A thermal-mechanical-phase field coupling model simulation method for concrete meso thermal cracking

Abstract

The application discloses a kind of thermal-mechanical-phase field coupling simulation methods of concrete mesoscopic thermal cracking, the present application is based on the numerical simulation method of mesoscopic concrete thermal cracking based on phase field theory, comprising the following steps: establishing mesoscopic concrete model, mainly including aggregate mortar and interface transition zone;According to the thermodynamic parameters of each component of mesoscopic concrete obtained according to data reference or physical test;Determine the thermodynamic boundary condition according to the environment of concrete;Free triangle or quadrilateral element is used to disperse mesoscopic concrete model, and determine solving error;Thermal-mechanical-phase field damage coupling model is used to solve the cracking process of concrete model under the action of temperature.The application considers the heterogeneous structure of concrete, establishes a thermal-mechanical damage coupling model based on the phase field fracture theory, and the coupling mode is more in line with engineering practice, which can provide calculation basis for concrete cracking caused by temperature and provide guiding opinions for safety evaluation of hydraulic concrete structure.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of water conservancy and hydropower engineering, in particular to a thermal-mechanical-phase field coupling model simulation method for concrete meso thermal cracking. BACKGROUND

[0002] With the construction of high concrete dams, the dam height is developing to 300m level, and the temperature cracking problem of concrete dams is prominent. A large amount of hydration heat is generated during the pouring of concrete dams, and the temperature change during the operation period will also generate a large temperature stress in the concrete, which will cause cracks on the surface and inside of the concrete dam. If the cracks appear on the surface of the dam, the dam will suffer serious durability problems such as ion erosion and freeze-thaw cycle, thereby affecting the service life of the dam. If the cracks are through, it will cause hidden dangers to the structural safety of the dam. Therefore, it is of great significance to use appropriate methods to evaluate whether the concrete will crack under the action of temperature and how the cracks develop.

[0003] The existing method for constructing a multi-field coupling model of concrete, such as Chinese patent publication No. CN116502492A, can simulate the deformation and stress characteristics of hydraulic concrete caused by temperature and humidity factors, and provide support for durability judgment, safety evaluation, and life prediction of hydraulic concrete structures. However, at the meso level, concrete contains aggregates, mortar, and interface transition zones, and is a multiphase heterogeneous material. Due to the existence of the interface transition zone, the cracking of concrete first occurs at the interface, and then connects along the interfaces around different aggregates until the cracks penetrate the entire concrete. At present, many calculations of concrete temperature cracking treat concrete as a homogeneous material, which cannot reflect the real meso structure of concrete. Therefore, it is of great significance to establish a meso concrete model that conforms to the real structure to study the crack propagation under the action of temperature. In view of this, in-depth research is conducted on the above-mentioned problems, and the present application is produced. SUMMARY

[0004] The purpose of the present application is to provide a thermal-mechanical-phase field coupling model simulation method for concrete meso thermal cracking, which can quickly and accurately obtain the crack development process of concrete under the action of temperature.

[0005] To achieve the above-mentioned purpose, the present application provides the following technical solution: a thermal-mechanical-phase field coupling model simulation method for concrete meso thermal cracking; the simulation method is based on numerical simulation of meso concrete thermal cracking according to the phase field theory, and includes the following steps:

[0006] Step 1. Establish a meso concrete model, mainly including aggregates, mortar, and interface transition zones;

[0007] Step 2. Obtain the thermodynamic parameters of each component of the meso concrete according to the reference data or physical test;

[0008] Step3. Determine the thermodynamic boundary conditions according to the environment of the concrete;

[0009] Step4. Discretize the meso-concrete model using free triangle or quadrilateral elements and determine the solution error;

[0010] Step5. Solve the cracking process of the concrete model under temperature action using the thermal phase field damage coupling model.

