Implementation method of SAC solder joint thermal coupling constitutive model in electronic package

Abstract

The present application relates to the technical field of electronic packaging, and particularly relates to an implementation method of a SAC solder joint thermal coupling constitutive model in electronic packaging. The method introduces thermal strain correction, temperature-dependent inelastic flow, follow-up hardening, dynamic recovery and internal structure variable evolution in the same constitutive model, and can be used to describe the multi-mechanism coupling response of SAC solder under temperature cycle loading. The method uses temperature-dependent flow index and temperature-dependent internal variable evolution parameters, so that the inelastic strain increment and internal structure variable update change with the current temperature, and is suitable for solder joint simulation under temperature cycle conditions. The method saves the elastic strain, inelastic strain, back stress and internal structure variable through UMAT state variables, so that the material integration point can record the loading history and continue to update in the subsequent increment step. The present application can establish a finite element model according to the actual packaging sample size, and is used for SAC solder joint thermal cycle mechanical response calculation and reliability analysis under different packaging structures and reinforcement modes.

Description

Technical Field

[0001] This invention relates to the field of electronic packaging technology, specifically to a thermo-mechanically coupled constitutive model and finite element method for the mechanical behavior of SAC solder under temperature cyclic loading. Background Technology

[0002] SAC solder is a Sn-Ag-Cu based lead-free solder widely used in electronic packaging structures. Solder joints serve both as the mechanical connection between the chip and the substrate, and as the electrical interconnection and load transfer function. During temperature cycling service or temperature cycling tests, the coefficients of thermal expansion of different components such as the chip, substrate, solder, and dispensing materials are not consistent. The packaging structure is subjected to repeated thermal deformation constraints, and cyclic thermal stress and inelastic strain accumulation easily occur inside the solder joints.

[0003] SAC solder has a low melting point and exhibits significant time- and temperature-dependent mechanical behavior within the common service temperature range of electronic packaging. Under temperature cycling, the solder joint response is not a single elastic, single plastic, or single creep process, but rather the result of the combined effects of mechanisms such as thermoelastic deformation, viscoplastic or creep flow, kinematic hardening, dynamic recovery, and internal structural evolution. If only one mechanism is considered in the constitutive model, it is difficult to reflect the stress relaxation, strain accumulation, and evolution of critical regions of the solder joint during thermal cycling.

[0004] Currently, in finite element analysis of electronic packaging solder joint reliability, empirical fatigue models, single creep models, or simplified elasto-plastic models are commonly used to describe the behavior of SAC solder. These models can obtain the local stress or strain distribution of the solder joint, but they are usually difficult to simultaneously handle thermal strain correction, temperature-dependent inelastic flow, back stress evolution, and the accumulation of material history variables. Therefore, in temperature cycling simulations, the calculated locations of critical solder joints, stress-strain trends, or failure risk areas may deviate from the metallographic morphology, crack locations, or failure trends after the experiment.

[0005] On the other hand, while finite element software such as Abaqus provides various built-in material models, these models are difficult to directly implement multi-mechanism coupled constitutive relations for the thermal cycling behavior of SAC solder. To store elastic strain, inelastic strain, back stress, and internal structural variables at the material integration point level and update them with temperature, stress, and inelastic deformation history in each increment step, a user material subroutine is required.

[0006] Therefore, a method for constructing a thermo-mechanically coupled constitutive model for SAC solder joints in electronic packaging is needed. This model should be converted into a UMAT user material subroutine, allowing it to be embedded into the finite element model of actual packaged samples for calculating the stress, strain, and internal variable evolution of solder joints under temperature cyclic loading. The thermo-mechanical coupling referred to in this invention refers to the influence of temperature changes on the mechanical response of SAC solder joints through thermal strain, temperature-dependent inelastic flow indices, and temperature-dependent internal structural variable evolution parameters. Summary of the Invention

[0007] The purpose of this invention is to overcome the shortcomings of existing technologies by providing an implementation method for a thermo-mechanical coupling constitutive model of SAC solder joints in electronic packaging, providing a thermo-mechanical coupling constitutive model of SAC solder integrating multiple physical mechanisms and a corresponding finite element simulation method, and applying it to solder joint material research in the field of electronic packaging; exploring the mechanical behavior and intrinsic mechanism of SAC solder under temperature cyclic loading, and providing simulation basis and technical support for the reliability design of electronic packaging solder joints.

