Relative Lagrangian Formulation of Finite Thermoelasticity

The motion of a body can be expressed relative to the present configuration of the body, known as the relative motion description, besides the classical Lagrangian and the Eulerian descriptions. When the time increment from the present state is small enough, the nonlinear constitutive equations can be linearized relative to the present state so that the resulting system of boundary value problems becomes linear. This formulation is based on the well-known ``small-on-large'' idea, and can be implemented for solving problems with large deformation in successive incremental manner. In fact, the proposed method is a process of repeated applications of the well-known “small deformation superposed on finite deformation” in the literature. This article presents these ideas applied to thermoelastic materials with a brief comment on the exploitation of entropy principle in general. Some applications of such a formulation in numerical simulations are briefly reviewed and a numerical result is shown.

Modeling of Cavitation Bubble Cloud with Discrete Lagrangian Tracking

In this paper, a Lagrangian-Eulerian (LE) two-way coupling model is developed to numerically study the cavitation bubble cloud. In this model, the gas-liquid mixture is treated directly as a continuous and compressible fluid and the governing equations are solved by methods in Eulerian descriptions. An isobaric closure exhibiting better consistency properties is applied to evaluate the pressure of gas-liquid mixture. The dispersed gas/vapor bubbles are tracked in a Lagrangian fashion, and their compression and expansion are described by a modified Rayleigh-Plesset equation, which considers the close-by flow properties other than these of the infinity for each bubble. The performance of the present method is validated by a number of benchmark tests. Then, this model is applied to study how the bubble cloud affects the shape and propagation of a pressure wave when the pressure pulse travels through. In the end, a three-dimensional simulation of a vapor cloud’s Rayleigh collapse is carried out, and the induced extreme pressure is discussed in detail. The total bubble number’s influence on the extreme collapse pressure and the size distribution of bubbles during the collapse are also analyzed.

On nonlinear dilatational strain gradient elasticity

We call nonlinear dilatational strain gradient elasticity the theory in which the specific class of dilatational second gradient continua is considered: those whose deformation energy depends, in an objective way, on the gradient of placement and on the gradient of the determinant of the gradient of placement. It is an interesting particular case of complete Toupin–Mindlin nonlinear strain gradient elasticity: indeed, in it, the only second gradient effects are due to the inhomogeneous dilatation state of the considered deformable body. The dilatational second gradient continua are strictly related to other generalized models with scalar (one-dimensional) microstructure as those considered in poroelasticity. They could be also regarded to be the result of a kind of “solidification” of the strain gradient fluids known as Korteweg or Cahn–Hilliard fluids. Using the variational approach we derive, for dilatational second gradient continua the Euler–Lagrange equilibrium conditions in both Lagrangian and Eulerian descriptions. In particular, we show that the considered continua can support contact forces concentrated on edges but also on surface curves in the faces of piecewise orientable contact surfaces. The conditions characterizing the possible externally applicable double forces and curve forces are found and examined in detail. As a result of linearization the case of small deformations is also presented. The peculiarities of the model is illustrated through axial deformations of a thick-walled elastic tube and the propagation of dilatational waves.

Modelling fluid deformable surfaces with an emphasis on biological interfaces

Fluid deformable surfaces are ubiquitous in cell and tissue biology, including lipid bilayers, the actomyosin cortex or epithelial cell sheets. These interfaces exhibit a complex interplay between elasticity, low Reynolds number interfacial hydrodynamics, chemistry and geometry, and govern important biological processes such as cellular traffic, division, migration or tissue morphogenesis. To address the modelling challenges posed by this class of problems, in which interfacial phenomena tightly interact with the shape and dynamics of the surface, we develop a general continuum mechanics and computational framework for fluid deformable surfaces. The dual solid–fluid nature of fluid deformable surfaces challenges classical Lagrangian or Eulerian descriptions of deforming bodies. Here, we extend the notion of arbitrarily Lagrangian–Eulerian (ALE) formulations, well-established for bulk media, to deforming surfaces. To systematically develop models for fluid deformable surfaces, which consistently treat all couplings between fields and geometry, we follow a nonlinear Onsager formalism according to which the dynamics minimizes a Rayleighian functional where dissipation, power input and energy release rate compete. Finally, we propose new computational methods, which build on Onsager’s formalism and our ALE formulation, to deal with the resulting stiff system of higher-order partial differential equations. We apply our theoretical and computational methodology to classical models for lipid bilayers and the cell cortex. The methods developed here allow us to formulate/simulate these models in their full three-dimensional generality, accounting for finite curvatures and finite shape changes.

