Yield Point Phenomenon Induced by Twinning Accompanied by Abnormal Grain Growth in Fine-Grained Copper

Abstract

In this study, an equiaxed fine-grained copper specimen was fabricated by powder injection molding, followed by hot isostatic pressing. The specimen showed a yield point phenomenon with a Lüders-type deformation during a uniaxial tensile test. This study aimed to identify the cause of the yield point phenomenon in the fine-grained copper specimen. First, the Lüders band was designed to partially propagate within the gauge section through an interrupted uniaxial tensile test equipped with a digital image correlation system. The microstructures of the Lüders and non-Lüders band regions were compared by electron backscatter diffraction. It was observed that self-annealing was accelerated by the applied stress in the Lüders band region. Consequently, the microstructure of the Lüders band region transformed into a bimodal structure composed of fine grains and abnormally coarse grains containing numerous large twins. Identification of active twin variants in the abnormally coarse grains suggested that a significant tensile strain can be accommodated by this twinning. Based on these results, it was confirmed that the strain burst generated by the sudden occurrence of numerous large twins induces the yield point phenomenon in the fine-grained copper.

1. Introduction

Grain size reduction is an effective method for improving the toughness of metallic materials.^1–3) Therefore, methods for fabricating fine-grained materials with high strength and toughness have been extensively studied over the past few years.^4–8) Severe plastic deformation (SPD) is a well-known process for generating fine-grained structures by subjecting a metallic material to a very large plastic deformation.^9–13) Among the various SPD processes, equal-channel angular pressing (ECAP) and high-pressure torsion (HPT) are most commonly used.¹⁴⁾ ECAP generates a significant plastic strain in a material by repeatedly applying a severe shear deformation by extrusion. This process has a cost/time disadvantage because numerous extrusion processes are required to obtain fine-grained microstructures.¹⁵⁾ HPT is a process of subjecting a small disc-shaped specimen to simultaneous compressive and torsional deformations using a punch. In HPT, a significant plastic strain can be applied to a material in a single operation. However, because it is applicable only to small disc-shaped specimens, it is difficult to manufacture large bulk materials.^10,16) In contrast, powder injection molding (PIM) is a process of manufacturing materials by sintering a powder mixed with a binder, without imposing a plastic deformation, different from SPD processes. PIM is a suitable process for fabricating fine-grained materials with equiaxed grain structure because it can use a fine metal powder, followed by sintering and post-annealing treatments.⁸⁾ PIM saves time and is cost-effective because it does not require repetitive processes. Moreover, it has major advantages over ECAP and HPT because materials of various sizes and shapes can be produced depending on the mold shape.¹⁷⁾

In fine-grained materials, it is important to maintain the obtained microstructure as well as to refine uniformly the grains. However, it is known that fine-grained copper can be self-annealed when left at room temperature for a long time, because of its low thermal stability due to the large fraction of grain boundaries. Observations of the microstructure of fine-grained copper specimens produced by ECAP and HPT after long-term storage at room temperature showed occurrence of recrystallization and grain growth.^8–22) In addition, a decrease in the hardness was observed in the region where self-annealing occurred.²²⁾ Therefore, when dealing with fine-grained copper, its self-annealing behavior, which significantly affects the mechanical properties, is an important factor to consider.

It has been reported that self-annealing kinetics is accelerated when a stress and/or strain is applied. For example, in a thin copper film, the stress induced by the thermal mismatch between a thin copper film and a substrate is known to accelerate its self-annealing kinetics.^23–26) Concurrently, in the case of Mo and Ta bulk materials, a tensile stress and/or strain can induce abnormal grain growth at a much lower temperature and faster rate than static annealing.^27–30) However, there has been no study on self-annealing and its related mechanical behavior that occurs when a tensile stress and/or strain is applied to bulk fine-grained copper specimens.

