Prediction method of material refinement crack

Abstract

The present invention provides a prediction method for refined cracks of materials, including: obtaining relevant mechanical parameters of materials; modeling according to the actual situation of the object to be simulated, and substituting the parameters obtained in step 1 into the established model; establishing the fracture theory of the phase field model under complex failure modes so that the phase field model can effectively simulate the fracture conditions of various failure modes; establishing a refined process theory to accurately simulate the crack path and expansion process; applying the actual load process to the model; solving the cracking process of the model according to the above established theory. The present invention can implicitly obtain the cracking and failure process of the simulated object under various external force conditions without adding additional judgment criteria. It can provide a theoretical basis and guidance for whether the material cracks under a certain environment and how to stop cracking after cracking, and solve the problem that the traditional method can only perform crack fitting rather than crack prediction, and the problem that the simulated crack is too rough.

Description

Technical Field

[0001] The invention belongs to the technical field of material mechanics research, and in particular relates to a method for predicting fine cracks in materials. Background Art

[0002] In recent years, a variety of numerical fracture simulation methods have been developed, including the extended finite element method, the boundary element method, the meshless method, etc., but these methods must explicitly track the crack path, that is, additional judgment criteria must be used to determine whether the material is cracked. In addition, most of the existing crack simulation theories are still in the stage of crack fitting, that is, before the crack path is obtained, it is impossible to refine the mesh of the finite element of the numerical model, and it is necessary to repeat the calculation to manually improve the mesh division process. Moreover, since the value of the fracture energy directly affects the calculation results, it is difficult to accurately measure the fracture energy in physical experiments, which affects the parameter acquisition and reduces the efficiency of numerical simulation.

[0003] In summary, the existing methods and technologies for simulating fracture have the following defects: (1) The simulated failure mode is single and cannot simulate complex failure processes; (2) The crack path must be explicitly solved through judgment criteria; (3) The finite elements in the numerical model cannot be automatically and reasonably refined, which limits the calculation accuracy; (4) It is difficult to accurately measure the fracture energy. Summary of the invention

[0004] The purpose of the present invention is to address the deficiencies of the prior art and to provide a method for predicting fine cracks in materials. The method can accurately simulate fine cracks in brittle and quasi-brittle materials such as rocks and metals, while improving the accuracy of crack paths and calculation results.

[0005] In order to solve the above technical problems, the present invention adopts the following technical solutions:

[0006] A method for predicting fine cracks in a material comprises the following steps:

[0007] Step 1: Perform a tensile test to obtain relevant mechanical parameters of the material;

[0008] Step 2: Model the object according to its actual condition and substitute the parameters obtained in step 1 into the established model;

[0009] Step 3: Establish the fracture theory of the phase field model under complex failure modes so that the phase field model can effectively simulate the fracture conditions of various failure modes;

[0010] Step 4: Establish a refined process theory to accurately simulate the crack path and propagation process;

[0011] Step 5: Apply actual load process to the model;

[0012] Step 6: Solve the cracking process of the model based on the theory established in steps 3 and 4.

[0013] Furthermore, the parameters include elastic modulus, Poisson's ratio, axial tensile fracture energy, tensile strength, and the ratio of critical shear failure strength to critical tensile failure strength.

[0014] Furthermore, the axial tensile fracture energy of the material can be obtained by the following formula:

[0015]

[0016] G c is the fracture energy, dU is the energy dissipated during the fracture of the specimen, and dA is the fracture area.

[0017] Furthermore, the axial tensile fracture energy of the material can be obtained by the following formula: For the fracture area dA, an ultrasonic transmitter is used to monitor the crack propagation state and crack area in real time, and the monitoring results are input into a computer for data analysis to obtain the actual crack state and area at any loading moment.

