Finite element simulation of localized plastic deformation

Summary A rational finite element algorithm for capturing localized plastic deformation due to softening and/or non-associated plastic flow is devised. The incremental relations are based on implicit integration. In each increment it is important to use a carefully designed modified Newton iteration procedure in conjunction with a start solution whose calculation is based on a diagnostic bifurcation analysis. The algorithm performs successfully for a finite element mesh that is biased in the sense that the element sides are prealigned with the anticipated localization zone which is demonstrated for a slope stability problem.

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Finite-Element-Berechnung von örtlichen plastischen Verformungen

Übersicht Die Konstruktion eines Algorithmus, der im Zusammenhang mit der Methode der Finiten Elemente örtliche plastische Verformungen als Folge von Materialentfestigung und/oder nichtassoziiertem plastischen Fließen erfaßt, wird beschrieben. Die inkrementellen Gleichungen ergeben sich aus der Verwendung eines impliziten Integrationsverfahrens. In jedem Inkrement wird ein sorgfältig entwickeltes, modifiziertes Newton-Iterationsverfahren verwendet, dessen erste Lösungsabschätzung auf einer diagnostischen Analyse des Verzweigungsproblems beruht. Am Beispiel einer Böschungsstabilitätsungtersuchung wird gezeigt, daß der Algorithmus erfolgreich ist, wenn das verwendete Finite-Elemente-Netz so ausgerichtet ist, daß die Elementseiten der erwarteten Versagenzone folgen.

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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     

Dynamic large deformation response of rigid plastic arches under impact

This paper presents a theoretical analysis of the dynamic large deformation response of highly circular arches under impact. By the Instantaneous Configuration Method (ICM), the solutions of the entire large deformation response process of the problem are obtained. The influences of the mass ratio, the energy ratio and the supported condition on the final deformation, the response time and the occurrence of plastic deformation are discussed in detail. The necessary condition for occurrence of the local reverse bending phenomenon has been found. An approximate method is provided to describe the phenomenon. The numerical results are in good agreement with experimental data.     

Macromolecular deformation in periodic extensional flows

The response of a dumbbell model for dilute polymer solutions is examined for periodic sequences of homogeneous flows chosen to approximate the unsteady kinematics appropriate for flow through porous media. The evolution of particle shape depends on the kinematic history of the flow and on microrheological properties of the dumbbell such as a variable friction factor and a nonlinear spring. The effect of vorticity on macromolecular stretching is found to differ qualitatively from results for steady flows, and entrance effects that persist over timescales much larger than the intrinsic macromolecular relaxation time are observed.     

Large deformation extensional rheology of bread dough

The stress–strain curves of bread dough were derived under uniaxial compression, uniaxial tension and equi-biaxial tension loading conditions. In uniaxial compression, a lubricant was used to eliminate frictional effects between the loading platens and the sample. In uniaxial tension, cylindrical samples with thin flat discs at both ends (‘I’ samples) were tested. The discs at both ends were allowed to air-dry and were subsequently glued onto the loading platens. In equi-biaxial tension, a thin disc of dough was inflated into a bubble using pressurised air. The thickness at the top of the bubble was measured by shining a light through the walls of the bubble and recording the change in light intensity as the wall becomes thinner. All methods ensured that uniform deformation was obtained. Stress and strain were accurately evaluated using image analysis techniques. The tests were performed at various strain rates and speeds that defined the time dependence of the material. A non-linear viscoelastic model based on the Prony series and Van der Waals hyperelasticity was used to predict all test data. The model had a total of five material parameters and two time constants, which were set to represent the actual time scales of the experiments. A reasonable agreement between the experimental data and the chosen material model was observed.     

Stability of an elastic arch in a rigid cavity

A thin, elastic circular arch, clamped at its ends and constrained in a rigid cavity, is subjected to a uniformly distributed parallel loading. Stability of equilibrium in the large “snap-through” is investigated both analytically and experimentally and good agreement is found to exist between the two modes of investigation. It is concluded that the set of simplifying assumptions introduced in the analysis is justified. Simple analytical results obtained in the paper allow the design of underground shelters under this type of loading.     

Transient deformation of an inviscid inclusion in a viscoelastic extensional flow

The transient deformation of a bubble in a viscoelastic extentional flow is analyzed by means of a finite element algorithm for viscoelastic moving boundary problems. Using the Oldroyd-B constitutive model, we find that bubbles in a viscoelastic fluid deform to the same steady-state configurations as bubbles in a Newtonian fluid at equal values of the far-field extensional stresses (corresponding to different stretch rates). Vapor bubbles in a developed extensional flow collapse more readily in the viscoelastic liquid than bubbles in Newtonian fluids because of the large compressive stresses associated with the viscoelastic liquid.     

