A method for predicting interface fracture of quasicrystal composite electrode materials for lithium batteries

Abstract

The present invention discloses a method for predicting interface fracture of quasi-crystal composite electrode materials for lithium batteries, which relates to the field of quasi-crystal processing technology, and includes the following steps: S1, constructing an interface surface density function for the quasi-crystal composite electrode material; S2, calculating the total potential energy functional of the quasi-crystal composite electrode material according to the interface surface density function; S3, determining a phase field control model based on the total potential energy functional of the quasi-crystal composite electrode material; S4, numerically implementing the phase field control model to obtain a prediction result. The present invention introduces phase field variables to describe the damage state of the material, and sharp cracks are regularized into diffuse damage areas through crack surface density functions, so the transition from the crack area to the intact area becomes smooth. The present invention can more easily and naturally combine the multi-physical field coupling effect while predicting the crack propagation path; and the comprehensive influence of phonon-phase sub-heat-electric-ion concentration-damage-interface coupling field is considered.

Description

Technical Field

[0001] The present invention relates to the technical field of quasicrystal processing, and in particular to a method for predicting interface fracture of quasicrystal composite electrode materials for lithium batteries. Background Art

[0002] Quasicrystals are a special and complex class of metal alloys with nonclassical rotational symmetry and quasi-periodic translational symmetry. Compared with crystals and amorphous materials, quasicrystals have many special properties due to their special atomic arrangement. The special properties make quasicrystals have high application potential in many engineering fields, such as: lithium-ion battery electrode materials, energy storage materials, surface protective coatings, composite biomaterials and thermoelectric materials. However, macroscopic quasicrystal structures have been experimentally proven to be inherently brittle and highly sensitive to defects at room temperature, which makes it very difficult to prepare large-scale quasicrystal structures.

[0003] The brittleness of quasicrystals can be completely changed at the microscopic scale, so quasicrystals are usually prepared in the form of powders or particles and applied as functional phases or reinforcement phases in metal or polymer-based composites. Quasicrystal composites exhibit extremely attractive properties, which bypass the inherent brittleness of quasicrystals and have good yield strength, hardness and corrosion resistance. Quasicrystal composites are widely used in the field of batteries.

[0004] The phase field fracture model shows a strong ability in simulating crack initiation, propagation, branching and coalescence. In the phase field fracture model, an order parameter called phase field variable is introduced to describe the material damage state, and the sharp crack is regularized into a diffuse damage area through the crack surface density function, so that the transition from the crack area to the intact area becomes smooth. As a variational fracture method, the phase field fracture model does not require preset crack paths and tracking crack surfaces, and can automatically capture the crack evolution process. Due to the mathematical structure of the variational principle adopted in the phase field fracture model, it allows the natural combination of multi-physics coupling effects. So far, the phase field fracture model has been extended to study various fracture problems, such as cohesive fracture, dynamic fracture, ductile fracture, fatigue fracture and interface fracture, which involve different media, such as porous media, piezoelectric materials, composite materials, thin films and battery materials. In addition, the phase field fracture model has been extended to study the fracture behavior of quasicrystals. The present invention develops an improved multi-phase field model to study the fracture behavior of quasicrystal composite electrode materials in lithium batteries, in which the comprehensive effects of temperature field, electric field and lithium ion concentration field, interface and phase subelastic field are considered. The developed phase-field model and numerical implementation method can provide a convenient tool for predicting interface failure and evaluating the safety of quasicrystal composite electrode materials for lithium batteries in engineering. Summary of the invention

[0005] In order to solve the above problems, the present invention proposes a method for predicting interface fracture of quasicrystal composite electrode materials for lithium batteries.

[0006] The technical solution of the present invention is: a method for predicting interface fracture of lithium battery quasi-crystal composite electrode materials comprises the following steps:

[0007] S1. Constructing the interface surface density function for the quasicrystalline composite electrode material;

[0008] S2. Calculate the total potential energy functional of the quasicrystal composite electrode material based on the interface surface density function;

[0009] S3. Determine the phase field control model based on the total potential energy functional of the quasicrystal composite electrode material;

[0010] S4. Perform numerical implementation processing on the phase field control model to obtain prediction results.

