# Grain scale simulation of local deformation behavior

- Tung Van Phan
^{1, 2}Email author

**1**:10

**DOI: **10.1186/s40540-014-0010-7

© Phan; licensee Springer 2014

**Received: **31 December 2013

**Accepted: **29 May 2014

**Published: **15 July 2014

## Abstract

In this work, a full-field finite element simulation of a heterogeneous DC04 steel microstructure identified from two-dimensional (2D) electron backscatter diffraction (EBSD) data is performed under a macroscopic tensile deformation. After discretization procedure by finite elements, the EBSD microstructure is subjected to homogeneous displacement boundary conditions approximately describing a large strain uniaxial tensile test. A crystal plasticity model applied on integration points of FE method is used to simulate the deformation behavior and the grain orientation evolution. The simulated grain orientation fields are compared to experimental measurements of the specimen after the tensile test at different deformation levels.

## Background

Most of the metals used in industrial processes are polycrystalline materials. They are aggregates of approximately single crystals in grains with different crystal orientations. The anisotropic plasticity of polycrystalline materials is mainly caused by non-uniform distributions of crystal orientations [1]. Therefore, the analysis of the crystallographic texture, i.e., preferred crystal orientations, plays an important role when investigating the macroscopic material behavior. Crystallographic texture data allow for the characterization and prediction of the anisotropic plasticity in heterogeneous materials by using statistical models. Some representative studies of different polycrystalline materials with focus on the texture evolution are given, e.g., by [2]-[10]. In the intensive increase of computer powerful tool, modelling and simulation have contributed to predict the mechanical behavior and to facilitate the material design with particular properties. This issue often requires a computational framework based on the linking between the constitutive law describing physical phenomenon and the experimental information at the mesoscopic and microscopic level. The experimental electron backscatter diffraction (EBSD) technique, known as scanning electron microscope (SEM)-based technique, has become a major tool in measuring crystal orientations from a polycrystalline structure. A common application is the use of orientation data at every Gauss integration point in finite element (FE) simulations of crystal plasticity models. Hence, a grain scale simulation of the polycrystalline structure could be performed to describe the grain orientation evolution.

In this paper, a heat-treated DC04 steel microstructure identified from two-dimensional (2D) EBSD data is considered. A grain structure model is constructed based on the DC04 steel microstructure identified by a Matlab toolbox MTEX. After discretized by finite elements with one element over thickness, a grain scale simulation of the polycrystalline DC04 steel sample cut from a tensile specimen is performed within the finite strain crystal plasticity framework. The crystal plasticity model accounting for specified grain orientations is applied at Gauss integration points of finite elements of the corresponding grains simultaneously. The experimentally observed local grain orientations and reorientations in the polycrystalline sample will be predicted and evaluated at different deformation states. Subsequently, the simulated grain reorientation fields are compared to experimental measurements of the DC04 sample after the tensile test at different deformation levels.

*Notation.* Throughout the text, a second-order tensor and a fourth-order tensor are A =*A*_{
i
j
}e_{
i
}⊗e_{
j
} and $\mathbb{A}={A}_{\mathit{\text{ijkl}}}{\mathit{e}}_{i}\otimes {\mathit{e}}_{j}\otimes {\mathit{e}}_{k}\otimes {\mathit{e}}_{l}$, respectively, where {e_{
i
}} represents an orthonormal basis of the three-dimensional (3D) Euclidean space. Symmetric and traceless tensors are designated by a prime, e.g., A^{′}. The set of proper orthogonal second-order tensors is specified by *S* *O*(3). The scalar product, the dyadic product, and the Frobenius norm are denoted by A ·B =tr(A^{
⊤
}B )=tr(AB^{
⊤
}), A ⊗B , and ∥A ∥=(A ·A )^{1/2}, respectively. Here, tr(·) represents the trace of a second-order tensor. A linear mapping of second-order tensors is written as $\mathit{A}=\u2102\left[\phantom{\rule{0.3em}{0ex}}\mathit{B}\right]$.