[0011] Preferably, the simulation method comprises generation of a three-phase model, mainly comprising the following three steps:

[0012] A. Generate polygonal aggregates by percentage using a random algorithm, calculate the area percentage of each graded aggregate by the Walaven formula and aggregate content, then randomly generate a circle in the graded particle size interval, randomly take points on the circle to generate polygonal aggregates, and judge the interference between the generated aggregates and the existing aggregates, if the generated aggregates do not intersect with the existing aggregates, generate the next aggregate, otherwise regenerate the aggregate;

[0013] B. Extend the coordinates of the aggregate points outward by 1mm to obtain an interface transition zone model;

[0014] C. Subtract the aggregate model and the interface transition zone model from the total model to obtain a mortar model.

[0015] The establishment of the thermal phase field damage coupling model mainly includes:

[0016] 1) Phase field model:

[0017] If the body force is ignored, the total energy on the calculation domain and the crack surface can be represented as:

[0018]

[0019] Where G f is the fracture energy density of the material, s is the phase field variable, s=0 represents no damage, s=1 represents complete damage, γ(s) is the crack density function, t is the surface force on the boundary , u is the displacement field, ε is the strain tensor, and ψ(ε(u), s) is the improved energy density function.

[0020] The variational formula (1) can be obtained as follows:

[0021]

[0022] Where n is the normal vector on the boundary, and the stress tensor can be represented as:

[0023]

[0024] The mechanical equilibrium equation and the phase field governing equation can be obtained as follows:

[0025]

[0026] where the equal and unequal signs are taken when δs>0 and δs=0, respectively, and the boundary conditions can be expressed as:

[0027]

[0028] In the phase field theory, the crack density function can be expressed as:

[0029]

[0030] where α(s) is the crack geometry function, l0is the crack width characteristic parameter, and:

[0031]

[0032] The crack geometry function can be expressed as:

[0033]

[0034] Substituting equation (8) into equation (4) gives:

[0035]

[0036] In the simulation, only the damage caused by tension is considered, so the improved energy density function can be expressed as:

[0037]

[0038] where ω(s) is the energy degradation function, is the initial elastic strain energy, which can be expressed as:

[0039]

[0040] where λ and μ are the material Lame constants, and the tensile strain energy in the simulation is where σ1is the maximum principal stress, E is the material elastic modulus, and <σ1> = max(σ1, 0), substituting equation (10) into equation (9) gives:

[0041]

[0042] In the phase field model, the energy degradation function can be expressed as:

[0043]

[0044] where p > 2, a1> 0 is a material property related parameter, and the expressions of a1, a2and a3are as follows:

[0045]

[0046] where f t is the tensile strength of the material, ξ is a parameter related to the crack geometry function, β k and β w are the initial slope and the final crack opening of the softening curve, respectively, and some commonly used softening curves are as follows:

[0047]

[0048] Since concrete is a quasi-brittle material, linear softening is used in the simulation.

[0049] To satisfy the irreversibility of the crack, a history variable H is introduced to record the maximum crack driving force, which can be expressed as:

[0050]

[0051] Therefore, the final governing equation can be expressed as:

[0052]

[0053] 2) Heat conduction:

[0054] The governing equation of heat conduction in solid is as follows:

[0055]

[0056] where p and c are the density and specific heat capacity of the material, θ is the temperature, Q int is the heat source, k is the heat conduction tensor, is the heat flow, is the given boundary heat flow, to consider the effect of cracks on heat conduction, the same degradation function is used to degrade the heat conduction performance, which can be expressed as:

[0057] k = ω(s)k0I (21)

[0058] where k0is the initial thermal conductivity of the material, and I is the second-order unit tensor.

[0059] The thermal strain caused by temperature change can be expressed as:

[0060] ε th = a(θ - θ ref )I (22)

[0061] where ε th is the thermal strain tensor, a is the thermal expansion coefficient of the material, θref T0 is the reference temperature.

[0062] Thus the total strain can be expressed as:

[0063] ε = ε th + ε σ (23)

[0064] where ε σ is the stress-induced strain tensor.

[0065] The final stress-strain relationship can be expressed as:

[0066]

[0067] where is the initial elastic matrix of the material.

[0068] For simplicity, the effect of temperature on material parameters is not considered and the displacement field has no effect on the temperature field, thus the construction of the phase-field thermo-mechanical coupled model is completed.

