Characterization of size-affected non-uniform deformation of polycrystalline copper

Highlights

• The interaction between the microstructure-induced and the specimen shape-induced non-uniform deformation was evaluated from experimental and numerical studies of the uniaxial tensile tests of the polycrystalline copper specimen with a curved gauge section.

• The deformation bands, in which the width and length are characterized by grain size, develop during the uniaxial tensile test of the polycrystalline specimen.

• We proposed a constitutive equation for the plastic strain as a function of the local stress and finite volume averaged stress.

Abstract

In polycrystalline materials, with the increase in grain size relative to the macrostructure, the collective behavior of crystal grains affects their macroscopic deformation field. To investigate the effect of relative size on the mechanical behavior of polycrystalline materials, the interaction between the microstructure-induced non-uniform deformation and the specimen shape-induced non-uniform deformation was evaluated based on experimental and numerical studies of uniaxial tensile tests of polycrystalline copper specimens with a curved gauge section. The effects of the macroscopic stress gradient and grain size on the strain field of the specimen were evaluated using specimens with different curvature radii obtained from different thermal treatment conditions. The development of strain distribution was measured using the digital image correlation method. A high strain concentration was observed at the minimum cross-section region in the specimen with smaller grains, whereas such strain concentration was relaxed in the specimen with larger grains because a random strain distribution occurred owing to the polycrystalline structure. A full-scale crystalline plasticity finite element method simulation, under conditions similar to those in the experiment, was then performed. Deformation concentrated zone, in which the length and width depended on the grain size, occurred in the polycrystalline specimen. The cross-section of the specimen was locally reduced when the deformation concentrated zone reached the free surface, and the tensile force became smaller for the specimen with larger grains. To discuss the relative specimen size effect, the plastic strain was divided into local and nonlocal plastic strains. Both the experimental and simulation results clarified that the nonlocal plastic strain gradient evaluated in the finite volume region increased with the region-averaged stress. From these results, we proposed a constitutive equation for the plastic strain as a function of local stress and finite volume averaged stress. The nonlocal plastic work for the evaluation region, which is estimated using the nonlocal strain gradient, increased with the stress during the strain-hardening stage in both the simulation and experimental results.

Introduction

Polycrystalline metals typically show non-uniform deformation in their polycrystalline structures even under macroscopically uniform deformation. With the increase in grain size relative to the macrostructure, the collective behavior of the crystal grains affects the macroscopic deformation field. Experimental quantification and theoretical modeling of the development of such micro- to macroscopic non-uniform deformation are important for optimizing the manufacturing process of miniaturized metal parts in various small-sized devices. Recently, the strain fields in various materials have been accurately measured using the digital image correlation (DIC) method for a wide range of scales. The author's research group also evaluated the non-uniform deformation of polycrystalline copper, polymer, and gel [36], [37], [39], [40]. These recent contributions of experimental studies will aid in the understanding of non-uniform deformation characterized by the material microstructure.

To predict the non-uniform deformation accurately, that is, the strain gradient, understanding the size effect on the mechanical properties is important. The size effect of the polycrystalline metal is categorized into grain size and specimen size effects [26]. The grain size effect is derived based on the kinetics of the dislocation, which is the main mechanism of the plastic deformation of a crystal grain. To represent the grain size effect, the framework of strain gradient plasticity (SGP) has been proposed and widely investigated. Earlier, Aifantis [1,2] and Fleck and Hutchinson [9,10]conducted pioneering studies of the SGP, and developed models have been proposed, for example, the microstress model [17,19,20], implicit gradient plasticity model [8,31] and the micromorphic model [13]. Recently, Voyiadjis and Song [42] comprehensively reviewed numerous studies on the SGP framework.

One of the main objectives of investigating the SGP is to predict the strain localization independent of the mesh division in the finite element method (FEM) simulation. The virtual work caused by the higher-order gradients of the strain is introduced to demonstrate a unique strain distribution under the given boundary condition. A computational simulation of the shear band formation in the softening material has been performed in several studies [4,8,25,45]. Another important objective of SGP is to represent the size effect on the flow stress, which is represented by the Hall-Petch effect and indentation size effect. Such effects are typically modeled by introducing the kinetics of the dislocation of crystalline materials [9], [10], [11], [14], [21]. In SGP models, a material parameter with a characteristic length scale should be included to represent the size-dependent mechanical behavior. The physical aspects of such a parameter were discussed by Liu and Dunstan [27] and Dahlberg and Boåsen [6]. The adequate determination of the characteristic length scale improves the prediction results of the size effects on the plastic deformation behavior of crystalline materials.

