Multi-scale micro-fissure zone method for predicting damage and fracture of quasi-brittle material

Abstract

The invention provides a multi-scale micro-fissure zone method for predicting damage and fracture of a quasi-brittle material. Comprising the following steps: proposing a microfissure zone hypothesis with distribution characteristics according to a fracture process zone concept, reconstructing an energy-minimization variational equation by adopting microfissure distribution equivalent transformation, and deducing a stress balance equation and a microfissure density parameter evolution rule under a thermodynamic framework. And associating macroscopic elastic rigidity and microfracture density parameters based on a homogenization method, forming a material softening relationship by an equivalent constitutive function, processing irreversible damage and asymmetric tension-compression stress response, and calculating and verifying representative example numerical values. According to the method, the phase field model and the micro-mechanical damage model are combined, the value contradiction that the length scale parameter approaches 0 can be overcome, the energy degradation process is effectively calibrated, the real damage distribution characteristic and the energy degradation behavior in the crack propagation process are reflected, and the accuracy of the crack propagation is improved. The method has important significance and remarkable progress in predicting the damage and cracking phenomena of the engineering structure.

Description

Technical Field

[0001] The present invention relates to the field of computational mechanics, and in particular to a multi-scale microcrack zone method for predicting damage and fracture of quasi-brittle materials. Background Art

[0002] Accurately predicting the damage and fracture behavior of quasi-brittle materials such as concrete and rock is crucial for the safe design of engineering structures and the prevention of catastrophic failures. However, damage mechanics models based on damage mechanics theory often do not consider non-local effects and lack internal length scale parameters, which leads to mesh sensitivity problems and distorted numerical results. Cohesive element method and extended finite element method based on fracture mechanics theory often require cracks to propagate between meshes, so there are mesh arrangement sensitivity problems. In addition, these methods often require explicit setting of crack propagation paths, which significantly increases the computational complexity (especially in the case of complex crack paths) and easily leads to non-convergence problems.

[0003] In comparison, the phase field method has shown significant superiority in both theoretical basis and numerical implementation. It can effectively capture the evolution of damage while spontaneously tracking the cracking path, and can predict complex cracking behaviors such as crack bifurcation and merging. However, the phase field fracture model still has some shortcomings. The first is the contradiction in the value of the length scale parameter. Specifically, the variational framework of the phase field model requires that the length scale parameter approaches 0 to ensure that the diffuse crack area is equal to the discrete crack area, while in practical applications, the length scale parameter needs to reflect the actual damage diffusion width. Another shortcoming is that the energy degradation process in the phase field model is very arbitrary and lacks an effective calibration method. This results in differences between the crack propagation process and the actual situation, even though the predicted final cracking path is the same as the experiment. Summary of the invention

[0004] The above problems can be attributed to the deficiencies in the variational framework of the phase field model and the lack of low-scale physical mechanisms. In order to solve the above problems in the prior art, the present invention provides a multi-scale microcrack zone method for predicting damage and fracture of quasi-brittle materials, comprising the following steps:

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

[0006] Among them, the microcracks are coin-shaped cracks, and the mathematical expression is:

[0007]

[0008] Where a is the radius of the microcrack surface, c is the half-opening of the microcrack, and x is 1 ,x 2 ,x 3 are the spatial coordinates of the local coordinate system of microcracks.

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

[0010] S3. The variational framework of microcrack evolution is reconstructed by using equivalent transformation of microcrack distribution, explicit expressions of volume strain energy and crack surface energy are given, and the stress equilibrium equation and the evolution law of microcrack density parameters 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. Clarify the energy degradation function, use the micromechanical homogenization method to establish the mapping relationship between macro elastic stiffness and low-scale microcrack density parameters, and calibrate the energy degradation process.

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

[0014]

[0015] In the formula, is the effective stress, δ n is the displacement increment caused by damage, E is the elastic modulus of the material, G c is the fracture energy, l is the numerical parameter for adjusting the width of microcrack distribution, b is the equivalent width determined by the microcrack geometry function and parameter l, ω m is the allowable microcrack density parameter, i.e., the maximum value of the microcrack density parameter, η is the average energy consumption ratio of generating unit microcrack density parameter before and after equivalent treatment, α(ω) is the microcrack geometry function, φ(ω) is the internal function related to the material softening law, ω * is the peak value of the microcrack density parameter distribution at any time.

[0016] Furthermore, it also includes:

[0017] S7, the historical field variable method and the equivalent stress method are used to ensure the irreversible damage condition and the asymmetric response of tension-compression stress.

[0018] Among them, the historical field variable H is defined as:

[0019]

[0020] In the formula, is the initial value of the damage driving force, is the damage driving force at time n, is the equivalent effective stress.

[0021] Furthermore, it also includes:

[0022] S8. Use the UMAT subroutine in ABAQUS for modeling and calculation, and use the alternating solution Newton-Raphson algorithm for stable solution.

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

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

[0025]

[0026] Where ω is the microcrack density parameter, ∈ = c / a is the shape ratio of the microcracks, Represents the total number of microcracks per unit volume.

