Tractable calculation of the Green’s tensor for shear wave propagation in an incompressible, transversely isotropic material

Abstract

Assessing material properties from observations of shear wave propagation following an acoustic radiation force impulse (ARFI) excitation is difficult in anisotropic materials because of the complex relations among the propagation direction, shear wave polarizations, and material symmetries. In this paper, we describe a method to calculate shear wave signals using Green's tensor methods in an incompressible, transversely isotropic (TI) material characterized by three material parameters. The Green's tensor is written as the sum of an analytic expression for the SH propagation mode, and an integral expression for the SV propagation mode that can be evaluated by interpolation within precomputed integral functions with an efficiency comparable to the evaluation of a closed-form expression. By using parametrized integral functions, the number of requried numerical integrations is reduced by a factor of 10 2 − 10 9 depending on the specific problem under consideration. Results are presented for the case of a point source positioned at the origin and a tall Gaussian source similar to an ARFI excitation. For an experimental configuration with a tilted material symmetry axis, results show that shear wave signals exhibit structures that are sufficiently complex to allow measurement of all three material parameters that characterize an incompressible, TI material.

Wave Speeds and Green’s Tensors for Shear Wave Propagation in Incompressible, Hyperelastic Materials with Uniaxial Stretch

“…E. Calculation of the Green's tensor Neglecting the P propagation mode, components G in (⃗ r, ω) of the Green's tensor in the temporal frequency domain can be written a sum of the SH and SV propagation modes as [5]…”

“…where P SH and P SV are polarization vectors, and the phase velocities v SH and v SV are given by (7) and (9), respectively. For the SH propagation mode, it can be shown [5], [9] that G SH in (⃗ r, ω) is given by the closed-form expression…”

“…We have not found a closed-form expression for G SV (⃗ r, ω). Instead, it can be shown [5] that G SV in (⃗ r, ω) are given by…”

“…More complicated geometries using a tilted symmetry axis as shown in Fig. 1(b) observe both the SH and SV propagation modes and are sensitive to all three material parameters that characterize an incompressible, TI material [4], [5].…”

“…However, this special case is not sufficiently general to fully describe the shear wave signals that can be measured with the tilted symmetry axis in Fig. 1(b) [5] .…”