Analysis method for defect evolution induced by fracture and electromigration based on phase field method

Abstract

The embodiment of the present application provides a method for analyzing the evolution of defects caused by fracture and electromigration based on the phase field method, determining the parameter information of the defective specimen and the multi-physical field load information, and establishing a simplified model; wherein, the multi-physical field load information includes: the load and boundary conditions of the force-thermal-electric field; based on the simplified model, a defect evolution phase field model is constructed; the defect evolution phase field model includes an electromigration phase field model coupled with the force-thermal-electric field and a thermoelastic fracture phase field model; the defect evolution phase field model is solved. In this solution, for each time step in the electromigration phase field evolution process, the electrostatic model, the electrothermal model, the thermoelastic solid mechanics model, and the thermoelastic fracture phase field model are solved for the defect morphology, so as to classify the crack penetrating the defective specimen into the final defect morphology and obtain the complete process of the evolution of the defect morphology over time.

Description

Technical Field

[0001] This application relates to the field of electromigration of components and an analysis method for defect evolution caused by fracture and electromigration based on the phase-field method. Background Art

[0002] A large number of engineering practices have shown that the performance degradation and failure of printed circuit board assemblies are caused by the fracture of intermetallic compounds at the solder joint interface, which means that the solder joint is crucial for the reliability of printed circuit board assemblies. During service, the solder joint is subjected to multi-physical field loads such as electricity, heat, and force. Under these loads, internal defects such as microcracks and microvoids in the intermetallic compound will gradually evolve and expand, eventually forming through-cracks. Some studies have found that the mechanisms driving crack evolution include not only electromigration but also fracture mechanics.

[0003] Currently, the research on defect evolution in solder joints mainly includes two categories. One category assumes that defect evolution is only controlled by solid-state diffusion induced by electromigration, ignoring the possibility of cracks emerging at stress concentration points. The other category only considers crack propagation caused by fracture mechanics, ignoring the evolution of hole morphology caused by electromigration, resulting in an incomplete and objective understanding of defect evolution inside the solder joint.

[0004] Therefore, it is necessary to study an analysis method for defect evolution that simultaneously considers the two defect evolution mechanisms and studies the interaction between the final fracture of the solder joint and the mass diffusion caused by electromigration to make the defect evolution inside the solder joint more complete. Summary of the Invention

[0005] The purpose of the embodiments of this application is to provide an analysis method for defect evolution caused by fracture and electromigration based on the phase-field method, which combines two different physical processes of electromigration and fracture to simulate the complete failure process of the intermetallic compound at the solder joint interface from early electromigration mass diffusion to final fracture failure. The specific technical solutions are as follows:

[0006] In the first aspect of the embodiments of this application, the analysis method for defect evolution caused by fracture and electromigration based on the phase-field method includes:

[0007] Determine the parameter information of the defective specimen and the multi-physical field load information, and establish a simplified model; wherein, the multi-physical field load information includes: the loads and boundary conditions of force-thermal-electric fields;

[0008] Based on the simplified model, construct a defect evolution phase-field model; the defect evolution phase-field model includes an electromigration phase-field model coupled with force-thermal-electric fields and a thermoelastic fracture phase-field model;

[0009] Solve the defect evolution phase-field model.

[0010] Optionally, the electro-migration phase field model with force-thermal-electric field coupling includes: an electrostatic model, a thermoelectric model, a thermo-elastic solid mechanics model, and an electro-migration solid state diffusion model;

[0011] In the electro-migration phase field model with force-thermal-electric field coupling, in the simulation domain a phase field variable ψ (ψ ∈ [-1, 1]) is introduced to distinguish the metal material from the pores, where ψ = -1 represents the pore defect region, ψ = 1 represents the metal region; ψ ∈ (-1, 1) represents the pore interface, and the simulation domain contains a pore region;

[0012] In the thermo-elastic fracture phase field model, in the metal region of the specimen a fracture phase field variable is introduced to distinguish the metal material from the crack, and respectively represent the states of complete material failure and integrity; represents the states of different degrees of material damage, and the metal region of the specimen does not contain a pore region.

