Formulation of finite element method using viscoplasticity

Abstract

To perform establishment by a finite element method using a macroscopic yield function capable of expressing a Bauschinger effect.SOLUTION: Provided is a finite element method using a constitutive expression in which a macroscopic stress is decomposed into components of a shear stress of a slip system by a Taylor factor to obtain a strain rate. In the crystal plasticity, since the Bauschinger effect was exhibited even without rotating the crystal, it was confirmed that the Bauschinger effect was exhibited by the back stress. Therefore, the Taylor factor is set to a constant. This is a program of a finite element method accompanied by a macroscopic constitutive expression obtained by improving a constitutive expression of crystal plasticity. The finite element method is composed of a constitutive expression, a yield function, and an expression representing plastic strain. This corresponds to the heart of the finite element method.SELECTED DRAWING: None

Description

[Technical Field]

[0001] This invention expresses a single tension-compression test using a viscoplastic relationship between stress and strain (hereinafter referred to as the viscoplastic constitutive equation). This viscoplastic constitutive equation focuses on the fact that it can be used to derive the relationship between the equivalent plastic strain rate and the equivalent stress rate and formulate the finite element method. The plastic strain portion of the viscoplastic finite element method is an extension of the rigid plastic finite element method. Furthermore, to take into account the Bauschinger effect, which occurs when a material is stretched and then compressed, the yield point when compressed is lower than the yield point when stretched, the introduction of back stress is an inventive step because the Bauschinger effect does not occur with isotropic hardening alone. Furthermore, we believe that the creation of the viscoplastic constitutive equation from the perspective of the crystal plasticity constitutive equation constitutes an inventive step of this invention. [Background technology]

[0002] Previously, there have been finite element methods that use isotropic hardening such as rigid plasticity and elastoplasticity. However, these constitutive equations for stress and strain that only involve isotropic hardening were experimentally derived from tensile tests, and did not take into account the relationship between stress and strain during compression, so it was thought that Bauschinger hardening would not occur. Introducing back stress into the constitutive equation for crystal plasticity would result in the Bauschinger effect, but this time, we established a constitutive equation that would produce the Bauschinger effect by using a finite element method that uses a yield function from a new macroscopic perspective, rather than a microscopic one. [Prior art documents] [Non-patent literature]

[0003] [Non-Patent Document 1] Chugoku-Shikoku Branch General Meeting / Lecture Proceedings of the Seminar Cyclic Plasticity Analysis Using Crystal Plasticity Summary of the Invention [Problem to be solved by the invention]

[0004] In the case of rigid plasticity, the finite element method cannot be expressed without three things: an expression of equivalent stress using a yield function, an empirical formula for equivalent stress and equivalent strain obtained from a single tensile test, and an equation relating plastic strain and stress. In the case of rigid plasticity, the formula is an empirical formula derived from a single tension test, so it cannot express hardening in the compression stage. In this study, a constitutive formula for equivalent stress and equivalent strain that takes back stress into account was used, so the Bauschinger effect can be expressed. In addition, because a new constitutive formula was devised for this purpose, it is novel and progressive, and was established using the finite element method using a macroscopic yield function that can express the Bauschinger effect. [Means for solving the problem]

[0005] Viscoplastic constitutive equation TIFF2025134678000001.tif831 formula 1 TIFF2025134678000002.tif43 is the strain rate constant, M is the Taylor factor, Y is the initial yield stress, D is the initial critical shear stress in the transition strain direction, m is a constant, and R represents isotropic hardening. Y is the initial yield point in a uniaxial tensile test. D is the initial critical shear stress of the slip plane obtained from a tensile test of a single crystal with known crystal orientation. This constitutive formula is When TIFF2025134678000003.tif867TIFF2025134678000004.tif617 exceeds the critical shear stress D, set m to about 0.001, TIFF2025134678000005.tif811 is the first to be greater than 1, The value is entered into the equivalent plastic strain rate, TIFF2025134678000006.tif43. TIFF2025134678000007.tif43, m can be any value. TIFF2025134678000008.tif10124 formula 2 TIFF2025134678000009.tif64 shows the equivalent stress. TIFF2025134678000010.tif178 shows the deviatoric stress. TIFF2025134678000011.tif1811 shows the back stress. Equation 2 shows the yield function. Next, we will show the plastic strain. TIFF2025134678000012.tif2132 formula 3 TIFF2025134678000013.tif209 shows the plastic strain rate. TIFF2025134678000014.tif43 is the equivalent strain rate. first, TIFF2025134678000015.tif4772TIFF2025134678000016.tif268 is the elastic strain rate. TIFF2025134678000017.tif6155 Up to the initial yield stress Y TIFF2025134678000018.tif1195 TIFF2025134678000019.tif64 calculates the equivalent force stress and substitutes it into equation 1. Above the initial yield stress Y, TIFF2025134678000020.tif10121 TIFF2025134678000021.tif64 calculates the equivalent force stress and substitutes it into equation 1. Also, plastic strain TIFF2025134678000022.tif2513 shows that after the initial yield stress Y is reached, TIFF2025134678000023.tif2132Extract from equation 3. In other words, there is a judgment of elasticity and plasticity. According to the principle of least work, TIFF2025134678000024.tif68148du is the displacement, B is the B matrix, and f is the nodal force. If you know TIFF2025134678000025.tif53, TIFF2025134678000026.tif2771 can be solved. TIFF2025134678000027.tif726TIFF2025134678000028.tif2239TIFF2025134678000029.tif8458 As mentioned before, Y is the initial yield stress, which is obtained from a uniaxial tensile test. D is the initial critical shear stress of the slip plane, which is obtained from a tensile test of a single crystal with known crystal orientation. [Effects of the Invention]

[0006] The back stress material constant C is used in a numerical simulation to predict springback using the crystal plasticity constitutive equation of the finite element method, and the press working conditions are determined, allowing press working to deal with actual springback. In addition, since a new constitutive equation has been devised in Equation 1, TIFF2025134678000030.tif63 is required. DETAILED DESCRIPTION OF THE INVENTION

[0007] From Equation 1, Equation 2, and Equation 3, TIFF2025134678000031.tif63 is found, TIFF2025134678000032.tif2771 appears. Therefore, the element-wise composite matrix can be solved. By creating an overall stiffness matrix, the displacement du or nodal force f can be obtained using the finite element method, such as the modified Cholesky method. [Industrial Applicability]

[0008] Unlike the crystal plasticity finite element method, this method allows for numerical simulations to be performed at the design and development stage before press processing or forging has been performed, allowing for the accumulation of know-how when determining the die mechanism for design and development. It can also be applied to springback.

Claims

[Claim 1] Viscoplastic constitutive equation Formula 1 is the strain rate constant, M is the Taylor factor, Y is the initial yield stress, D is the initial critical shear stress in the transition strain direction, m is a constant, and R represents isotropic hardening. Y is the initial yield point in a uniaxial tensile test. D is the initial critical shear stress of the slip plane obtained from a tensile test of a single crystal with known crystal orientation. This constitutive formula is Formula 2 The yield function in Eq. Formula 3 Considering the relationship between deviatoric stress, back stress and plastic strain in Equation 3, I was able to solve This is a program that derives the relationship between displacement and nodal force.

Images