On strain localization analysis of elastoplastic materials at finite strains

The problem of strain localization into planar bands of rate-independent elastoplastic solids with smooth yield surface and plastic potential is analyzed reconsidering the work of Rice and Rudnicki in 1980. It is shown that strain localization with elastic unloading on one side of the band first becomes possible either at localization in the comparison solid corresponding to the loading branch of the constitutive equation or at the snap-back threshold. The elastic unloading is shown to start from the condition of neutral loading, occurring in fact at the onset of localization. The case of localization with elastic unloading into the band and plastic loading outside that was not considered by Rice and Rudnicki is taken into account.     

A note on strain localization for a class of non-associative plasticity rules

Summary Explicit solutions for the formation of discontinuity bands are obtained, for a class of non-associative flow rules. Specialization to particular yield functions for pressure sensitive, dilatant or compactive materials is given.

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Bemerkungen zur Lokalisierung der Verformungen für eine Klasse von nicht assoziierten plastischen Fließgesetzen

Übersicht Es werden explizite Lösungen für die Bildung von Unstetigkeitsflächen bei einer Klasse von nicht assoziierten plastischen Fließgesetzen hergeleitet. Diese werden insbesondere für einige spezielle Fließgesetze diskutiert, die zur Beschreibung von druckempfindlichen, dilatierenden oder kontrahierenden Materialien geeignet sind.

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Presented at the workshop on Numerical Methods for Localization and Bifurcation of Granular Bodies, held at the Technical University of Gdansk (Poland), September 25–30, 1989     

On the interplay between strain rate and strain rate sensitivity on flow localization in the dynamic expansion of ductile rings

In this work a stability analysis on flow localization in the dynamic expansion of ductile rings is conducted. Within a 1-D theoretical framework, the boundary value problem of a radially expanding thin ring is posed. Based on a previous work, the equations governing the stretching process of the expanding ring are derived and solved using a linear perturbation method. Then, three different perfectly plastic material constitutive behaviours are analysed: the rate independent material, the rate dependent material showing constant logarithmic rate sensitivity and the rate dependent material showing non-constant and non-monotonic logarithmic rate sensitivity. The latter allows to investigate the interaction between inertia and strain rate sensitivity on necking formation. The main feature of this work is rationally demonstrate that under certain loading conditions and material behaviours: (1) decreasing rate sensitivity may not lead to more unstable material, (2) increasing loading rate may not lead to more stable material. This finding reveals that the relation between rate sensitivity and loading rate controls the unstable flow growth. Additionally a finite element model of the ring expansion problem is built in ABAQUS/Explicit. The stability analysis properly reflects the results obtained from the numerical simulations. Both procedures, perturbation analysis and numerical simulations, allow for emphasizing the interplay between rate sensitivity and inertia on strain localization.     

On the linear and nonlinear stability analysis in the theory of thin elastic shells

Summary In the frame of the geometrically nonlinear theory of thin elastic shells with moderate rotations a set of consistent equations for the nonlinear stability analysis is derived by application of energy criteria. Some methods of functional analysis are used which enable to prove the symmetry of the stability equations and to calculate bifurcation buckling from linear and nonlinear equilibrium branches and also snap-through buckling loads by variational approximating procedures.

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Über die lineare und nichtlineare Stabilitätsberechnung in der Theorie dünner elastischer Schalen

Übersicht Im Rahmen einer geometrisch-nichtlinearen Theorie dünner Schalen mit moderaten Rotationen werden konsistente Gleichungen zur nichtlinearen Stabilitätsberechnung hergeleitet, wobei von Energiekriterien ausgegangen wird. Die Benutzung einiger Methoden der Funktionalanalysis ermöglicht den Nachweis der Symmetrie der Stabilitätsgleichungen und die Berechnung des Verzweigungs-Beulproblems bei linearen und nichtlinearen Gleichgewichtszuständen sowie die Bestimmung der kritischen Last beim Durchschlagproblem mit Hilfe variationeller Näherungsverfahren.

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Lecture: XVth Inter. Congr. Theor. Appl. Mech., Toronto/Canada, 17.–23. Aug. 1980

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The author is indepted to Docent Dr. habil. W. Pietraszkiewicz, Institute of Fluid-Flow Machinery Gdask, for valuable remarks and the Polish Academy of Science for continuous support.     

The mathematically linear analysis of strain in continua

Summary One-one correspondence is postulated for two coordinate continua. One continuum is regarded as the initially undeformed state of a currently deformed continuum. The two continua are orthogonal trasformations each to the other. The square root of the quadratic metric, when the appropriate self-conjugate stretch dyadic is expressed in its principal form, gives the mathematically linear form to the analysis. The self-conjugate dyadic is expressed as =grad grad P in terms of a scalar potential function P. The physical and mathematical continuity of the strain dyadic is ensured by curl =0.A 4-vector analysis is evolved from a 4-vector quinternion analysis. The 4-vector analysis is of the same form as the usual 3-vector analysis that evolved from Hamilton's quaternion analysis.