[0008] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A method for implementing a constitutive model of SAC solder joint thermal coupling in electronic packaging includes the following steps: Step 1: Establish a thermo-mechanical coupled incremental constitutive model for SAC solder. The thermo-coupled incremental constitutive model includes thermal strain correction relations, elastic prediction relations, overstress criterion considering back stress, temperature-dependent inelastic flow relations, back stress evolution relations, and tensor-type damage-related internal structural variable evolution relations; the temperature-dependent flow index is determined based on the temperature at the start of the material integration point increment. Temperature-related internal structure variables evolution parameters The temperature-dependent flow index is used. The inelastic flow response is determined using the temperature-dependent internal structure variable evolution parameters. Update the tensor-type damage-related internal structure variable components and adjust the equivalent values ​​of the internal structure variables. D Participate in the calculation of inelastic flow resistance; Step 2: Compile the aforementioned thermo-coupling incremental constitutive model into the Abaqus user material subroutine UMAT. Set the input / output variables, model parameters, and state variables according to the interface format of the Abaqus user material subroutine UMAT, and establish the material integral point incremental update process in the UMAT. In each finite element increment step, UMAT reads the stress and state variables from the previous increment step, as well as the strain increment, initial temperature, temperature increment, and time increment of the current increment step. It then sequentially performs thermal strain correction, elastic prediction, and overstress judgment. When the overstress is greater than zero, it calculates the inelastic strain increment, corrects the stress, updates the back stress and tensor-type damage-related internal structural variable components, and recalculates their equivalent values ​​based on the updated internal structural variable components. Finally, it returns the updated stress, material stiffness matrix, and state variables to Abaqus. Step 3: Perform finite element analysis of temperature cycling of SAC solder joints in electronic packaging using the UMAT. The UMAT is assigned to the SAC solder joint region in the electronic packaging finite element model, and a temperature cycling load is applied to the electronic packaging finite element model. In each finite element increment step, Abaqus calls the UMAT and passes the material integration point variables. The UMAT completes the incremental update of the material integration points and returns the update results to obtain the evolution results of stress, strain, inelastic strain, back stress, and tensor damage-related internal structural variables of the SAC solder joint region.