Penetration of a rigid cone into a plastic orthotropic half space

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 159–163, September–October, 1991.     

Free rigid/plastic plane bending of a slender beam

Summary Large rigid/plastic plane bending of a slender structural element as e.g., a beam or a sheet metal strip is examined, in view of its applications to metal forming. Loads, i.e. forces and moments, are admitted only at the ends of the element; the loads or the corresponding displacements and rotations may be prescribed. Using the normal force, the transversal force and the bending moment as generalized stresses acting at the cross sections the differential equations of the problem are set up for an arbitrary, strain-hardening and/or rate-sensitive material. As an example a homogeneous, ideally plastic beam is considered to be plastified by means of the bending moment only. It is shown that it can be brought into any shape provided the end conditions are adequately controlled. Circular bending as a special case becomes possible in two different ways i.e., by pure bending (without end forces) or by localized bending (generated by a moving yield hinge).

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Ebenes, freies biegen von schlanken, starrplastischen trägern

Übersicht Es wird die große, ebene, starrplastische Biegung schlanker Träger, also beispielsweise von Balken oder Blechen, im Hinblick auf Anwendungen in der Umformtechnik von Metallen untersucht. Lediglich an den Enden der Träger greifen Lasten (Kräfte, Momente) an; diese Lasten oder die entsprechenden Verschiebungen und Neigungen kann man vorgeben. Mit der Normalkraft, der Querkraft und dem Biegemoment als generalisierte Spannungen in den Querschnitten werden die Differentialgleichungen des Problems für ein beliebig verfestigendes und/oder geschwindigkeitsabhängiges Material formuliert und speziell auf einen homogenen, idealplastischen Träger angewendet, dessen Plastifizierung nur vom Biegemoment abhängt. Es wird gezeigt, daß man ihn in jede beliebige Gestalt bringen kann, vorausgesetzt, die Bedingungen an den Enden werden angemessen gesteuert. Eine kreisförmige Biegung erreicht man zum Beispiel auf zwei verschiedenen Wegen: Durch reine Biegung (ohne Kräfte an den Enden) oder durch lokalisierte Biegung infolge eines wandernden Fließgelenkes).

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Herrn Prof. Dr. Dr. h.c. H.-P. Stüwe, Montan-Universität Leoben, zum 60. Geburtstag am 14. Sept. 1990 gewidmet.     

Steady penetration of a rigid cone into pressure-dependent plastic material

The objective of the present paper is to find a semi-analytical axisymmetric solution for steady penetration of a rigid cone into pressure-dependent plastic material obeying the double-shearing model. As expected, the solution is singular near the maximum friction surface. It is important to mention that the singularity is not due to the geometry of the problem but the friction law. The type of the singularity is the same as in plane-strain solutions based on the double-shearing model and in classical plasticity. This allows for calculating the strain rate intensity factor. The solution is illustrated by a numerical example.     

Closed-form solution of a shear deformable,extensional ring in contact between two rigid surfaces

Contact of a circular ring with a flat, rigid ground is considered using curved beam theory and analytical methods. Applications include tires, springs, and stiffeners, among others. The governing differential equations are derived using the principle of virtual work and the formulation includes deformations due to bending, transverse shear and circumferential extension. The three associated stiffness quantities, EI, GA and EA, respectively, remain as independent parameters in the differential equations. This allows the special cases such as an inextensible Timoshenko beam (EI and GA) or an extensible Euler beam (EI and EA) to be obtained directly by the appropriate limits. The effect of these three stiffness parameters on the contact pressure solution is studied, which shows how those fundamental parameters can be selected for the purpose of the application. Although the formulation is for small displacement theory, both radial and circumferential distributed loads are considered, which allows the pressure in the deformed state to be vertical rather than radial, which is shown to be important. Closed form expressions for all force and displacement quantities are obtained in terms of the angular location of the edge of contact, which must be determined numerically. Extensibility complicates the analytical expressions within the contact region, and a series solution is proposed in this case. A two-term asymptotic expression for the stiffness of the ring is determined analytically. Finally, all solutions are validated using the commercial finite element software ABAQUS, with attention to non-linear behavior and the range of validity of these solutions.     

