A multiscale microcrack band method for predicting damage and fracture of quasi-brittle materials

Abstract

The present application provides a kind of multi-scale microcrack zone method for predicting damage and fracture of quasi-brittle materials, including proposing microcrack zone hypothesis with distribution characteristics based on the concept of fracture process zone, reconstructing variational equation of energy minimization by using microcrack distribution equivalent transformation, deriving stress balance equation and microcrack density parameter evolution law under the framework of thermodynamics, correlating macroscopic elastic stiffness and microcrack density parameter based on homogenization method, constituting material softening relationship by equivalent constitutive function, irreversible damage and tensile-compressive stress asymmetry response processing, and numerical calculation verification of representative examples.The present application can overcome the value contradiction of length scale parameter approaching to 0 by combining phase field model and micro-mechanical damage model, effectively calibrate energy degradation process, reflect real damage distribution characteristics and energy degradation behavior in crack propagation process, and has important significance and significant progress for predicting damage and cracking phenomenon of engineering structure.

Description

Technical Field

[0001] This invention relates to the field of computational mechanics, specifically a method for predicting multi-scale microcrack zones in quasi-brittle materials. Background Technology

[0002] Accurate prediction of the damage and fracture behavior of quasi-brittle materials such as concrete and rock is crucial for the safe design and catastrophic failure prevention of engineering structures. However, damage mechanics models based on damage mechanics theory often fail to consider nonlocal effects and lack internal length-scale parameters, resulting in mesh sensitivity issues and distorted numerical results. Methods based on fracture mechanics theory, such as the cohesive element method and the extended finite element method, often require crack propagation between meshes, leading to mesh arrangement sensitivity issues. Furthermore, these methods often require explicit setting of crack propagation paths, significantly increasing computational complexity (especially in cases of complex crack paths) and increasing the likelihood of non-convergence.

[0003] In comparison, the phase-field method demonstrates significant advantages in both theoretical foundation and numerical implementation. It can effectively capture damage evolution while spontaneously tracking crack paths, and can predict complex cracking behaviors such as crack bifurcation and merging. However, the phase-field fracture model still has some shortcomings. Firstly, there is a contradiction in the value of the length-scale parameter. Specifically, the variational framework of the phase-field model requires the length-scale parameter to approach 0 to ensure that the area of ​​diffuse cracks is equal to the area of ​​discrete cracks. However, in practical applications, the length-scale parameter needs to reflect the actual damage dispersion width. Another shortcoming is that the energy degradation process in the phase-field model is very arbitrary, lacking an effective calibration method. This can lead to discrepancies between the predicted final crack path and the actual crack propagation process, even if the predicted final crack path is the same as the experimental result. Summary of the Invention

[0004] The aforementioned problems can be attributed to the deficiencies in the variational framework of the phase-field model and the lack of low-scale physical mechanisms. To address these issues in the prior art, this invention provides a multi-scale microcrack zone prediction method for quasi-brittle material damage and fracture, comprising the following steps:

[0005] S1. Propose the hypothesis of low-scale microcrack zone and establish a multi-scale matrix-microcrack research system.

[0006] Among them, the micro-cracks are coin-shaped cracks, and their mathematical expression is as follows:

[0007]

[0008] In the formula, a is the radius of the microcrack surface, c is the half-opening of the microcrack, and x1, x2, x3 are the spatial coordinates of the local coordinate system of the microcrack.

[0009] S2. Based on the law of micro-force equilibrium and the law of thermodynamics, the linear constitutive relation, micro-stress equilibrium equation and micro-force equilibrium equation are derived within the framework of thermodynamics.

[0010] S3. The variational framework of microcrack evolution is reconstructed by equivalent transformation of microcrack distribution, and explicit expressions of volumetric strain energy and crack surface energy are given. The stress balance equation and evolution rule of microcrack density parameter are derived.

[0011] S4. Define the geometric distribution function as α(ω)=ω, where ω is the microcrack density parameter, and give the microcrack distribution form.