[0009] Furthermore, the method for establishing the thermo-coupled incremental constitutive model in step one includes: (1) Determine the parameters required for the thermo-mechanical coupling incremental constitutive model of SAC solder. The material parameters of the SAC solder thermo-coupling incremental constitutive model include the elastic modulus. E Poisson's ratio ν Coefficient of thermal expansion αT Initial flow resistance R 0. Internal resistance parameters D 0. Kinematic hardening strength parameters C Dynamic response parameters Evolution index of internal structural variables N proportionality coefficient p Temperature-related flow index and temperature-related internal structural variables evolution parameters The temperature-related flow index is mentioned above. Temperature-related internal structure variables evolution parameters The temperature at the start of the material integration point increment is used respectively. T The function representation; (2) Establish temperature-dependent flow index Temperature-related internal structure variables evolution parameters ; To describe the variation of the inelastic flowability of SAC solder with temperature, a temperature-dependent quadratic function is used to define the flow index. : (1) To describe the evolution rate of internal structural variables as a function of temperature, a temperature-dependent quadratic function is used to define the evolution parameters of the internal variables. : (2) in, T The temperature at the start of the material integration point increment is given, and the temperature unit is consistent with the parameter calibration process. These are temperature-related parameters calibrated based on SAC solder mechanical response data; The above quadratic function form can be calibrated based on the mechanical response data of SAC solder at different temperatures, existing material parameters, or simulation inversion results; its function is to change the evolution rate of the inelastic flow index and internal structural variables with temperature changes, thereby adapting to temperature cycling conditions. (3) Based on the small deformation thermoelastic theory and the isotropic Hooke's law, establish the thermal strain correction relationship and elastic prediction relationship; Furthermore, the aforementioned thermal strain correction relationship and elasticity prediction relationship include: The total strain increment is decomposed into mechanical strain increment and thermal strain increment. The mechanical strain increment of the normal strain component is: (3) In the formula, This represents the total strain increment. For temperature increment, For mechanical strain increment, The coefficient of thermal expansion is used; the shear component is not corrected for thermal strain. Based on elastic modulus E Poisson's ratio and mechanical strain increment Construct an isotropic elastic stiffness matrix Ce And calculate the elastic predicted stress for the current increment step: (4) in, For elastically predicted stress, For the isotropic elastic stiffness tensor, This is the stress of the previous increment step; (4) The von Mises equivalent stress relationship considering back stress is used to establish the overstress criterion. The temperature-dependent inelastic flow relationship is established based on the Perzyna-type overstress viscoplasticity concept and the associated flow rule. The stress correction relationship and back stress evolution relationship are established based on the inelastic strain increment. Furthermore, the establishment of the overstress criterion based on the von Mises equivalent stress relationship considering back stress refers to... When overstress At that time, the inelastic strain increment is calculated, and the stress and back stress are corrected and updated based on the inelastic strain increment. The specific relationship is as follows: Based on elasticity, predict stress and the back stress of the previous incremental step Determine the relative deviatoric stress tensor : (5) The von Mises equivalent stress considering back stress is: (6) in, To account for the relative deviatoric stress tensor after considering the back stress, ":" indicates the tensor double dot product; Then define hyperstress for: (7) when When, the material updates according to its elastic response; when When, the material enters the inelastic flow calculation; when At that time, according to the equivalent stress Determine the direction n of the inelastic flow, and based on the overstress... Temperature-related flow index Equivalent values ​​of internal structural variables D Calculate the increment of inelastic equivalent strain : (8) (9) in, For the equivalent value of the internal structural variable, For the relative deviatoric stress tensor, For internal resistance parameters, This is the current increment step time increment. The inelastic flow direction is determined by the deviatoric stress state considering back stress; the above relationship treats overstress as the driving force for inelastic flow, and uses the equivalent value of internal structural variables. D and internal resistance parameters As an inelastic flow resistance term, and through The effect of temperature on the sensitivity of inelastic flow is described; a stress correction relationship is established based on the inelastic strain increment, and the back stress evolution relationship is established using the Armstrong-Frederick type kinematic hardening concept; The inelastic strain increment is: (10) The stress correction relationship is as follows: (11) The back stress tensor is updated to: (12) In the formula, The stress is updated in the current increment step. For the back stress of the previous incremental step, The back stress is updated in the current increment step. For hardening strength parameters, For dynamic response parameters, This is the direction of inelastic flow; (5) Introduce damage-related internal structural variables and establish the evolution relationship of internal structural variables; Furthermore, the tensor-type damage-related internal structure variable is used to characterize the evolution of the material's internal structure related to damage accumulation during the cyclic inelastic deformation of the weld joint, and it is not directly used as a criterion for unit failure. The internal structure variables mentioned The equivalent value D characterizes the internal structure evolution of SAC solder under temperature cycling and inelastic deformation. (13) The equivalent value of the total strain increment is: (14) The evolution direction components of the internal structural variables are: (15) The equivalent value of the evolution direction of internal structural variables is: (16) Internal structural variable components Update in the following format: (17) The temperature-related evolution coefficient is: (18) (19) For the previous incremental step Each internal structural variable component For the current incremental step Each internal structural variable component Shear modulus For the evolution direction component of internal structural variables This is the equivalent value of the evolution direction component. The evolution index, This is the proportionality coefficient. These are the evolution parameters for temperature-related internal structure variables.

