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

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

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

The inhomogeneous mechanical behaviour of Ascending Thoracic Aortic Aneurism (ATAA)

Surgical management of ascending thoracic aortic aneurysms (aTAAs) relies on maximum diameter, growth rate, and presence of connective tissue disorders. The surgical decision however is often not considering that dissection and rupture do occur in patients who do not meet criteria for surgical repair [1,2]. In this study the authors aim to investigate the mechanical properties of aTAAs to be implemented in computational biomechanics models for a preclinical risk evaluation. Additionally, in some recent studies, some data about the biomechanical properties of the aTAAs have been reported [3], but without any relation to bicuspidal or tricuspidal aTAA. The aim of this study was to investigate aTAA mechanical properties using a biaxial system to compare the circumferential and axial stress-strain relations for bicuspidal and tricuspidal aTAAs

Power-Laws hereditariness of biomimetic ceramics for cranioplasty neurosurgery

We discuss the hereditary behavior of hydroxyapatite-based composites used for cranioplasty surgery in the context of material isotropy. We classify mixtures of collagen and hydroxiapatite composites as biomimetic ceramic composites with hereditary properties modeled by fractional-order calculus. We assume isotropy of the biomimetic ceramic is assumed and provide thermodynamic of restrictions for the material parameters. We exploit the proposed formulation of the fractional-order isotropic hereditariness further by means of a novel mechanical hierarchy corresponding exactly to the three-dimensional fractional-order constitutive model introduced

Buckling soft tensegrities: Fickle elasticity and configurational switching in living cells

Tensegrity structures are special architectures made by floating compressed struts kept together by a continuous system of tensed cables. The multiplicity of shapes that tensegrity structures can assume and their intrinsic capability to be deployable and assembled, so storing (and releasing) elastic energy, have motivated their success as paradigm -pioneeringly proposed by Donald E. Ingber- to explain some underlying mechanisms regulating dynamics of living cells. The interlaced structure of the cell cytoskeleton, constituted by actin and intermediate filaments and microtubules which continuously change their spatial organization and pre-stresses through polymerization/depolymerization, seems to steer migration, adhesion and cell division by obeying the tensegrity construct. Even though rough calculations lead to estimate discrepancies when comparing axial stiffness of actin filaments and microtubules and recent works have shown bent microtubules, no one has yet tried to remove the hypothesis of rigid struts in tensegrities when used to idealize the cytoskeleton mechanics. With reference to the 30-element tensegrity cell paradigm, we introduce both compressibility and bendability of the struts and rewrite the theory to take into account nonlinear elasticity of both tendons and bars, so abandoning the classical linear stress-strain assumptions. By relaxing the hypothesis of rigidity of the struts, we demonstrate that some quantitative confirmations and many extreme and somehow counterintuitive mechanical behaviors actually exploited by cells for storing/releasing energy, resisting to applied loads and deforming by modulating their overall elasticity and shape through pre-stress changes and instability-guided configurational switching, can be all theoretically found

Small-on-large fractional derivative-based single-cell model incorporating cytoskeleton prestretch

In the last years, experimental evidences have suggested important direct implications of vis-coelasticity of human cells and cell cytoskeleton dynamics on some relevant collective and at single cell behaviors such as migration, adhesion and morphogenesis. As a consequence, the me-chanical properties of single cells as well as how cells respond to mechanical stimuli have been –and currently are– at the center of a vivid debate in the scientific community. By making reference to important experimental findings from the literature which have shown that human metastatic tumor cells are about 70% softer than benign cells, independently from the cell lines examined, the present authors have very recently theoretically demonstrated that these differences in stiffness might be exploited to mechanically discriminate healthy and cancer cells, for example through low intensity therapeutic ultrasound. In particular, by means of a general-ized viscoelastic paradigm combining classical and fractional derivative-based models, it has been found that selected frequencies (from tens to hundreds kHz) are associated to resonance-like phe-nomena that are prevailing on thermal fluctuations and that could be hence, at least in principle, helpfully utilized for both targeting and selectively attacking tumor cells. With the aim of investigating the effect of the prestress –for instance induced in protein filaments during cell adhesion– on the overall cell stiffness and, in turn, on its in-frequency response, a simple multiscale scheme is here proposed to bottom-up enrich the spring-pot-based viscoelastic single-cell models, by incorporating finite elasticity and in this way determining, through sensitivity analyses, the role played by the stretched state of the cytoskeletal elements on the cell vibration