[0012] S5. Define the energy degradation function, establish the mapping relationship between macroscopic elastic stiffness and low-scale microcrack density parameters using the micromechanical homogenization method, and calibrate the energy degradation process.

[0013] S6. Combining the geometric distribution function and the energy degradation function, the parameterized constitutive function of the quasi-brittle material is derived. The parameterized constitutive function of the quasi-brittle material is:

[0014]

[0015] In the formula, For effective stress, δ n Let E be the displacement increment caused by damage, and G be the elastic modulus of the material. c Let ω be the fracture energy, l be a numerical parameter that adjusts the width of the microcrack distribution, b be the equivalent width determined by the microcrack geometry and parameter l, and ω be the fracture energy. m To allow for the maximum value of the microcrack density parameter, η is the average energy consumption ratio per unit microcrack density parameter generated before and after equivalent treatment, α(ω) is the microcrack geometric function, φ(ω) is the internal function related to the material softening law, and ω * The peak value of the microcrack density parameter distribution at any given time.

[0016] Furthermore, it also includes:

[0017] S7 is processed using the historical field variable method and the equivalent stress method respectively to ensure the irreversible damage condition and the asymmetric response of tensile-compressive stress.

[0018] The historical field variable H is defined as follows:

[0019]

[0020] In the formula, The initial value of the damage driving force, Let n be the damage driving force. This is the equivalent effective stress.

[0021] Furthermore, it also includes:

[0022] S8. In ABAQUS, the UMAT subroutine is used for modeling and calculation, and the Newton-Raphson algorithm with alternating solutions is used for stable solution.

[0023] S9. Several numerical examples are used to verify the crack path and global mechanical response capability of type I and mixed fracture in two-dimensional and / or three-dimensional scenarios.

[0024] Furthermore, in S1, the volume fraction of microcracks for:

[0025]

[0026] In the formula, ω is the microcrack density parameter, and ∈=c / a is the shape ratio of the microcracks. This indicates the total number of microcracks per unit volume.

[0027] Furthermore, in S2, the elastic constitutive relation is:

[0028]

[0029] The micro-stress equilibrium equation is:

[0030]

[0031] The equations for equilibrium of small forces are:

[0032]

[0033] In the formula, σ is the macroscopic stress tensor, ρ is the material density, ε is the macroscopic strain tensor, and ψ is the macroscopic strain tensor. e Let H be the elastic strain energy of the material, H be the micro-stress related to the micro-crack density parameter, K be the micro-force related to the micro-crack density parameter, ▽ be the gradient operator, and ψ be the micro-force. f The surface energy of the crack required to form a crack surface.

[0034] Furthermore, in S3, based on Griffith theory and the variational principle of energy minimization, the specific form of the variational formula for energy minimization constructed using the equivalent transformation of microcrack distribution is as follows:

[0035] ω=Arg{inf ω∈W I(ω)}withI(ω)=∫ Ω {α(ω)+l 2 ω′ 2}dV

[0036] In the formula, the domain is W = [0, ω]. m ],ω mTo allow for the maximum value of the microcrack density parameter, I(ω) represents the sum of the microcrack density parameters in the system, ω(x) is the distribution function of the microcrack density parameter when the energy in the system reaches the minimum value, α(ω) is the microcrack geometric function, and l is a numerical parameter for adjusting the microcrack distribution width.

[0037] Furthermore, the mathematical expression for the equivalent transformation of the microcrack distribution is as follows:

[0038] g c I(ω(x))=2g c ′ω m bΓdx

[0039] In the formula, g c and g c ' represents the average energy consumed to generate a unit microcrack density parameter before and after the equivalent transformation of the microcrack distribution, b is the equivalent width determined by the microcrack geometric function and parameter l, and Γ is the macroscopic crack area.

[0040] Furthermore, in S3, the volumetric strain energy and crack surface energy are expressed as follows:

[0041]

[0042] In the formula, m(ω) is the energy degradation function. G is the fourth-order elastic stiffness tensor. c For fracture energy, The average energy consumption ratio of the unit microcrack density parameter before and after equivalent treatment.

