A crack dispersion characteristic size coordinated fracture phase field method

Abstract

The application discloses a kind of crack dispersion characteristic size coordination's fracture phase field method, the method includes the following steps: step one, the theoretical framework of classic brittle fracture phase field method is established;Step two, the energy degradation function of crack dispersion characteristic size adjustable is established;Step three, the limit strength solution based on one-dimensional rod softening analysis;Step four, the adjustment of specific crack dispersion characteristic size;Step five, the finite element implementation of crack dispersion characteristic size adjustable fracture phase field method.The application introduces adjustable scaling factor s in the energy degradation function in phase field method, to make the mapping relationship between material parameters and crack dispersion characteristic size adjustable, so as to realize the coordination between crack dispersion characteristic scale and structure size, unit size;Based on finite element method, the numerical implementation is carried out to the model, compared with classic fracture mechanics phase field method, the ability of the method to obtain coordinated crack propagation path and the effectiveness of dealing with different size structure fracture problem are verified.

Description

Technical Field

[0001] This invention belongs to the field of fracture mechanics technology and relates to a fracture phase field method, specifically a fracture phase field method that allows arbitrary adjustment of the crack dispersion characteristic size to coordinate with the structural size and unit size. Background Technology

[0002] Most lightweight, high-strength materials commonly used in aerospace are brittle. During their fabrication and service, brittle materials, under complex loads and multiphysics coupling effects, often experience crack initiation and propagation, leading to significant reductions in aircraft reliability and service life. Studying the fracture mechanics of brittle materials is crucial for damage tolerance design of aircraft. Solid fracture is a highly nonlinear physical process, and the fracture of brittle materials is often a sudden, instantaneous failure. Simulating the brittle fracture process has long been a challenging engineering problem.

[0003] Classical simulation methods treat cracks as discontinuous geometric features within the solid domain, such as crack surfaces in three-dimensional space or crack curves in two-dimensional plane problems. These geometrically discontinuous cracks are also known as discrete cracks. Numerical calculation methods for fracture mechanics based on discrete cracks include extended finite element methods and remeshing methods. When simulating crack initiation and propagation, these methods require additional fracture criteria as conditions for crack initiation and propagation. However, the applied fracture criteria are often non-universal and somewhat empirical.

[0004] The phase-field method in fracture mechanics overcomes the need for additional fracture criteria in simulating crack initiation and propagation. It disperses the classical discrete crack and uses the characteristic size parameter of crack dispersion to characterize the size of the dispersed crack region, giving the crack the same dimension as the problem under study. Furthermore, the phase-field method satisfies thermodynamic consistency; its governing equations are derived thermodynamically, eliminating the need for additional fracture criteria. The simulation of crack initiation and propagation can be automatically achieved by solving the phase-field governing equations on a finite element mesh. In the phase-field method, the energy degradation function characterizes the material damage behavior. The quadratic energy degradation function in the classical brittle fracture phase-field method leads to a fixed mapping relationship between material parameters and the characteristic size of crack dispersion. This mapping relationship is obtained from the one-dimensional uniform solution of a system of partial differential equations. To accurately predict the ultimate strength of the material, the characteristic size parameter of crack dispersion in the phase-field method needs to be constant according to the mapping relationship, which limits the characteristic size of the dispersed crack region. This limitation on the characteristic size of crack dispersion leads to the limitation of the fracture phase-field method in calculating small-sized structures, where the size of the dispersed crack region is too large to identify the crack initiation location and crack propagation path. For large-sized structures, the fixed crack characteristic length often means a huge number of meshes and computational costs, making the fracture phase-field method unsuitable for large-sized structures. In addition, because the size of the dispersed crack is too small relative to the structure size, it is also difficult to identify the crack propagation path. This inconsistency between the crack dispersion characteristic scale and the element size or structure size greatly limits the application of the fracture phase-field method in structures of arbitrary size. Summary of the Invention