[0011] Furthermore, in S1, the interface surface density function of the quasicrystalline composite electrode material is The expression is:

[0012] ;

[0013] In the formula, represents the interface field order parameter, represents the gradient of the interface field order parameter, represents the interface field length scalar parameter, represents the gradient operator.

[0014] Furthermore, in S2, the total potential energy functional of the quasicrystal composite electrode material is The expression is:

[0015] ;

[0016] In the formula, represents the spatial region, represents the damage phase field variable, represents the gradient operator, represents the phonon Cauchy stress, represents the phonon elastic strain tensor, represents the electric displacement vector, represents the electric field vector, represents the interface field order parameter, represents the damage field length scalar parameter, represents the equivalent critical energy release rate, represents the gradient of the damage phase field, represents the energy release rate of interface fracture, represents the interface field length scalar parameter, represents the gradient of the interface field order parameter, represents the phonon force, represents the phonon displacement, represents the electric field force, represents the electric potential, represents the reference chemical potential, represents the lithium ion concentration, is the gas constant, Indicates temperature, represents the reference concentration, represents a subregion of the quasicrystal, Indicates area, Indicates volume, represents the phase sub-Cauchy stress, represents the phase subelastic strain tensor, represents the phase sub-force, represents the phase shift, represents the Newman part of the phonon field boundary of the quasicrystal composite material, represents the fixed phonon pulling force, represents the Newman part of the quasicrystal phase subfield boundary, represents the fixed phase sub-traction force, represents the Newman part of the electric field boundary of the quasicrystal composite material, represents the surface free charge density.

[0017] Furthermore, S3 includes the following sub-steps:

[0018] S31, perform first-order variation of the total potential energy functional of the quasicrystalline composite electrode material;

[0019] S32. Set the total potential energy functional after the first-order variation to zero to obtain a phase field control model.

[0020] Furthermore, in S32, the phase field control model includes phonon field control equations, phase subfield control equations, electric field control equations, lithium element mass conservation control equations, damage field control equations and interface field control equations.

[0021] Furthermore, the expression of the phonon field control equation is:

[0022] ;

[0023] In the formula, represents the gradient operator, represents the material damage degradation function, represents the phonon elastic tensor, represents the phonon elastic strain tensor, represents the phonon piezoelectric coefficient, represents the phonon-phaseon coupling elastic tensor, represents the electric field vector, represents the phonon force, Indicates temperature, represents the damage phase field variable, represents the phase subelastic strain tensor, represents double dot product;

[0024] The expression of the phase subfield control equation is:

[0025] ;

[0026] In the formula, represents the phase subelasticity tensor, represents the phase sub-piezoelectric coefficient, Represents the phase sub-force;

[0027] The expression of the electric field control equation is:

[0028] ;

[0029] In the formula, represents the permittivity parameter tensor, Represents the electric field force.

[0030] Furthermore, the expression of the lithium mass conservation control equation is:

[0031] ;

[0032] In the formula, represents the derivative of lithium ion concentration with respect to time, represents the diffusion coefficient, represents the gradient of lithium ion concentration, represents the gradient of the hydrostatic stress, represents the gradient operator, represents the lithium ion concentration, represents the partial molar volume of lithium ions, represents the hydrostatic stress, is the gas constant, Indicates temperature.

[0033] Furthermore, the damage field control equation is expressed as:

[0034] ;

[0035] In the formula, represents the interface field order parameter, represents the equivalent critical energy release rate, represents the damage phase field variable, represents the damage field length scalar parameter, represents the gradient operator, represents the energy release rate of interface fracture, represents the interface field length scalar parameter, represents the gradient of the interface field order parameter, Represents a history function.

[0036] Furthermore, the expression of the interface field control equation is:

[0037] ;

[0038] In the formula, represents the interface field order parameter, represents the interface field history function, represents the energy release rate of interface fracture, represents the interface field length scalar parameter, represents the Laplace operator, represents the gradient operator.

[0039] The beneficial effects of the present invention are:

[0040] (1) The present invention describes the evolution of cracks by introducing phase field variables instead of directly defining cracks geometrically. Therefore, the formation and propagation of cracks can be handled naturally, avoiding the complexity of handling discontinuous boundaries. The present invention allows cracks to develop automatically during loading without specifying the crack path or position in advance, making the prediction of cracks under complex loading conditions more flexible and reliable.