[0010] Furthermore, step two, which involves writing the thermo-coupled incremental constitutive model into the Abaqus user material subroutine UMAT, includes: (2) Establish the UMAT subroutine interface; According to the interface format of Abaqus user material subroutine UMAT, set input and output variables such as stress, strain increment, temperature, temperature increment, time increment, material parameters, state variables and material stiffness matrix so that UMAT can be called by Abaqus in each finite element increment step. (2) Determine the method for transferring material parameters; elastic modulus E Compared to Poisson ν As the material parameter input for Abaqus users; include the coefficient of thermal expansion. αT Initial flow resistance R 0. Internal resistance parameters D 0. Hardening strength parameter C, dynamic recovery parameter γ, evolution index N proportionality coefficient p Temperature-related flow index m(T) Temperature-related internal variable evolution parameters These model parameters are used in the calculation as model parameters; the above model parameters can also be extended to Abaqus user material parameter inputs as needed. (3) Define UMAT state variables; The UMAT state variables include elastic strain, inelastic strain, back stress, internal structural variables, and equivalent values ​​of internal structural variables, which are used to store the historical response of the material integration point during temperature cyclic loading. (4) Establish the UMAT incremental integration process; In each increment step, UMAT reads the stress and state variables from the previous increment step, as well as the strain increment, initial temperature, temperature increment, and time increment for the current increment step; calculates the mechanical strain increment based on the thermal strain correction relationship; calculates the elastic predicted stress; determines whether inelastic flow has occurred; when inelastic flow occurs, updates the inelastic strain, stress, back stress, and internal structural variables; and finally returns the updated stress, material stiffness matrix, and state variables to Abaqus.

[0011] Furthermore, the UMAT is used for three-dimensional solid stress state and sets 25 state variables, of which the 1st to 6th state variables are used to store elastic strain components, the 7th to 12th state variables are used to store inelastic strain components, the 13th to 18th state variables are used to store back stress components, the 19th to 24th state variables are used to store tensor-type damage-related internal structural variable components, and the 25th state variable is used to store the equivalent value of the internal structural variables.

[0012] Furthermore, step three involves using the UMAT to perform finite element analysis of the temperature cycling of the SAC solder joints in the electronic package, specifically including: (1) In one embodiment, the dimensions of an actual electronic package sample are measured, and a finite element model of the electronic package containing SAC solder joints is established based on the measurement results; (2) Assign the SAC solder joint area to the UMAT user material model in step two; (3) Apply a temperature cycling load to the electronic packaging finite element model, the temperature cycling load including the minimum temperature, the maximum temperature, the heating time, the cooling time, the holding time, and the number of cycles; (4) Output the stress, strain, inelastic strain, back stress and internal structural variable evolution results of the SAC weld area.

[0013] The beneficial effects of this invention are as follows: This invention provides an implementation method for a thermo-mechanically coupled constitutive model of SAC solder joints in electronic packaging. By introducing thermal strain correction, temperature-dependent inelastic flow, kinematic hardening, dynamic recovery, and internal structural variable evolution into the same constitutive model, it can be used to describe the multi-mechanism coupled response of SAC solder under temperature cyclic loading. By using temperature-dependent flow index and temperature-dependent internal variable evolution parameters, the inelastic strain increment and internal structural variable update change with the current temperature, making it suitable for solder joint simulation under temperature cyclic conditions. By saving elastic strain, inelastic strain, back stress, and internal structural variables through UMAT state variables, the material integration point can record the loading history and continue to update in subsequent incremental steps. This invention can establish a finite element model based on the actual package sample size for calculating the thermo-mechanical response and reliability analysis of SAC solder joints under different packaging structures and reinforcement methods. Attached Figure Description

[0014] Figure 1 This is a flowchart of the technical solution of the present invention; Figure 2 These are actual samples from the embodiments of the present invention; Figure 3 This is a schematic diagram of the finite element geometric model established based on measured dimensions in an embodiment of the present invention; Figure 4This is the temperature cyclic load curve in an embodiment of the present invention; Figure 5 This is the X-direction inelastic strain curve of the SAC weld joint in this embodiment of the invention. Detailed Implementation

[0015] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.

[0016] To achieve the above objectives, the present invention provides the following specific embodiments: Figure 1 As shown, a method for implementing a thermo-coupling constitutive model of SAC solder joints in an electronic package includes the following steps: First, a thermo-mechanical coupling incremental constitutive model of SAC solder is established.

