The application of the extremum principles for a Bingham solid

Summary An extension of the extremum principle concerned with statically admissible stress fields for boundary value problems of an incompressible rigid visco-plastic (Bingham) solid is derived. This extension is an inequality that gives a lower bound for the rate of work of the unknown surface tractions in certain visco-plastic boundary value problems. Lower bounds are obtained for the torque required to twist a prismatic visco-plastic bar of square crossection and these lower bounds are presented along with upper bounds obtained from an inequality that was derived in a previous paper¹).Nomenclature k yield stress in pure shear - coefficient of viscosity - _ij stress tensor referred to rectangular cartesian axes Ox _i - s _ij stress deviator tensor - T _i surface traction - J (1/2s _ij s _ij )^1/2 - v _i velocity vector - _ij = rate of deformation tensor - n _i outward unit normal to surface - I (2 _{ij ij} )^1/2 - [v] magnitude of a velocity discontinuity in flow of a rigid perfectly plastic solid - S surface - V volume - S _t part of surface upon which T _i is prescribed - S _v part of surface upon which v _i is prescribed - S _d surface of velocity discontinuity in flow of a rigid plastic solid - x, y, z rectangular cartesian coordinates - u, v, w velocity components in the x, y and z directions respectively - rate of twist per unit length - T torque     

Rate variational extremum principles for finite elastoplasticity

Dual variational extremum principles for rate problems of classical elastoplasticity at finite deformation are studied in Updated Lagrangian rate forms. It is proved that the convexity of the variational functionals are closely related to a so-called gap function, which plays an important role in nonlinear variational problems.     

Some asymptotic extremum principles for magnetohydrodynamic pipe flow

Upper and lower bounds are obtained for the mass flow rate for the steady flow of a conducting liquid along a pipe of arbitrary cross section under the influence of a uniform transverse magnetic field. The extrema are constructed by first establishing an identity which is then used in some simple inequalities. It is shown how these bounds can be used to derive an asymptotic expansion for the flow rate at large Hartmann number. Two illustrations are included.     

Fluid boundary of a viscoplastic Bingham flow for finite solid deformations

The modelling of viscoplastic Bingham fluids often relies on a rheological constitutive law based on a “plastic rule function” often identical to the yield criterion of the solid state. It is also often assumed that this plastic rule function vanishes at the boundary between the solid and fluid states, based on the fact that it is true in the limit of small deformations of the solid state or for simple yield criteria. We show that this is not the case for finite deformations by considering the example of a two state flow on a tilted plane where the solid state is described by a Neo-Hookean model with a Von Mises yield criterion. This opens new approaches for the modelling and the computation of the fluid state boundaries.     

Dual extremum principles for geometrically exact finite strain beams

Both necessary and sufficient conditions for the existence of two complementary-dual extremum principles for geometrically exact finite strain (one-dimensional) beam models are investigated by means of two different approaches. One is based on the results published by Gao and Strang, and the other relies on the approach proposed by Noble and Sewell. While the former is limited to beam models restricted to moderate large deformations, the latter is valid for arbitrarily large deformations (and strains). The numerical implementation of the complementary-dual extremum principles can lead to simple true global upper bounds of the error of the approximate solutions.     

Stability and extremum principles for post yield analysis of finite plasticity

The post yield behavior of rigid-perfectly plastic solids at the collapse load is studied based on the finite deformation theory. By using the general duality theory developed by Gao-Strang (1989), a global stability criteria is proposed and a pair of dual extremum principles, expressed in terms of displacements, displacement rates and the Kirchhoff stresses are established for plastic collapse analysis. It is proved that under large deformations, the existence of the plastic limit state at the collapse load depends on the directional derivative of a so-called complementary gap function. The application to the nonlinear plastic collapse theory yields a pair of dual bounding theorems for limit loading factor associated with any transient displacement of the deformed body when the global extremum criteria are satisfied. Dedicated to Professor Y.K. Cheung on the occasion of his 60th birthday     

Dual extremum principles in the dynamics of rigid perfectly-plastic bodies undergoing finite deformations

