Elastic field inversion

The problem of elastic field inversion is much more complicated than acoustic or vector wavefield inversion, considered in the previous sections of the book. However, the fundamental principles of elastic inversion resemble those discussed above for more simple models of seismic waves. I will present in this section a brief overview of the basic ideas underlining the elastic field inversion. 

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Thus, the corrections to the velocities Cpb (r) and Csb (r) on the first iteration can be obtained by correlating the back-propagated scattered elastic field with the derivatives of the incident field U (r,t). This transformation is similar to wavefield migration described previously. Thus we see that elastic field inversion can be constructed by iterative migration of the residual elastic data. 

I. Viscoelastic Solutions in Terms of Elastic Solutions. The fundamental result is the Classical Correspondence Principle. It is based on the observation that the time Fourier transform (FT) of the governing equations of Linear Viscoelasticity may be obtained by replacing elastic constants by corresponding complex moduli in the FT of the elastic field equations. It follows that, whenever those regions over which different types of boundary conditions are specified do not vary with time, viscoelastic solutions may be generated in terms of elastic solutions that satisfy the same boundary conditions. In practical terms this method is largely restricted to the non-inertial case, since then a wide variety of elastic solutions are available and transform inversion is possible. 

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

This scattering mode occurs only in piezoelectric materials, i.e., in crystals without inversion symmetry, and is caused by the electric field associated with acoustical phonons. Zinc oxide exhibits very high electro-mechanical coupling coefficients P (see later), exceeding that of quartz [42,59], which is one of the mostly used piezoelectric materials. Zook [60] has calculated the piezoelectrically limited mobility on the basis of the elastic and piezoelectric constants as (see also Rode [54]) ... 

As in the previous chapter our discussion will be restricted mostly to the fields satisfying wave equations. However, in the last section I will outline the basic principles of nonlinear elastic inversion. 

We see that the sub-matrix is not the Inverse of the sub-matrix hence C the elastic moduli at constant electric field and constant temperature are different from the moduli at constant electric polarization and constant entropy. The specific heat (at constant strain and polarization) is commonly called (specific heat at constant volume), whereas (specific heat at... 

The domain flow theory of LCP assumes a balance between the alignment tendency under a velocity field and the elastic resistance to deformation of the director field [Marrucci, 1984]. The average value of the Eriksen distortion stress, as, was taken as proportional to the elastic constant, K, and inversely proportional to the domain size. The flow behavior should depend on the local orientation for high velocity in the region where the orientation director and velocity vector are parallel to each other, with low velocity for the opposite direction. As a result, the relation between the stress and the deformation rate might be scaled by the domain size ... 

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