As another characteristic, it has been reported that a fine-grained copper specimen undergoes Lüders-type deformation accompanied by a yield point phenomenon.^31,32) Lüders-type deformation can cause plastic instability, which can lead to unexpected premature failure during the forming process.^33–35) Therefore, understanding the cause of the yield point phenomenon in fine-grained copper is important in material design to avoid unexpected premature failure. In most studies, the cause of the yield point phenomenon in fine-grained materials has been explained by sudden dislocation multiplication.^{31,32,36–42)} For a copper single crystal, especially, which also shows a yield point phenomenon, a lamellar structure consisting of twinned and untwinned regions has been reported to be formed in the Lüders band region.⁴³⁾

In this study, an equiaxed fine-grained copper specimen with a grain size of less than 1 µm was fabricated by PIM followed by hot isostatic pressing (HIP). The Lüders band was designed to partially propagate within the gauge section through an interrupted uniaxial tensile test equipped with a digital image correlation (DIC) system. The microstructure and mechanical properties of the Lüders and non-Lüders band regions within the gauge region were compared by electron backscatter diffraction (EBSD), Vickers indentation, and nanoindentation. In addition, the strain accommodated by the twinning within the Lüders band region was calculated from a phenomenological approach, considering the active twin variants. Accordingly, the cause of the yield point phenomenon in fine-grained copper was investigated and analyzed.

2. Materials and Method

In this study, a commercial micro-sized copper powder (99.9%) manufactured by gas atomization was used. The average powder size was approximately 644 nm with a spherical morphology. The feedstock for the PIM was prepared by mixing copper powder with a binder composed of paraffin wax and polyethylene. The feedstock was melted, injected into the mold cavity of a molding machine, cooled, and solidified. After removing the binder by solvent and thermal debinding processes, a highly porous pure copper was obtained. A fine-grained pure copper specimen, with plate dimensions of 100 mm × 100 mm × 2.7 mm, was fabricated by sintering the formed highly porous pure copper at 850°C for 2 h in a hydrogen atmosphere, followed by HIP treatment. The HIP process was performed at 780°C for 2 h under a pressure of 1000 bar in an argon atmosphere to increase the density of the fine-grained pure copper specimen by reducing the number of closed pores. The relative density of the fine-grained pure copper specimen after HIP treatment was 8.89 g/cc, i.e., 99.3% theoretical density.

An ASTM E8 sub-size uniaxial tensile test sample with a gauge length of 25 mm, gauge width of 6 mm, and thickness of 2.7 mm was fabricated from the HIP-treated specimen, for the analysis of the uniaxial tensile behavior of fine-grained copper. To enable DIC analysis, random speckle patterns were formed by spraying a black paint on the white background of the specimen surface. Uniaxial tensile tests were performed using a tensile testing machine (5582, INSTRON) at a strain rate of 10⁻³ s⁻¹ under displacement-controlled conditions at room temperature of 25°C. Digital images obtained during the uniaxial tensile test were analyzed using the ARAMIS software to produce global and local tensile strain distribution maps.

Microstructure analysis, Vickers indentation, and nanoindentation experiments were conducted on the specimen after mechanical polishing using a diamond suspension followed by electropolishing using D2 solution at 4 V for 10 s. A field-emission scanning electron microscope (SU70, Hitachi) equipped with an EBSD system (EDAX/TSL, Hikari) and operated at an accelerating voltage of 15 kV was used to observe the microstructure of the specimen. A step size of 0.4 µm was used for the EBSD data, which were analyzed using the TSL OIM Analysis 7 software. The average grain size and grain size distribution of the specimen were measured using the line-intercept method with a critical misorientation angle of 5°. The Vickers hardness profile along the tensile direction in the gauge section was measured by applying a 0.1 kgf load for 5 s at 0.5 mm intervals. To increase the reliability of the data, the experiment was repeated five times in each area.

Nanoindentation tests were conducted at room temperature of 25°C using a Hysitron Tribolab nanoindentation system with a diamond Berkovich indenter tip. The nanoindentation tests were conducted in a load control mode with constant loading and unloading rates of ±400 µN s⁻¹ and a maximum load of 2000 µN. A total of 100 independent indentations (10 × 10 grids with intervals of 10 µm) were performed for each region of interest in the gauge section of the interrupted tensile specimen.

