Crystal Plasticity Finite-element Simulation of Non-uniform Deformation Behavior at Grain Level of Ultralow Carbon Steel

Abstract

In this study, deformation behaviors at the grain level of coarse-grained ultralow carbon steel subjected to uniaxial tension and simple shear were simulated by using a crystal plasticity finite-element method. Heterogeneity of strain distributions appeared at the early stage and remained almost unchanged in the following deformation. Localized strain bands occurred at the grain level, but the directions of the bands depended on the deformation mode. These trends agreed well with experimental results reported in a previous paper [Hama et al., ISIJ Int., 61 (2021), 1971]. The mechanisms that the direction of the localized strain bands depended on the deformation mode were studied on the basis of the slip activities. The activities of slip systems roughly followed the Schmid factor, and the slip directions of the most active slip systems were consistent with the directions of localized strain bands, suggesting that the direction of localized strain bands were determined primarily by the Schmid factor.

1. Introduction

Understanding macroscopic deformation behavior is essential to properly form metallic materials into structural components. The microscopic deformation of each grain determines the macroscopic deformation behavior of polycrystalline materials. The grain-level deformation is highly nonuniform, even under uniform macroscopic deformation, because of the heterogeneous grain shape, crystal misorientation, and crystalline anisotropy. Therefore, to comprehensively understand the macroscopic deformation behavior of polycrystalline materials, the grain-level deformation should be understood; hence, studies have been actively conducted.^{1,2,3,4,5,6,7,8,9,10,11,12,13)} The details of the previous studies were reviewed by Efstathiou et al.⁴⁾ and Hama et al.¹⁴⁾

Metallic materials used as structural components usually consist of crystal grains with sizes of less than several tens of micrometers. Scanning electron microscopy (SEM) has been performed to observe grain-level microscopic deformations of these materials, which imposes experimental limitations. Therefore, most previous studies were limited to observations under uniaxial tension, although some studies^15,16,17) investigated deformations under biaxial tension. Because coarse-grained materials enable detailed observations of grain-level deformations without a microscope, they have been often used in studies.^{1,6,18,19,20,21,22,23,24,25,26,27)} In a previous study,¹⁴⁾ we conducted uniaxial tensile and simple shear tests using a coarse-grained ultralow-carbon steel sheet with an average grain size of ~0.47 mm and showed that the nonuniform deformation behavior at the grain level could be evaluated in detail using the digital image correlation (DIC) method without using a microscope. In particular, the findings revealed that under uniaxial tension, strain bands appeared oblique to the tensile direction, and this trend is similar to that of materials with grain sizes of less than several tens of micrometers.^1,4,28) By contrast, under simple shear, the strain bands appeared parallel or perpendicular to the shear direction, indicating for the first time that the generation of strain bands depended on the macroscopic deformation mode.

In a previous study, the authors investigated these aspects based on the macroscopic stress states and concluded that the strain band directions were correlated with the planes of maximum/minimum shear stress. However, although grain-level nonuniform deformation should be induced by slip activities in each grain, a detailed discussion of the relationship between slip activities and nonuniform deformation has not been conducted. The crystal plasticity finite-element method (CPFEM) was useful for further study. Delaire et al.¹⁸⁾ conducted uniaxial tension simulations of coarse-grained copper and described the mechanisms by which strain localization occurs at the grain-boundary triple points based on the slip activities and crystal rotations in each grain. Baudoin et al.²³⁾ performed experiments and CPFEM simulations of the uniaxial tension of coarse-grained pure titanium and showed that the experimentally observed slip lines correlated well with the slip planes of the most active slip systems obtained in the simulations. Demir et al.²⁷⁾ compared the experimental and simulation results of the uniaxial tensile deformation behavior of coarse-grained aluminum and showed that the strain gradient affects the work hardening near the grain boundaries. Kubo et al.²⁹⁾ studied the evolution of the surface roughening of an IF steel sheet under biaxial tension using CPFEM and revealed the mechanisms by which the surface-roughening evolution depends on the strain ratio based on slip activities. However, to the best of our knowledge, the generation of strain bands across multiple crystal grains and their deformation-mode dependency have not been investigated using CPFEM.