[0013] Optionally, the electrostatic model includes:

[0014]

[0015] where ζ(ψ) = h(ψ)·ζ is the effective conductivity, and the material property parameter interpolation function satisfies h(1) = 1 in the metal region and h(-1) = 0 in the pore region, ζ is the metal conductivity, and φ is the electric potential field.

[0016] Optionally, the thermoelectric model includes:

[0017]

[0018] where k(ψ) = h(ψ)·k is the effective thermal conductivity, k is the metal thermal conductivity, and T is the temperature field.

[0019] Optionally, the thermo-elastic solid mechanics model includes:

[0020]

[0021] where C(ψ) = h(ψ)·C is the effective fourth-order elastic tensor, C is the metal's fourth-order elastic tensor, and ε e is the elastic strain tensor.

[0022] Optionally, the electro-migration solid state diffusion model includes:

[0023]

[0024] Among them, the electromigration chemical potential μ is:

[0025]

[0026] Among them, t is time, ε is the phase field interface control parameter, σ is the stress tensor, Ω is the atomic volume, γ s is the surface energy density, represents the effective charge number of surface electromigration, e represents the electron charge amount, represents the heat transferred by surface electromigration, and the surface diffusivity M can be written as:

[0027]

[0028] Among them, 1 - ψ 2 characterizes that the restricted diffusion only occurs on the pore surface, and the solid-state diffusion coefficient D s is:

[0029]

[0030] In the formula, D s is the pre-exponential factor, Q s is the surface diffusion activation energy, δ s represents the diffusion layer width, k B refers to the Boltzmann constant.

[0031] Optionally, the thermoelastic fracture phase field model includes a control equation set for crack initiation and propagation, and the control equation set for crack initiation and propagation is:

[0032]

[0033] Among them, l0 is the characteristic width of the phase field model that controls the crack dispersion degree, G c is the critical energy release rate, the constant p << 1 to avoid numerical singularity, and to prevent crack self-healing, the elastic strain energy density history variable is introduced:

[0034]

[0035] The above formula represents the historical maximum value of the elastic strain energy density function at in the time period [0, t].

[0036] Optionally, the electrostatic model, the electrothermal model, the thermoelastic solid mechanics model, and the thermoelastic fracture phase field model are all steady-state models without time terms.

[0037] Optionally, solving the defect evolution phase field model includes:

[0038] For the simulation domain Discretize in the time and space dimensions, input the initial phase field variable ψ and the electric potential boundary conditions into the electrostatic model to obtain the electric potential field φ output by the electrostatic model; input the initial phase field variable ψ, the temperature boundary conditions, and the output electric potential field φ into the electrothermal model to obtain the temperature field T output by the electrothermal model; input the displacement boundary conditions and the output temperature field T into the thermoelastic solid mechanics model to obtain the displacement field u output by the thermoelastic solid mechanics model; input the initial phase field variable ψ and the obtained electric potential field φ, temperature field T, and displacement field u information into the electromigration solid-state diffusion model to obtain the phase field ψ at the next time layer; perform mapping on the output phase field ψ Output the mapped phase field Input the displacement boundary conditions, temperature boundary conditions, and the output phase field information into the elastic fracture phase field model to obtain the phase field variables after fracture For the output phase field Perform the inverse mapping ψ(r) and output the inverse mapped phase field ψ; perform the above steps on the output phase field ψ in a loop until the crack penetrates. In terms of the numerical method for solving the model, the electrostatic model, electrothermal model, and thermoelastic solid mechanics model all adopt the finite volume method, the electromigration solid-state diffusion model adopts the finite difference method, and the thermoelastic fracture phase field model adopts the finite element method.

[0039] In another aspect of the implementation of the present application, an electronic device is provided, including a processor, a communication interface, a memory, and a communication bus. Among them, the processor, communication interface, and memory complete communication with each other through the communication bus;

[0040] The memory is used to store a computer program;

[0041] The processor, when executing the program stored on the memory, realizes the defect evolution analysis method based on the phase field method caused by fracture and electromigration.

