Peter Hähner

On the Portevin¿Le Chatelier effect: theoretical modeling and numerical results

Abstract A new model is presented for a physically-consistent description of plastic material instabilities referred to as Portevin–Le Chatelier (PLC) effect, namely the oscillatory plastic flow that may be observed in metallic alloys subjected to load-or displacement-controlled plastic deformation in a certain range of strain, strain rate and temperature. The model is conceived to describe the kinetics of Dynamic Strain Ageing (DSA), that is the dynamic interaction between mobile dislocations and diffusing solute atoms which is known to be the primary mechanism inducing the PLC effect. The model is coupled in time and (one-dimensional) space and introduces two intrinsic time scales in the evolution equations and a characteristic length scale through a diffusion-like term with spatial second-order gradient. Approximate analytical solutions are first derived for the boundaries of the PLC range and for the strain localization characteristics defining the kinematics of PLC deformation bands. Numerical results are then obtained through Finite Differences solutions of the space–time coupled equations: a considerable wealth of features is discovered, including the appearance of various PLC band types depending on the applied strain rate. Phenomenological characteristics (i.e. stress–strain curves) are presented together with the corresponding space–time patterns of strain localization. The obtained results are in agreement with the analytical solutions developed here and with the available experimental observations on the PLC effect.

On the characteristics of Portevin–Le Chatelier bands in aluminum alloy 5182 under stress-controlled and strain-controlled tensile testing

Abstract Plastic instabilities associated with the Portevin–Le Chatelier (PLC) effect are investigated in an Al-based alloy of the 5000 series, in a wide range of temperatures and loading rates. The domains of occurrence of the PLC effect are explored and compared for constant stress rate and constant strain rate tensile testing. In both cases, experimental results show that the PLC domains are bounded towards high temperatures and loading rates. The critical strain for the onset of repeated yielding increases with increasing loading rate at low temperatures and high loading rates (normal behaviour), while it decreases at high temperatures and low loading rates (inverse behaviour), with different types of serration associated with the normal and the inverse behaviour. A simple model is proposed to explain this relation between the PLC type and the behaviour of the critical strain. Moreover, the specimen surface finish is found to have a significant influence on the stress–strain curves and the band propagation velocities. Finally, the experimentally observed PLC domain is shown to be in good agreement with a physical model of the dynamic strain ageing associated with the PLC effect.

Modelling the spatiotemporal aspects of the Portevin-Le Châtelier effect

Abstract Based on a phenomenological treatment of dynamic strain ageing, the spatiotemporal dynamics of Portevin-Le Châtelier band propagation is modelled. To this end, the efficiencies of several potential propagation mechanisms, i.e. cross-slip, long-range dislocation interactions, and incompatibility stresses, are compared. For constant-stress-rate testing conditions, pulse-like solutions are constructed using a piecewise linear caricature of the stress-strain rate characteristic. Expressions for the propagation velocity and the strain-burst amplitude are derived. For constant-strain-rate testing, control parameters are given for (1) the occurrence of non-propagative type C bands and (2) the transition (Hopf bifurcation) from the continuous propagation of a type A band to the hopping motion of a type B band. The results are compared with experimental findings, and future experiments are proposed to check theoretical predictions.

Fractal analysis of deformation-induced dislocation patterns

The paper reports extensive analyses of the fractal geometry of cellular dislocation structures observed in Cu deformed in multiple-slip orientation. Several methods presented for the determination of fractal dimensions are shown to give consistent results. Criteria are formulated which allow the distinguishing of fractal from non-fractal patterns, and implications of fractal dislocation patterning for quantitative metallography are discussed in detail. For an interpretation of the findings a theoretical model is outlined according to which dislocation cell formation is associated to a noise-induced structural transition far from equilibrium. This allows relating the observed fractal dimensions to the stochastic properties of deformation by collective dislocation glide.

On the kinematics of Portevin-Le Chatelier bands: theoretical and numerical modelling

A model is presented for the description of the Portevin–Le Chatelier (PLC) effect, namely the repetitive plastic yielding that may be observed in metallic alloys for certain ranges of temperature and applied stress or strain rates. The model reflects the underlying microstructural dynamic strain ageing (DSA), i.e. the dynamic interaction between mobile dislocations and diffusing solute atoms. Focus is made on PLC instabilities of Type A, that is PLC bands that nucleate and propagate continuously throughout the tensile specimen as solitary plastic waves. The kinematics of these PLC bands is studied analytically and validated numerically by a Finite Differences integration of the model equations. The characteristics of the PLC bands, that is band speed, band width and band plastic strain, are determined from the space–time fields of plastic activity. The band parameters exhibit very good matching with the theoretical results and order-of-magnitude agreement with the experimental observation of the PLC effect.

