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On the null condition for nonlinearly elastic solids

Domański W., Ogden R. W.

Archives of Mechanics

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2006

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Vol. 58, nr 4-5

339-361

EN

Smooth solutions to the Cauchy problem for the equations of nonlinear elastodynamics exist typically only locally in time. However, under the assumption of small initial data and an additional restriction, the so-called null condition, global existence and uniqueness of a classical solution can be proved. In this paper, we examine this condition for the elastodynamic equations and study its connection with the property of genuine nonlinearity as well as its relation with the phenomenon of self-resonance of nonlinear elastic waves. Using a special structure of plane waves elastodynamics [13], we provide an alternative and simple formulation of the null condition. This condition is then evaluated for some examples of elastic constitutive laws in order to determine the nature of the restrictions that it imposes.

2

Formulas for the Rayleigh wave speed in orthotropic elastic solids

Vinh P. Chi, Ogden R. W.

Archives of Mechanics

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2004

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Vol. 56, nr 3

247-265

EN

Formulas for the speed of Rayleigh waves in orthotropic compressible elastic materials are obtained in explicit form by using the theory of cubic equations. Different formulas are obtained by using different forms of the (cubic) secular equation. Each formula is expressed as a continuous function of three dimensionless material parameters, which are the ratios of certain elastic constants. It is interesting to note that one of the formulas includes as a special case the formula obtained recently by Malischewsky for isotropic materials.

3

Material instabilities in fiber-reinforced nonlinearly elastic solids under plane deformation

Merodio J., Ogden R. W.

Archives of Mechanics

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2002

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Vol. 54, nr 5-6

525-552

EN

Material instabilities in fiber-reinforced nonlinearly elastic solids are examined under plane deformation. In particular, the materials under consideration are isotropic nonlinearly elastic models augmented by a function that accounts for the existence of a unidirectional reinforcing. This function describes the anisotropic (transversely isotropic) character of the material and is referred to as a reinforcing model. The onset of failure is signalled by the loss of ellipticity of the governing differential equations. Previous work has dealt with the analysis of specific reinforcing models and has established that the loss of ellipticity for such augmented isotropic materials requires contraction in the reinforcing direction. The loss of ellipticity was related to fiber kinking. Here we generalize these results and establish sufficient conditions for the ellipticity of the governing equations of equilibrium for more general reinforcing models to be guaranteed. We also establish necessary conditions for failure of ellipticity. The incipient loss of ellipticity is interpreted in terms of fiber kinking, fiber de-bonding, fiber splitting and matrix failure in fiber-reinforced composite materials. Attention is restricted to incompressible materials in this paper.