Ceramic multi-field response and crack evolution calculation method under laser loading

Abstract

The invention discloses a ceramic multi-field response and crack evolution calculation method under laser loading, and relates to the field of laser damage multi-physical field calculation theories, and the method comprises the following steps: building a mathematical model for ceramic multi-field response under laser loading according to basic physical property parameters of laser and ceramic materials; building a fracture phase field model of the thermal-mechanical coupling effect of the ceramic material under the laser effect; and the phase field model is numeralized, and a UEL subprogram is written based on Abaqus, so that prediction of multi-physical field response and fracture evolution of the ceramic material under the action of laser is realized. By the adoption of the ceramic multi-field response and crack evolution calculation method under laser loading, calculation and prediction of multi-physical field mapping and crack evolution of the ceramic material under the laser loading effect are achieved.

Description

Technical Field

[0001] The invention relates to the field of multi-physics field calculation theory of laser damage, in particular to a calculation method for multi-field response and crack evolution of ceramics under laser loading. Background Art

[0002] Ceramic materials are widely used in aerospace, energy, machinery manufacturing and other fields due to their excellent high temperature strength, hardness, corrosion resistance and wear resistance. However, the brittle characteristics of ceramic materials make it easy for cracks to propagate under complex loading conditions, especially under laser loading conditions. The generation of thermal stress and the evolution of cracks are key factors affecting material performance and reliability.

[0003] Laser loading technology is a high-energy-density local heating technology that is widely used in the fields of material surface processing, welding, marking, and failure analysis. When the laser beam is irradiated on the ceramic surface, a significant temperature gradient will be generated, leading to thermal expansion and thermal stress, which may induce the generation and expansion of cracks. During the laser loading process, ceramic materials are very susceptible to thermal stress due to their low thermal expansion coefficient and high brittleness, resulting in cracks and damage.

[0004] Although some studies have explored the thermal response and crack behavior of ceramic materials under laser loading, existing technologies mainly focus on the analysis of a single physical field, such as the study of thermal field or mechanical stress field, and lack comprehensive consideration of the coupling of multiple physical fields. Most of the existing crack growth models are based on classical linear fracture mechanics theory, which cannot accurately describe the complex behavior of ceramic materials under laser loading conditions, especially the crack evolution process under the combined action of thermal stress and mechanical stress.

[0005] In addition, existing numerical simulation methods usually ignore the dynamic crack growth process under laser loading, or have low calculation accuracy, making it difficult to handle the brittleness of ceramic materials and the nonlinear characteristics of crack growth. These deficiencies have led to significant limitations in the existing research results in engineering applications, and they cannot meet the high-precision requirements for performance evaluation and failure prediction of ceramic materials under laser loading.

[0006] Patent CN118468742A discloses a crack evolution theory and simulation method in the manufacturing process. The patent couples the fracture phase field model with the mathematical model of the additive manufacturing process to solve the fracture. The coupling model is heat-fluid-solid coupling, which mainly faces the prediction of the problem of thermal cracks at the bottom of the molten pool during the additive manufacturing SLM process. However, there is no attempt to couple electromagnetic + thermal + mechanical coupling.

[0007] In summary, there are still many problems in the multi-field coupling response and crack propagation of ceramic materials under laser loading. Existing numerical calculation methods are often limited in efficiency, accuracy and physical effects considered, and it is difficult to meet the needs of ceramic material performance evaluation, failure analysis and reliability design in actual engineering.

[0008] Therefore, there is an urgent need for a new computational method that can accurately predict the multi-field coupling effects of ceramic materials under laser loading and effectively simulate the crack evolution process to address the deficiencies in the existing technology. Summary of the invention

[0009] In order to solve the above problems, this application proposes a calculation method for multi-field response and crack evolution of ceramics under laser loading, calculates the process of laser energy deposition in the medium, thereby obtaining the temperature at the bottom of the ablation pit, and calculates the impact force at the bottom through the Puff EOS equation. It serves as the basic theory of the thermal-mechanical coupling fracture phase field method. The coupling model is electromagnetic + thermal + mechanical coupling. It is aimed at the process of damage and fracture of ceramic materials under the irradiation of strong lasers. It is used to improve the deficiencies in the research on multi-field coupling response and crack evolution of ceramic materials under laser loading conditions. This method is particularly suitable for the prediction and analysis of the mechanical response, thermal response and crack propagation behavior of ceramic materials under laser loading conditions.

[0010] The method for calculating the multi-field response and crack evolution of ceramics under laser loading mentioned in this application includes the following steps:

[0011] S1. Establish a mathematical model for the multi-field response of ceramics under laser loading based on the basic physical parameters of laser and ceramic materials;

[0012] S2. Based on the mathematical model of multi-field response of ceramics under laser loading, a fracture phase field model of thermal-mechanical coupling of ceramic materials under laser action is constructed;

[0013] S3. Numerize the phase field model and write a UEL subroutine based on Abaqus to predict the multi-physical field response and fracture evolution of ceramic materials under laser action.

