László S. Tóth

A Dislocation-Based Model for All Hardening Stages in Large Strain Deformation

A new model is presented to describe the hardening behaviour of cell-forming crystalline ma- terials at large strains. Following previous approaches, the model considers a cellular dislocation structure consisting of two phases: the cell walls and the cell interiors. The dislocation density evolution in the two phases is considered in conjunction with a mechanical analysis for the cell structure in torsional defor- mation in which the cell walls are lying at 458 with respect to the macroscopic shear plane and are strongly elongated in the direction perpendicular to the applied shear direction. Guided by recent results on the volume fraction of cell walls (Muller, Zehetbauer, Borbely and Ungar, Z. Metallk. 1995, 86, 827), the cell- wall volume fraction is considered to decrease as a function of strain. Within a single formulation, all stages of large strain behaviour are correctly reproduced in an application for copper torsion. Moreover, strain rate and temperature eAects are accounted for correctly and the predicted dislocation densities are in accord with experimental measurements. It is suggested that the factor responsible for the occurrence of hardening Stages IV and V is a continuous decrease of the volume fraction of the cell walls at large strains. A significant eAect of the deformation texture variation on strain hardening is also discussed. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.

Effect of rate sensitivity on the stability of torsion textures

Abstract The yield stress potentials pertaining to the rate sensitive deformation of cubic crystals are described; they are shown to be strictly convex, with shapes that depend on the rate sensitivity exponent, m. A four -variable maximization procedure is presented, which permits the stress state associated with full constraint conditions to be found rapidly. The rigid body, lattice and glide rotation rates are clearly distinguished and specified; the lattermost rate decreases as m is increased, going to zero when m = 1. It is demonstrated that none of the torsion ideal orientations can be permanently stable under rate sensitive conditions of flow. An orientation stability parameter is introduced, which is used for the construction of orientation stability maps (OSMs) in Euler space. The OSMs are employed to characterize the tubes associated with fixed end testing. Finally, a “cloud” model of texture development is proposed; it is employed to account for the initial anti-shear rotations of experimental textures and for their subsequent migration parallel to the rigid body rotation, i.e. to the applied shear. With the aid of an axial stress map in Euler space, the cloud model is also used to account for the development of compressive, followed by tensile, axial stresses.

Strain Hardening at Large Strains as Predicted by Dislocation Based Polycrystal Plasticity Model

A recent strain hardening model for late deformation stages (Estrin, Y., Toth, L.S., Molinari, A., and Brechet, Y., Acta Materialia, 1998, A dislocation-based model for all hardening stages in large strain deformation, Vol. 46, pp. 5509-5522) was generalized for the 3D case and for arbitrary strain paths. The model is based on a cellular dislocation arrangement in which a single- phase material is considered as a composite of a hard skeleton of cell walls and soft cell interiors. An important point in the approach is the evolution of the volume fraction of the cell walls which decreases with the deformation and gives rise to a plateau-like behavior (Stage IV) followed by a drop-off (Stage V) of the strain hardening rate observed at large strains. The hardening model was implemented into the viscoplastic self-consistent polycrystal model to predict hardening curves corresponding to different proportional loading paths. The calculated curves were evaluated to elucidate the path dependence of hardening.

Stress response and persistence characteristics of the ideal orientations of shear textures

Abstract The stress states associated with the ideal orientations for the shear deformation of f.c.c. polycrystals are derived; the persistence characteristics of these orientations are also investigated. First, a classical, rate independent analysis is implemented to deduce analytical expressions for the stress states, active slip systems and lattice rotation rates for single crystals lying along the ideal orientation fibres. These solutions exhibit the well-known ambiguity characteristics of crystal plasticity theory. A rate sensitive analysis, which is expectedly free from these ambiguity problems, is then carried out. Some closed-form analytical solutions are obtained, and it is shown that the classical ambiguities are non-existent in the rate independent limiting built of the rate sensitive solution. The persistence characteristics of the ideal single crystal orientations are also analyzed under rate sensitive conditions in the form of orientation persistence maps (OPMs). To infer the texture evolution characteristics of polycrystals , the rotation fields in the vicinity of the ideal orientations are examined, together with the associated divergence maps in Euler space. Finally, polycrystal model predictions are presented in ODF form and the corresponding texture evolution is explained in terms of the OPM and divergence maps.

