Structural analysis of viscoelastic materials

A 'FEW years ago, the author drew attention to the growing importance of viscoelastic analysis in evaluating the structural integrity of solid propellant rocket motors 1 2 Although viscoelastic materials have been incorporated as structural elements in many engineering situations, as, for example, in asphalt pavements, their greatest recent application, which has provided the stimulus for accompanying analyses, has certainly been due to the increasing use of viscoelastic solid fuels in rockets As described in the pre ceding papers, this fuel is characteristically of two types: double-base or composite The former is essentially an equal-part mixture of solvent, nitroglycerin, and nitrocellulose cast or extruded into the desired shape, whereas the latter is composed of three-quarters relatively hard, but small, oxidizer particles embedded in the remaining one-quarter rubber fuel matrix In both cases, the elastomeric content is sufficiently large to cause the mixture to possess significant time-dependent properties At the outset it should be recognized that the medium is not the assumed isotropic-homogeneous continuum on the microscale Nevertheless, for analysis purposes, it will be treated as such in much the same way as concrete aggregate, asphalt, or, for that matter, metals In the gross sense, such an assumption is permissible with important exceptions, such as stress-induced anisotropy and fracture origins, treated separately The difference between viscoelastic media and more common elastic ones lies essentially in the relation between stress and strain Whereas normal elastic analyses are based upon a (spring) constant proportionality between the two, with Young's modulus as the proportionality constant, the more general viscoelastic relation must allow for time or strainrate effects Conceptually, it turns out that the easiest way to incorporate rate effects into the mathematical model is to provide dashpot elements in addition to the elastic spring elements In this way, an arbitrary stress-strain time relation can be represented by increasing numbers of springdashpot elements having responses over a wide spectrum of spring or rate sensitivities The least amount of analytical complexity is introduced if the material is linearly viscoelastic Such behavior implies that, at any instant, the magnitude of a time-dependent response is directly proportional to the magnitude of the applied time-dependent input Fortunately, this assumption has proved sufficiently precise for most solid propellant materials Thus, one is permitted to assume that the spring and dashpot characteristics depend at most on temperature, and that no nonlinear coupling occurs in the stress-strain law Although the use of discrete spring and dashpot element representation is conceptually convenient, quantitative accuracy over large time ranges usually requires an inordinately large number of such elements that algebraically complicate the analysis As an alternative, therefore, one can develop a method for approximating the broad-band time response, wherein detailed accuracy at any one time is sacrificed in order to obtain reasonable representation over the complete time scale between the short-time (glassy) behavior and long-time (rubbery) behavior This approach leads to what is ordinarily