In the current study, CPFEM simulations that imitated the uniaxial tension and simple shear of coarse-grained ultralow-carbon steel, as reported in a previous study, were conducted. The simulation results were validated by comparison with the experimental results. Thereafter, the mechanisms that generate strain bands were discussed based on slip activities. Moreover, numerical experiments were conducted to study the effects of the simulation parameters on the results and relationship between the active slip systems and Schmid factors.

2. CPFEM

2.1. Crystal-plasticity Model

The CPFEM program developed by Hama et al.^30,31,32,33) was used in this study. The basic formulations are explained in detail in the literature; therefore, this paper provides only an overview. Because the ultralow-carbon steel used in this study was a body-centered cubic (BCC) metal, the {110}<111> and {112}<111> slip systems were considered active slip systems.

The slip rate ${\dot{\gamma }}^{\alpha }$ of the slip system α is given by the following viscoplastic power law³⁴⁾

  

$$\frac{{\dot{\gamma }}^{\alpha }}{{\dot{\gamma }}_0}={\left|\frac{{\tau }^{\alpha }}{{\tau }_{\mathrm{Y}}^{\alpha }}\right|}^{\frac{\text{1}}{m}}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left({\tau }^{\alpha }\right),$$(1)

where ${\dot{\gamma }}_0$ is the reference slip rate, and m is the rate-sensitivity exponent. τ^α is the resolved shear stress calculated as follows:

  

$${\tau }^{\alpha }=\mathit{\sigma }:({\mathit{s}}^{\alpha }\otimes {\mathit{m}}^a),$$(2)

where σ is the Cauchy stress tensor, and s^α and m^a are the unit vectors representing the slip direction and normal to the slip plane, respectively. ${\tau }_Y^{\alpha }$ is the slip resistance, whose initial value corresponds to the critical resolved shear stress τ₀. The evolution of ${\tau }_Y^{\alpha }$ is as follows:

  

$${\dot{\tau }}_{\mathrm{Y}}^a=\sum _{\beta }{h}_{\alpha \beta }\left|{\dot{\gamma }}^{\beta }\right|.$$(3)

The hardening moduli h_αβ is given by the following equation,^35,36) which considers the dislocation density:

  

$$h_{\alpha \beta }=\frac{\mu }{2}{g}_{\alpha \beta }{\left(\sum _{\kappa }{g}_{\alpha \kappa }{\rho }^{\kappa }\right)}^{-\frac{1}{2}}\left[\frac{1}{L^{\beta }}-2y_c{\rho }^{\beta }\right],$$(4)

where ρ^α is the dislocation density, μ is the shear modulus, y_c is the characteristic length representing the annihilation of dislocation dipoles, and g_αβ is the interaction matrix. L^α represents the mean-free path of the dislocation and is given as follows:

  

$$L^{\alpha }=\frac{K}{\sqrt{\sum _{\kappa }{g}_{\alpha \kappa }{\rho }^{\kappa }}},$$(5)

where K is a parameter representing the effect of forest dislocation. Notably, the physical meaning of g_αβ in Eq. (5) differs from Eq. (4). However, to simplify the problem, the same parameters were used. The evolution of the dislocation density is given by:

  

$${\dot{\rho }}^{\alpha }=\frac{1}{b}\left(\frac{1}{L^{\alpha }}-2y_c{\rho }^{\alpha }\right)\left|{\dot{\gamma }}^{\alpha }\right|,$$(6)

where b denotes the magnitude of the Burgers vector.

The aforementioned crystal-plasticity model was incorporated into the FEM. Explicit time integration³⁷⁾ was employed. To avoid an excessive increase in nonequilibrium between the external and internal forces, the increment size was controlled using the r_min method. Moreover, the rate tangent modulus method³⁸⁾ was used to achieve stable time integration under relatively large time increments Δt.

When BCC metals are modeled, characteristic properties, such as the difference in activity between the {110}<111> and {112}<111> slip systems,^32,39) interactions between slip systems,^40,41) and non-Schmid effects,^42,43,44) should be considered, as described in a recent review paper.³³⁾ However, although theoretical modeling of these characteristic properties was proposed,^45,46) it has not been widely used because of difficulties in parameter identification; thus, their validity has not been verified thoroughly. Therefore, in this study, we used the relatively simple and classical model explained earlier, which is widely used for BCC metals.^47,48,49,50) The effects of the interaction matrix on the simulation results are discussed in Section 4.4.