[0018] Furthermore, the governing equation of the phase field model established in step 3 is:

[0019]

[0020] Where: s∈[0,1] is the crack phase field variable, l0 is the crack diffusion width, is the tensile stress perpendicular to the failure surface, is the equivalent shear stress parallel to the failure surface, n is the direction vector perpendicular to the failure surface, m is the direction vector parallel to the failure surface, ψ0 is the energy density before degradation, ψ 0II is the sheared part of ψ0, ω I (s) and ω II (s) are two degenerate functions. A prime on a variable represents the first-order derivative of the variable, such as ω I ′(s) is ω I (s) is the first-order derivative with respect to s, is the Hamiltonian operator, σ is the degenerate stress tensor, D is the degenerate elastic matrix, ε is the strain tensor, α(s) is the geometric crack function, E is the elastic modulus, μ is the shear modulus, G cI and G cII are the fracture energy of mode I crack and mode II crack, respectively, τ s and σ t are the ultimate tensile stress and the ultimate shear stress, respectively, and define the variable χ=τs / σ t ,and

[0021] ω I (s),ω II The relationship between (s) and the phase field variable s is written as:

[0022]

[0023]

[0024] Q(s)=a1s+a1a2s 2 +a1a2a3s 3 +a1a2a3a4s 4 +…

[0025] Where: b I , b II It is used to control the ability of the degradation function, and its value is in [0,1]. Under the pure tensile failure state, b I =1, b II =0; in pure shear state, b I =0, b II =1; in general, b can also be used I =b II = 1, while ensuring the calculation accuracy, the iteration efficiency is improved. φ(s) and Q(s) are polynomials of the phase field variable s, which are used to determine the form of the degradation function, p and a i is an undetermined parameter, which is obtained through material parameters and a specific softening curve.

[0026] Furthermore, the refinement process theory established in step 4 is to enable the model to obtain the most reasonable finite element distribution through the mesh refinement algorithm. The specific theory is as follows:

[0027] A multi-node quadrilateral element is established, where the number of nodes in the element varies between 4 and 8, and its shape function is constructed as follows:

[0028]

[0029] Among them, N n (ξ,η) is the shape function of any point in the unit at node n. For nodes 5, 6, 7, and 8, when they do not exist, the corresponding shape function value becomes 0; (ξ,η) is the local coordinate of any point in the unit, and the local coordinates of each node are:

[0030]

[0031] Furthermore, the refinement process in step 4 also includes establishing a set of refinement criteria. The specific strategies are as follows:

[0032] When initially dividing the units, all quadrilateral 4-node units are used. When the phase field value of a node reaches a certain critical value, a node is added on the boundary of the unit adjacent to the unit. When the number of nodes of any unit reaches 8, the unit is divided into 4 small 4-node units.

[0033] For the node phase field critical value, a five-level division of 0.2 / 0.4 / 0.6 / 0.8 / 1.0 is adopted, and each level is divided once. At the same time, it is stipulated that the difference between adjacent units cannot exceed two levels.

[0034] Furthermore, in step 6, an iterative method or a step-by-step decoupling algorithm is used to solve the cracking process of the model.

[0035] Furthermore, the step-by-step decoupling algorithm is used to solve the phase field control equation in each load step. After the phase field control equation is solved, the phase field model control equation and the mesh refinement algorithm need to be cycled until the mesh refinement level meets the simulation accuracy requirements, and the crack propagation state and stress-displacement data of this step are obtained. The data are then substituted into the next load step for analysis until the specimen is completely broken. The data obtained in each load step are read and processed to obtain the crack propagation process and distribution state.

[0036] Compared with the prior art, the present invention has the following beneficial effects:

[0037] 1) The present invention can implicitly obtain the cracking and destruction process of the simulated object under various external force conditions without adding additional judgment criteria. It can provide a theoretical basis and guidance for whether the material will crack under a certain environment and how to stop the crack after cracking, and solve the problem that the traditional method can only perform crack fitting but not crack prediction, and the problem that the simulated crack is too rough;

[0038] 2) The universal phase field method for simulating different failure modes of brittle materials of the present invention divides the energy equation into three parts: tension, shear and compression for simulation, adopts a double degenerate function, which respectively represents the contribution of normal stress and tangential stress to crack extension, adopts a unified fracture criterion, incorporates the relationship between the compressive strength and the tensile strength of the material into the theory, and creates a hybrid model, thereby achieving a significant improvement in iteration efficiency while ensuring the calculation accuracy; a method for accurately calculating the crack direction is created, which can solve the problem that the traditional method is difficult to accurately predict the crack direction;

