Parameterization of the stress–strain relation for modeling wave propagation in nearly incompressible transversely isotropic materials

The stress–strain relation in a transversely isotropic (TI) material is described by five independent parameters. In the incompressible limit, only three parameters are required to describe shear wave propagation. Existing material parameterization models are not ideal for the analysis of wave propagation in the nearly incompressible TI (NITI) regime due to difficult-to-interpret parameters, complicated forms of the stiffness matrix elements, or the lack of five independent parameters. This study describes a new parameterization model for a general, TI material that uses the bulk modulus K, shear moduli μT and μL, a modulus-like term μE, and a new parameter η. In the proposed parameterization model, each parameter has a clear interpretation related to compressibility and shear wave propagation. The incompressible limit is represented by the limit K → ∞. Wave speeds and polarizations are derived and evaluated in both incompressible and NITI regimens. First-order NITI corrections are shown to be inversely proportional to the ratio of bulk modulus to shear moduli. In biological soft tissues, this ratio is approximately 106. NITI corrections depend on all five independent parameters; however, the small scale of these corrections validates previous studies that have assumed particular values for the parameter η.