Method for establishing efficient LPBF block-by-block finite element model based on thermal-mechanical coupling

Abstract

The invention belongs to the technical field of metal additive manufacturing numerical simulation, and particularly relates to a method for establishing an efficient LPBF block-by-block finite element model based on thermal-mechanical coupling, which comprises the following steps of: establishing an LPBF block-by-block finite element model on the basis of ANSYS software; constructing a three-dimensional heat transfer calculation model and a three-dimensional stress calculation model of the metal LPBF by utilizing an APDL language; the ANSYS software program is used for carrying out grid division on the target workpiece according to a block-by-block modeling strategy, and temperature distribution calculation is carried out according to thermophysical parameters of a target workpiece material; and applying the calculated temperature distribution as a load to the three-dimensional stress calculation model, and performing stress field distribution calculation according to the temperature distribution and the mechanical property parameters of the target workpiece material at different temperatures to obtain stress distribution. The method provided by the invention realizes the prediction of the temperature field and the stress field, can reduce the time cost, and improves the simulation efficiency and the simulation precision.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of numerical simulation of metal additive manufacturing, and particularly relates to a method for establishing an efficient LPBF block-by-block finite element model based on thermal-mechanical coupling. BACKGROUND

[0002] As one of the fastest and most concerned technologies in the current additive manufacturing field, Laser Powder Bed Fusion (LPBF) has been widely used in many industries such as aerospace and biomedicine. As an advanced powder near-net forming process, LPBF not only breaks through the bottleneck of traditional manufacturing methods in complex structure forming and difficult material adaptability, but also meets the urgent needs of modern manufacturing for high precision and rapid forming. Compared with traditional forging process, LPBF can realize rapid manufacturing of complex parts without mold, greatly improving the design freedom and part precision, and effectively avoiding the problems of dendritic segregation and composition unevenness commonly seen in traditional processes. However, due to the high energy density and rapid solidification characteristics in the LPBF forming process, large residual stress is often generated inside the part, which may lead to deformation or even cracking of the workpiece. To accurately reveal the distribution of residual stress and explore its regulation mechanism, it is of great significance to establish a numerical simulation model with high precision and high efficiency.

[0003] Currently, in the numerical simulation research of LPBF forming process, two strategies of "point-by-point modeling" and "layer-by-layer modeling" are mainly used. Point-by-point modeling can simulate the formation of each powder point in detail and obtain higher simulation accuracy, but due to the extremely detailed grid division, the calculation amount is huge and the efficiency is low. Layer-by-layer modeling takes the layer as the unit, relaxes the requirement for grid, and has higher calculation efficiency, but it ignores the influence of laser scanning strategy on temperature and stress distribution, and it is difficult to meet the higher demand for result accuracy.

[0004] Therefore, the current two strategies have the outstanding problem of being difficult to balance simulation efficiency and accuracy, and it is urgent to develop a new modeling method to solve the contradiction between the two. SUMMARY

[0005] The main purpose of the present application is to provide a method for establishing an efficient LPBF block-by-block finite element model based on thermal-mechanical coupling. The method provided by the present application can realize the prediction of temperature field and stress field, reduce time cost, and improve simulation efficiency and simulation accuracy.

[0006] The present application provides a method for establishing an efficient LPBF block-by-block finite element model based on thermal-mechanical coupling, comprising the following steps:

[0007] S1, based on ANSYS software, a three-dimensional heat transfer calculation model of metal LPBF is constructed by using APDL language;

[0008] S2, based on ANSYS software, uses APDL language to construct a three-dimensional stress calculation model for metallic LPBF;

[0009] S3, Temperature field calculation: Using the ANSYS software program and according to the corresponding settings, the target part is meshed using a block-by-block modeling strategy, and the temperature distribution is calculated based on the thermal properties of the target part material to obtain the temperature distribution of the three-dimensional heat transfer calculation model.

[0010] S4, Stress field calculation: The temperature distribution is applied as a load to the three-dimensional stress calculation model. The stress field distribution is calculated based on the temperature distribution and the mechanical property parameters of the target part material at different temperatures to obtain the stress distribution of the three-dimensional stress calculation model.

[0011] In some embodiments of the present invention, in step S3, the target part is meshed using a block-by-block modeling strategy, and the thermal property parameters of the target part material corresponding to each mesh node are obtained sequentially. The temperature of each mesh node is obtained based on the thermal property parameters, and the temperature distribution of the three-dimensional heat transfer calculation model is obtained based on the temperature of all mesh nodes.

[0012] In some embodiments of the present invention, step S3 includes setting initial and boundary conditions: setting the heat source as a uniformly distributed volume heat source, wherein the heat flux density q of the volume heat source is... b The expression is: Where A is the absorption rate of the powder to the laser, which is set to 0.3; P is the laser power; D is the spot diameter, which is set to 150 μm; H is the scanning interval; tL is the thickness of the powder layer; and mb is the heat source coefficient of the block-by-block model, which is set to 0.05.

[0013] In some embodiments of the present invention, in step S3, the temperature field calculation is discretized using Solid 70 elements.

[0014] In some embodiments of the present invention, in step S4, the stress field calculation uses Solid 185 elements.

[0015] In some embodiments of the present invention, in step S4, the temperature of each grid node is obtained sequentially using the ANSYS software program, and corresponding mechanical performance parameters are assigned according to the state properties of the target part material. The stress of the grid node is obtained according to the mechanical performance parameters, and the stress distribution of the three-dimensional stress calculation model is obtained according to the stress of each grid node.

[0016] In some embodiments of the present invention, step S1, the method for constructing the three-dimensional heat transfer calculation model includes:

[0017] S1-1, Set reasonable assumptions for the temperature field;

[0018] S1-2, Set the heat transfer control equation inside the part material: The Fourier heat transfer equation is used to describe the heat conduction process inside the part material during the LPBF forming process.

[0019] S1-3, set the initial and boundary conditions, where the initial condition equation is: T| t＝0 =T0(x, y, z, t); where T is the initial temperature of the powder bed and the substrate; T0 is the preheating temperature; t is the time; x, y, z represent spatial coordinates, x and y are two planar directions, and z is the height direction.

