A rationally based yield criterion for work hardening materials

Summary The paper is concerned with the development of a rationally based yield criterion for work hardening materials. In an attempt to construct a single analytical function to represent yield behaviour the mathematics of continuum mechanics is combined with a statistical argument to produce a yield criterion which takes into account the inherent inhomogeneity of stress and strain distribution throughout a polycrystalline aggregate.A mathematical model is proposed which consists of spherical, anisotropic inclusions embedded in an elastic-plastic matrix. The yield criterion derived from this model is based on three main considerations, (a) arguments involving the local value of shear strain energy density, (b) Eshelby's solution for the local stress acting on a spherical inclusion in an infinite matrix and (c) the statistics of extreme values.An example is given on the fitting of the yield criterion to the results of thin-walled aluminium tubes prestrained in tensiontorsion space.

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Sommario Si considera lo sviluppo di una razionale condizione di plasticità per materiali con incrudimento. Nel tentativo di costruire una sola funzione analitica per rappresentare il comportamento plastico del materiale, la matematica della meccanica dei continui è combinata con considerazioni statiche onde definire una condizione di plasticità che tenga conto dell'implicita inomogeneità nella distribuzione di sforzi e deformazioni in un aggregato policristallino.Si propone un modello matematico che prende in considerazione inclusioni sferiche anisotrope avvolta da materiale elastoplastico. La condizione di plasticità derivata da questo modello è basata su tre considerazioni principali, (a) argomenti che involvono il valore locale della densità di energia di deformazione al taglio, (b) la soluzione di Eshelby per lo sforzo locale che agisce su ogni inclusione sferica in una matrice infinita e (c) la statistica dei valori estremi.Si dà un esempio sulla capacità del criterio di plasticità di interpretare i risultati ottenuti su tubi di alluminio sottile precaricato nello spazio tensione-torsione.

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Phenomenological theory of failure criterion for composite materials

In this paper, some current anisotropic failure criteria in the forms of tensor polynomials are investigated. In order to determine the interaction coefficients of the failure criterion, a non-linear optimization method is proposed. The results obtained by different theories as well as the optimization method are compared with the test data of some composite materials. The comparison shows that the optimization method is effective.     

基于微面有效应力矢量的各向异性屈服准则

基于微面模型,定义损伤变量为微面上有效承载面积的减少. 将Kachanov的一维有效 应力概念推广到三维,提出微面有效应力矢量的概念. 根据微面的有效应力矢量,将无损材 料的宏观应力张量及不变量与微面应力矢量的积分关系拓展到有损材料,得到了有损材料的 宏观有效应力张量及其不变量与宏观名义应力张量、微面面积损伤组构张量之间的关系. 将 无损材料的以应力张量不变量表示的Drucker-Prager准则推广到有损材料,建立了含缺陷 材料的各向异性屈服准则. 对有损材料,宏观有效应力张量与Murakami的有效应力张量具 有相同的形式,各向异性强度准则与Liu等提出的扩展Hill准则有相同的形式,当不考虑 静水应力对屈服的影响时,它与Hill准则具有相同的形式.     

A general yield criterion for engineering materials,depending on void growth

Summary Previously introduced yield criteria, taking into consideration the influence of internal dilation of the materials during yielding, were based on a conventional assumption of the influence of the hydrostatic component of stresses on the yielding process. They were fitted to the actual macroscopic behaviour of the materials and they were proved experimentally to predict the overall plastic behaviour of a great number of substances. In this paper a yield criterion was tested, which was based on the theory of void growth and coalescence in the vicinity of internal discontinuities of the material during yielding and fracture. The inverse dependence of yielding on the hydrostatic tension was incorporated, in the criterion and it was shown that an increase of the hydrostatic component creates a rapid decrease of fracture ductility.