[0042] Beneficial effects of the embodiments of the present application:

[0043] The defect evolution analysis method based on the phase field method caused by fracture and electromigration provided by the embodiments of the present application determines the parameter information and multi-physical field load information of the defective specimen and establishes a simplified model; among them, the multi-physical field load information includes: the load and boundary conditions of the force-thermal-electric field; based on the simplified model, a defect evolution phase field model is constructed; the defect evolution phase field model includes an electromigration phase field model coupled with the force-thermal-electric field and a thermoelastic fracture phase field model; solve the defect evolution phase field model. In this solution, for each time step in the electromigration phase field evolution process, the electrostatic model, electrothermal model, thermoelastic solid mechanics model, and thermoelastic fracture phase field model are solved for the defect morphology, so as to classify the crack penetrating the defective specimen into the final defect morphology and obtain the complete process of the defect morphology evolving with time. Description of the Drawings

[0044] To more clearly illustrate the technical solutions in the embodiments of the present application or in the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art.

[0045] Figure 1 It is a flowchart of the method for analyzing the evolution of defects caused by fracture and electromigration based on the phase field method in the embodiments of the present application.

[0046] Figure 2 It is a schematic diagram of the evolution mechanism of a specimen with a hole defect under the drive of a dual physical mechanism, which is subjected to a high-density current and a uniaxial tensile load in the present application;

[0047] Figure 3 It is a schematic diagram of the geometric structure and boundary conditions of the specimen with defects in the present application.

[0048] Figure 4 It is a simulation flowchart of the brittle fracture of the specimen caused by the evolution of holes induced by electromigration in the present application;

[0049] Figure 5 It is a result diagram of the crack initiation and evolution process of a specimen with a circular hole defect under horizontal uniaxial tensile and electric field loads in the present application. Detailed Embodiments

[0050] The following will clearly and completely describe the technical solutions in the embodiments of the present application with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all of them. Based on the embodiments of the present application, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the scope of protection of the present application.

[0051] To solve the problems in the prior art, the present application provides a method for analyzing the evolution of defects caused by fracture and electromigration based on the phase field method, which combines two different physical processes of electromigration and fracture to simulate the complete failure process of the intermetallic compound at the solder joint interface from the early electromigration mass diffusion to the final fracture failure.

[0052] It should be noted that the method for analyzing the evolution of defects caused by fracture and electromigration based on the phase field method provided in the embodiments of the present application can be applied to electronic devices. In practical applications, the electronic device can be: a smart phone, a tablet computer, a notebook computer, etc., which are all reasonable.

[0053] The following will first introduce the method for analyzing the evolution of defects caused by fracture and electromigration based on the phase field method provided in the embodiments of the present application.

[0054] As Figure 1As shown in the figure, the method for analyzing defect evolution caused by fracture and electromigration based on the phase field method provided by the embodiments of the present application may include the following steps:

[0055] S101. Determine the parameter information of the defective specimen and the multi-physical field load information, and establish a simplified model; wherein, the multi-physical field load information includes: the loads and boundary conditions of the force-thermal-electric field.

[0056] Among them, the parameter information of the defective material may include the defect geometry and position information, and may also include the force-thermal-electrical material parameter information of the material. Exemplarily, the defect geometry may include the width and height of the defect, as well as the diameter of the defect, etc. In addition, the defective specimen may be any defective specimen in the pre-collected specimen library, or any specified defective specimen.

[0057] The loads of the force-thermal-electric field may be uniaxial loads in a single direction, for example: a uniaxial tensile load in the horizontal direction, and the boundary conditions may be natural boundary conditions, etc.

[0058] S102. Based on the simplified model, construct a defect evolution phase field model; the defect evolution phase field model includes an electromigration phase field model coupled with the force-thermal-electric field, and a thermoelastic fracture phase field model.