## Methods

The first subsection introduces the constitutive equations of a crystal plasticity material model applied at Gauss integration points of FE model. The constitutive model is described via an ABAQUS user subroutine UMAT allowing to define the mechanical behavior of material. The EBSD measurement technique to obtain the crystallographic texture data set is discussed in the second subsection. By using this experimental EBSD technique, the different EBSD data sets representing the deformation states of a tensile specimen of the heat-treated DC04 steel are shown. In addition, a Matlab toolbox MTEX allowing to import the EBSD data formats and to obtain an image of the grain structure is introduced. In the third subsection, the microstructural image of the tensile specimen at the initial state is discretized by finite elements. The FE mesh is used as an input data for performing crystal plasticity simulations on the grain scale into the ABAQUS/CAE software.

### Constitutive model on a single crystal

#### Elastic law

_{ e }and a plastic part F

_{ p }[11],[12]

_{ p }is the plastic contribution from crystallographic slips. The elastic deformation F

_{ e }accounts for the lattice distortion, which is inherently elastic. As the elastic strains are assumed to be small, a linearized relation between a conjugate pair of stress and strain measures is applicable for the description of the elastic behavior. Here, the elastic law is assumed to be given by

_{ α }is given by

The components ${\stackrel{~}{C}}_{\mathrm{\alpha \beta}}$ are defined by ${\stackrel{~}{C}}_{\mathrm{\alpha \beta}}={\mathit{B}}_{\alpha}\xb7\stackrel{~}{\u2102}\phantom{\rule{0.3em}{0ex}}\left[{\mathit{B}}_{\beta}\right]$. The orthonormal base tensors B_{
α
} used are given by [13]. Due to the cubic material under consideration, the stiffness tensor $\stackrel{~}{\u2102}$ has three independent elastic constants.

#### Flow rule and hardening law

_{ p }of F

*m*quantifies the strain rate sensitivity of the material, ${\stackrel{\u0307}{\gamma}}_{0}$ is a reference rate, and ${\stackrel{~}{\mathit{M}}}_{\alpha}$ is the Schmid tensor.

*τ*

^{ C }denotes the critical resolved shear stress. A rate-dependent Kocks-Mecking hardening model see, e.g., [7],[14]

*Θ*

_{0}. The rate of the accumulated plastic slip is computed by

*J*=det(F

_{ e }) is the determinant of F

_{ e }. The Schmid or slip system tensors are rank-one tensors, which are defined in terms of the slip direction ${\stackrel{~}{\mathit{d}}}_{\alpha}$ and slip plane normal ${\stackrel{~}{\mathit{n}}}_{\alpha}$ in the undistorted configuration

_{ e }(0)=Q (

*t*=0)∈

*S*

*O*(3) and the initial critical resolved shear stress ${\tau}^{C}\left(0\right)={\tau}_{0}^{C}$. The crystal orientation is given by a proper orthogonal tensor Q (

*t*)=g

_{ i }(

*t*)⊗e

_{ i }, where the vectors g

_{ i }and e

_{ i }denote the orthonormal lattice vectors and the fixed orthonormal basis, respectively. The initial orientation of the single crystal Q (

*t*=0)=g

_{ i }(0)⊗e

_{ i }is defined in terms of the orthonormal lattice vectors g

_{ i }(0) at the time

*t*=0. As shown in [15], intrinsic characteristics of body-centered cubic (BCC) crystals are revealed by using a proper parameter identification method. The authors applied a BCC crystal plasticity model to perform uniaxial tension simulations at the material point level for different types of BCC single crystals and compare these with experiments. The results indicate that {110} and {112} planes are identified as intrinsic slip systems of BCC crystals, but not the {123} plane. Therefore, in this work, the attention is focused on a combination of {110}〈111〉 and {112}〈111〉 slip system families (Figure 1). There are two slip directions in each of the slip planes along the main diagonals of the cube. In total, there are 24 slip systems shown in Tables 1 and 2.