[0017] The first step is to determine the model parameters. The SAC solder used in this example is SAC305. The required parameters for the constitutive model include the elastic modulus. E Poisson's ratio ν Coefficient of thermal expansion αT Initial flow resistance R 0. Internal resistance parameters D 0. Kinematic hardening strength parameters C Dynamic response parameters Evolution index of internal structural variables N proportionality coefficient p Temperature-related flow index and temperature-related internal structural variables evolution parameters .

[0018] The model parameters used in the embodiment are shown in the table below. The stress parameters are expressed in MPa, the time in seconds, and the T in the temperature function in °C.

[0019]

[0020] The second step is to establish temperature-related parameters, including the temperature-dependent flow index. Temperature-related internal structure variables evolution parameters Determined according to the aforementioned formula. In this embodiment, temperature T is the material integration point temperature at the start of the current increment, and its temperature unit is consistent with the parameter calibration process. The temperature function used is: (1) (2) The third step is to establish the relationship between thermal strain correction and elastic prediction. Based on the small-deformation thermoelasticity theory, the total strain increment is decomposed into mechanical strain increment and thermal strain increment, and thermal strain correction is performed according to the temperature increment of the current increment step: (3) The elastic predicted stress is calculated based on the isotropic elastic stiffness matrix, the stress from the previous increment step, and the mechanical strain increment: (4) The fourth step is to establish the relationships between inelastic flow, stress correction, and back stress evolution. This involves considering back stress. von Mises equivalent stress relation establishes overstress criterion, overstress for: (5) when When, the material updates according to its elastic response; when When the material enters inelastic flow, calculate the direction of the inelastic flow: (6) The result obtained in step two Substituting the relationship between the inelastic equivalent variable increments: (7) Therefore, the inelastic strain increment is calculated and the stress is corrected: (8) (9) The back stress is updated using a relationship that includes both kinematic hardening and dynamic recovery terms: (10) Step 5: Establish the evolutionary relationships of damage-related internal structural variables. Introduce tensor-type damage-related internal structural variables, whose equivalent values ​​are: (11) Calculate its equivalent value based on the total strain increment of the current increment step: (12) The evolution direction components of internal structural variables are determined based on stress components, the direction of total strain increment, and the equivalent value of total strain increment: (13) The equivalent value of the evolution direction of internal structural variables is: (14) The result obtained in step two Substituting the temperature-related evolution coefficients, update each component of the internal structural variables according to the following formula: (15) The constitutive model will now be written as the Abaqus user material subroutine UMAT.

[0021] Step 6: Establish the UMAT subroutine interface. The established thermo-coupled incremental constitutive model is written as the Abaqus user material subroutine UMAT using Fortran. Following the Abaqus UMAT interface format, the input and output relationships for stress, state variables, total strain increment, time increment, temperature at the start of the increment, temperature increment, user material parameters, and the material stiffness matrix are set. This embodiment uses Abaqus version 2023, and the Fortran compilation environment is Microsoft Visual Studio 2019 and Intel Fortran Compiler 2020.

[0022] Step 7: Determine the model parameter transfer method. In this embodiment, the elastic modulus of the SAC solder is input through the user material parameter PROPS. E Compared to Poisson ν The coefficient of thermal expansion, initial flow resistance, internal resistance parameters, kinematic hardening strength parameters, dynamic recovery parameters, internal structural variable evolution index, proportional coefficient, and temperature function parameters are calculated using preset parameters within the program.

[0023] Material integration point temperature at the start of the current increment The temperature increment corresponding to the TEMP input from the UMAT interface. Corresponding to DTEMP.

[0024] Step 8: Define and initialize the UMAT state variables. The UMAT state variables include elastic strain, inelastic strain, back stress, tensor damage-related internal structure variables, and equivalent values ​​of these internal structure variables, used to store the historical response of the material's integration points during temperature cycling. In the first increment of the first analysis step, all back stress components are initialized to 0, and the three positive components and three shear components of the tensor damage-related internal structure variables are initialized to 1. This embodiment focuses on the inelastic strain components stored in state variables 7 through 12.

