On the mathematical and geometrical structure of the determining equations for shear waves in nonlinear isotropic incompressible elastodynamics

Abstract

Using the theory of 1+1 hyperbolic systems we put in perspective the mathematical and geometrical structure of the celebrated circularly polarized waves solutions for isotropic hyperelastic materials determined by Carroll in Acta Mechanica 3 (1967) 167-181. We show that a natural generalization of this class of solutions yields an infinite family of linear solutions for the equations of isotropic elastodynamics. Moreover, we determine a huge class of hyperbolic partial differential equations having the same property of the shear wave system. Restricting the attention to the usual first order asymptotic approximation of the equations determining transverse waves we provide the complete integration of this system using generalized symmetries.

Systems of conservation laws with third-order Hamiltonian structures

“…Remark. In the particular case ξ = 0, equations (29), (30), (33) were obtained in [29]. If ξ = 0, the corresponding Hamiltonian density h has explicit x-dependence.…”

“…Systems of conservation laws appear in a wide range of applications in continuum mechanics and mathematical physics, see e.g. [21,25,35,32,34,33]. Following the geometric approach of [1,2], with system (1) we associate a congruence (that is, n-parameter family of lines),…”

Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids

“…Such nonphysical behavior is a result of exceeding the physical applicability limits of the incompressible non-dissipative model. A similar phenomenon of shock formation has been observed in [39], where a generalization of Carroll's circularly polarized wave solutions [8] is developed.…”

Ut vis sic tensio

“…where is the constant density and is the generalized shear modulus. For all the generalized Mooney-Rivlin (3.3) we have that ≡ the infinitesimal shear modulus [18]. Therefore for such a class of materials the equations (3.4) are uncoupled and linear: this fact seems to be not confirmed by experiments.…”