It is 100 years since the well-know Mohr-Coulomb strength theory was established in 1900. A considerable amount of theoretical and experimental research on strength theory of materials under complex stress state was done in the 20th Century. This review article presents a survey of the advances in strength theory (yield criteria, failure criterion, etc) of materials (including matellic materials, rock, soil, concrete, ice, iron, polymers, energetic material, etc) under complex stress, discusses the relationship among various criteria, and gives a method of choosing a reasonable failure criterion for applications in research and engineering. Three series of strength theories, the unified yield criterion, the unified strength theory, and others are summarized. This review article contains 1163 references regarding the strength theories. This review also includes a biref discussion of the computational implementation of the strength theories and multi-axial fatigue.

Advances in strength theories for materials under complex stress state in the 20th Century

Ratcheting in cyclic plasticity, part II: Multiaxial behavior

Up to now, a number of models have been proposed and discussed to describe a wide range of inelastic behaviours of materials. The fatal problem of using such models is however the existence of model errors, and the problem remains inevitably as far as a material model is written explicitly. In this paper, the authors define the implicit constitutive model and propose an implicit viscoplastic constitutive model using neural networks. In their modelling, inelastic material behaviours are generalized in a state-space representation and the state-space form is constructed by a neural network using input–output data sets. A technique to extract the input–output data from experimental data is also described. The proposed model was first generated from pseudo-experimental data created by one of the widely used constitutive models and was found to replace the model well. Then, having been tested with the actual experimental data, the proposed model resulted in a negligible amount of model errors indicating its superiority to all the existing explicit models in accuracy. © 1998 John Wiley & Sons, Ltd.

Implicit constitutive modelling for viscoplasticity using neural networks

A new concept based on the use of a function expressed as a (complete) polynomial expansion in terms of the three invariants of the stress tensor is proposed for deriving yield, failure and plastic potential functions for use in plasticity based constitutive laws. A mathematical interpretation and physical meaning of the proposed concept are provided by using the idea of the singular nature of constiutive matrices in incremental hypoelastic laws. It is suggested that the proposed function and (polynomial) forms of material moduli can be synonymous. A number of specialized forms of the general function are adopted and their values at failure from advanced three-dimensional tests for a number of (geological) media are evaluated. The results indicate the possibility that there exist invariant numbers associated with the functions(s) that may apply to a wide range of materials. Some ideas on implementation of the proposed concept are also presented.

A general basis for yield, failure and potential functions in plasticity

A review of certain aspects of the Bauschinger effect in metals

By modifying the translational rule in a non-linear, kinematic hardening model, it is shown that some phenomena occuring during cyclic loading can be simulated: stress relaxation for cycles with prescribed strain amplitude superposed upon fixed strain, axial strain accumulation for asymetric stress cycle, cyclic hardening or softening. It is also shown that the variation of hardening modulus along the yield surface after plastic prestrain is fairly well described by the present model. The accumulation and relaxation phenomena depend on additional material function which can be determined from uniaxial loading-unloading or cyclic loading tests.

A non-linear hardening model and its application to cyclic loading

The work-hardening of an initially isotropic material is mathematically described by a hardening rule which specifies the modification of the initial yield condition in the process of continued plastic deformation. Although such rules have been proposed by various investigators and shown to be in good agreement with the experimental results, they are based on different specific assumptions. In the present work it is shown that all the existing hardening rules can be obtained from a general mathematical concept which is based upon the isotropic representation of a scalar valued function which is a function of two symmetric 3 × 3 matrices. A more general form has then been derived which accounts for the second-order effect relating to axial strain accumulation in cyclic torsion.



Die Verfestigung eines anfangs isotropen Stoffes wird mathematisch durch ein Verfestigungsgesetz beschrieben, das die Modifikation der ursprunglichen Fliesbedingung wahrend des Prozesses der plastischen Deformation darstellt. Obwohl solche Gesetze von verschiedenen Experimentatoren vorgeschlagen worden sind und Ubereinstimmung mit experimentellen Resultaten gezeigt haben, fehlt noch die mathematische Begrundung dieser Gesetze. In der vorliegenden Arbeit wird gezeigt, das alle bisherigen Verfestigungsgesetze aus einer allgemeinen mathematischen Idee abgeleitet werden konnen, welche auf der isotropen Darstellung einer Skalarfunktion von zwei symmetrischen 3 × 3 Matrizen beruht. Daruber hinaus wird eine allgemeinere Form des Verfestigungsgesetzes abgeleitet, welche die Wirkungen zweiter Ordnung bezuglich der Akkumulation der axialen Verformungen bei zyklischer Torsion berucksichtigt.

Yield Criterion and the Hardening Rule for a Plastic Solid

A yield criterion in planestress state is derived here based on isotropic representation of a scalar valued function depending upon symmetric 2×2 stress and strain matrices. The material has been assumed to be incompressible. In particular, for tension-torsion loading the yield surface is nonsymmetric with respect to the torsional stress axis. Due to the non-symmetry, the yield condition describes the second-order effect relating to axial-strain accumulation in cyclic torsion, and at the same time it has got a very simple form compared to other yield conditions describing this effect.

Yield Criterion and Second-Order Effects in Plane-Stress

Yield Condition and Hardening Rule for Density Varying Materials

The main objective of this paper is to develop a criterion of tensile plastic instability which should be sufficiently general to include both the Bauschinger effect and anisotropy. The criterion so developed reduces to Moore and Wallace's condition for anisotropic materials, in case the Bauschinger effect is either completely neglected or it is negligible at the onset of deformation process. However, this analysis shows significant change in instability condition if the material exhibits the Bauschinger effect initially.



It is also shown that localised neck can also take place in biaxial loading for stress ratios greater than half. This would depend upon the anisotropy coefficients and the degree of the Bauschinger effect initially exhibited by the material.



Das Hauptanliegen dieser Arbeit ist die Entwicklung eines Kriteriums fur Instabilitat bei plastischer Verformung unter Zugbeanspruchung, das in seiner Allgemeinheit den Bauschinger-Effekt und Anisotropie einschliest. Das so entwickelte Kriterium reduziert sich auf die Moorschen und die Wallaceschen Bedingungen fur anisotrope Materialien, wenn der Bauschinger-Effekt entweder vollig vernachlassigt wird oder er vernachlassigbar bei Beginn der Instabilitat ist. Jedoch zeigt diese Analyse eine wesentliche Anderung in den Instabilitatsbedingungen, wenn das Material anfanglich den Bauschinger-Effekt aufweist.



Es wird ebenso gezeigt, das jener lokal begrenzte Ansatz auch bei biaxialen Belastungen fur Spannungsverhaltnisse groser als einhalb Anwendung findet. Das hangt ab von den Anisotropiekoeffizienten und dem Grade des Bauschinger-Effektes, den das Material anfanglich aufweist.

H. P. Shrivastava

Papers

A non-linear hardening model and its application to cyclic loading

Yield Criterion and the Hardening Rule for a Plastic Solid

Yield Criterion and Second-Order Effects in Plane-Stress

Yield Condition and Hardening Rule for Density Varying Materials

Kinematic Hardening and Plastic Instability under Plane Stress