[0020] In some embodiments of the present invention, in steps S1-2, the heat transfer control equation inside the material of the part is:

[0021]

[0022] Where ρ is density; c is specific heat capacity; T is temperature; t is time; kx, ky, and kz are the thermal conductivity of the material along the three coordinate axes, respectively; and q is heat flux density.

[0023] In some embodiments of the present invention, in steps S1-3, the boundary conditions are set as follows: when the temperature at the boundary is determined, the boundary condition equation is expressed as: T|=T(x,y,z,t); where T is the initial temperature of the powder bed and the substrate; t is time; x, y, and z represent spatial coordinates, x and y are two planar directions, and z is the height direction.

[0024] In some embodiments of the present invention, when the heat flux density at the boundary is determined, the boundary condition equation is expressed as follows:

[0025] Where k is the thermal conductivity, n is the outer normal direction of the surface; T is the initial temperature of the powder bed and the substrate; q is the heat flux density; t is time; x, y, and z represent spatial coordinates, x and y are two planar directions, and z is the height direction.

[0026] In some embodiments of the present invention, when convective heat transfer and thermal radiation occur at the boundary, the boundary condition equation is expressed as: h comb =σε(T+T0)(T+T0) 2 ); where h comb σ represents the coefficients for combining convective and radiative boundary conditions; σ is the Boltzmann constant, taken as 5.67 × 10⁻⁶. -8 W / m 2 ·K -4 ε is the emissivity, and a negative value indicates that the heat flow and temperature gradient are in opposite directions; T is the initial temperature of the powder bed and the substrate; T0 is the preheating temperature.

[0027] In some embodiments of the present invention, step S2, the method for constructing the three-dimensional stress calculation model includes:

[0028] S2-1, Set reasonable assumptions about the stress field;

[0029] S2-2, using the strengthening criterion and the thermo-elastic-plastic equation, the stress-strain relationship in the model is set;

[0030] S2-3, Set initial and boundary conditions: The bottom surface of the substrate does not undergo displacement in any direction, UX=UY=UZ=0; where: UX, UY and UZ are the displacements of the bottom surface of the substrate in the X, Y and Z directions, respectively; no boundary conditions are applied at other positions, and free deformation is allowed.

[0031] In some embodiments of the present invention, in step S2-2, the setting of the stress-strain relationship includes: when the stress exceeds the yield limit, the stress-strain curve does not change linearly, and the stress-strain relationship is expressed as: {σ}=[D]{ε e}; where: [D] is the elastic or plastic matrix; {ε e} represents the elastic strain vector.

[0032] In some embodiments of the present invention, for isotropic materials, the stress-strain relationship is expressed in a coordinate system as follows:

[0033]

[0034] Where: E and G are the elastic modulus and tangent modulus, respectively; γ xy γ yz γ xz The shear strains are in three directions, τ. xy τ yz τ xz These represent shear stresses in three directions. These represent the plastic shear strain in three directions.

[0035] The present invention provides a method for establishing an efficient block-by-block finite element model of LPBF based on thermo-mechanical coupling, which aims to improve the simulation of low-precision and low-efficiency LPBF metal material forming process. It adopts a method based on ANSYS APDL language combined with thermo-structure indirect coupling to establish a model of temperature field and stress field. This model is used to predict the temperature field and molten pool evolution law, as well as the dynamic stress and residual stress distribution during the forming process of different structural parts. This solves the problem that existing LPBF simulation technology cannot accurately and efficiently simulate the residual stress and deformation distribution, and reduces the time and financial costs of prediction and simulation calculations.

[0036] The above description is merely an overview of the technical solution of the present invention. In order to better understand the technical means of the present invention and to implement it in accordance with the contents of the specification, and in order to make the above and other objects, features and advantages of the present invention more apparent and understandable, specific embodiments of the present invention are described below. Attached Figure Description

[0037] Various other advantages and benefits will become apparent to those skilled in the art upon reading the following detailed description of preferred embodiments. The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. In the drawings:

[0038] Figure 1 This is a flowchart of the method for establishing an efficient LPBF block-by-block finite element model based on thermo-mechanical coupling in an embodiment of the present invention;

[0039] Figure 2 This is a structural design and mesh generation diagram of the TA15 component in an embodiment of the present invention; wherein, Figure 2 In the diagram, (a) shows the part structure and mesh generation under different process parameters. Figure 2 (b) shows the part structure and mesh generation under different scanning strategies. Figure 2 (c) in the figure represents the component structure and mesh division under different support structures;

[0040] Figure 3 This is a diagram showing the thermophysical properties of TA15 titanium alloy in an embodiment of the present invention; wherein, Figure 3 (a) to (d) in the figure are the relationship between thermal conductivity, density, specific heat, volume enthalpy and temperature, respectively.

[0041] Figure 4 This is a schematic diagram of three scanning strategies in Embodiment 2 of the present invention; wherein, Figure 4 (a) in the text represents a single-item scan. Figure 4 (b) in the diagram represents a reciprocating scan. Figure 4 (c) in the image represents a reciprocating scan with 90° rotation between layers;

[0042] Figure 5 This is a comparison diagram of the simulation and experimental results of residual stress in an embodiment of the present invention; wherein, Figure 5 (a) in the diagram represents the test location. Figure 5 Figures (b) through (d) show the comparison of results under different process parameters, scanning strategies, and support structures, respectively.

[0043] Figure 6 This is a comparison diagram of the simulation and experimental results of deformation in an embodiment of the present invention; wherein, Figure 6 (a) in the diagram represents the test location. Figure 6 Figures (b) through (d) show the comparison of results under different process parameters, scanning strategies, and support structures, respectively.

