Asymptotic behavior of Structures made of Plates

The aim of this work is to study the asymptotic behavior of a structure made of plates of thickness $2\delta$ when $\delta\to 0$. This study is carried on within the frame of linear elasticity by using the unfolding method. It is based on several decompositions of the structure displacements and on the passing to the limit in fixed domains. We begin with studying the displacements of a plate. We show that any displacement is the sum of an elementary displacement concerning the normal lines on the middle surface of the plate and a residual displacement linked to these normal lines deformations. An elementary displacement is linear with respect to the variable $x$3. It is written $U(^x)+R(^x)\land x3e3$ where U is a displacement of the mid-surface of the plate. We show a priori estimates and convergence results when $\delta \to 0$. We characterize the limits of the unfolded displacements of a plate as well as the limits of the unfolded of the strained tensor. Then we extend these results to the structures made of plates. We show that any displacement of a structure is the sum of an elementary displacement of each plate and of a residual displacement. The elementary displacements of the structure (e.d.p.s.) coincide with elementary rods displacements in the junctions. Any e.d.p.s. is given by two functions belonging to $H1(S;R3)$ where S is the skeleton of the structure (the plates mid-surfaces set). One of these functions : U is the skeleton displacement. We show that U is the sum of an extensional displacement and of an inextensional one. The first one characterizes the membrane displacements and the second one is a rigid displacement in the direction of the plates and it characterizes the plates flexion. Eventually we pass to the limit as $\delta \to 0$ in the linearized elasticity system, on the one hand we obtain a variational problem that is satisfied by the limit extensional displacement, and on the other hand, a variational problem satisfied by the limit of inextensional displacements

Continuum Electromechanical Modeling of Protein-Membrane Interaction

A continuum electromechanical model is proposed to describe the membrane curvature induced by electrostatic interactions in a solvated protein-membrane system. The model couples the macroscopic strain energy of membrane and the electrostatic solvation energy of the system, and equilibrium membrane deformation is obtained by minimizing the electro-elastic energy functional with respect to the dielectric interface. The model is illustrated with the systems with increasing geometry complexity and captures the sensitivity of membrane curvature to the permanent and mobile charge distributions.Comment: 5 pages, 12 figure

On the numerical approximation of p-biharmonic and ∞-biharmonic functions

In [KP16] (arXiv:1605.07880) the authors introduced a second-order variational problem in L∞. The associated equation, coined the ∞-Bilaplacian, is a \emph{third order} fully nonlinear PDE given by Δ2∞u:=(Δu)3|D(Δu)|2=0. In this work we build a numerical method aimed at quantifying the nature of solutions to this problem which we call ∞-Biharmonic functions. For fixed p we design a mixed finite element scheme for the pre-limiting equation, the p-Bilaplacian Δ2pu:=Δ(|Δu|p−2Δu)=0. We prove convergence of the numerical solution to the weak solution of Δ2pu=0 and show that we are able to pass to the limit p→∞. We perform various tests aimed at understanding the nature of solutions of Δ2∞u and in 1-d we prove convergence of our discretisation to an appropriate weak solution concept of this problem, that of -solutions

Identification of nonlinearity in conductivity equation via Dirichlet-to-Neumann map

We prove that the linear term and quadratic nonlinear term entering a nonlinear elliptic equation of divergence type can be uniquely identified by the Dirichlet to Neuman map. The unique identifiability is proved using the complex geometrical optics solutions and singular solutions

A simplicial gauge theory

We provide an action for gauge theories discretized on simplicial meshes, inspired by finite element methods. The action is discretely gauge invariant and we give a proof of consistency. A discrete Noether's theorem that can be applied to our setting, is also proved.Comment: 24 pages. v2: New version includes a longer introduction and a discrete Noether's theorem. v3: Section 4 on Noether's theorem has been expanded with Proposition 8, section 2 has been expanded with a paragraph on standard LGT. v4: Thorough revision with new introduction and more background materia

Relativistic Elastostatics I: Bodies in Rigid Rotation

We consider elastic bodies in rigid rotation, both nonrelativistically and in special relativity. Assuming a body to be in its natural state in the absence of rotation, we prove the existence of solutions to the elastic field equations for small angular velocity.Comment: 25 page

The density functional theory of classical fluids revisited

We reconsider the density functional theory of nonuniform classical fluids from the point of view of convex analysis. From the observation that the logarithm of the grand-partition function $\log \Xi [\phi]$ is a convex functional of the external potential $\phi$ it is shown that the Kohn-Sham free energy ${\cal A}[\rho]$ is a convex functional of the density $\rho$. $\log \Xi [\phi]$ and ${\cal A}[\rho]$ constitute a pair of Legendre transforms and each of these functionals can therefore be obtained as the solution of a variational principle. The convexity ensures the unicity of the solution in both cases. The variational principle which gives $\log \Xi [\phi]$ as the maximum of a functional of $\rho$ is precisely that considered in the density functional theory while the dual principle, which gives ${\cal A}[\rho]$ as the maximum of a functional of $\phi$ seems to be a new result.Comment: 10 page

Dissipative-particle dynamics simulations of flow over a stationary sphere in compliant channels

Dissipative-particle dynamics (DPD), a particle-based fluid-simulation approach, is employed to simulate isothermal pressure-driven flow across a sphere in compliant cylindrical channels. The sphere is represented by frozen DPD particles, while the surrounding fluid is modeled using simple fluid particles. The channel walls are made up of interconnected finite extensible nonlinear elastic bead-spring chains. The wall particles at the inlet and outlet ends of the channel are frozen so as to hinge the channel. The model is assessed for accuracy by computing the drag coefficient CD in shear flow past a uniform sphere in unbounded flow, and comparing the results with those from correlations in literature. The effect of the aspect ratio λ of the channel, i.e., the ratio of the sphere diameter d to the channel diameter D, on the drag force FD on the sphere is investigated, and it is found that FD decreases as λ decreases toward the values predicted by the correlations as λ approaches zero. The effect of the elasticity of the wall is also studied. It is observed that as the wall becomes more elastic, there is a decrease in FD on the sphere.Harinath Reddy and John Abraha

Relativistic Elasticity

Relativistic elasticity on an arbitrary spacetime is formulated as a Lagrangian field theory which is covariant under spacetime diffeomorphisms. This theory is the relativistic version of classical elasticity in the hyperelastic, materially frame-indifferent case and, on Minkowski space, reduces to the latter in the non-relativistic limit . The field equations are cast into a first -- order symmetric hyperbolic system. As a consequence one obtains local--in--time existence and uniqueness theorems under various circumstances.Comment: 23 page