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The surface-impedance matrix method is used to study interfacial waves polarized in a plane of symmetry of anisotropic elastic materials. Although the corresponding Stroh polynomial is a quartic, it turns out to be analytically solvable in quite a si mple manner. A specific application of the result concerns the calculation of the speed of a Stoneley wave, polarized in the common symmetry plane of two rigidly bonded anisotropic solids. The corresponding algorithm is robust, easy to implement, and gives directly the speed (when the wave exists) for any orientation of the interface plane, normal to the common symmetry plane. Through the examples of the couples (Aluminum)-(Tungsten) and (Carbon/epoxy)-(Douglas pine), some general features of a Stoneley wave speed are verified: the wave does not always exist; it is faster than the slowest Rayleigh wave associated with the separated half-spaces.
Mechanical characterization of brain tissue has been investigated extensively by various research groups over the past fifty years. These properties are particularly important for modelling Traumatic Brain Injury (TBI). In this research, we present t he design and calibration of a High Rate Tension Device (HRTD) capable of performing tests up to a maximum strain rate of 90/s. We use experimental and numerical methods to investigate the effects of inhomogeneous deformation of porcine brain tissue during tension at different specimen thicknesses (4.0-14.0 mm), by performing tension tests at a strain rate of 30/s. One-term Ogden material parameters (mu = 4395.0 Pa, alpha = -2.8) were derived by performing an inverse finite element analysis to model all experimental data. A similar procedure was adopted to determine Youngs modulus (E= 11200 Pa) of the linear elastic regime. Based on this analysis, brain specimens of aspect ratio (diameter/thickness) S < 1.0 are required to minimise the effects of inhomogeneous deformation during tension tests.
Finite Element simulations of rubbers and biological soft tissue usually assume that the material being deformed is slightly compressible. It is shown here that in shearing deformations the corresponding normal stress distribution can exhibit extreme sensitivity to changes in Poissons ratio. These changes can even lead to a reversal of the usual Poynting effect. Therefore the usual practice of arbitrarily choosing a value of Poissons ratio when numerically modelling rubbers and soft tissue will, almost certainly, lead to a significant difference between the simulated and actual normal stresses in a sheared block because of the difference between the assumed and actual value of Poissons ratio. The worrying conclusion is that simulations based on arbitrarily specifying Poissons ratio close to 1/2 cannot accurately predict the normal stress distribution even for the simplest of shearing deformations. It is shown analytically that this sensitivity is due to the small volume changes which inevitably accompany all deformations of rubber-like materials. To minimise these effects, great care should be exercised to accurately determine Poissons ratio before simulations begin.
Acousto-elasticity is concerned with the propagation of small-amplitude waves in deformed solids. Results previously established for the incremental elastodynamics of exact non-linear elasticity are useful for the determination of third- and fourth-o rder elastic constants, especially in the case of incompressible isotropic soft solids, where the expressions are particularly simple. Specifically, it is simply a matter of expanding the expression for $rho v^2$, where $rho$ is the mass density and v the wave speed, in terms of the elongation $e$ of a block subject to a uniaxial tension. The analysis shows that in the resulting expression: $rho v^2 = a + be + ce^2$, say, $a$ depends linearly on $mu$; $b$ on $mu$ and $A$; and $c$ on $mu$, $A$, and $D$, the respective second-, third, and fourth-order constants of incompressible elasticity, for bulk shear waves and for surface waves.
The classical flexure problem of non-linear incompressible elasticity is revisited assuming that the bending angle suffered by the block is specified instead of the usual applied moment. The general moment-bending angle relationship is then obtained and is shown to be dependent on only one non-dimensional parameter: the product of the aspect ratio of the block and the bending angle. A Maclaurin series expansion in this parameter is then found. The first-order term is proportional to $mu$, the shear modulus of linear elasticity; the second-order term is identically zero, because the moment is an odd function of the angle; and the third-order term is proportional to $mu(4beta -1)$, where $beta$ is the non-linear shear coefficient, involving third-order and fourth-order elasticity constants. It follows that bending experiments provide an alternative way of estimating this coefficient, and the results of one such experiment are presented. In passing, the coefficients of Rivlins expansion in exact non-linear elasticity are connected to those of Landau in weakly (fourth-order) non-linear elasticity.
In the theory of weakly non-linear elasticity, Hamilton et al. [J. Acoust. Soc. Am. textbf{116} (2004) 41] identified $W = mu I_2 + (A/3)I_3 + D I_2^2$ as the fourth-order expansion of the strain-energy density for incompressible isotropic solids. Su bsequently, much effort focused on theoretical and experimental developments linked to this expression in order to inform the modeling of gels and soft biological tissues. However, while many soft tissues can be treated as incompressible, they are not in general isotropic, and their anisotropy is associated with the presence of oriented collagen fiber bundles. Here the expansion of $W$ is carried up to fourth-order in the case where there exists one family of parallel fibers in the tissue. The results are then applied to acoustoelasticity, with a view to determining the second- and third-order nonlinear constants by employing small-amplitude transverse waves propagating in a deformed soft tissue.
A block of rubber eventually buckles under severe flexure, and several axial wrinkles appear on the inner curved face of the bent block. Experimental measurements reveal that the buckling occurs earlier ---at lower compressive strains--- than expecte d from theoretical predictions. This paper shows that if rubber is modeled as being bimodular, and specifically, as being stiffer in compression than in tension, then flexure bifurcation happens indeed at lower levels of compressive strain than predicted by previous investigations (these included taking into account finite size effects, compressibility effects, and strain-stiffening effects.) Here the effect of bimodularity is investigated within the theory of incremental buckling, and bifurcation equations, numerical methods, dispersion curves, and field variations are presented and discussed. It is also seen that Finite Element Analysis software seems to be unable to encompass in a realistic manner the phenomenon of bending instability for rubber blocks.
266 - Michel Destrade , Yibin Fu 2008
It is shown that in the Love-Kirchhoff plate theory, an edge wave can travel in a circular thin disk made of an isotropic elastic material. This disk edge wave turns out to be faster than the classic flexural acoustic wave in a straight-edged, semi-i nfinite, thin plate, a wave which it mimics when the curvature radius becomes very large compared to the wavelength.
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