Balance Of Linear Momentum Research Articles

A Multilayer Shallow Water Approach for Polydisperse Sedimentation with Sediment Compressibility and Mixture Viscosity

A three-dimensional multilayer shallow water approach to study polydisperse sedimentation and sediment transport in a viscous fluid is presented. The fluid is assumed be loaded with finely dispersed solid particles that belong to a finite number of species that differ in density and size. The model formulation allows one to recover the global mass and linear momentum balance laws of the mixture. The model incorporates compressibility of the sediment and viscosity of the mixture through a viscous stress tensor. As a consequence of a dimensional analysis applied to the global mass conservation and linear momentum balance equations, the horizontal components of the compression term and the horizontal terms of the viscous stress tensor may be neglected. This results in a final model that is vertically consistent with the classical one-dimensional vertical model. Numerical simulations illustrate the coupled solids volume fraction and flow fields in various scenarios and the effect of the compressibility and viscosity terms. Various bottom topographies give rise to recirculation of the fluid and high solids volume fractions on the bottom.

Stability analysis of flow of active extensile fibers in confined domains.

In this article, we study shear flow of active extensile filaments confined in a narrow channel. They behave as nematic liquid crystals that we assumed are governed by the Ericksen-Leslie equations of balance of linear and angular momentum. The addition of an activity source term in the Leslie stress captures the role of the biofuel prompting the dynamics. The dimensionless form of the governing system includes the Ericksen, activity, and Reynolds numbers together with the aspect ratio of the channel as the main driving parameters affecting the stability of the system. The active system that guides our analysis is composed of microtubules concentrated in bundles, hundreds of microns long, placed in a narrow channel domain, of aspect ratios in the range between 10-2 and 10-3 dimensionless units, which are able to align due to the combination of adenosine triphosphate-supplied energy and confinement effects. Specifically, this work aims at studying the role of confinement on the behavior of active matter. It is experimentally observed that, at an appropriately low activity and channel width, the active flow is laminar, with the linear velocity profile and the angle of alignment analogous to those in passive shear, developing defects and becoming chaotic, at a large activity and a channel aspect ratio. The present work addresses the laminar regime, where defect formation does not play a role. We perform a normal mode stability analysis of the base shear flow. A comprehensive description of the stability properties is obtained in terms of the driving parameters of the system. Our main finding, in addition to the geometry and magnitude of the flow profiles, and also consistent with the experimental observations, is that the transition to instability of the uniformly aligned shear flow occurs at a threshold value of the activity parameter, with the transition also being affected by the channel aspect ratio. The role of the parameters on the vorticity and angular profiles of the perturbing flow is also analyzed and found to agree with the experimentally observed transition to turbulent regimes. A spectral method based on Chebyshev polynomials is used to solve the generalized eigenvalue problems arising in the stability analysis.

A novel linear elastic constitutive model for continuum-kinematics-inspired peridynamics

Javili et al. recently reported a continuum-kinematics-inspired peridynamic model, in which theoretical aspects regarding the balance of linear and angular momentum and other conservation principles are considered. However, the analytical formulation of the model constants and the microelastic potential energy functions and numerical implementation were not defined. In this paper, a novel linear elastic constitutive model is proposed for the continuum-kinematics-inspired peridynamics by introducing specific expressions for various interaction potentials. The one-neighbor interaction potential equivalent to conventional bond-based interaction potential is utilized to account for the constitutive relationship within line elements between two material points. In contrast, the two- and three-neighbor interaction potentials are employed to consider the areal and volumetric effects under general mechanical loads. Three relevant material parameters are introduced and derived from energy equivalence to a classical linear elastic continuum mechanics model. Equipped with the three types of interaction potentials, the novel continuum-kinematics-inspired peridynamics is extended from classical bond-based peridynamics, wherein the two interaction force vectors within a bond are unequal and not parallel to the bond direction, can be regarded as an alternative version of non-ordinary state-based peridynamics. The proposed model is numerically demonstrated to be effective in absolutely eliminating the restriction of the fixed Poisson’s ratio in classical bond-based peridynamics, notably improving the effectiveness of the other enriched bond-based peridynamics in reproducing the elastic deformation of solids subjected to heterogeneous deformation fields and completely removing the numerical oscillations in non-ordinary state-based peridynamics.