[0014] Preferably, the basic physical property parameters of the selected laser and ceramic material described in S1 include the physical property characteristic parameters of the laser and the basic physical property parameters of light, force and heat of the selected ceramic material;

[0015] The physical characteristic parameters of the selected laser include laser power P, laser wavelength λ, laser beam diameter d, laser pulse duration τ, and laser energy absorption rate α;

[0016] Among them, the basic physical properties of the selected ceramic materials include the optical properties of the ceramic materials, namely the laser energy absorption rate α, reflectivity R, and transmittance T, and the mechanical properties of the ceramic materials, namely Young's modulus E, Poisson's ratio υ, and fracture toughness K. Ic 、Coefficient of thermal expansion α T , yield strength σ y The thermal properties of ceramic materials are specific heat capacity Cp, thermal conductivity k, thermal diffusivity D, melting point T m .

[0017] Preferably, the mathematical model for establishing multi-field response of ceramics under laser loading in S1 includes an optical transfer model, an energy accumulation model, and a temperature conversion model;

[0018] The expression of the optical transfer model is:

[0019]

[0020] ε=ε 0 ·ε r ,μ=μ 0 ·μ r ;

[0021] In the formula, I represents the light intensity, H * represents the magnetic field intensity, ε and μ represent the dielectric constant and magnetic permeability, E c represents the electric field strength, ε 0 =8.5342x10 -12 F·m -1 , ε r =·1.49 2 , μ 0 =4πx10 -7 H·m -1 , μ r =1, RE[·] means taking the real part of the complex number;

[0022] Energy accumulation model includes E p The expression of the modulated energy at and the E inside the material P Expression for the energy absorbed at ;

[0023] Among them, E p The modulated energy E p The expression is:

[0024] E p =[2n 1 / (n 1 +n 2 )]·E 0

[0025] Among them, n 1 ,n 2Represent the refractive index of air and ceramic materials, E 0 represents the normalized electric field intensity;

[0026] E inside the material P The energy absorbed at E The expression is:

[0027] E E =Q abs ·πa 2 I abs =Q abs ·πa 2 I d LIEF;

[0028] LIEF=I E / I P ;

[0029] In the formula, I E is the laser intensity inside a surface-perfect material crystal, I P is the maximum laser intensity inside the crystal after the incident laser is modulated by the convex defect, LIEF is the light intensity enhancement factor, Q abs is the laser absorption efficiency, I abs is the internal energy of the crystal modulated by the protrusion defect under laser irradiation, a is the spot radius of the laser, I d is the incident light intensity;

[0030] According to the E inside the material P The energy absorbed at the point is used to obtain the distribution diagram of laser energy deposition of the material, and the position where the energy maximum is located is defined as the detonation point;

[0031] The temperature conversion model is a calculation model for the formation of the detonation point and the temperature and pressure;

[0032] The E absorbed by the protrusion defect modulation P Unit mass e E The expression of the energy size is:

[0033] e E =E E / (ρ 0 V)=C v (TT 0 );

[0034] Where ρ 0 =2338kg / m 3 is the initial density of the material, V is E P Volume. T 0 =300K is the initial temperature of the material, and T is the temperature of the local point.

[0035] Preferably, the state equation PUFF EOS of the phase field fracture model is expressed as:

[0036] P=P H +Γ 0 ρ 0 e E

[0037] In the equation:

[0038]

[0039] Γ 0 =αc 0 2 / C v

[0040] Among them, c 0 represents the speed of sound in the crystal, ρ is the density, and ρ 0 is the initial density, α=2.9×10 -5 represents the thermal expansion coefficient of the material, Γ 0 is the Gruneisen coefficient, P is the laser power, v 0 is the reference speed, v is the true speed of light propagating in the medium, s is the coefficient related to the material, C v is the specific heat of the material;

[0041] When the protrusion defect modulates the absorbed E P Unit mass e E The energy is much greater than the sublimation energy of the material crystal, so the phase field fracture model is applied.

[0042] Preferably, the coupling effect described in S2 is calculated by transmission boundary conditions;

[0043] The boundary conditions of the fracture phase field model of thermal-mechanical coupling of ceramic materials under laser action in S2 are the temperature and impact force at the bottom of the ablation "pit", which are calculated by the mathematical model of multi-field response of ceramics under laser loading in S1;

[0044] That is, the fracture evolution of S3 is stimulated by the temperature and force in S2.

