  A body force concentrated at a point and moving at a high speed can induce
shear-wave Mach cones in dusty-plasma crystals or soft materials, as observed
experimentally and named the elastic Cherenkov effect (ECE). The ECE in soft
materials forms the basis of the supersonic shear imaging (SSI) technique, an
ultrasound-based dynamic elastography method applied in clinics in recent
years. Previous studies on the ECE in soft materials have focused on isotropic
material models. In this paper, we investigate the existence and key features
of the ECE in anisotropic soft media, by using both theoretical analysis and
finite element (FE) simulations, and we apply the results to the non=invasive
and non-destructive characterization of biological soft tissues. We also
theoretically study the characteristics of the shear waves induced in a
deformed hyperelastic anisotropic soft material by a source moving with high
speed, considering that contact between the ultrasound probe and the soft
tissue may lead to finite deformation. On the basis of our theoretical analysis
and numerical simulations, we propose an inverse approach to infer both the
anisotropic and hyperelastic parameters of incompressible transversely
isotropic (TI) soft materials. Finally, we investigate the properties of the
solutions to the inverse problem by deriving the condition numbers in
analytical form and performing numerical experiments. In Part II of the paper,
both ex vivo and in vivo experiments are conducted to demonstrate the
applicability of the inverse method in practical use.