[0005] To address the shortcomings of existing research, this invention provides a fracture phase-field method for coordinating the characteristic size of crack dispersion. This method establishes an adjustable mapping relationship between material parameters and the characteristic size of crack dispersion, thereby achieving coordination between structures of arbitrary size and the characteristic size of crack dispersion. The coordinated crack dispersion characteristic scale ensures that the size of the dispersed crack region is neither too large nor too small, thus avoiding problems such as excessive mesh size or difficulty in identifying crack propagation paths. This invention introduces an adjustable scaling factor s into the energy degradation function in the phase-field method to make the mapping relationship between material parameters and the characteristic size of crack dispersion adjustable, thereby achieving coordination between the characteristic size of crack dispersion and the structural and element sizes. Based on the finite element method, the model is numerically implemented, and compared with the classical fracture mechanics phase-field method, verifying the method's ability to obtain coordinated crack propagation paths and its effectiveness in handling fracture problems of structures of different sizes.

[0006] The objective of this invention is achieved through the following technical solution:

[0007] A fracture phase field method with coordinated crack dispersion characteristic dimensions includes the following steps:

[0008] Step 1: Establish the theoretical framework of the classical brittle fracture phase-field method:

[0009] Based on Griffith's fracture theory, a regularized fracture free energy is introduced to establish the energy functional of the brittle fracture phase-field method. The general governing equations of the brittle fracture phase-field method are then derived using variational principles. The specific steps are as follows:

[0010] Step 11: Dispersive the energy functional of classical Griffith fracture mechanics to establish the energy functional of the brittle fracture phase-field method:

[0011]

[0012] Where ε is the total potential energy functional; u is the displacement field of the elastic body; The gradient of the fracture phase field is represented by Ω; the solution domain is Ω; and g(d) is the energy degradation function. This refers to the strain energy of the tensile portion; For the compressible strain energy; G c The critical fracture energy release rate is indicated by c0, which is the geometric coefficient; l0 is the fracture length scale; α(d) is the geometric function; b refers to the body force; t refers to the surface force; dx and ds refer to the volume element and area element, respectively.

[0013] Steps 1 and 2: Obtain the governing equations applicable to the three-dimensional brittle fracture problem using the variational method:

[0014] Mechanical equilibrium equations and stress boundary conditions:

[0015]

[0016] Crack phase field evolution equation and phase field boundary conditions:

[0017]

[0018] Step 2: Establish an energy degradation function with adjustable crack dispersion characteristic size:

[0019] Under the premise of strictly meeting the requirements of the fracture phase-field method for the degradation function, a scaling factor s is introduced into the degradation function, and an amplified energy degradation function g is designed to meet the actual needs of amplifying and reducing the characteristic size of crack dispersion. up (s,d) and the reduced energy degradation function g sd (s,d):

[0020]

[0021]

[0022] Step 3: Ultimate strength solution based on one-dimensional rod softening analysis:

[0023] The one-dimensional homogeneous solution of the fracture phase field method using the energy degradation function established in step two is derived to obtain the analytical or numerical expression of the mapping relationship between the crack dispersion characteristic size and material parameters and scaling factors. The specific steps are as follows:

[0024] Step 3.1. Considering the energy degradation function established in Step 2, the governing equations in Step 1 are degenerated into a one-dimensional case, resulting in:

[0025]

[0026]

[0027] In the formula, σ x For axial stress, ε x Here, E represents the axial strain, and E represents Young's modulus. For strain energy;

[0028] Step 3.2: Solve the above one-dimensional governing equations to obtain the analytical solution for axial stress and strain:

[0029]

[0030] Step 3: By performing a one-dimensional stability analysis on the stress solution, the critical phase field damage value d corresponding to the maximum axial stress is determined. c ∈[0,1);

[0031] Steps 3 and 4: After obtaining the critical damage value, substitute it into the analytical expression for axial stress to obtain the ultimate strength, where:

[0032] For the scaled energy degradation function, the critical damage value d c = 0 always holds true, and the expression for the ultimate strength is:

[0033]

[0034] For a reduced energy degradation function, if a non-zero positive solution exists, then the critical damage value d needs to be calculated based on first-order stability analysis. c The corresponding governing equations are:

[0035]

[0036] After obtaining the critical damage value, substituting it into the axial stress solution, the expression for the ultimate strength is:

[0037]

[0038] Step 4: Adjusting the characteristic size of specific crack dispersion:

[0039] Based on the stability analysis of the one-dimensional uniform solution, the scaling factor s value corresponding to the crack dispersion characteristic size at any scaling factor is determined:

[0040]

[0041] In the formula, d c The critical damage value is l0. * N represents the scaled crack dispersion feature size, where N is a positive number.