[0025] Step nine involves establishing the UMAT material integration point incremental update process. In each incremental step, UMAT reads the stress and state variables from the previous incremental step, as well as the strain increment, initial temperature, temperature increment, and time increment for the current incremental step; calculates the mechanical strain increment based on the thermal strain correction relationship; calculates the elastic predicted stress; determines whether inelastic flow has occurred; if inelastic flow occurs, updates the inelastic strain, stress, back stress, and internal structural variables; finally, returns the updated stress, material stiffness matrix, and state variables to Abaqus. The UMAT source program file is set as a user subroutine file and submitted for calculation along with the finite element model.

[0026] Finite element analysis of actual electronic packaging samples was performed using UMAT.

[0027] Step 10: Measure the actual electronic packaging sample and establish a finite element geometric model. This embodiment selects BGA and CCGA packaging devices as research objects. The electronic packaging sample mainly includes: chip, SAC solder joints, substrate, printed circuit board, reinforcement materials, etc. Figure 2 As shown, the packaged sample was measured three times, and the average value was used as the finite element modeling dimension. A three-dimensional finite element model of the electronic package was established using Abaqus 2023 based on the measurement results, as shown below. Figure 3 As shown, the SAC solder joint geometry consists of solder balls and solder pillars. Simplified forms are used for local details that have little impact on the temperature cycling response of the solder joint.

[0028] Step 11: Set material properties and assign UMAT to the SAC solder joint area. Assign the written UMAT to the SAC solder joint area in the finite element model. Besides the SAC solder joints, the other constituent materials of the electronic packaging model in this embodiment are shown in the table below:

[0029] Step 12: Mesh generation and condition setting. Based on the structure of the actual electronic packaging sample and the constraints of the temperature cycling test, the finite element model is meshed, and the mesh is locally refined in the SAC solder joint area. Solid elements suitable for three-dimensional stress analysis are used, and boundary constraints are set according to the actual installation state of the sample, eliminating rigid body displacement of the model. Automatic time increment is used during the analysis, with Abaqus calling UMAT in each time increment step to update the SAC solder joint material integration point state.

[0030] Step 13: Apply temperature load and submit the calculation. For example... Figure 4 As shown, a temperature cyclic load was applied to the finite element model of the electronic package. The cyclic loading was performed within a temperature range of -55℃ to 125℃, with a temperature change rate of 5℃ / min, and the temperature was held for 15 minutes at both the high and low temperature stages.

[0031] Step fourteen: Output and analyze the finite element results. After completing the temperature cycling calculation, output the stress, total strain, inelastic strain, back stress, damage-related internal structural variable components, and their equivalent values ​​for the SAC weld area. This embodiment focuses on the inelastic strain of the weld area. The state variables of the inelastic strain components stored in the 7th to 12th segments of the weld array are extracted for analysis, such as... Figure 5 As shown.

[0032] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for implementing a thermo-mechanical coupling constitutive model of SAC solder joints in electronic packaging, characterized in that, Includes the following steps: Step 1: Establish a thermo-mechanically coupled incremental constitutive model for SAC solder. The thermo-coupled incremental constitutive model includes thermal strain correction relations, elastic prediction relations, overstress criterion considering back stress, temperature-dependent inelastic flow relations, back stress evolution relations, and tensor-type damage-related internal structural variable evolution relations; the temperature-dependent flow index is determined based on the temperature at the start of the material integration point increment. Temperature-related internal structure variables evolution parameters ; Temperature-dependent flow index was used. Determining the inelastic flow response using temperature-dependent internal structure variable evolution parameters Update the tensor-based damage-related internal structure variable components to achieve the equivalent values ​​of the internal structure variables. D Participate in the calculation of inelastic flow resistance; Step 2: Write the thermo-coupled incremental constitutive model as the Abaqus user material subroutine UMAT. Set the input / output variables, model parameters, and state variables according to the interface format of the Abaqus user material subroutine UMAT, and establish the incremental update process of material integration points in UMAT; In each finite element increment step, UMAT reads the stress and state variables from the previous increment step, as well as the strain increment, initial temperature, temperature increment, and time increment of the current increment step. It then sequentially performs thermal strain correction, elastic prediction, and overstress assessment. When the overstress is greater than zero, it calculates the inelastic strain increment, corrects the stress, updates the back stress and tensor damage-related internal structural variable components, and recalculates their equivalent values ​​based on the updated internal structural variable components. Finally, it returns the updated stress, material stiffness matrix, and state variables to Abaqus. Step 3: Perform finite element analysis of temperature cycling of SAC solder joints in electronic packaging using UMAT. UMAT is applied to the SAC solder joint region in the electronic package finite element model, and a temperature cycling load is applied to the electronic package finite element model. In each finite element increment step, Abaqus calls UMAT and passes the material integration point variables. UMAT completes the incremental update of the material integration points and returns the update results to obtain the evolution results of stress, strain, inelastic strain, back stress, and tensor damage-related internal structural variables in the SAC weld area.

