...

As with the elastic solid we can see that as the stress is applied the strain increases up to a time t = t. Once the stress is removed we see partial recovery of the strain. Some of the strain has been dissipated in viscous flow. Laboratory measurements often show a high frequency oscillation at short times after a stress is applied or removed just as is observed with the stress relaxation experiment. We can replace a Kelvin model by a distribution of retardation times ... 

In order to proceed with the evaluation of the time-dependent Poisson ratio v(0, both sets of relaxation behaviour are required. Now from Chapter 2 we know the Poisson ratio is the ratio of the contractile to the tensile strain and that for an incompressible fluid the Poisson ratio v = 0.5. Suppose we were able to apply a step deformation as we did for a shear stress relaxation experiment. The derivation then follows the same course as that to Equation (4.69) ... 

The time-temperature superposition principle has practical applications. Stress relaxation experiments are practical on a time scale of 10 to 10 seconds (10 to 10 hours), but stress relaxation data over much larger time periods, including fractions of a second for impacts and decades for creep, are necessary. Temperature is easily varied in stress relaxation experiments and, when used to shift experimental data over shorter time intervals, can provide a master curve over relatively large time intervals, as shown in Figure 5.65. The master curves for several crystalline and amorphous polymers are shown in Figure 5.66. 

Figure H3.3.2 A stress relaxation experiment where a small constant strain (y) is applied to a sample. After the strain is applied, the stress (o) of the material is measured as a function of time.

Figure H3.3.4 Mechanical models are often used to model the response of foods in creep or stress relaxation experiments. The models are combinations of elastic (spring) and viscous (dashpot) elements. The stiffness of each spring is represent by its compliance (J= strain/stress), and the viscosity of each dashpot is represent by a Newtonian viscosity (ri). The form of the arrangement is often named after the person who originally proposed the model. The model shown is called a Burgers model. Each element in the middle—i.e., a spring and dashpot arranged in parallel—is called a Kelvin-Voigt unit.

Figure 2.50 The stress-relaxation experiment. Reprinted with permission from J. E. Mark, Physical Chemistry of Polymers, ACS Audio Course C-89, American Chemical Society, Washington, DC, 1986. Copyright 1986, American Chemical Society.

Stress relaxation experiments were performed on Instron 1122 tensile tester. Dog-bone-shaped specimens were prepared in accordance to ASTM D1708-66. The specimens were 22.25 mm long (linear section of the dog-bone-shaped specimen),... 

With physical aging at 140 °C in nitrogen/dark atmosphere, the dynamic storage modulus is very sensitive to aging time. The modulus increased from 13 GPa (10 minutes-aged) to 18 GPa for samples aged up to 105 minutes at 140 °C (see Fig. 16). These results agree with observations made in the stress relaxation experi-... 

The temperature dependence of the relaxation modulus at 500 seconds of polycarbonate (7), polystyrene (8), and their blends (75/25, 50/50, and 25/75) was obtained from stress-relaxation experiments (Figure 4, full lines). In the modulus-temperature curves of the blends, two transition regions are generally observed in the vicinity of the glass-rubber transitions of the pure components. The inflection temperatures Ti in these transition domains are reported in Table I they are almost independent on composition. The presence of these two well-separated transitions is a confirmation of the two-phase structure of the blends, deduced from microscopic observations. 

In a stress relaxation experiment the Maxwell-element is subjected to an instantaneous deformation sQ which is held constant. It means that ... 

This is an important point, so let s beat it to death with an example. Imagine that we perform a simple tensile creep experiment where a stress oQ is applied to a sample and after 10 hours the strain, (10) is measured. Now let s take an identical sample and perform a stress relaxation experiment where the sample is stretched instantaneously to give... 

Because you are probably brain dead after working your way through all of those mechanical models, we will first remind you of the type of data you observe in, for example, a stress relaxation experiment as a function of time (shown schematically in Figure 13-83) compared to an equivalent experiment conducted as a function of temperature (see Figure 13-82) there is an obvious equivalence. 

Stress relaxation experiments correspond to the situations in which the deformations sketched in Fig. 11-12 are imposed suddenly and held fixed while the resulting stresses are followed with time. The tensile relaxation modulus Y(t) is then obtained as... 

The Maxwell body is appropriate for the description of stress relaxation, while the Voigt element is more suitable for creep deformation. In a stress relaxation experiment, a strain yo is imposed atr = Oand held constant thereafter (dy/r// = 0) while r is monitored as a function of t. Under these conditions, Eq. (11-29) for a Maxwell body behavior becomes... 

From data in a stress relaxation experiment (Chapter 3), where the strain is constant and stress is measured as a function oftime, a If), the relaxation time may be estimated from the time necessary for [cr(r)/cr(0)] to become (1/e) = 0.368. Typically, several Maxwell elements are used to fit experimental data, a t). For the Kelvin-Voigt element (Figure 1-8, right) under stress, the equation is ... 

Figure 3-45 Typical Stress versus Time Curve Obtained from a Stress Relaxation Experiment (Ccanby etal., 1985).

Figure 5,7 Sketch illustrating the transition from linear to nonlinear behavior in shear relaxation experiments. Note that the data must be obtained by a series of stress relaxation experiments.

Strictly speaking, there are no static viscoelastic properties as viscoelastic properties are always time-dependent. However, creep and stress relaxation experiments can be considered quasi-static experiments from which the creep compliance and the modulus can be obtained (4). Such tests are commonly applied in uniaxial conditions for simphcity. The usual time range of quasi-static transient measurements is limited to times not less than 10 s. The reasons for this is that in actual experiments it takes a short period of time to apply the force or the deformation to the sample, and a transitory dynamic response overlaps the idealized creep or relaxation experiment. There is no limitation on the maximum time, but usually it is restricted to a maximum of 10" s. In fact, this range of times is complementary, in the corresponding frequency scale, to that of dynamic experiments. Accordingly, to compare these two complementary techniques, procedures of interconversion of data (time frequency or its inverse) are needed. Some of these procedures are discussed in Chapters 6 and 9. 

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