2.2. Finite-element Model

Figure 1 shows an example of an inverse pole figure (IPF) map¹⁴⁾ of the gauge section of a specimen measured by electron backscattered diffraction (EBSD). This IPF map shows the distribution of the crystal orientations along the direction normal to the sheet surface of the specimen. The average grain size is ~0.47 mm. Figure 2 shows the corresponding pole figures. The peak value is small, indicating that the crystal orientations are randomly distributed.

Fig. 1. Inverse pole figure map of a steel sheet obtained from the experiment.¹⁴⁾ The area designated by the black box was modeled in the simulation. (Online version in color.)

Fig. 2. Pole figures of the steel sheet. (Online version in color.)

In CPFEM, an area of 14.0×4.3 mm², designated by the black box shown in Fig. 1, was extracted and discretized. Only a part of the entire area was modeled to reduce the computational cost, and this area was sufficient to capture the characteristic nonuniform deformation behavior.

Baudoin et al.²³⁾ used two different finite-element models in CPFEM simulations of coarse-grained pure titanium; the reference model had a mesh size of 0.113 mm, which was determined based on a subset size in the DIC method, whereas the coarse model had a mesh size of 0.225 mm. They demonstrated that the overall strain distribution is nearly independent of the model. Therefore, assuming that similar trends also hold in this study, a mesh size was set to 0.1 mm based on the subset size of the DIC method used in the experiment.¹⁴⁾ As shown in Fig. 1, the grain size largely differs depending on the grain, and several grains have sizes smaller than 0.1 mm. Because the thickness of the specimens used in the experiments was 0.4 or 0.6 mm, it is presumed that several grains were not columnar and did not penetrate through the thickness. By contrast, when a finite-element model is created based on the IPF map shown in Fig. 1, it is necessary to assume that all the grains are columnar and crystal orientations on the surface are maintained throughout the thickness, as described below. This assumption led to a large discrepancy between the simulation model and specimen used in the experiment, making a quantitative comparison between the experiment and simulation difficult. Therefore, this study was limited to comparing qualitative trends and introduced several assumptions to create an appropriate model. The modeling procedure is briefly described below. For more details, please refer to the literature.²³⁾

An area of 14.0×4.3 mm² was divided into 140×43 elements. Eight-node solid elements with selective reduced integration were used. The length of the side of each finite-element was 0.1 mm. Three layers were formed at this thickness. The number of divisions was determined from the viewpoints of predictive accuracy and computational cost, following the literature.²³⁾ Because the element size was 0.1 mm, only 565 grains with a grain size larger than 0.1 mm were considered in the model. Grain boundaries were determined based on an inverse-pole figure map. The grain boundaries were modified so that tiny grains with a grain size <0.1 mm were included in the neighboring coarse grains. Assuming that the crystal orientation was uniform within each grain, the crystal orientation at a representative point was assigned to all elements comprising a grain. Moreover, the same orientations were assigned to all three layers throughout the thickness.

Figure 3(a) shows the developed finite-element model and a schematic of the grain boundaries. The blue lines denote the grain boundaries that were slightly modified to ignore small grains, agreeing well with the exact boundaries indicated by the white lines determined from the IPF map. The grain boundaries indicated by the blue lines were discretized, as shown in Fig. 3(b), in the finite-element model, that is, the grain boundaries were moved to neighboring element lines when they crossed an element.

Fig. 3. Finite-element model. (a) Schematic of grain boundaries used in the model, and (b) schematic of grain-boundary discretization. In (a), white and blue lines denote grain boundaries extracted from the inverse pole figure map (Fig. 1) and those assigned to the model, respectively. In (b), blue-dotted and red-solid lines denote grain boundaries assigned to the model and those after discretization, respectively. (Online version in color.)

2.3. Boundary Conditions

Uniaxial tensile and simple shear simulations were performed using the finite-element model described in the previous section. In the uniaxial tensile simulations, displacements were fully fixed for the node on (x,y,z)=(0,0,0), as shown in Fig. 3(a); displacements in the z direction were fixed for the nodes on (x,y,z)=(14,0,0) and (x,y,z)=(0,4.3,0); displacements in the x direction were fixed for the nodes on plane x=0; and small displacement increments were given in the x direction for the nodes on plane x=14. The other planes were assumed to be traction-free.