Free Convection in a Vertical Slit Micro-channel with Super-hydrophobic Slip and Temperature Jump Conditions

The main objective of the convective incompressible fluid flow placed in a magnetic field, with vertical motion through an isothermally heated porous channel. The Super-hydrophobic the slip was applied on one wall along with temperature jump, while there is no slip on another wall. The analytical results are discussed qualitatively for the parameters which were on the mathematical formulation on the basis of the model equations designed for the linear momentum and energy balance. Decreasing the value of the slip parameter is noticed due to decreasing in temperature. The effects of heat source and porous medium on the micro channel flow is analysized in detail. Coparesion of current results is made with the earlier work. The effect of Dray parameter is to increase the fluid velocity in the channel.

Preconditioning strategies for vectorial finite element linear systems arising from phase-field models for fracture mechanics

Phase-field models are frequently adopted to simulate fracture mechanics problems in the context of the finite element method. To depict fracture, this method involves solving a coupled set of Helmholtz-like damage-field equation and augmented linear momentum balance equation. Solutions to these coupled equations are then used as descriptions of crack propagation phenomena within solids. However, this method imposes a constrain of using extremely fine meshing for properly predicting cracks. For practical problems of interest, this very often leads to linear systems with large sizes that have to be repetitively assembled and solved. As such, iterative solution procedures such as the Krylov subspace-based methods for solving these large linear systems within the framework of serial/parallel computing environments become mandatory to obtain results in a feasible time. In this work, the vectorial finite discretization for a hybrid phase-field formulation – a monolithic solving scheme – is presented. The underlying nonlinearity present in the coupled set of equations of the hybrid phase-field model is dealt through Picard iteration that helps to preserve the symmetry of the linearized system to solve. Due to the symmetric positive definite nature of the finite element linear systems obtained for this problem, the conjugate gradient method makes a standard choice of iterative solution algorithm. In this article, to improve convergence rates, consequently time to solution, of the conjugate gradient method applied to crack propagation problems, different preconditioning strategies are analyzed, tuned, and discussed. Brittle fracture benchmarks are used to measure the performance of preconditioners which are then applied to massively parallel simulations with millions of unknowns. A series of numerical experiments show that the algebraic multigrid preconditioner is well suited for solving the phase-field model for fracture, being superior to the Jacobi and the block Jacobi preconditioning in all regards: ease of solving the problem, iterations to converge, time to solution, and parallel scaling on more than a thousand processes.

A plausible extension of standard penalty, streamline upwind and immersed boundary techniques to the improved element-free Galerkin-based solution of incompressible Navier–Stokes equations

The present work has been conducted in order to propose the extension of standard penalty and stabilization techniques to the improved element-free Galerkin (IEFG) method, for the numerical solution of incompressible fluid flow problems. In principle, the numerical procedures to be implemented in this communication have been conceived for finite element method (FEM)-based solutions, and these include the reduced integration penalty method (RIPM), the streamline upwind Petrov–Galerkin (SUPG) scheme, and a penalty-based immersed boundary method (PBIBM) for the imposition of essential boundary conditions along internal fluid–solid interfaces. The linear momentum balance and mass conservation equations have been coupled via the RIPM, in order to obtain a global weak formulation where the IEFG model is entirely developed in terms of improved moving least squares (IMLS) approximations of the velocity field. A detailed explanation concerning the appropriate extension of both the RIPM and SUPG procedures to the context of IEFG formulations, has also been provided. The resulting formulation has been applied to the solution of two well-known benchmark problems: i) Lid-driven square cavity flow, and ii) Flow past a fixed cylinder. Regarding the flow past a fixed cylinder benchmark problem, the fluid–solid interaction has been imposed as an internal immersed boundary condition via the PBIBM. The feasibility and reliability of implementing the RIPM, SUPG and PBIBM procedures in the IEFG formulation, have been proven by comparison with experimental and mesh-based numerical results reported in the literature. The results obtained in this study have revealed that a proper extension of the aforementioned penalty and stabilization techniques to the IEFG formulation, allows the achievement of accurate and stable numerical results during the solution of incompressible fluid-dynamics problems.

