Coupled mesoscopic constitutive modelling and finite element simulation for plastic flow and microstructure of two-phase alloys

Abstract

A mesoscopic dislocation-based model was coupled with macro-scale finite element analysis for concurrent study of local plastic flow and microstructure of two-phase alloys during thermomechanical deformation. The model was implemented in the ABAQUS code to simulate the thermomechanical processing of a Ti–6Al–4V alloy in the (α + β) phase field, with consideration of the effects of local dislocation density variation, deformation heating and phase volume fraction. The simulation show that the intergranular interaction results in non-uniform distribution of dislocations within each grain, particularly in the initial stages of deformation. Phase boundaries pose stronger influence on deformation than grain boundaries. The onset of shear localization was strongly influenced by the strain rate sensitivity parameter, deformation heating, phase volume fraction, and the die/sample friction coefficient. Both deformation heating and phase transformation in the shear-localized region contributes to the flow-stress variation during processing. The phase volume fraction largely affects the microstructure, distribution of the equivalent stress, but not the equivalent strain.

Introduction

Many engineering alloys are two-phase materials. Their thermomechanical processing is often carried out in the two- (major) phase field to impart optimum microstructure and mechanical properties. Here, the major concerns are a balanced grain/phase structure, flow resistance and deformability, whereas the issue with preferred crystallographic orientation, or texturing, is not as important as it is in cold deformation [1]. Hence, it is of theoretical and practical significance to develop easily assessable models for the understanding of concurrent plastic flow and microstructural variation at the local level for the processing of such alloys. Computational mesoscopic modelling has become an indispensable tool for such purposes, but has not been widely applied to two-phase materials.

Deformation of single-phase polycrystalline materials has been investigated by a variety of classic elasto-plastic deformation models that consider the interaction among neighbouring grains [2], [3], [4], [5], [6], [7], [8], [9], [10]. These models treat differently the two essential deformation constraints of polycrystalline materials, i.e. intergranular force equilibrium and strain compatibility, but none has strictly satisfied both constraints simultaneously. For instance, the full-constraint Taylor model assumes an identical strain in all grains, but ignores the intergranular force equilibrium [2], [3]. The Sachs model satisfies the force equilibrium but the inter-grain compatibility is ignored [1], [7]. The self-consistent models consider the equilibrium and compatibility over an averaged field [4], [5], [6]. In the past two decades, finite element (FE) method has been widely used to study elasto-plastic deformation of crystalline materials. FE modelling is usually combined with classic crystalline elasto-plastic deformation theories, where the constraints of intergranular force equilibrium and strain compatibility can be strictly satisfied and specific deformation around grain boundaries and in the grain interior can be readily simulated [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24].

Various constitutive models for FE analysis of elasto-plastic deformation of crystalline materials have been developed [16], [25], [26], [27], [28], [29]. Most of the models consider deformation of single-phase crystalline materials. For two- or multi-phase materials, the mechanical properties and deformation behaviour are strongly influenced by phase volume fraction, phase distribution, properties of the constituent phases and interactions across phase boundaries, particularly when there is a large difference in their mechanical properties. Although various forms of the law of mixtures have been proposed to predict the mechanical properties of two- and multi-phase materials [30], [31], [32], these can only represent a macroscopic approximation of the macroscopic mechanical properties from the properties of the constituent phases, and are unable to show detailed microstructural deformation and interactions at grain and phase boundaries.

Friction at die/sample interfaces plays an important role for stress–strain relationships at medium to high levels of plastic deformation [38], [33], particularly at the macroscopic scale [34], [35], [36]. In an example, friction-induced rapid hardening for single crystal deformation can lead to 25% increment of the flow stress [37]. So it is essential to consider friction in the analysis of polycrystalline deformation.

FE modelling is a powerful tool for simulation of plastic deformation of two- or multi-phase materials. The influence of phase volume fraction, phase distribution, size and properties of constituent phases on the deformation behaviour of two- or multi-phase crystalline materials can be investigated with great insight if FE modelling is coupled with more elaborate constitutive models for polycrystalline deformation.

Several FE investigations have been carried out for two- or multi-phase composites and crystalline materials. Karlsson and Lindén used FE modelling to study the yield and work hardening of ferrite–pearlite aggregates with a continuous ferrite matrix [39]. Ankem et al. [40], [41], [42], [43] used FE modelling to investigate the stress–strain relationship of two-phase materials at different particle sizes and phase distributions. According to their results, most of the strain is carried by the softer phase and the stress by the harder phase, and transverse stresses are generated as a result of the interaction between the phases. The stress and strain distribution and the magnitude of the transverse stresses depend on the phase volume fraction and the strength difference between the two phases. Bolmaro et al. [44] used FE analysis to study the influence of volume fraction, geometry, phase distribution, strain hardening and yield stress ratio on the deformation behaviour of two-phase materials. Steinkopff et al. [45], [46], [47] developed a reasoning algorithm for net-adaptation and the multiphase element method for the FE modelling of multiphase composites, such as Ag–Ni fiber composites and Ag–Ni particulate composites. The phase boundaries are put inside the elements instead of the normal element edges, and the phase properties are designated to the integration points. Grujicic and Sankaran [48] developed a constitutive model which describes transformation plasticity accompanying stress-assisted martensitic transformation in two-phase materials consisting of a stable matrix and a transforming dispersed phase, and the model was used to analyse the uniaxial tensile behaviour of the two-phase systems. All these studies that use FE modelling of the deformation characteristics of two- or multi-phase materials have considered the influence of phase volume fraction and phase distribution, but none of these has incorporated the influence of mesoscopic dislocation density variation on the deformation behaviour. In practice, microstructural evolution of two- or multi-phase materials during deformation is closely associated with dislocation activities as well as the phase ratio and distribution. The effect of interfacial friction was only studied for either single crystals or at the mesoscopic scale [34], [35], [36].

In this paper, a meso-scale dislocation-based constitutive model was integrated with a micro-scale rate-dependent plastic flow equation, and then coupled with finite element analysis for accurate study of both the stress–strain relationship and microstructural evolution during thermomechanical deformation of two-phase alloys, with due consideration of thermal softening as a result of deformation heating. Die/sample interface friction has also been considered for both sticking and slipping frictions, and the coefficient of slipping friction varies from 0.1 to 0.01. The model was implemented into the commercial FEM package ABAQUS through the user subroutine UMAT. The influence of thermomechanical processing parameters, such as strain rate, temperature, and phase volume fraction on the deformation characteristics of an (α + β) Ti–6Al–4V alloy was studied as an example. The simulated stress–strain curves and grain structure were compared with experimental results for the Ti–6Al–4V alloy deformed in the (α + β) phase field.