[0059] Exemplarily, in the electromigration phase field model coupled with the force-thermal-electric field, in the simulation domain introduce a phase field variable ψ (ψ ∈ [-1, 1]) to distinguish the metal material from the hole, where ψ = -1 represents the hole defect region, ψ = 1 represents the metal region; ψ ∈ (-1, 1) represents the hole interface, and the simulation domain contains the hole region; introduce a phase field and consider the degradation of the interface material performance parameters, and the above models are as follows:

[0060] (1) Electrostatic model:

[0061]

[0062] Among them, ζ(ψ) = h(ψ)·ζ is the effective conductivity, the material property parameter interpolation function h(ψ) = (1 + ψ) 3 [20 - 15(1 + ψ) + 3(1 + ψ) 2 16, which satisfies h(1) = 1 in the metal region and h(-1) = 0 in the hole region, ζ is the metal conductivity, and φ is the electric potential field.

[0063] (2) Electrothermal model:

[0064]

[0065] Among them, \(k(\psi)=h(\psi)\cdot k\) is the effective thermal conductivity, \(k\) is the metal thermal conductivity, and \(T\) is the temperature field.

[0066] (3) Thermoelastic solid mechanics model:

[0067]

[0068] Among them, \(C(\psi)=h(\psi)\cdot C\) is the effective fourth-order elastic tensor, \(C\) is the fourth-order elastic tensor of the metal, and \(\varepsilon\) e is the elastic strain tensor.

[0069] (4) Electromigration solid-state diffusion model:

[0070]

[0071] Among them, the electromigration chemical potential \(\mu\) is:

[0072]

[0073] Among them, \(t\) is the time, \(\varepsilon\) is the phase-field interface control parameter, \(\sigma\) is the stress tensor, \(\Omega\) is the atomic volume, and \(\gamma\) s is the surface energy density, represents the effective charge number of surface electromigration, \(e\) represents the electron charge, represents the heat transferred by surface electromigration, and the surface diffusivity \(M\) can be written as:

[0074]

[0075] Among them, \(1 - \psi\) 2 limits the diffusion to occur only on the pore surface, and the solid-state diffusion coefficient \(D\) s is:

[0076]

[0077] In the formula, \(D\) s is the pre-exponential factor, \(Q\) s is the surface diffusion activation energy, \(\delta\) s represents the diffusion layer width, and \(k\) B refers to the Boltzmann constant.

[0078] Exemplarily, in the thermoelastic fracture phase-field model, the fracture phase-field variable is introduced in the specimen metal region to distinguish the metal material from the crack, and respectively represent the states of complete destruction and integrity of the material; represents the damage state of the material to different degrees, and the specimen metal region does not include the pore region.

[0079] Exemplarily, the thermoelastic fracture phase-field model includes a control equation set for crack initiation and propagation, and the control equation set for crack initiation and propagation is as follows:

[0080]

[0081] where l0 is the characteristic width of the phase-field model that controls the crack dispersion degree, G c is the critical energy release rate, the constant p << 1 to avoid numerical singularities, and to prevent crack self-healing, an elastic energy density history variable is introduced:

[0082]

[0083] The above equation represents the historical maximum value of the elastic strain energy density function at within the time period [0, t].

[0084] Exemplarily, when constructing the defect evolution phase-field model, except for the electromigration solid-state diffusion model, it can be assumed that the electrostatics model, the electrothermal model, the thermoelastic solid mechanics model, and the thermoelastic fracture phase-field model are all steady-state models, that is, the control equations are steady-state equations (without time terms). Then, both defect evolution mechanisms involve two-way coupling: the displacement field, the temperature field, and the electric field jointly regulate the evolution of defects, and the change in the defect morphology redistributes these three fields in the reverse direction.

[0085] S103, solve the defect evolution phase-field model.

