Dynamics of Composites
These notes provide a brief account of how to analyze the propagation of ultrasonic waves in a composite; the methods and theory apply equally well to problems on the terrestrial scale, such as the propagation of seismic waves through regions containing arrays of cracks. The methods are less advanced, and the results less precise, than those available for the statics of composites. Furthermore, a wider class of phenomena needs to be addressed. Here, attention is restricted to methods that have their counterparts in statics, to which they reduce in the limit of very slow deformations.
The basic equations of elastodynamics are first introduced, together with some fundamental concepts: plane waves, Green’s functions, and variational principles. Then, waves in random media are introduced, using perturbation theory, valid for weak variations in material properties. This is not the case of most practical interest, but it is one for which “honest” information can be generated, so allowing some of the phenomena to be expected in all randomly inhomogeneous media to be recognised. Later sections treat dilute suspensions, and then the general case, for which the best that can be hoped for is a strategy for developing systematic approximations. Approximations based on variational principles are advocated, both for their intrinsic merit and because they reproduce, in the static limit, results of proven utility, developed elsewhere in this course.
Final sections discuss waves in composites which display nonlinear behaviour. Research in this area has hardly been started, so only some preliminary indications can be given. The main difficulty is that the nonlinearity of the composite, which is not known a priori, exerts a crucial influence on the wave that is excited by the applied loading: this cannot be assumed to be time-harmonic, for example, so the whole problem has to be approached by a “bootstrap” procedure, in which mean constitutive relations and the mean wave that is propagated through the medium have somehow to be determined concurrently. One example, worked out very recently, is discussed for illustration. This concerns a linearly elastic medium, containing cracks. The response of the cracks is nonlinear, because the cracks are open during tension but closed during compression. The entire medium is thereby rendered nonlinear, and all of the challenges alluded to above have to be confronted.
KeywordsVariational Principle Elastic Medium Ensemble Average Random Medium Momentum Density
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