[0042] Step 5: Finite element implementation of the fracture phase field method with adjustable crack dispersion characteristic size:

[0043] By introducing the crack irreversibility condition through the penalty function method, an augmented Lagrangian form of the system energy is established. This enables a fracture phase-field method with adjustable crack dispersion characteristic size within the MOOSE finite element calculation environment. The specific steps are as follows:

[0044] Step 51: Based on the energy functional established in Step 1, derive the equivalent weak form of the phase-field method integral equation for brittle fracture:

[0045]

[0046] In the formula, and It is a trial function;

[0047] Step 52: To enforce the irreversibility of the crack phase field, an additional penalty function is introduced to establish the augmented Lagrangian form:

[0048]

[0049] In the formula, γ>>1 represents the Lagrange multiplier;

[0050] Step 53: Using an alternating solution scheme, solve the equilibrium equations and phase field evolution equations sequentially, i.e.:

[0051]

[0052] Step 54: Finite element implementation of the fracture phase field method:

[0053] Based on the finite element method, the finite element interpolation expressions for the displacement field and the fracture phase field are as follows:

[0054]

[0055]

[0056] Among them, {u i} represents the displacement column vector, {u i}=[u x u y ]T M is the number of nodes in a unit; N u N is the nodal displacement shape function matrix; d The shape function matrix of the fracture phase field at the node;

[0057] The corresponding gradient matrix of the shape function is:

[0058]

[0059]

[0060] Step 55: Substitute the finite element interpolation expressions into the weak form integral equations from Step 51 to obtain the following nodal force equilibrium equations and nodal fracture phase field driving force equilibrium equations:

[0061] Nodal force equilibrium equation: K uu {u I}=F u ;

[0062] The equilibrium equation of the driving force of the phase field during nodal fracture: K dd {d I}=F d ;

[0063] Among them, K uu K is the displacement-dependent stiffness matrix. dd F is the fracture phase field stiffness matrix; u For nodal forces; F d The driving force of the fracture phase field at the node; the expressions for each physical quantity are as follows:

[0064]

[0065]

[0066]

[0067]

[0068] Steps 5 and 6: Write finite element programs in the open-source finite element calculation environment MOOSE to implement the fracture phase field method with adjustable crack dispersion characteristic size within the MOOSE finite element calculation environment.

[0069] Compared with the prior art, the present invention has the following advantages:

[0070] 1. For the first time, a fracture phase field method with arbitrarily adjustable crack dispersion characteristic size was proposed, breaking the limitation of fixed mapping between material properties and crack dispersion characteristic size, so that for the fracture problem of the same material, arbitrary crack dispersion characteristic size can be used for simulation.

[0071] 2. The method of this invention can be coordinated with the structural dimensions and mesh size by adjusting the crack dispersion characteristic size. The mesh number and the resolution of the dispersed cracks are adjusted according to actual calculation requirements.

[0072] 3. The method of this invention has good applicability and stability, is easily implemented in the finite element method, and achieves adjustment of the crack dispersion characteristic size for structural phase field fracture problems under two-dimensional plane strain. This method can be naturally extended to three-dimensional problems and can be developed into a commercial program to flexibly adapt to changes in the required problem. Attached Figure Description

[0073] Figure 1 A flowchart of the fracture phase field method for coordinating the characteristic size of crack dispersion;

[0074] Figure 2 This is a schematic diagram of dispersed cracks in the fracture phase field method.

[0075] Figure 3 The graph shows the degenerate function.