**12 BCC slip systems of** {110}〈111〉 **[**[16]**]**

Slip system(α) | ||||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

$\sqrt{2}\left[{\xf1}_{i}^{\alpha}\right]$ | [1,−1,0] | [1,−1,0] | [1,0,−1] | [1,0,−1] | [0,1,−1] | [0,1,−1] |

$\sqrt{3}\left[{\stackrel{~}{d}}_{i}^{\alpha}\right]$ | [1,1,1] | [1,1,−1] | [1,1,1] | [1,−1,1] | [1,1,1] | [−1,1,1] |

7 | 8 | 9 | 10 | 11 | 12 | |

$\sqrt{2}\left[{\xf1}_{i}^{\alpha}\right]$ | [1,1,0] | [1,1,0] | [1,0,1] | [1,0,1] | [0,1,1] | [0,1,1] |

$\sqrt{3}\left[{\stackrel{~}{d}}_{i}^{\alpha}\right]$ | [−1,1,1] | [1,−1,1] | [−1,1,1] | [1,1,−1] | [1,−1,1] | [1,1,−1] |

**12 BCC slip systems of** {112}〈111〉 **[**[15]**]**

Slip system(α) | ||||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

$\sqrt{6}\left[{\xf1}_{i}^{\alpha}\right]$ | [1,1,2] | [−1,1,2] | [1,−1,2] | [1,1,−2] | [1,2,1] | [−1,2,1] |

$\sqrt{3}\left[{\stackrel{~}{d}}_{i}^{\alpha}\right]$ | [1,1,−1] | [1,−1,1] | [−1,1,1] | [1,1,1] | [1,−1,1] | [1,1,−1] |

7 | 8 | 9 | 10 | 11 | 12 | |

$\sqrt{6}\left[{\xf1}_{i}^{\alpha}\right]$ | [1,−2,1] | [1,2,−1] | [2,1,1] | [−2,1,1] | [2,−1,1] | [2,1,−1] |

$\sqrt{3}\left[{\stackrel{~}{d}}_{i}^{\alpha}\right]$ | [1,1,1] | [−1,1,1] | [−1,1,1] | [1,1,1] | [1,1,−1] | [1,−1,1] |

### Experimental identification based on EBSD data of DC04 steel tensile samples

#### EBSD measurement

*X*and

*Y*positions of the measurement points. Their dimensional unit is micrometer [

*μ*m]. The rotation is commonly parameterized by a triple of Euler angles

*ϕ*

_{1},

*Φ*,

*ϕ*

_{2}and is described through a 3×3 orthogonal matrix given by

where the three Euler angles *ϕ*_{1},*Φ*, and *ϕ*_{2}, shown schematically [20] in Figure 2 (right), are used to describe the crystallographic orientation of the crystals in relation to a reference coordinate system. In the database, these three Euler angles are recorded in the fourth, fifth, and sixth columns, respectively. The other parameters are neglected for the consideration.

#### Identification of microstructures and orientation information

An experimental tensile specimen of heat-treated low-carbon DC04 steel [21] is discussed here. The specimen geometry is width *w*=5 mm, gauge length *L*_{0}=15 mm, and thickness *t*=1 mm. From such specimen, a tiny sample was cut by laser rays with the same thickness. Through the EBSD technique in the scanning electron microscope, an initial raw database of such a sample is obtained.

^{−3}s

^{−1}is applied to the tensile specimen in loading direction. During the tensile test, several EBSD databases have been determined experimentally, and thereby, the evolution of the crystallographic texture is measured at different elongation states (5%, 10%, 15%, and 20%).

*X*and

*Y*positions (in

*μ*m) and three Euler angles (in degree) of the measurement points. In addition, the grain to which the measurement points belong is indicated in the database. The order numbering of grains and the identification of 574 grains are shown in Figure 5 at the initial state. Additionally, 2D images of the raw EBSD data at different strain states (5%, 10%, 15%, and 20%) are shown in Figure 6. In all microstructural images, each color indicates the lattice orientation in each grain at different states.

Some conclusive descriptions of the raw EBSD data are summarized. In the raw EBSD database of the initial state, the total number of columns and rows is 7 and 105,000, respectively. After neglecting measurement errors, in such an EBSD database remain 103,671 rows (or the number of pixels) and 7 columns. The number of measurement errors is 1,329 pixels, i.e., approximately 1.26% of the area fraction. The map size of the raw grain structure in Figure 4a is 349×299 *μ* m. The area of one measurement point is 1 *μ* m^{2}. For the subsequent strain states, the measurement errors correspond to area fractions of 3.6%, 8.5%, 9.4%, and 18.5%, respectively.