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Sommario Si postula una corrispondenza biunivoca per due continui coordinati. Un continuo viene considerato come lo stato inizialmente non deformato di un continuo attualmente deformato.I due continui sono trasformazioni ortogonali uno dell'altro. La radice quadrata della metrica quadratica, quando la relativa diade di dilatazione autoconiugata venga espressa nella sua forma principale dà la forma matematicamente lineare alla analisi.La diade coniugata viene espresse come =grad grad P in termini di una funzione potenziale scalare P. La continuità fisica e matematica della diade di deformazione è assicurata da rot =0.Una analisi tetravettoriale viene sviluppata da analisi di quinternioni tetravettoriali. L'analisi tetravettoriale è della stessa forma dell'analisi trivettoriale che viene vsiluppata dalla analisi quaternionica trivettoriale di Hamilton.

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A generalisation that, when particularised, applies to the finite or infinitesimal straining of elastic bodies, incremental straining of plastic or fluidic bodies in 3-dimensional continua and to the space-time continuum as a 4-dimensional continuum.     

Vibration localization in one-dimensional linear and nonlinear lattices: discrete and continuum models

The phenomenon of vibration localization plays an important role in the dynamics of inhomogeneous and nonlinear materials and structures. The vibration localization can occur in the case of inhomogeneity under the following conditions: (i) the frequency spectrum of the periodic structure includes stopbands, (ii) a perturbation of periodicity is present, and (iii) the eigenfrequency of the perturbed element falls into a stopband. Under these conditions, the energy can be spatially localized in the vicinity of the defect with an exponential decay in the infinity. The influence of nonlinearity can shift frequency into the stopband zone. In the present paper, the localization of vibrations in one-dimensional linear and nonlinear lattices is investigated. The localization frequencies are determined and the attenuation factors are calculated. Discrete and continuum models are developed and compared. The limits of the applicability of the continuum models are established. Analysis of the linear problem has allowed a better understanding of specifics of the nonlinear problem and has led to developing a new approach for the analysis of nonlinear lattices alternative to the method of continualization.     

Instability and localization of plastic flow in shear at high strain rates

Shear band formation in a thermal viscoplastic heat conducting material is described in a simple shear test at high strain rate with inertia effects. The classical perturbation method is discussed, and a new relative perturbation method accounting for non-steadiness of plastic flow is presented. They respectively provide instability and localization criteria which are compared. Furthermore both are compared to available nonlinear exact results and to experimental data. The influence of material parameters, initial imperfections, and boundary conditions is described.     

Bipotential versus return mapping algorithms: Implementation of non associated flow rules

Numerical implementation of constitutive laws involves specific incremental methods. The “return mapping” (Simo and Hughes, 1998) and the “bipotential” (de Saxcé, 1992) are one of those, associated respectively to two different classes of materials: the General Standard Materials (GSM) for the return mapping and the Implicit Standard Materials (ISM) for the bipotential.The objective of this paper is then to compare the implementation of those both methods in the case of non associated flow rules in plasticity.In the first section, the properties of the different previous material classes will be recalled and the methods of “return mapping” and “bipotential” will be detailed. The comparison of both methods is realised on the non linear kinematic hardening rule of Armstrong–Frederick (Armstrong and Frederick, 1966) in a second section and the details are given in a third part. The numerical implementation is realised in Abaqus/Standard 6.11 by the means of a UMat subroutine and the practical simple case of tension–compression is analysed in a last section.     

Boundary regimes with peaking and localization in nonlinear non-Newtonian flow through porous media

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 91–97, July–August, 1991.     

On linear and nonlinear aspects of dynamic mode decomposition

The approximation of reduced linear evolution operator (propagator) via dynamic mode decomposition (DMD) is addressed for both linear and nonlinear events. The 2D unsteady supersonic underexpanded jet, impinging the flat plate in nonlinear oscillating mode, is used as the first test problem for both modes. Large memory savings for the propagator approximation are demonstrated. Corresponding prospects for the estimation of receptivity and singular vectors are discussed. The shallow water equations are used as the second large‐scale test problem. Excellent results are obtained for the proposed optimized DMD method of the shallow water equations when compared with recent POD‐based/discrete empirical interpolation‐based model reduction results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Electroviscous potential flow in nonlinear analysis of capillary instability

We study the nonlinear stability of electrohydrodynamic of a cylindrical interface separating two conducting fluids of circular cross section in the absence of gravity using electroviscous potential flow analysis. The analysis leads to an explicit nonlinear dispersion relation in which the effects of surface tension, viscosity and electricity on the normal stress are not neglected, but the effect of shear stresses is neglected. Formulas for the growth rates and neutral stability curve are given in general. In the nonlinear theory, it is shown that the evolution of the amplitude is governed by a Ginzburg–Landau equation. When the viscosities are neglected, the cubic nonlinear Schrödinger equation is obtained. Further, it is shown that, near the marginal state, a nonlinear diffusion equation is obtained in the presence of viscosities. The various stability criteria are discussed both analytically and numerically and stability diagrams are obtained. It is also shown that, the viscosity has effect on the nonlinear stability criterion of the system, contrary to previous belief.     