In the simple shear simulations, the displacements were fully fixed for the nodes on plane y=0. For the nodes on plane y=4.3, small displacement increments were provided in the x direction, whereas the displacements in the y and z directions were fixed. The deformation in the z direction was more strongly constrained in the simple shear simulations than in the uniaxial tensile simulations. This was performed to avoid deformation defects, such as out-of-plane buckling during deformation. An advantage of numerical simulations, difficult to achieve experimentally, is that the deformation behaviors under uniaxial tension and simple shear can be separately evaluated on identical materials with the same grain geometries and crystal orientation distributions.

2.4. Material Parameters Identification

Owing to the strong elastic anisotropy at the crystalline level in BCC metals, considering this property in crystal-plasticity simulations is preferable. Experimental measurement of the anisotropic elasticity parameters of the currently used material is difficult; hence, another option is to employ literature values. However, the stress–strain curves calculated using the literature values do not necessarily match the experimental results. Therefore, an isotropic elasticity was assumed to simplify this problem. The Young’s modulus was set to 81.9 GPa. The reasons for this are described later. The Poisson’s ratio was set to 0.3. Shear modulus μ in Eq. (4) was calculated using the Young’s modulus and Poisson’s ratio. The reference slip rate ${\dot{\gamma }}_0$, rate-sensitivity exponent m, and magnitude of the Burgers vector b were set to 0.001 s⁻¹, 0.02, and 2.48×10⁻⁷ mm, respectively, and because the initial dislocation density ${\rho }_0^{\alpha }$ was not measured experimentally, it was set to 2.7×10⁵ mm⁻² regardless of the slip system, based on the literature.^32,51) Notably, the value was slightly modified from that in the literature when determining other hardening parameters, as described later. For simplicity, all the components of the interaction matrix g_αβ were set to 0.1. Other hardening parameters were determined to fit the stress–strain curve under uniaxial tension. Because deformation up to an equivalent strain of ~0.01 is mainly discussed in this study, the hardening parameters were determined to fit the stress–strain curve in this strain range. Consequently, the parameters were τ₀=43.0 MPa, y_c=4.2×10⁻⁶ m, and K=2.0. For simplicity, the same parameters were assigned to the {110}<111> and {112}<111> slip systems.

Figure 4 shows the true stress-logarithmic strain curves obtained from the simulation and experiment. The true stress in the experiment was calculated from the tensile force and the cross-sectional area determined from the volume constancy. In the simulation, it was defined as the average of the true stress at all integration points. The logarithmic strain was defined as the average over the entire gauge section in both the experiment and simulation. The simulation results reproduced the experimental results in the region from macroscopic yielding to a strain of ~0.01. By contrast, the simulation results deviated significantly from the experimental results in the macroscopic elastic region. In particular, in the experimental results, the slope from the beginning of the deformation to a stress of ~30 MPa was represented by a typical Young’s modulus of 210 GPa, as indicated by the dashed line in the right figure, whereas an apparently nonlinear curve was exhibited in the following deformation. By contrast, in the simulation results, the stress–strain relationship was approximately linear up to macroscopic yielding. Moreover, the simulation results overestimated the stresses at strains >0.01.

Fig. 4. Average longitudinal stress-longitudinal strain curves under uniaxial tension obtained from the simulation and the experiment. The enlarged figure in the strain range to 0.01 is shown on the right.

The large difference between the experimental and simulation results presumably occurred because of the assumption of two-dimensional columnar grains in the simulation and the fact that the simulation did not properly evaluate the effects of the nonuniform strain distribution on the macroscopic stress–strain curves. These results suggest that when the material parameters are determined for coarse-grained materials, it is important to accurately predict not only the macroscopic stress–strain curve but also the nonuniform strain distribution. Herein, to evaluate the strain distributions at strain levels similar to the experimental results reported in the literature,¹⁴⁾ the macroscopic elastic deformation was approximated using the virtual Young’s modulus, i.e., 81.9 GPa. More accurate predictions of the stress–strain curve, including the macroscopic elastic and large-strain regions, will be the subject of our future work. Notably, the deformations at average equivalent strains of 0.01 or less will be primarily discussed in this study. Therefore, the deviations at large strains do not have a significant effect on the following discussion.