Microscale diffusion-mechanics model for a polymer-based solid-state battery cathode

A non-linear microscale diffusion-mechanics model combining mass transport and linear momentum balance equations, with elasto-viscoplastic polymer constitutive law and interfacial traction-separation law is proposed to provide a new insight into the effects of viscoplasticity and interfacial damage on the in situ diffusive-mechanical behaviour of a polymer-based cathode for a solid-state battery (SSB). Diffusion and mechanics are coupled through two mechanisms: (1) active particle (AP) volumetric change dependence on Li concentration, and (2) interfacial flux dependence on mechanical opening. The model is resolved for a simple cathode microstructure using an axisymmetric unit cell concept, and integrated with the non-linear finite-element solver ABAQUS with the help of its user subroutines (UMAT and UINTER). Finite-element simulations reveal that plastic deformations of the polymer due to volumetric changes of the AP reduce the value of the interfacial opening displacement, which is desirable for maintaining interfacial flux. The results also demonstrate that slower battery charging rates may lead to a softer polymer response, and thus a smaller interfacial gap. Moreover, a comparison between the linear elastic and current elasto-viscoplastic models for the polymer electrolyte shows that even 5% volumetric shrinkage of the AP leads to an overprediction of the interfacial opening with the linear elastic material law, which limits its validity in modelling polymer-based SSBs.

Brinkman equation in reactive flow: Contribution of each term in carbonate acidification simulations

This article studies the individual contribution of each term in the Navier-Stokes-Darcy (Brinkman) equation to the fluid linear momentum balance and the consequences of opting for the Darcy or Brinkman equations in 3D reactive flow simulations of a vuggy carbonate. A two-scale continuum model was used to simulate the acidification of a carbonate sample using the Darcy and Brinkman equations. Rock heterogeneity is considered in the initial porosity and permeability fields, which are calculated directly from micro-CT images. Simulations showed that the Darcy equation predicts the acid breakthrough earlier in most cases. The convective acceleration term in the Brinkman equation plays an important role in describing the fluid momentum balance at high injection velocities, and this is a clear evidence against the use of the Darcy equation to model reactive flow in such conditions.

Asymptotic analysis of compression sensing in ionic polymer metal composites: The role of interphase regions with variable properties

Ionic Polymer Metal Composites (IPMCs) consist of two noble metal electrodes plating an electroactive polymeric membrane, referred to as ionomer, which is electroneutralised by a solvent including mobile ions. The IPMC manufacturing leads to thin interphase regions next to the electrodes, the so-called Composite Layers (CLs), in which metal atoms occupy interstitial sites within the ionomer. In this work we extend previous efforts of our group on IPMC compression sensing to include the important effect of CLs, where large variations of the electrochemical properties occur. In IPMC compression sensing the application of a through-the-thickness displacement leads to a shortcircuit electric response, here assumed to be governed by a linearised <i>modified</i> Poisson-Nernst-Planck (PNP) system of partial differential equations (PDEs), to be solved for the time-evolving electric potential and mobile ions concentration as functions of the displacement field evaluated through the linear momentum balance. The variation of material properties in the CLs requires the simultaneous integration of the governing system of PDEs in three regions: the membrane and the two CLs. To this purpose, we resort to the perturbative method of matched asymptotic expansions. Except for a numerical inverse Laplace transform, this allows us to obtain an analytical solution through which we establish an equivalent circuit model elucidating the main features of the IPMC sensing behaviour. We validate and discuss the analytical solution through comparison with finite element analyses, whereby we also numerically solve the nonlinear modified PNP systems fully coupled with the linear momentum balance accounting for the electrochemical stresses. We finally provide some insight into the role of CLs in the IPMC sensing behaviour, by assessing its sensitivity to some key parameters. We expect the obtained results to aid the design of optimised IPMC sensors.