[0045] Preferably, the specific content of constructing the fracture phase field model of thermal-mechanical coupling of ceramic materials under laser action described in S2 is:

[0046] The total energy expression of the fracture phase field model is:

[0047]

[0048] Among them, Ψ(u,φ,T) represents the total energy of the system, ψ thermalrepresents the energy due to temperature;

[0049] The displacement variable u and the fracture phase field variable are varied respectively, and the control equations of the displacement field and fracture phase field are obtained respectively;

[0050]

[0051] in, is the Nabol operator, σ is stress, ε is strain, f is body force, φ is the fracture phase field, l 0 represents crack characteristic parameters, H is the maximum historical energy;

[0052] in,

[0053] G c (T) represents the definition of the correction term for the critical energy release rate due to temperature damage, G c0 is the fracture energy at the reference temperature, b 1 and b 2 is a temperature dependent parameter, is the energy density term in the crack zone; represents the gradient term of the crack, T represents the temperature of the local point, T ref Indicates the reference temperature.

[0054] Preferably, S3's digitizing the phase field model and writing a UEL subroutine based on Abaqus refers to deducing the equilibrium equations, phase field equations and heat transfer equations from strong form to weak form, extracting the specific contents of the unit stiffness matrix and the nodal force vector, and then updating the relevant variables and organizing them to obtain the discretized total equation.

[0055] Preferably, the relationship between the data transfer between the UEL subroutine and the ABAQUS main program is:

[0056] The main functions of the Abaqus main program are to control the time step, assemble the overall stiffness matrix, solve the global equations, store and transfer the results. The main function of the UEL subroutine is to obtain the temperature and stress values ​​inside the unit through shape function interpolation. Calculate the unit stiffness moment and right-hand side terms, etc. and update the state variables. The variables between the main program and the subroutine are mainly transferred through the icopy mechanism in the kstatevar subroutine. When icopy is defined as 0, the value of the global state variable is converted to the local state variable. When icopy is defined as 1, the local state variable is passed to the global state variable.

[0057] Preferably, the discretized total equation is:

[0058]

[0059] In the formula, K uu , K φφ , K TT are the stiffness matrices of displacement variables, phase field variables, and temperature variables respectively; u e ,φ e and T e Respectively represent the displacement, phase field and temperature of the unit; F u 、F φ 、F T denote the right-hand side terms of the evolution of displacement, phase field and temperature variables respectively.

[0060] In summary, compared with the traditional technology, the method for calculating the multi-field response and crack evolution of ceramics under laser loading of the present invention introduces a multi-physics field coupling model of thermal field, stress field and crack extension, comprehensively considers the local thermal expansion, thermal stress effect and dynamic expansion behavior of cracks caused by laser heating, thereby realizing accurate simulation and prediction of ceramic materials under laser loading. The calculation method provided by the present invention can accurately describe the multi-field coupling response of ceramic materials under laser loading, especially the initiation and expansion behavior of cracks, and provides theoretical support for the design, performance optimization and failure analysis of ceramic materials under laser loading environment. Through this method, the reliability and service life of ceramic materials under extreme conditions such as high temperature and laser loading can be improved, providing strong technical support for applications in aerospace, laser processing and other fields.

[0061] The technical method of the present invention is further described in detail below through the drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1 This is a flow chart of the method for calculating multi-field response and crack evolution of ceramics under laser loading of the present invention;

[0063] Figure 2 A diagram of the data interaction mechanism between the main program and the subprograms of the present invention;

[0064] Figure 3 Schematic diagram of energy deposition during fracture formation. DETAILED DESCRIPTION

[0065] The technical method of the present invention is further described below by means of the accompanying drawings and embodiments. It should be noted that unless otherwise specifically stated, the relative arrangement of the components and steps, numerical expressions and numerical values ​​described in these embodiments do not limit the scope of the present application.

[0066] The following description of at least one exemplary embodiment is merely illustrative in nature and is in no way intended to limit the present application, its application, or uses.

[0067] Technologies, systems, and devices known to ordinary technicians in the relevant art may not be discussed in detail, but where appropriate, the technologies, systems, and devices should be considered part of the specification.

[0068] In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not limiting. Therefore, other examples of the exemplary embodiments may have different values.

[0069] Unless otherwise defined, technical or scientific terms used in the present invention shall have the common meanings understood by one having ordinary skills in the field to which the present invention belongs.

[0070] The present invention provides a method for calculating the multi-field response and crack evolution of ceramics under laser loading, such as Figure 1 As shown, the following steps are included:

[0071] S1. Establish a mathematical model for the multi-field response of ceramics under laser loading based on the basic physical parameters of laser and ceramic materials;

[0072] The basic physical property parameters of the selected laser and ceramic material described in S1 include the physical property characteristic parameters of the laser and the basic physical property parameters of light, force and heat of the selected ceramic material;

[0073] The physical characteristic parameters of the selected laser include laser power P, laser wavelength λ, laser beam diameter d, laser pulse duration τ, and laser energy absorption rate α;

[0074] Among them, the basic physical properties of the selected ceramic materials include the optical properties of the ceramic materials, namely the laser energy absorption rate α, reflectivity R, and transmittance T, and the mechanical properties of the ceramic materials, namely Young's modulus E, Poisson's ratio υ, and fracture toughness K. Ic 、Coefficient of thermal expansion α T , yield strength σ y The thermal properties of ceramic materials are specific heat capacity Cp, thermal conductivity k, thermal diffusivity D, melting point T m .