A pair of dual extremum principles is derived for a rigid perfectly-plastic body subjected to dynamic loading. No restrictions are placed on the magnitude of the deformation. A Lagrangian formulation is used, and the dual principles are obtained in a systematic manner by using a saddle-shaped functional and a pair of operators adjoint to each other, to generate the governing equations and inequalities. This procedure was developed by B. Noble and M. J. Sewell (1972) in their study of dual extremum principles in applied mathematics. It is shown that the principles derived here are closely connected to those given by M. Capurso (1972c) for the case of small strains and large displacements.     

The derivation of dual extremum and complementary stationary principles in geometrical non-linear shell theory

Summary In non-linear elasticity dual extremum principles can be formulated for some class of elastic deformations, for which uniqueness of the solution is assured. These results are used in the present paper to derive extremal variational principles for geometrical non-linear shells with moderate rotations. Furthermore two complementary variational principles are considered, which are stationary principles without any extremum property. The proposed theorems are valid also for the special cases of linear plates and shells, for the non-linear von Kármán plate theory and for non-linear Donnell-Marguerre type shells.

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Übersicht In der nichtlinearen Elastizitätstheorie lassen sich duale Extremalprinzipe für solche elastische Verformungen herleiten, für die Eindeutigkeit der Lösung gewährleistet ist. Diese Resultate werden in der vorliegenden Arbeit benutzt, um für geometrisch-nichtlineare Schalen mit moderaten Rotationen Extremalprinzipe zu erhalten. Darüber hinaus werden zwei komplementäre Variationstheoreme angegeben, die Stationaritätsprinzipe ohne Extremaleigenschaft sind. Die vorgeschlagenen Verfahren gelten auch für die Sonderfälle der linearen Platten- und Schalentheorie, für die nichtlineare von Kármánsche Plattentheorie sowie für die nichtlineare Donnell-Marguerresche Schalentheorie.

     

An extremum variational principle and error estimate procedure for radial flow of a viscous fluid between porous discs

A global extremum variational principle for radial flow of a viscous fluid between porous discs is presented. For approximate solutions an error estimate procedure, based on the value of the functional, is developed. An approximate solution via the Ritz method is obtained.     

Start-up Flow of a Bingham Fluid in a Pipe

We present an analytical solution of axisymmetric motion for a Bingham fluid initially at rest subjected to a constant pressure gradient applied suddenly. Using the Laplace transform, we obtain expressions which allow the calculation of the instantaneous velocity, plug radius and rate of flow as a function of time. We also give a relation for the shear stress in the plug and in the region where the behaviour of the fluid is Newtonian.     

Numerical Flow Simulation for Bingham Plastics in a Single-Screw Extruder

Numerical simulations have been performed concerning the operation of a single-screw extruder, pumping a Bingham plastic under isothermal, developed flow conditions. Under the assumption of sufficiently low Reynolds numbers, inertia effects are neglected. The singular rheological behavior of the Bingham plastic is considered as the limiting case within a class of generalized Newtonian liquids with smooth constitutive equations. The validation of this regularization process is shown for a related flow problem where the Bingham solution is known analytically. A mixed finite-element method is applied to the flow in the screw-extruder to reduce the equations of motion, the continuity equation, and the regularized constitutive equation to a set of nonlinear algebraic equations, which are solved using a Newton method. In particular, the pumping characteristics of a given screw geometry are extracted from the finite-element calculations, i.e., the dependence of the volumetric flow rate and of the power requirement on the axial pressure drop, on the screw speed, and on the rheological parameters. Calculated flow fields clearly show the size and position of regions in the extruder channel where the Bingham plastic behaves like a solid. Received: 12 December 1995 and accepted 12 November 1996     

An implicit solution of bi-penalty approximation with orthogonality projection for the numerical simulation of Bingham fluid flow

