Direction Of Strain Increment Research Articles

An investigation of the strain path dependence of the forming limit curve

The dependence of the forming limit curve on the deformation history is examined by considering two specific examples of sheet metal forming, representative of two classes of non-proportional strain path: one example is an axisymmetric punch stretching problem, which gives rise to a gradual variation in the strain path from a proportional path; the other example consists of a sheet with residual strains subjected to a proportional stretching so that the complete deformation history, if represented in the principal in-plane strain coordinates, takes a sharp turn. Results are presented for these examples for each of three plasticity theories: flow theory, J2-deformation theory and the recently proposed corner theory. By assuming a plastic potential increment dependent on the direction and magnitude of the incipient plastic strain increment the corner theory gives results which agree qualitatively with experimental observations and therefore seems to be favored for the bifurcation analysis of sheet metal forming problems involving significant strain path variations.

Experimental Investigation on History-dependence of Plastic Behaviour of Brass under Combined Loading

Experimental investigations about the plastic deformation behaviours of brass at an initially isotropic state are performed along the strain trajectories with one or two corners starting at the zero-strain state in a vector space corresponding to strain deviators. Experimental results along the trajectories with one corner show that the history effect of a corner having a corner-angle less than 120° disappears with an increase of deformation along the trajectory after the corner. This feature verifies the existence of the so-called "Fading Memory". From the experimental results along thee trajectories with two corners, it is found that the effect of the first corner on the deformation behaviour after the second one disappears for arc length between both corners longer than a certain value, if the trajectory does not contain a reverse in the direction of strain increment. On the contrary, the deformation behaviour along the trajectory with a reverse is affected by a generalized Bauschinger effect.

The conditions for failure of a face-centered cubic array of uniform rigid spheres

The general solution for the strength of a face-centred cubic array of uniform rigid spheres is presented. The applied stress system is allowed three degrees of rotational freedom and the complete stress range σ3 ≤ σ2 ≤ σ1 is considered. The strength is found to be dependent on the relative orientation of the applied stresses and the optimum solutions for various degrees of rotational freedom are identified. The solution for the case in which the stress and strain–increment tensors are coaxial extends the previous work of Dantu (1961), Rowe (1962) and Leussink and Wittke (1963) to generahzed stress conditions. The optimum solution for freedom to rotate about the intermediate principal strain–increment direction is identical to Parkin's (1965) solution except that the angles of deviation between the stress and strain–increment directions are correctly prescribed. None of the remaining solutions contained in the analysis have been presented before. The analysis shows that a variety of failure envelopes are possible and suggests that various experimentally obtained failure envelopes for dense sand may be valid representations of material behaviour, assuming that the different apparatuses used produce different distributions of the local deviation angles associated with microscopic deformation processes. L'article présente la solution générale destinée au calcul de la resistance d'un réseau cubique à faces centrées de sphéres rigides uniformes. Le système de contraintes appliquées a trois degrés de liberté de rotation, et la gamme compléte de contraintes envisagée est σ2 ≤ σ3 ≤ σ1. Les résultats montrent que la résistance est fonction de l′orientation relative des contraintes appliquées, et les solutions optima portant sur divers degrés de liberté de rotation sont identifiées. La solution trouvée dans le cas de tenseurs de contraintes et d'incréments de déformation coaxiaux permet d'étendre les travaux antérieurs de Dantu (1961), Rowe (1962), et Leussink et Wittke (1963) à des conditions de contraintes généralisées. La solution optimum relative à la liberté de rotation autour de l'axe d'incréments de déformation principale intermédiaire est identique à celle de Parkin (1965) excepté que les angles de déviation séparant les axes de contraintes et d'incréments de déformation sont correctement prescrits. Aucune des autres solutions de I'analyse n'a déjà été présentée auparavant. L'analyse montre que différentes enveloppes de rupture sont possibles et suggére que diverses enveloppes de rupture obtenues expémentalement pour du sable dense pourraient constituer des représentations valables du comportement du matériau en supposant que les différents appareils utilisés produisent différentes distributions des angles de déviation locaux associés aux processus de déformation microscopique.

A Mechanical Model for The Stress-Strain Behaviour of Normally Consolidated Cohesive Soil

ABSTRACT A mechanical model on the stress-strain behaviour of normally consolidated cohesive soil is presented in this paper based upon the experimental results. Theoretical equations are derived based on the theory of elastoplasticity. Time effect is not taken into consideration. In the analysis, strain increment is divided into two components, i. e., elastic and plastic ones. The plastic component in strain increment is further divided into two components termed here as ƞ-component and p-component. These components of plastic strain increment occur due to effective stress increments in the directions of constant mean effective principal stress and constant stress ratio, respectively. Corresponding to two components of plastic strain increment, two sets of plastic potential, yield function and hardening function are proposed, and nonassociated flow rule is formulated. The strain increments due to the effective stress increments in various directions are calculated by this theory. The model thus obtained for the stress-strain behaviour of soil reasonably explains the dependency of the direction of plastic strain increment on that of effective stress increment. This model is applied to the analyses of triaxial drained shear tests along various stress paths and satisfactory result is obtained. The stress-strain behaviour by this model is compared with those obtained by theories by Roscoe et al. (1963, 1968) and Calladine (1971). Discussion on the stress-strain behaviour of soil predicted by these theories including the author’s one is given.

