Crystal Plasticity Finite Element Simulation of Crystal Plasticity Finite Element Simulation of...

Crystal Plasticity Finite Element Simulation of Crystal Plasticity Finite Element Simulation of Nano/Micro
Crystal Plasticity Finite Element Simulation of Crystal Plasticity Finite Element Simulation of Nano/Micro
Crystal Plasticity Finite Element Simulation of Crystal Plasticity Finite Element Simulation of Nano/Micro
Crystal Plasticity Finite Element Simulation of Crystal Plasticity Finite Element Simulation of Nano/Micro
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Transcript of Crystal Plasticity Finite Element Simulation of Crystal Plasticity Finite Element Simulation of...

  • 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

    αα γ&Pd p and ( ) ( )∑ =

    = 12

    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} 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)* .

    ( ) ( ) ( ) ( ) ( )( )**** 2 1 ααααα smmsP += . (4)

    ( ) ( ) ( ) ( ) ( )( )**** 2 1 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

    gg a α

    α

    α

    α αα 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

    β αβ

    α γ&& ∑ =

    = 12

    1 hg . (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, th