Lagrangian Configuration Research Articles

Transformation of surface [formula omitted]-forces in high grade elasticity

The expression of the elastic energy, postulated for a given material, specifies through the principle of virtual work which external loadings turn out to be admissible. For the elastic bodies, the energy of which depends on the gradients of the placement higher than (H+1) (or, equivalently, on the derivatives of the deformation gradient higher than H), external force densities can be prescribed over the boundary surface, which do work on the H-th derivative of the virtual displacements along the normal direction. Such generalized forces with H=1,2 have been dealt with in the literature, referred to as double and triple forces: when H is increased, however, the analytical evaluation of these work terms may become prohibitive. Each Eulerian H-force density of this kind, acting on the current boundary surface, turns out to be equivalent to a set of diverse actions defined in the Lagrangian configuration. These actions include: (i) surface terms doing work on the virtual displacement vector; (ii) surface terms doing work on the higher order derivatives of the virtual displacement along the normal direction; (iii) edge terms, doing work on the virtual displacement or on its higher order gradients, which may be further reduced. In this study, a recursive algorithm is utilized to generate all these Lagrangian contributions for any H. Such an approach rests on the binomial expansion of the tensor products of complementary surface projectors, symmetrized in the work duality, and on the integration by parts combined with the generalized divergence theorem.

Numerical design and experimental validation of shockless plate impact configurations for the Lagrangian analysis of armor ceramics

Ceramic materials are widely used in armor or protective structures, providing weight saving at equivalent performance in comparison to their steel counterparts. Plate-impact experiments are commonly used to investigate the dynamic behavior of ceramics under compressive loading. Using the particle velocity measured at the back of the target, some mechanical properties such as the Hugoniot elastic limit (HEL) as well as the Hugoniot curve of the material can be deduced. Nevertheless, these tests do not provide a direct measurement of the plastic hardening (post-HEL) behavior of the target. In the present work, an experimental shockless plate-impact configuration was developed and implemented to conduct a Lagrangian analysis. This configuration relies on the use of a wavy-machined flyer plate impacting a target made of a buffer, two ceramic plates of different thicknesses, and two window plates as backing. First, the use of wavy flyer plates to generate a loading ramp was validated by considering the impact of the wavy-machined flyer plate against a target, both made of 316L steel. A numerical analysis of this test was developed to confirm the pulse-shaping effect observed experimentally. Next, a ceramic, F99.7 alumina was subjected to the shockless plate impact test in Lagrangian configuration considering the same steel as a buffer and flyer plate material. These tests coupled with Lagrangian analysis enable the curve of axial-stress vs axial-strain beyond the isentropic elastic limit (IEL) to be deduced. The experimental data allow identifying the parameters of an elastoplastic with strain-hardening model to describe the behavior of the tested alumina.

Equilibrium stability in the triangular restricted four-body problem with non-spherical primaries

This paper investigates the Lagrangian configuration of the restricted four-body problem in which the three primaries are non-spherical, specifically either prolate or oblate. By using various standard numerical methods, the positions of equilibrium points and their linear stability and dynamical type were determined. The impact of mass and shape of the primaries on the system’s equilibrium points and their linear stability were systematically explored by discretizing the parameter space for the non-sphericity parameter within a specified interval. The study revealed that the system always has an even number of equilibrium points, ranging from 8 to 22. Linearly stable points always exist, except for the case where there are 10 equilibrium points, where all the points are unstable.

Dynamics of a small planetoid in Newtonian gravity field of Lagrangian configuration of three primaries

Dynamics of a small planetoid in Newtonian gravity field of Lagrangian configuration of three primaries

Path integrals as analysis on path space by time slicing approximation

We explain the Lagrangian (configuration space) path integrals in Section 1 and the Hamiltonian (phase space) path integrals in Section 2. The two sections are independent of each other. Readers can read from either section.

Approach à la Piola for the equilibrium problem of bodies with second gradient energies. Part II: Variational derivation of second gradient equations and their transport

After the wide premise of Part I, where the equations for Cauchy’s continuum were retrieved through the energy minimization and some differential geometric perspectives were specified, the present paper as Part II outlines the variational derivation of the equilibrium equations for second gradient materials and their transformation from the Eulerian to the Lagrangian form. Volume, face and edge contributions to the inner virtual work were provided through integration by parts and by repeated applications of the divergence theorem extended to curved surfaces with border. To sustain double forces over the faces and line forces along the edges, the role of the third rank hyperstress tensor was highlighted. Special attention was devoted to the edge work, and to the evaluation of the variables discontinuous across the edge belonging to the contiguous boundary faces. The detailed expression of the contact pressures was provided, including multiple products of normal vector components, their gradient and a combination of them: in particular, the dependence on the local mean curvature was shown. The transport of the governing equations from the Eulerian to the Lagrangian configuration was developed according to two diverse strategies, exploiting novel differential geometric formulae and revealing a coupling of terms transversely to the involved domains.