[0039] 3) Based on the relationship between the tensile, compressive and shear properties of materials, the present invention establishes a universal phase field method for simulating different failure modes of brittle materials. By applying the unified failure theory, the constraint that the phase field theory can only simulate the failure mode under tensile conditions is broken, and the failure process and crack path of brittle materials under complex boundary conditions under other possible failure forms (pure shear, tensile shear, and compressive shear) are accurately simulated; the parameters obtained from uniaxial tensile tests and uniaxial compression tests are used to predict the crack propagation direction under any loading path;

[0040] 4) The present invention has a simple implementation process, strong practicability and a wide range of applications. It can provide theoretical and technical support for the simulation of crack propagation of brittle materials and quasi-brittle materials under different failure modes, and promote the accurate simulation of brittle solid materials such as rocks, metals, and ceramics by finite element and phase field models. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 This is a flow chart of a method for predicting fine cracks in materials according to an embodiment of the present invention;

[0042] Figure 2 This is a schematic diagram of the size and boundary conditions of a uniaxial compression specimen according to an embodiment of the present invention;

[0043] Figure 3 This is the actual crack distribution diagram measured by the indoor uniaxial compression test of the embodiment of the present invention;

[0044] Figure 4 It is a monitoring roadmap of ultrasonic infrared technology of an embodiment of the present invention;

[0045] Figure 5 This is a schematic diagram of the simulation results of the crack propagation path of a uniaxially compressed specimen according to an embodiment of the present invention;

[0046] Figure 6 Schematic diagram of material fine cracks predicted by an embodiment of the present invention. DETAILED DESCRIPTION

[0047] The technical solutions in the embodiments of the present invention will be described clearly and completely below in combination with the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0048] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

[0049] The present invention will be further described below in conjunction with specific embodiments, but the present invention is not limited thereto.

[0050] See also Figure 1 The present invention provides a method for predicting fine cracks in a material, comprising the following steps:

[0051] Step 1: Perform a tensile test to obtain relevant mechanical parameters of the material. In this embodiment, the simulation of the compression failure process of two uniaxially compressed concrete specimens is taken as an example. The specimen size is as follows: Figure 2 As shown in the figure, there are three initial cracks in the center area of ​​the specimen. Let θ = 45°, S = 12.7mm, C = 0mm. This is actually the test process in which the compression shear and tension shear failure modes coexist. The test results are shown in Figure 3 shown.

[0052] The material parameters required by this method were obtained by conducting an indoor uniaxial tensile test. The following parameters were obtained after measuring the material: (1) elastic modulus E = 5.96 GPa, (2) Poisson's ratio ν = 0.15, (3) uniaxial tensile fracture energy G c =15N / m, (4) tensile strength f t =3.2MPa, (5) the ratio of critical shear failure strength to critical tensile failure strength χ = 5.0.

[0053] Among them, the tensile fracture energy of the material can be obtained by the following formula:

[0054]

[0055] G c is the fracture energy, dU is the energy dissipated during the fracture of the specimen, and dA is the fracture area.

[0056] Among them, the fracture area dA is difficult to measure the true value due to the uneven fracture surface. Figure 4 , the ultrasonic infrared method is used to monitor the crack propagation state and crack area in real time. Specifically, the ultrasonic transmitter monitors and obtains the crack propagation of the material in real time, and inputs the monitoring results into the computer for data analysis, so as to obtain the actual cracking state and area at any loading moment.

[0057] Step 2: According to the actual situation of the object to be simulated, proportional modeling is performed in numerical simulation software. Commercial software such as ANSYS, ABAQUS or self-programming software can be used, and the parameters obtained in step 1 are substituted into the model.