[0076] Figure 4 The curve showing the mapping relationship between material parameters and crack dispersion characteristic size as a function of scaling factor s;

[0077] Figure 5 To calculate the phase field evolution diagram of a type I crack using the classical fracture phase field method;

[0078] Figure 6 Phase field evolution of type I cracks was calculated using the fracture phase field method to determine the characteristic size of crack dispersion. Detailed Implementation

[0079] The technical solution of the present invention will be further described below with reference to the accompanying drawings, but it is not limited thereto. Any modifications or equivalent substitutions to the technical solution of the present invention that do not depart from the spirit and scope of the technical solution of the present invention should be covered within the protection scope of the present invention.

[0080] This invention provides a fracture phase field method for coordinating the size of crack dispersion characteristics, such as... Figure 1 As shown, the method includes the following steps:

[0081] Step 1: Establish the theoretical framework of the classical brittle fracture phase-field method:

[0082] Based on Griffith's fracture theory, a regularized fracture free energy is introduced to establish the energy functional of the brittle fracture phase field method. The general governing equations of the brittle fracture phase field method are then derived using variational principles. The energy functional is a regularized form of the classical Griffith fracture mechanics theory, and the AT1 method is used for regularization. To distinguish the contributions of tensile and compressive loads to crack phase field evolution, strain spectrum decomposition is employed, ensuring that only the strain energy from the tensile portion contributes to crack evolution, while the strain energy from the compressive portion does not. The specific steps are as follows:

[0083] Step 11: Energy functional of the brittle fracture phase-field method:

[0084] Classical Griffith fracture theory states that linear elastic fracture mechanics problems ( Figure 2 a) The energy functional form satisfied is as follows:

[0085]

[0086] Where ε is the total potential energy functional; u is the displacement field of the elastic body; Ω refers to the crack surface (discrete crack); Ω refers to the solution domain. The solution domain excluding the discrete crack surface; The boundary of the solution domain; ψ e (ε) refers to the elastic strain energy density. For engineering strain; G c 'b' refers to the critical fracture energy release rate; 't' refers to the volume force; 't' refers to the surface force; 'dx' and 'ds' refer to the volume element and area element, respectively. It can be seen that the crack surface... As an independent variable, and affecting the solution domain of the integral equation, the minimization problem of this energy functional is difficult to solve.

[0087] Bourdin, Miehe, and others introduced an order parameter d (d = 1, indicating complete material damage; d = 0, indicating the material is intact) to characterize the degree of material damage, and proposed to disperse discrete cracks (see...). Figure 2 b) A regularized fracture energy functional was obtained:

[0088]

[0089] Where g(d) is the energy degradation function, which describes the evolution of material damage. In classical theory, g(d) = (1-d). 2 α(d) is a geometric function; this invention selects the classical geometric function α(d) = d. Other forms of geometric functions can be studied within the theoretical framework of this invention. c0 is the geometric coefficient, corresponding to the classical geometric function. l0 is the fracture length scale; This represents the gradient of the fracture phase field.

[0090] Although the crack surface can no longer withstand tensile loads, it can still withstand compressive loads. To distinguish between the degradation of tensile strain energy and compressive strain energy, a tensile-compressive energy decomposition method needs to be introduced into the strain energy component of the phase-field method, namely:

[0091]

[0092] in, For the strain energy of the tensile portion, This refers to the compression strain energy. Common methods for decomposing tension and compression energy include volume decomposition and strain spectrum decomposition, and it is foreseeable that commonly used energy decomposition methods are all within the theoretical framework of this invention. Taking strain spectrum decomposition as an example, the elastic strain energy is decomposed into:

[0093]

[0094] Where λ and μ are Lamé coefficients. trε=ε ii The trace represents the strain tensor, i.e., volumetric strain. Let be the three principal strains of the strain tensor. The angle brackets in the equation denote the unit ramp function in the forward and reverse directions. ± = (a ± |a|) / 2.