[0041] (2) The present invention can automatically simulate the formation and expansion of cracks during loading, and can naturally handle complex crack behaviors without the need for re-meshing when cracks expand. The present invention describes cracks through phase field variables and does not introduce displacement discontinuities, thereby showing better robustness in complex structures.

[0042] (3) The present invention introduces phase field variables to describe the damage state of the material. The sharp crack is regularized into a diffuse damage area through the crack surface density function, so the transition from the crack area to the intact area becomes smooth. While predicting the crack propagation path, the present invention can more easily and naturally combine the multi-physical field coupling effect; and considers the comprehensive influence of phonon-phase sub-heat-electricity-ion concentration-damage-interface coupling field. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 Flow chart of the method for predicting interface fracture of quasicrystal composite electrode materials for lithium batteries. DETAILED DESCRIPTION

[0044] The embodiments of the present invention will be further described below in conjunction with the accompanying drawings.

[0045] like Figure 1As shown, the present invention provides a method for predicting interface fracture of quasi-crystal composite electrode materials for lithium batteries, comprising the following steps:

[0046] S1. Constructing the interface surface density function for the quasicrystalline composite electrode material;

[0047] S2. Calculate the total potential energy functional of the quasicrystal composite electrode material based on the interface surface density function;

[0048] S3. Determine the phase field control model based on the total potential energy functional of the quasicrystal composite electrode material;

[0049] S4. Perform numerical implementation processing on the phase field control model to obtain prediction results.

[0050] In an embodiment of the present invention, it is assumed that the matrix material is isotropic, and the quasicrystal is used as a particle reinforcement phase and a functional phase of the composite material. First, based on the phase field method, an order parameter (i.e., a phase field variable) is introduced to describe the degree of damage of the material, and the discontinuous sharp cracks are regularized. Secondly, the interface field variable is introduced to regularize the interface in the composite material. In addition, in order to characterize the influence of the interface on crack propagation, an equivalent critical energy release rate is introduced. According to the Griffith fracture theory, a total potential energy functional considering the crack evolution of the interface and the phase sub-elastic field of the quasicrystal composite electrode material is constructed. The first-order variation of the total potential energy functional is calculated, and the phonon field, phase sub-field, electric field, and phase field control equations are established. The evolution of the interface field is not considered, so the interface problem can be regarded as a static crack problem. The interface field control equation is similar to the phase field control equation. Considering the asymmetric characteristics of the fracture behavior of brittle materials under tensile and compressive loads, the elastic energy of the material is decomposed into tension and compression. Select appropriate energy to drive the crack evolution in the quasicrystal composite material to avoid unrealistic fracture modes. In order to avoid unrealistic crack recovery during compression or unloading, a monotonically increasing history function with respect to time is introduced into the phase field evolution equation to replace the driving force of crack extension. Finally, the crack extension and interface debonding in quasicrystal composite electrode materials can be predicted by solving the problem using a separated (interleaved) iterative algorithm.

[0051] In the embodiment of the present invention, in S1, the interface surface density function of the quasi-crystalline composite electrode material The expression is:

[0052] ;

[0053] In the formula, represents the interface field order parameter, represents the gradient of the interface field order parameter, represents the interface field length scalar parameter, represents the gradient operator.

[0054] In the embodiment of the present invention, in S2, the total potential energy functional of the quasicrystal composite electrode material The expression is:

[0055] ;

[0056] In the formula, represents the spatial region, represents the damage phase field variable, represents the gradient operator, represents the phonon Cauchy stress, represents the phonon elastic strain tensor, represents the electric displacement vector, represents the electric field vector, represents the interface field order parameter, represents the damage field length scalar parameter, represents the equivalent critical energy release rate, represents the gradient of the damage phase field, represents the energy release rate of interface fracture, represents the interface field length scalar parameter, represents the gradient of the interface field order parameter, represents the phonon force, represents the phonon displacement, represents the electric field force, represents the electric potential, represents the reference chemical potential, represents the lithium ion concentration, is the gas constant, Indicates temperature, represents the reference concentration, represents a subregion of the quasicrystal, Indicates area, Indicates volume, represents the phase sub-Cauchy stress, represents the phase subelastic strain tensor, represents the phase sub-force, represents the phase shift, represents the Newman part of the phonon field boundary of the quasicrystal composite material, represents the fixed phonon pulling force, represents the Newman part of the quasicrystal phase subfield boundary, represents the fixed phase sub-traction force, represents the Newman part of the electric field boundary of the quasicrystal composite material, represents the surface free charge density.