Riemannian and Euclidean material structures in anelasticity

In this paper, we discuss the mechanics of anelastic bodies with respect to a Riemannian and a Euclidean geometric structure on the material manifold. These two structures provide two equivalent sets of governing equations that correspond to the geometrical and classical approaches to non-linear anelasticity. This paper provides a parallelism between the two approaches and explains how to go from one to the other. We work in the setting of the multiplicative decomposition of deformation gradient seen as a non-holonomic change of frame in the material manifold. This allows one to define, in addition to the two geometric structures, a Weitzenböck connection on the material manifold. We use this connection to express natural uniformity in a geometrically meaningful way. The concept of uniformity is then extended to the Riemannian and Euclidean structures. Finally, we discuss the role of non-uniformity in the form of material forces that appear in the configurational form of the balance of linear momentum with respect to the two structures.

Evaluation of dynamic behavior of a falling porous magnesium particle over the ignition and combustion processes

In this research study, combustion of micron-sized porous magnesium particle which freely falls into an infinite hot oxidizer medium is investigated. To examine the particle behavior during the process, acceleration and all forces acting on it including mass, buoyancy and drag forces are considered. The effects of produced magnesium oxide and both types of porosity consisting of surface and volume porosities are applied in mathematical modeling. The governing equations including magnesium particle mass continuity, linear momentum balance and energy conservation are numerically solved. Afterward, the impacts of important parameters on combustion characteristics are studied. Results show that by considering both types of surface and volume porosities, combustion time decreases compared to the cases in which one of these parameters is employed. With increasing the particle diameter and its porosity factor, velocity and acceleration enhance. Moreover, during the combustion process, mass and drag forces of magnesium oxide and its radius variations have the most effective contributions in total acceleration with the shares of 39.8%, 30.07% and 12.8%, respectively. Also, contribution of magnesium oxide in total acceleration is 4.8 times greater than that of magnesium.

Thence the moment of momentum

AbstractFifty years have passed since Truesdell's seminal paper on the origin and status of the balance for the moment of momentum was published in ZAMM. It is time to take stock: Important new developments in the theory of generalized continua with internal degrees of freedom and some fascinating fundamental applications need to be pointed out. Is there new evidence from classical papers regarding its independence from the balance of linear momentum? Can micropolar theory be used to “explain” electromagnetism? How is the conservation of the moment of momentum viewed in today's physics textbooks? In this paper an attempt is made to answer these and many more interesting questions.

High-Order Accurate Methods for the Numerical Analysis of a Levitation Device

This paper establishes different axisymmetric and two-dimensional models for a levitation device. Therein, the Maxwell equations are combined with the balance of linear momentum. Different possible formulations to describe the Maxwell equations are presented and compared and discussed in the example. A high order finite element discretization using Galerkin’s method in space and the generalized Newmark $$-\alpha$$ method in time are developed for the electro-magneto-mechanical approach. Several studies on spatial and temporal discretization with respect to convergence will be investigated. In addition, the boundary influences and the domain size with respect to the levitation device are also examined.

A Shell Description of Beams Incorporating Transverse Thickness Strain Energies for Receding Contact

Abstract The geometrically exact nonlinear deflection of a beamshell is considered here as an extension of the formulation derived by Libai and Simmonds (1998, The Nonlinear Theory of Elastic Shells, Cambridge University Press, Cambridge, UK) to include deformation through the thickness of the beam, as might arise from transverse squeezing loads. In particular, this effect can lead to receding contact for a uniform beamshell resting on a smooth, flat, rigid surface; traditional shell theory cannot adequately such behavior. The formulation is developed from the weak form of the local equations for linear momentum balance, weighted by an appropriate tensor. Different choices for this tensor lead to both the traditional shell equations corresponding to linear and angular momentum balance, as well as the additional higher-order representation for the squeezing deformation. In addition, conjugate strains for the shell forces are derived from the deformation power, as presented by Libai and Simmonds. Finally, the predictions from this approach are compared against predictions from the finite element code abaqus for a uniform beam subject to transverse applied loads. The current geometrically exact shell model correctly predicts the transverse shell force through the thickness of the beamshell and is able to describe problems that admit receding contact.