[0058] Step 3: Establish the fracture theory of complex failure modes of the phase field model so that the phase field model can effectively simulate the fracture conditions of various failure modes. In order to implicitly obtain the crack propagation path, the recently developed phase field fracture model is used, but the traditional phase field model can only simulate the tensile failure process. Here, a phase field model that can simulate tensile and shear failure modes is used, and the criterion that can simulate tensile and shear failure in the paper "Unified tensile fracture criterion" is introduced. The results in the paper "A unified phase-field theory for the mechanics of damageand quasi-brittle failure" are applied to eliminate the instability of the calculation results caused by the diffusion width of the phase field model. The specific theoretical derivation is as follows:

[0059] By decomposing the energy density function into three parts, different failure modes can be considered, and different degradation functions can be used to degrade each part:

[0060]

[0061] Where: ψ is the energy density function after degradation, ψ0 is the energy density before degradation, is the tensile part of ψ0, ψ 0II is the sheared part of ψ0, is the other part of ψ0, And 0I and ψ 0II The definition is

[0062]

[0063]

[0064] in: is the tensile stress perpendicular to the failure surface, is the equivalent shear stress parallel to the failure surface, n is the direction vector perpendicular to the failure surface, and m is the direction vector parallel to the failure surface; ω I (s) and ω II (s) correspond to and ψ 0II Two degradation functions of , s∈[0,1] is the crack phase field variable, which is used to indicate the degree of material damage, E is the elastic modulus, and μ is the shear modulus.

[0065] In order to improve the computational efficiency, a hybrid mode is proposed: I (s),ω II The relationship between (s) and the phase field variable s is written as:

[0066]

[0067]

[0068] Q(s)=a1s+a1a2s 2 +a1a2a3s 3 +a1a2a3a4s 4 +…

[0069] Where: b I , b II It is used to control the ability of the degradation function, and its value is in [0,1]. Under the pure tensile failure state, b I =1, b II =0; in pure shear state, b I =0, b II =1; in general, b can also be used I =b II = 1, while ensuring the calculation accuracy, the iteration efficiency is improved. φ(s) and Q(s) are polynomials of the phase field variable s, which are used to determine the form of the degradation function, p and a i It is an undetermined parameter and can be obtained through material parameters and specific softening curve.

[0070] Among them, in the constitutive relationship, in order to improve the convergence and convergence efficiency of the calculation, a modified hybrid model is used to calculate the degraded stress tensor and the degraded elastic matrix. The calculation method is:

[0071]

[0072]

[0073] Where: σ is the degenerate stress tensor, D is the degenerate elastic matrix, ε is the strain tensor, and the other symbols have the same meanings as above.

[0074] Among them, in order to simulate the failure form of compression and shear, the following crack evolution equation is established:

[0075]

[0076] Where: G cI and G cII are the fracture energy of mode I crack and mode II crack, respectively, l0 is the diffusion width of the crack, α(s) is the geometric crack function, α′(s) is the first-order derivative of α(s) with respect to s, and ω I ′(s) is ω I (s) is the first-order derivative with respect to s, is the Hamiltonian operator.

[0077] If you want to calculate the crack propagation direction of tensile shear failure or compressive shear failure, you can solve it by the new function defined below:

[0078]

[0079] Where: θ can be obtained by the following formula:

[0080]

[0081] Where: Parameter and are the first and second principal normal stresses, respectively, and θ is Direction and first principal normal stress The angle between the directions, and n = [cos(θ+θ0), sin(θ+θ0)], θ0 is The angle between the axis and the coordinate axis x1, the crack direction m is perpendicular to n, and n is Direction, τ s and σ t are the ultimate tensile stress and the ultimate shear stress, respectively, and define the variable χ=τ s / σ t .

[0082] In order to solve the crack evolution equation smoothly and avoid crack healing, the following two historical field variables corresponding to the tensile energy and shear energy are defined: and

[0083]

[0084] Where: m is the mth load step, is the historical field variable of the mth load step. In the initial load step,

[0085]

[0086] in:

[0087]

[0088] in:

[0089]

[0090] in: is the fracture energy of the uniaxial tensile test, and k1 is defined as follows:

[0091]

[0092] For b I =1, b II = 0 and b I =0, b II = 1, the governing equations for tensile failure and shear failure can be obtained respectively. These two cases have been widely used. I =b II =1.0, which can greatly expand the scope of application of the phase field model and obtain the phase field model control equation that can calculate the damage under the simultaneous action of shear and tension:

[0093]

[0094] Step 4: Establish a refined process theory to accurately simulate the crack path and propagation process.