[0069] 3) Numerical solution:

[0070] The weak form of equations (19) and (20) can be expressed as:

[0071]

[0072] where:

[0073]

[0074] The finite element method is used to solve equation (25), the same shape function matrix is used to discretize the displacement field, temperature field and phase field variables, for a given point x we can get:

[0075]

[0076]

[0077] where are the displacement, phase field and temperature vectors of the element, N is the shape function matrix, B u ,B s ,B θ is the shape function derivative matrix.

[0078] Thus equation (25) can be expressed as:

[0079]

[0080] Therefore, the formula (31) can be solved by using an incremental iteration method, time is divided into N time steps, Newton iteration method is used to solve each variable in each time step, and backward Euler method is used to solve the variable related to the next time step.

[0081] Compared with the prior art, the concrete meso thermal cracking thermal-mechanical-phase field coupling model simulation method has the beneficial effects that:

[0082] 1. The non-homogeneous structure of the concrete is considered, the polygon aggregate is generated according to the grading curve of the concrete, the interface transition zone is obtained by extending the aggregate outward by 1mm, and a large number of meso concrete numerical calculation models can be generated.

[0083] 2. The thermal-mechanical damage coupling model is established based on the phase field fracture theory, the influence of the fracture on the heat conduction is considered, and the coupling mode is more in line with the engineering practice.

[0084] 3. The finite element method is used to discretize the multi-field coupling model, Newton iteration and backward Euler method are used to solve the variables at different times.

[0085] In summary, the method can provide a calculation basis for the concrete cracking caused by temperature, and provide guiding opinions for the safety evaluation of the hydraulic concrete structure. BRIEF DESCRIPTION OF DRAWINGS

[0086] Figure 1 It is a flow chart of the meso concrete thermal cracking numerical simulation method based on the phase field theory of the application;

[0087] Figure 2 It is a schematic diagram of the phase field, temperature field and displacement field solving process;

[0088] Figure 3 It is a schematic diagram of three-phase meso concrete with different aggregate contents;

[0089] Figure 4 It is a schematic diagram of the thermodynamic boundary conditions of the example model;

[0090] Figure 5 It is a temperature distribution diagram of the example model at different times;

[0091] Figure 6 It is a crack distribution diagram of the example model at different times;

[0092] Figure 7 It is a schematic diagram of the boundary conditions of the comparative example model;

[0093] Figure 8 It is a temperature distribution diagram of the comparative example model at different times;

[0094] Figure 9 The crack distribution diagram of the comparative example model at different times is shown in the following table:

[0095] Figure 10 The comparative diagram of the final crack distribution of the comparative example model and the simulation results of other calculation methods is shown in the following table. DETAILED DESCRIPTION

[0096] The technical solutions in the embodiments of the present application will be clearly and completely described 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 the embodiments. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without creative work fall within the protection scope of the present application.

[0097] Please refer to Figures 1-10 The present application provides a technical solution: a thermal-mechanical-phase field coupling model simulation method for mesoscopic thermal cracking of concrete. The simulation method is based on numerical simulation of mesoscopic thermal cracking of concrete according to the phase field theory, and includes the following steps:

[0098] Step 1. Establish a mesoscopic concrete model, mainly including aggregate mortar and interface transition zone.

[0099] The model aggregate content is 30%, and the size is 100mm*100mm. The generated effect diagram is shown in the following table: Figure 3

[0100] Step 2. Obtain the thermodynamic parameters of each component of the mesoscopic concrete according to the reference data or physical test.

[0101] The thermodynamic parameters of the three-phase components in the concrete in different literatures are shown in the following tables 1 and 2.

[0102] Table 1: Mechanical parameters of mesoscopic concrete

[0103]

[0104]

[0105] Table 2: Thermal parameters of mesoscopic concrete

[0106]

[0107] Step 3. Determine the thermodynamic boundary conditions according to the environment of the concrete.