[0086] In one implementation, solving the defect evolution phase-field model may include:

[0087] Discretize the simulation domain in the time and space dimensions, input the initial phase-field variable ψ and the electric potential boundary condition into the electrostatics model to obtain the electric potential field φ output by the electrostatics model; input the initial phase-field variable ψ, the temperature boundary condition, and the output electric potential field φ into the electrothermal model to obtain the temperature field T output by the electrothermal model; input the displacement boundary condition and the output temperature field T into the thermoelastic solid mechanics model to obtain the displacement field u output by the thermoelastic solid mechanics model; input the initial phase-field variable ψ and the obtained electric potential field φ, temperature field T, and displacement field u information into the electromigration solid-state diffusion model to obtain the phase-field ψ at the next time layer; perform a mapping on the output phase-field ψ Output the mapped result phase-field Input the displacement boundary condition, the temperature boundary condition, and the output phase-field information into the elastic fracture phase-field model to obtain the phase-field variable after fracture Input the output phase-field Perform the inverse mapping ψ(r) and output the inverse mapped result phase-field ψ; perform the above steps on the output phase-field ψ in a loop until the crack penetrates.

[0088] Exemplarily, the numerical methods for solving the phase field model may include:

[0089] The finite volume method is adopted for the electrostatic model, electrothermal model, and thermoelastic solid mechanics model, the finite difference method is adopted for the electromigration solid-state diffusion model, and the finite element method is adopted for the thermoelastic fracture phase field model.

[0090] In the embodiments of the present application, parameter information of the defective specimen and multi-physical field load information are determined, and a simplified model is established; wherein, the multi-physical field load information includes: loads and boundary conditions of force-thermal-electric fields; based on the simplified model, a defect evolution phase field model is constructed; the defect evolution phase field model includes an electromigration phase field model coupled with force-thermal-electric fields and a thermoelastic fracture phase field model; the defect evolution phase field model is solved. In this solution, the electrostatic model, electrothermal model, thermoelastic solid mechanics model, and thermoelastic fracture phase field model are solved for the defect morphology at each time step in the electromigration phase field evolution process, so as to classify the crack penetrating the defective specimen into the final defect morphology, and obtain the complete process of the evolution of the defect morphology over time.

[0091] To further illustrate the solution of the present application, the following introduces, with specific examples, the defect evolution analysis method based on the phase field method of the present application.

[0092] As Figures 2 - 5 shown, the defect evolution analysis method based on the phase field method for fracture and electromigration may include the following steps:

[0093] As Figure 2 shown, when the defective specimen is simultaneously subjected to a potential difference ΔV and uniaxial tension , there is not only electromigration but also crack initiation at the edge of the hole. The model in this embodiment simultaneously considers two physical processes: mass migration caused by electromigration and crack propagation induced by stress concentration at the edge of the hole, and can simulate the complete failure process of the intermetallic compound at the solder joint interface from early electromigration mass diffusion to final fracture failure. It reveals the mechanism of electromigration of hole defects on the fracture of the interface intermetallic compound along the hole.

[0094] Step 1: Determine the specimen parameters and boundary conditions. Specifically, the specimen geometry and boundary conditions are as Figure 3 shown. The specimen has a width L = 5 μm, a height H = 5 μm, and a circular hole defect with a radius R = 0.5 μm at its center. It is subjected to a uniaxial tensile load with a magnitude of in the horizontal direction. The left boundary of the specimen is grounded, a uniform potential ΔV = 5×10 -4 V is applied to the right boundary, while the top and bottom boundaries are insulated. The phase field ψ and Natural boundary conditions are adopted. The temperatures of all boundaries of the specimen are fixed at 473 K. The simulated metallic material is AuAl2, and its material parameters and phase-field model parameters are shown in Table 1.

[0095] Table 1 Model parameters

[0096]

[0097] Step 2: Construct a phase-field model driven by a dual physical mechanism for defect evolution, including an electromigration phase-field model with force-thermal-electric multi-field coupling and a thermoelastic fracture phase-field model, and numerically solve the electromigration-thermoelastic fracture phase-field model.