A refined theory of elastic thick plates for extensional deformation

Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification. It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised. In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein.     

A 1-D theory for extensional deformation of a viscoelastic filament under exponential stretching

We consider a viscoelastic filament placed between two coaxial discs, with the bottom plate fixed and the top plate pulled at an exponential rate. Using a slender rod approximation, we derive a one-dimensional (1-D) model which describes the deformation of a viscoelastic filament governed by the Oldroyd-B constitutive model. It is assumed that the flow is axisymmetric and that inertia and gravity are negligible. One solution of the model equations corresponds to ideal uniaxial elongation. A linear stability analysis shows that this solution is unstable for a Newtonian fluid and for viscoelastic filaments with small Deborah number (De ≤ 0.5). For Deborah number greater than 0.5, ideal uniaxial elongation is linearly stable. Numerical solution of the nonlinear equations confirms the result of the linear stability analysis. For initial conditions close to ideal uniaxial flow, our results show that if De > 0.5, the central portion of the filament undergoes considerable strain hardening. As a result, the sample remains almost cylindrical and the deformation approaches pure uniaxial extension as the Hencky strain increases. For De ≤ 0.5, the Trouton ratio based on the effective extension rate at the mid-plane radius gives a much better approximation to the true extensional viscosity than that based on the imposed stretch rate.     

圆柱壳的塑性膨胀和变形

根据材料不可压缩假设,适当选取材料塑性本构关系,将圆柱壳塑性膨胀运动简化为以圆柱壳内 半径为因变量的常微分方程初值问题。对3种初始内外半径比纯铜厚壁圆柱壳膨胀过程的计算表明,应变率 硬化阻碍圆柱壳的塑性膨胀运动,应变硬化和温度效应的影响可以忽略,而与圆柱壳的厚度无关。     

Motion of a rigid plastic cantilever in a damping medium under transverse impact

The motion of a rigid plastic cantilever beam which is surrounded by a damping medium and struck transversely at the tip by a moving mass is studied. The elementary theory, which disregards effects due to rate of straining and geometry changes is used. The governing equations of motion are integrated numerically. For comparison the case of discrete damping provided at the tip only is also solved. Results are presented for a wide range of parameters.     

Specific features of microparticle deformation upon impact on a rigid barrier

Based on experimental data and numerical modeling, it is shown that a lamina of melted metal of thickness of order0.01 d, in which the temperature is close to the melting point of the particle material, can be formed upon high-speed impact (v ₀≈500–1200 m/sec) of a fine metal particle (d=1–50 μm) on a rigid undeformable barrier near the contact surface. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 204–209, January–February, 2000.     

The plastic deformation of non-homogeneous polycrystals

To describe the work hardening process of polycrystals processed using various thermomechanical cycles with isochronal annealing from 500 to 900 °C, a dislocation based strain hardening model constructed in the basis of the so-called Kocks–Mecking model is proposed. The time and temperature dependence of flow stress is accounted via grain boundary migration, and the migration is related to annihilation of extrinsic grain boundary dislocations (EGBD’s) by climb via lattice diffusion of vacancies at the triple points. Recovery of yield stress is associated with changes in the total dislocation density term ρ^T. A sequence of deformation and annealing steps generally result in reduction of flow stress via the annihilation of the total dislocation density ρ^T defined as the sum of geometrically necessary dislocations ρ^G and statistically stored dislocations ρ^S. The predicted variation of yield stress with annealing temperature and cold working stages is in agreement with experimental observations. An attempt is made to determine the mathematical expressions which best describe the deformation behaviour of polycrystals in uniaxial deformation.     

Dynamic plastic deformation of hexagonal frames

The dynamic plastic response of a hexagonal frame to an internal pressure pulse of arbitrary shape is analyzed, including large-deformation geometric effects that result in redistribution of the bending and membrane reactions. Peak pressures of several multiples of the static yield load are considered, and the frame material is assumed to be rigid perfectly plastic. The effect of pulse shape on final plastic deformation is determined by numerically solving the governing sets of differential equations for a variety of parameter combinations. In the small deformation range, the permanent plastic deformation is shown to be dependent on an effective pressure, defined in terms of the first moment of the pressure pulse; the response duration is proportional to the pulse duration. In the large deformation range, the permanent plastic deformation is a function of the average pressure applied during the response, and the response duration depends on a characteristic time constant which is a function of material properties and hexagon size.