[0108] According to the actual working environment of the model, the temperature boundary and displacement constraint and load condition are applied to simulate the actual cracking process of the concrete. The model is free around the periphery without constraint; the initial temperature of the model is 20℃, and the temperature boundary condition is applied around the periphery, and the temperature curve is: θ=20+345log​10 (8t+1), the heating time is 1 minute, and the thermodynamic boundary conditions are as shown in the accompanying Figure 4

[0109] Step 4. Discretize the meso-concrete model by using free triangles or quadrilateral elements, and determine the solution error;

[0110] Since the thickness of the interface is 1 mm, the model is discretized by using a 0.5 mm grid size, and the solution error is set to 0.001.

[0111] Step 5. Solve the cracking process of the concrete model under the action of temperature by using the thermal phase field damage coupling model.

[0112] Establish the thermal damage control equation under the action of temperature, discretize the calculation region by using the finite element method, and solve the temperature and crack distribution at different time steps by using the Newton iteration method and the backward Euler method, as shown in the accompanying Figure 5 and the accompanying Figure 6 .

[0113] Preferably, the simulation method comprises generation of a three-phase model, mainly including the following three steps:

[0114] A. Generate polygonal aggregates by percentage using a random algorithm, calculate the area percentage of each graded aggregate by using the Walaven formula and aggregate content, then randomly generate a circle in the graded particle size interval, randomly take points on the circle to generate polygonal aggregates, and perform interference judgment on the generated aggregates and the already generated aggregates, if the generated aggregate does not intersect with the existing aggregate, generate the next aggregate, otherwise, regenerate the aggregate;

[0115] B. Extend the coordinates of the aggregate points outward by 1 mm to obtain an interface transition zone model;

[0116] C. Subtract the aggregate model and the interface transition zone model from the total model to obtain a mortar model.

[0117] Establishment of the thermal phase field damage coupling model, which mainly includes:

[0118] 1) Phase field model:

[0119] If the body force is ignored, the total energy on the calculation domain and the crack surface can be represented as:

[0120]

[0121] Where G f is the fracture energy density of the material, s is the phase field variable, s=0 represents no damage, s=1 represents complete damage, γ(s) is the crack density function, and t is the boundary ​where u is the displacement field, ε is the strain tensor, and ψ(ε(u), s) is the modified energy density function.

[0122] The variational derivative of equation (1) gives the following expression:

[0123]

[0124] where n is the normal vector on the boundary, and the stress tensor can be expressed as:

[0125]

[0126] The mechanical equilibrium equation and the phase field governing equation can then be obtained as follows:

[0127]

[0128] where the equal and unequal signs are taken when δs > 0 and δs = 0, respectively, and the corresponding boundary conditions can be expressed as:

[0129]

[0130] In the phase field theory, the crack density function can be expressed as:

[0131]

[0132] where α(s) is the crack geometry function, l0is the crack width characteristic parameter, and:

[0133]

[0134] The crack geometry function can be expressed as:

[0135]

[0136] Substituting equation (8) into equation (4) gives:

[0137]

[0138] In the simulation, only the damage caused by tension is considered, so the modified energy density function can be expressed as:

[0139]

[0140] where ω(s) is the energy degradation function, is the initial elastic strain energy, which can be expressed as:

[0141]

[0142] where λ and μ are the material Lame constants, and in the simulation the tensile strain energy is where σ1 is the maximum principal stress, E is the elastic modulus of the material, and <σ1> = max(σ1, 0), substituting equation (10) into equation (9) gives:

[0143]

[0144] In the phase field model, the energy degradation function can be expressed as:

[0145]

[0146] where p > 2, a1 > 0 is a parameter related to the material properties, and the expressions of a1, a2 and a3 are as follows:

[0147]

[0148] where f t is the tensile strength of the material, ξ is a parameter related to the crack geometry function, β k and β w are the initial slope and the final crack opening of the softening curve, respectively, and some commonly used softening curves are as follows:

[0149]

[0150] Since concrete is a quasi-brittle material, linear softening is used in the simulation.