3. Results and Discussion

Figure 1 shows the microstructure measured by EBSD and the uniaxial tensile stress–strain curve of a fine-grained copper specimen fabricated by PIM, followed by HIP. ND denotes the surface normal direction of the specimen and ID denotes the direction of the powder injection. As shown in the inverse pole figure (IPF) map in Fig. 1(a), the microstructure is composed of equiaxed fine grains with an almost random texture. The average grain size of the specimen was approximately 0.9 µm. Figure 1(b) shows the grain orientation spread (GOS) map. Most of the grains have a GOS value of less than 2°, indicating that the specimen contained a few dislocations in the grains.^44,45) The red line in Fig. 1(c) shows a noticeable high fraction of Σ3 twin boundaries, i.e., annealing twins are generated during PIM supplemented with HIP processes.⁴⁴⁾ Figure 1(d) shows the stress–strain curve of a specimen during its uniaxial tensile test. A yield point phenomenon, in which the stress decreases from the upper to lower yield point, is clearly observed. The specimen has an upper yield strength of 231 MPa, lower yield strength of 208 MPa, tensile strength of 251 MPa, and total elongation of 54%. A stress plateau appears after the stress drop at the onset of yielding, indicating that the specimen presents a Lüders-type deformation behavior. The Lüders strain, which corresponds to the end of the stress plateau, is approximately 6%. A DIC analysis was performed to characterize the Lüders deformation behavior of a specimen.

Fig. 1

(a)–(c) EBSD maps of fine-grained copper with grain size d ∼ 0.9 µm: (a) IPF ND map, (b) GOS map. Grains with GOS values smaller and larger than 2° are marked in blue and white, respectively in (b). (c) Grain boundary map with image quality map. Low-angle boundaries with misorientations between 2° and 5°, high-angle boundaries with misorientations larger than 5°, and Σ3 twin boundaries are drawn in blue, black, and red lines, respectively in (c). (d) Stress–strain curve of fine-grained copper during uniaxial tensile deformation at room temperature and strain rate 10⁻³ s⁻¹.

Figure 2(a) shows the local tensile strain distribution map of a specimen interrupted at a global tensile strain of 3%, which is smaller than the total Lüders strain. The Lüders band gradually propagates and sweeps half of the gauge area. Figure 2(b) presents the Vickers hardness profile along the tensile direction measured at the gauge section of a specimen interrupted at 3% global tensile strain. Regs. 1–3 correspond to the non-Lüders band region where the Lüders band has not yet propagated, Lüders front region, and Lüders band region where the Lüders band has propagated. The corresponding average Vickers hardness values of the regions were 80, 77, and 89 HV. In between Regs. 2 and 3, a remarkable increase in the Vickers hardness is observed, probably due to strain hardening after the Lüders band propagation. In contrast, the Vickers hardness value in Reg. 2 is lower than that in Reg. 1. Specifically, even though a plastic strain is applied to the Lüders front region, strain hardening does not occur; instead, softening occurs. The cause of this phenomenon is discussed subsequently in this paper.

Fig. 2

(a) Local tensile strain distribution map of tensile specimen interrupted at 3% global tensile strain. (b) Vickers hardness profile from Reg. 1 to Reg. 3 in Fig. 2(a).

