Crystal Plasticity Finite Element Simulation of Nano/Micro Plastic Forming for Metallic Material

Akinori YAMANAKA1, Tsuyoshi KAWANISHI2 and Masahiko YOSHINO3

1 Graduate School of Science and Engineering, Tokyo Institute of Technology, Japan, ayamanaka@mes.titech.ac.jp 2 Graduate School of Science and Engineering, Tokyo Institute of Technology, Japan,tkawanishi@yoshino.mes.titech.ac.jp 3 Graduate School of Science and Engineering, Tokyo Institute of Technology, Japan, myoshino@mes.titech.ac.jp

Abstract:

The three-dimensional crystal plasticity finite element simulation of the nano plastic forming (NPF) to a single crystal copper (Cu) specimen is performed. In order to understand plastic deformation behavior of the specimen during the NPF, the NPF process to (001) surface of the specimen is simulated, and we investigate the evolution of stress and strain inside the specimen. The simulation results clearly describe the mechanism of the plastic deformation and shape change of the specimen due the NPF. Therefore, the proposed finite element simulation of the NPF is effective way to control the NPF process perfectly. Keywords: Nano plastic forming, Crystal plasticity finite element simulation, Single crystal copper, Plastic deformation

1. Introduction

The nano/micro forming has been actively studied as an important way to produce small parts for the micro-electro-mechanical-system (MEMS) device [1]. The author’s group has been proposed a flexible nano/micro plastic forming method based on the nano/micro imprinting [2-3]. This method is called as the nano plastic forming (NPF) and can be applied to the micro fabrication of various materials, such as hard brittle materials (silicon, grass and ceramic) and metallic materials [4]. In order to develop the parts for MEMS device made of the metallic materials by using the NPF, the shape change of the material during the NPF process must be controlled in the scale of a single crystal. Therefore, the elastic- and plastic deformation behaviors of the single crystal during the NPF should be predicted precisely.

The authors have been experimentally investigated the plastic deformation behavior of a single crystal copper (Cu) during the NPF with the knife edge type diamond tool. In their experiment, the change of crystal orientation of the Cu specimen due to the NPF has been revealed by using scanning electron microscope observation and electron backscatter diffraction analysis [5]. However, since the plastic deformation behavior of the metallic material depends on the crystal orientation and so on, it is difficult to understand the complex elastoplastic deformation behavior and the distribution of stress and strain inside the specimen only by experiments. Thus, numerical simulation of the NPF is essential tool to understand the plastic deformation behavior of the specimen during the NPF and control the NPF process perfectly. In this study, the plastic deformation behavior of the single crystal Cu during the NPF is clarified by the numerical simulation and the experimental investigation. In this paper, we demonstrate that the crystal plasticity finite element simulation is useful tool to understand the

deformation behavior of the single crystal Cu specimen during the NPF process. Employing the crystal plasticity theory [6-8], it is possible to simulate not only the evolution of stress and strain in the specimen, but also the change of the crystal orientation. The plastic deformation behavior and the shape change of the Cu specimen for different depths of the indentation are studied in detail. 2. Crystal Plasticity Theory

As mentioned in the previous section, to study the plastic deformation of the single crystal Cu during the NPF, the crystal plasticity finite element method is employed. In the following we briefly explain the essential feature of the crystal plasticity theory [6-8].

The total deformation gradient F can be decomposed into elastic and plastic components as

pFFF ⋅= * . (1) Here, F* describes the elastic deformation and rigid body rotation. On the other hand, Fp describes the plastic deformation due to slip deformation on slip plane. By differentiating Eqn. (1) with respect to time, the following equation is obtained.

