Feynman diagram technique in averaging of motion equations for random nonhomogeneous elastic composite material

We consider nonlinear dynamics of random nonhomogeneous elastic medium. By random nonhomogeneous media we mean composite materials, granular materials, porous rocks with chaotic components distribution. To describe such medium we need to use Lagrangian coordinates instead of Eulerian coordinates. In this case Piola–Kirchhoff tensor should be used as strain tensor. It is asymmetrical tensor and defined as derivative of an energy with respect to dilatation, i.e. gradient of displacement vector. Here we use the most simple approach to Lagrangian representation which was developed in the Landau–Lifshitz model.
The Landau–Lifshitz approach is generalized here for nonlinear random nonhomogeneous elastic medium. So, equations motion of contain random coordinate dependent coefficients. In this article we considered effect of inherent stresses and finite deformations on medium oscillations near areas with high inherent stresses. Equations of wave propagation near stressed area are derived. As a result of medium random nonhomogeneity these equations describe not only wave propagation but also all multiple reflections from nonhomogeneities.
For averaging in this work we have used the Feynman diagram technique. This technique makes it possible to derive precise equation for average elastic field, which characterizes coherent propagation of waves subject to multiple reflections. This equation is integro-differential. It's kernel (correlation operator) contains contributions from random nonhomogeneities correlation functions of any order. This operator directly defines velocities of $P$- and $S$-waves in random nonhomogeneous elastic medium. These velocities depends on inherent stresses and our approach allows approximate calculation of this dependence. In inverse case one can use experimental velocities of sound in areas with stresses near to critical for material breaking. Using these velocities state of stressed medium can be defined and it's effective parameters. This article doesn't cover inverse case. We only derive basic equation here which make it possible to state the inverse problem.