On the viscous Cahn-Hilliard equation with singular potential and inertial term

We consider a relaxation of the viscous Cahn-Hilliard equation induced by the second-order inertial term~$u_{tt}$. The equation also contains a semilinear term $f(u)$ of "singular" type. Namely, the function $f$ is defined only on a bounded interval of ${\mathbb R}$ corresponding to the physically admissible values of the unknown $u$, and diverges as $u$ approaches the extrema of that interval. In view of its interaction with the inertial term $u_{tt}$, the term $f(u)$ is difficult to be treated mathematically. Based on an approach originally devised for the strongly damped wave equation, we propose a suitable concept of weak solution based on duality methods and prove an existence result.Comment: 11 page

A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint

We consider a system of two viscoelastic bodies attached on one edge by an adhesive where a delamination process occurs. We study the dynamic of the system subjected to external forces, suitable boundary conditions, and an unilateral constraint on the jump of the displacement at the interface between the bodies. The constraint arises in a graph inclusion, while the delamination coefficient evolves in a rate-independent way. We prove the existence of a weak solution to the corresponding system of PDEs

A variational approach to statics and dynamics of elasto-plastic systems

We prove some existence results for dynamic evolutions in elasto-plasticity and delamination. We study the limit as the data vary very slowly and prove convergence results to quasistatic evolutions. We model dislocations by mean of currents, we introduce the space of deformations in the presence of dislocations and study the graphs of these maps. We prove existence results for minimum problems. We study the properties of minimizers

A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint

We consider a system of two viscoelastic bodies attached on one edge by an adhesive where a delamination process occurs. We study the dynamic of the system subjected to external forces, suitable boundary conditions, and an unilateral constraint on the jump of the displacement at the interface between the bodies. The constraint arises in a graph inclusion, while the delamination coeficient evolves in a rate-independent way. We prove the existence of a weak solution to the corresponding system of PDEs

Existence and regularity of solutions for an evolution model of perfectly plastic plates

We continue the study of a dynamic evolution model for perfectly plastic plates, recently derived from three-dimensional Prandtl-Reuss plasticity. We extend the previous existence result by introducing non-zero external forces in the model, and we discuss the regularity of the solutions thus obtained. In particular, we show that the first derivatives with respect to space of the stress tensor are locally square integrable

On the strongly damped wave equation with constraint

A weak formulation for the so-called "semilinear strongly damped wave equation with constraint" is introduced and a corresponding notion of solution is defined. The main idea in this approach consists in the use of duality techniques in Sobolev-Bochner spaces, aimed at providing a suitable "relaxation" of the constraint term. A global in time existence result is proved under the natural condition that the initial data have finite "physical" energy.Comment: 21 page

Existence, energy identity and higher time regularity of solutions to a dynamic visco-elastic cohesive interface model

We study the dynamics of visco-elastic materials coupled by a common cohesive interface (or, equivalently, {two single domains separated by} a prescribed cohesive crack) in the anti-plane setting. We consider a general class of traction-separation laws featuring an activation threshold on the normal stress, softening and elastic unloading. In strong form, the evolution is described by a system of PDEs coupling momentum balance (in the bulk) with transmission and Karush-Kuhn-Tucker conditions (on the interface). We provide a detailed analysis of the system. We first prove existence of a weak solution, employing a time discrete approach and a regularization of the initial data. Then, we prove our main results: the energy identity and the existence of { solutions} with acceleration in $L^\infty (0,T; L^2)$

Modeling Mercury Capture by Powdered Activated Carbon in a Fluidized Bed Reactor

A steady state model of mercury capture on activated carbon in a bubbling fluidized bed of inert material is presented. The model takes into account the fluidized bed fluid-dynamics, the presence of both free and adhered carbon in the reactor as well as mass transfer limitations and mercury adsorption equilibrium. The activated carbon adsorption parameters and the relative amount of free versus adhered carbon in the reactor have been estimated with purposely designed experiments. Model results are compared with results from mercury capture experiments conducted with commercial powdered activated carbon at 100°C in a lab-scale pyrex fluidized bed of inert particles. The role of free versus adhered carbon in determining the overall mercury capture efficiency is discussed

Relaxed area of graphs of piecewise Lipschitz maps in the strict BV-convergence

We compute the relaxed Cartesian area in the strict BV-convergence on a class of piecewise Lipschitz maps from the plane to the plane, having jump made of several curves allowed to meet at a finite number of junction points. We show that the domain of this relaxed area is strictly contained in the domain of the classical L1-relaxed area