( ) ( )pp ωdωdFF +++=⋅ − **1& . (2)

Here, d* and ω* are the rate of elastic stretching and elastic spin tensor, respectively. And, the rate of plastic strain dp and plastic spin tensor ωp are related to the rate of plastic shear strain on each slip system as

( ) ( )∑

=

=12

1α

αα γ&Pd p and ( ) ( )∑=

=12

1α

αα γ&Wω p . (3)

Here, ( )αγ& is the plastic shear strain rate and α is the number of slip system. In this study, since {111}<110> slip system of a FCC crystal shown in Fig. 1 is considered, α varies from 1 to 12. The tensors P(α) and W(α) are given by a function of a unit slip direction vector s(α)* and a unit

Figure 1: Slip system for FCC crystal.

normal vector of the slip plane m(α)* .

( ) ( ) ( ) ( ) ( )( )****

21 ααααα smmsP += . (4)

( ) ( ) ( ) ( ) ( )( )****

21 ααααα smmsW −= . (5)

The change of the slip direction and the slip plane normal are described by using the elastic part of the deformation gradient F* and initial slip direction vector s(α) and initial slip normal vector of the slip plane m(α)* as ( ) ( )αα sFs ⋅= ** and ( ) ( ) 1** −⋅= Fmm αα . (6)

The Jaumann rate of Kirchhoff stress tensor ∇

S is related to the rate of Cauchy stress tensor σ& as

σωωσσS ⋅−⋅+=∇

& , (7)

where ω is the total spin tensor. The elastic part of Eqn. (7) is also given by the following equation,

σωωσσS ⋅−⋅+=∇

*** & . (8)

Considering the relation *ωωω −=p and Eqn. (3), we can obtain the following equation from Eqns. (7) and (8) as

( ) ( )( ) ( )∑=

∇∇

⋅−⋅+=12

1

*

α

ααα γ&WσσWSS . (9)

Assuming the elastic part of Kirchhoff stress tensor S* is not affected by the plastic deformation, Eqn. (9) can be written as

( )pee ddDdDS −==∇

:: ** . (10)

Substituting Eqn. (10) into Eqn. (9) leads to the final form of the constitutive equation, which is used in the crystal plasticity finite element simulation.

( ) ( )∑=

∇

−=12

1:

α

αα γ&RdDS e , (11)

( ) ( ) ( ) ( )αααα WσσWPDR ⋅−⋅+= :e . (12) The plastic shear strain rate on each slip system ( )αγ&

is given by the rate-dependent power law relation, which was suggested by Pan and Rice [8] as

Figure 2: Simulation model for the NPF with the wedge-type tool and finite element mesh.

Figure 3: Configuration of crystal orientation of the Cu

specimen and the tool.

( ) ( )( )

( )

( )

( )

11−

=m

gga α

α

α

ααα ττγ& . (13)

Here, a(α) is the reference shear strain rate. τ (α) is the resolved shear stress on the α th slip system, which can be evaluated by

( ) ( )σP αατ = . (14) And, g(α) is the critical resolved shear stress and its evolution law has the following form [9].

( ) ( )β

βαβ

α γ&& ∑=

=12

1hg . (15)

Here, hαβ is the hardening coefficient given by,

αβαβαβ βκδ Hh += . (16)

κ, β and δij are hardening parameters and the Kronecker’s delta, respectively. Hαβ denotes the hardening matrix of dislocation interaction defined by Bassani and Wu [10]. 3. Simulation Model

The constitutive model for a single crystal Cu described in the previous section is implemented into the commercial finite element simulation code, ABAQUS/Explicit ver.6.7, by the means of the user- defined subroutine, VUMAT. Figure 2 shows the three-

Figure 4: Distributions of equivalent stress (a) for the whole specimen and (b-d) for a set of successive (010) cross sections at indentation depth d = 1.5 µm.

