Lattice strain partitioning in a two-phase alloy and its redistribution upon yielding

Abstract

Load sharing in two-phase alloys is investigated by means of finite element simulations of polycrystalline aggregates subjected to uni-axial compression. Attention is focused on the redistribution of stress as the overall load level increases and the material progresses from a purely elastic state to eventual yielding of both phases. Virtual specimens having two phases, one of iron and the other of copper, were created and subjected to compression. The lattice (elastic) strains, which are directly proportional to the stress, were examined over the course of the compression test for a system having equal volume fractions of each phase. Comparisons were made first with measurements by neutron diffraction to confirm that the simulated strains followed the observed behavior. The computed lattice strain tensors then were examined in terms of the changes in their principal directions as the overall load increased. The redirection of the stress signals a change in the relative stiffnesses of the phases, from which follows a repartitioning of stress between the phases.

Introduction

Most bulk metallic materials are polycrystalline. These materials often are composed of multiple phases to enhance their properties by combining merits from the different phases. To understand the properties of a multi-phase material fully, it is necessary to understand how the phases interact when placed under load. In multi-phase systems, how applied loads are carried by the material depends on interactions between grains of the different phases. The distribution of stress over a multi-phase system may also be thought of in terms of the partitioning of lattice strains, since lattice (elastic) strains are linearly proportional to the stress. As a multi-phase material is loaded to stress levels that cause plastic deformation, the relative contributions of the various to the overall load-carrying capacity changes. This is evident in the distinct stages of behavior reported by Harjo et al. [1] for $\alpha$ –$\gamma$ Fe–Cr–Ni alloys. Lattice strain evolution in a two-phase material shows three distinct stages [1]. These stages also are evident in the behavior observed in the experiments of a iron–copper specimen shown schematically in Fig. 1. Stage I is characterized by elastic behavior of both phases; in Stage II, the weaker (Cu) phase has yielded, but the stronger phase (Fe) remains elastic; in Stage 3, both phases have yielded. Although all three stages were not observed distinctly, the transition from Stage I to Stage II in TRIP steel was reported by Furnémont et al. [2]. Similar lattice strain partitioning over several stages also was reported by Stone et al. [3] for a two phase Waspalloy. Typically, the transverse lattice strain of the weaker phase in crystals may show a particularly interesting behavior: the sign of the slope of the transverse direction lattice strain reverses during State II. As will be demonstrated, this behavior coincides with a redirection of the stress rather than a reduction in its magnitude.

The behavior presented in Fig. 1 is obtained from diffraction data on diffraction volumes containing many crystals. Neutron and X-ray diffraction techniques provide the ability to measure in situ lattice strain in bulk polycrystalline solids. Examples are published by Turner et al. [4], Holden et al. [5], Clausen and Lorentzen [6], Clausen et al. [7], [8]. The diffraction volumes typically are sufficiently large to ensure averaging over many crystals. Recently, experimental investigations on micro-mechanical behavior of multi-phase polycrystalline solids, of which measurements are performed on individual grains Lienert et al. [9], have been reported.

The modeling tool for interpreting multi-phase diffraction experiments has been primarily the self-consistent method, which is based on a fundamental solution by Eshelby [10] of an elastic strain field induced by an ellipsoidal inclusion in a homogeneous matrix. For example, Dye et al. [11] formulated a two-phase elastic-plastic self-consistent model and validated the model with lattice strains from diffraction experiments. Commentz et al. [12] performed texture and macroscopic stress–stain analyses on iron–copper two-phase materials with a range of volume fractions using a viscoplastic self-consistent model. An advantage of a self-consistent model is that it provides a mechanism for grain interactions without becoming too complicated or requiring long execution times. As more information such as grain shape, grain size, and phase distribution information is added to the self-consistent model, the complexity increases and much of the advantage is mitigated. If local responses at the grain or sub-grain level are of interest, explicit representation of grain geometry is required and using the finite element method is more suitable. Finite element analyses with a restricted slip model for modeling lattice strain evolution from diffraction experiments have been performed for single phase polycrystalline solids [9], [13], but analyses of two-phase polycrystalline solid using the restricted slip model are scarce in the literature.

The focus of this paper is on grain interactions in a uniformly distributed, iron–copper, two-phase polycrystalline alloy in the small macroscopic strain regime. Grain interactions control the partitioning of strain among the constituent crystals where the material is placed under load. As the load increases to the point that one phase yields, the partitioning changes. This coincides with a change in the straining direction in that phase, as seen by relative changes in the slopes of the lattice (elastic) strains as functions of applied load. The progression of this process was investigated using a finite element based model of polyphase materials which naturally accounts for the grain interactions, as well as for the spatial distributions and connectivities of two phases. The investigation focuses on a system with each volume fractions of the two phases, 50% Fe and 50% Cu (Fe50–Cu50), and has been guided by experimental data available for physical specimens of the approximately the same phase volume fractions [14]. Simulations also were performed for samples with different combinations of the volume fractions, namely, pure copper (Cu100) and 67% iron and 33% copper (Fe67–Cu33), to determine constitutive model parameters. Specimens used in all of these experiments were extracted from material produced by the sintering of high purity powders of copper and iron. A thorough description of these materials is given in Commentz et al. [12]. Here we mention that by fabricating the materials from the same powders, the materials differed principally in the connectivities of the phases. The grain sizes in the physical specimens ranged from 1 to 15 $\mu$m regardless of the volume fraction (a uniform grain size was assumed in the simulations). Lattice strains were measured on the Spectrometer for Materials Research at Temperature and Stress (SMARTS) at the Los Alamos Neutron Science Center (LANSCE) using in situ loading in a manner similar to that described by Carter and Bourke [15]. Independent tests were performed to ascertain the stress–strain response of the bulk material under similar conditions of temperature and strain rate to the in situ diffraction tests Mataya [16].

The intent of the present paper is to offer a mechanistic understanding of the multistage behavior described by Harjo et al. [1] based on the results of the simulations. The current work differs from the analyses of Commentz et al. [12] and of Hartig and Mecking [17] on iron–copper polycrystals in several respects. Commentz et al. [12] discuss the accuracy of a Taylor-type linking assumption and of a self-consistent model for predicting textures over large deformation in light of tests on these materials. Hartig and Mecking [17] extend this investigation by examining the ability of a finite element model to predict the evolution of the textures across the suite of phase volume fractions. They also compare measured and computed lattice strains on average. Here, we employ a finite element model to analyze the three stages in the lattice strain behaviors under small deformation observed from the diffraction experiment. Stress histories were extracted for different sets of grains whose lattice orientations satisfy particular Bragg conditions. The finite element simulations offer a more complete explanation for the interesting macro- and micro-mechanical behaviors observed from the experiment mentioned previously.

In the next section, the constitutive model and the finite element formulation used for the simulations are described briefly. The two-phase, polycrystalline, virtual samples are then presented. This is followed by a summary of the procedure used to determine the constitutive model parameters. Next we confirm that the simulations reproduce trends observed for measured lattice strains under in situ loading. Then we examine the simulation results in greater detail to explain the transition of load sharing among the phases over the course of the stages in the loading.