[0044] Figure 7 These are thermal stress-time curves for the entire processing of two models and three support structures in Embodiment 3 of the present invention; wherein, Figure 7 (a) to (c) in the model are the support structures 1 to 3 under the block-by-block strategy. Figure 7 (d) to (f) in the model are the support structures 1 to 3 under the layer-by-layer strategy;

[0045] Figure 8 These are temperature cloud maps at different energy densities in Embodiment 1 of the present invention; wherein, Figure 8 (a) to (c) in the figure are temperature contour maps at energy densities 1 to 3 under the pointwise strategy. Figure 8 In the diagram, (d) to (f) are temperature contour maps at energy densities 1 to 3 under the block-by-block strategy. Figure 8 (g) to (i) in the figure are temperature contour maps of energy densities 1 to 3 under the layer-by-layer strategy;

[0046] Figure 9 These are residual stress cloud diagrams at different energy densities in Embodiment 1 of the present invention; wherein, Figure 9 In the diagrams (a) to (c), the residual stress contour plots are shown for energy densities 1 to 3 under the pointwise strategy. Figure 9 In the diagram, (d) to (f) are residual stress contour plots at energy densities 1 to 3 under the block-by-block strategy. Figure 9 (g) to (i) are residual stress cloud diagrams at energy densities 1 to 3 under the layer-by-layer strategy;

[0047] Figure 10 These are deformation vector cloud maps under different energy densities in Embodiment 1 of the present invention; wherein, Figure 10 In the diagram, (a) to (c) are the deformed vector cloud maps for energy densities 1 to 3 under the pointwise strategy. Figure 10 In the diagram, (d) to (f) are the deformation vector cloud maps for energy densities 1 to 3 under the block-by-block strategy. Figure 10 In the diagram, (g) to (i) are the deformed vector cloud maps of energy densities 1 to 3 under the layer-by-layer strategy;

[0048] Figure 11 These are temperature cloud maps under different scanning strategies in Embodiment 2 of the present invention; wherein, Figure 11 Images (a) to (c) in the figure are temperature contour maps under scanning strategies 1 to 3 of the point-by-point strategy. Figure 11 (d) to (f) are temperature contour maps under scan strategies 1 to 3 of the block-by-block strategy;

[0049] Figure 12 These are residual stress cloud diagrams under different scanning strategies in Embodiment 2 of the present invention; wherein, Figure 12In the diagrams (a) to (c), the residual stress cloud maps are for scanning strategies 1 to 3 under the point-by-point strategy. Figure 12 (d) to (f) are residual stress cloud diagrams under scanning strategies 1 to 3 in the block-by-block strategy;

[0050] Figure 13 These are deformation vector cloud maps under different scanning strategies in Embodiment 2 of the present invention; wherein, Figure 13 In the diagram, (a) to (c) are the deformed vector cloud maps under scanning strategies 1 to 3 of the point-by-point strategy. Figure 13 In the diagram, (d) to (f) are the deformed vector cloud maps under scan strategies 1 to 3 under the block-by-block strategy;

[0051] Figure 14 These are temperature contour maps under different support structures in Embodiment 3 of the present invention; wherein, Figure 14 (a) to (c) in the figure are temperature contour maps of support structures 1 to 3 under the block-by-block strategy. Figure 14 (d) to (f) are temperature contour maps under scanning strategies 1 to 3 of the layer-by-layer strategy;

[0052] Figure 15 These are residual stress cloud diagrams under different support structures in Embodiment 3 of the present invention; wherein, Figure 15 In the diagrams (a) to (c), the residual stress cloud diagrams are for support structures 1 to 3 under the block-by-block strategy. Figure 15 (d) to (f) are residual stress cloud diagrams under scanning strategies 1 to 3 in the layer-by-layer strategy;

[0053] Figure 16 These are deformation vector cloud diagrams under different support structures in Embodiment 3 of the present invention; wherein, Figure 16 In the diagram, (a) to (c) are the deformation vector cloud diagrams of support structures 1 to 3 under the block-by-block strategy. Figure 16 In the diagram, (d) to (f) are the deformed vector cloud maps under scanning strategies 1 to 3 of the layer-by-layer strategy. Detailed Implementation

[0054] Exemplary embodiments of the present invention will now be described in more detail with reference to specific examples. It should be understood that the present invention can be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that this invention will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art.

[0055] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains; the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the invention; the terms “comprising” and “having”, and any variations thereof, in the specification, claims, and foregoing description of the invention, are intended to cover non-exclusive inclusion.

[0056] In the description of the embodiments of the present invention, the technical terms "first" and "second" are used only to distinguish different objects and should not be construed as indicating or implying relative importance or implicitly indicating the number, specific order or primary and secondary relationship of the indicated technical features.

[0057] In this document, the term "embodiment" means that a particular feature, structure, or characteristic described in connection with an embodiment may be included in at least one embodiment of the invention. The appearance of this phrase in various places throughout the specification does not necessarily refer to the same embodiment, nor is it a separate or alternative embodiment mutually exclusive with other embodiments. It will be explicitly and implicitly understood by those skilled in the art that the embodiments described herein can be combined with other embodiments.

[0058] In the description of the embodiments of this invention, the term "and / or" is merely a description of the association relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A exists, A and B exist simultaneously, and B exists. Additionally, the character " / " in this document generally indicates that the preceding and following related objects have an "or" relationship.

[0059] In the description of the embodiments of the present invention, the term "multiple" refers to two or more (including two), similarly, "multiple sets" refers to two or more (including two sets), and "multiple pieces" refers to two or more (including two pieces).

[0060] In the description of the embodiments of the present invention, the technical terms "center," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," "counterclockwise," "axial," "radial," and "circumferential" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing the embodiments of the present invention and simplifying the description, and are not intended to indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the embodiments of the present invention.

[0061] In the description of the embodiments of the present invention, unless otherwise explicitly specified and limited, the technical terms such as "installation," "connection," "joining," and "fixing" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral part; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal communication of two components or the interaction between two components. Those skilled in the art can understand the specific meaning of the above terms in the embodiments of the present invention according to the specific circumstances.

[0062] This invention provides a method for establishing an efficient block-by-block finite element model of a metal LPBF based on thermo-mechanical coupling. The key to this method lies in constructing a three-dimensional heat transfer calculation model and a three-dimensional stress calculation model of the metallic LPBF using ANSYS software, followed by the calculation of the temperature and stress fields. Specifically, the temperature field calculation involves using ANSYS to mesh the target part according to settings, sequentially obtaining the thermophysical parameters of the material corresponding to each mesh node, obtaining the temperature of each mesh node based on the thermophysical parameters, and then obtaining the temperature distribution of the three-dimensional heat transfer calculation model based on the temperatures of all mesh nodes. The stress field calculation involves applying the temperature distribution as a load to the three-dimensional stress calculation model, sequentially obtaining the temperature of each mesh node using ANSYS, assigning corresponding mechanical property parameters based on the material state properties of each mesh node, obtaining the stress of that mesh node based on the mechanical property parameters, and finally obtaining the stress distribution of the entire three-dimensional stress calculation model based on the stress of each mesh node.

