Inhomogeneous ‘longitudinal’ plane waves in a deformed elastic material

By definition, a homogeneous isotropic compressible Hadamard material has the property that an infinitesimal longitudinal homogeneous plane wave may propagate in every direction when the material is maintained in a state of arbitrary finite static homogeneous deformation. Here, as regards the wave, homogeneous means that the direction of propagation of the wave is parallel to the direction of eventual attenuation; and longitudinal means that the wave is linearly polarized in a direction parallel to the direction of propagation. In other words, the displacement is of the form u = ncos k(n · x – ct), where n is a real vector. It is seen that the Hadamard material is the most general one for which a longitudinal inhomogeneous plane wave may also propagate in any direction of a predeformed body. Here, inhomogeneous means that the wave is attenuated, in a direction distinct from the direction of propagation; and longitudinal means that the wave is elliptically polarized in the plane containing these two directions, and that the ellipse of polarization is similar and similarly situated to the ellipse for which the real and imaginary parts of the complex wave vector are conjugate semi-diameters. In other words, the displacement is of the form u = {S exp i(S · x – ct)}, where S is a complex vector (or bivector). Then a Generalized Hadamard material is introduced. It is the most general homogeneous isotropic compressible material which allows the propagation of infinitesimal longitudinal inhomogeneous plane circularly polarized waves for all choices of the isotropic directional bivector. Finally, the most general forms of response functions are found for homogeneously deformed isotropic elastic materials in which longitudinal inhomogeneous plane waves may propagate with a circular polarization in each of the two planes of central circular section of the ⁿ -ellipsoid, where is the left Cauchy-Green strain tensor corresponding to the primary pure homogeneous deformation.     

In General There Are No Unsheared Tetrads

Suppose the principal stretches are all different at a point P in a deformed body. In this case, it has been shown [1] that generally there is an infinity of non coplanar infinitesimal material line elements at P which remain unsheared following the deformation – that is, the angle between the arms of each pair of material line elements forming the triad remains unchanged. Here it is shown that in this case when all three principal stretches at P are different, there is no set of four infinitesimal material line elements, no three of which are coplanar, and such that the angle between each pair of the six pairs of material line elements is unchanged following the deformation. It is only when all three principal stretches at P are equal to each other, that there are unsheared tetrads at P, and in that case all tetrads are unsheared. This revised version was published online in June 2006 with corrections to the Cover Date.     

On energy flux and group velocity

Inhomogeneous plane wave solutions to the wave equations for a linear isotropic elastic solid and a linear isotropic dielectric are shown to possess energy flux velocity vectors which are non-coincident with corresponding group velocity vectors.In contrast to free surface waves, these examples imply a driving constraint and have an associated non-zero Lagrangian energy density.     

Finite amplitude waves superimposed on pseudoplanar motions for Mooney-Rivlin viscoelastic solids

This paper deals exclusively with finite amplitude motions in viscoelastic materials for which the stress is the sum of a part corresponding to the classical Mooney-Rivlin incompressible isotropic elastic solid and of a dissipative part corresponding to the classical viscous incompressible fluid. Of particular interest is a finite pseudoplanar elliptical motion which is an exact solution of the equations of motion. Superposed on this motion is a finite shearing motion. An explicit exact solution is presented. It is seen that the basic pseudoplanar motion is stable with respect to the finite superposed shearing motion. Particular exact solutions are obtained for the classical neo-Hookean solid and also for the classical Navier-Stokes equations. Finally, it is noted that parallel results may be obtained for a basic pseudoplanar hyperbolic motion.     

On Shearing, Stretching and Spin

An analysis is presented of stretching, shearing and spin of material line elements in a continuous medium. It is shown how to determine all pairs of material line elements at a point x, at time t, which instantaneously are not subject to shearing. For a given pair not subject to shearing, a formula is presented for the determination of a third material line element such that all three form a triad not subject to shearing, instantaneously. It is seen that there is an infinity of such triads not subject to shearing. A new decomposition of the velocity gradient L is introduced. In place of the classical decomposition of Cauchy and Stokes, L=d+w, where d is the stretching tensor and w is the spin tensor, the new decomposition is L=?+ℳ, where ?, called the ldquo;modified” stretching tensor, is not symmetric, and ℳ, called the “modified” spin tensor, is skew-symmetric – the tensor ? being chosen so that it has three linearly independent real right (and left) eigenvectors. The physical interpretation of this decomposition is that the material line elements along the three linearly independent right eigenvectors of ? instantaneously form a triad not subject to shearing. They spin as a rigid body with angular velocity μ (say) associated with ℳ. Also, for each decomposition L=?+ℳ, there is a decomposition L=? ^T +, where is also skew-symmetric. The triad of material line elements along the right eigenvectors of ? ^T (the set reciprocal to the right eigenvectors of ?) is also instantaneously not subject to shearing and rotates with angular velocity (say) associated with . It is seen that the vorticity vector ω is the mean of the two angular velocities μ and , ω =(μ+)/2. For irrotational motion, ω =0, so that μ=-; any triad of material line elements suffering no shearing rotates with angular velocity equal and opposite to that of the reciprocal triad of material line elements. It is proved that provided d is not spherical, there is an infinity of choices for ? and ℳ in the decomposition L=?+ℳ. Two special types of decompositions are introduced. The first type is called “CCS-decomposition” (where CCS is an abbreviation for Central Circular Section). It is associated with the infinite family of triads (not subject to shearing) with a common edge along the normal to one plane of central circular section of an ellipsoid ? associated with the stretching tensor, and the two other edges arbitrary in the other plane of central circular section of ?. There are two such CCS-decompositions. The second type is called “triangular decomposition”, because, in a rectangular cartesian coordinate system, ? has three off-diagonal zero elements. There are six such decompositions. Received 14 November 2000 and accepted 2 August 2001     