[0043] Furthermore, in S3, the stress balance equation and the evolution rule of the microcrack density parameter are as follows:

[0044]

[0045] The corresponding boundary conditions are:

[0046]

[0047] In the formula, Δ is the Laplace operator. Let x be the displacement boundary and x be the stress boundary, respectively, and n(x) be the normal vector at point x.

[0048] Furthermore, in S5, the energy degradation function takes the following form:

[0049]

[0050] In the formula, the parameter E is the elastic modulus, f t P(ω) represents the tensile strength and is an internal function related to the material softening law.

[0051] This invention, based on Griffith theory and the variational principle of energy minimization, reconstructs a variational framework using an equivalent transformation of microcrack distribution and establishes a quantitative mapping relationship between macroscopic elastic stiffness and microcrack density parameters using a homogenization method. This model avoids the problem of length-scale parameters approaching zero and calibrates the energy degradation process, reflecting the true damage distribution and energy degradation during crack propagation. It overcomes the technical shortcomings of existing models proposed in the background art in practical applications. Attached Figure Description

[0052] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0053] Figure 1 This is a schematic diagram of the technical route of the present invention;

[0054] Figure 2 This is a schematic diagram of the research system of the present invention;

[0055] Figure 3 This is a schematic diagram of the equivalent transformation of the microcrack distribution in this invention;

[0056] Figure 4 This is the equivalent softening relationship diagram of the present invention;

[0057] Figure 5 The figure shows the numerical calculation results of this invention. Detailed Implementation

[0058] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0059] like Figure 1-5 As shown, the method in this embodiment includes the following steps:

[0060] S1. Taking the fracture process zone that objectively exists in quasi-brittle materials such as concrete and rock as a starting point, it proposes the basic assumption that macroscopic cracks are formed by the aggregation of low-scale microcracks and that microcracks that follow a specific distribution are diffused around macroscopic cracks, thus establishing a multi-scale matrix-microcrack research system.

[0061] Among them, the microcracks are coin-shaped cracks, which can be approximated as an ellipsoid, and their mathematical expression is as follows:

[0062]

[0063] In the formula, a is the radius of the microcrack surface, and c is the half-opening of the microcrack; using This represents the total number of microcracks per unit volume, and the corresponding microcrack volume fraction. It can be represented as:

[0064]

[0065] In the formula, ω is the microcrack density parameter, and ∈=c / a is the shape ratio of the microcrack.

[0066] S2. Based on the law of micro-force equilibrium and the laws of thermodynamics, the elastic constitutive relation, the micro-stress equilibrium equation and the micro-force equilibrium equation were rigorously derived within the framework of thermodynamics.

[0067] The elastic constitutive relation is as follows:

[0068]

[0069] The micro-stress equilibrium equation is:

[0070]

[0071] The equations for equilibrium of small forces are:

[0072]

[0073] In the formula, σ is the macroscopic stress tensor, ρ is the material density, ε is the macroscopic strain tensor, and ψ is the macroscopic strain tensor. e Let H be the elastic strain energy of the material, H be the micro-stress related to the micro-crack density parameter, K be the micro-force related to the micro-crack density parameter, ▽ be the gradient operator, and ψ be the micro-force. f The surface energy of the crack required to form a crack surface.

[0074] S3. Based on Griffith theory and the variational principle of energy minimization, the variational framework of microcrack evolution was reconstructed by using the equivalent transformation of microcrack distribution. The explicit expression of volumetric strain energy and crack surface energy was given, and the stress balance equation and the evolution law of microcrack density parameters were derived.

[0075] The variational formula based on energy minimization takes the following specific form:

[0076] ω=Arg{inf ω∈W I(ω)}withI(ω)=∫ Ω {α(ω)+l 2 ω′ 2}dV

[0077] In the formula, the domain is W = [0, ω]. m ],ω m To allow for the maximum value of the microcrack density parameter, I(ω) represents the sum of the microcrack density parameters in the system, ω(x) is the distribution function of the microcrack density parameter when the energy in the system reaches the minimum value, α(ω) is the microcrack geometric function, and l is a numerical parameter for adjusting the microcrack distribution width.