[0151] To satisfy the irreversibility of the crack, a history variable H is introduced to record the maximum crack driving force, which can be expressed as:

[0152]

[0153] Therefore, the final governing equation can be expressed as:

[0154]

[0155] 2) Heat conduction:

[0156] The governing equation of solid heat conduction is as follows:

[0157]

[0158] where ρ and c are the density and specific heat capacity of the material, θ is the temperature, Q int is the heat source, k is the heat conduction tensor, is the heat flow, is the given boundary heat flow, to consider the effect of cracks on heat conduction, the same degradation function is used to degrade the heat conduction performance, which can be expressed as:

[0159] k = ω(s)k0I (21)

[0160] where k0 is the initial thermal conductivity of the material, I is the second order identity tensor.

[0161] The thermal strain caused by the temperature change can be expressed as:

[0162] ε th = a(θ - θ ref )I (22)

[0163] where ε th is the thermal strain tensor, a is the thermal expansion coefficient of the material, θ ref is the reference temperature.

[0164] Therefore the total strain can be expressed as:

[0165] ε = ε th + ε σ (23)

[0166] where ε σ is the stress-induced strain tensor.

[0167] The final stress-strain relationship can be expressed as:

[0168]

[0169] where is the initial elastic matrix of the material.

[0170] For simplicity, the effect of temperature on material parameters is not considered and the displacement field has no effect on the temperature field, thus the construction of the phase-field thermo-mechanical coupled model is completed.

[0171] 3) Numerical solution:

[0172] The weak form of equations (19) and (20) can be expressed as:

[0173]

[0174] where:

[0175]

[0176] The finite element method is used to solve equation (25), the same shape function matrix is used to discretize the displacement field, temperature field and phase field variables, for a given point x we can get:

[0177]

[0178] where is the displacement, phase field and temperature vector of the element, N is the shape function matrix, B u ,B s ,B θDerivative matrix for shape function.

[0179] Thus, equation (25) can be expressed as:

[0180]

[0181] Therefore, the incremental iteration method can be used to solve equation (31), and the calculation time is divided into N time steps, and the Newton iteration method is used to solve each variable in each time step, and the backward Euler method is used to solve the variables related to the next time step. Due to the complexity of the equation, a separate equation is used to solve each field in each time step, and the specific solving process is shown in the following table. Figure 2

[0182] Further, a comparative example is provided for verification. The comparative example verifies the simulation effect of the meso-concrete thermal cracking based on the phase field theory in the above embodiment through a concrete temperature cracking test, and compares with other existing models.

[0183] The experimental test environment is INTEL I7 10700 CPU@3.20GHz, RAM 32GB, and windows10 operating system. The model size is 150mm*150mm, the aggregate content is 30%, the circular hole size is 15mm*15mm, the upper and lower ends of the model are constrained in the normal direction, the initial temperature is 20℃, the circular hole adopts a heating boundary, and the heating rate is 0.1℃ per minute. The simulation results of the present application are compared with the results obtained by C.Yan using the particle flow simulation method and the test results of D.P.Jansen.

[0184] From the attached Figure 10 It can be seen that the crack propagation mode obtained by simulation of the present application is consistent with the test and simulation results of other methods. Since the present application considers the meso-structure of concrete, the crack mainly develops along the interface transition zone to the hole wall, and the calculation parameters have clear physical meaning, the mesh division is simple, and the present application provides an efficient and accurate numerical calculation method for concrete temperature cracking simulation.

[0185] The contents not described in detail in the specification belong to the prior art known to those skilled in the art, although the embodiments of the present application have been shown and described, it can be understood by those skilled in the art that various changes, modifications, replacements and variations can be made to these embodiments without departing from the principles and spirits of the present application, the scope of the present application is defined by the appended claims and their equivalents.​

