Time-dependent stress relaxation modulus

Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

Because of equipment limitations in measuring stress and strain in polymers, the time-temperature superposition principle is used to develop the viscoelastic response curve for real polymers. For example, the time-dependent stress relaxation modulus as a function of time and temperature for a PMMA resin is shown in... 

Substituting the relation between relaxation time and mode wavelength [Eq. (8.109)] into the expression for modulus [Eq. (8.116)] leads to the time-dependent stress relaxation modulus that decays as the - 3/4 power of time ... 

In a stress relaxation test a sample is quickly placed under a strain that is then held constant, and the resulting stress is recorded as a function of time. The response of an ideal elastic solid in stress relaxation is a stress that remains constant with time, while an ideal liquid responds with an immediate return to zero stress as soon as the test strain is imposed. Viscoelastic materials respond with a stress that decays with time. Stress relaxation data are commonly reported as a time-dependent stress relaxation modulus ... 

Equations analogous to those for s(t) above, can be written for the time-dependent stress relaxation modulus, Gy(t), of a linearly viscoelastic body subjected to successive shear strains,... 

For experiments performed in shear, there is a rather complicated relation between the time-dependent stress relaxation shear modulus G(t) defined by Equation 3.19 and the time-dependent creep compliance J t) defined by Equation 3.21. But if the slope of log G(r) versus log r is — m, then, to a good approximation. 

In polymers the time dependence of an modulus plays a more important role than in metals. If polymers are loaded with a constant stress they undergo a deformation e, which increases with time. This process is named creep. Conversely, if a test specimen is elongated to a certain amount and kept under tension, the initial stress s decreases with time. This decay is called stress relaxation. 

The data are not usually reported as a stress/time plot, but as a modulus/time plot. This time-dependent modulus, called the relaxation modulus, is simply the time-dependent stress divided by the (constant) strain (Equation 13-71) ... 

Consider imposing a step strain of magnitude 7 at time t = 0 (see Fig. 7.20). If the material between the plates is a perfectly elastic solid, the stress will jump up to its equilibrium value Gj given by Hooke s law [Eq. (7.98)] and stay there as long as the strain is applied. On the other hand, if the material is a Newtonian liquid, the transient stress response from the jump in strain will be a spike that instantaneously decays to zero. For viscoelastic materials, the stress after such a step strain can have some general time dependence a(t). The stress relaxation modulus G(t) is defined as the ratio of the stress remaining at time t (after a step strain was applied at time t = 0) and the magnitude of this step strain 7 ... 

For viscoelastic liquids, the Maxwell model can be used to qualitatively understand the stress relaxation modulus. In the step strain experiment, the total strain 7 is constant and Eqs (7.101)-(7.103) can be combined to give a first order differential equation for the time-dependent strain in the viscous element ... 

The time dependence of the stress relaxation modulus in semidilute unentangled solution is sketched in Fig. 8.10. Experimental verification of Rouse dynamics for frequencies smaller than 1/r was shown in Fig. 8.5, for a semidilute unentangled polyelectrolyte solution. 

The second important consequence of the relaxation times of all modes having the same temperature dependence is the expectation that it should -bp possible to superimpose linear viscoelastic data taken at different temperatures. This is commonly known as the time-temperature superposition principle. Stress relaxation modulus data at any given temperature Tcan be superimposed on data at a reference temperature Tq using a time scale multiplicative shift factor uj- and a much smaller modulus scale multiplicative shift factor hf. 

The stress relaxation modulus decays as a power law in timb with exponent 3/(1 -I- 2) = 0.66. This power law dependence continues up to the longest relaxation time r of the characteristic branched polymer. At the gel point this power law extends forever because t diverges [see Eq. (8.139)]. 

The time-dependent viscoelastic response of polymers is broken down into individual modes that relax on the scale of subsections of the chain with Njp monomers. The Rouse and Zimm models have different structure of their mode spectra, which translates into different power law exponents for the stress relaxation modulus G t) ... 

The full-time dependence of the stress relaxation modulus ot randomly branched unentangled polymers is best derived from the fractal dynamics of Section 8.8 using the relaxation rate spectrum P( ) ... 

Doi was the first to point out that the decrease of tube length due to these fluctuations leads to partial relaxation of stress. The stress relaxation modulus G(t) is not quite constant in the rubbery plateau, but decreases -slightly with time. The weak time dependence of the stress relaxation... 

Figure 2. Time dependence of the linear stress relaxation modulus for the Si02 - PBA50K hybrid at a strain of 0.02 is shown. The stress relaxation data for strain values below 0.04 were identical and exhibited solid-like behavior for 80,000 s after application of the step strain.

Thus viscoelasticity is characterized by dependencies on temperature and time, the complexities of which may be considerably simplified by the time-temperature superposition principle. Similarly the response to successively loadings can be simply represented using the applied Boltzmann superposition principle. Experimentally viscoelasticity is characterized by creep compliance quantified by creep compliance (for example), stress relaxation (quantified by stress relaxation modulus), and by dynamic mechanical response. 

Opposite to the regulations of the standard ISO 899-1 the compression-creep modulus is named with the symbol Fee- The frequently used values in the following Table are the modulus at 1 hour Ecd, 100 hours Fccioo and at 1000 hours Fcciooo- In the case of stress relaxation experiments the compression-relaxation modulus Ere can be determined from the time dependent stress constant strain level Sco-... 

A polymeric cantilever spring of length /, whose cross-section has a second moment of area /, is held to a constant deflection Sg by a force F applied at its free end. Show that the time-dependent decay of F is given as follows in terms of the stress relaxation modulus Eif) ... 

Where /o is the imposed fixed strain, (p(t) is the relaxation function decreasing from 0) = 1 at f = 0 to (p(°°) = 0att = °°, and is the equilibrium modulus, which is finite for a viscoelastic solid and zero for viscoelastic liquid. The time-dependent stresses arising from different molecular mechanisms are not additive, and hence it is difficult if not impossible to isolate and characterize each one of them individually. Nevertheless, G(t) is connected to J(t) by the convolution integral equation, SoG s)J(t - s)ds = 1, from which one function can be calculated from the other by a numerical procedure [22]. 

Viscoelastic characteristics of polymers may be measured by either static or dynamic mechanical tests. The most common static methods are by measurement of creep, the time-dependent deformation of a polymer sample under constant load, or stress relaxation, the time-dependent load required to maintain a polymer sample at a constant extent of deformation. The results of such tests are expressed as the time-dependent parameters, creep compliance J t) (instantaneous strain/stress) and stress relaxation modulus Git) (instantaneous stress/strain) respectively. The more important of these, from the point of view of adhesive joints, is creep compliance (see also Pressure-sensitive adhesives - adhesion properties). Typical curves of creep and creep recovery for an uncross-Unked rubber (approximated by a three-parameter model) and a cross-linked rubber (approximated by a Voigt element) are shown in Fig. 2. 

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