Figure 3 shows the microstructures of each region in Fig. 2(a) as observed by EBSD. Figure 3(a) shows the IPF map of Reg. 1. The microstructure is composed of equiaxed fine grains with almost random textures, and the average grain size is approximately 1.1 µm, which is similar to the initial microstructure of the specimen shown in Fig. 1(a). This similarity is explained by the nonpropagation of the Lüders band into Reg. 1. Figure 3(b) shows the IPF map of Reg. 2. Interestingly, an abnormal grain growth accompanied by numerous large twins is observed in the Lüders front region. This occurs abruptly at the moment of Lüders band formation with the occurrence of numerous twins at the onset of yielding. This abrupt occurrence of numerous large twins produces a strain burst, which causes a stress drop during the displacement-controlled uniaxial tensile test.^43,46) Here, occurrence of numerous large twins during very short time, i.e., the moment of Lüders band formation, indicates that twins grow at very rapid rates as soon as they are nucleated. This is feasible when the stress required for twin growth is much smaller than that for twin nucleation.⁴⁷⁾ Therefore, difference between stress required for twin nucleation and that for twin growth may cause the stress drop in stress-strain curve. As another characteristic, such abnormal grain growth is known to be related to the self-annealing behavior of fine-grained pure copper.^18–26,48) Apparently, the self-annealing kinetic seems to be accelerated by the applied stress during the tensile test; therefore, the abnormal grain growth occurs very rapidly at the moment of Lüders band formation.^23–30) Also, recrystallization and grain coarsening due to the mechanically induced self-annealing in Reg. 2 cause softening, which is consistent with the Vickers hardness drop in Reg. 2 shown in Fig. 2(b). Figure 3(c) shows the IPF map of Reg. 3. A twinned microstructure accompanied with abnormal grain growth is also observed in this region. Different from Reg. 2, some color gradient is observable in the abnormally coarse grains of Reg. 3. This suggests that a dislocation substructure is formed when an additional plastic strain after Lüders band formation is applied to Reg. 3. The formation of the dislocation substructure causes dislocation hardening, which is consistent with the increase in the Vickers hardness of Reg. 3, as shown in Fig. 2(b). Figures 3(d) and (e) present the grain size distributions measured from EBSD data of Regs. 1 and 3, respectively. Reg. 1 shows a unimodal distribution, whereas for Reg. 3, there is a bimodal distribution owing to the abnormal grain growth. To distinguish fine grains from abnormally coarse grains, 6.6 µm, the largest grain size based on the grain size distribution of Reg. 1, was set as the critical grain size. Specifically, grains smaller and larger than 6.6 µm were defined as fine and coarse grains, respectively.

Fig. 3

(a)–(c) EBSD IPF ND maps of all regions in Fig. 2(a): (a) Reg. 1, (b) Reg. 2, and (c) Reg. 3. (d), (e) Grain size distribution measured from EBSD data of (d) Reg. 1 and (e) Reg. 3 in Fig. 2(a).

Figure 4 shows the kernel average misorientation (KAM) maps for each area shown in Fig. 2(a). Figure 4(a) shows that the KAM degree of the fine grains in Reg. 2 is slightly increased compared to that in Reg. 1. The increase is due to the plastic deformation of the fine grains in Reg. 2, resulting in the increase in their KAM degree.^49–51) Concurrently, Fig. 4(d) presents that the coarse grains in Reg. 2 are mostly colored blue, i.e., their KAM degree is very low, and they have a very low dislocation density. This occurs because the coarse grains are recrystallized by self-annealing during the tensile test and the twinning functions as a deformation mechanism at the onset of yielding. Figures 4(a) and (b) show that the fine grains in Reg. 3 have a slightly higher KAM degree than those in Reg. 2, whereas the coarse grains in Reg. 3 have a significantly higher KAM degree than those in Reg. 2. This is because, as shown in Fig. 4(d), the dislocation density of the coarse grains in Reg. 2 is relatively lower than that of the fine grains. Therefore, the coarse grains can relatively easily accommodate the deformation than the fine grains, resulting in a strain concentration. It is also confirmed from Fig. 4(e) that there are many points with a high KAM degree, colored in green, inside the coarse grains.

Fig. 4

(a), (b) KAM value distributions of all regions in Fig. 2(a): (a) fine grains and (b) coarse grains. (c)–(e) EBSD KAM maps of all regions in Fig. 2(a): (c) Reg. 1, (d) Reg. 2, and (e) Reg. 3.