[0078] The mathematical expression for the equivalent transformation of microcrack distribution is:

[0079] g c I(ω(x))=2g c ′ω m bΓdx

[0080] In the formula, g c and g c ' represents the average energy consumed to generate a unit microcrack density parameter before and after the equivalent transformation of the microcrack distribution, b is the equivalent width determined by the microcrack geometric function and parameter l, and Γ is the macroscopic crack area.

[0081] The volumetric strain energy and crack surface energy are expressed as follows:

[0082]

[0083] In the formula, m(ω) is the energy degradation function. G is the fourth-order elastic stiffness tensor. c For fracture energy, The average energy consumption ratio of the unit microcrack density parameter before and after equivalent treatment.

[0084] The stress balance equation and the evolution rule of microcrack density parameters are as follows:

[0085]

[0086] The corresponding boundary conditions are:

[0087]

[0088] In the formula, is the Laplace operator.

[0089] S4. The geometric distribution function is defined, the microcrack distribution form is given, and the length scale parameter can be calibrated according to the distribution law of the fracture process zone measured by tests such as three-point bending of concrete.

[0090] Preferably, the geometric distribution function takes the following form:

[0091] α(ω)=ω

[0092] S5. The energy degradation function was defined, and the mapping relationship between macroscopic elastic stiffness and low-scale microcrack density parameters was established using the micromechanical homogenization method, thus calibrating the energy degradation process.

[0093] The energy degradation function takes the following form:

[0094]

[0095] In the formula, the parameter E is the elastic modulus, f t P(ω) represents the tensile strength and is an internal function related to the material softening law.

[0096] S6. By combining the geometric distribution function and the energy degradation function, the parameterized constitutive function of quasi-brittle materials is derived. The softening behavior of different quasi-brittle materials can be adjusted by the parameters in the equivalent constitutive function.

[0097] The parameterized constitutive function for quasi-brittle materials is:

[0098]

[0099] In the formula, For effective stress, δ n This represents the displacement increment caused by damage.

[0100] S7. The historical field variable method and the equivalent stress method were used respectively to ensure the irreversible damage condition and the tensile-compressive stress asymmetric response, and the irreversible damage condition of the model was successfully verified.

[0101] Among them, historical field variables Defined as:

[0102]

[0103] In the formula, The initial value of the damage driving force, Let n be the damage driving force at time n. This is the equivalent effective stress.

[0104] S8. The proposed multi-scale microcrack zone damage model was modeled and calculated using the UMAT subroutine in the large-scale commercial software ABAQUS, and a stable solution was obtained by using the alternating Newton-Raphson algorithm.

[0105] S9. Based on the ABAQUS post-processing module, a damage distribution cloud map is plotted, providing crack path prediction results consistent with experimental results. Force-displacement curves unaffected by scale parameters and mesh size are also provided. The prediction results are shown below. Figure 5 .