[0098] Discretization of the solution space of the field variables in the phase-field model. Since the governing equations in the model contain terms related to time, not only the simulation domain needs to be discretized in space but also in time. Specifically, a uniform grid with a step size of Δx = Δy = 0.01 μm is adopted for spatial discretization of the simulation region ; time is discretized into multiple equal time intervals with a time step of Δt = 1×10 -2 s.

[0099] The governing equation of the phase-field variable ψ in the electromigration phase-field model is a non-steady-state equation (the governing equation contains terms related to time), while the governing equations of the electrostatic model, electrothermal model, thermoelastic solid mechanics model, and thermoelastic fracture phase-field model reach equilibrium relaxation in a time much shorter than the diffusion time, so they are all steady-state equations, which means that during the evolution of the phase-field variable ψ with time, other physical processes always maintain an equilibrium state.

[0100] Figure 4 Shows the specific process of model solution. Solve the electrostatic model, electrothermal model, thermoelastic solid mechanics model, and thermoelastic fracture phase-field model for the distribution of the phase-field ψ updated at each time step. The field variables (electric potential field φ, temperature field T, displacement field u, and fracture phase-field ) related to the phase-field ψ in the model are also updated accordingly. The governing equation of the electromigration phase-field variable ψ is solved by the finite difference method, using the forward difference format; the governing equations of the electrostatic model, electrothermal model, and thermoelastic solid mechanics model are solved by the finite volume method; the fracture phase-field variable is solved by the finite element method, using a one-step staggered iteration format; before executing the thermoelastic fracture model, the phase-field variable ψ of the electromigration model needs to be mapped:

[0101]

[0102] If no crack initiation occurs, the phase-field ψ before inputting into the fracture model is directly inherited into the phase-field ψ evolution equation of the next time step; otherwise, the fracture phase-field variable after executing the fracture phase-field model is mapped:

[0103]

[0104] Step 3. Simulation results. Figure 5 shows the evolution process of a circular hole under uniaxial tension with a stress magnitude of . 5(a) shows the initial defect morphology. As can be seen from Figure 5 (b), the void drifts along the electric field direction. In addition, due to the elastic strain energy concentration and local high temperature caused by the "current crowding" effect, sharp corners appear at the top and bottom of the hole respectively. However, at the vertices of the sharp corners, the fracture mechanism does not play a role because the crack propagation criterion is not yet satisfied. To better characterize the stress accumulation effect during electromigration, a scaling factor based on the first principal stress can be defined where and represent the maximum first principal stresses of the intermetallic compound at times t = t n and t = 0 respectively. This scaling factor starts from 1.0 at t = 0 and gradually increases with the development of mass diffusion, reaching 1.417 at 80000 s, which means that the maximum first principal stress increases by more than 40%. As the mass diffusion caused by electromigration proceeds, the development of the two sharp corners amplifies the stress concentration at the top and bottom of the void. As shown in Figure 5 (c), electromigration drives the two sharp corners to continuously extend in the vertical direction, eventually leading to the initiation and propagation of cracks here. The time scale of the brittle fracture process is much smaller than the time scale of the mass diffusion induced by electromigration. In the simulation, when t = 80100 s, the initiation and propagation of cracks are completed within one time step. The crack propagation process within this time step is shown in Figure 5 (d)-(f). It can be seen from the figure that the cracks start from the upper and lower two sharp corners formed during electromigration and propagate upward and downward respectively. This is consistent with the experience of fracture mechanics, that is, when subjected to uniaxial tensile loading, cracks tend to propagate perpendicular to the loading direction.

[0105] The embodiment of the present application also provides an electronic device, including a processor, a communication interface, a memory, and a communication bus, wherein the processor, the communication interface, and the memory complete communication with each other through the communication bus,

[0106] the memory is used to store a computer program;

[0107] the processor is used to implement the defect evolution analysis method based on the phase field method provided by the embodiment of the present application when executing the program stored in the memory.