Nonlinear Stability of Equilibrium Points in the Planar Equilateral Restricted Mass-Unequal Four-Body Problem

In this paper, we investigate the stability of equilibrium points for the planar restricted equilateral four-body problem in the case that one particle of negligible mass is moving under the Newtonian gravitational attraction of three positive masses [Formula: see text], [Formula: see text] and [Formula: see text] (called primaries). These always lie at the vertices of an equilateral triangle (Lagrangian configuration) and move with constant angular velocity in circular orbits around their center of masses. We consider the case where all the primaries have unequal masses, and investigate the nonlinear stability (in the sense of Lyapunov) of the elliptic equilibrium for the specific values of the mass [Formula: see text] and [Formula: see text] of the primary, fixed on the horizontal axis. Moreover, the [Formula: see text][Formula: see text]:[Formula: see text][Formula: see text] four-order resonant cases are determined and the stability is investigated. In this study, Markeev’s theorem and Arnold’s theorem become key ingredients.

A Lagrangian meshfree mesoscale simulation of powder bed fusion additive manufacturing of metals

AbstractWe present a powder‐scale computational framework to predict the microstructure evolution of metals in powder bed fusion additive manufacturing (PBF AM) processes based on the hot optimal transportation meshfree (HOTM) method. The powder bed is modeled as discrete and deformable three‐dimensional bodies by integrating statistic information from experiments, including particle size and shape, and powder packing density. Tractions in Lagrangian framework are developed to model the recoil pressure and surface tension. The laser beam is applied to surfaces of particles and substrate dynamically as a heat flux with user‐specified beam size, power, scanning speed, and path. The linear momentum and energy conservation equations are formulated in the Lagrangian configuration and solved simultaneously in a monolithic way by the HOTM method to predict the deformation, temperature, contact mechanisms, and fluid‐structure interactions in the powder bed. The numerical results are validated against single track experiments. Various powder bed configurations, laser powers, and speed are investigated to understand the influence of dynamic contact and inelastic material behavior on the deformation, heat transfer, and phase transition of the powder bed. The formation of defects in the microstructure of 3D printed metals, including pores, partially, and unmelted particles, is predicted by the proposed computational scheme.

On the isogeometric analysis of geometrically nonlinear shell structures with the consideration of surface energies

Presented in this article is a size-dependent analysis which is aimed to discover the surface stress effects on the large deformation characteristics of small-scaled shell structures. Consideration of surface energies in the micro/nanodimensions, and also in the cases of structures with high ratios of surface to volume like shells, significantly affects the predicted responses. To take the surface stress effects into account, it is assumed that the body is surrounded by thin surface layers at the top and bottom. The first-order shear deformation shell theory with seven parameters is adopted in the Lagrangian configuration system for the bulk part of structure. Also, the effects of surface stresses are captured based on the surface elasticity theory. Because of its unique features, IGA (isogeometric analysis) solution methodology is implemented to solve the governing equations. In this regard, the matrix–vector form of nonlinear relations, including the constitutive equations and energy functionals, is presented to directly utilize IGA for the problem. As case studies and also to show the main contribution of present investigation, i.e., studying the effects of surface energies on the mechanical behavior of geometrically nonlinear shells, the well-known benchmarks in the literature are studied in micro/nanodimensions with the consideration of surface influences.

Heterogeneous primaries in CR4BP

This paper investigates the motion of the massless body moving under the influence of the gravitational forces of the three equal heterogeneous oblate spheroids placed at Lagrangian configuration. After determining the equations of motion and the Jacobian constant of the massless body, we have illustrated the numerical work (Stationary points, zero-velocity curves, regions of motion, Poincare surfaces of section and basins of attraction). And then we have checked the linear stability of the stationary points and found that all the stationary points are unstable.Â

Combined effect of Stokes drag, oblateness and radiation pressure on the existence and stability of equilibrium points in the restricted four-body problem

This paper studies numerically the existence of collinear and non-collinear equilibrium points and their linear stability in the frame work of photogravitational circular restricted four-body problem with Stokes drag acting as a dissipative force and considering the first primary as a radiating body and the second primary as an oblate spheroid. The mass of the fourth body is assumed to be infinitesimal and does not affect the motion of the three primaries which are always at the vertices of an equilateral triangle (Lagrangian configuration). It is found that at constant dissipative force, and a simultaneous increase in both radiation pressure and oblateness coefficients, the curves that define the path of the motion of the infinitesimal body are found to shrink due to shifts in the positions of equilibrium points. All the collinear and non-collinear equilibrium points are found to be linearly unstable under the combined effect of radiation pressure, oblateness and Stokes drag. The energy integral is seen to be time dependent due to the presence of the drag force. More so the dynamic property of the system is investigated with the help of Lyapunov characteristic exponents (LCEs). It is found that the system is chaotic as the trajectories locally diverge from each other and the equilibrium points are chaotic attractors. We justified the relevance of the model in astronomy by applying it to a stellar system (Gliese 667C) and another found that both the existence and stability of the equilibrium points of any restricted few body system greatly depend of the value of the mass parameter.