[0057] In this embodiment of the present invention, S3 includes the following sub-steps:

[0058] S31, perform first-order variation of the total potential energy functional of the quasicrystalline composite electrode material;

[0059] S32. Set the total potential energy functional after the first-order variation to zero to obtain a phase field control model.

[0060] In an embodiment of the present invention, in S32, the phase field control model includes phonon field control equations, phase subfield control equations, electric field control equations, lithium element mass conservation control equations, damage field control equations and interface field control equations.

[0061] In the embodiment of the present invention, the expression of the phonon field control equation is:

[0062] ;

[0063] In the formula, represents the gradient operator, represents the material damage degradation function, represents the phonon elastic tensor, represents the phonon elastic strain tensor, represents the phonon piezoelectric coefficient, represents the phonon-phaseon coupling elastic tensor, represents the electric field vector, represents the phonon force, Indicates temperature, represents the damage phase field variable, represents the phase subelastic strain tensor, represents double dot product;

[0064] The expression of the phase subfield control equation is:

[0065] ;

[0066] In the formula, represents the phase subelasticity tensor, represents the phase sub-piezoelectric coefficient, Represents the phase sub-force;

[0067] The expression of the electric field control equation is:

[0068] ;

[0069] In the formula, represents the permittivity parameter tensor, Represents the electric field force.

[0070] In the embodiment of the present invention, the expression of the lithium element mass conservation control equation is:

[0071] ;

[0072] In the formula, represents the derivative of lithium ion concentration with respect to time, represents the diffusion coefficient, represents the gradient of lithium ion concentration, represents the gradient of the hydrostatic stress, represents the gradient operator, represents the lithium ion concentration, represents the partial molar volume of lithium ions, represents the hydrostatic stress, is the gas constant, Indicates temperature.

[0073] In the embodiment of the present invention, the damage field control equation is expressed as:

[0074] ;

[0075] In the formula, represents the interface field order parameter, represents the equivalent critical energy release rate, represents the damage phase field variable, represents the damage field length scalar parameter, represents the gradient operator, represents the energy release rate of interface fracture, represents the interface field length scalar parameter, represents the gradient of the interface field order parameter, Represents a history function.

[0076] In the embodiment of the present invention, the expression of the interface field control equation is:

[0077] ;

[0078] In the formula, represents the interface field order parameter, represents the interface field history function, represents the energy release rate of interface fracture, represents the interface field length scalar parameter, represents the Laplace operator, represents the gradient operator.

[0079] In an embodiment of the present invention, after establishing a phase-field fracture model of crack propagation in a quasi-crystalline composite material, it is numerically implemented based on a standard finite element method. Nowadays, the phase-field method can be implemented through a variety of software platforms, such as ABAQUS, Comsol Multiphysics, FEniCS, MOOSE, DEAL.II, etc. Comsol Multiphysics software has a variety of built-in physical modules, and different physical modules can be automatically coupled. A simple and flexible method for numerically implementing the phase-field fracture model in the commercial software Comsol Multiphysics. The method is based on only one module, namely the weak form partial differential equation, and is suitable for materials with complex constitutive relations. This method does not require the development of subroutines like ABAQUS, and can directly provide a visual image of the crack path, or it can provide crack path data and draw it into an image with the help of other programs such as MATLAB and ORIGIN.

[0080] During the numerical implementation, material constants and model parameters are created and defined by global definitions. Physical variables are defined and stored in variables defined under components. State variables in the variable tool are used to calculate and update the history field variables. The integration branch is used to calculate the loading forces on the boundaries. The history function takes the larger value of the previous solution and the tensile elastic energy, and records the previous solution through a special function in the solver. In the study branch, the multi-field coupled system is solved using a segregated (staggered) solution, and an iteration or tolerance option is selected in the termination technique. The given relative tolerance and maximum number of iterations can be adjusted according to the computational workload. In addition, when the crack starts to grow, the incremental step size can be reduced to ensure computational efficiency and accuracy, which can be set in the transient solver branch of the study.