[0095] Steps 1 and 2: Thermodynamically Consistent Variational Principle

[0096] Based on the second law of thermodynamics, for a closed, isothermal thermodynamic system, the variation of the total energy functional of the system is always no greater than zero, that is:

[0097]

[0098] Applying the divergence theorem, we can obtain:

[0099]

[0100] Wherein, the stress tensor σ satisfies According to the basic principles of the variational method, for any small perturbation δu, the following holds:

[0101]

[0102] This is the mechanical equilibrium equation and stress boundary condition. Due to the irreversibility of the crack, the small perturbation δd is always not less than 0, i.e., δd≥0. Therefore, the phase field evolution satisfies:

[0103]

[0104] Simultaneously solving the mechanical equilibrium equations and the crack phase field evolution equations allows for the automatic simulation of complex fracture mechanics behaviors such as crack initiation, propagation, bifurcation, and convergence. The geometric and degenerative functions in the above formulas are undetermined; however, they can be designed under certain conditions. This invention does not require additional design of the geometric functions, but instead uses the geometric functions of the classic AT1 model, i.e., α(d) = d.

[0105] Step 2: Establish an energy degradation function with adjustable crack dispersion characteristic size:

[0106] Under the premise of strictly meeting the requirements of the fracture phase-field method for the degradation function, a scaling factor s is introduced into the degradation function, and an amplified energy degradation function g is designed to meet the actual needs of amplifying and reducing the characteristic size of crack dispersion. up (s,d) and the reduced energy degradation function g sd (s,d). The energy degradation function must strictly satisfy the mathematical constraints imposed on the degradation function by the fractured phase field method to ensure thermodynamic consistency. The specific steps are as follows:

[0107] The degradation function is a special function in the phase-field method that characterizes the degradation form of material stiffness. Generally, to ensure the correctness and convergence of the phase-field model, the degradation function should satisfy g(0) = 1, g(1) = 0, g(d) ≥ 0 ∩ g′(d) ≤ 0, g′(1) = 0, and g′(0) < 0. The classic quadratic degradation function curve is g = (1-d). 2 While meeting the above requirements, this function lacks adjustable parameters and can only generate a mapping relationship between material parameters and crack dispersion characteristic size. To achieve adjustability of the crack dispersion characteristic size, this invention introduces an additional scaling factor s into the degenerate function. To make the crack dispersion characteristic size adjustable, the designed degenerate function is as follows:

[0108]

[0109] To achieve adjustable crack dispersion feature size, the degeneracy function is designed as follows:

[0110]

[0111] The introduced scaling factor s∈(0,1) ensures that the degradation functions in both designs meet the requirements of the fracture phase-field method. When s→0, both degradation functions are consistent with the classical quadratic degradation function; when s→1, the two degradation functions can respectively increase and decrease the characteristic size of crack dispersion (see step three for specific adjustment methods). The curves of the two degradation functions are shown in... Figure 3 .

[0112] Step 3: Ultimate strength solution based on one-dimensional rod softening analysis:

[0113] Consider a one-dimensional rod under uniaxial tensile stress, assuming both the stress field and the damage phase field are uniform. Establish the corresponding set of governing equations for the fracture phase field to obtain the analytical solution for the uniform field case. Determine the limit stress value in the one-dimensional uniform solution. To obtain the limit stress in the model, perform a stability analysis on the one-dimensional uniform solution to obtain the algebraic equations for the critical phase field value and scaling factor s corresponding to the limit stress. The algebraic equations are solved using Newton's iteration method. The specific steps are as follows:

[0114] In the classical fracture phase-field method, the crack dispersion characteristic size is considered a parameter related to material properties, and the quantitative relationship between the crack dispersion characteristic size and material parameters is obtained by analyzing the analytical solution of a one-dimensional rod under tensile softening. To achieve the ability of this invention to adjust the crack dispersion characteristic size, it is first necessary to analyze the analytical solution of a one-dimensional rod under axial tension. Degrading the governing equations of the fracture phase-field method to the one-dimensional case, we obtain:

[0115]

[0116]

[0117] In the formula, σ x For axial stress, ε x Here, E represents the axial strain, and E represents Young's modulus. Let be the strain energy. Solving the above one-dimensional governing equations yields the analytical solution for axial stress and strain:

[0118]

[0119] By performing a one-dimensional stability analysis on the stress solution, the critical phase field damage value d corresponding to the maximum axial stress (i.e., ultimate strength) is determined. c ∈[0,1), that is After obtaining the critical damage value, it can be substituted into the analytical expression for axial stress to obtain the ultimate strength. In this invention, for the enlarged degradation function, the critical damage value d... c =0 always holds true, therefore the expression for the limiting strength is:

[0120]

[0121] For a reduced degradation function, the critical damage value corresponding to the ultimate strength may have a non-zero positive solution. If a non-zero positive solution exists, the critical damage value d needs to be calculated based on first-order stability analysis. c The corresponding governing equations are:

[0122]

[0123] After obtaining the critical damage value, substituting it into the axial stress solution, the expression for the ultimate strength can be obtained:

[0124]

[0125] It can be seen that regardless of the degradation function used, the expression for the ultimate strength always includes the crack dispersion characteristic size l0. If we consider Young's modulus E and critical fracture release rate G... c and ultimate strength σ c If these are considered material property parameters, then the value of l0 will also depend on these material parameters. This invention, by introducing an additional scaling factor s, makes the value of l0 adjustable. For material characteristic length, Figure 4 The curves showing the variation of the crack dispersion characteristic size with the scaling factor s are presented.

[0126] Step 4: Methods for adjusting the characteristic size of specific crack dispersion:

[0127] Based on the stability analysis of the one-dimensional uniform solution, the scaling factor s value corresponding to the crack dispersion characteristic size at any scaling factor is determined. The specific steps are as follows:

[0128] The value of the crack propagation feature size can be adjusted by modifying the scaling factor s in the degradation function. To reduce computational cost and obtain a clear crack propagation path, a crack propagation feature size that is compatible with the structure size and mesh size needs to be selected. Let's assume the value of l0 is adjusted to... N is a positive number. To adjust the specific crack dispersion characteristic size, a specific scaling factor s needs to be determined. For the case of a shrinking degradation function, the critical damage value may not be zero as s changes, therefore, the critical damage value needs to be solved separately. The first-order stability analysis equation is solved simultaneously with the equation to determine the critical damage value d. c The value of the scaling factor s is related to:

[0129]

[0130] Solving the above system of equations allows us to determine the value of the scaling factor s corresponding to a specific crack dispersion characteristic size.

[0131] Step 5: Finite element implementation of the fracture phase field method with adjustable crack dispersion characteristic size:

[0132] By introducing the penalty function method, an augmented Lagrangian form of the basic energy functional is established to realize the irreversibility of the crack phase field. To ensure the convexity of the energy functional, the solution of the fracture phase field governing equations adopts an alternating Newton iteration scheme. The specific steps are as follows:

[0133] Step 51: Equivalent Weak Form of the Integral Equation in the Fracture Phase Field Method

[0134] The functional form of the fracture phase-field method is as follows:

[0135]

[0136] Introducing trial functions and Then, the equivalent weak form is as follows:

[0137]

[0138] To enforce the irreversibility of the crack phase field, an additional penalty function is introduced, resulting in the augmented Lagrangian form:

[0139]

[0140] In the formula, γ>>1 represents the Lagrange multiplier. Due to the non-convexity of the system's energy functional, directly solving the above weak form presents a convergence problem. Therefore, an alternating solution scheme is adopted, solving the equilibrium equation and the phase field evolution equation sequentially, i.e.:

[0141]

[0142] Step 5.2: Finite Element Implementation of the Fracture Phase Field Method

[0143] This invention primarily presents the finite element discretization form of the planar problem. Based on the finite element method, the finite element interpolation expressions for the displacement field and the fracture phase field are as follows:

[0144]

[0145]

[0146] Among them, {u i} represents the displacement column vector, {u i}=[u x u y ] T M is the number of nodes in a unit; N u N is the nodal displacement shape function matrix; d Let be the shape function matrix of the nodal fracture phase field.