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

[0028]

[0029] The microstress equilibrium equation is:

[0030]

[0031] The micro force balance equation is:

[0032]

[0033] Where σ is the macroscopic stress tensor, ρ is the material density, ε is the macroscopic strain tensor, and ψ e is the elastic strain energy of the material, H is the microstress related to the microcrack density parameter, K is the microforce related to the microcrack density parameter, ▽ is the gradient operator, ψ f The crack surface energy required to form a crack surface.

[0034] Furthermore, in S3, according to Griffith theory and the variational principle of energy minimization, the variational formula for energy minimization constructed by 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 W=[0,ω m ],ω mis the maximum value of the allowed microcrack density parameter, i.e., 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 geometry function, and l is the numerical parameter for adjusting the microcrack distribution width.

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

[0038] g c I(ω(x))=2g c ′ω m b Γdx

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

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

[0041]

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

[0043] Furthermore, in S3, the stress balance equation and the evolution law of microcrack density parameters are:

[0044]

[0045] The corresponding boundary conditions are:

[0046]

[0047] Where Δ is the Laplace operator, are the displacement boundary and stress boundary respectively, and n(x) is the normal vector at point x.

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

[0049]

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

[0051] Based on Griffith theory and variational principle of energy minimization, the present invention reconstructs the variational framework by using equivalent transformation of microcrack distribution, and establishes the quantitative mapping relationship between macro elastic stiffness and microcrack density parameters by homogenization method. The model can avoid the problem of length scale parameter approaching 0 and calibrate the energy degradation process, and can reflect the real damage distribution and energy degradation in the crack propagation process, overcoming the technical defects of the existing models proposed in the background technology in actual use. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

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

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

[0055] Figure 3 It is a schematic diagram of equivalent transformation of microcrack distribution of the present invention;

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

[0057] Figure 5 This is a diagram of numerical calculation results of the present invention. DETAILED DESCRIPTION

[0058] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in 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.

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

[0060] S1. Taking the fracture process zone objectively existing in quasi-brittle materials such as concrete and rock as the starting point, the basic assumption that macro cracks are formed by the aggregation of low-scale micro cracks and that micro cracks with specific distribution are diffused around macro cracks is proposed, thus establishing a multi-scale matrix-micro crack research system.

[0061] Among them, the shape of the microcrack is a coin-shaped crack, which can be approximated as an ellipsoid. The mathematical expression is:

[0062]

[0063] In the formula, a is the radius of the microcrack surface, c is the half opening of the microcrack; Represents the total number of microcracks per unit volume, and the corresponding microcrack volume fraction It can be expressed as:

[0064]

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

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

[0067] The elastic constitutive relation is:

[0068]

[0069] The microstress equilibrium equation is:

[0070]

[0071] The micro force balance equation is:

[0072]

[0073] Where σ is the macroscopic stress tensor, ρ is the material density, ε is the macroscopic strain tensor, and ψ e is the elastic strain energy of the material, H is the microstress related to the microcrack density parameter, K is the microforce related to the microcrack density parameter, ▽ is the gradient operator, ψ f The crack surface energy required to form a crack surface.

[0074] S3. According to Griffith theory and the variational principle of energy minimization, the variational framework of microcrack evolution is reconstructed by adopting the equivalent transformation of microcrack distribution, the explicit expression of volume strain energy and crack surface energy is given, and the stress equilibrium equation and the evolution law of microcrack density parameters are derived.

[0075] Among them, the specific form of the variational formula based on energy minimization is as follows:

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

[0077] In the formula, the domain W=[0,ω m ],ω m is the maximum value of the allowed microcrack density parameter, i.e., 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 geometry function, and l is the numerical parameter for adjusting the microcrack distribution width.

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

[0079] g c I(ω(x))=2g c ′ω m b Γdx

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

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

[0082]

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

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

[0085]

[0086] The corresponding boundary conditions are:

[0087]

[0088] Where is the Laplace operator.

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

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

[0091] α(ω)=ω

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

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

[0094]

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

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

[0097] Among them, the parameterized constitutive function of quasi-brittle material is:

[0098]

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

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

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

[0102]

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

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

[0105] S9. Based on the ABAQUS post-processing module, a damage distribution cloud map is drawn to give a crack path prediction result consistent with the test results, and a force-displacement curve that is not affected by scale parameters and grid size is provided. The prediction results are shown in Figure 5 .

[0106] Obviously, those skilled in the art can make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if these modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include these modifications and variations.