[0106] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A method for predicting damage and fracture of quasi-brittle materials using multi-scale microcrack zones, characterized in that, Includes the following steps: S1. Propose the hypothesis of low-scale microcrack zone and establish a multi-scale matrix-microcrack research system; Among them, the micro-cracks are coin-shaped cracks, and their mathematical expression is as follows: In the formula, The radius of the microcrack surface. For microcrack half-opening, These are the spatial coordinates of the local coordinate system of the microcrack; Microcrack volume fraction for: In the formula, For microcrack density parameters, The shape ratio of the microcracks. This indicates the total number of microcracks per unit volume; S2. Based on the law of micro-force equilibrium and the law of thermodynamics, the linear constitutive relation, micro-stress equilibrium equation and micro-force equilibrium equation are derived within the framework of thermodynamics. S3. The variational framework of microcrack evolution is reconstructed by equivalent transformation of microcrack distribution, and explicit expressions of volumetric strain energy and crack surface energy are given. The stress balance equation and the evolution law of microcrack density parameters are derived. The volumetric strain energy and crack surface energy are expressed as follows: In the formula, Let be the energy degradation function. It is a fourth-order elastic stiffness tensor. For fracture energy, The average energy consumption ratio of the unit microcrack density parameter before and after equivalent treatment; S4. Define the geometric distribution function as follows: , Given the microcrack density parameter, the microcrack distribution pattern is given; S5. Define the energy degradation function, establish the mapping relationship between macroscopic elastic stiffness and low-scale microcrack density parameters using the micromechanical homogenization method, and calibrate the energy degradation process; S6. Combining the geometric distribution function and the energy degradation function, the parameterized constitutive function of the quasi-brittle material is derived. The parameterized constitutive function of the quasi-brittle material is: In the formula, For effective stress, The displacement increment caused by damage The elastic modulus of the material. Let be the fracture energy, l be a numerical parameter that adjusts the width of the microcrack distribution, and b be the equivalent width determined by the microcrack geometry and parameter l. To allow for the maximum value of the microcrack density parameter, The average energy consumption ratio of the unit microcrack density parameter before and after equivalent treatment. For microcrack geometry function, An internal function related to the material softening law. The peak value of the microcrack density parameter distribution at any given time; S7 is processed using the historical field variable method and the equivalent stress method respectively to ensure the irreversible damage condition and the asymmetric response of tensile-compressive stress; Among them, historical field variables Defined as: In the formula, The initial value of the damage driving force, Let n be the damage driving force at time n. This is the equivalent effective stress; S8. In ABAQUS, the UMAT subroutine is used for modeling and calculation, and the Newton-Raphson algorithm with alternating solutions is used for stable solutions. S9. Several numerical examples are used to verify the crack path and global mechanical response capability of type I and mixed fracture in two-dimensional and / or three-dimensional scenarios.

2. The method for predicting multi-scale microcrack zones of quasi-brittle materials according to claim 1, characterized in that, In S2, the elastic constitutive relation is: The micro-stress equilibrium equation is: The equations for equilibrium of small forces are: In the formula, For macroscopic stress tensor, For material density, For macroscopic strain tensor, The elastic strain energy of the material For micro-stress related to microcrack density parameters, For microforces related to microcrack density parameters, For gradient operators, The surface energy of the crack required to form a crack surface.

3. The method for predicting damage and fracture of quasi-brittle materials according to claim 1, characterized in that, In S3, based on Griffith theory and the variational principle of energy minimization, the variational formula for energy minimization constructed using the equivalent transformation of microcrack distribution takes the following specific form: In the formula, the domain , To allow for the maximum value of the microcrack density parameter, Represents the sum of microcrack density parameters within the system. Let be the distribution function of microcrack density parameters when the energy within the system reaches its minimum value. is the geometric function of the microcracks, and l is a numerical parameter that adjusts the width of the microcrack distribution.

4. The method for predicting damage and fracture of quasi-brittle materials according to claim 3, characterized in that, The mathematical expression for the equivalent transformation of microcrack distribution is: In the formula, and , representing the average energy consumed to generate a unit microcrack density parameter before and after the equivalent transformation of the microcrack distribution, respectively; b is the equivalent width determined by the microcrack geometry function and parameter l. This represents the area of ​​the macroscopic crack.

5. The method for predicting damage and fracture of quasi-brittle materials according to claim 1, characterized in that, In S3, the stress balance equation and the evolution rule of the microcrack density parameter are as follows: The corresponding boundary conditions are: In the formula, For the Laplace operator, These are displacement boundaries and stress boundaries, respectively. Let x be the normal vector at point x.

6. The method for predicting damage and fracture of quasi-brittle materials according to claim 1, characterized in that, In S5, the energy degradation function takes the following form: In the formula, the parameter , For elastic modulus, For tensile strength, This is an internal function related to the material softening law.

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