[0075] Specifically, the physical properties of the selected laser and ceramic materials should include the following aspects:

[0076] (1) Physical parameters of laser

[0077] Laser power (P): The output power of a laser beam, usually measured in watts (W). Laser power directly affects the intensity of laser heating of ceramic materials.

[0078] Laser wavelength (λ): The wavelength of the laser determines the interaction between the laser and the material, affecting the efficiency of energy absorption. Lasers of different wavelengths have different transmittance and absorption rates on ceramic materials.

[0079] Laser beam diameter (d): The diameter of the laser beam determines the distribution range of the laser energy and also affects the expansion area of ​​the thermal field.

[0080] Laser pulse duration (τ): The pulse time of the laser affects the rate of change of the material surface temperature, which in turn affects the generation of thermal stress.

[0081] Laser energy absorption rate (α): refers to the energy ratio of the laser beam absorbed on the surface of the ceramic material. Different materials have different absorption rates for lasers of different wavelengths.

[0082] (2) Optical properties of ceramic materials:

[0083] Laser energy absorption rate (α): The absorption characteristics of ceramic materials to lasers determine the degree of energy absorption of the material under laser irradiation. The laser energy absorption rate is related to the laser wavelength. Different ceramic materials have different laser energy absorption rates for lasers of different wavelengths.

[0084] Reflectivity (R): The ability of the ceramic material surface to reflect laser light. Reflectivity affects the effective use of laser energy.

[0085] Transmittance (T): For thin or highly transparent ceramic materials, the laser may pass through the material. Transmittance refers to the proportion of laser light that passes through the material.

[0086] (3) Mechanical properties of ceramic materials:

[0087] Young's modulus (E): The elastic modulus of a material, which determines the degree of deformation of the material when subjected to force. Young's modulus affects the mechanical response of the material during laser loading.

[0088] Poisson's ratio (υ): describes the ratio of the strain in the perpendicular direction to the strain in the tensile direction when the material is subjected to force in one direction. It has an important influence on the distribution of the stress field and crack evolution.

[0089] Fracture toughness (K Ic ): The fracture toughness of ceramic materials is an important indicator for evaluating their ability to resist crack growth. Under laser loading conditions, crack growth is closely related to fracture toughness.

[0090] Thermal expansion coefficient (α T ): Describes the expansion characteristics of ceramic materials under temperature changes. The thermal expansion coefficient affects the thermal stress and thermal strain of the material.

[0091] Yield strength (σ y ): The stress value when the material begins to undergo plastic deformation, which affects the mechanical response of the ceramic material during laser loading.

[0092] (4) Thermal properties of ceramic materials:

[0093] Specific heat capacity (Cp): The specific heat capacity of a material determines the energy required for a unit temperature change and affects the rate of temperature change of the ceramic material during laser heating.

[0094] Thermal conductivity (k): Thermal conductivity determines the internal heat transfer rate of ceramic materials and directly affects the thermal field distribution.

[0095] Thermal diffusivity (D): Thermal diffusivity is a combination of thermal conductivity and specific heat capacity, and describes how fast heat spreads in a material.

[0096] Melting point (T m ): The melting point of a ceramic material is the temperature at which it begins to melt at high temperatures and has an important influence on the material behavior during thermal loading.

[0097] The mathematical model established for the multi-field response of ceramics under laser loading as described in S1 includes an optical transfer model, an energy accumulation model, and a temperature conversion model;

[0098] Specifically, the optical transfer model assumes that a perfect magnetic conductor is added at the boundary parallel to the laser incident direction to ensure that the plane wave propagates infinitely in the lateral direction. The simplified Maxwell equations are simplified to obtain the Helmholtz equations,

[0099] Due to the relationship between light intensity and laser damage mechanism, the light intensity I is further introduced using the equation,

[0100] ε=ε 0 ·ε r ,μ=μ 0 ·μ r ;

[0101] In the formula, I represents the light intensity, H * represents the magnetic field intensity, ε and μ represent the dielectric constant and magnetic permeability, E c represents the electric field strength, ε 0 =8.5342x10 -12 F·m -1 , ε r =·1.49 2 , μ 0 =4π×10 -7 H·m -1 , μ r =1, RE[·] means taking the real part of the complex number.

[0102] The model is verified by the Fresnel reflection law. The average value of the model is calculated by the Fresnel reflection law. The electric field intensity calculated by the numerical method is averaged and compared with the result obtained by the Fresnel reflection law to verify the calculation results of optical propagation.