[0063] In this embodiment of the invention, the method for establishing an efficient LPBF block-by-block finite element model based on thermo-mechanical coupling is specifically carried out according to the following steps.

[0064] S1, based on ANSYS software, uses APDL language to construct a three-dimensional heat transfer calculation model of metal LPBF.

[0065] In some embodiments, based on ANSYS software, the relevant parameters and equations are set in the software using the APDL language to establish a temperature field simulation model and obtain a three-dimensional heat transfer calculation model of the metal LPBF.

[0066] In some embodiments, the method for constructing the three-dimensional heat transfer calculation model in step S1 includes:

[0067] S1-1, Set reasonable assumptions about the temperature field.

[0068] In some embodiments, reasonable assumptions about the temperature field include treating the powder bed as a homogeneous and continuous material and ignoring the deformation behavior of the metal powder during melting.

[0069] In some embodiments, reasonable assumptions about the temperature field include that the increase in heat caused by laser irradiation is equivalent to a volumetric heat source.

[0070] In some embodiments, reasonable assumptions about the temperature field include ignoring energy changes caused by the angle between the laser and the forming surface.

[0071] In some embodiments, reasonable assumptions about the temperature field include not taking into account the vaporization effect of metal powder melting at high temperatures.

[0072] In some embodiments, reasonable assumptions about the temperature field include the assumption that the surface of the part remains flat, i.e., that the powder has the same thermophysical properties before and after melting.

[0073] S1-2, Set the heat transfer control equation inside the part material: The Fourier heat transfer equation is used to describe the heat conduction process inside the part material during the LPBF forming process.

[0074] In some embodiments, the Fourier heat transfer equation (or understood as the heat transfer control equation within the material of the component) is expressed as:

[0075]

[0076] Where ρ is density; c is specific heat capacity; T is temperature; t is time; kx, ky, and kz are the thermal conductivity of the material along the three coordinate axes, respectively; and q is heat flux density.

[0077] S1-3 sets the initial and boundary conditions.

[0078] In some embodiments, the initial condition equation is: T| t＝0 =T0(x, y, z, t). Where T is the initial temperature of the powder bed and the substrate; T0 is the preheating temperature; t is the time; x, y, z represent spatial coordinates, x and y are two planar directions, and z is the height direction.

[0079] In some embodiments, for the setting of boundary conditions, when the temperature at the boundary is determined, the boundary condition equation is expressed as: T|=T(x,y,z,t). Where T is the initial temperature of the powder bed and the substrate; t is time; x, y, and z represent spatial coordinates, x and y are two planar directions, and z is the height direction.

[0080] In some embodiments, for the setting of boundary conditions, when the heat flux density at the boundary is determined, the boundary condition equation is expressed as:

[0081] Where k is the thermal conductivity, n is the outer normal direction of the surface; T is the initial temperature of the powder bed and the substrate; q is the heat flux density; t is time; x, y, and z represent spatial coordinates, x and y are two planar directions, and z is the height direction.

[0082] In some embodiments, for the setting of boundary conditions, when convective heat transfer and thermal radiation occur at the boundary, the boundary condition equation is expressed as: h comb =σε(T+T0)(T+T0) 2 ).

[0083] Among them, h comb σ represents the coefficients for combining convective and radiative boundary conditions; σ is the Boltzmann constant, taken as 5.67 × 10⁻⁶. -8 W / m 2 ·K -4 ε is the emissivity, and a negative value indicates that the heat flow and temperature gradient are in opposite directions; T is the initial temperature of the powder bed and the substrate; T0 is the preheating temperature.

[0084] S2, based on ANSYS software, uses the APDL language to construct a three-dimensional stress calculation model for metallic LPBF.

[0085] In some embodiments, based on ANSYS software, the relevant parameters and equations are set in the software using the APDL language to establish a stress field simulation model and obtain a three-dimensional stress calculation model of the metal LPBF.

[0086] In some embodiments, the method for constructing the three-dimensional stress calculation model in step S2 includes:

[0087] S2-1, Set reasonable assumptions about the stress field.

[0088] In some embodiments, reasonable assumptions about the stress field include using the Von Mises yield criterion to determine whether the part material undergoes plastic deformation.

[0089] In some embodiments, reasonable assumptions about the stress field include not considering the interaction of powders, performing failure treatment on selected elements using the "element birth and death technique" in the ANSYS simulation platform, and assuming that the stress-strain behavior of the part material follows the flow criterion and the isotropic hardening criterion.

[0090] In some embodiments, reasonable assumptions about the stress field include that the temperature dependence changes approximately linearly over small time steps.

[0091] S2-2, using the strengthening criterion and the thermo-elastic-plastic equation, sets the stress-strain relationship in the model.

[0092] In some embodiments, the stress-strain relationship is set such that when the stress exceeds the yield limit, the stress-strain curve does not change linearly, and the stress-strain relationship is expressed as: {σ}=[D]{ε e}

[0093] Where: [D] is the elastic or plastic matrix; {ε e} represents the elastic strain vector.

[0094] In some embodiments, the stress-strain relationship is set as follows: for isotropic materials, the stress-strain relationship is represented in a coordinate system as:

[0095]

[0096]

[0097] Where: E and G are the elastic modulus and tangent modulus, respectively; γ xy γ yz γ xz The shear strains are in three directions, τ. xy τ yz τ xz These represent shear stresses in three directions. These represent the plastic shear strain in three directions.

[0098] S2-3, Set initial and boundary conditions.

[0099] In some embodiments, during the LPBF forming process, the powder is melted and formed layer by layer on a substrate, with the substrate in a fixed state. Therefore, in steps S2-3, the boundary conditions can be defined as the substrate bottom surface not undergoing any displacement change in any direction, UX = UY = UZ = 0. Wherein: UX, UY, and UZ are the displacements of the substrate bottom surface in the X, Y, and Z directions, respectively; no boundary conditions are applied at other locations, allowing for free deformation.