Claims

1. A multi-scale microcrack zone method for predicting damage and fracture of quasi-brittle materials, characterized in that: The following steps are involved: S1. Propose a low-scale microcrack zone hypothesis and establish a multi-scale matrix-microcrack research system; Among them, the microcracks are coin-shaped cracks, and the mathematical expression is: Where a is the radius of the microcrack surface, c is the half-opening of the microcrack, and x1, x2, and x3 are the spatial coordinates of the local coordinate system of the microcrack respectively; S2. Based on the micro-force balance law and the laws of thermodynamics, the linear constitutive relations, micro-stress balance equations and micro-force balance equations are deduced in the framework of thermodynamics; S3. Use the equivalent transformation of microcrack distribution to reconstruct the variational framework of microcrack evolution, give the explicit expression of volume strain energy and crack surface energy, and derive the stress equilibrium equation and the evolution law of microcrack density parameters; S4. Define the geometric distribution function as α(ω)=ω, where ω is the microcrack density parameter and give the microcrack distribution form; S5. Define the energy degradation function, use the micromechanical homogenization method to establish the mapping relationship between macro elastic stiffness and low-scale microcrack density parameters, and calibrate the energy degradation process; S6. Combining the geometric distribution function with the energy degradation function, the parametric constitutive function of the quasi-brittle material is derived. The parametric constitutive function of the quasi-brittle material is: In the formula, is the effective stress, δ n is the displacement increment caused by damage, E is the elastic modulus of the material, G c is the fracture energy, l is the numerical parameter for adjusting the width of microcrack distribution, b is the equivalent width determined by the microcrack geometry function and parameter l, ω m is the allowable microcrack density parameter, i.e., the maximum value of the microcrack density parameter, η is the average energy consumption ratio of generating unit microcrack density parameter before and after equivalent treatment, α(ω) is the microcrack geometry function, φ(ω) is the internal function related to the material softening law, ω * is the peak value of the microcrack density parameter distribution at any time.

2. The multi-scale microcrack zone method for predicting damage and fracture of quasi-brittle materials according to claim 1, characterized in that: Also includes: S7, the historical field variable method and the equivalent stress method were used to ensure the irreversible damage condition and the asymmetric response of tension-compression stress; Among them, historical field variables Defined as: In the formula, is the initial value of the damage driving force, is the damage driving force at time n, is the equivalent effective stress.

3. The multi-scale microcrack zone method for predicting damage and fracture of quasi-brittle materials according to claim 2, characterized in that: Also includes: S8. Use the UMAT subroutine in ABAQUS for modeling and calculation, and use the alternating solution Newton-Raphson algorithm for stable solution; S9. Several examples are used to verify the crack paths and global mechanical response capabilities of type I and mixed fractures in two-dimensional and / or three-dimensional scenarios.

4. The multi-scale microcrack zone method for predicting damage and fracture of quasi-brittle materials according to claim 1, characterized in that: In S1, the volume fraction of microcracks for: Where ω is the microcrack density parameter, ∈ = c / a is the shape ratio of the microcracks, Represents the total number of microcracks per unit volume.

5. The multi-scale microcrack zone method for predicting damage and fracture of quasi-brittle materials according to claim 1, characterized in that: In S2, the elastic constitutive relation is: The microstress equilibrium equation is: The micro force balance equation is: Where σ is the macroscopic stress tensor, ρ is the material density, ε is the macroscopic strain tensor, and ψ e is the elastic strain energy of the material, H is the microstress related to the microcrack density parameter, K is the microforce related to the microcrack density parameter, ▽ is the gradient operator, ψ f The crack surface energy required to form a crack surface.

6. The multi-scale microcrack zone method for predicting damage and fracture of quasi-brittle materials according to claim 1, characterized in that: In S3, according to Griffith theory and the variational principle of energy minimization, the variational formula for energy minimization constructed by equivalent transformation of microcrack distribution is as follows: ω=Arg{inf ω∈W I(ω)}withI(ω)=∫ Ω {a(ω)+l 2 oh' 2 }dV Where, the domain W=[0,ω m ],ω m is the maximum value of the allowed microcrack density parameter, i.e., 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 geometry function, and l is the numerical parameter for adjusting the microcrack distribution width.

7. The multi-scale microcrack zone method for predicting damage and fracture of quasi-brittle materials according to claim 6, characterized in that: The mathematical expression of the equivalent transformation of microcrack distribution is: g c I(ω(x))=2g c Oh m bΓdx In the formula, g c and g c ′ represents the average energy consumed to generate unit microcrack density parameter before and after the equivalent transformation of microcrack distribution, b is the equivalent width determined by the microcrack geometry function and parameter l, and Γ is the macro crack area.

8. The multi-scale microcrack zone method for predicting damage and fracture of quasi-brittle materials according to claim 6, characterized in that: In S3, the volume strain energy and crack surface energy are expressed as follows: Where m(ω) is the energy degradation function, is the fourth-order elastic stiffness tensor, G c is the fracture energy, The average energy consumption ratio of unit microcrack density parameters before and after equivalent treatment is generated.

9. The multi-scale microcrack zone method for predicting damage and fracture of quasi-brittle materials according to claim 1, characterized in that: In S3, the stress equilibrium equation and the evolution law of microcrack density parameters are: The corresponding boundary conditions are: Where Δ is the Laplace operator, are the displacement boundary and stress boundary respectively, and n(x) is the normal vector at point x.

10. The multi-scale microcrack zone 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 parameters E is the elastic modulus, f t is the tensile strength, and P(ω) is the internal function related to the softening law of the material.

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