[0147]

[0148] N d =[N1...N i ...N N ];

[0149] The corresponding gradient matrix of the shape function is:

[0150]

[0151]

[0152] Substituting the finite element interpolation expressions into the weak form integral equations in step 5.1, we obtain the following nodal force equilibrium equations and nodal fracture phase field driving force equilibrium equations:

[0153] Nodal force equilibrium equation: K uu {u I}=F u ;

[0154] The equilibrium equation of the driving force of the phase field during nodal fracture: K dd {d I}=F d ;

[0155] Among them, K uu K is the displacement-dependent stiffness matrix. dd F is the fracture phase field stiffness matrix; u For nodal forces; F d This represents the driving force of the fracture phase field at the node. The expressions for each physical quantity are as follows:

[0156]

[0157]

[0158]

[0159]

[0160] The finite element program was written in the open-source finite element computing environment MOOSE, the variational inequality solver used PETSC, and the post-processing visualization used Paraview.

[0161] Taking a two-dimensional plane strain case as an example, calculation examples of type I cracks in three plates of small, medium, and large size are given. Material parameters are: Young's modulus E = 210 GPa, Poisson's ratio v = 0.3, and critical fracture release rate G... c =2.7 N / mm, ultimate strength σ c =4611MPa. The side lengths D of the small, medium, and large plates are calculated to be 0.1mm, 1mm, and 10mm, respectively. In the classical fracture phase-field method, to maintain the ultimate strength property unchanged, the crack dispersion characteristic size needs to be fixed at l0 = 0.01mm. Figure 5 The crack phase field evolution calculated using the classic AT1 fracture phase field method suffers from several drawbacks. Due to the fixed dispersion feature length, the crack dispersion is too high in small plates, making accurate crack location identification impossible, while in large plates, the crack dispersion is too low, resulting in excessive computation. This invention adjusts the scaling factor s introduced in this invention to adjust the crack dispersion feature size l0 while maintaining the ultimate strength. Using the crack dispersion feature size of a medium-sized plate as a reference (considered to be compatible with the structural and mesh dimensions), the crack dispersion feature size l0 of a plate with a side length D of 0.1 mm is adjusted to 0.001 mm, and the crack dispersion feature size l0 of a plate with a side length D of 1 mm is adjusted to 0.1 mm, respectively, by applying a reduced-type degradation function and a method-type degradation function. The calculated crack evolution cloud diagram is shown below. Figure 6 As shown. Comparison Figure 5 and Figure 6 The calculation results show that this invention overcomes the shortcomings of the fracture phase-field method in calculating small-sized plates, which suffers from low crack resolution and cannot accurately determine crack propagation paths. Simultaneously, for large-sized plates, the method proposed in this invention significantly reduces the number of finite element meshes (from 66790 to 21389, approximately 68%), thereby greatly reducing the computational load. The method proposed in this invention improves the feasibility of applying the fracture phase-field method to structures of arbitrary sizes, obtains size-coordinated crack propagation paths, and it is foreseeable that the phase-field method with adjustable crack dispersion characteristic size proposed in this invention is applicable to three-dimensional fracture mechanics problems.

Claims

1. A fracture phase field method for coordinating the size of crack dispersion characteristics, characterized in that... The method includes the following steps: Step 1: Establish the theoretical framework of the classical brittle fracture phase-field method: Based on Griffith's fracture theory, a regularized fracture free energy is introduced to establish the energy functional of the brittle fracture phase-field method. The general governing equations of the brittle fracture phase-field method are then derived using variational principles. The specific steps are as follows: Step 11: Dispersive the energy functional of classical Griffith fracture mechanics to establish the energy functional of the brittle fracture phase-field method: ； in, The total potential energy functional; For the displacement field of the elastic body; Represents the gradient of the fracture phase field; The solution domain; The boundary of the solution domain; It is the energy degradation function; This refers to the strain energy of the tensile portion; To compress part of the strain energy; The critical fracture energy release rate; Geometric coefficients; The fracture length is the scale. It is a geometric function; Physical strength; Finger force; and These refer to volume micro-elements and area micro-elements, respectively. Steps 1 and 2: Obtain the governing equations applicable to the three-dimensional brittle fracture problem using the variational method: Mechanical equilibrium equations and stress boundary conditions: ； Crack phase field evolution equation and phase field boundary conditions: ； Step 2: Establish an energy degradation function with adjustable crack dispersion characteristic size: Under the premise of strictly satisfying the requirements of the fracture phase-field method for the degenerate function, a scaling factor is introduced into the degenerate function. To address the practical need to amplify and reduce the size of crack dispersion features, an amplified energy degradation function is designed. and reduced energy degradation function : ； ； Step 3: Ultimate strength solution based on one-dimensional rod softening analysis: The one-dimensional homogeneous solution of the fracture phase field method using the energy degradation function established in step two is derived to obtain the analytical or numerical expression of the mapping relationship between the crack dispersion characteristic size and material parameters and scaling factors. The specific steps are as follows: Step 3.