Generalized Riemann waves and their adjoinment through a shock wave

Generalized simple waves of the gas dynamics equations in Lagrangian and Eulerian descriptions are studied in the paper. As in the collision of a shock wave and a rarefaction wave, a flow becomes nonisentropic. Generalized simple waves are applied to describe such flows. The first part of the paper deals with constructing a solution describing their adjoinment through a shock wave in Eulerian coordinates. Even though the Eulerian form of the gas dynamics equations is most frequently used in applications, there are advantages for some problems concerning the gas dynamics equations in Lagrangian coordinates, for example, of being able to be reduced to an Euler–Lagrange equation. Through the technique of differential constraints, necessary and sufficient conditions for the existence of generalized simple waves in the Lagrangian description are provided in the second part of the paper.

Relative Lagrangian Formulation of Finite Thermoelasticity

Besides the Lagrangian and the Eulerian descriptions, the motion of a body can also be expressed relative to the present configuration of the body, known as the relative motion description. It is interesting to consider such a relative motion description in general to formulate the basic system of field equations for solid bodies. In doing so, when the time increment from the present state is small enough, the nonlinear constitutive equations can be linearized relative to the present state so that the resulting system becomes linear. This will be done for thermoelastic materials with a brief comment on the exploitation of entropy principle in general. Relative Lagrangian formulation is based on the well-known ``small-on-large'' idea, and can be implemented for solving problems with large deformation in successive incremental manner. Some applications of such a formulation in numerical simulations are briefly reviewed.

Neutralization of a spray of electrically charged droplets by a corona discharge

The neutralization of a dilute spray of electrically charged droplets by ions of the opposite polarity generated by a corona discharge at a wire ring is analysed numerically. A Lagrangian description of the spray and Eulerian descriptions of the gas and the ions are used to deal with this two-way coupled problem. A model of the corona consisting of a line of charge and a distribution of ion sources is proposed. In the configuration that is analysed, neutralization usually begins at the shroud of the spray and extends to inner regions when the corona current increases. The number density of droplets is large at the shroud due to neutralized droplets that are no longer pushed by the electric field. These droplets can be dragged towards a collector surface by a weak forced flow that overcomes the ionic wind due to the force of the ions on the gas. The fraction of the spray charge that is neutralized increases with the corona current, but the value of this current required for full neutralization is several times larger than the inlet electric current of the spray owing to loss of ions to the boundaries of the system. The electric field induced by the charge of the droplets opposes the field due to the voltage applied between the wire ring and the extractor through which the droplets are injected, and thus reduces the threshold voltage of the corona and significantly affects its current–voltage characteristic, which may become multivalued. In turn, the electric field due to the applied voltage and the space charge of the ions affects the shape of the spray and the velocity of the droplets.

IMPLEMENTATION OF THE EULERIAN AND THE ARBITRARY LAGRANGIAN–EULERIAN DESCRIPTIONS IN FINITE ELEMENT SIMULATION OF EXTRUSION PROCESSES

In this paper, backward and forward–backward-radial extrusion processes of aluminum have been simulated using finite element method. Due to the extreme deformation of the workpiece and the restrictions of the Lagrangian approach to simulate such problems, the arbitrary Lagrangian–Eulerian (ALE) and the Eulerian descriptions have been implemented in backward and forward–backward-radial extrusion processes, respectively. Operator-split method is used to solve the coupled governing equations of the Eulerian and the ALE formulations. To validate the finite element simulations, the results have been compared with experimental data in terms of extrusion load and geometry of final products. A good agreement has been seen between the results demonstrating the capability of the Eulerian and the ALE methods on finite element simulation of extrusion processes.