A thermodynamically consistent model of magneto-elastic materials under diffusion at large strains and its analysis

The theory of elastic magnets is formulated under possible diffusion and heat flow governed by Fick's and Fourier's laws in the deformed (Eulerian) configuration, respectively. The concepts of nonlocal nonsimple materials and viscous Cahn-Hilliard equations are used. The formulation of the problem uses Lagrangian (reference) configuration while the transport processes are pulled back. Except the static problem, the demagnetizing energy is ignored and only local non-selfpenetration is considered. The analysis as far as existence of weak solutions of the (thermo)dynamical problem is performed by a careful regularization and approximation by a Galerkin method, suggesting also a numerical strategy. Either ignoring or combining particular aspects, the model has numerous applications as ferro-to-paramagnetic transformation in elastic ferromagnets, diffusion of solvents in polymers possibly accompanied by magnetic effects (magnetic gels), or metal-hydride phase transformation in some intermetalics under diffusion of hydrogen accompanied possibly by magnetic effects (and in particular ferro-to-antiferromagnetic phase transformation), all in the full thermodynamical context under large strains.

A monolithic Lagrangian meshfree scheme for Fluid–Structure Interaction problems within the OTM framework

We present a monolithic Lagrangian meshfree solution for Fluid–Structure Interaction (FSI) problems within the Optimal Transportation Meshfree (OTM) framework. The governing equations of the fluid and structure are formulated in the Lagrangian configuration and solved simultaneously in a monolithic way. Mainly, the fully discretized equations are constructed by leveraging on the OTM method to address the challenges in the Lagrangian description of the fluid domain. In this approach, the fluid–structure interface becomes an internal surface of the entire field, and the continuity and force equilibrium on the interface are automatically satisfied without any extra computations. The monolithic Lagrangian solution provides enhanced stability comparing to partitioning approaches and eliminates the problem of free surface and material interface tracking. The presented method enables a Direct Numerical Simulation (DNS) of the fluid flow with the absence of the convective terms. The accuracy and robustness of the OTM FSI approach are systematically investigated by the classical Blasius solution of the boundary layer problem. Furthermore, we illustrate the range and scope of the method through two examples: the impact of a rigid body on the fluid domain in a container and the interaction between the fluid and highly flexible structures in an open channel.

Lattice continuum and diffusional creep.

Diffusional creep is characterized by growth/disappearance of lattice planes at the crystal boundaries that serve as sources/sinks of vacancies, and by diffusion of vacancies. The lattice continuum theory developed here represents a natural and intuitive framework for the analysis of diffusion in crystals and lattice growth/loss at the boundaries. The formulation includes the definition of the Lagrangian reference configuration for the newly created lattice, the transport theorem and the definition of the creep rate tensor for a polycrystal as a piecewise uniform, discontinuous field. The values associated with each crystalline grain are related to the normal diffusional flux at grain boundaries. The governing equations for Nabarro-Herring creep are derived with coupled diffusion and elasticity with compositional eigenstrain. Both, bulk diffusional dissipation and boundary dissipation accompanying vacancy nucleation and absorption, are considered, but the latter is found to be negligible. For periodic arrangements of grains, diffusion formally decouples from elasticity but at the cost of a complicated boundary condition. The equilibrium of deviatorically stressed polycrystals is impossible without inclusion of interface energies. The secondary creep rate estimates correspond to the standard Nabarro-Herring model, and the volumetric creep is small. The initial (primary) creep rate is estimated to be much larger than the secondary creep rate.

Nonlinear stability analysis in a equilateral restricted four-body problem

We consider the problem of the stability of equilibrium points in the restricted equilateral four-body problem where a particle of negligible mass is moving under the Newtonian gravitational attraction of three positive masses (called primaries) which move on circular orbits around their center of masses, such that their configuration is always an equilateral triangle (Lagrangian configuration). We consider the case of two bodies of equal mass, which in adimensional units is the parameter of the problem. We investigate the Birkhoff normal form around the linearly stable equilibrium points L 3 for μ∈(0,0.0027096302) and, L 5 and L 6 for μ∈(0,0.018858526). The quadratic part of the normalized Hamiltonian has no definite sign, so we examine a function which depends on the coefficients of the fourth order normal form and the stability follows from KAM theory if this function is not zero, otherwise the normal form has to be taken to sixth order or higher. This is the case for L 5 and L 6 for one value of μ. We calculate the sixth order normal form and establish the stability of the points in this case of the failure of the fourth order analysis. We also discuss the stability at the 2:1 resonances for the equilibrium point L 3 and the symmetrical L 5 and L 6 points.

