Dispersion Characteristics Of Elastic Waves In Biological Soft Tissues With The Growth-mechanical Coupling

The growth-mechanical effect,a unique property of biological tissues,is widespread in soft tissues such as skin,blood vessels,mucous membranes,etc.Such effect often leads to changes in the morphology,mechanical property and mechanical behavior of biological soft tissues.It is therefore of great significance to investigate the mechanical responses of biological tissues with the growth-mechanical coupling effect in medical diagnoses and treatments.This work examined the propagation of elastic waves in finitely pre-deformed biological tissues with the growth-mechanical coupling effects under the steady growth conditions,focusing on the frequency dispersion characteristics of elastic waves propagation and the influence of related parameters on wave dispersion.Also it aims to establish the method to distinguish the growth pre-deformation and the elastic pre-deformation.Research outcomes are expected to provide the theoretical basis for the applications of elastic waves in the medical fields.The research work covers:(1)The basic acoustoelastic equations were established for biological soft tissues considering the growth-mechanical coupling effect within the framework of continuum mechanics,including the kinematics equations of deformation,the constitutive equations,the equilibrium equations and the boundary conditions.In particular,the incremental constitutive equations which involve the growth evolution and the constitutive parameters related to the growth evolution were deduced.In this work,the growth effect is introduced through deformation gradient decomposition,and the growth variation is featured by the growth evolution equations which may be different for various biological tissues.Analysis on the basic equations showed that the impact of the growth-mechanical coupling on the mechanical responses of biological soft tissues is closely associated with the growth pre-deformation and the growth evolution.When the growth evolution is ignored,such as in medical elastography,a biological soft tissue exhibits different material properties and mechanical responses from an ordinary soft material similarly pre-deformed due to the existence of the initial growth deformation.(2)The propagation of the anti-plane waves(SH waves)in the structures which consist of two layers of finitely deformed biological soft tissues was investigated under the steady growth conditions.The frequency dispersion equation was firstly deduced from which the corresponding frequency dispersion curves were obtained using the numerical methods.The effects of the tensile and shear pre-deformations and the thickness of biological tissue layers on SH waves dispersion of the zero and the first modes were then discussed.The results showed that SH waves of the zero mode are non-dispersive while SH waves of the first mode display obvious dispersion features,especially within the low frequency range.The SH wave phase velocities of the above two modes increase with the increase of the stretch ratio,and the phase velocities first decrease and then increase with the increase of the tensile index and the shear pre-deformationsk₁,when the shear pre-deformationsk₂ increases,the phase velocities change from slight increase to rapid decrease and to continuous increase.Moreover,increasing the ratio of the layer thicknesses does not change the phase velocity of the zero mode while gives rise to higher phase velocity of the first mode.In addition,the analytic expressions of SH wave phase velocities of the first two modes were derived from the frequency dispersion equation when the wavenumber tends to 0 and infinity.It is found that the variation of such wave velocity limits with the aforementioned factors had strong correlation with the changes of frequency dispersion.(3)The propagation of plane waves(Lamb waves)in the structures which consist of two layers of biological soft tissues with either finite stretches or finite simple shear deformation was investigated under the steady growth conditions.Firstly,the frequency dispersion equations were deduced and numerically solved.For the cases under the finite tensile pre-deformation,Lamb wave phase velocities of the first and the second modes when the wavenumber approaches to 0,as well as the phase velocity of the first mode when the wavenumber tends to infinity,were analytically deduced.Then,for the structures where two layers are dissimilar or are identical,the effects of the tensile or shear pre-deformations,the initial external load and the layer thickness(only for structures with two different layers)on Lamb waves dispersion of the first and the second modes were discussed respectively.Our results showed that the existence of the above three wave velocity limits depends on the values of the pre-deformations,the initial external load and the thickness ratio.Combining the influence of the presence of wave velocity limits,the property dissimilarity of two layers and the pre-deformation type,some special phenomena occured in Lamb waves dispersion of the first and the second modes,such as frequency cut-off,frequency blocking,bandgap,band-pass and mode vanishment.Furthermore,with the increase of the above three types of parameters,the variation of Lamb wave phase velocity is generally not monotonous and dependent on the frequency/wavenumber of Lamb waves.Finally,a method was proposed to distinguish the elastic pre-deformation from the growth pre-defornation in terms of the wave velocity limits deduced in this work.