Figures 5(a)–(c) show the cumulative probability distributions of the nanohardness values for the fine and coarse grains measured in each region presented in Fig. 2(a). Figure 5(d) shows the average nanohardness values for each region. The average nanohardness value of the fine grains in Reg. 2 is higher than that in Reg. 1. This is because of the increase in the dislocation density due to the plastic deformation applied to Reg. 2. Concurrently, the average nanohardness value of the coarse grains is significantly lower than that of the fine grains in Reg. 2. This is attributed to the coarse grains not only accommodating the plastic strain by twinning, instead of dislocation glides and multiplication during the Lüders band formation, but also to the self-annealing acceleration by the applied stress during the tensile test. Different from Reg. 2, it can be confirmed that the average nanohardness of the coarse grains in Reg. 3 is larger than that of the fine grains. Based on the KAM and nanohardness analysis, it was confirmed that twinning, not dislocation glides and multiplication, is the initial yielding mechanism of the abnormally coarse grains during the Lüders band formation, and subsequently, strain is concentrated on the soft coarse grains after the Lüders band formation.

Fig. 5

Cumulative distributions of nanohardness measurements of fine (black square) and coarse grains (red square) of all regions in Fig. 2(a): (a) Reg. 1, (b) Reg. 2, and (c) Reg. 3. (d) Average nanohardness of each region in Fig. 2(a).

In this study, the strain accommodated by twinning within the Lüders band region was calculated from a phenomenological approach, considering active twin variants based on the measured EBSD data. Figure 6 shows the results of identifying the type of twin system and active variants in the framed area in Fig. 3(b). Figure 6(a) presents the microstructure of the framed area consisting of mint-colored matrix grains (M) and red-colored twins (T). The interface between two grains is classified as a Σ3 twin boundary. Figure 6(b) shows the orientation of the matrix grains and twins measured from the region shown in Fig. 6(a) as a {111} pole figure. The four poles corresponding to the four {111} planes of the matrix grains and twins are marked in blue and red, respectively. The orientations measured from the matrix grains and twins share the $(11\bar{1})$ pole, which suggests that $(11\bar{1})$ is the twinning plane of a twin formed in a matrix grain. Table 1 lists the Schmid factors of possible twin variants calculated under the assumption that the stress direction in a matrix grain is identical to the external tensile loading direction. It was determined that the $(11\bar{1})$ $[2\bar{1}1]$ twin variant was activated, based on its Schmid factor being the largest. The displacement gradient tensor has a simple form in the twin reference frame, x–y–z, where x, y, and z are parallel to the twinning direction, twin plane normal direction, and direction of the cross product of x and y, respectively.   

\begin{equation} S_{ij} = \begin{pmatrix} 0 & s & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix} , \end{equation} (1)

where $\text{s} = 1/\sqrt{2} $ for the twin of a face-centered cubic crystal.⁵²⁾ By coordinate transformation from the twin reference frame to the crystal coordinate system, the displacement gradient tensor expressed in the crystal coordinate system can be calculated as follows:   

\begin{equation} e_{ij} = \begin{pmatrix} \dfrac{1}{3} & \dfrac{1}{3} & -\dfrac{1}{3}\\ -\dfrac{1}{6} & -\dfrac{1}{6} & \dfrac{1}{6}\\ \dfrac{1}{6} & \dfrac{1}{6} & -\dfrac{1}{6} \end{pmatrix} . \end{equation} (2)

Fig. 6

(a) EBSD IPF map of framed area in Fig. 3(b) consisting of matrix grains (M) and twins (T). (b) {1 1 1} pole figure showing orientations of matrix grains and twins in Fig. 6(a). LD and TD correspond to tensile loading and transverse directions, respectively.

Table 1 Schmid factors for $(11\bar{1})$ twin variants of matrix grains in Fig. 6(a).