3. Simulation Results

3.1. Uniaxial Tensile Deformation

Figure 5 shows the equivalent strain distributions at average equivalent strains of ~0.0019 and 0.042, which represent the deformation immediately after macroscopic yielding and that after plastic deformation progressed, respectively. For reference, the grain boundaries are indicated by white lines. Notably, these grain boundaries resulted from image processing of the grain boundaries determined from the initial inverse pole figure map in accordance with the subsequent tensile deformation; thus, they may not exactly match the actual grain boundaries. The maximum strain in each colored bar was set to twice the average equivalent strain. A nonuniform strain distribution appeared immediately after macroscopic yielding, that is, at an average equivalent strain of ~0.0019, where regions with strains larger or smaller than the average equivalent strain were mixed. The strain bands appeared oblique to the tensile direction. Several strain-band boundaries appeared along the grain boundaries, whereas intragranular nonuniform deformation also occurred in some grains. Hereafter, the qualitative trend of the strain distribution is termed the nonuniformity of the strain distribution. The angles between the strain bands and tensile direction were evaluated as described in a previous study.¹⁴⁾ They ranged from ~24–60°, which was qualitatively consistent with the experimental results, ranging from 40.5 to 73.9°. The distribution trends remained nearly unchanged at an equivalent strain of 0.042, showing that the nonuniformity of the strain distribution was already determined at the initial stage of deformation. To study this result quantitatively, the evolution of the local equivalent strains at five representative points, that is, points A–E in Fig. 5(a), chosen from inside and outside the strain bands, is shown in Fig. 6. The strains increased nearly linearly regardless of the point, confirming that the nonuniformity of the strain distribution remained approximately unchanged from the beginning of deformation over this strain range. For reference, Fig. 7(a) shows the experimental results of the equivalent strain distribution under uniaxial tension at an average equivalent strain of 0.04, reported in a previous study.¹⁴⁾ The simulation results qualitatively agreed with the experimental results.

Fig. 5. Equivalent strain distributions at the average equivalent strains of (a) 0.0019 and (b) 0.042 under uniaxial tension. The maximum value of the strain range is set to be 2 times larger than the average strain in each figure. (Online version in color.)

Fig. 6. Evolution of local equivalent strains as a function of average equivalent strain measured at points A to E shown in Fig. 5(a). (Online version in color.)

Fig. 7. Experimental results of equivalent strain distributions (a) under uniaxial tension at the average equivalent strain of 0.04 and (b) under simple shear at the average equivalent strain of 0.036.¹⁴⁾ (Online version in color.)

3.2. Simple Shear Deformation

Figure 8 shows the equivalent strain distributions at average equivalent strains of ~0.0015 and 0.03. Apparent nonuniform distributions appeared from the early stage of deformation, as in the case of uniaxial tension, and strain bands appeared parallel or perpendicular to the shear direction. The nonuniformity of the strain distribution was nearly independent of the average equivalent strain. The evolution of the local strain at each point was nearly linear, although detailed results are not provided to avoid redundancy. These results are similar to the experimental results shown in Fig. 7(b).¹⁴⁾

Fig. 8. Equivalent strain distributions at the average equivalent strains of (a) 0.0015 and (b) 0.03 under simple shear. The maximum value of the strain range is set to be 1.5 times larger than the average strain in each figure. (Online version in color.)

The simulation results qualitatively reproduced the experimental results regardless of the deformation mode, thereby validating the simulations. Moreover, these results showed that the nonuniformity of the strain distribution depended on the deformation mode, even for materials with exactly the same grain shapes and crystallographic orientations, confirming that the difference in the specimen did not play a significant role in the difference in the nonuniformity between the deformation modes observed in the experiments. In the following section, the deformation mechanisms are studied based on simulation results. Because the nonuniformity of the strain distribution was already determined at the early stage of deformation, the results at the early stage are focused on in the following discussion.

