Philippe Boulanger

Poisson's Ratio for Orthorhombic Materials

An example of an orthorhombic material is constructed such that even though the strain-energy density is positive definite, Poissons ratio may take an arbitrarily large positive value for one pair of orthogonal directions and take an arbitrarily small negative value for another pair of orthogonal directions.

Unsheared triads and extended polar decompositions of the deformation gradient

Abstract In this paper, the concept of unsheared triads of material line elements at a point X is introduced. We find that there is an infinity of unsheared triads. More precisely, it is shown that, in general, for any given unsheared pair at X , a unique third material line element at X may be found such that the three material line elements form an unsheared triad. Special cases are analyzed in detail. A link between unsheared triads and new decompositions of the deformation gradient, is exhibited. These decompositions generalize the classical polar decomposition F = RU = VR of the deformation gradient F , in which R is a proper orthogonal tensor and U , V are positive-definite symmetric. Associated with any unsheared (oblique) triad is a new decomposition F = QG = HQ , in which Q is a proper orthogonal tensor, but G and H are no longer symmetric, but have three positive eigenvalues and three linearly independent right eigenvectors. Because there is an infinity of unsheared triads, there is an infinity of such decompositions. We call them “extended polar decompositions”. Several examples of unsheared triads and extended polar decompositions are presented.

On young's modulus for anisotropic media

If a piece of homogeneous anisotropic elastic material is subject to simple tension along a direction n for which Young’s modulus E(n ) is an extremum, then the corresponding strain field is coaxial with the simple tension stress field. An appropriate set of rectangular cartesian coordinate axes may be introduced such that three of the elastic compliances are zero. In this coordinate system the displacement field may be written explicitly and corresponds to a pure homogeneous deformation.

Special inhomogenous plane waves in cubic elastic materials

Abstract. The purpose of this paper is to present new special explicit inhomogeneous plane wave solutions of the linearized equations of motion for elastic cubic crystals. It is based upon the “directional-ellipse” method which leads to an eigenvalue problem for the complex symmetric acoustical tensor. The solutions are obtained by considering a special case for which the determination of the three complex eigenvalues of this tensor reduces to finding the three complex cubic roots of a real positive number. Explicit simple expressions are presented for the slowness and amplitude bivectors.

Inhomogeneous plane waves in viscous fluids

The propagation of elliptically polarised inhomogeneous plane waves in a linearly viscous fluid is considered. The angular frequency and the slowness vector are both assumed to be complex. Use is made throughout of Gibbs bivectors (complex vectors). It is seen that there are two types of solutions—the zero pressure solution, for which the increment in pressure due to the propagation of the wave is zero, and a universal solution which is independent of the viscosity.Since the waves are attenuated in time, the usual mean energy flux vector is not a suitable way of measuring energy flux. A new energy flux vector, appropriate to these waves is defined, and results relating it with energy dissipation and energy density are obtained. These results are related to a result derived directly from the balance of energy equation.

Finite-Amplitude Waves in Mooney-Rivlin and Hadamard Materials

These lectures deal with the the propagation of finite amplitude plane waves in Mooney-Rivlin and Hadamard elastic materials which are maintained in a state of arbitrary static homogeneous deformation. Exact plane wave solutions are presented for arbitrary propagation direction. The energy properties of these waves are investigated.

Energy flux for damped inhomogeneous plane waves in viscoelastic fluids

Abstract The propagation of inhomogeneous plane waves in the context of the linearized theory of incompressible viscoelastic fluids is considered. The angular frequency and the slowness vector are both assumed to be complex. As in incompressible purely viscous fluids, two kinds of waves may propagate: A “zero pressure wave” for which the increment in pressure due to the wave is zero, and a “universal wave” which is independent of the viscoelastic relaxation modulus. The balance of energy is written using a decomposition of the stress power into a reversible component and a dissipative component proposed namely by P.W. Buchen [J. R. Astr. Soc. 23 (1971) 531–542]. For the inhomogeneous waves, a “weighted mean” energy flux vector, “weighted mean” energy density and “weighted mean” energy dissipation are introduced. It is shown that they satisfy two modulus independent relations. These generalize to the case of viscoelasticity relations previously obtained in other contexts.

Shear, shear stress and shearing

Abstract New simpler formulae are derived for the shear of a pair of material elements within the context of infinitesimal strain and finite strain. Also, new formulae are derived for shear stress based on the (symmetric) Cauchy stress and for the rate of shear of a pair of material elements within the rate of strain theory. These formulae are exploited to obtain results and to derive new simpler proofs of familiar classical results. In particular, a very simple short derivation is presented of the classical result of Coulomb and Hopkins on the maximum orthogonal shear stress.

Wave propagation in sheared rubber

SummaryThe speeds of propagation and polarization amplitudes are presented for finite amplitude plane shear waves propagating in rubber which is maintained in a state of static finite simple shear. The Mooney-Rivlin form of the stored-energy function is used to model the mechanical behaviour of the material. General relations are obtained between the speed of propagation of the fastest and slowest waves and the speed of propagation of the finite amplitude circularly polarized waves which may propagate along the acoustic axes. The slowness and ray surfaces are also presented.

Toy models: The jumping pendulum

We consider a simple pendulum consisting of a mass attached to an inextensible string of negligible mass. For small or large initial velocities, the motion of the pendulum is along a circle. When given sufficient but not too large an initial velocity, the mass will reach a certain height and leave the circle. After such a jump, it will follow a parabolic path until the string is again fully extended and the motion is again constrained by the string. We assume that the radial component (along the string) of the velocity of the mass instantaneously vanishes when the string becomes taut and that the mass loses some of its energy in the shock and resumes its circular motion. What is the dynamics of such a pendulum? Can it jump more than once? How many times can it jump?