[0100] S3, Temperature Field Calculation: Using the ANSYS software program and according to the corresponding settings, the target part is meshed using a block-by-block modeling strategy, and the temperature distribution is calculated based on the thermal properties of the target part material to obtain the temperature distribution of the three-dimensional heat transfer calculation model.

[0101] In some embodiments, the method for calculating the temperature field includes:

[0102] S3-1, Set initial and boundary conditions.

[0103] In some embodiments, the initial and boundary condition settings include: setting the heat source as a uniformly distributed volumetric heat source to ensure simulation accuracy while achieving high-efficiency simulation shaping, and the heat flux density q of the volumetric heat source. b The expression is:

[0104]

[0105] Where A is the absorption rate of the powder to the laser, taken as 0.3; P is the laser power; D is the spot diameter, set to 150 μm; H is the scanning interval; t L For the thickness of the powder layer; m bThe heat source coefficient for the block-by-block model is set to 0.05.

[0106] S3-2 involves consulting multiple literature sources to collect the thermophysical parameters of the target component material for temperature field calculations. During the calculation, the program imports these thermophysical parameters into the heat transfer equations for calculation and outputs the results.

[0107] S3-3 uses a block-by-block modeling strategy to mesh the target part and calculates the temperature distribution based on the thermal properties of the target part material.

[0108] In some embodiments, meshing the target part using a block-by-block modeling strategy simplifies multi-layer powder into a single equivalent layer. The appropriate number of equivalent layers needs to be selected based on the size of the target part, balancing simulation accuracy and efficiency, to achieve a balance between computational efficiency and result accuracy.

[0109] Specifically, the target part is meshed using a block-by-block modeling strategy. The thermal properties of the target part material corresponding to each mesh node are obtained sequentially. The temperature of each mesh node is obtained based on the thermal properties. The temperature distribution of the three-dimensional heat transfer calculation model is obtained based on the temperature of all mesh nodes.

[0110] In some embodiments, the temperature field calculation is discretized using Solid 70 elements during mesh generation.

[0111] S4, Stress field calculation: The temperature distribution obtained in step S3 is applied as a load to the three-dimensional stress calculation model. The stress field distribution is calculated based on the temperature distribution and the mechanical property parameters of the target part material at different temperatures to obtain the stress distribution of the three-dimensional stress calculation model.

[0112] In some embodiments, the method for calculating the stress field includes:

[0113] S4-1 involves consulting multiple literature sources to collect mechanical property parameters of the target component material at different temperatures for stress field calculation. During the calculation, the mechanical property parameters are imported into a three-dimensional stress calculation model via a program for calculation and result output.

[0114] S4-2, Calculate the corresponding stress field distribution based on the temperature distribution and related mechanical property parameters obtained in step S3.

[0115] Specifically, the temperature of each grid node is obtained sequentially using the ANSYS software program, and corresponding mechanical property parameters are assigned according to the state properties of the target part material. The stress of the grid node is obtained based on the mechanical property parameters, and the stress distribution of the three-dimensional stress calculation model is obtained based on the stress of each grid node.

[0116] In some embodiments, the stress field calculation uses the same mesh generation mode as the temperature field, but the element type is changed to Solid185.

[0117] Unless otherwise defined, the technical terms used in the following embodiments have the same meaning as commonly understood by those skilled in the art. Unless otherwise specified, the experimental reagents used in the following embodiments are all conventional biochemical reagents; the raw materials, instruments, and equipment used in the following embodiments can all be obtained commercially or through existing methods; unless otherwise specified, the amounts of experimental reagents used are the amounts used in conventional experimental operations; unless otherwise specified, the experimental methods are conventional methods. It should be further noted that the following description is merely exemplary and not a specific limitation of the present invention.

[0118] It should be noted that the following Examples 1 to 3 exemplify the method for establishing an efficient LPBF block-by-block finite element model based on thermo-mechanical coupling provided by the present invention. See [link to example]. Figure 1 As shown, the construction methods of the three-dimensional heat transfer calculation model in step S1 and the three-dimensional stress calculation model in step S2 are the same, serving as general steps in Examples 1 to 3.

[0119] S1. Using the APDL language in ANSYS software, relevant parameters and equations are set in the software to establish a three-dimensional heat transfer calculation model.

[0120] Specifically, it is assumed that the powder bed is considered as a uniform and continuous material, and the deformation behavior of the metal powder during melting is ignored; the heat increase caused by laser irradiation is equivalent to a volume heat source; the energy change caused by the angle between the laser and the forming surface is ignored; the vaporization effect of the metal powder melting at high temperature is not considered; it is assumed that the surface of the part remains flat, that is, the powder has the same thermophysical properties before and after melting.

[0121] Meanwhile, the internal heat transfer control equation of the component material is set as follows:

[0122]

[0123] Where ρ is density; c is specific heat capacity; T is temperature; t is time; k x k y k z q represents the thermal conductivity of the material along the three coordinate axes, respectively; q is the heat flux density.

[0124] Next, set the initial and boundary conditions.

[0125] The initial condition equation is: T| t＝0=T0(x, y, z, t). Where T is the initial temperature of the powder bed and the substrate; T0 is the preheating temperature; t is the time; x, y, z represent spatial coordinates, x and y are two planar directions, and z is the height direction.

[0126] For setting the boundary conditions, when the temperature at the boundary is determined, the boundary condition equation is expressed as: T|=T(x,y,z,t). Where T is the initial temperature of the powder bed and the substrate; t is time; x, y, and z represent spatial coordinates, x and y are two planar directions, and z is the height direction.

[0127] When the heat flux density is fixed at the boundary, the boundary condition equation is expressed as:

[0128]

[0129] Where k is the thermal conductivity, n is the outer normal direction of the surface; T is the initial temperature of the powder bed and the substrate; q is the heat flux density; t is time; x, y, and z represent spatial coordinates, x and y are two planar directions, and z is the height direction.

[0130] When convective heat transfer and thermal radiation occur at the boundary, the boundary condition equation is expressed as: h comb =σε(T+T0)(T+T0) 2 ).

[0131] Among them, h comb σ represents the coefficients for combining convective and radiative boundary conditions; σ is the Boltzmann constant, taken as 5.67 × 10⁻⁶. -8 W / m 2 ·K -4 ε is the emissivity, and a negative value indicates that the heat flow and temperature gradient are in opposite directions; T is the initial temperature of the powder bed and the substrate; T0 is the preheating temperature.