[0095] The refinement process theory enables the numerical model to obtain the most reasonable finite element distribution through the mesh refinement algorithm. The specific theory is as follows:

[0096] A multi-node quadrilateral element is established (the number of nodes in the element varies between 4 and 8), and its shape function is constructed as follows:

[0097]

[0098] Among them, N n (ξ,η) is the shape function of any point in the unit at node n. For nodes 5, 6, 7, and 8, when they do not exist, the corresponding shape function value becomes 0; (ξ,η) is the local coordinate of any point in the unit, and the local coordinates of each node are

[0099]

[0100] When the units are initially divided, all quadrilateral 4-node units are used. When the phase field value of a node reaches a certain critical value, a node is added on the boundary of the unit adjacent to the unit. When the number of nodes of any unit reaches 8, the unit is divided into 4 small 4-node units.

[0101] For the node phase field critical value, a five-level division of 0.2 / 0.4 / 0.6 / 0.8 / 1.0 is adopted, and each level is divided once. At the same time, it is stipulated that the difference between adjacent units cannot exceed two levels.

[0102] Step 5: Apply the actual loading process to the numerical model. To simulate the static loading process, a sufficiently small load step is used. Each displacement load step is set to Δu = 5.0 × 10 -5 mm, gradually apply displacement, and establish a crack recognition system. The boundary and initial crack of the model are processed separately, and the crack boundary is divided into two to prevent crack healing and affect the simulation results.

[0103] Step 6: Solve the cracking process of the model according to the theory established in steps 3 and 4. Use the iterative method or the step-by-step decoupling algorithm to solve the cracking process of the simulated object. In the iterative method, the phase field control equation is repeatedly calculated until the result of the last iteration meets the error requirement. This method achieves higher solution accuracy, but the calculation efficiency is slower. The distributed decoupling algorithm does not need to repeatedly calculate the phase field control equation. This method is not as accurate as the iterative method, but when the load step is small enough, it can still guarantee the calculation accuracy, but the calculation efficiency is very high. This example uses the step-by-step decoupling algorithm for solution. In each load step, after solving the phase field control equation, the crack propagation area needs to be refined. In order to ensure the calculation accuracy after refinement, the phase field model control equation and mesh refinement algorithm need to be cycled until the mesh refinement degree meets the simulation accuracy requirements, and the crack propagation state and stress displacement data of this step are obtained. Then the data is substituted into the next load step for analysis until the specimen is completely broken. The data obtained in each load step is substituted into commercial software such as ABAQUS for reading and processing, and the crack propagation process and distribution state can be obtained. Figure 5 and Figure 6 It is a schematic diagram of the crack evolution process during the numerical simulation, as well as the adaptive grid distribution during the crack evolution process. It can be seen that the crack path calculated by the present invention is basically consistent with the actual test, and the effect is significant.

[0104] In summary, the present invention can provide theoretical and technical support for the crack propagation simulation of brittle materials and quasi-brittle materials under different failure modes, and promote the accurate simulation of brittle solid materials such as rocks, metals, and ceramics by finite element and phase field models.