DISCRETIZATION TECHNIQUES FOR ORIENTATION DISTRIBUTION FUNCTIONS

Two methods for the discrete orientation representation of continuous orentation distribution functions (ODFs) are presented. The first one is based on the cumulative ODF, while the second one uses a minimum orientation distance criterion. The properties of these new techniques are discussed and contrasted with each other as well as with an earlier method which is based on cutting below certain limiting intensity. Four kinds of tests have been carried out on these techniques: i. their performance in reproducing the ODF, ii. prediction of physical parameters, as R and M values, iii. deformation texture predictions, iv. rediscretizations during deformation texture modelling. The results of these tests show the good applicability of the proposed two new discretization techniques for approximating the ODF, to calculate physical parameters and for deformation texture modelling, even at relatively low number of orientations. The cutting technique, however, found to be unprecise, even at large number of orientations. On the basis of the results obtained during the rediscretization tests, a new technique for modelling twinning in deformation texture codes has been proposed.

Tuning a self consistent viscoplastic model by finite element results—I. Modeling

Abstract The self consistent model of Molinari et al.[Acta metall.35, 2983 (1987)] is modified with the help of the finite element results of Gilormini and Germain [Int. J. Solids Struct., 23, 413 (1987)]. The modification implies the introduction of a new scalar parameter in the interaction law of the self consistent model. In this first part, the model is established and the new parameter is tuned so that the self consistent predictions and the results of the finite element predictions for a spherical inclusion in an infinite matrix nearly coincide. A simple relation between the new parameter and the value of the strain rate sensitivity has been found.

Development of ferrite rolling textures in low- and extra low-carbon steels

The texture changes that affect the main ideal orientations displayed by rolled low-and extra low-carbon steels are examined. Predictions are made regarding the stability of these components using a rate-dependent theory for mixed 112 (111) and 110 (111) slip. Both full constraint (Taylor) and relaxed constraint (lath and pancake) grain interaction models are employed. It is shown that the experimentally observed texture changes can be reproduced by adopting the deformation modes which require the least plastic work. The orientation dependences of the preferred deformation modes are described together with the relative stabilities of the expected end textures.

Dislocation structure and work hardening in polycrystalline ofhc copper rods deformed by torsion and tension

Abstract The dislocation structure during work hardening in copper deformed by torsion and tension is investigated by X-ray line broadening and TEM. In the case of torsion the equivalent tensile flow stress and the rate of work hardening is lower than that obtained in tensile experiments. At the same time the dislocation density at the same equivalent strain is considerably larger in the torsionally deformed material and the TEM microstructure indicates a double-slip deformation mechanism. In the tensile deformed samples asymmetric X-ray line broadening indicates long-range internal stresses with relatively lower dislocation densities. In this mode of deformation multiple slip takes place. The lower equivalent flow stress and the smaller rate of work hardening in the torsionally deformed material is correlated with the restricted number of slip systems and the absence of strong long-range internal stresses, which leads to the relative ease of generating high dislocation densities.

On the stability of the ideal orientations of rolling textures for F.C.C. polycrystals

Abstract The stability of the ideal orientations of rolling textures for f.c.c. metals (cube, Goss, brass, copper, Taylor and S) and their influence on texture formation are investigated. The characteristics of the three-dimensional lattice rotation fields at and in the vicinity of these orientations, as well as the development of preferred orientations during deformation, are simulated numerically using a rate-sensitive crystal plasticity model. Three types of boundary conditions are considered: plane strain compression, “lath” compression, and “pancake” compression. The investigation shows that, with the help of the three-dimensional rotation fields, the rotation characteristics of an orientation can be described by its rate of change in Euler space, the corresponding gradients and the relative rate of change of its ODF intensity. The formation of rolling textures depends on the characteristics of the α (Goss-brass) and β fibres. Boundary conditions affect the composition of β and the flow velocity of orientations towards and along α and β, and consequently result in different texture components after large deformation.

Qualitative population divergence in proximate determination of a sexually selected trait in the collared flycatcher

We examined proximate determination of sexually selected forehead patch size in a Central‐European population of Ficedula albicollis, the collared flycatcher, using a 9‐year database, and compared our results with those obtained in other populations of the same and the sister species. Between‐individual variation of forehead patch size was large, its repeatability larger than, and heritability similar to the Swedish population. Unlike in the other populations, the trait proved unaffected by body condition, and only very slightly influenced by age. There was no relationship between forehead patch size and breeding lifespan, and a marginal negative association with survivorship in adult males. Our results suggest that additive genetic variance of the trait in this population is large, but genes act independently of body condition, and there is no viability indicator value of the trait. This is the first report of a qualitative intraspecific difference in proximate determination of a sexually selected trait.