Abstract：The fractional acoustic wave models that describe the ultrasound propagation in biological tissues can be constructed from the perspectives of time-fractional derivative or space-fractional integral viscoelastic constitutive relations. These two modeling perspectives treat the biological tissues as non-local viscoelastic material, respectively. The first perspective has been proposed in literature whereas the second one is rarely reported. The paper mainly concerns the second modeling perspective.
A new stress-strain constitutive relation in the form of Riesz potential is derived after replacing the exponential kernel function in the conventional Eringen nonlocal model by the power kernel function. Using the new relation, the conventional viscoelastic elements, i.e., spring and dashpot, are extended to their nonlocal counterparts. Through the series and parallel connections of such elements, the Kelvin and the Maxwell non-local constitutive models in arbitrary spatial dimensions are obtained, and the creep compliance and the relaxation modulus of one-dimensional models are also given. The Riesz potential order and the material internal characteristic length are two primary parameters of the nonlocal models, and the retardation time of both the Kelvin and the Maxwell non-local constitutive models gets larger as the internal characteristic length increases. Data fitting of the experiment data of a kind of soft soil implies that the nonlocal models could give better descriptions of the viscoelastic behaviors of complex media, say, creep of soft soil, ultrasound in biological tissues, and vibration of nano-composites.
The non-local constitutive models of different series and parallel connections of nonlocal spring and dashpot, complemented with the momentum conservation and the displacement-strain relations, lead to a family of space-fractional wave equations, which describe the power-law frequency-dependent wave dissipation and dispersion in biological tissues. This is a potential application of the non-local constitutive models.
2017, Vol. 38 
