Fractional Hereditariness of Lipid Membranes: Instabilities and Linearized Evolution

In this work lipid ordering phase changes arising in planar membrane bilayers is investigated both accounting for elas- ticity alone and for effective viscoelastic response of such assemblies. The mechanical response of such membranes is studied by minimizing the Gibbs free energy which penalizes perturbations of the changes of areal stretch and their gradients only [1]. As material instabilities arise whenever areal stretches characterizing homogeneous configurations lie inside the spinoidal zone of the free energy density, bifurcations from such configurations are shown to occur as oscillatory perturbations of the in-plane displacement. Experimental observations [2] show a power-law in-plane viscous behavior of lipid structures allowing for an effective viscoelastic behavior of lipid membranes [3], which falls in the framework of Fractional Hereditariness. A suitable generalization of the variational principle invoked for the elasticity is applied in this case, and the corresponding Euler-Lagrange equation is found together with a set of bound- ary and initial conditions. Separation of variables allows for showing how Fractional Hereditariness owes bifurcated modes with a larger number of spatial oscillations than the corresponding elastic analog. Indeed, the available range of areal stresses for material instabilities is found to increase with respect to the purely elastic case. Nevertheless, the time evolution of the perturbations solving the Euler-Lagrange equation above exhibits time-decay and the large number of spatial oscillation slowly relaxes, thereby keeping the features of a long-tail type time-response.Comment: 21 pages, 11 figures, special issu

Growth, Development, and Vertebrate and Invertebrate Herbivory of the Federally Endangered Spiranthes parksii Correll and Sympatric Congener Spiranthes cernua

ABSTRACT Spiranthes parksii Correll, a terrestrial orchid protected under the Endangered Species Act, and its congener Spiranthes cernua (L.) Rich, were studied in the Post Oak Savanna Ecoregion of Central Texas in 2014 and 2015. The species are sympatric and each produces a single inflorescence in the fall with emergence of a basal rosette during flower senescence or early spring. Objectives of this study were to 1) assess variation in annual and seasonal growth 2) determine the impact of vertebrate and invertebrate herbivores on the rosette and flower phases, and 3) identify invertebrate herbivores that utilize S. parksii and S. cernua. To assess variation in annual growth patterns between years an analysis of precipitation, demographic (presence or absence), and growth data (leaf area and inflorescence height) was performed. From 2014 to 2015 there was a reduction in precipitation, plants present, plant height, and the number of flowering plants that survived to seed production. To determine the difference between vertebrate and invertebrate herbivores, a 2 x 3 factorial experiment was conducted. Plants were randomly assigned to one of five treatments: Control (accessible to Vertebrates and Invertebrates), Insecticide with no cage (Vertebrate Only), Cage with no insecticide (Invertebrate Only), Caged with insecticide (Cage+Insecticide; no vertebrate or invertebrate), and cage with mesh cover and no insecticide (Mesh; access by only small invertebrates). During the flower season, herbivory was visually estimated for plant stalk and inflorescence by 5 percent increments. For rosettes, herbivory was visually estimated for each leaf in 5 percent increments and averaged over the whole rosette. During the first flowering season, vertebrates consumed more reproductive tissue (46%) than invertebrates (3%), while in the second season there was no significant difference between the two at 19% and 2%, respectively. There was no significant difference in percent herbivory of rosettes by vertebrates or invertebrates at 9% and 14% in 2014 and 16% and 11% in 2015. Invertebrates that were observed consuming Spiranthes sp. inflorescences and rosettes were armyworms (Order Lepidoptera: Family Noctuidae), grasshoppers (Family Acrididae), and an unidentified member of the Actiinae subfamily. This experiment confirms that vertebrates have a direct effect on Spiranthes sp. fitness through removal of reproductive tissue and an indirect impact by consuming rosettes. In addition, it documents that invertebrate herbivores can have a similar effect on inflorescence and rosettes. This knowledge can be important in understanding the influence of plant-herbivore interactions on conservation and management plans for S. parksii

A mechanical picture of fractional-order Darcy equation

In this paper the authors show that fractional-order force-flux relations are obtained considering the flux of a viscous fluid across an elastic porous media. Indeed the one-dimensional fluid mass transport in an unbounded porous media with power-law variation of geometrical and physical properties yields a fractional-order relation among the ingoing flux and the applied pressure to the control section. As a power-law decay of the physical properties from the control section is considered, then the flux is related to a Caputo fractional derivative of the pressure of order 0 ≤ β≤1. If, instead, the physical properties of the media show a power-law increase from the control section, then flux is related to a fractional-order integral of order 0 ≤ β≤1. These two different behaviors may be related to different states of the mass flow across the porous media