[0081] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A method for predicting interface fracture of quasi-crystal composite electrode materials for lithium batteries, characterized in that: The following steps are involved: S1. Constructing the interface surface density function for the quasicrystalline composite electrode material; S2. Calculate the total potential energy functional of the quasicrystal composite electrode material based on the interface surface density function; S3. Determine the phase field control model based on the total potential energy functional of the quasicrystal composite electrode material; S4, numerically implement the phase field control model to obtain prediction results; The S3 comprises the following sub-steps: S31, perform first-order variation of the total potential energy functional of the quasicrystalline composite electrode material; S32, setting the total potential energy functional after the first-order variation to zero to obtain a phase field control model; In the S32, the phase field control model includes a phonon field control equation, a phase subfield control equation, an electric field control equation, a lithium element mass conservation control equation, a damage field control equation, and an interface field control equation; The expression of the phonon field control equation is: ； In the formula, represents the gradient operator, represents the material damage degradation function, represents the phonon elastic tensor, represents the phonon elastic strain tensor, represents the phonon piezoelectric coefficient, represents the phonon-phaseon coupling elastic tensor, represents the electric field vector, represents the phonon force, Indicates temperature, represents the damage phase field variable, represents the phase subelastic strain tensor, represents double dot product; The expression of the phase subfield control equation is: ； In the formula, represents the phase subelasticity tensor, represents the phase sub-piezoelectric coefficient, Represents the phase sub-force; The expression of the electric field control equation is: ； In the formula, represents the permittivity parameter tensor, represents the electric field force; The expression of the lithium element mass conservation control equation is: ； In the formula, represents the derivative of lithium ion concentration with respect to time, represents the diffusion coefficient, represents the gradient of lithium ion concentration, represents the gradient of the hydrostatic stress, represents the gradient operator, represents the lithium ion concentration, represents the partial molar volume of lithium ions, represents the hydrostatic stress, is the gas constant, Indicates temperature; The damage field control equation is expressed as: ； In the formula, represents the interface field order parameter, represents the equivalent critical energy release rate, represents the damage phase field variable, represents the damage field length scalar parameter, represents the gradient operator, represents the energy release rate of interface fracture, represents the interface field length scalar parameter, represents the gradient of the interface field order parameter, represents the history function; The expression of the interface field control equation is: ； In the formula, represents the interface field order parameter, represents the interface field history function, represents the energy release rate of interface fracture, represents the interface field length scalar parameter, represents the Laplace operator, represents the gradient operator.

2. The method for predicting interface fracture of lithium battery quasicrystal composite electrode materials according to claim 1, characterized in that: In S1, the interface surface density function of the quasicrystalline composite electrode material The expression is: ； In the formula, represents the interface field order parameter, represents the gradient of the interface field order parameter, represents the interface field length scalar parameter, represents the gradient operator.

3. The method for predicting interface fracture of lithium battery quasicrystal composite electrode materials according to claim 1, characterized in that: In S2, the total potential energy functional of the quasicrystal composite electrode material The expression is: ； ； ； ； In the formula, represents the spatial region, represents the damage phase field variable, represents the gradient operator, represents the phonon Cauchy stress, represents the phonon elastic strain tensor, represents the electric displacement vector, represents the electric field vector, represents the interface field order parameter, represents the damage field length scalar parameter, represents the equivalent critical energy release rate, represents the gradient of the damage phase field, represents the energy release rate of interface fracture, represents the interface field length scalar parameter, represents the gradient of the interface field order parameter, represents the phonon force, represents the phonon displacement, represents the electric field force, represents the electric potential, represents the reference chemical potential, represents the lithium ion concentration, is the gas constant, Indicates temperature, represents the reference concentration, represents a sub-region of the quasicrystal, Indicates area, Indicates volume, represents the phase sub-Cauchy stress, represents the phase subelastic strain tensor, represents the phase sub-force, represents the phase shift, represents the Newman part of the phonon field boundary of the quasicrystal composite material, represents the fixed phonon pulling force, represents the Newman part of the quasicrystal phase subfield boundary, represents the fixed phase sub-traction force, represents the Newman part of the electric field boundary of the quasicrystal composite material, represents the surface free charge density.

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