Tensile and compressive simulations using viscoplasticity

Abstract

In the crystal plasticity finite element method, it was necessary to represent the crystal orientation in the data in order to determine the anisotropy caused by differences in crystal orientation. Furthermore, many finite element methods using yield functions exhibit isotropic hardening. [Solution] The crystal orientation was changed to represent the Taylor factor M. In other words, for a face-centered cubic lattice, the data for the 12-slip system could be rewritten in terms of the Taylor factor M. Furthermore, by including the deviated back stress, the Bauschinger effect could be expressed. This is not possible for rigid-plastic properties.

Description

Technical Field

[0001] The present invention expresses a single tensile compression test using a relational expression of stress and viscoplastic strain (hereinafter referred to as a constitutive equation of viscoplasticity), and creates a program that can be used for tensile compression simulation. Also, since the plastic strain rate can be obtained, a stiffness matrix of the finite element method can be formed.

Background Art

[0002] Currently, in my case, the formulation of the crystal plasticity finite element method has been done. Due to the crystal orientation, anisotropy, kinematic hardening, and isotropic hardening can be achieved. However, a method of using kinematic hardening and isotropic hardening in a simpler way has been invented. Here, a method of representing kinematic hardening in a constitutive equation, which is a relational expression of stress and strain, is used. By doing this, it is possible to create a tensile compression simulation that can also consider the Bauschinger effect and the initial elastic region. Also, since the plastic strain rate can be obtained, it is considered applicable to the finite element method.

Prior Art Documents

Non-Patent Documents

[0003]

Non-Patent Document 1

Summary of the Invention

Problems to be Solved by the Invention

[0004] In the crystal plasticity finite element method, due to the use of crystal orientation, anisotropy can be achieved. Also, kinematic hardening such as the Bauschinger effect can be achieved by introducing a back stress. Isotropic hardening can also be achieved. This time, it was examined whether isotropic hardening and kinematic hardening can be achieved using a macroscopic yield function. Kinematic hardening introduces a back stress. Here, the reliability of the constitutive equation is obtained in a tensile compression simulation. Since the crystal orientation can be omitted, the finite element method and the tensile compression simulation can be simplified.

Means for Solving the Problems

[0005] Constitutive formula for viscoplasticity The constitutive equations for stress and strain are shown below. This formula The product of the initial yield stress D and the Taylor factor M is If TIFF2026123132000001.tif76 exceeds a certain value, it indicates a significant increase in distortion. This results in the file TIFF2026123132000002.tif85. This is a uniaxial tensile test. TIFF2026123132000003.tif65 represents equivalent stress. The following is the formula. TIFF2026123132000004.tif2229TIFF2026123132000005.tif106 are material constants, and 0.01 is preferred. Let m be a constant (for example, 0.005). D = 60 MPa. M = 2.3. The macroscopic yield function is derived using isotropic hardening and kinematic hardening. TIFF2026123132000006.tif146 is a unit vector and determines the direction.

[0006] Here, we represent the yield function due to mobile hardening. TIFF2026123132000007.tif47134 Let f be the plastic strain rate potential. The direction of the plastic strain rate increment h, which includes dynamic hardening, coincides with the partial derivative of the plastic potential with respect to stress. Using this perpendicularity law, we obtain the following equation. TIFF2026123132000008.tif1294 TIFF2026123132000009.tif72142 Similarly TIFF2026123132000010.tif5168TIFF2026123132000011.tif4495TIFF2026123132000012.tif7473TIFF2026123132000013.tif6574 The shear stress components are TIFF2026123132000014.tif2334 Therefore, TIFF2026123132000015.tif2760 Therefore, the plastic strain rate is TIFF2026123132000016.tif3542 (1.4)

[0007] Let me explain mobile hardening. TIFF2026123132000017.tif3940(1.5) I will now explain isotropic hardening. Isotropic hardening increment TIFF2026123132000018.tif64 is TIFF2026123132000019.tif1428(1.6) H is around 100-400.