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Sommario I criteri di snervamento precedentemente introdotti, che considerano l'influenza della dilatazione interna dei materiali sullo snervamento, erano basati sull'assunzione convenzionale che la componente idrostatica del tensore degli sforzi avesse influenza sul processo di plasticizzazione. In accordo con l'attuale comportamento macroscopico del materiale, è stato provato sperimentalmente che essi predicono il comportamento plastico totale di un gran numero di sostanze. In questo articolo viene proposto un criterio basato sulla teoria di crescita dei vuoti e di coalescenza in vicinanza di discontinuità interne del materiale durante lo scorrimento plastico e la frattura. Nel criterio è contenuta la dipendenza inversa della plasticizzazione dalla tensione idrostatica, e viene dimostrato che l'aumento della componente idrostatica crea una rapida diminuzione della duttilità della frattura.

     

Finite element analysis of elastic-plastic plane stress problems based upon Tresca yield criterion

Summary A finite element technique for the elastic-plastic analysis of two dimensional structures subjected to conditions of plane stress and monotonically increasing loads is presented. The complete load deformation history as well as the propagation of the yield zones in the structure up to plastic collapse are studied. The material is assumed to be elastic-perfectly plastic and yielding is governed by Tresca yield condition. Plastic stress-strain relations for the sides and corners of Tresca yield condition are derived in terms of the components of the stresses and strains along a fixed reference coordinate system and the direction of the principal stress. The load is applied in small increments and the principal stress direction for each plastic element during a load increment is determined by an interpolation technique which leads to stresses that satisfy the yield condition. Numerical examples are given to illustrate the accuracy of the results obtained by the proposed method.

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Übersicht Es wird eine finite element-Methode vorgeschlagen zur Berechnung zweidimensionaler elastischplastischer Baukonstruktionen, die einem ebenen Spannungszustand mit monoton wachsender Last unterworfen sind. Das Belastungs-Verformungs-Verhalten und die Ausbreitung der Fließbereiche bis zum plastischen Zusammenbruch werden untersucht. Das Material soll der Fließbedingung von Tresca genügen. Spannungs-Verformungs-Berechnungen für die Ränder und Ecken des Tresca-Bereiches werden in Komponenten der Spannungen und Dehnungen längs der Achsriclitungen eines festen Bezugssystems und der Hauptspannungsrichtungen ausgedrückt. Die Belastung wird in kleinen Stufen aufgebracht, und die Hauptspannungsrichtungen werden für jedes plastische Teilchen während des Lastaufbringens durch Interpolation bestimmt. Die so erhaltenen Spannungen genügen der Fließbedingung. Durch numerische Beispiele wird die Genauigkeit des vorgeschlagenen Verfahrens demonstriert.

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On leave from Northwestern University, Evanston, Illinois     

A material model for cementitious composite materials with an exterior point Eshelby microcrack initiation criterion

A micromechanical model for cementitious composite materials is described in which microcrack initiation, in the interfacial transition zone between aggregate particles and cement matrix, is governed by an exterior-point Eshelby solution. The model assumes a two-phase elastic composite, derived from an Eshelby solution and the Mori–Tanaka homogenization method, to which circular microcracks are added. A multi-component rough crack contact model is employed to simulate normal and shear behaviour of rough microcrack surfaces. The development of the microcrack initiation criterion and the rules adopted for microcrack evolution are a particular focus of the paper. Finally, it is shown, on the basis of several numerical simulations, that the model captures key characteristics of the behaviour of cementitious composites such as concrete.     

Limit analysis of plates with piecewise linear yield surface

Summary Rigid-plastic plates having a piecewise linear yield surface are studied by limit analysis. Extremum principles for the evaluation of specific dissipation power are defined by means of linear programming concepts. The law governing the plastic collapse of plates is formulated on the basis of the well-known kinematic theorem of limit analysis. A general procedure for the approximate determination of the collapse load is proposed. The paper ends with a brief numeric investigation of the uniformly loaded square plate.