[0103] The specific calculation results of Fresnel's reflection law, E p =[2n 1 / (n 1 +n 2 )]·E 0 =0.8032V / m.

[0104] In fact, the complexity and inhomogeneity of the material crystals will reduce the laser's actual focus inside the material. The laser-induced damage threshold is reduced. Therefore, the light intensity enhancement factor is introduced to determine the defect at E p The energy modulated at (Explosion Point).

[0105] E p The modulated energy E p The expression is:

[0106] E p =[2n 1 / (n 1 +n 2 )]·E 0

[0107] Among them, n 1 ,n 2 Represent the refractive index of air and ceramic materials, E 0 represents the normalized electric field intensity;

[0108] E inside the material P The energy absorbed at E The expression is:

[0109] E E =Q abs ·πa 2 I abs =Q abs ·πa 2 I d LIEF;

[0110] LIEF=I E / I P ;

[0111] In the formula, I E is the laser intensity inside a surface-perfect material crystal, I P is the maximum laser intensity inside the crystal after the incident laser is modulated by the convex defect, LIEF is the light intensity enhancement factor, Qabs is the laser absorption efficiency, I abs is the internal energy of the crystal modulated by the protrusion defect under laser irradiation, a is the spot radius of the laser, I d is the incident light intensity. In this way, the distribution diagram of laser energy deposition of the material can be obtained. The location of the maximum energy is considered to be the detonation point.

[0112] The specific model for the formation of the detonation point and the calculation model of temperature and pressure is as follows: the initial energy deposition will lead to E P The high temperature and high pressure characteristics of the zone. The process of damage must involve a high degree of thermal expansion.

[0113] The E absorbed by the protrusion defect modulation P Unit mass e E The energy size can be calculated by the following formula.

[0114] e E =E E / (ρ 0 V)=C v (TT 0 )

[0115] Where ρ 0 =2338kg / m 3 is the initial density of the material, V is E P Volume. T 0 =300K is the initial temperature of the material, and T is the temperature of the local point.

[0116] Need to meet the E absorbed by the convex defect modulation P Unit mass e E The energy is much greater than the sublimation energy of the material crystal. This is a judgment condition. Only when this judgment is met can the PUFF EOS equation theory be used.

[0117] The state equation PUFF EOS applied to the phase field fracture model can be expressed as:

[0118] P=P H +Γ 0 ρ 0 e E ;

[0119] In the equation:

[0120]

[0121] Γ 0 =αc 0 2 / C v ;

[0122] Among them, c0 represents the speed of sound in the crystal, ρ is the density, and ρ 0 is the initial density, α=2.9x10 -5 represents the thermal expansion coefficient of the material, Γ 0 is the Gruneisen coefficient, P is the laser power, v 0 is the reference speed, v is the true speed of light propagating in the medium, s is the coefficient related to the material, C v is the specific heat of the material.

[0123] When the protrusion defect modulates the absorbed E P Unit mass e E The energy is much greater than the sublimation energy of the material crystal. Apply the phase field fracture model

[0124] S2. Build a fracture phase field model of thermal-mechanical coupling of ceramic materials under laser action;

[0125] The coupling effect described in S2 is calculated by transport boundary conditions;

[0126] The boundary conditions of the fracture phase field model of thermal-mechanical coupling of ceramic materials under laser action in S2 are the temperature and impact force at the bottom of the ablation "pit", which are calculated by the mathematical model of multi-field response of ceramics under laser loading in S1;

[0127] That is, the fracture evolution of S3 is stimulated by the temperature and force in S2.

[0128] The total energy expression of the fracture phase field model is:

[0129]

[0130] Among them, Ψ(u,φ,T) represents the total energy of the system, ψ thermal represents the energy due to temperature;

[0131] By varying the displacement variable u and the fracture phase field variable respectively, we can obtain the control equations of the displacement field and fracture phase field respectively;

[0132]

[0133]

[0134] in,

[0135] G c (T) represents the definition of the correction term for the critical energy release rate due to temperature damage, G c0 is the fracture energy at the reference temperature, b 1 and b2 is a temperature dependent parameter, is the energy density term in the crack zone; represents the gradient term of the crack, T represents the temperature of the local point, T ref Indicates the reference temperature.

[0136] Preferably, Figure 3 As shown, step S2 of building a fracture phase field model of thermal-mechanical coupling of ceramic materials under laser action mainly includes the following contents.

[0137] Under the action of laser, after the explosion point and ablation retreat, thermal cracks will be generated at the bottom of the ablation pit, so the fracture problem under the action of laser is attributed to the fracture problem under the action of thermal-mechanical coupling. The total energy functional of the phase field method includes elastic energy, crack surface energy and thermal energy, which can usually be expressed as:

[0138] Ψ(u,d,T)=∫ Ω (ψ elastic +ψ fracture +ψ thermal )dV;

[0139] Represents the elastic energy density, which usually describes the deformation energy of a material.