[0008] TIFF2026123132000020.tif5148 This is a uniaxial tensile test, TIFF2026123132000021.tif1628 From the previous step Proceed through TIFF2026123132000022.tif1257 steps. TIFF2026123132000023.tif832 and Create a graph for TIFF2026123132000024.tif65. In the stress-strain curve, if C=2000, H=120000, and H=200000, the following curve can be drawn. Although qualitative, kinematic hardening can be determined from compression and tensile tests of stress-strain curves. The vertical axis represents stress. TIFF2026123132000025.tif830 and the horizontal axis The image will be plotted as TIFF2026123132000026.tif65. This is a tensile and compressive simulation with strains ranging from 0.2 to -0.2. [Effects of the Invention]

[0009] TIFF2026123132000027.tif4099 Take TIFF2026123132000028.tif860du, TIFF2026123132000029.tif1077(1.7) The B matrix is determined by the interpolation function of the elements. The D matrix is the elastic matrix. Equation (1.7) is the element stiffness matrix. TIFF2026123132000030.tif435

[0010] TIFF2026123132000031.tif83103 TIFF2026123132000032.tif3977 TIFF2026123132000033.tif4385

[0011] TIFF2026123132000034.tif6463

[0012] It shows the relationship between the shape function of the element, strain, and displacement. TIFF2026123132000035.tif48116 TIFF2026123132000036.tif549

[0013] (x, y, z) coordinates and Coordinate transformation of TIFF2026123132000037.tif88 and determination of elements by determinant F The B matrix is Since it is a function of TIFF2026123132000038.tif410, it is not a function of (x, y, z). Therefore, What is obtained by partially differentiating TIFF2026123132000039.tif43 with respect to x, y, and z Should be represented by TIFF2026123132000040.tif88. Here, there is the following method. TIFF2026123132${{000}}041.tif4788 TIFF2026123132000042.tif3357

[0014] The Gaussian numerical integration of eight nodes is as follows. TIFF2026123132000043.tif47113 Here, TIFF2026123132000044.tif8113 Substituting the numerical integral values ​​of Gauss from earlier, we get the following: Therefore, solve the others in the same way. TIFF2026123132000045.tif5114TIFF2026123132000046.tif8654TIFF2026123132000047.tif1627The following B matrix is ​​generated. TIFF2026123132000048.tif26110

[0015] The elastic matrix D is shown. [D]= TIFF2026123132000049.tif3978E is Young's modulus. ν is Poisson's ratio. For steel, E = 203 GPa ν = 0.33 It's about that much.

[0016] The method for solving systems of linear equations using the finite element method is modified to Cholesky. TIFF2026123132000050.tif6092TIFF2026123132000051.tif6049TIFF2026123132000052.tif2256 [Modes for carrying out the invention]

[0017] The program for uniaxial tensile test simulation was calculated using equations (1.1), (1.2), (1.4), (1.5), and (1.6). A finite element method program using equations (1.1), (1.2), (1.4), (1.5), and (1.6), with equation (1.7) being used. [Industrial applicability]

[0018] It can be used to simplify the formulation of the finite element method. [Brief explanation of the drawing]

[0019] Figure 1 shows stress on the vertical axis and strain on the horizontal axis. Equations (1.1), (1.2), (1.4), (1.5), and (1.6) are shown. It is expressed using [this method]. In conclusion, the type of hardening—isotropic or kinematic—is determined by the back stress coefficient C, which is the back stress. If back stress is included in the constitutive equation, the yield function will be isotropic hardening, but kinematic hardening will also be present. Figure 2 shows the anisotropy resulting from a change in the Taylor factor. The Taylor factor is the reciprocal of the Schmidt law and simplifies the crystal orientation. Figure 3 shows the node numbers for one element. There are eight nodes in this element.

Claims

[Claim 1] constitutive formula （１．１） and （１．２） and （１．４） （１．５） and （１．６） and （１．７） A finite element method program for stress and strain using [a specific method / tool].

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