Endochronic inelasticity and incremental plasticity

Endochronic and non-associated plastic formulations are compared by introducing an “inelastic stiffness locus”, defined as the locus of all strain increments in the strain space which give the same magnitude of inelastic strain increments. For classical plasticity the locus is a straight line, while for endochronic formulations it is a circle, sphere or a quadratic surface (ellipsoid). Similarly to vertex hardening models and the deformation theory of plasticity, endochronic theory gives inelastic strain for strain increments tangential to the current loading surface, while plasticity gives perfectly elastic response. However, in contrast to vertex hardening, the endochronic inelastic strain for tangential strain increments is normal to the loading surface. Consequently, endochronic theory is stiffer than vertex hardening for this loading direction and is less prone to indicate instability. However, it is softer than plasticity. Among all possible constitutive relations, plasticity (without yield vertex) is least prone to indicate material instability, and so it is the least safe model to assume if test data are inconclusive as far as the type of constitution law is concerned.Tangential linearization of the endochronic inelasticity is presented. The tensor of tangential moduli, with all of its components, depends continuously on the strain increment direction in the strain space. Endochronic analogs of the loading surface and of kinematic and isotropic hardening rules are indicated, and stress-induced anisotropy of the quadratic form defining intrinsic time increments is formulated. It is shown that for proportional loading an endochronic formulation can be readily converted to an equivalent plasticity formulation. The fracturing material theory in which the loading function depends on strain rather than stress is also analyzed and it is shown that its inelastic stiffness locus is similar as for plasticity.Implications for material instability, and especially for stability of the response to pulsating loads of small amplitude, are discussed. By contrast to plasticity, but similarly to viscoplasticity, the endochronic inelasticity violates Liapunov-type stability conditions, but it meets a proper continuity condition. Refinements to satisfy both are possible, but questionable if one deals with materials such as geological materials, which are unstable or exhibit strain softening. Introducing unloading and reloading criteria and a certain type of kinematic hardening, the endochronic formulation may be refined so as to model cyclic strain accumulation yet satisfy Drucker's postulate for the hysteresis loops.

Nonisothermal Kinematic Hardening Law in Plasticity

The incremental stress-strain relations have been formulated for nonisothermal kinematic hardening materials in which the yield surface is assumed to change its size with temperature. By following Prager’s assumption, the yield surface undergoes a translation in the direction of the plastic strain increment during the plastic flow. As pointed out by Shield and Ziegler, the yield surface with Prager’s hardening rule, in general, does not satisfy the invariance requirement with respect to a reduction in dimensions in the stress space. Nevertheless, if the von Mises criterion is adopted, such a discrepancy would not arise. If, however, other types of criterion are used, such as Tresca criterion, Ziegler’s modification must be incorporated.

Experimental and Theoretical Investigations of a Passive Earth Pressure Problem

Synopsis The Paper presents data from the experimental investigation of the passive failure of an initially vertical, rough, plane wall which is rotated about its toe into a mass of dry sand with an unloaded horizontal surface. The sand is constrained to deform in plane strain. Measurements are made of the distribution of the normal and shear stresses on the wall while strain fields are determined from observations by an X-ray technique of the positions of lead shot buried in the sand mass. The form of the rupture surface mechanism in dense sand is determined. The strain data are used to investigate the φ= constant assumption of the Sokolovski (1965) method and Sokolovski solutions are compared with the experimental data for two cases. There is marked agreement between the predicted principal compressive stress directions and the observed principal compressive strain increment directions. Cet Exposé présente des données provenant de recherches expérimentales portant sur la rupture de butée d'un mur plan, rugueux, vertical au départ auquel on imprime un mouvement de rotation autour de son pied dans une masse de sable sec à surface horizontale ne supportant pas de charge. Le sable est contraint de se déformer selon une déformation en plan. On prend des mesures de la répartition des contraintes normales et de cisaillement subies par le mur tandis que des champs de déformation sont filig;xés à partir d'observations au moyen d'une technique à rayons X des positions de billes de plomb enterées dans la masse de sable. On établit le genre du mécanisme de la surface de rupture. On utilise les données relatives à la déformation pour examiner la supposition portant sur la constante φ de la méthode de Sokolovski (1965) et l'on compare des solutions de Sokolovski aux données expérimentales relativement à deux cas. Il se présente un accord notable entre les directions des contraintes de compression principales prévues et les directions des accroissements de déformation de compression principales observées.