On the relation for plastic spin

Summary Within the framework of the theory of finite elastoplasticity based on the known concept of the local, relaxed, isoclinic configuration the large plastic deformations with strain induced anisotropy and emerging plastic spin are studied. The derivation of the relation for plastic spin, based on the analysis on the simple shear problem for rigid-plastic material with kinematic hardening is provided and the experimental verification of the calculated axial elongation in the unconstrained large-strain shear problem is presented. The applicability of the proposed model is discussed.

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Zur Ermittlung der Stoffgesetze für den plastischen Spin

Übersicht Im Rahmen der Theorie endlicher elastoplastischer Formänderungen unter Verwendung des bekannten Konzepts lokaler, relaxierender, isokliner Konfigurationen werden große plastische Deformationen mit durch Dehnung hervorgerufener Anisotropie und daraus resultierendem plastischen Spin untersucht. Die Gleichung für den plastischen Spin wird mit dem Modell der einfachen Scherung hergeleitet. Dabei wird starr-plastisches, kinematisch verfestigendes Material angenommen. Das Modell des reinen Schubes wird numerisch behandelt und mit experimentellen Ergebnissen tordierter dünnwandiger Zylinder mit unbehinderter axialer Dehnung verglichen. Die Anwendbarkeit des vorgeschlagenen Modells wird diskutiert.

     

Fully plastic crack problems: The center-cracked strip under plane strain

Crack problems are formulated for solids characterized by a pure power hardening relation between the stresses and the strains. For such problems there are simple functional relationships between the amplitude of the dominant crack-tip singularity, as measured by the path-independent J-integral, and the applied load, the load point displacement, and the crack opening displacement. The solutions are valid for both incremental and deformation theories of plasticity; they also apply to problems involving steady-state creep. Numerical results are presented for the center-cracked strip of finite width under plane strain conditions. A preliminary discussion is given of the applicability of the solutions to large scale yielding fracture mechanics.     

An incremental theory of large strain and large displacement problems and its finite element formulation

Summary This paper is concerned with the development of an incremental finite'element theory for the large strain and the large displacement problems, referred to the current configuration of the body. Using the convected coordinate system which is embedded in the body, the incremental representations of strain and stress tensors and the energy relations are presented, and then the general procedure to construct the so-called element stiffness matrix in incremental form is considered. The finite element formulation is developed for a typical constitutive relation and it is shown that some correction matrices, some of which have been omitted in the previous works, are to be added to the element stiffness matrix. Finally the method to assemble the equations of the element to the global system is discussed and a simple finite element model satisfying the compatibility condition is presented.The finite element theory developed in this paper is able to be extended to the problems for the general thermodynamical process of a broad class of nonlinear materials.

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Übersicht Mit Hilfe der Methode der finiten Elemente wird eineZuwachstheorie zurBehandlung von Problemen mit endlicher Verformung abgeleitet. Dabei wird ein im Körper eingebettetes, der momentanen Form angepaßtes Bezugssystem verwendet. Es werden Ausdrücke für die Energie sowie für die Änderungen der Spannungs- und Verformungs-Tensoren abgeleitet und es wird ein Verfahren zur Konstruktion der Steifigkeitsmatrix für ein Element angegeben. Ein typisches Stoffgesetz wird dabei zugrundegelegt. Dabei zeigt es sich, daß einige in früheren Arbeiten vernachlässigte Korrektur-Matrizen zu der Steifigkeits-Matrix des Elementes hinzugefügt werden müssen. Die Möglichkeiten der Zusammenfassung der für die Elemente geltenden Gleichungen zu einem globalen Gleichungssystem werden diskutiert und es wird ein den Verträglichkeitsbedingungen genügendes Elemente-Modell angegeben.Das angegebene Verfahren kann für allgemeine thermodynamische Prozesse in einer breiten Klasse nichtlinearer Materialien erweitert werden.

     

A comparison between crystal and isotropic strain gradient plasticity theories with accent on the role of the plastic spin

In the small deformation range, we consider crystal and isotropic “higher-order” theories of strain gradient plasticity, in which two different types of size effects are accounted for: (i) that dissipative, entering the model through the definition of an effective measure of plastic deformation peculiar of the isotropic hardening function and (ii) that energetic, included by defining the defect energy (i.e., a function of Nye's dislocation density tensor added to the free energy; see, e.g., [Gurtin, M.E., 2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32]). In order to compare the two modellings, we recast both of them into a unified deformation theory framework and apply them to a simple boundary value problem for which we can exploit the Γ-convergence results of [Bardella, L., Giacomini, A., 2008. Influence of material parameters and crystallography on the size effects describable by means of strain gradient plasticity. J. Mech. Phys. Solids 56 (9), 2906–2934], in which the crystal model is made isotropic by imposing that any direction be a possible slip system. We show that the isotropic modelling can satisfactorily approximate the behaviour described by the isotropic limit obtained from the crystal modelling if the former constitutively involves the plastic spin, as in the theory put forward in Section 12 of [Gurtin, M.E., 2004. A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin. J. Mech. Phys. Solids 52, 2545–2568]. The analysis suggests a criterium for choosing the material parameter governing the plastic spin dependence into the relevant Gurtin model.     