Correlation of temporal instabilities and spatial localization during Portevin–LeChatelier deformation of Cu–10 at.% Al and Cu–15 at.% Al

Abstract The Portevin–LeChatelier (PLC) effect with characteristic temporal instabilities in the recorded stress is closely associated with local inhomogeneities of deformation. Measurements on Cu–10 at.% Al and 15 at.% Al polycrystals by means of a novel laser scanning extensometer, which provides quasi-simultaneous spatial resolution along the main part of the specimen, have been used for an unambiguous characterization of the PLC deformation bands of type A (continuous propagation of single band along the specimen), type B (discontinuous band propagation) and type C (stochastic nucleation of single bands along the specimen), supplementing and correcting the common characterization from the load serrations. Surprising transitions between the types have been detected in the effective stress versus temperature (or versus strain rate) diagram. The parameters of type A bands (local band strain, band width, propagation rate) have been measured at various temperatures in dependence of grain size, specimen thickness and imposed deformation rate, and are compared with predictions of a recent theoretical treatment.

Portevin-LeChatelier effect in strain and stress controlled tensile tests

Abstract The nucleation and propagation of Portevin–LeChatelier (PLC) deformation bands in Cu–15at.%Al have been studied by means of a novel multi-zone scanning laser extensometer providing information on the development of local strain along the main part of the specimen, in addition to the conventional measurement of stress serrations. By means of tensile tests with imposed constant strain rate independent data have been determined on propagation velocity, concentrated strain and width of the bands appearing in three different types of PLC-deformation. The tensile test machine has been softened to permit measurements with imposed constant stress rate. First results will be presented and compared to the strain-controlled ones on Cu–15at%Al.

Modelling of propagative plastic instabilities

The present paper gives a brief account of recent progress concerning the modeling aspects of phenomena characterized by the nucleation and propagation of localized modes of plastic deformation in a tensile test. Emphasis is laid on the various physical mechanisms that govern the spatio-temporal dynamics of different types of propagative instabilities. In view of the diversity of phenomena observed, it is attempted to present a comprehensive survey of those aspects that proved to be important in solving long-standing problems related to the propagation of plastic waves. The results presented may stimulate further experimental investigations in order to check the theoretical findings.

The flow stress of fractal dislocation arrangements

Abstract In metals deforming under multislip conditions, fractal dislocation arrangements may develop. The composite model of the flow stress of a dislocation cell structure is generalized for dislocation arrangements which exhibit a broad spectrum of local hardness. Both forest strengthening (cell walls) and the Orowan stress (cell interiors) are taken into account. Relations between the flow stress and the characteristic parameters of the dislocation arrangement such as fractal dimension, maximum and minimum cell size are established. The model is used to calculate stress dependences of fractal dimension and cell wall volume fraction of dislocation patterns in high-symmetry oriented f.c.c. single crystals. The results are in good agreement with experimental observations on Cu and Cu-rich Cu–Mn.

Dislocation dynamics and work hardening of fractal dislocation cell structures

The dislocation dynamics during multiple slip deformation is formulated in terms of a simple stochastic model for the evolution of the densities of mobile and immobile dislocations. Randomness results in modified effective dislocation multiplication and reaction rates which account for the topology of the evolving microstructure. Depending on the intensity of local strain-rate fluctuations, two types of solution can be distinguished: (1) At low noise levels homogeneous dislocation structures develop which are described by a single characteristic length scale, i.e. the mean dislocation spacing. This is the case of b.c.c. metals deformed at low temperature. (2) Above a critical noise level self-similar dislocation cell patterns are found which are characterized by a lower cut-off length, i.e. the minimum dislocation spacing in the cell walls, and scale invariance beyond that cut-off. This case refers to rate-insensitive f.c.c. metals, where fractal dislocation structures have been identified recently [P. Hahner, K. Bay, M. Zaiser, Phys. Rev. Lett. 81 (1998) 2470]. The model yields critical deformation conditions for fractal dislocation patterning and enables one to establish relations between the evolution of the fractal dimension of the cell structure, the strain-hardening behaviour, and the underlying dislocation dynamics. This is achieved without postulating a priori that the dislocation microstructure be heterogeneous.