[0132] S2. By using the APDL language in ANSYS software to set relevant parameters and equations, a three-dimensional stress calculation model is established.

[0133] Specifically, the Von Mises yield criterion is used to determine whether the part material has undergone plastic deformation; without considering the interaction of powders, the selected elements are subjected to failure treatment through the "element birth and death technique" in the ANSYS simulation platform; it is assumed that the stress-strain behavior of the part material follows the flow criterion and the isotropic hardening criterion; the temperature correlation is approximately linear within a small time step.

[0134] Meanwhile, the stress-strain relationship is set as follows: when the stress exceeds the yield limit, the stress-strain curve does not change linearly, and the stress-strain relationship is expressed as: {σ}=[D]{ε e}

[0135] Where: [D] is the elastic or plastic matrix; {ε e} represents the elastic strain vector.

[0136] For isotropic materials, the stress-strain relationship is expressed in a coordinate system as follows:

[0137]

[0138] Where: E and G are the elastic modulus and tangent modulus, respectively; γ xy γ yz γ xz The shear strains are in three directions, τ. xy τ yz τ xz These represent shear stresses in three directions. These represent the plastic shear strain in three directions.

[0139] Next, set the initial and boundary conditions.

[0140] In the LPBF forming process, the powder is melted and formed layer by layer on the substrate, which remains in a fixed state. Therefore, the boundary conditions can be defined as follows: the bottom surface of the substrate does not undergo displacement in any direction, UX = UY = UZ = 0. Where: UX, UY, and UZ are the displacements of the bottom surface of the substrate in the X, Y, and Z directions, respectively; no boundary conditions are applied at other locations, allowing for free deformation.

[0141] Example 1

[0142] Taking the influence of process parameters on residual stress and deformation in the LPBF forming of TA15 parts as an example, the method of establishing an efficient block-by-block finite element model of LPBF based on thermo-mechanical coupling is as follows:

[0143] Construct a three-dimensional heat transfer calculation model and a three-dimensional stress calculation model according to the general steps S1 and S2 described above.

[0144] S3 calculates the temperature field.

[0145] Initial and boundary condition settings include: setting the heat source as a uniformly distributed volumetric heat source to ensure simulation accuracy while achieving high-efficiency modeling; and setting the heat flux density q of the volumetric heat source. b The expression is:

[0146]

[0147] Where A is the absorption rate of the powder to the laser, taken as 0.3; P is the laser power; D is the spot diameter, set to 150 μm; H is the scanning interval; t L For the thickness of the powder layer; m b The heat source coefficient for the block-by-block model is set to 0.05.

[0148] In Example 1, the effective number of layers was set to 2. The temperature field under different process parameters was simulated using a block-by-block modeling strategy. The target part design and mesh generation were as follows: Figure 2 As shown in Figure (a), the different process parameters used are shown in Table 1. The mesh size of the process parameter model is 0.06 mm. 3 ×0.06mm 3 ×0.06mm 3 .

[0149] Collect the thermophysical parameters of TA15 titanium alloy, such as Figure 3 As shown, the temperature field is calculated based on the mesh generation, and the temperature field calculation results are obtained, namely the temperature contour map of the three-dimensional heat transfer calculation model. See [link / reference]. Figure 8 As shown.

[0150] Table 1

[0151]

[0152] S4 calculates the stress field.

[0153] The mechanical properties of TA15 titanium alloy at different temperatures are shown in Table 2.

[0154] Table 2

[0155]

[0156]

[0157] The stress field calculation results are obtained based on the temperature field calculation results. The residual stress field simulation results are as follows: Figure 9 The simulation results of deformation are as follows Figure 10 .

[0158] To compare the computational performance of different modeling strategies, point-by-point and layer-by-layer strategies were used for modeling. The solution efficiency and solution accuracy of the three strategies are shown in Table 3.

[0159] Table 3

[0160]

[0161] As shown in Table 3, in terms of accuracy, the block-by-block strategy is close to the point-by-point strategy and higher than the layer-by-layer strategy. In terms of solution efficiency, the block-by-block strategy significantly improves the solution efficiency compared to the point-by-point strategy. This indicates that the model established by the method provided in this invention achieves an optimal balance between accuracy and efficiency.

[0162] A simulated actual part was created using the studied model. The simulated residual stress and deformation were read at the same location and compared with the measured residual stress and deformation of the actual part, which has a size of 40 mm. 3 ×3mm 3 ×20mm 3 The thin-walled component was prepared with the parameters set as shown in Table 1. Residual stress was tested using X-ray diffraction. The X-ray spot diameter was 2 mm, the working voltage was 30 kV, and the working current was 6.7 mA. The measurement positions were as follows: Figure 5 In (a), the comparison results are as follows: Figure 5 (b) The deformation was measured using a digital vernier caliper with a precision of 5 μm. The measurement position was as follows: Figure 6 In (a), the comparison results are as follows: Figure 6 In (b), the deviation between the simulation and the actual part is less than 10%, indicating that the simulation has high accuracy.

[0163] Example 2

[0164] Taking the influence of scanning strategy on residual stress and deformation in simulating LPBF forming of TA15 parts as an example, the method of establishing an efficient block-by-block finite element model of LPBF based on thermo-mechanical coupling includes:

[0165] Construct a three-dimensional heat transfer calculation model and a three-dimensional stress calculation model according to the general steps S1 and S2 described above.

[0166] S3 calculates the temperature field.

[0167] Initial and boundary condition settings include: setting the heat source as a uniformly distributed volumetric heat source to ensure simulation accuracy while achieving high-efficiency modeling; and setting the heat flux density q of the volumetric heat source. b The expression is:

[0168]

[0169] Where A is the absorption rate of the powder to the laser, taken as 0.3; P is the laser power; D is the spot diameter, set to 150 μm; H is the scanning interval; t L For the thickness of the powder layer; m b The heat source coefficient for the block-by-block model is set to 0.05.