A STATE-SPACE APPROACH TO DYNAMIC STABILITY OF FRACTIONAL-ORDER SYSTEMS: THE EXTENDED ROUTH-HURWITZ THEOREM

This paper considers the case of Beck’s column, a linear elastic cantilever column subjected to a constant follower load at its free end. The column foundation is modeled as bed of hereditary elements that react with a vertical force distributed along the beam axis. The reacting supports are modeled with spring-pot element that is a two parameters mechanical elements (C; ) with an intermediate behavior between spring and dashpot. The constitutive equation of the spring-pot involves the so called fractional order derivatives and dynamic stability problem in presence of fractional-order operator must be faced for the Beck’s column. In this study , the authors generalize Routh-Hurwitz theorem of stability on the fractional order differential equation (FODE), system that governs the dynamic stability. Some numerical examples has been reported in the paper for two-degree of freedom system

Structured Deformations of Continua: Theory and Applications

The scope of this contribution is to present an overview of the theory of structured deformations of continua, together with some applications. Structured deformations aim at being a unified theory in which elastic and plastic behaviours, as well as fractures and defects can be described in a single setting. Since its introduction in the scientific community of rational mechanicists (Del Piero-Owen, ARMA 1993), the theory has been put in the framework of variational calculus (Choksi-Fonseca, ARMA 1997), thus allowing for solution of problems via energy minimization. Some background, three problems and a discussion on future directions are presented.Comment: 11 pages, 1 figure, 1 diagram. Submitted to the Proceedings volume of the conference CoMFoS1

Free energy and states of fractional-order hereditariness

Complex materials, often encountered in recent engineering and material sciences applications, show no complete separations between solid and fluid phases. This aspect is reflected in the continuous relaxation time spectra recorded in cyclic load tests. As a consequence the material free energy cannot be defined in a unique manner yielding a significative lack of knowledge of the maximum recoverable work that can extracted from the material. The non-uniqueness of the free energy function is removed in the paper for power-laws relaxation/creep function by using a recently proposed mechanical analogue to fractional-order hereditariness

Modeling of curved cantilever dielectric elastomer actuator using universal solution in finite bending

This study presents a model of a curved cantilever dielectric elastomer actuator (DEA) containing liquid-phase metal electrodes utilizing universal solutions from finite nonlinear elasticity. The DEA comprises a compliant capacitor which has been prestrained some prescribed amount, affixed to a substrate, and bonded to a secondary layer of unstrained bulk elastomer. Upon release of the cured layers, internal stresses cause a bending moment and force the final configuration into a beam with some initial curvature. Application of a voltage across the electrodes creates an electrostatic pressure, inducing compressive Maxwell stresses across the dielectric layer. This relieves some of the internal moment and forces actuation of the device in the form of beam straightening. We assume incompressibility and isotropic, neo-Hookean behavior of our bulk elastomeric material. Employing simplifying assumptions such as constant curvature across the length of the beam (i.e., a perfectly circular arc) and plane strain in the plane of actuation, we utilize the Universal Solution in finite bending to represent the kinematics and implement Maxwell’s equations to describe beam deflection as a function of applied voltage. We use the principal of minimum potential energy to solve for beam deflection (represented by curvature of the device) as a function of electric potential across the electrodes after considering energy contributions due to elasticity, electrostatics and expended electric work. This model is then compared to experimental data obtained from testing multiple fabricated devices. The emerging fields of soft robotics and wearable computing require new classes of soft and elastically deformable electronics, which unlike traditional electronic components, must be flexible and/or stretchable. Thus it is important to develop predictive and comprehensive models describing their behavior

On the asymptotic behavior of the quasi-static problem for a linear viscoelastic fluid

In this paper we study the quasi-static problem for a viscoelastic fluid by means of the concept of minimal state. This implies the use of a different free energy defined in a wider space of data. The existence and uniqueness is proved in this new space and the asymptotic decay for the problem with non vanishing supplies is obtained for a large class of memory kernels, including those presenting an exponential or polynomial decay.Comment: 6 page