In this paper special Blatz-Ko nonlinear elastic materials are considered, which are characterized by a constitutive constant and a constitutive function. We deal with the propagation of finite-amplitude inhomogeneous plane waves in such materials subjected to an arbitrary static homogeneous deformation. Linearly polarized transverse “damped” inhomogeneous plane wave solutions are explicitly obtained. Such waves are attenuated (or amplified) both in space and time (time-harmonic inhomogeneous plane waves obtained previously by Destrade appear as a special case). The properties of the energy flux vector and energy density associated with these wave solutions are investigated. With an appropriate concept of mean, it is seen that the “mean” energy-flux vector and the “mean” energy density satisfy two relations which are independent of the constitutive constant and constitutive function of the model, and of the homogeneous static deformation of the material. These relations are the same as those obtained by Hayes in the general context of time-harmonic inhomogeneous plane waves in linear systems. However, here, the theory is nonlinear, and the finite-amplitude waves are not time-harmonic.
The uniqueness of several 2D inverse problems for incompressible nonlinear hyperelasticity is studied. These problems are motivated by elastography, in which one is given a measured deformation field in a 2D domain and seeks to reconstruct the pointwise distribution of material parameters within. Two classes of models are considered. The simpler class is material models characterized by a single material parameter exemplified by the Neo-Hookean model. The second class of material models considered is characterized by two material parameters, and includes a simplified Veronda-Westmann model, a Blatz model and a modified Blatz model. Consistent with the results in linear elasticity, we find that significantly fewer data are required to determine the material properties under plane stress conditions than under plane strain conditions. The results show that, roughly speaking, one needs one measured deformation for each material parameter sought under plane stress conditions, and twice as much data for plane strain conditions.
In this paper, special Blatz—Ko nonlinear elastic materials are considered, which are characterized by a constitutive constant and a constitutive function. We here deal with the propagation of finite-amplitude waves in such materials subjected to an arbitrary static homogeneous deformation. In a previous paper, it was shown that linearly polarized transverse damped inhomogeneous plane waves may propagate. The orthogonal propagation and polarization directions are arbitrary. The special Blatz—Ko materials are compressible so that homogeneous longitudinal waves may also propagate. Here it is shown that the superposition of a transverse damped inhomogeneous wave and of a longitudinal wave is also a solution, in the case when the propagation direction of the longitudinal wave is orthogonal to the polarization direction of the transverse wave. Also, results are obtained for the energy density and the energy flux of the superposition of these waves.
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