On the use of electrical-resistance metallic foil strain gages for measuring large dynamic plastic deformation

The use of electrical-resistance metallic foil strain gages for measuring large plastic strain in dynamic experiments in studied. The maximum nominal strains obtained in this investigation are 35 percent in compression, 25 percent in tension. A linear variation of gage factor with strain is found in this range. The corrected maximum strains are in excellent agreement with permanent strains measured after the tests. Thus foil strain gages can be effectively used to measure the large dynamic plastic strains.     

Localized plastic strain in polycrystalline materials with hole and notches

Analyzed numerically are the localized strain of polycrystalline materials subjected to quasi-static loading. The objective is to study the peculiarities associated with the deformation process close to the stress concentrators such as holes, notches and interfaces of internal structure. Analytical results show that geometry and/or heterogeneous internal structure of material together with the action of maximum shear result in the development of a system of plastically deformed shear bands. Shears and rotations in the regions of strain localization are found to be higher than in other parts of the specimen while rotations are more sensitive to localization.     

Analytical solutions of mixed mode plane strain crack problems in elastic perfectly plastic materials

In the present paper analytical solutions concerning the stress state at the tip of a crack in an elastic-perfectly plastic body, subjected to mixed mode loadings under plane strain conditions, are presented. Analytical solutions of the nonlinear ordinary differential equations are obtained and the dominant singularity is completely determined with the aid of suitable boundary conditions. The obtained results are in perfect agreement with those given by other investigators, both analytical and numerical. The novel aspect here is the methodology used for the solution, as well as the direct determination of the plastic zones. As a consequence, the resulting analytical solutions cover many more problems in the mathematical theory of plasticity compared to similar existing methods and they may be proved of importance in various applications.     

Roles of plastic spin in shear banding

In this paper, the effects of plastic spin on shear banding and simple shear are examined systematically. Three types of plastic constitutive model with plastic spin are considered: (i) a non-coaxial model in which the direction of the plastic strain rate depends on that of the stress rate; (ii) a strain-softening model based on the J₂ flow theory; and (iii) the pressure-sensitive porous plasticity model. All the constitutive models are formulated in viscoplastic forms and in conjunction with non-local concepts that have been recently focused and discussed. First, behavior in simple shear is examined by numerical analysee with the aforementioned constitutive models. Moreover, some experimental evidences for stress response to simple shear are shown; that is, several large torsion tests of metal tubes and bars are carried out. Next, finite element simulations of shear banding in plane strain tension are performed. A critical effect of plastic spin on shear banding is observed for the noncoaxial model, while an almost negligible effect is observed for the porous model. The identical effects of plastic spin are observed, whether nonlocality exists or not. Finally, we discuss the relationship between the behavior in simple shear and the shear band formation. It is emphasized that this is a critical issue in predicting shear banding in macroscopic grounds.     

A FE formulation for elasto-plastic materials with planar anisotropic yield functions and plastic spin

In this article a stress integration algorithm for shell problems with planar anisotropic yield functions is derived. The evolution of the anisotropy directions is determined on the basis of the plastic and material spin. It is assumed that the strains inducing the anisotropy of the pre-existing preferred orientation are much larger than subsequent strains due to further deformations. The change of the locally preferred orientations to each other during further deformations is considered to be neglectable. Sheet forming processes are typical applications for such material assumptions. Thus the shape of the yield function remains unchanged. The size of the yield locus and its orientation is described with isotropic hardening and plastic and material spin.The numerical treatment is derived from the multiplicative decomposition of the deformation gradient and thermodynamic considerations in the intermediate configuration. A common formulation of the plastic spin completes the governing equations in the intermediate configuration. These equations are then pushed forward into the current configuration and the elastic deformation is restricted to small strains to obtain a simple set of constitutive equations. Based on these equations the algorithmic treatment is derived for planar anisotropic shell formulations incorporating large rotations and finite strains. The numerical approach is completed by generalizing the Return Mapping algorithm to problems with plastic spin applying Hill’s anisotropic yield function. Results of numerical simulations are presented to assess the proposed approach and the significance of the plastic spin in the deformation process.     

Minimum theorems for displacement and plastic strain rate histories in structural elastoplasticity

Summary The finite element method approach is used to obtain formulations of analysis problems relative to elastic-plastic structures when subjected to prescribed programmes of loads, and under the restrictive hypotheses:a) the yielding surfaces are piecewise linearized, andb) the plastic flow-laws are supposed to be of holonomic type within a single “finite” time interval. For mulations are given as linear complementarity problems and quadratic programming problems: one pair of formulations in terms of velocity and plastic multiplier rate histories, and another pair in terms of plastic multiplier rate histories only. The solutions are shown to be characterized by two minimum principles for displacement and plastic strain rate histories. After some general remarks about computational procedures, the paper is concluded with some suggestions for future developments.

