The correlation between plastic strain and anisotropy strain in aluminium alloy polycrystals

Abstract

Samples of fine-grained Al alloy reinforced with SiC particles were subjected to in situ deformation in four-point bending, and multiple-peak diffraction patterns were collected along a line traversing the sample bend axis, using time-of-flight neutron diffraction at the ISIS pulsed source, at the Rutherford Appleton Laboratory, Oxford. The patterns were analysed in order to determine the average macroscopic lattice (elastic) strain, and also the so-called anisotropy strain. It was assumed that the deformation condition in the sample corresponded to a simple uniaxial stress state, and that the entire composite plastic strain was accommodated by matrix plasticity. This allowed comparisons to be made with the prediction of some models, namely, a simple two-phase uniform strain (Voigt) model, and an elasto-plastic self-consistent polycrystalline deformation model. A direct correlation is observed between the matrix plastic strain and the anisotropy strain determined from the diffraction spectra.

Introduction

Diffraction of penetrating radiation is a unique non-destructive tool for strain measurement in crystalline materials [1]. The technique is remarkably versatile, and allows a wide variety of structural and deformation parameters to be evaluated in individual grains or single crystals, as well as in powders or polycrystalline aggregates. The use of diffraction for strain measurement deep inside polycrystalline specimens is particularly interesting from the engineering viewpoint, since it presents novel opportunities for detailed non-destructive characterisation of stress and damage levels in real components at sub-millimetre spatial resolution.

The mechanical properties of polycrystalline aggregates and composites are influenced by internal stress under monotonic and cyclic thermo-mechanical loading. Internal stresses develop during treatment processes, such as consolidation of powders, solution heat treatment and quenching, extrusion, forging and rolling, shot peening and bending. For the purposes of analysis, internal stresses are usually classified into macro- and micro-stress components, which refers to the length scales over which the stress varies [2]. Traditionally, structural engineers are primarily concerned with average macrostress values, which they use in lifing calculations. Microstresses, on the other hand, exist on a finer spatial scale, and represent the mechanical interaction between microscopic components, e.g. different neighbouring grains or groups of grains, grains belonging to different phases, lattice defects and defect structures, such as dislocation cells and grain boundaries.

While the above classification is helpful as far as internal stress analysis is concerned, it may also lead to a mistaken belief that macroscopic and microscopic stresses can be physically completely de-coupled. This, however, is not the case, as most thermal or mechanical processes are likely to affect both types of stress at the same time. For example deformation processing in particular generates residual macrostresses. However, the attendant evolution of microstresses, and intergranular stresses in particular, is less studied and not well understood. Stress-relieving heat treatments are often used to bring the macroscopic stress to negligible or reduced levels. However such a treatment is accompanied by a change in the microstresses, due to thermal expansion anisotropy and microstructural processes, such as grain growth, precipitation, etc. It is our view [3] that intergranular stresses must play an important role in determining the strength and fatigue endurance of polycrystalline components, and that their origins and evolution should be correlated with macroscopic deformation history, on the one hand, and residual strength or life, on the other. In the present paper, we address the former aspect of this problem.

Most metallic alloys are employed in modern structural engineering applications in polycrystalline form. Schmid's law [4] provides a well-established basis for understanding single crystal plasticity, and can be used as the starting point for modelling an ensemble of differently oriented grains. However, the single crystal deformation mode considered by Schmid is unconstrained; in fact the solution to the solid mechanics problem of co-operative deformation of a grain ensemble within a polycrystal must simultaneously satisfy the kinematic (strain compatibility) and dynamic (stress equilibrium) conditions. A solution satisfying only one of the requirements, while the other one is discarded will furnish an upper or lower bound to the original problem.

The task of determining the stress–strain partitioning between grains in polycrystals was first systematically addressed by Sachs [5]. The crucial assumption made by Sachs was that uniform stress (Reuss) conditions prevail over the whole body, while each grain deforms to the strain predicted by Schmid's law. Strain compatibility conditions are violated under this approach, and little agreement with experimental observations can be achieved.

A possible alternative is to use the assumption of uniform uniaxial strain (Voigt). Now the stresses may vary from grain to grain, introducing the concept of intergranular stress. In the simplest case, a single slip system is assumed to be active in each grain. However, this leads to different transverse strains for different grains. It thus becomes clear that this solution does not maintain a compatible deformation field. However, the approach is attractive due to its remarkable simplicity and the insight that it can offer into the evolution of intergranular stress for an imposed macrostrain. This formulation will be used in the present paper for a simple two-component elastic-ideally plastic model which usefully illustrates the correlation between plastic strain (macroscopic), and intergranular, or anisotropy strains (microscopic).

Taylor [6] recognised that to achieve a shape change described by six independent components of the strain tensor requires a minimum of five independent slip systems to be active simultaneously. Thus, kinematic compatibility conditions are satisfied firstly, while the microstress in a group or family of grains that possess a certain orientation can be determined from Schmid's law. Taylor [6] and Hill and Bishop [7] considered each grain to be ideally plastic, and successfully derived the relationship between the critical resolved shear stress and macroscopic uniaxial tensile stress for FCC polycrystals. While this result has been of major importance for predicting material strength, it is not open to validation using diffraction techniques, since it ignores the elastic component of grain deformation.

As a further step, an elasto-plastic self-consistent (EPSC) modelling approach was developed by Hill [8] using the ideas of Eshelby [9], and Kröner [10], and was first applied by Hutchinson [11]. In the present study, we use codes implemented by Turner and Tomé [12] to predict the lattice strain in grains of certain orientation, and to compare these predictions with strains determined from the shift of the corresponding diffraction peaks. In brief, a population of grains is chosen with a distribution of orientations and volume fractions that match the measured texture. Each grain in the model is treated as an ellipsoidal inclusion and is attributed anisotropic elastic constants and slip mechanisms characteristic of a single crystal of the material under study. Interactions between individual grains and the surrounding medium, (which has properties of the average of all the grains) are performed using an elasto-plastic Eshelby type formulation. Since the properties of the medium derive from the average response of all the grains, it is initially undetermined and must be solved by iteration. Small total deformations are assumed (typically less than 4%), and no lattice rotation or texture development is incorporated in the model.