### FE modeling and full-field simulation

*E*=200 GPa) and Poisson’s ratio (

*ν*=0.3) are assigned to the ideal von Mises plastic region. For the plastic behavior, the flow parameters

*σ*

_{F 0}=180 MPa,

*σ*

_{ F ∞ }=303 MPa, describing the linear hardening, are estimated from the experimental tensile stress-strain curve for 0° to the RD (Figure 9a).

**Set of material parameters**

_{ κ }(X ,

*t*) is the current position at time

*t*. The displacement gradient H can be given in terms of the deformation gradient F

^{−1}. The special form of the velocity gradient implies a displacement in the

*X*-

*Y*plane in the reference configuration and a constant volume during the simulation. From Equation 20, the displacement on the boundary is obtained

*t*in terms of the deformation gradient in Equation 23, so that the displacement becomes

*%*elongation. The FE results are evaluated at different total times such as 65, 130, and 195 s corresponding to 5%, 10%, and 15% elongation, respectively. The field of critical resolved shear stress

*τ*

_{ c }is depicted in Figure 16 at the aforementioned different states of the FE simulation. These results show a strongly inhomogeneous field caused by the evolution of individual grain orientations in the microstructure during deformation.

*α*(

*α*=1…24) in BCC slip systems {110}〈111〉+{112}〈111〉 is computed by integrating the slip rate ${\stackrel{\u0307}{\gamma}}_{\alpha}$ over the time during the grain scale simulation. The plastic slip is described as follows:

## Results and discussion

_{0}represents the crystal orientation at the initial state of deformation and Q represents the crystal orientation of the same point at different states of deformation. Both Q

_{0}and Q are parameterized by Euler angles lying in the same fundamental zone. During the simulation, Q is extracted by the polar decomposition F

_{ e }=R

_{ e }U

_{ e }, where R

_{ e }=Q is the elastic rotation and U

_{ e }is the elastic stretch tensor. The computed reorientation is the minimum relative orientation distance between the initial and actual crystal orientation. By comparing the color distribution representing the values in the legend (Figure 24), the reorientations in the numerical simulations agree well with the experimental results. In addition, the computed reorientations of local grains #345 and #296 are in good agreement with the experiment. However, the computed reorientation of local grain #357 is lower than in the experiment. This can probably be explained by neglecting the beneath grain interaction in the ND due to the lack of the experimental 2D EBSD data.

## Conclusions

In the paper, a finite strain crystal plasticity model has been presented. The constitutive equations applied on the grain scale to model the elasto-viscoplastic behavior of BCC single crystals have been described in the context of large deformations. In particular, the crystal plasticity model is rate dependent and takes into account hardening effects on the microscale. The material parameters of the DC04 steel used in the crystal plasticity model were identified in the work of [25].

In addition, the experimental EBSD technique to obtain the microstructural information has been introduced. By using the EBSD measurement technique, a data set of a DC04 steel specimen in the cold formed and the heat-treated processes has been presented. In order to analyze and evaluate quantitatively the experimental texture data, a Matlab toolbox MTEX has been introduced and used. The MTEX allowed to import the different EBSD data formats and to obtain a 2D grain structure with corresponding crystal orientations. These EBSD data sets have been processed to identify the grain information and to obtain the clustered data.

A verification of the material model has been carried out. Based on EBSD data, the grain structure was modeled by a FE model. The FE mesh has been imported into the ABAQUS/CAE software for the performance of the grain scale simulation. This FE simulation has been used for the identified DC04 steel material microparameters, and two families of the BCC slip systems were assumed to possibly act simultaneously, namely {110}〈111〉+{112}〈111〉. In addition, a procedure for mapping the initial grain orientations into the fundamental zone has been implemented. The micromechanical behavior was analyzed in terms of the evolution of grain orientations. The grain orientations and reorientations are compared to the experiment for a group of local grains. The numerical results of local grain reorientations seem to underestimate the heterogeneity compared to the experimental results. This issue can be probably explained by the neglect of the 3D microstructure and interaction of the beneath microstructure in the grain scale simulation. The influence of 3D interaction needs to be further investigated in further studies.

## Declarations

### Acknowledgements

The author acknowledges gratefully T. Böhlke (ITM, KIT) for his constructive contribution. The author also thanks S. Schreijäg and R. Mönig (IAM, KIT) for the supply of the EBSD data of DC04 steel in this research work.

## Authors’ Affiliations

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