[0140] represents the crack surface energy, which is described by the crack growth energy of the phase field model.

[0141] It represents the thermal energy density and describes the thermal energy contribution of the temperature distribution to the material.

[0142] The elastic energy density function is expressed as: ; The damage degradation function is expressed as: ; Initial elastic energy density, ; The fracture energy density can be expressed as ; The definition of the correction term for the critical energy release rate due to temperature-induced damage is: ; in, is the fracture energy at the reference temperature, and is a temperature dependent parameter.

[0143] is the energy density term in the crack zone;

[0144] It represents the gradient term of the crack, controls the width of the crack band, and makes the crack distribution smooth.

[0145] Total system energy contributed by thermal energy

[0146] Heat accumulation term It indicates the rate of change of heat energy per unit time, that is, the rate of accumulation or dissipation of heat energy. If this item is positive, it means the heat increases; if it is negative, it means the heat decreases.

[0147] Heat conduction term Here k(d) represents the thermal conductivity of the material, which can change with the damage parameter d.

[0148] k(d)=(1-d) 2 k 0 It means that as the damage increases, the thermal conductivity decreases. According to Fourier's heat conduction theorem, the heat flux density Therefore, the heat conduction term represents the divergence of heat flux density, that is, the rate of change of thermal energy per unit volume due to heat conduction. It describes the transfer of heat in space driven by temperature gradient.

[0149] The internal heat source term Q represents the additional heat source or heat sink inside the system, with the unit of W / m3, which can come from the heat generation inside the material, chemical reaction, external heating, etc. When Q>0, it means that there is additional heat generated in the system (such as friction heat), which will heat the system; when Q<0, it means that there is a heat sink (such as a cooling device) in the system, which will reduce the heat in the system.

[0150] In summary, the total energy expression of the system is:

[0151]

[0152] By varying the displacement variable u and the fracture phase field variable respectively, we can obtain the control equations of the displacement field and fracture phase field.

[0153]

[0154] S3. Numerize the phase field model and write a UEL subroutine based on Abaqus to predict the multi-physical field response and fracture evolution of ceramic materials under laser action.

[0155] S3 states that digitizing the phase field model and writing a UEL subroutine based on Abaqus means deducing the equilibrium equations, phase field equations and heat transfer equations from strong form to weak form, extracting the specific contents of the unit stiffness matrix and the nodal force vector, and then updating the relevant variables and organizing them to obtain the discretized total equation.

[0156] Preferably, the step S3 of digitizing the phase field model and writing a UEL subroutine based on Abaqus means deducing the equilibrium equations, phase field equations and heat transfer equations from strong form to weak form, extracting the specific contents of the unit stiffness matrix and the vector of the node force, and then updating the relevant variables.

[0157] (1) For the stress field,

[0158] The equilibrium equation of elasticity,

[0159] Introducing the stress solution of the damage variable, σ=g(d)C:(∈-α(TT 0 )I);

[0160] Introducing the imaginary displacement variable and integrating it, we obtain the weak form of the equilibrium equation:

[0161]

[0162] Through shape function interpolation,

[0163] Substituting into the weak form equation, we obtain the weak form of the equilibrium equation:

[0164]

[0165] The left side of the equation can be rearranged to obtain,

[0166]

[0167] Element stiffness matrix, K ue =∫ Ω B T g(d)CBdV;

[0168] The additional thermal stress caused by thermal strain, F thermal =∫ Ω B T g(d)Cα(TT 0 )IdV; after expansion, we get:

[0169]

[0170] Contributions from external body and boundary forces, F f =∫ Ω N T fdV;

[0171] The total right-hand side term can be recorded as, F = F f +F thermal ;

[0172]

[0173] (2) For the fracture phase field,

[0174] By introducing imaginary variables and integrating, we can obtain the weak form of the fracture phase field equation:

[0175]

[0176] Further expansion can be obtained,

[0177]

[0178] Through shape function interpolation,

[0179] d≈Nd e ,δd≈Nδd e , Here we define the area of ​​this item as 0;

[0180]

[0181] Furthermore,

[0182]

[0183] Move the item to get,

[0184]

[0185] The final stiffness matrix of the fracture phase field is expressed as:

[0186]

[0187] K dd d e =F d ;

[0188] (3) For the temperature field,

[0189]

[0190] By introducing the imaginary variable δT, we can obtain the weak form of the heat conduction equation:

[0191]

[0192] Boundary conditions;

[0193]

[0194] Substituting in, we get,

[0195]

[0196] Through the shape function, we can get

[0197] T≈NTe ,δT≈NδT e ,

[0198] The final heat transfer equation is,

[0199] Extract the heat capacity matrix, C T =∫ Ω N T ρcNdV;