Nature and magnitude of operating forces in a horizontal bend conveying gas-liquid slug flows

Operating forces and magnitude of loads from gas-liquid slug flows exerted on a horizontally orientated 90o bend are investigated. The distributed forces are either Newtonian, associated with the fluids motion or Configurational, inherent to the internal distributions of the phases. The forces are derived through the conventional balances of mass and linear momentum arising from the volume of fluid (VOF) description of gas-liquid flows. The study uses the integral form of the momentum balance to estimate the operating forces budget. Invoking dynamical time scales separation discloses the connection of the Lamb vector (vortex-force) to the local time rate of momentum. An interesting outcome being an explicit expression for Favre-Reynolds stress that reveals the contribution of void fraction fluctuations in the redistribution of the stress across the interface. Numerical simulations are performed to determine the magnitude of Newtonian loads on bend using a segmented domain technique to represent the fully established slug flow regime. The time-dependent traces of the relevant flow variables such as liquid hold-up, flow rates and resultant forces on the bend are recorded and analysed. Compared to the isotropic component, the deviatoric stresses are shown to have a marginal contribution to the total forces. It is also shown that loading cycles on bends are much higher than slugging cycles; this is an important feature for the structural integrity assessment of pipelines with bends.

Chemo-mechanical coupling and material evolution in finitely deforming solids with advancing fronts of reactive fluids

A new stabilized method is presented for coupled chemo-mechanical problems involving chemically reacting fluids flowing through deformable elastic solids. A mixture theory model is employed wherein kinematics is represented via an independent set of balance laws for each of the interacting constituents. A significant feature of the mixture model is the interactive force field in the momentum balance equations that couples the constituents implicitly at the level of the governing system of equations. The constitutive relations for the constituents in the mixture model are based on maximization of the rate of entropy production. Since each constituent is not discretely modeled and the interactive effects are mathematically coupled at the local continuum level, the resulting system serves as a physics-based reduced-order model for the complex microstructure of the material system. When constitutive equations are substituted into the balance laws, they give rise to a system of coupled nonlinear PDEs. Evolving nonlinearity and coupled chemo-mechanical effects give rise to spatially localized phenomena, namely boundary layers, shear bands, and steep gradients that appear at the reaction fronts. For large reaction rates, the balance of mass of the fluid becomes a singularly perturbed equation (reaction-dominated), which may exhibit boundary and/or internal layers. Likewise, for large reaction rates and/or low diffusivity, the balance of linear momentum for the fluid constituent also becomes a singularly perturbed PDE. Presence of these features in the solution requires stable numerical methods, and we present a variational multiscale (VMS)-based stabilized finite element method for the initial-boundary value problem. Mathematical attributes of the method are investigated via a range of numerical test cases that involve diffusion of chemically reacting fluids through nonlinear elastic solids. Enhanced stabilization features and higher spatial accuracy of the models and the methods are highlighted.

FFT-based solver for higher-order and multi-phase-field fracture models applied to strongly anisotropic brittle materials

This paper presents the application of a fast Fourier transform (FFT) based method to solve two phase field models designed to simulate crack growth of strongly anisotropic materials in the brittle regime. By leveraging the ability of the FFT-based solver to generate solutions with higher-order and global continuities, we design two simple algorithms to capture the complex fracture patterns (e.g. sawtooth, and curved crack growth) common in materials with strongly anisotropic surface energy via the multi-phase-field and high-order phase-field frameworks. A staggered operator-split solver is used where both the balance of linear momentum and the phase field governing equations are formulated in the periodic domain. The unit phase field of the initial failure region is prescribed by the penalty method to alleviate the sharp material contrast between the initial failure region and the base material. The discrete frequency vectors are generalized to estimate the second and fourth order gradients such that the Gibbs effect near shape interfaces or jump conditions can be suppressed. Furthermore, a preconditioner is adopted to improve the convergence rate of the iterative linear solver. Three numerical experiments are used to systematically compare the performance of the FFT-based method in the multi-phase-field and high-order phase-field frameworks.