2. The method for implementing the SAC solder joint thermo-coupling constitutive model in electronic packaging as described in claim 1, characterized in that, The method for establishing the thermo-coupled incremental constitutive model in step one includes: (1) Determine the parameters required for the thermo-mechanical coupling incremental constitutive model of SAC solder. The material parameters of the SAC solder thermo-coupling incremental constitutive model include elastic modulus. E Poisson's ratio ν Coefficient of thermal expansion αT Initial flow resistance R 0. Internal resistance parameters D 0. Kinematic hardening strength parameters C Dynamic response parameters Evolution index of internal structural variables N proportionality coefficient p Temperature-related flow index and temperature-related internal structural variables evolution parameters ; Among them, temperature-related flow index Temperature-related internal structure variables evolution parameters The temperature at the start of the material integration point increment is used respectively. T The function representation; (2) Establish temperature-dependent flow index Temperature-related internal structure variables evolution parameters ; To describe the variation of the inelastic flowability of SAC solder with temperature, a temperature-dependent quadratic function is used to define the flow index. : （1） To describe the evolution rate of internal structural variables as a function of temperature, a temperature-dependent quadratic function is used to define the evolution parameters of the internal variables. : （2） in, T The temperature at the start of the material integration point increment is given, and the temperature unit is consistent with the parameter calibration process. These are temperature-related parameters calibrated based on SAC solder mechanical response data; (3) Based on the small deformation thermoelasticity theory and the isotropic Hooke's law, establish the thermal strain correction relationship and elastic prediction relationship, specifically including: The total strain increment is decomposed into mechanical strain increment and thermal strain increment. The mechanical strain increment of the normal strain component... for: （3） In the formula, This represents the total strain increment. For temperature increment, The coefficient of thermal expansion is used; the shear component is not corrected for thermal strain. Based on elastic modulus E Poisson's ratio and mechanical strain increment Construct an isotropic elastic stiffness matrix Ce And calculate the elastic predicted stress for the current increment step. for: （4） in, For the isotropic elastic stiffness tensor, This is the stress of the previous increment step; (4) The von Mises equivalent stress relationship considering back stress is used to establish the overstress criterion. The temperature-dependent inelastic flow relationship is established based on the Perzyna-type overstress viscoplasticity concept and the associated flow rule. The stress correction relationship and back stress evolution relationship are established based on the inelastic strain increment. Among them, establishing the overstress criterion based on the von Mises equivalent stress relationship considering back stress refers to... When overstress At that time, the inelastic strain increment is calculated, and the stress and back stress are corrected and updated based on the inelastic strain increment. The specific relationship is as follows: Based on elasticity, predict stress and the back stress of the previous incremental step Determine the relative deviatoric stress tensor for: （5） The von Mises equivalent stress considering back stress is: （6） in, To account for the relative deviatoric stress tensor after considering the back stress, ":" indicates the tensor double dot product; Then define hyperstress for: （7） when When, the material updates according to its elastic response; when When, the material enters the inelastic flow calculation; when At that time, according to the equivalent stress Determine the direction n of the inelastic flow, and based on the overstress... Temperature-related flow index Equivalent values ​​of internal structural variables D Calculate the increment of inelastic equivalent strain : （8） （9） in, For the equivalent value of the internal structural variable, For the relative deviatoric stress tensor, For internal resistance parameters, This is the current increment step time increment. The inelastic flow direction is determined by the deviatoric stress state considering the back stress. Inelastic strain increment for: （10） The stress correction relationship is as follows: （11） The back stress tensor is updated to: （12） In the formula, The stress is updated in the current increment step. For the back stress of the previous incremental step, The back stress is updated in the current increment step. For hardening strength parameters, For dynamic response parameters, This is the direction of inelastic flow; (5) Introduce damage-related internal structural variables and establish the evolutionary relationship of internal structural variables, specifically as follows: Use internal structure variables The equivalent value D characterizes the internal structure evolution of SAC solder under temperature cycling and inelastic deformation. （13） Equivalent value of total strain increment for: （14） Internal structural variable evolution direction components for: （15） Equivalent value of the evolution direction of internal structural variables for: （16） Internal structural variable components Update in the following format: （17） Among them, the temperature-related evolution coefficient , They are respectively: （18） （19） In the formula, For the previous incremental step Each internal structural variable component For the current incremental step Each internal structural variable component Shear modulus For the evolution direction component of internal structural variables This is the equivalent value of the evolution direction component. The evolution index, This is the proportionality coefficient. These are the evolution parameters for temperature-related internal structure variables.