4. Discussions

4.1. Distribution of Schmid Factors

Figure 9 shows the Schmid factor distribution under uniaxial tension at an average equivalent strain of ~0.0019. The Schmid factors were calculated assuming all integration points were subjected to uniaxial tension in the x direction. Notably, Fig. 9 shows the Schmid factors for the most active of the 24 slip systems at each integration point. Compared to the equivalent strain distribution shown in Fig. 5(a), the regions with Schmid factors <0.4, which are shown in blue, had small strains and were located outside the strain bands. By contrast, regions with large Schmid factors had large strains, exhibiting a strong correlation between the Schmid factor and equivalent strain distributions.

Fig. 9. Distribution of the Schmid factor of the most active slip system in each element under uniaxial tension. (Online version in color.)

Figure 10 shows the Schmid factor distribution under simple shear at an average equivalent strain of ~0.0015. In the case of multiaxial stress states, including simple shear, calculating the Schmid factor in the same way as for uniaxial tension is difficult.^52,53,54) In this study, the principal stress direction was calculated at each integration point, and then the Schmid factor was calculated by assuming that each integration point was subjected to uniaxial tension in the principal stress direction. Because the absolute values of the maximum and minimum principal stresses were similar, the average of the Schmid factors for the maximum and minimum principal-stress directions was used in this study.

Fig. 10. Distribution of the Schmid factor of the most active slip system in each element under simple shear. (Online version in color.)

Compared to the equivalent strain distribution shown in Fig. 8(a), the regions with small Schmid factors clearly have small strains, as in the case of uniaxial tension, exhibiting a strong correlation between the equivalent strain and Schmid factor distributions under simple shear. These results suggest that the nonuniformity of the strain distribution can be explained in terms of the Schmid factor, regardless of the deformation mode.

4.2. Mechanism of Strain Bands Generation

The mechanisms that determine the direction of the strain bands are discussed for uniaxial tension based on the simulation results. Figure 11(a) shows the relative frequency distribution of the angles between the slip direction vector and the uniaxial tensile direction (x direction) for all the most active slip systems at each integration point. Notably, the slip direction vectors that projected onto the sheet surface, that is, the x–y plane, were used to calculate the angles. The slip systems with the slip direction inclined 30–40°from the tensile direction were the most frequent, more than 30%, and those with the slip direction inclined 20–50°were dominant, more than 70%. The average angle over the entire region was ~34.6°. These results are consistent with the aforementioned results, in which the strain bands were inclined 24–60°from the tensile direction.

Fig. 11. Relative frequency distributions of angles between the slip direction vector and the uniaxial tensile direction. The results are for (a) all the most active slip systems and (b) those with Schmid factors larger than 0.45. (Online version in color.)

Next, Fig. 12 shows the relative frequency distribution of the Schmid factors of the most active slip systems under uniaxial tension. This result was obtained from Fig. 9. More than 75% of the most active slip systems had the Schmid factors >0.45. Figure 11(b) shows the relative frequency distribution of the angles between the slip direction vector and uniaxial tensile direction for slip systems with Schmid factors >0.45. The distribution trend was similar to that of the most active slip systems (Fig. 11(a)). More rigorously, the slip systems with a slip direction inclined 20–50°were more dominant; their frequency was >80%.

Fig. 12. Relative frequency distribution of the Schmid factors of the most active slip systems under uniaxial tension. (Online version in color.)

The following hypothesis is based on the aforementioned results: In this material, slip systems with a Schmid factor >0.45 are the most active in most of the grains, and the slip directions of these slip systems are generally inclined 20–50°from the tensile direction. Consequently, plastic deformation easily propagated in these directions, generating strain bands inclined at 24–60°from the tensile direction.

Similar analyses were conducted for the simple shear results shown in Fig. 13(a). The Schmid factors distributed more widely under simple shear than that of uniaxial tension, but ~60% of the most active slip systems had Schmid factors greater than 0.4. Figure 13(b) shows the relative frequency distribution of the angles between the slip direction vector and shear direction for slip systems with Schmid factors >0.4. Slip systems with a slip direction inclined 0–20°or 70–90°from the shear direction were dominant. This result is consistent with the finding that the strain bands were parallel or perpendicular to the shear direction, showing that a hypothesis similar to that of the uniaxial tension holds for simple shear.

Fig. 13. Results of simple shear. (a) Relative frequency distribution of the Schmid factors of the most active slip systems and (b) relative frequency distribution of angles between the slip direction vector and the shear direction for the most active slip systems with Schmid factors larger than 0.4. (Online version in color.)