[0200] The transient term can be discretized into equations such as,

[0201] The diffusion term is discrete,

[0202] The stiffness matrix of the diffusion term is given by

[0203] The equation for the diffusion term,

[0204] The heat source term is discrete,

[0205] Heat source term, F T =∫ Ω N T QdV;

[0206] The equation for the heat source term,

[0207] The discretized equation

[0208] Substitute the discretized equation using the backward Euler method,

[0209] Substitute into the discretized matrix heat transfer equation,

[0210] After reconstructing the equation, we get

[0211] Define the equivalent right-hand side term and stiffness matrix,

[0212]

[0213] Therefore, the equivalent discretized equation is, K eff T e n+1 =F eff ;

[0214] Furthermore, the relationship between the data transfer between the UEL subroutine and the ABAQUS main program is:

[0215] like Figure 2 As shown, the main functions of the Abaqus main program are to control the time step, assemble the overall stiffness matrix, solve the global equations, store and transfer the results. The main function of the UEL subroutine is to obtain the temperature and stress values ​​inside the unit through shape function interpolation. Calculate the unit stiffness moment and right-hand side terms, etc. and update the state variables. The variables between the main program and the subroutine are mainly transferred through the icopy mechanism in the kstatevar subroutine. When icopy is defined as 0, the value of the global state variable is converted to the local state variable. When icopy is defined as 1, the local state variable is passed to the global state variable.

[0216] Preferably, by arranging the equations of the three physical fields, the discretized total equation is:

[0217]

[0218] In the formula, K uu , K φφ , K TT are the stiffness matrices of displacement variables, phase field variables, and temperature variables respectively; u e ,φ e and T e Respectively represent the displacement, phase field and temperature of the unit; F u 、F φ 、F T denote the right-hand side terms of the evolution of displacement, phase field and temperature variables respectively.

[0219] Preferably, step S4 achieves accurate prediction of the multi-physical field response and fracture evolution of ceramic materials under laser action.

[0220] Finally, it should be noted that the above embodiments are only used to illustrate the technical method of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical method of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical method to deviate from the spirit and scope of the technical method of the present invention.

Claims

1. The method for calculating multi-field response and crack evolution of ceramics under laser loading is characterized by: The following steps are involved: S1. Establish a mathematical model for the multi-field response of ceramics under laser loading based on the basic physical parameters of laser and ceramic materials; S2. Based on the mathematical model of multi-field response of ceramics under laser loading, a fracture phase field model of thermal-mechanical coupling of ceramic materials under laser action is constructed; S3. Numerize the phase field model and write a UEL subroutine based on Abaqus to predict the multi-physical field response and fracture evolution of ceramic materials under laser action.

2. The method for calculating multi-field response and crack evolution of ceramics under laser loading according to claim 1, characterized in that: The basic physical property parameters of the selected laser and ceramic material described in S1 include the physical property characteristic parameters of the laser and the basic physical property parameters of light, force and heat of the selected ceramic material; The physical characteristic parameters of the selected laser include laser power P, laser wavelength λ, laser beam diameter d, laser pulse duration τ, and laser energy absorption rate α; Among them, the basic physical properties of the selected ceramic materials include the optical properties of the ceramic materials, namely the laser energy absorption rate α, reflectivity R, and transmittance T, and the mechanical properties of the ceramic materials, namely Young's modulus E, Poisson's ratio υ, and fracture toughness K. Ic 、Coefficient of thermal expansion α T , yield strength σ y The thermal properties of ceramic materials are specific heat capacity Cp, thermal conductivity k, thermal diffusivity D, melting point T m .

3. The method for calculating multi-field response and crack evolution of ceramics under laser loading according to claim 1, characterized in that: The mathematical model established for the multi-field response of ceramics under laser loading as described in S1 includes an optical transfer model, an energy accumulation model, and a temperature conversion model; The expression of the optical transfer model is: ε=ε0·ε r ,μ=μ0·μ r ; In the formula, I represents the light intensity, H * represents the magnetic field intensity, ε and μ represent the dielectric constant and magnetic permeability, E c Represents the electric field strength, ε0=8.5342x10 -12 F·m -1 , ε r =·1.49 2 ,μ0=4π×10 -7 H·m -1 , μ r =1, RE[·] means taking the real part of the complex number; Energy accumulation model includes E p The expression of the modulated energy at and the E inside the material P Expression for the energy absorbed at ; Among them, E p The modulated energy E p The expression is: HAVE BEEN p =[2n1 / (n1+n2)]·E0 Where n1, n2 represent the refractive index of air and ceramic material respectively, and E0 represents the normalized electric field intensity; E inside the material P The energy absorbed at E The expression is: E E =Q abs ·πa 2 ·I abs =Q abs ·πa 2 ·I d ·LIEF; LIEF=I E / I P ; In the formula, I E is the laser intensity inside a surface-perfect material crystal, I P is the maximum laser intensity inside the crystal after the incident laser is modulated by the convex defect, LIEF is the light intensity enhancement factor, Q abs is the laser absorption efficiency, I abs is the internal energy of the crystal modulated by the protrusion defect under laser irradiation, a is the spot radius of the laser, I d is the incident light intensity; According to the E inside the material P The energy absorbed at the point is used to obtain the distribution diagram of laser energy deposition of the material, and the position where the energy maximum is located is defined as the detonation point; The temperature conversion model is a calculation model for the formation of the detonation point and the temperature and pressure; The E absorbed by the protrusion defect modulation P Unit mass e E The expression of the energy size is: e E =E E / (ρ0V)=C v (T-T0); Where ρ0=2338kg / m 3 is the initial density of the material, V is E P volume, T0 = 300K is the initial temperature of the material, T is the temperature of the local point, C v is the specific heat of the material.