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Sommario Si usa il metodo degli elementi finiti per formulare problemi di analisi relativi a strutture elasto-plastiche soggette a prescritti programmi di carico, sotto le ipotesi restrittive:a) le superfici di plasticizzazione sono linearizzate a tratti, eb) la legge del flusso plastico è olonoma all'interno del singolo intervallo di tempo “finito”. Si danno formulazioni come problemi di complementarità lineare e come problemi di programmazione quadratica: due formulazioni sono in termini di storia delle velocità e dei coefficienti di attivazione plastica, altre due sono in termini di storia dei coefficienti di attivazione plastica soltanto. Si dimostra che le soluzioni sono caratterizzate da due principi di minimo per la storia delle velocità di deformazione. Dopo alcune osservazioni generali sui procedimenti di calcolo, il lavoro si conclude con dei suggerimenti per futuri sviluppi.

     

An elasto-plastic constitutive model with plastic strain rate potentials for anisotropic cubic metals

An elasto-plastic constitutive model with the plastic strain rate potential was developed for finite element analysis. In the model, isotropic-kinematic hardening was incorporated under the plane stress condition for anisotropic sheet cubic metal forming analysis. The formulation is general enough for any homogeneous plastic strain rate potential (with the first-order homogeneous effective strain rate) but the plastic strain rate potential Srp2004-18p was considered here. Attention was focused on the development of the elasto-plastic transition criterion and the effective stress update algorithm. Also, to assure the quadratic convergence rate in Newton’s method, the elasto-plastic tangent modulus was analytically derived. Accuracy and convergence of the stress update algorithm were assessed by the iso-error maps, whereas stability of the algorithm was confirmed by analytical procedure. Validations were performed for the examples of the circular cup drawing, 2D draw-bending and unconstrained cylindrical bending tests, utilizing aluminum sheet alloys.     

Optimum plastic design for multiple sets of loads

Summary We study optimum plastic design of structures made up, or conceived as assemblies of finite elements, each having an elemental piece-wise linear rigid-plastic behaviour. Since cost function linearly dependent on design variables are considered, optimization problems in linear programming are encountered. Allowance is made for design dependent mass forces, and for some technological constraints. The design growing process is studied in the case of various sets of alternative applied loads, and the optimality conditions are written in a proper geometrical form which leads to a generalization of the concept of Foulkes mechanism.

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Sommario Si studia il progetto plastico ottimale di strutture formate di elementi finiti, ciascuno avente un comportamento globale rigido-plastico linearizzato a tratti. Si considerano funzioni costo dipendenti linearmente dalle variabili di progetto, e pertanto si va incontro a problemi di ottimizzazione nell'ambito della programmazione lineare. Tenendo conto delle forze di massa e di eventuali vincoli tecnologici, viene analizzato il processo di crescita del progetto nel caso in cui la struttura è sottoposta a diversi sistemi alternativi di carichi, e si scrivono le condizioni di ottimalità in una appropriata forma geometrica, la quale conduce a una generalizzazione del concetto di meccanismo di Foulkes.

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Capital bold-face letters are matrices, small bold-face letters are vectors. 0 is a vector having only zero entries. The tilde posed upon a vector or a matrix means “traspose of”. Other meanings are:     

含主应力轴旋转的土体平面应变问题弹塑性数值模拟

在含主应力轴旋转的土体本构关系研究的基础上,通过含主应力轴旋转的土体平面应变问题的弹塑性数值模拟结果分析,总结主应力轴旋转对土体应力分布与应形影响的规律,得出需要考虑主应力轴旋转的条件及影响的相对大小。     

Evolving internal length scales in plastic strain localization for granular materials

A numerical model in the Cosserat continuum for strain localization phenomena in granular materials is developed and proposed in this paper. The model assumes a constant internal length scale that is used to describe the shear band thickness. However, it is observed that the internal length scales need to change to accommodate the possible change in the contact surface between the particles, damage of the particles or/and any change in the local void ratio within the domain, which will change the shear band thickness. The mathematical formulations used in the present numerical model were equipped with evolution equations for the length scales through the Micropolar theory, those formulations are proposed and discussed in this paper. The evolution equations of the internal length scales describe any possible change in the contact surface between the particles, damage of the particles if exists and/or any change in the local void ratio within the domain. Hence, the strain localization described by the enhanced model with evolving internal length scales is more accurate and closer to the real solution. The solution for the shear bands thickness shows more accurate correlation with the experimental results and less dependency on the mesh size when such evolution equations are used. Moreover, the shear band thickness and inclination evolve during the deformation process.