It is unclear whether these results hold true for other materials with different crystal structures and crystallographic orientation distributions. This will be investigated in future studies.

4.3. Relationship between Active Slip Systems and Schmid Factors

In the aforementioned discussions, we focused on the most active slip systems and their Schmid factors. In this section, the relationship between the most active slip systems and slip systems with the largest Schmid factors is studied for uniaxial tension. In particular, we studied whether the slip system with the largest Schmid factor matched the most active slip system at each integration point. The results are shown in Fig. 14. The blue elements indicate that the slip systems with the largest Schmid factor exhibited the largest slip activity, whereas the red elements indicate that the slip systems with the largest Schmid factor did not exhibit the largest activity. In the early stage of deformation, as shown in Fig. 14(a), most slip systems with the largest Schmid factors exhibited the largest activity. By contrast, they did not show the highest activity in ~25% of the areas, and most of these areas appeared near the grain boundaries. In some areas, this disagreement occurred over the entire grain. As deformation progressed, this trend became more pronounced, particularly near the grain boundaries, as shown in Fig. 14(b). These results suggest that the stress states near the grain boundaries exhibited a different evolution from that of the macroscopic stress state. This result is consistent with the trend in which strain bands tend to appear along grain boundaries. Moreover, these results indicate that the evolution trend of the local strain could change as the deformation progresses, suggesting that the linear evolution of the local strains shown in Fig. 6 is no longer valid at large strains. Therefore, it is considered that the Schmid factors determined from the macroscopic stress state are effective in qualitatively discussing the deformation at the early stage, as in the previous sections; however, the nonuniformity of the stress distribution should be considered when the deformations near grain boundaries or at large strains are evaluated in detail.

Fig. 14. Correlations between the Schmid factors and the most active slip systems at the average equivalent strains of (a) 0.0019 and (b) 0.042 under uniaxial tension. Blue elements denote that the slip systems with the largest Schmid factor showed the largest activity among the 24 systems, whereas red elements denote that they did not. (Online version in color.)

In a previous study, we hypothesized that the direction of the strain bands was correlated with the planes where the macroscopic maximum and minimum shear stresses occurred. Figure 14(a) indicates that in the early stage of deformation, the stress states in each grain were close to the uniaxial tensile state; thus, the planes where the macroscopic maximum and minimum shear stresses occurred agreed well with the slip planes with large Schmid factors.

4.4. Effect of Interaction Matrix

In this study, all components of the interaction matrix g_αβ were set to 0.1 to simplify the problem. In reality, the component values differ depending on the combination of slip systems. For instance, Madec and Kubin^40,41) evaluated the component values using discrete dislocation dynamics simulations. Hama et al.³²⁾ conducted crystal-plasticity simulations of the biaxial tension of a mild steel sheet and suggested that the anisotropic work-hardening behavior of the contour of equal plastic work depends significantly on the components of the interaction matrix. Given that the active slip systems differ depending on the grain, the strain distribution at the grain level may also depend on the components of the interaction matrix.

Therefore, a uniaxial tensile simulation was performed using the interaction matrix components reported in the literature,³²⁾ and the effect of the interaction matrix on the strain distribution was studied. The components of the interaction matrix used in the simulations are listed in Table 1. Although the details are not described in Table 1, the components are classified into 17 categories following the report by Madec and Kubin,^40,41) and their values are shown. The reader is referred to a previous study^40,41) for further details.

Table 1. Interaction matrix moduli³²⁾ used in the simulation to examine the effect of interaction matrix moduli on the strain distribution.

h(1) 0.1	h(2) 0.1	h(3) 0.45	h(4) 5.5	h(5) 0.4	h(6) 0.6	h(7) 5.5	h(8) 0.1	h(9) 0.1
h(10) 5.5	h(11) 0.1	h(12) 0.1	h(13) 0.1	h(14) 0.1	h(15) 0.1	h(16) 0.1	h(17) 0.1

Figure 15 shows the simulation results obtained using the parameters listed in Table 1. The average equivalent strains differed slightly from those in Fig. 5, although the results were obtained for the same total elongation, as shown in Fig. 5. The magnitudes of the local strains in each region differed largely from those shown in Fig. 5, although the qualitative trend of the nonuniformity of the strain distribution remained nearly unchanged. These results suggest that developing a finite-element model that accurately represents the specimen and determining accurately the material parameters, including the interaction matrix, are essential to reproduce the strain distributions at the grain level with good accuracy.