[0170] In Example 2, the effective number of layers is set to 2. The temperature field under different scanning strategies is simulated using a block-by-block modeling strategy. The target part design and mesh generation are as follows: Figure 2 As shown in (b) above, the different scanning strategies used are as follows: Figure 4 As shown, Figure 4In the diagram, (a) to (c) are set as scanning strategies 1 to 3, respectively, and the mesh size of the scanning strategy model is 0.03 mm. 3 ×0.03mm 3 ×0.03mm 3 .

[0171] Collect the thermophysical parameters of TA15 titanium alloy, such as Figure 3 As shown, the temperature field is calculated based on the mesh division, and the temperature field calculation result is obtained, namely the temperature contour map of the three-dimensional heat transfer calculation model, as shown. Figure 11 As shown.

[0172] S4 calculates the stress field.

[0173] The mechanical properties of TA15 titanium alloy at different temperatures are shown in Table 2 of Example 1.

[0174] The stress field calculation results are obtained based on the temperature field calculation results. The residual stress field simulation results are as follows: Figure 12 The simulation results of deformation are as follows Figure 13 .

[0175] To compare the computational performance of different modeling strategies, point-by-point and layer-by-layer strategies were used for modeling. The solution efficiency and solution accuracy of the three strategies are shown in Table 4.

[0176] Table 4

[0177]

[0178] As shown in Table 4, in terms of accuracy, the block-by-block strategy is close to the point-by-point strategy and higher than the layer-by-layer strategy. In terms of solution efficiency, the block-by-block strategy significantly improves the solution efficiency compared to the point-by-point strategy. This indicates that the model established by the method provided in this invention achieves an optimal balance between accuracy and efficiency.

[0179] A simulated actual part was created using the studied model. The simulated residual stress and deformation were read at the same location and compared with the measured residual stress and deformation of the actual part, which has a size of 40 mm. 3 ×3mm 3 ×20mm 3 The scanning strategy settings for thin-walled wall components are as follows: Figure 4 As shown, residual stress was tested using X-ray diffraction. The X-ray spot diameter was 2 mm, the operating voltage was 30 kV, and the operating current was 6.7 mA. The measurement positions were as follows. Figure 5 In (a), the comparison results are as follows: Figure 5 (c) The deformation was measured using a digital vernier caliper with a precision of 5 μm. The measurement position was as follows: Figure 6 In (a), the comparison results are as follows:Figure 6 In (c), the deviation between the simulation and the actual part is less than 10%, indicating that the simulation has high accuracy.

[0180] Example 3

[0181] Taking the influence of the support structure on residual stress and deformation in the LPBF-formed TA15 part as an example, the method of establishing an efficient block-by-block finite element model of LPBF based on thermo-mechanical coupling is as follows:

[0182] Construct a three-dimensional heat transfer calculation model and a three-dimensional stress calculation model according to the general steps S1 and S2 described above.

[0183] S3 calculates the temperature field.

[0184] Initial and boundary condition settings include: setting the heat source as a uniformly distributed volumetric heat source to ensure simulation accuracy while achieving high-efficiency modeling; and setting the heat flux density q of the volumetric heat source. b The expression is:

[0185]

[0186] Where A is the absorption rate of the powder to the laser, taken as 0.3; P is the laser power; D is the spot diameter, set to 150 μm; H is the scanning interval; t L For the thickness of the powder layer; m b The heat source coefficient for the block-by-block model is set to 0.05.

[0187] In Example 3, the effective number of layers was set to 2. The temperature field under different scanning strategies was simulated using a block-by-block modeling strategy. The target part design and mesh generation were as follows: Figure 2 As shown in (c) in the figure, the different scanning strategies used are shown in Table 5, and the mesh size of the support structure model is 0.09 mm. 3 ×0.09mm 3 ×0.09mm 3 .

[0188] Collect the thermophysical parameters of TA15 titanium alloy, such as Figure 3 As shown, the temperature field is calculated based on the mesh generation, and the temperature field calculation results are obtained, namely the temperature contour map of the three-dimensional heat transfer calculation model. See [link / reference]. Figure 14 As shown.

[0189] Table 5

[0190] Support structure code Tooth height (mm) Tooth top width (mm) Tooth root width (mm) 1 # ]] 0.1 0.12 0.14 2 # ]] 0.12 0.12 0.14 3 # ]] 0.14 0.12 0.14

[0191] S4 calculates the stress field.

[0192] The mechanical properties of TA15 titanium alloy at different temperatures are shown in Table 2 of Example 1.

[0193] The stress field calculation results are obtained based on the temperature field calculation results. The residual stress field simulation results are as follows: Figure 15 The simulation results of deformation are as follows Figure 16 .

[0194] To compare the computational performance of different modeling strategies, a layer-by-layer strategy was used to build the model, and the thermal stress-time curves of the entire manufacturing process were compared. Figure 7 As shown, the results demonstrate that the model established by the block-by-block strategy can accurately simulate the first stress cycle, and the stress value after cooling to room temperature increases with the increase of tooth height, which is consistent with the actual forming law. However, the model established by the layer-by-layer strategy does not reflect these laws, indicating that the model established in this paper can more accurately reflect the LPBF forming process, and the established model achieves high solution accuracy.

[0195] A real part was simulated using the studied model. The simulation results of residual stress and deformation were read at the same location and compared with the measured results of residual stress and deformation of the actual part, which has a size of 40 mm. 3 ×3mm 3 ×20mm 3 The thin-walled component was prepared with the parameters set as shown in Table 6. Residual stress was tested using X-ray diffraction. The X-ray spot diameter was 2 mm, the working voltage was 30 kV, and the working current was 6.7 mA. The measurement positions were as follows: Figure 5 In (a), the comparison results are as follows: Figure 5 (d) The deformation was measured using a digital vernier caliper with a precision of 5 μm. The measurement position was as follows: Figure 6 In (a), the comparison results are as follows: Figure 6 In (d), the deviation between the simulation and the actual part is less than 10%, indicating that the simulation has high accuracy.

