Convariant q-Differential Calculus and its Deformations at qN = 1

We construct the generalized version of covariant Z₃-graded differential calculus introduced by one of us (R.K.) and then extend it to the case of arbitrary Z_N grading. Here our main purpose is to establish the recurrence formulae for the Nth power of covariant q-differential D_q=d_q + A and to analyze more closely the particular case of q being an Nth primitive root of unity. The generalized notions of connection and curvature are introduced and several examples of realization are displayed for N=3 and 4. Finally we briefly discuss the idea of infinitesimal deformations of the parameter q in the complex plane.     

Peculiar features in the rheological behavior of electrorheological suspensions

Studies have been made of concentrated (up to 60%) diatomite suspensions in transformer oil, the structure and theological properties of which depend on an applied electric field. Studies have been conducted of steady-state and transient regimes of straining involving continuous and periodic shear. The structure in such suspensions is formed in the presence of an electric field of 10^–3 –10² duration. The suspensions under continuous stationary strain behave as non-Newtonian fluids with a yield stress dependent on electric intensity. Under periodic deformation conditions the test suspensions exhibit elasticity which abruptly diminishes with increasing deformation amplitude.     

The Effects of Compressibility on Inhomogeneous Deformations for a Class of Almost Incompressible Isotropic Nonlinearly Elastic Materials

Experimental data for simple tension suggest that there is a power–law kinematic relationship between the stretches for large classes of slightly compressible (or almost incompressible) non-linearly elastic materials that are homogeneous and isotropic. Here we confine attention to a particular constitutive model for such materials that is of generalized Varga type. The corresponding incompressible model has been shown to be particularly tractable analytically. We examine the response of the slightly compressible material to some nonhomogeneous deformations and compare the results with those for the corresponding incompressible model. Thus the effects of slight compressibility for some basic nonhomogeneous deformations are explicitly assessed. The results are fundamental to the analytical modeling of almost incompressible hyperelastic materials and are of importance in the context of finite element methods where slight compressibility is usually introduced to avoid element locking due to the incompressibility constraint. It is also shown that even for slightly compressible materials, the volume change can be significant in certain situations.      

Differential realizations of polynomial algebras in finite-dimensional spaces of monomials

We give a complete classification of at most second-order differential ladder operators preserving finite-dimensional spaces of monomials and closing under the Lie bracket to give a cubic polynomial of the diagonal generators.     

Evidence for multiple band terminations in Pd

The level structure of ¹⁰²Pd has been investigated using data collected with the Eurogam 2 array. Several cascades of γ-rays have been established up to high spins. Termination of rotational bands has been observed at I^π = 28⁻ and 32⁺, and tentatively at I^π = 38⁺ and 42⁺. The nucleus ¹⁰²Pd is the first case where rotational bands built on valence space configurations are followed from spin close to zero up to termination and, at higher spins, a smooth rotational band which appears to terminate is built on core excited configurations.     

Universal Deformations for a Class of Compressible Isotropic Hyperelastic Materials

Differential conditions are derived for a smooth deformation to be universal for a class of isotropic hyperelastic materials that we regard as a compressible variant (a notion we make precise) of Mooney–Rivlin’s class, and that includes the materials studied originally by Tolotti in 1943 and later, independently, by Blatz. The collection of all universal deformations for an incompressible material class is shown to contain, modulo a uniform dilation, all the universal deformations for its compressible variants. As an application of this result, by searching the known families of universal deformations for all incompressible isotropic materials, a nontrivial universal deformation for Tolotti materials is found. This revised version was published online in August 2006 with corrections to the Cover Date.     

Maximal Classes of Stored Energies Compatible with Cylindrical Inflations

We consider a family of deformations describing cylindrical inflations within the context of finite, compressible, isotropic elasticity. We pose the problem of finding the maximal class of materials for which these deformations are possible at equilibrium under surface tractions only. We solve this problem for families of cylindrical inflations whose principal strain invariants have a special dependence on the radius. These families comprise and extend all cases considered by Murphy [2]. This revised version was published online in August 2006 with corrections to the Cover Date.     

Azimuthal Shear in Compressible Finite Elasticity

In an earlier paper, the broadest classes of compressible isotropic strain energies that support irrotational universal deformations were identified and the problems of cylindrical and spherical inflation or compaction were solved in closed form for all of these strain energies. Similar closed form solutions of the problem of azimuthal shear are presented here.      

Invariant star-products on symplectic manifolds

Let (M,F) be a symplectic manifold and consider a Lie subalgebra $\text{G}$ of its Lie algebra of symplectic vector fields. We prove that every one-differentiable deformation of order k of the Poisson Lie algebra of M, which is invariant with respect to $\text{G}$, extends to an invariant one-differentiable deformation of infinite order. If M admits a $\text{G}$-invariant linear connection, a similar result holds true for differentiable deformations and for star-products. In particular, if M admits a $\text{G}$- -invariant linear connection, there always exists a $\text{G}$-invariant star-product.     

Time derivative obtained by applying the Riemannian manifold of Riemannian metrics to kinematics of continuaDérivée temporelle obtenue en utilisant la variété riemannienne des métriques riemanniennes aux cinématiques des milieux continus

The infinite dimensional Riemannian geometry of Riemannian metrics is employed to propose novel objective time derivative by means of covariant derivative. To cite this article: Z. Fiala, C. R. Mecanique 332 (2004).