Analytical study on the motions around equilibrium points of restricted four-body problem

We investigate the motions around the equilibrium points of restricted four-body problem, where the three primaries with unequal masses constitute a Lagrangian configuration which is linearly stable. About the dynamical model studied, there are eight non-collinear equilibrium points, three of them are stable and the remaining ones are unstable. The linear dynamics of these equilibrium points state that there are center and hyperbolic manifolds in the vicinity of unstable equilibrium points, and there are long, short and vertical periodic orbits around stable equilibrium points. Based on the nonlinear equations of motion, the general solutions around equilibrium points are expanded as formal series of several amplitude parameters. Lissajous orbits around unstable equilibrium points are expressed as formal series of the in-plane and out-of-plane amplitudes. Invariant manifolds around unstable equilibrium points are expanded as formal series of four amplitudes, two of them correspond to hyperbolic dynamics and the remaining ones correspond to center dynamics. The motions around stable equilibrium points are expressed as formal series of long, short and vertical periodic amplitudes. By means of Lindstedt–Poincaré method, series solutions are constructed up to a certain order. The advantage of the series solutions constructed lies in that the motions around equilibrium points can all be parameterized. At last, the practical convergence has been computed in order to check the validity of the series expansions constructed.

Transport orbits in an equilateral restricted four-body problem

In this paper we consider a restricted equilateral four-body problem where a particle of negligible mass is moving under the Newtonian gravitational attraction of three masses (called primaries) which move on circular orbits around their center of masses such that their configuration is always an equilateral triangle (Lagrangian configuration). We consider the case of two bodies of equal masses, which in adimensional units is the parameter of the problem. We study numerically the existence of families of unstable periodic orbits, whose invariant stable and unstable manifolds are responsible for the existence of homoclinic and heteroclinic connections, as well as of transit orbits traveling from and to different regions. We explore, for three different values of the mass parameter, what kind of transits and energy levels exist for which there are orbits with prescribed itineraries visiting the neighborhood of different primaries.

An integrated approach to shape optimization and mesh adaptivity based on material residual forces

Abstract This contribution is concerned with the coupling of finite element based shape optimization to methods of mesh adaptivity. Therein, the nodal points of a finite element mesh serve as design variables in an optimization problem that aims to minimize a cost functional with respect to different constraints. In order to avoid the occurrence of oscillating boundaries in the optimal design trials, we generate the respective design updates through solving a series of fictitious boundary value problems on an updated Lagrangian configuration. The design update process is accompanied by adaptive mesh refinement to obtain more accurate results for the structural analysis and the design sensitivity analysis. For the mesh adaptation process, we consider r -adaptive node relocation, h -adaptive mesh refinement and a combination of both approaches. In each case, the derivations rest upon the notion of material residual forces induced by finite element discretization and the coupling of shape optimization and mesh adaptivity is of intermittent type, i.e. mesh adaptivity is invoked once a certain number of design updates has been generated. We examine the benefits of the proposed method on the basis of some numerical examples in comparison to the same shape optimization method not involving adaptive mesh refinement where we evaluate the corresponding numerical costs, the gain in accuracy and the effects on the optimal shapes being obtained.

DARK HALOS AND ELLIPTICAL GALAXIES AS MARGINALLY STABLE DYNAMICAL SYSTEMS

The origin of equilibrium gravitational configurations is sought in terms of the stability of their trajectories, as described by the curvature of their Lagrangian configuration manifold of particle positions --- a context in which subtle spurious effects originating from the singularity in the two body potential become particularly clear. We focus on the case of spherical systems, which support only regular orbits in the collisionless limit, despite the persistence of local exponential instability of $N$-body trajectories in the anomalous case of discrete point particle representation even as $N \rightarrow \infty$. When the singularity in the potential is removed, this apparent contradiction disappears. In the absence of fluctuations, equilibrium configurations generally correspond to positive scalar curvature, and thus support stable trajectories. A null scalar curvature is associated with an effective, averaged, equation of state describing dynamically relaxed equilibria with marginally stable trajectories. The associated configurations are quite similar to those of observed elliptical galaxies and simulated cosmological halos, and are necessarily different from the systems dominated by isothermal cores, expected from entropy maximization in the context of the standard theory of violent relaxation. It is suggested that this is the case because a system starting far from equilibrium does not reach a 'most probable state' via violent relaxation, but that this process comes to an end as the system finds and (settles in) a configuration where it can most efficiently wash out perturbations. We explicitly test this interpretation by means of direct simulations.

Periodic orbits in the restricted four-body problem with two equal masses

The restricted (equilateral) four-body problem consists of three bodies of masses m 1, m 2 and m 3 (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh’s critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them.