[0196] Table 6

[0197] Support structure code Tooth height (mm) Tooth top width (mm) Tooth root width (mm) 1 # ]] 1 1.2 1.4 2 # ]] 1.2 1.2 1.4 3 # ]] 1.4 1.2 1.4

[0198] The present invention provides a method for constructing step-by-step, block-by-block finite element models of temperature and stress fields in the ANSYS APDL environment based on thermo-mechanical indirect coupling. This method effectively improves the computational efficiency of LPBF process simulation while ensuring simulation accuracy. Using the method described in this invention, the evolution and distribution of temperature and stress in different structural components during the LPBF forming process can be predicted efficiently and accurately. Experimental results show that the present invention significantly reduces computation time while maintaining computational accuracy, and can accurately predict the residual stress distribution and deformation characteristics of the formed parts.

[0199] This invention provides a reliable technical means and theoretical basis for the numerical simulation of LPBF process, and is applicable to the simulation of efficient and accurate manufacturing process of complex structural parts.

[0200] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for establishing an efficient block-by-block finite element model of LPBF based on thermo-mechanical coupling, characterized in that, Includes the following steps: S1, based on ANSYS software, uses APDL language to construct a three-dimensional heat transfer calculation model of metal LPBF; S2, based on ANSYS software, uses APDL language to construct a three-dimensional stress calculation model for metallic LPBF; S3, Temperature field calculation: Using the ANSYS software program and according to the corresponding settings, the target part is meshed using a block-by-block modeling strategy, and the temperature distribution is calculated based on the thermal properties of the target part material to obtain the temperature distribution of the three-dimensional heat transfer calculation model. S4, Stress field calculation: The temperature distribution is applied as a load to the three-dimensional stress calculation model. The stress field distribution is calculated based on the temperature distribution and the mechanical property parameters of the target part material at different temperatures to obtain the stress distribution of the three-dimensional stress calculation model.

2. The method as described in claim 1, characterized in that, In step S3, the target part is meshed using a block-by-block modeling strategy. The thermal properties of the target part material corresponding to each mesh node are obtained sequentially. The temperature of each mesh node is obtained based on the thermal properties. The temperature distribution of the three-dimensional heat transfer calculation model is obtained based on the temperature of all mesh nodes.

3. The method as described in claim 1, characterized in that, In step S3, the corresponding settings include the initial and boundary conditions: the heat source is set as a uniformly distributed volumetric heat source, and the heat flux density q of the volumetric heat source is... b The expression is: Where A is the absorption rate of the powder to the laser, taken as 0.3; P is the laser power; D is the spot diameter, set to 150 μm; H is the scanning interval; t L For the thickness of the powder layer; m b The heat source coefficient for the block-by-block model is set to 0.

05.

4. The method as described in claim 1, characterized in that, In step S3, the temperature field calculation uses Solid 70 elements for discretization; and / or, In step S4, the stress field calculation uses Solid 185 elements.

5. The method as described in claim 1, characterized in that, In step S4, the temperature of each grid node is obtained sequentially using the ANSYS software program, and the corresponding mechanical property parameters are assigned according to the state properties of the target part material. The stress of the grid node is obtained based on the mechanical property parameters, and the stress distribution of the three-dimensional stress calculation model is obtained based on the stress of each grid node.

6. The method as described in claim 1, characterized in that, In step S1, the method for constructing the three-dimensional heat transfer calculation model includes: S1-1, Set reasonable assumptions for the temperature field; S1-2, Set the heat transfer control equation inside the part material: The Fourier heat transfer equation is used to describe the heat conduction process inside the part material during the LPBF forming process. S1-3, set the initial and boundary conditions, where the initial condition equation is: T| t＝0 =T0(x, y, z, t); where T is the initial temperature of the powder bed and the substrate; T0 is the preheating temperature; t is the time; x, y, z represent spatial coordinates, x and y are two planar directions, and z is the height direction.

7. The method as described in claim 6, characterized in that, In step S1-2, the heat transfer control equation inside the part material is: Where ρ is density; c is specific heat capacity; T is temperature; t is time; k x k y k z q represents the thermal conductivity of the material along the three coordinate axes, respectively; q is the heat flux density.

8. The method as described in claim 1, characterized in that, In steps S1-3, the boundary conditions are set as follows: When the temperature is fixed at the boundary, the boundary condition equation is expressed as: T|=T(x,y,z,t); where T is the initial temperature of the powder bed and the substrate; t is time; x, y, and z represent spatial coordinates, x and y are two planar directions, and z is the height direction; or, When the heat flux density is fixed at the boundary, the boundary condition equation is expressed as: Where k is the thermal conductivity, n is the outer normal direction of the surface; T is the initial temperature of the powder bed and the substrate; q is the heat flux density; t is time; x, y, z represent spatial coordinates, x and y are two planar directions, and z is the height direction; or, When convective heat transfer and thermal radiation occur at the boundary, the boundary condition equations are expressed as follows: h comb =σε(T+T0)(T+T0) 2 ); where h comb σ represents the coefficients for combining convective and radiative boundary conditions; σ is the Boltzmann constant, taken as 5.67 × 10⁻⁶. -8 W / m 2 ·K -4 ε is the emissivity, and a negative value indicates that the heat flow and temperature gradient are in opposite directions; T is the initial temperature of the powder bed and the substrate; T0 is the preheating temperature.

9. The method as described in claim 1, characterized in that, In step S2, the method for constructing the three-dimensional stress calculation model includes: S2-1, Set reasonable assumptions about the stress field; S2-2, using the strengthening criterion and the thermo-elastic-plastic equation, the stress-strain relationship in the model is set; S2-3, Set initial and boundary conditions: The bottom surface of the substrate does not undergo displacement in any direction, UX=UY=UZ=0; where: UX, UY and UZ are the displacements of the bottom surface of the substrate in the X, Y and Z directions, respectively; no boundary conditions are applied at other positions, and free deformation is allowed.

10. The method as described in claim 9, characterized in that, In step S2-2, the setting of the stress-strain relationship includes: When the stress exceeds the yield limit, the stress-strain curve does not change linearly, and the stress-strain relationship is expressed as: {σ}=[D]{ε e }; where: [D] is the elastic or plastic matrix; {ε e } represents the elastic strain vector; or, For isotropic materials, the stress-strain relationship is expressed in a coordinate system as follows: Where: E and G are the elastic modulus and tangent modulus, respectively; γ xy γ yz γ xz The shear strains are in three directions, τ. xy τ yz τ xz These represent shear stresses in three directions. These represent the plastic shear strain in three directions.

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