This displacement gradient tensor can be expressed in the macroscopic sample coordinate system, x′–y′–z′, where x′, y′, and z′ are parallel to the tensile loading, specimen width, and specimen thickness directions, respectively, by the coordinate transformation. The displacement gradient tensor expressed in the macroscopic sample coordinate system can be calculated as   

\begin{equation} e_{ij}' = \begin{pmatrix} 0.341 & -0.361 & -0.0737\\ 0.333 & -0.353 & -0.0719\\ -0.0560 & 0.0594 & 0.0121 \end{pmatrix} , \end{equation} (3)

where $e'_{11}$, $e'_{22}$, and $e'_{33}$ in eq. (3) denote the normal strains in the x′, y′, and z′ directions produced by twinning, respectively. It can be seen that $e'_{11}$ representing the strain produced in the macroscopic tensile loading direction by twinning is positive. This means that twinning produces a plastic strain in the macroscopic tensile loading direction. Subsequently, a strain burst can be produced by the abrupt creation of numerous large twins during the Lüders band formation, as shown in Fig. 3(b). This strain burst induces a yield point phenomenon in displacement-controlled uniaxial tensile test and corresponding upper and lower yield points may be related to the stress required for twin nucleation and that for twin growth, respectively.^43,46,47) Thus, this investigation suggests that the abrupt occurrence of significant twinning in abnormally coarse grains was the cause of the yield point phenomenon in the fine-grained copper specimen.

4. Conclusion

To investigate the origin of the yield point phenomenon in a fine-grained copper specimen fabricated by PIM supplemented with HIP, the microstructures and mechanical properties of the Lüders and non-Lüders band regions were compared. In the Lüders band region, significant twinning accompanied by abnormal grain growth occurred. The abnormal grain growth was induced by the acceleration of the self-annealing kinetics of the fine-grained copper specimen by the applied stress during the tensile test. The KAM and nanohardness of the fine and coarse grains in the Lüders front (Reg. 2 in Fig. 2(a)) and inner Lüders band regions (Reg. 3 in Fig. 2(a)) were compared. The results showed that twinning of the abnormally coarse grains was the yielding mechanism during the Lüders band formation. By identifying active twin variants in the abnormally coarse grains, a strain burst could be produced by the abrupt occurrence of significant twinning during the Lüders band formation. From the above results, it was found that the yield point phenomenon of the fine-grained copper specimen was induced by twinning accompanied by abnormal grain growth.

Acknowledgments

Byeong-Seok Jeong and Keunho Lee contributed equally to this work. This study was supported by the National Research Foundation of Korea (NRF) grants funded by the Ministry of Science and ICT (Grant Numbers: 2020R1A5A6017701, 2021M3H4A6A01045764, and 2021R1A2C3005096). K.L. and L.P. were supported by Defense Research Programs funded by the Agency for Defense Development (ADD) of the Republic of Korea (Grant Numbers – 211555-912440203). The Institute of Engineering Research at Seoul National University provided the research facilities for this study.

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Appendix

The coordinate transformation matrix, R, from the twin reference frame to the crystal coordinate system is obtained as follows:   

\begin{equation} R = \begin{pmatrix} 0.817 & -0.408 & 0.408\\ 0.577 & 0.577 & -0.577\\ 0 & 0.707 & 0.707 \end{pmatrix} . \end{equation} (A1)

Subsequently, e_ij, which is the displacement gradient tensor expressed in the crystal coordinate system in eq. (2), can be calculated as e_ij = R^T · S_ij · R.

The coordinate transformation matrix, R′, from the crystal coordinate system to the macroscopic sample coordinate system is determined as follows:   

\begin{equation} R' = \begin{pmatrix} -0.972 & -0.152 & 0.180\\ -0.234 & 0.717 & -0.657\\ -0.0313 & -0.680 & -0.732 \end{pmatrix} . \end{equation} (A2)

Subsequently, $e'_{ij}$, which is the displacement gradient tensor expressed in the macroscopic sample coordinate system in eq. (3) can be calculated as $e'_{ij}$ = R′^T · e_ij · R′.