A phase-field model for fracture of unidirectional fiber-reinforced polymer matrix composites

This study presents a crack phase-field approach for anisotropic continua to model, in particular, fracture of fiber-reinforced matrix composites. Starting with the variational formulation of the multi-field problem of fracture in terms of the deformation and the crack phase fields, the governing equations feature the evolution of the anisotropic crack phase-field and the balance of linear momentum, presented for finite and small strains. A recently proposed energy-based anisotropic failure criterion is incorporated into the model with a constitutive threshold function regulating the crack initiation in regard to the matrix and the fibers in a superposed framework. Representative numerical examples are shown for the crack initiation and propagation in unidirectional fiber-reinforced polymer composites under Mode-I, Mode-II and mixed-mode bending. Model parameters are obtained by fitting to sets of experimental data. The associated finite element results are able to capture anisotropic crack initiation and growth in unidirectional fiber-reinforced composite laminates.

Thermo‐hydro‐mechanical modeling of unsaturated soils using isogeometric analysis: Model development and application to strain localization simulation

SummaryThis study presents a thermo‐hydro‐mechanical (THM) model of unsaturated soils using isogeometric analysis (IGA). The framework employs Bézier extraction to connect IGA to the conventional finite element analysis (FEA), featuring the current study as one of the first attempts to develop an IGA‐FEA framework for solving THM problems in unsaturated soils. IGA offers higher levels of interelement continuity making it an attractive method for solving highly nonlinear problems. The governing equations of linear momentum, mass, and energy balance are coupled based on the averaging procedure within the hybrid mixture theory. The Drucker‐Prager yield surface is used to limit the modified effective stress where the model follows small strain, quasi‐static loading conditions. Temperature dependency of the surface tension is implemented in the soil‐water retention curve. Nonuniform rational B‐splines (NURBS) basis functions are used in the standard Galerkin method and weak formulations of the balance equations. Displacement, capillary pressure, gas pressure, and temperature are four independent quantities that are approximated by NURBS in spatial discretization. The framework is used to simulate strain localization in an undrained dense sand subjected to plane strain biaxial compression under different temperatures and displacement velocities. Results show that an increase in the displacement rate leads to reduction in the equivalent plastic strain while an increase in the temperature leads to an increase in the equivalent plastic strain. The findings suggest that the proposed IGA‐based framework offers a viable alternative for solving THM problems in unsaturated soils.

A mixed least‐squares finite element formulation with explicit consideration of the balance of moment of momentum, a numerical study

AbstractImportant conditions in structural analysis are the fulfillment of the balance of linear momentum (vanishing resultant forces) and the balance of angular momentum (vanishing resultant moment), which is not a priori satisfied for arbitrary element formulations. In this contribution, we analyze a mixed least‐squares (LS) finite element formulation for linear elasticity with explicit consideration of the balance of angular momentum. The considered stress‐displacement (σ − u) formulation is based on the squared L2(ℬ)‐norm minimization of the residuals of a first‐order system of differential equations. The formulation is constructed by means of two residuals, that is, the balance of linear momentum and the constitutive equation. Motivated by the crucial point of weighting factors within LS formulations, a scale independent formulation is constructed. The displacement approximation is performed by standard Lagrange polynomials and the stress approximation with Raviart‐Thomas functions. The latter ansatz functions do not a priori fulfill the symmetry of the Cauchy stress tensor. Therefore, a redundant residual, the balance of angular momentum ((x − x0) × (divσ + f) + axl[σ − σT]), is introduced and the results are discussed from the engineering point of view, especially for coarse mesh discretizations. However, this formulation shows an improvement compared to standard LS σ − u formulations, which is considered here in a numerical study.