3. The method for implementing the SAC solder joint thermo-coupling constitutive model in electronic packaging as described in claim 1, characterized in that, Step two involves writing the thermo-coupled incremental constitutive model into the Abaqus user material subroutine UMAT, which includes: (1) Establish the UMAT subroutine interface; According to the interface format of Abaqus user material subroutine UMAT, set input and output variables such as stress, strain increment, temperature, temperature increment, time increment, material parameters, state variables and material stiffness matrix so that UMAT can be called by Abaqus in each finite element increment step. (2) Determine the method for transferring material parameters; elastic modulus E Compared to Poisson ν As the material parameter input for Abaqus users; include the coefficient of thermal expansion. αT Initial flow resistance R 0. Internal resistance parameters D 0. Hardening strength parameter C, dynamic recovery parameter γ, evolution index N proportionality coefficient p Temperature-related flow index m(T) Temperature-related internal variable evolution parameters These model parameters are used in the calculation as model parameters; the above model parameters can also be extended to Abaqus user material parameter inputs as needed. (3) Define UMAT state variables; The UMAT state variables include elastic strain, inelastic strain, back stress, internal structural variables, and equivalent values ​​of internal structural variables, which are used to store the historical response of the material integration point during temperature cyclic loading. (4) Establish the UMAT incremental integration process; In each increment step, UMAT reads the stress and state variables from the previous increment step, as well as the strain increment, initial temperature, temperature increment, and time increment for the current increment step; calculates the mechanical strain increment based on the thermal strain correction relationship; calculates the elastic predicted stress; determines whether inelastic flow has occurred; when inelastic flow occurs, updates the inelastic strain, stress, back stress, and internal structural variables; and finally returns the updated stress, material stiffness matrix, and state variables to Abaqus.

4. The method for implementing the SAC solder joint thermo-coupling constitutive model in electronic packaging as described in claim 3, characterized in that; The UMAT is used for three-dimensional solid stress state and sets 25 state variables, of which the 1st to 6th state variables are used to store elastic strain components, the 7th to 12th state variables are used to store inelastic strain components, the 13th to 18th state variables are used to store back stress components, the 19th to 24th state variables are used to store tensor-type damage-related internal structural variable components, and the 25th state variable is used to store the equivalent value of the internal structural variables.

5. A method for implementing a constitutive model of SAC solder joint thermo-coupling in electronic packaging as described in any one of claims 1-4, characterized in that, Step three involves using the UMAT to perform finite element analysis of the temperature cycling of the SAC solder joints in the electronic package, specifically including: (1) Dimensional measurements were taken of the actual electronic packaging sample, and a finite element model of the electronic packaging containing SAC solder joints was established based on the measurement results; (2) Assign the SAC solder joint area to the UMAT user material model in step two; (3) Apply a temperature cycling load to the electronic packaging finite element model, the temperature cycling load including the minimum temperature, the maximum temperature, the heating time, the cooling time, the holding time, and the number of cycles; (4) Output the stress, strain, inelastic strain, back stress and internal structural variable evolution results of the SAC weld area.

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