Figure 5: Distributions of plastic shear strain for the slip system 1 at different indentation depths (a) d = 0.5 µm, (b) d = 1.0 µm and (c) d = 1.5 µm.

dimensional simulation model for the NPF to the single crystal Cu specimen with a wedge-type tool. [100], [010] and [001] directions of the Cu crystal is set to be parallel to x, y and z axes of the computational domain. The tool is parallel to the [010] direction. The radius and angle of the tool tip are set as r = 0.6 µm and θ = 100 degree, respectively. The size of the specimen is 20 µm × 10 µm × 10 µm and the specimen is divided by the regular eight-node brick element with reduced integration. The total number of the element is 16000. Since the specimen is largely deformed during the NPF, we employ the arbitrary Lagrange-Euler adaptive mesh refinement (ALE) algorism. The initial size of an regular mesh is 0.5 µm × 0.5 µm × 0.5 µm. The tool is modeled by the analytical rigid surface. The friction between the surface of the tool and the specimen is ignored, i.e. the friction coefficient is chosen as zero. As the boundary condition, the fixed boundary condition is applied for the bottom surface of the specimen, and the symmetric boundary condition is applied for two (010) surfaces. The elastic constants for the single crystal Cu are C11 = 168 GPa, C12 = 121.4 GPa and C44 = 75.4 GPa [11]. The initial resolved shear stress τ0 is chosen as 35 MPa. The material and hardening parameters are a(α) = 0.001 s-1, m = 0.1, κ = 4.0 and β = 0.6, respectively.

As shown in Fig.3, the NPF to (001) surface of the specimen is simulated. The depth of the indentation d is 1.5 µm along [00-1] direction. In this paper, by considering the symmetry of the slip system of Cu crystal, the plastic deformation on the slip planes (111) and (-111) is closely investigated.

4. Results and Discussion Figure 4 (a) shows the simulated distribution of the

equivalent stress of the whole specimen for the indentation depth d = 1.5 µm. The result shows the pile-up is produced around the groove. It can be also observed that the stress concentrates under the tool. In order to investigate the distributions of the stress inside the specimen, the stress distribution for a set of successive three (010) planes (Plane A, B and C) are shown in Fig. 4 (b)-(c). In all planes, the same distribution of the stress is observed. That is, the high stress distributes along 40 degree with respect to [00-1] direction. This direction depends on the angle of the tool tip. These results indicate that the two-dimensional plastic deformation of the specimen occurs in this NPF condition. Hereafter, we focus on the plastic deformation behavior in the Plane B. The shape change of the specimen due to the NPF induces the large plastic deformation. Figure 5 shows the distributions of plastic shear strain for the slip system 1 at different indentation depths, d = 0.5, 1.0 and 1.5 µm. It can be seen that the plastic shear strain increases with increasing the indentation depth d. Furthermore, we can observe that the large plastic shear deformation occurs beneath the groove and near the surface of the specimen. To reveal the relationship between the plastic deformation and shape change of the specimen in detail, Figure 6 shows the distributions of the resolved shear stress on the slip systems 1, 2, 7 and 8 for the indentation depth d = 1.0 µm. It is found that the stress distribution for the slip systems 1 and 7 exhibits the symmetry with respect to the indent direction [00-1]. On the other hand, the distributions of the stress for the slip system 2 and 8

Figure 5: Distributions of the resolved shear stress on the slip systems 1, 2, 7 and 8 for indentation depth d = 1.0 µm

Figure 6: Schematic illustration of slip direction on (111) and (-111) slip planes under the tool.

are almost same. From these simulated distributions of the resolved shear stress, we can coarsely interpret the plastic deformation behavior of the Cu specimen during the NPF as illustrated in Fig. 6. That is, in the regions A and F, the material elevates along (111) and (-111) slip planes. This behavior promotes the pile-up at the indented surface. While, in the regions C and D, since the compressive deformation is occurred under the tool, the material exhibits sink-in behavior along (111) and (-111) planes. And, in the regions B and E, the material translation changes from upward flow to downward flow. 5. Conclusion In this study, the three-dimensional crystal plasticity finite element simulation of the NPF to the single crystal Cu specimen is conducted to investigate the plastic deformation behavior of the specimen. The simulation results clearly describe the stress and strain distributions inside the specimen and clarify the mechanism of the shape change of the specimen. Therefore, by using the proposed numerical simulation method and the experimental investigation of the NPF, it is enabled us to predict the optimum condition of the NPF, e.g., crystal orientation of the specimen, the indentation depth and the shape of the tool, to control the final shape of the specimen perfectly. Acknowledgement

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