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Sommario Si studiano attraverso l'analisi limite le piastre costituite di materiale rigido-plastico aventi superficie di snervamento di tipo poliedrico. I principi di estremo riguardanti la valutazione della potenza specifica di dissipazione vengono definiti attraverso i concetti della programmazione lineare. Attraverso il noto teorema cinematico dell'analisi limite, si formula la legge che regola il collasso plastico delle piastre. Viene proposto un procedimento per la determinazione approssimata del carico di collasso del tutto generale. Si conclude infine con una breve indagine numerica relativa al caso di piastra quadrata con carico uniforme.

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The results presented in the paper form part of a Research supported by the National Research Council (C.N.R.).     

Limit analysis for a general class of yield conditions

The paper describes a generalisation of the programming method described by Ponter and Carter (1997) for the evaluation of optimal upper bounds on the limit load of a body composed of a rigid/perfectly plastic material. The method is based upon similar principles to the `Elastic Compensation' method which has been used for design calculations for some years but re-interpreted as a non-linear programming method. A sufficient condition for convergence is derived which relates properties of the yield surface to those of the linear solutions solved at each iteration. The method is demonstrated through an application to a Drucker–Prager yield condition in terms of the Von Mises effective stress and the hydrostatic pressure. Implementation is shown to be possible using the user routines in a commercial finite element code, ABAQUS. The examples chosen indicate that stable convergent solutions may be obtained. There are, however, limits to the application of the method if isotropic linear solutions are used for an isotropic yield surface. In an accompanying paper (Ponter and Engelhardt, 2000) the method is extended to shakedown and related problems.     

A generating function for the yield criterion of isotropic and anisotropic polycrystalline materials

Summary A yield criterion for elastic pure-plastic polycrystalline materials is generated under simplified conditions by assuming that for yielding a certain fraction Q _c of the total number of slip planes in the material has to be active. This fraction Q _c is called the critical active quantity. We suppose Q _c to be independent of the state of stress. The yield criterion is mathematically expressed as an integral, which is a function of Q _c. This criterion can also be used for anisotropic materials.For isotropic materials the ratio (r) of the yield stress in torsion to that in tension is calculated as a function of Q _c. We find 0.5r0.61.The value r=0.5 (Tresca's criterion) is obtained for Q _c=0 and Q _c=1. The value r=0.577 (von Mises criterion) is obtained for Q _c=0.34 and Q _c=0.79. The difference between two criteria with the same r is the magnitude of the yield stress. We think the value Q _c=0.79 corresponds to the experiments for f.c.c. materials, since a rough estimation gives Q _c>0.75 for yielding.The independence of Q _c on the state of stress brings on that r>0.5 is more probable. This is caused by the slower increase to Q _c in torsion compared with the case of tension.From the theory follows that in the general case (Q _c0) the middle principal stress has influence on yielding.In this paper we don't determine Q _c, but adapt its value to the experimental results. However, a rough estimation of Q _c is given for isotropic materials.     

Thermographic stress analysis of composite materials

Several critical aspects of stress measurements in composite materials by thermographic stress analysis (TSA; also SPATE method) have been investigated. The emphasis is on the observed effects of thermal-expansion coefficients with positive and negative signs, thickness of surface coating, and absolute temperature increases in the material due to cyclic loading. Heat transfer and mean stress effects are also discussed.     

The strain energy density ratio criterion for predicting cracking direction in composite materials

The strain energy density ratio criterion for predicting cracking direction incomposite materials is proposed.The Tsai-Hill criterion and Norris criterion ofcomposite materials are extended to predict the cracking direction in composites.Thethree criteria are used to analyse the crack propagation problem of the unidirectionalfibre composite sheet with various fibre directions.The predicted results are comparedwith those of the existing normal stress ratio criterion and strain energy densitycriterion.     