Fig. 15. Equivalent strain distributions under uniaxial tension at the average equivalent strains of (a) 0.0019 and (b) 0.04 obtain by using different interaction matrix moduli. The maximum value of the strain range is set to be 2 times larger than the average strain in each figure. (Online version in color.)

The material parameters are usually determined based on the limited experimental results of the stress–strain curves and evolution of the r-value. Therefore, the unique identification of parameters is difficult if the number of parameters is considerably larger than the available experimental conditions, as in the case of the components of the interaction matrix. To solve this problem, it would be effective to use two-dimensional information, such as strain distributions at the grain level, instead of one-dimensional information, such as stress–strain curves, for parameter determination. It is expected that this procedure enables parameter determination based on notably larger experimental data than the conventional method because all results of local strain evolution in the measurement field of view can be used for comparisons between experiments and simulations. Baudoin et al.²³⁾ conducted uniaxial tensile simulations of coarse-grained pure titanium and studied the effect of the critical resolved shear stress of the basal slip on the predictive accuracy of the strain distribution. They reported the possibility of parameter determination with improved accuracy based on strain distribution. Grilli et al.²⁴⁾ proposed a parameter identification procedure that refers to the strain distribution under uniaxial tension for coarse-grained α uranium. However, the simulation results for strain distribution reported in these previous studies agreed only qualitatively with experimental results. To achieve better predictive accuracy and quantitative agreement of strain distributions using this procedure, it is important to process and efficiently use enormous two-dimensional information for parameter identification. Therefore, data-driven scientific applications are effective. We are currently developing a parameter-identification method that combines the deformation of coarse-grained materials with data-driven science. This will be reported elsewhere.

Although a simple finite-element model was used in this study, the simulation results of the local strain evolution qualitatively reproduced the experimental results, validating the simulation model. However, to achieve quantitative agreement with the experimental results, further improvements to not only the crystal-plasticity model but also the finite-element model are necessary, as described earlier. Three-dimensional crystal-plasticity simulations were widely performed using finite-element models that considered grain shapes.^55,56,57) Some studies^58,59) have used finite-element models that accurately mimic the three-dimensional grain shapes of specimens using serial sectioning. By contrast, in most studies, experimental and simulation results were compared only for global properties, such as average strain evolution over the entire region or local deformation behavior in very limited regions. However, some studies^20,21,23,27) also investigated the distribution and evolution of local strains over an entire region using columnar-grained specimens in experiments. However, the predictive accuracy of these simulation results remains insufficient. Further improvements in predictive accuracy will be investigated in our future work.

5. Conclusions

A crystal plasticity finite-element method was used to simulate the deformation behavior of a coarse-grained ultralow-carbon steel sheet under uniaxial tension and simple shear, and strain evolution at the grain level was studied. The results of this study are as follows:

(1) Localized strain bands appeared in the 24–60° direction inclined from the loading direction under uniaxial tension, whereas they appeared perpendicular or parallel to the shear direction under simple shear, consistent with the experimental results reported in the literature.

(2) In the early stage of uniaxial tension, slip systems with Schmid factors of 0.45 or higher showed the largest activity in most grains, and their slip directions were inclined 20–50° from the tensile direction. Consequently, plastic deformation easily propagated in these directions, generating strain bands inclined at 24–60°from the tensile direction.

(3) In the early stages of deformation, the stress states in each grain were close to the macroscopic stress state; thus, the planes where the macroscopic maximum and minimum shear stresses occurred agreed well with the slip planes with large Schmid factors.

(4) The stress states near the grain boundaries exhibited a different evolution from that of the macroscopic stress state, and this tendency became more pronounced as deformation progressed. Therefore, the nonuniformity of the stress distribution should be considered when deformations near grain boundaries or at large strains are studied in detail.

Acknowledgements

The authors thank Mr. Sohei Uchida of the Osaka Prefectural Institute of Technology for cooperating with the EBSD measurements. This study was partially supported by JSPS KAKENHI (grant number 20H02480) and the Amada Foundation (grant number AF-2019004-A3).

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