1. Considering the energy degradation function established in Step 2, the governing equations in Step 1 are degenerated into a one-dimensional case, resulting in: ； ； In the formula, For axial stress, For axial strain, For Young's modulus, For strain energy; Step 3.2: Solve the above one-dimensional governing equations to obtain the analytical solution for axial stress and strain: ； Step 3: By performing a one-dimensional stability analysis on the stress solution, the critical phase field damage value corresponding to the maximum axial stress is determined. ; Steps 3 and 4: After obtaining the critical damage value, substitute it into the analytical expression for axial stress to obtain the ultimate strength, where: For the scaled-up energy degradation function, the critical damage value If this holds true consistently, the expression for the ultimate strength is: ； For a reduced energy degradation function, if a non-zero positive solution exists, the critical damage value needs to be calculated based on first-order stability analysis. The corresponding governing equations are: ； After obtaining the critical damage value, substituting it into the axial stress solution, the expression for the ultimate strength is: ； Step 4: Adjusting the characteristic size of specific crack dispersion: Based on the stability analysis of the one-dimensional uniform solution, the scaling factor s value corresponding to the crack dispersion characteristic size at any scaling factor is determined: ； In the formula, d c The critical damage value is l0. * This refers to the scaled-down crack dispersion feature size, where N is a positive number; Step 5: Finite element implementation of the fracture phase field method with adjustable crack dispersion characteristic size: By introducing the crack irreversibility condition through the penalty function method, an augmented Lagrangian form of the system energy is established. This enables a fracture phase-field method with adjustable crack dispersion characteristic size within the MOOSE finite element calculation environment. The specific steps are as follows: Step 51: Based on the energy functional established in Step 1, derive the equivalent weak form of the phase-field method integral equation for brittle fracture: ； In the formula, and It is a trial function; Step 52: To enforce the irreversibility of the crack phase field, an additional penalty function is introduced to establish the augmented Lagrangian form: ； In the formula, For Lagrange multipliers; Step 53: Using an alternating solution scheme, solve the equilibrium equations and phase field evolution equations sequentially, i.e.: ； Step 54: Finite element implementation of the fracture phase field method: Based on the finite element method, the finite element interpolation expressions for the displacement field and the fracture phase field are as follows: ； ； in, Represents the displacement column vector. M represents the number of nodes in a single unit. The matrix represents the shape functions of the nodal displacements; The shape function matrix of the fracture phase field at the node; The corresponding gradient matrix of the shape function is: ； ； Step 55: Substitute the finite element interpolation expressions into the weak form integral equations from Step 51 to obtain the following nodal force equilibrium equations and nodal fracture phase field driving force equilibrium equations: Nodal force equilibrium equations: ; Equilibrium equations for driving forces in the phase field at nodal fracture: ; in, This is the displacement-dependent stiffness matrix; Here is the stiffness matrix of the fracture phase field; For nodal forces; The driving force of the fracture phase field at the node; Steps 5 and 6: Write finite element programs in the open-source finite element calculation environment MOOSE to implement the fracture phase field method with adjustable crack dispersion characteristic size within the MOOSE finite element calculation environment.

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