[0108] The communication bus mentioned in the above terminal may be a Peripheral Component Interconnect (PCI) bus, an Extended Industry Standard Architecture (EISA) bus, or the like. The communication bus can be divided into an address bus, a data bus, a control bus, etc. For the sake of convenience of representation, only a thick line is used in the figure, but it does not mean that there is only one bus or one type of bus.

[0109] The communication interface is used for communication between the above terminal and other devices.

[0110] The memory may include a Random Access Memory (RAM), or may also include a non-volatile memory, such as at least one disk memory. Optionally, the memory may also be at least one storage device located far from the aforementioned processor.

[0111] The above-mentioned processor may be a general-purpose processor, including a Central Processing Unit (CPU), a Network Processor (NP), etc.; it may also be a Digital Signal Processor (DSP), an Application Specific Integrated Circuit (ASIC), a Field-Programmable Gate Array (FPGA), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components.

[0112] In another embodiment provided by the present application, a computer-readable storage medium is further provided. A computer program is stored in the computer-readable storage medium, and when the computer program is executed by a processor, the defect evolution analysis method based on the phase field method described in any one of the above embodiments is implemented.

[0113] In another embodiment provided by the present application, a computer program product containing instructions is further provided. When it runs on a computer, the computer is caused to execute the defect evolution analysis method based on the phase field method described in any one of the above embodiments.

[0114] In the above embodiments, it can be implemented in whole or in part by software, hardware, firmware, or any combination thereof. When implemented using software, it can be implemented in whole or in part in the form of a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, the processes or functions described in the embodiments of the present application are generated in whole or in part. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable devices. The computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, the computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired (such as coaxial cable, optical fiber, digital subscriber line (DSL)) or wireless (such as infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium that can be accessed by a computer or a data storage device such as a server or data center that includes one or more integrated available media. The available medium can be a magnetic medium (such as a floppy disk, hard disk, magnetic tape), an optical medium (such as a DVD), or a semiconductor medium (such as a solid state disk (SSD)).

[0115] It should be noted that, in this document, relational terms such as first and second are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the term "comprising", "including" or any other variation thereof is intended to cover non-exclusive inclusion, so that a process, method, article or device comprising a series of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article or device. Without further limitation, an element defined by the statement "comprising an..." does not exclude the existence of additional identical elements in the process, method, article or device comprising the element.

[0116] Each embodiment in this specification is described in a related manner. The same or similar parts between the embodiments can be referred to each other, and the differences between each embodiment and other embodiments are emphasized. In particular, for the device embodiments, since they are basically similar to the method embodiments, the description is relatively simple, and the relevant parts can be referred to the description of the method embodiments.

[0117] The above are only the preferred embodiments of the present application and are not intended to limit the protection scope of the present application. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present application are all included in the protection scope of the present application.

