The constitutive tensor of linear elasticity: its decompositions, Cauchy relations, null Lagrangians, and wave propagation

In linear anisotropic elasticity, the elastic properties of a medium are described by the fourth rank elasticity tensor C. The decomposition of C into a partially symmetric tensor M and a partially antisymmetric tensors N is often used in the literature. An alternative, less well-known decomposition, into the completely symmetric part S of C plus the reminder A, turns out to be irreducible under the 3-dimensional general linear group. We show that the SA-decomposition is unique, irreducible, and preserves the symmetries of the elasticity tensor. The MN-decomposition fails to have these desirable properties and is such inferior from a physical point of view. Various applications of the SA-decomposition are discussed: the Cauchy relations (vanishing of A), the non-existence of elastic null Lagrangians, the decomposition of the elastic energy and of the acoustic wave propagation. The acoustic or Christoffel tensor is split in a Cauchy and a non-Cauchy part. The Cauchy part governs the longitudinal wave propagation. We provide explicit examples of the effectiveness of the SA-decomposition. A complete class of anisotropic media is proposed that allows pure polarizations in arbitrary directions, similarly as in an isotropic medium.Comment: 1 figur

Wolbachia in the flesh: symbiont intensities in germ-line and somatic tissues challenge the conventional view of Wolbachia transmission routes

Symbionts can substantially affect the evolution and ecology of their hosts. The investigation of the tissue-specific distribution of symbionts (tissue tropism) can provide important insight into host-symbiont interactions. Among other things, it can help to discern the importance of specific transmission routes and potential phenotypic effects. The intracellular bacterial symbiont Wolbachia has been described as the greatest ever panzootic, due to the wide array of arthropods that it infects. Being primarily vertically transmitted, it is expected that the transmission of Wolbachia would be enhanced by focusing infection in the reproductive tissues. In social insect hosts, this tropism would logically extend to reproductive rather than sterile castes, since the latter constitute a dead-end for vertically transmission. Here, we show that Wolbachia are not focused on reproductive tissues of eusocial insects, and that non-reproductive tissues of queens and workers of the ant Acromyrmex echinatior, harbour substantial infections. In particular, the comparatively high intensities of Wolbachia in the haemolymph, fat body, and faeces, suggest potential for horizontal transmission via parasitoids and the faecal-oral route, or a role for Wolbachia modulating the immune response of this host. It may be that somatic tissues and castes are not the evolutionary dead-end for Wolbachia that is commonly thought

Mechanics of Reversible Unzipping

We study the mechanics of a reversible decohesion (unzipping) of an elastic layer subjected to quasi-static end-point loading. At the micro level the system is simulated by an elastic chain of particles interacting with a rigid foundation through breakable springs. Such system can be viewed as prototypical for the description of a wide range of phenomena from peeling of polymeric tapes, to rolling of cells, working of gecko's fibrillar structures and denaturation of DNA. We construct a rigorous continuum limit of the discrete model which captures both stable and metastable configurations and present a detailed parametric study of the interplay between elastic and cohesive interactions. We show that the model reproduces the experimentally observed abrupt transition from an incremental evolution of the adhesion front to a sudden complete decohesion of a macroscopic segment of the adhesion layer. As the microscopic parameters vary the macroscopic response changes from quasi-ductile to quasi-brittle, with corresponding decrease in the size of the adhesion hysteresis. At the micro-scale this corresponds to a transition from a `localized' to a `diffuse' structure of the decohesion front (domain wall). We obtain an explicit expression for the critical debonding threshold in the limit when the internal length scales are much smaller than the size of the system. The achieved parametric control of the microscopic mechanism can be used in the design of new biological inspired adhesion devices and machines

Limiting problems for a nonstandard viscous Cahn--Hilliard system with dynamic boundary conditions

This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice and was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp.105--118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the Laplace-Beltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested to some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions to the corresponding solutions for the limit problem is proven. Then, we study the long-time behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omega-limit set in both cases

Drosophila Lipophorin Receptors Mediate the Uptake of Neutral Lipids in Oocytes and Imaginal Disc Cells by an Endocytosis-Independent Mechanism

Lipids are constantly shuttled through the body to redistribute energy and metabolites between sites of absorption, storage, and catabolism in a complex homeostatic equilibrium. In Drosophila, lipids are transported through the hemolymph in the form of lipoprotein particles, known as lipophorins. The mechanisms by which cells interact with circulating lipophorins and acquire their lipidic cargo are poorly understood. We have found that lipophorin receptor 1 and 2 (lpr1 and lpr2), two partially redundant genes belonging to the Low Density Lipoprotein Receptor (LDLR) family, are essential for the efficient uptake and accumulation of neutral lipids by oocytes and cells of the imaginal discs. Females lacking the lpr2 gene lay eggs with low lipid content and have reduced fertility, revealing a central role for lpr2 in mediating Drosophila vitellogenesis. lpr1 and lpr2 are transcribed into multiple isoforms. Interestingly, only a subset of these isoforms containing a particular LDLR type A module mediate neutral lipid uptake. Expression of these isoforms induces the extracellular stabilization of lipophorins. Furthermore, our data indicate that endocytosis of the lipophorin receptors is not required to mediate the uptake of neutral lipids. These findings suggest a model where lipophorin receptors promote the extracellular lipolysis of lipophorins. This model is reminiscent of the lipolytic processing of triglyceride-rich lipoproteins that occurs at the mammalian capillary endothelium, suggesting an ancient role for LDLR–like proteins in this process

Sequence and expression pattern of the germ line marker vasa in honey bees and stingless bees

Queens and workers of social insects differ in the rates of egg laying. Using genomic information we determined the sequence of vasa, a highly conserved gene specific to the germ line of metazoans, for the honey bee and four stingless bees. The vasa sequence of social bees differed from that of other insects in two motifs. By RT-PCR we confirmed the germ line specificity of Amvasa expression in honey bees. In situ hybridization on ovarioles showed that Amvasa is expressed throughout the germarium, except for the transition zone beneath the terminal filament. A diffuse vasa signal was also seen in terminal filaments suggesting the presence of germ line cells. Oocytes showed elevated levels of Amvasa transcripts in the lower germarium and after follicles became segregated. In previtellogenic follicles, Amvasa transcription was detected in the trophocytes, which appear to supply its mRNA to the growing oocyte. A similar picture was obtained for ovarioles of the stingless bee Melipona quadrifasciata, except that Amvasa expression was higher in the oocytes of previtellogenic follicles. The social bees differ in this respect from Drosophila, the model system for insect oogenesis, suggesting that changes in the sequence and expression pattern of vasa may have occurred during social evolution