The grain size effect is characterized by the intrinsic length scale of the crystal grain, whereas the specimen size effect is characterized by the relative grain size of the specimen, void, second phase, etc. This size effect is derived from the interaction between the polycrystalline structure-induced (hereafter, referred to as microstructure-induced) non-uniform deformation and the external factor-induced non-uniform deformation (e.g., non-uniform deformation caused by geometrical heterogeneity, boundary conditions, and strength differences between the matrix and inclusion). In addition to the grain size effect, the specimen size effect is important for understanding the mechanical properties and performance of engineering materials. For example, the development of the surface roughness, which is an important factor for predicting the fatigue and formability limit of a material, is directly characterized by the strain localization in the polycrystalline structure [3,18,34]. In addition, the relative size of the polycrystal structure affects the strain field around the void [5] and the mechanical response of the bimodal polycrystalline metal [12]. Furthermore, our research group is investigating the effects of the polycrystalline structure on the strain field under a macroscopic stress gradient [36,37,41]. Our experimental results show that the strain concentration owing to the stress gradient is suppressed for specimens with larger crystal grains.

The understanding and accurate prediction of the specimen size effect is more significant in the microforming process [7, 15, 16, 32]. It is noteworthy that the flow stress of the polycrystalline metal decreases when the number of grains through the specimen thickness is small .[7], [22], [29], [43] To explain the decrease in the flow stress with the specimen size, a surface layer model was proposed and developed [7,16,26,33,44]. In the surface layer model, the material is divided into a surface layer with finite thickness and an interior layer. When the specimen size is small, the strength of the specimen decreases because of the increase in the volume fraction of the surface layer, which has a weaker constraint. Although the surface layer model can explain the effect of specimen size on the flow stress, the development of non-uniform deformation of the polycrystalline structure was not introduced in the model.

To investigate the interaction between the microstructure-induced non-uniform deformation and the external factor-induced non-uniform deformation by numerical simulation, the computational model should have a resolution of analysis smaller than the crystal grain size. When the difference in size between the grains and the whole model is large, a large amount of computational resources is required. The homogenization method enables the parallel simulation of non-uniform deformation in micro- and macrostructures separately. However, information about the material length scale is not shared at both scales in the conventional first-order homogenization method. Futher, the second-order homogenization method [23,24,28,41], which can evaluate micro- to macroscopic deformation behaviors depending on the difference in size between their scales, can predict the relative size effect between micro- and macrostructures.

Using the second-order homogenization method, we performed an FEM simulation of the tensile test of polycrystalline copper accompanying the macroscopic stress gradient, which is similar to the experimental study [41]. However, the computational simulation could not demonstrate the decrease in the strain concentration for the larger grain model observed in the experimental result. In the experiment, the microstructure-induced non-uniform deformation occurred at various sites of the specimen owing to the randomness of the polycrystalline structure. This relaxed the global strain concentration caused by the macroscopic stress gradient of the specimen. In the FEM model, the second-order homogenization method could not introduce the randomness of the microstructure because the macrostructure was modeled as a homogeneous material in the homogenization scheme. Therefore, we concluded that the experimental characterization and theoretical modeling of the microstructure-induced non-uniform deformation are necessary for predicting the specimen size effect of polycrystalline metals.

In this study, we investigate the interaction between the microstructure-induced non-uniform deformation and the specimen shape-induced non-uniform deformation. For that purpose, uniaxial tensile tests accompanying the stress gradient were performed using a polycrystalline copper specimen with a curved gauge section. The effects of the macroscopic stress gradient and grain size on the strain field on the specimen were evaluated using specimens with different curvature radii obtained from different thermal treatment conditions. Furthermore, full-scale crystalline plasticity finite element method (CPFEM) simulations with similar conditions to the experiment are performed. Based on the experimental and numerical results of the non-uniform deformation, we characterized the microstructure-induced non-uniform deformation. The future goal of the present study is theoretical modeling, which can predict the specimen size effect of the polycrystalline structure. For this, the physical quantities and their relations are discussed to establish the nonlocal constitutive equation of the material.