On periodic flow of non linear viscoelastic liquids

Summary After brief reference to energetical principles in order to find suitable constitutive equations for non-linear viscoelastic fluids, the behaviour of stress components is examined, for a simple shear motion with sinusoidal time time varying displacements of a liquid with an oscillating fading memory.

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Sommario Dopo aver richiamato brevemente un possibile procedimento per la determinazione della equazione costitutiva per liquidi viscoelastici nel caso di grandi deformazioni, si esamina il comportamento delle componenti del tensore degli sforzi nel caso di un moto periodico a filetti rettilinei e paralleli, supposto che il liquido abbia una memoria debole e oscillante nel tempo.

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Research supported by C.N.R., Gruppo nazionale per la fisica matematica.     

Energy analysis in a nonlinear hybrid system containing linear and nonlinear subsystems coupled by hereditary element

Energy transfer between subsystems coupled by standard light hereditary element in hybrid system is very important for different engineering applications, especially for dynamical absorption. An analytical study of the energy transfer between coupled linear and nonlinear oscillators in the free vibrations of a viscoelastically connected double-oscillator system as a new hybrid nonlinear system with two and half degrees of freedom is pointed out. The analytical study shows that the viscoelastic–hereditary connection between oscillators causes the appearance of like two-frequency regimes of subsystem's vibrations and that the energy transfer between subsystems appears. The Lyapunov exponents corresponding to each of two eigenmodes of the hybrid system, as well as to the subsystems are obtained and expressed by using energy of the corresponding eigentime components. The Lyapunov exponents are measures of the vibration processes stability in the hybrid system and in component subsystem vibrations. In Honor of Giuseppe Rega and Fabrizio Vestroni on the Occasion of their 60th Birthday.     

Flow pattern analysis of linear gradient flow distribution

This paper uses the Oseen transformation to solve the differential equations governing motion of the vertical linear gradient flow distribution close to a wall surface. The Navier-Stokes equations are used to consider the inertia term along the flow direction. A novel contour integral method is used to solve the complex Airy function. The boundary conditions of linear gradient flow distribution for finite problems are determined. The vorticity function, the pressure function, and the turbulent velocity profiles are provided, and the stability of particle trajectories is studied. An L_x-function form of the third derivative circulation is used to to simplify the solution. Theoretical results are compared with the experimental measurements with satisfactory agreement.     

Weakly nonlinear theory versus theory of secondary instability for plane poiseuille flow

In [1] the theory of secondary instability for plane Poiseuille flow was proposed. In this paper the weakly nonlinear theory is used to investigate the same problem. The results are consistent with those of the theory of secondary instability. It indicates that for plane Poiseuille flow there exists a certain intrinsic relation between the weakly nonlinear theory and the theory of secondary instability. This work is supported by the Science Fund of the Chinese Academy of Sciences.     

基于非局部塑性模型的应变局部化理论分析及数值模拟

通过求解一个第二类Fredholm方程,得到了基于非局部塑性软化模型的应变局部化问题理论解,结果表明,只有在当采用过非局部修正形式的非局部塑性软化模型才能得到应变局部化解,且得到的塑性应变分布和荷载响应依赖于所引入的特征长度及过非局部权参数。通过一维应变局部化有限元数值解,验证了非局部理论的引入能克服计算结果的网格敏感...     

STABILITY ANALYSIS OF LINEAR AND NONLINEAR PERIODIC CONVECTION IN THERMOHALINE DOUBLE－DIFFUSIVE SYSTEMS

ＳＴＡＢＩＬＩＴＹＡＮＡＬＹＳＩＳＯＦＬＩＮＥＡＲＡＮＤＮＯＮＬＩＮＥＡＲＰＥＲＩＯＤＩＣＣＯＮＶＥＣＴＩＯＮＩＮＴＨＥＲＭＯＨＡＬＩＮＥＤＯＵＢＬＥ－ＤＩＦＦＵＳＩＶＥＳＹＳＴＥＭＳＺｈａｎｇＤｉｍｉｎｇ（张涤明）;ＬｉＬｉｎ（李琳）;ＨｕａｎｇＨ...     

On the nonlinear interfacial instability of rotating core-annual flow

The interfacial stability of rotating core-annular flows is investigated. The linear and nonlinear effects are considered for the case when the annular region is very thin. Both asymptotic and numerical methods are used to solve the flow in the core and film regions which are coupled by a difference in viscosity and density. The long-time behavior of the fluid-fluid interface is determined by deriving its nonlinear evolution in the form of a modified Kuramoto-Sivashinsky equation. We obtain a generalization of this equation to three dimensions. The flows considered are applicable to a wide array of physical problems where liquid films are used to lubricate higher- or lower-viscosity core fluids, for which a concentric arrangement is desired. Linearized solutions show that the effects of density and viscosity stratification are crucial to the stability of the interface. Rotation generally destabilizes nonaxisymmetric disturbances to the interface, whereas the centripetal forces tend to stabilize flows in which the film contains the heavier fluid. Nonlinear effects allow finite-amplitude helically traveling waves to exist when the fluids have different viscosities.This research was partially supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18605 while the second author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665. This work was also supported by the Science and Engineering Research Council.