Claims

1. A method for predicting fine cracks in a material, characterized in that: The steps include: Step 1: Perform a tensile test to obtain relevant mechanical parameters of the material; Step 2: Model the object according to its actual condition and substitute the parameters obtained in step 1 into the established model; Step 3: Establish the fracture theory of the phase field model under complex failure modes so that the phase field model can effectively simulate the fracture conditions of various failure modes; Step 4: Establish a refined process theory to accurately simulate the crack path and propagation process; Step 5: Apply actual load process to the model; Step 6: Solve the cracking process of the model based on the theory established in steps 3 and 4; Among them, the refinement process theory established in step 4 is to enable the model to obtain the most reasonable finite element distribution through the mesh refinement algorithm. The specific theory is as follows: A multi-node quadrilateral element is established, where the number of nodes in the element varies between 4 and 8, and its shape function is constructed as follows: Among them, N n (ξ,η) is the shape function of any point in the unit at node n. For nodes 5, 6, 7, and 8, when they do not exist, the corresponding shape function value becomes 0; (ξ,η) is the local coordinate of any point in the unit, and the local coordinates of each node are: The refinement process in step 4 also includes establishing a set of refinement criteria. The specific strategies are as follows: When initially dividing the units, all quadrilateral 4-node units are used. When the phase field value of a node reaches a certain critical value, a node is added on the boundary of the unit adjacent to the unit. When the number of nodes of any unit reaches 8, the unit is divided into 4 small 4-node units. For the node phase field critical value, five levels of 0.2 / 0.4 / 0.6 / 0.8 / 1.0 are used, and each level is divided once. At the same time, it is stipulated that the difference between adjacent units cannot exceed two levels; The governing equation of the phase field model established in step 3 is: Where: s∈[0,1] is the crack phase field variable, l0 is the crack diffusion width, is the tensile stress perpendicular to the failure surface, is the equivalent shear stress parallel to the failure surface, n is the direction vector perpendicular to the failure surface, m is the direction vector parallel to the failure surface, ψ0 is the energy density before degradation, is the tensile part of ψ0, ψ 0II is the sheared part of ψ0, ω I (s) and ω II (s) are two degenerate functions. A prime on a variable represents the first-order derivative of the variable, such as ω I ′(s) is ω I (s) is the first-order derivative with respect to s, is the Hamiltonian operator, σ is the degenerate stress tensor, D is the degenerate elastic matrix, ε is the strain tensor, α(s) is the geometric crack function, E is the elastic modulus, μ is the shear modulus, G cI and G cII are the fracture energy of mode I crack and mode II crack, respectively, τ s and σ t are the ultimate tensile stress and the ultimate shear stress, respectively, and define the variable χ=τ s / σ t ,and Where: b I 、b II It is used to control the ability of the degradation function, and its value is in [0,1]. Under the pure tensile failure state, b I =1, b II =0; in pure shear state, b I =0, b II =1; in general, b can also be used I =b II = 1, while ensuring the calculation accuracy, the iteration efficiency is improved. φ(s) and Q(s) are polynomials of the phase field variable s, which are used to determine the form of the degradation function, p and a i is an undetermined parameter, which is obtained through material parameters and a specific softening curve.

2. The method for predicting material fine cracks according to claim 1, characterized in that: The parameters include elastic modulus, Poisson's ratio, axial tensile fracture energy, tensile strength, and the ratio of critical shear failure strength to critical tensile failure strength.

3. The method for predicting material fine cracks according to claim 2, characterized in that: The axial tensile fracture energy of the material can be obtained by the following formula: G c is the fracture energy, dU is the energy dissipated during the fracture of the specimen, and dA is the fracture area.

4. The method for predicting material fine cracks according to claim 3, characterized in that: The axial tensile fracture energy of the material can be obtained by the following formula: For the fracture area dA, an ultrasonic transmitter is used to monitor the crack propagation state and crack area in real time, and the monitoring results are input into a computer for data analysis to obtain the actual crack state and area at any loading moment.

5. The method for predicting material fine cracks according to claim 1, characterized in that: In step 6, an iterative method or a step-by-step decoupling algorithm is used to solve the cracking process of the model.

6. The method for predicting material fine cracks according to claim 5, characterized in that: The step-by-step decoupling algorithm is used to solve the phase field control equations in each load step. The phase field model control equations and the mesh refinement algorithm need to be cycled until the mesh refinement level meets the simulation accuracy requirements, and the crack propagation state and stress-displacement data of this step are obtained. The data are then substituted into the next load step for analysis until the specimen is completely broken. The data obtained in each load step are read and processed to obtain the crack propagation process and distribution state.

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