Isoelectric focusing in a drop

A novel approach to molecular separations is investigated using a technique termed droplet-based isoelectric focusing. Drops are manipulated discretely on a superhydrophobic surface, subjected to low voltages for isoelectric focusing, and split-resulting in a preparative separation. A universal indicator dye demonstrates the generation of stable, reversible pH gradients (3-10) in ampholyte buffers, and these gradients lead to protein focusing within the drop length. Focusing was visually characterized, spectroscopically verified, and assessed quantitatively by noninvasive light scattering measurements. It was found to correlate with a quantitative model based on 1D steady-state theory. This work illustrates that molecular separations can be deployed within a single open drop, and the differential fractions can be separated into new discrete liquid elements.     

Revision of the absolute configurations of bethosides B and C and their aglycone

The absolute stereochemistry of the steroidal saponins bethosides B and C was previously assigned as (22R,25R) on the basis of work that employed Horeau's method. Our studies of helosides A and B created doubt about both the original assignment and consequently our conclusion that relied upon it. The absolute configurations of bethosides B and C are revised to (22S,25R) following X-ray crystallographic analysis of their aglycone. Synthesis and full spectral characterization of both the 22R and 22S aglycones is reported to facilitate future stereochemical assignments in this series of saponins.     

Amphiphilicity determines nanostructure in protic ionic liquids

The bulk structure of the two oldest ionic liquids (ILs), ethylammonium nitrate (EAN) and ethanolammonium nitrate (EtAN), is elucidated using neutron diffraction. The spectra were modelled using empirical potential structure refinement (EPSR). The results demonstrate that EAN exhibits a long-range structure of solvophobic origin, similar to a bicontinuous microemulsion or disordered L(3)-sponge phase, but with a domain size of only 1 nm. The alcohol (-OH) moiety in EtAN interferes with solvophobic association between cation alkyl chains resulting in small clusters of ions, rather than an extended network.     

An in situ STM/AFM and impedance spectroscopy study of the extremely pure 1-butyl-1-methylpyrrolidinium tris(pentafluoroethyl)trifluorophosphate/Au(111) interface: potential dependent solvation layers and the herringbone reconstruction

The structure and dynamics of the interfacial layers between the extremely pure air- and water-stable ionic liquid 1-butyl-1-methylpyrrolidinium tris(pentafluoroethyl)trifluorophosphate and Au(111) has been investigated using in situ scanning tunneling microscopy, cyclic voltammetry, electrochemical impedance spectroscopy, and atomic force microscopy measurements. The in situ scanning tunnelling microscopy measurements reveal that the Au(111) surface undergoes a reconstruction, and at -1.2 V versus Pt quasi-reference the famous (22 × √3) herringbone superstructure is probed. Atomic force microscopy measurements show that multiple ion pair layers are present at the ionic liquid/Au interface which are dependent on the electrode potential. Upon applying cathodic electrode potentials, stronger ionic liquid near surface structure is detected: both the number of near surface layers and the force required to rupture these layers increases. The electrochemical impedance spectroscopy results reveal that three distinct processes take place at the interface. The fastest process is capacitive in its low-frequency limit and is identified with electrochemical double layer formation. The differential electrochemical double layer capacitance exhibits a local maximum at -0.2 V versus Pt quasi-reference, which is most likely caused by changes in the orientation of cations in the innermost layer. In the potential range between -0.84 V and -1.04 V, a second capacitive process is observed which is slower than electrochemical double layer formation. This process seems to be related to the herringbone reconstruction. In the frequency range below 1 Hz, the onset of an ultraslow faradaic process is found. This process becomes faster when the electrode potential is shifted to more negative potentials.     

Central Circular Sections of the Strain Ellipsoids

A simple direct proof is given of the known result that in any deformation x = x(X), the planes of the central circular sections of the material strain-ellipsoid at X are deformed into the planes of the spatial strain-ellipsoid at x. This revised version was published online in August 2006 with corrections to the Cover Date.