Claims

1. A method for analyzing defect evolution caused by fracture and electromigration based on the phase field method, characterized in that, Including: Determine the parameter information of the defective specimen and the multi-physical field load information, and establish a simplified model; wherein, the multi-physical field load information includes: the loads and boundary conditions of force-thermal-electric fields; Based on the simplified model, construct a defect evolution phase field model; the defect evolution phase field model includes an electromigration phase field model coupled with force-thermal-electric fields, and a thermoelastic fracture phase field model; Solve the defect evolution phase field model; Wherein, the electromigration phase field model coupled with force-thermal-electric fields includes: an electrostatic model, a thermoelectric model, a thermoelastic solid mechanics model, and an electromigration solid-state diffusion model; In the electro-migration phase-field model of the force-thermal-electric field coupling, in the simulation domain A phase-field variable ψ is introduced, where ψ ∈ [-1, 1] is used to distinguish the metal material from the holes. Here, ψ = -1 represents the hole defect region, ψ = 1 represents the metal region; ψ ∈ (-1, 1) represents the hole interface, and the simulation domain contains a hole region; In the thermo-elastic fracture phase-field model, in the specimen metal region a fracture phase-field variable is introduced to distinguish the metal material from the crack, and respectively represent the states of complete destruction and integrity of the material; represents the states of different degrees of damage of the material, and the specimen metal region does not contain a pore region; The solving of the defect evolution phase field model includes: For the analog domain Discretize in the time and space dimensions, input the initial phase field variable ψ and the electric potential boundary conditions into the electrostatic model to obtain the electric potential field φ output by the electrostatic model; input the initial phase field variable ψ, the temperature boundary conditions, and the output electric potential field φ into the electrothermal model to obtain the temperature field T output by the electrothermal model; input the displacement boundary conditions and the output temperature field T into the thermoelastic solid mechanics model to obtain the displacement field u output by the thermoelastic solid mechanics model; input the initial phase field variable ψ and the information of the obtained electric potential field φ, temperature field T, and displacement field u into the electromigration solid-state diffusion model to obtain the phase field ψ at the next time layer; perform a mapping on the output phase field ψ Output the mapped phase field Input the displacement boundary conditions, temperature boundary conditions, and the output phase field information into the elastic fracture phase field model. If no crack initiation occurs, directly inherit the phase field ψ before inputting into the fracture model into the phase field ψ evolution equation for the next time step; otherwise, perform the fracture phase field variable after the fracture phase field model Perform a mapping to obtain the phase field variable after fracture For the output phase field Perform the inverse mapping ψ(r) and output the inverse mapped phase field ψ; perform the above steps in a loop on the output phase field ψ until the crack penetrates; Among them, is the sample metal area.

2. The method according to claim 1, wherein The electrostatic model includes: Among them, ζ(ψ) = h(ψ)·ζ is the effective conductivity, and the material property parameter interpolation function h(ψ) = (1 + ψ) 3 [20 - 15(1 + ψ) + 3(1 + ψ) 2 / 16, which satisfies h(1) = 1 in the metal region and h(-1) = 0 in the hole region, ζ is the metal conductivity, and φ is the electric potential field.

3. The method according to claim 2, wherein The thermoelectric model includes: Wherein, k(ψ) = h(ψ)·k is the effective thermal conductivity, k is the metal thermal conductivity, and T is the temperature field.

4. The method according to claim 2, characterized in that, The thermoelastic solid mechanics model includes: Among them, C(ψ) = h(ψ)·C is the effective fourth-order elastic tensor, C is the fourth-order elastic tensor of the metal, and ε e is the elastic strain tensor.

5. The method according to claim 2, characterized in that, The electromigration solid-state diffusion model includes: Wherein, the electromigration chemical potential μ is: where t is time, ε is the phase-field interface control parameter, σ is the stress tensor, Ω is the atomic volume, γ s is the surface energy density, represents the effective charge number of surface electromigration, e represents the electron charge, represents the heat transported by surface electromigration, and the surface diffusivity M can be written as: Among them, 1 - ψ 2 represents that the restricted diffusion only occurs on the pore surface, and the solid-state diffusion coefficient D s is as follows: Wherein, D s is the pre-exponential factor, Q s is the activation energy of surface diffusion, δ s represents the diffusion layer width, k B refers to the Boltzmann constant.

6. The method according to claim 1, wherein The thermoelastic fracture phase field model includes a control equation set for crack initiation and propagation, and the control equation set for crack initiation and propagation is: where \(l_0\) is the characteristic width of the phase field model that controls the degree of crack dispersion, \(G\) c is the critical energy release rate, the constant \(p\ll1\) to avoid numerical singularities, and to prevent crack self-healing, the historical variable of elastic energy density is introduced: The above equation represents the historical maximum value of the elastic strain energy density function in the time period [0, t].

7. The method according to claim 1, wherein The electrostatic model, the thermoelectric model, the thermoelastic solid mechanics model, and the thermoelastic fracture phase field model are all steady-state models without time terms.

8. An electronic device, characterized in that, Including a processor, a communication interface, a memory, and a communication bus. Among them, the processor, the communication interface, and the memory complete mutual communication through the communication bus; The memory is used to store computer programs; The processor is used to implement the method steps described in any one of claims 1-7 when executing the programs stored on the memory.

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