4. The method for calculating multi-field response and crack evolution of ceramics under laser loading according to claim 1, characterized in that: The state equation of the phase field fracture model PUFF EOS laser energy absorption rate is expressed as: P=P H +G0r0e E ; In the equation: Γ0=αc0 2 / C v ; Where c0 is the speed of sound in the crystal, ρ is the density, ρ0 is the initial density, and α = 2.9 × 10 -5 represents the thermal expansion coefficient of the material, Γ0 is the Gruneisen coefficient, P is the laser power, v0 is the reference speed, v is the true speed of light propagating in the medium, s is the coefficient related to the material, and C v is the specific heat of the material; When the protrusion defect modulates the absorbed E P Unit mass e E The energy is much greater than the sublimation energy of the material crystal, so the phase field fracture model is applied.

5. The method for calculating multi-field response and crack evolution of ceramics under laser loading according to claim 1, characterized in that: The coupling effect described in S2 is calculated by transport boundary conditions; The boundary conditions of the fracture phase field model of thermal-mechanical coupling of ceramic materials under laser action in S2 are the temperature and impact force at the bottom of the ablation "pit", which are calculated by the mathematical model of multi-field response of ceramics under laser loading in S1; That is, the fracture evolution of S3 is stimulated by the temperature and force in S2.

6. The method for calculating multi-field response and crack evolution of ceramics under laser loading according to claim 1, characterized in that: The specific contents of constructing the fracture phase field model of thermal-mechanical coupling of ceramic materials under laser action as described in S2 are: The total energy expression of the fracture phase field model is: Among them, Ψ(u,φ,T) represents the total energy of the system, ψ thermal represents the energy due to temperature; The displacement variable u and the fracture phase field variable are varied respectively, and the control equations of the displacement field and fracture phase field are obtained respectively; in, is the Nabol operator, σ is stress, ε is strain, f is body force, φ is fracture phase field, l0 represents crack characteristic parameters, and H is the maximum historical energy; in, G c (T) represents the definition of the correction term for the critical energy release rate due to temperature damage, G c0 is the fracture energy at the reference temperature, b1 and b2 are temperature-dependent parameters, is the energy density term in the crack zone; represents the gradient term of the crack, T represents the temperature of the local point, T ref Indicates the reference temperature.

7. The method for calculating multi-field response and crack evolution of ceramics under laser loading according to claim 1, characterized in that: S3 states that digitizing the phase field model and writing a UEL subroutine based on Abaqus means deducing the equilibrium equations, phase field equations and heat transfer equations from strong form to weak form, extracting the specific contents of the unit stiffness matrix and the nodal force vector, and then updating the relevant variables and organizing them to obtain the discretized total equation.

8. The method for calculating multi-field response and crack evolution of ceramics under laser loading according to claim 7, characterized in that: The relationship between the data transfer between the UEL subroutine and the ABAQUS main program is: The main functions of the Abaqus main program are to control the time step, assemble the global stiffness matrix, solve the global equations, and store and transmit the results; The main function of the UEL subroutine is to obtain the temperature and stress values ​​inside the unit through shape function interpolation; Calculate element stiffness moment and right-hand side terms, etc. and update state variables; The variables between the main program and the subprogram are mainly transferred through the icopy mechanism in the kstatevar subprogram; When icopy is defined as 0, the value of the global state variable is converted to the local state variable. When icopy is defined as 1, the local state variable is passed to the global state variable.

9. The method for calculating multi-field response and crack evolution of ceramics under laser loading according to claim 7, characterized in that: The total equation after discretization is: In the formula, K uu , K φφ , K TT are the stiffness matrices of displacement variables, phase field variables, and temperature variables respectively; u e ,φ e and T e Respectively represent the displacement, phase field and temperature of the unit; F u 、F φ 、F T denote the right-hand side terms of the evolution of displacement, phase field and temperature variables respectively.

Images