1.IntroductionItisamajordiffct.encefi-omtheNewtonnuidflowthattheBinghammodelofNonNewtonfluidflowischaracterizedbytwoparameters:ayieldstressandaviscosity.WhenthestressoftheBinghalnmaterialbelowtheyieldstress,materialisrigidotherwisethequasiNewtolliannowresultstll:'71.Hence,therearesomeofthefloating"rigidcores"involvedintheBinghamfluidfloworsomeofthe'rigidcores"attachedtotheboundaries,inwhichthelocationsalldshapesofthese"rigidcores"maychangeforthetransientBinghamfluid,flow.ThisBingllammodelh…     

An extension of the TV-HLL scheme for multi-dimensional compressible flows

This paper investigates a very simple method to numerically approximate the solution of the multi-dimensional Riemann problem for gas dynamics, using the literal extension of the Toro Vazquez-Harten Lax Leer (TV-HLL) scheme as its basis. Indeed, the present scheme is obtained by following the Toro–Vazquez splitting, and using the HLL algorithm with modified wave speeds for the pressure system. An essential feature of the TV-HLL scheme is its simplicity and its accuracy in computing multi-dimensional flows. The proposed scheme is carefully designed to simplify its eventual numerical implementation. It has been applied to numerical tests and its performances are demonstrated for some two-dimensional and three-dimensional test problems.     

An exact analytical solution for creeping Dean flow of Bingham plastics through curved rectangular ducts

In this paper, an exact analytical solution for creeping flow of Bingham plastic fluid passing through curved rectangular ducts is presented for the first time. The closed form of axial velocity distribution, flow resistance ratio, and wall shear stress are derived using bounded Fourier transformation. An extensive investigation on mutual effects of Hedstrom number, curvature ratio, and aspect ratio is conducted. The results indicate that a drag reduction is caused in the flow field by increasing the Hedstrom number. It is shown that unlike the Newtonian creeping Dean flow, the critical aspect ratio (an aspect ratio in which the flow resistance ratio is independent from curvature ratio) does not exist at large enough Hedstrom numbers. Analytical solution also indicated that as Hedstrom number is increased, the value of Poiseuille number is enhanced, and unlike the Newtonian flows, the value of Poiseuille number is not zero at edges of cross section.     

Peristaltic flow of a Bingham fluid in a channel

We study the peristaltic transport of a Bingham fluid in a channel with small aspect ratio whose walls behave as a periodic traveling wave. The governing equations in the unyielded phase are obtained writing the integral formulation for the momentum balance. As shown in Fusi et al. (2015), this approach allows to overcome the so-called “lubrication paradox” which may arise in the thin film approximation. We consider the case in which the inlet flux is prescribed and the one in which the flow is driven by a given pressure drop. In both cases the solution of the problem is determined solving a nonlinear integral equation for the yield surface. We perform some numerical simulations to illustrate the behavior of the yield surface, assuming that the traveling wave describing the peristaltic motion has a sinusoidal shape.     

Deformation of a Bingham viscoplastic fluid in a plane confuser

A review is given to and comprehensive numerical-analytic study is carried out of the problem of steady Bingham viscoplastic flow in a plane confuser. The solution is constructed in the first approximation with the yield stress as a small parameter and the solution of the Jeffery-Hamel problem (steady radial motion of an incompressible viscous material in a plane confuser) as the zero-order approximation. The numerical analysis is based on the modified accelerated-convergence method proposed earlier by the authors. The bifurcations of the deformation pattern occurring when the parameters reach some critical values are discussed and commented on. The asymptotic boundaries of the rigid zones that appear at infinity upon perturbation of the yield stress are determined __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 4, pp. 3–45, April 2006.     

Bingham Viscoplastic as a Limit of Non-Newtonian Fluids

A new formulation is proposed for the equations of the Bingham viscoplastic. Global existence of x--periodic solutions is proved. A uniqueness theorem is established in the two-dimensional case. A relation with the G. Duvaut--J. L. Lions variational inequality is discussed, and a result on equivalence is obtained. The question of interaction between fluid-rigid phases is studied when the initial state is rigid. A free-boundary problem that describes two-phase one-dimensional flows is considered.