Claims

1. A thermo-mechanical-phase-field coupled model simulation method for microscopic thermal cracking in concrete, characterized in that: The simulation method is based on the phase-field theory for numerical simulation of thermal cracking in mesoscopic concrete, and includes the following steps: Step 1. Establish a micro-concrete model, mainly including aggregate mortar and interface transition zone; Step 2. Obtain the thermodynamic parameters of each component of the concrete by referring to relevant data or conducting physical experiments; Step 3. Determine the thermodynamic boundary conditions based on the environment in which the concrete is located; Step 4. Discretize the microscopic concrete model using free triangular or quadrilateral elements and determine the solution error; Step 5. Use a thermo-phase-field damage coupling model to solve the cracking process of the concrete model under temperature action; A thermodynamic damage control equation under temperature was established, the calculation region was discretized using the finite element method, and the temperature and crack distribution at different time steps were solved using the Newton iteration method and the backward Euler method. The thermo-phase-field damage coupling model mainly includes the phase-field model, heat conduction analysis, and numerical solution: (1) Phase field model: Based on phase field theory, mechanical equilibrium equations and phase field control equations are established that consider crack density function, energy degradation function and damage caused by tension, and historical variables are introduced to satisfy the irreversibility of cracks. (2) Heat conduction analysis: establish the solid heat transfer control equation, consider the degradation effect of cracks on heat conduction, and calculate the thermal strain caused by temperature change; (3) Numerical solution: The phase field variables, temperature field and displacement field are discretized by the finite element method and numerical solution is performed by the incremental iteration method.

2. The method for simulating the thermo-mechanical-phase-field coupling model of microscopic thermal cracking in concrete according to claim 1, characterized in that: The simulation method includes the generation of a three-phase model, which mainly includes the following three steps: A. A random algorithm is used to generate polygonal aggregates by percentage. The area percentage of each graded aggregate is calculated by the Varavan formula and aggregate content. Then, a circle is randomly generated within the gradation particle size range, and points are randomly selected on the circle to generate polygonal aggregates. The generated aggregates are compared with the already generated aggregates for interference judgment. If the generated aggregates do not intersect with the existing aggregates, the next aggregate is generated; otherwise, the aggregate is regenerated. B. Extend the coordinates of the aggregate points outward by 1mm to obtain the interface transition zone model; C. Subtract the aggregate model and interface transition zone model from the overall model to obtain the mortar model.

3. The method for simulating the thermo-mechanical-phase-field coupling model of microscopic thermal cracking in concrete according to claim 1, characterized in that: The phase-field model is established by first calculating and listing the total energy expression of the domain and the crack surface, listing the stress tensor therein, and then obtaining the mechanical equilibrium equation and the phase-field control equation.

4. The method for simulating the thermo-mechanical-phase field coupling model of microscopic thermal cracking in concrete according to claim 3, characterized in that: The phase-field model is established by listing the crack density function and crack geometry function through phase-field theory and substituting them into the mechanical equilibrium equation and phase-field control equation for calculation. At the same time, only the damage caused by tension is considered in the simulation, and the energy density function is improved accordingly.

5. The method for simulating the thermo-mechanical-phase-field coupling model of microscopic thermal cracking in concrete according to claim 4, characterized in that: In establishing the phase-field model, the energy degradation function needs to be listed. Since concrete is a quasi-brittle material, linear softening is used in the simulation. To satisfy the irreversibility of cracks, a historical variable is introduced to record the maximum crack driving force H, and the final governing equation is obtained.

6. The method for simulating the thermo-mechanical-phase-field coupling model of microscopic thermal cracking in concrete according to claim 1, characterized in that: The heat conduction analysis first requires listing the governing equations for heat transfer in the solid. To account for the influence of cracks on heat conduction, the same degradation function is used to degrade the heat conduction performance. The thermal strain tensor caused by temperature changes is calculated, and the final stress-strain relationship is obtained. For the sake of simplicity, the influence of temperature on material parameters was not considered, and the displacement field had no effect on the temperature field. Thus, the construction of the phase field thermodynamic coupling model was completed.

7. The method for simulating the thermo-mechanical-phase-field coupling model of microscopic thermal cracking in concrete according to claim 1, characterized in that: The numerical solution process first lists the formulas and their weak forms, then solves the formulas using the finite element method, and finally discretizes the displacement field u and temperature field using the same shape function matrix. And the phase field variable s.

8. The method for simulating the thermo-mechanical-phase field coupling model of microscopic thermal cracking in concrete according to claim 7, characterized in that: The numerical solution uses an incremental iterative method to solve the formula, dividing the computation time into N time steps. In each time step, Newton's iterative method is used to solve each variable, and backward Euler's method is used to solve the variables related to the next time step. Due to the complexity of the equation, a separate solution is used to solve each field in each time step.

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