Stress-strain curve program based on crystal plasticity constitutive equation

Abstract

To provide a method useful for determining a material constant of a constitutive equation in consideration of also a Bauschinger effect in a tension compression test, and in consideration of crystal slip, by using the constitutive equation of crystal plasticity.SOLUTION: By using a shear strain rate increment (formula 4) of a slip system in a constitutive equation, a boundary condition of joint force is provided from a principle of a virtual work, and a tension compression simulation is taken, and displacement u is outputted. Hereby, a back stress parameter C, or C and (a) are determined in combination with an actual tension compression stress strain curve.SELECTED DRAWING: None

Description

[Technical field]

[0001] This invention is a stress-strain relational equation (hereinafter referred to as constitutive equation) that focuses on the fact that plastic deformation (slip deformation) occurs when slip shear stress acts on a metal crystal above the critical shear stress on the slip system. In addition, we propose a program for the constitutive equation's stress-strain curve that takes into account back stress (the application of shear stress in the opposite direction to slip deformation at the grain boundary). [Background technology]

[0002] Previously, tension and compression tests were conducted by determining the stress-strain constitutive equation for macroscopic behavior using a viscoplastic constitutive equation. Here, we decided to use a crystal plasticity constitutive equation, take into account crystal slippage, and use a program to determine a method that will help determine the material constants of the constitutive equation that also takes into account the Bauschinger effect in tension and compression tests. In previous literature, non-patent literature, such as the Chugoku-Shikoku Branch General Meeting and Lecture Collection, "Repeated Plasticity Analysis Using Crystal Plasticity," the constitutive equation and work hardening law are described, but not much detail is given. Here, we decided to set up a crystal plasticity constitutive equation for each crystal. [Prior art documents] [Non-patent literature]

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

[0004] A numerical simulation is made that takes polycrystalline bodies into account using the constitutive equation of a simple tension-compression test. Here, we will show a method for determining the material parameter C of the back stress of the stress-strain curve using the results of a tension-compression test that shows the Bauschinger effect (the compression yield force is smaller than the tensile yield force) by drawing a stress-strain curve. By doing this, the Bauschinger effect can be matched with the results of a numerical simulation and a tension-compression test. In other words, a press working simulation can be performed by inputting this material constant C or material constant C and a into the finite element method. This numerical simulation can also analyze the numerical simulation of springback. In other words, it is possible to derive the work hardening law from anisotropy in the crystal orientation of each crystal, a single crystal tensile test, and perform a complex press working numerical simulation (with springback) to determine the processing conditions. [Means for solving the problem]

[0005] Crystal plasticity constitutive equation (Equation 1) TIFF0007599635000001.tif1236TIFF0007599635000002.tif1235r is the slip number of a single crystal TIFF0007599635000003.tif44 is the shear strain rate of the slip system with slip number r TIFF0007599635000004.tif44=0.001(constant) TIFF0007599635000005.tif44 is the shear stress of the slip system of slip number r TIFF0007599635000006.tif44 is the critical shear stress for slip system with slip number r TIFF0007599635000007.tif44 is r Slip number Back stress of slip system At m=0.001 (constant), Taylor expansion, TIFF0007599635000008.tif850 (Formula 2) Further refinement of (Equation 2) gives (Equation 3). TIFF0007599635000009.tif881 (Formula 3) Now, if we do linear interpolation, TIFF0007599635000010.tif444θ Linear Interpolation Constant Δt time increment TIFF0007599635000011.tif1250 (Formula 4) Where: TIFF0007599635000012.tif44= The file is TIFF0007599635000013.tif413. NS is the number of slip systems (body-centered cubic lattice 12) In the hardening law proposed by Pierce, TIFF0007599635000014.tif429TIFF0007599635000015.tif43 is 1 when r=s. r TIFF0007599635000016.tif44: 0 q=1.2 (constant) TIFF0007599635000017.tif840The material constants are TIFF0007599635000018.tif43=534[MPa], TIFF0007599635000019.tif43=0[MPa], TIFF0007599635000020.tif43=60[MPa], TIFF0007599635000021.tif42=108[MPa]. TIFF0007599635000022.tif412 (Prager's law) C is the material constant. or, TIFF0007599635000023.tif427In the above formula, C and a are material constants. Back stress increments can also be used. The material constants C and a can also be calculated by comparing the results of actual tension and compression tests using the finite element method. Here, the plastic strain rate of crystal plasticity is TIFF0007599635000024.tif44 is shown below. TIFF0007599635000025.tif4637TIFF0007599635000026.tif44, TIFF0007599635000027.tif44, TIFF0007599635000028.tif44 is the unit orientation vector (x, y, z) of the slip surface with slip number k. TIFF0007599635000029.tif44, TIFF0007599635000030.tif44, TIFF0007599635000031.tif44 is the (x, y, z) unit orientation vector of the slip direction of r slip number. Here, the following stiffness matrix is ​​obtained from the principle of virtual work: Using this formula, a global stiffness matrix is ​​created, and a finite element method is created. TIFF0007599635000032.tif6991 (Formula 6) The B matrix is ​​the shape function differentiated with respect to the local coordinate variables. The shape function and the interpolation function are the same, so isometric elements are used. The D matrix is ​​the elasticity matrix. Furthermore, the program uses numerical integration values. (Equation 6) is the increment in the stiffness matrix of one element (one crystal) in the crystal plasticity finite element method. This is converted to a global crystal plasticity matrix. It is solved using the modified Cholesky method. The displacement increment u can be obtained by applying the nodal forces of the boundary conditions. Effect 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 be performed in a way that deals with the actual springback. Back stress material constant C or a may also be used. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0007] A numerical simulation was performed using a computer program that uses the constitutive equations (Eq. 1), (Eq. 4), and (Eq. 6). The back stress is given by Use either TIFF0007599635000033.tif649TIFF0007599635000034.tif532. [Industrial Applicability]

[0008] At the design and development stage when press forming dies are not yet made, numerical simulations can be performed to determine the die mechanism for design and development, and know-how can be accumulated. It can also be applied to springback.

Claims

[Claim 1] "Constitutive formula (formula 1) (where r is the slip number of a single crystal, is the shear strain rate of the slip system with slip number r, is 0.001, is the shear stress of the slip system with slip number r, is the critical shear stress of the slip system with slip number r, is the back stress of the slip system with slip number r, m = 0.001) and (Formula 4) Where: ＝ Here, NS is the total number of slip systems. For example, for a face-centered cubic lattice, it is 12.) and (Formula 6) (where: 、 、 is the unit orientation vector (x, y, z) of the slip surface with slip number k, 、 、 is the unit orientation vector (x, y, z) of the slip direction of slip number k. In computer programming using In the following formula: (Prager's law) C is the material constant. Determine the back stress material constant C.

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