M-integral analysis for microcrack-damaged anisotropic composite materials

Summary  This paper presents an M-integral analysis for the microcracked anisotropic composite materials. By using an elementary solution derived for a single finite crack subjected to a concentrated force on crack faces, the problem of strong interacting, arbitrarily oriented and located microcracks in an anisotropic composite materials is reduced to a system of Fredholm integral equations. The crack-tip fracture parameters, such as the stress intensity factors, are evaluated from a numerical solution of the system of integral equations. Its dependence on the coordinate system, calculation, and physical interpretation of the M-integral are discussed in the interaction problem. Finally, a numerical example of the damage evaluation by the M-integral analysis is given. Received 24 September 1999; accepted for publication 8 February 2000     

Correction of the Gurson yield criterion

The correction in the derivation of the Gurson yield criterion is reported. A prerequisite kinematical condition ignored in Gurson's paper is imposed, and a parameter (C₁), which was unreasonably set to zero, is restored.     

A micromechanics based damage model for composite materials

The predictive capacity of ductile fracture models when applied to composite and multiphase materials is related to the accuracy of the estimated stress/strain level in the second phases or reinforcements, which defines the condition for damage nucleation. Second phase particles contribute to the overall hardening of the composite before void nucleation, as well as to its softening after their fracture or decohesion. If the volume fraction of reinforcement is larger than a couple of percents, this softening can significantly affect the resistance to plastic localization and cannot be neglected. In order to explicitly account for the effect of second phase particles on the ductile fracture process, this study integrates a damage model based on the Gologanu–Leblond–Devaux constitutive behavior with a mean-field homogenization scheme. Even though the model is more general, the present study focuses on elastic particles dispersed in an elasto-plastic matrix. After assessing the mean-field homogenization scheme through comparison with two-dimensional axisymmetric finite element calculations, an extensive parametric study is performed using the integrated homogenization-damage model. The predictions of the integrated homogenization-damage model are also compared with experimental results on cast aluminum alloys, in terms of both the fracture strain and overall stress–strain curves. The study demonstrates the complex couplings among the load transfer to second phase particles, their resistance to fracture, the void nucleation mode, and the overall ductility.     

GM准则解析受线性荷载简支圆板的极限载荷

用GM(几何中线)屈服准则,对受线性荷载作用下的简支圆板进行塑性极限分析,求得极限载荷的解析解.该解为圆板半径a、切向应力最大点半径r0以及极限弯矩的函数.与Tresca、Mises和TSS(双剪应力)屈服准则预测的极限载荷比较表明,Tresca屈服准则预测极限载荷的下限,TSS屈服准则预测极限载荷的上限,GM准则预测的极限载荷恰居二者中间,并靠近Mises解.圆板半径a与切向应力最大点半径r0的变化关系为r0随着a的增加而增加,满足线性关系,r0分别出现在r0=0.7710a和r0=0.5472a的位置上.     

Shakedown analysis of structures made of materials with temperature-dependent yield stress

In this work, the shakedown of structures made of materials with temperature-dependent yield stress is considered. Under some restrictions on the thermal loading condition the yield stress is linearized and shakedown theorems are established. Based on these linearized shakedown theorems, the shakedown limit is formulated as a problem of convex optimization. An algorithm is built to compute shakedown limits. Numerical tests show good agreement with analytic solutions and experimental data.     

Prediction of mixed-mode fracture based on an elasto–plastic energy balance criterion

The maximum energy release rate criterion, i.e., G _max criterion, is commonly used for crack propagation analysis. This fracture criterion is based on the elastic macroscopic strength of materials. In the present investigation, however, the G _max criterion has been modified in order to accommodate the consideration of plastic strain energy. This modified criterion is extended to study the fatigue crack growth characteristics of mixed-mode cracks. To predict crack propagation due to fatigue loads, a new elasto–plastic energy model is presented. This new model includes the effects of material properties such as strain hardening exponent n, yield strength σ _y , and fracture toughness and stress intensity factor ranges. The results obtained are compared with those obtained using the commonly employed crack growth law and the experimental data.