Viscoelastic tests/parameters stress relaxation

Stress relaxation experiments involve the measurement of the force required to maintain the deformation produced initially by an applied stress as a function of time. Stress relaxation tests are not performed as often as creep tests because many investigators believe they are less readily understood. The latter point is debatable, and it may only be that the practical aspects of creep measurements are simpler. As will be shown later, all the mechanical parameters are in theory interchangeable, and so all such measurements will contribute to the understanding of viscoelastic theory. Whereas stress relaxation measurements are useful in a general study of polymeric behavior, they are particularly useful in the evaluation of antioxidants in polymers, especially elastomers, because measurements on such systems are relatively easy to perform and are sensitive to bond rupture in the network. 

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. 

Dynamic testing of polymer blends at small amplimde is a relatively simple and reliable procedure. The resulting storage and loss shear moduli, G and G", respectively, should be first corrected for yield stress then the loss data can be fitted to Eq. 7.42 to determine the value of the four parameters, t)o, x, nii, and m2. Once these parameters are known, the Gross frequency relaxation spectrum, and as a result aU linear viscoelastic functirms, can be calculated (see Eqs. 7.85, 7.86, and 7.87). 

The dynamic tests at small amplitude in parallel plates or cone-and-plate geometry are simple and reproducible. From the experimental values of storage and loss shear moduli, G and G", respectively, first the yield stress ought to be extracted and then the characteristic four material parameters in Eq. (2.13), rjo, r, mi, and m2, might be calculated. Next, knowing these parameters one may calculate the Gross frequency relaxation spectrum (see Eqs. (2.31) and (2.32)) and then other linear viscoelastic functions. 

The non linear viscoelasticity of various particles filled rubber is addressed in range of studies. It is found that the carbon black filled-elastomer exhibit quasi-static and dynamic response of nonlinearity. Hartmann reported a state of stress which is the superposition of a time independent, long-term, response (hyperelastic) and a time dependent, short-term, response in carbon black filled-rubber when loaded with time-dependent external forces. The short term stresses were larger than the long term hyperelastic ones. The authors had done a comparative study for the non linear viscoelastic models undergoing relaxation, creep and hysteresis tests [20-22]. For reproducible and accurate viscoelastic parameters an experimental procedure is developed using an ad hoc nonlinear optimization algorithm. 

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. 

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