Optimization method for inhibiting local gradient damage model crack zone divergence
By modifying the damage parameters representing residual strength in the damage evolution equation into an exponential function and incorporating it into the local gradient damage model, the problem of fracture band divergence in the local gradient damage model is solved, which significantly improves the prediction accuracy of the damage damage process.
Patent Information
- Application Number
- CN202411851802.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-12-16
- Publication Date
- 2025-04-15
AI Technical Summary
When the local gradient damage model simulates the damage and failure process of quasi-brittle materials, the resulting crack zone has a large divergence and cannot accurately reflect the local damage characteristics of the concrete structure.
The damage parameters representing residual strength in the damage evolution equation are modified from a constant to an exponential function, and the damage evolution equation representing residual strength based on the exponential function is included in the local gradient damage model to obtain a new control equation.
The divergence of the crack zone is effectively avoided, greatly improving the accuracy of the local gradient damage model to predict damage and damage damage of quasi-brittle materials.
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Abstract
Description
Technical Field
[0001] The present invention belongs to the technical field of damage mechanics simulation, and in particular relates to an optimization method for suppressing crack zone divergence in a local gradient damage model simulation process of quasi-brittle material damage and destruction. Background Art
[0002] In the past two decades, my country has been in a stage of rapid development, and infrastructure such as high-speed railways, viaducts and high-rise buildings have been built on a large scale. Quasi-brittle materials such as concrete have played a vital role in this development process, but they are damaged and destroyed during use, and even fractures lead to structural collapse and other accidents. Therefore, it is particularly important to study the damage and failure process of quasi-brittle materials such as concrete. Developing a numerical model that can accurately predict the damage and failure process of quasi-brittle materials such as concrete has become the key to studying this problem.
[0003] Although the traditional gradient damage model can overcome the grid dependency problem, there is still a phenomenon of false damage growth. In order to solve the problem of false damage growth, many researchers have proposed corresponding solutions. One of the better methods is called the local gradient damage model. This method adopts a non-local enhancement theory in which the interaction domain gradually decreases as the damage increases, which reduces the phenomenon of false damage growth in the traditional gradient damage model to a certain extent. However, the crack zone obtained by the local gradient damage model in the analysis of the damage and failure process of quasi-brittle materials such as concrete still has a large divergence, which is inconsistent with the typical local failure characteristics of concrete structure cracks. The failure characteristics of quasi-brittle materials such as concrete are structural damage caused by highly localized non-uniform deformation in a narrow local area, and at the same time, the elastic unloading state is entered outside the local area. Therefore, the crack zone with large divergence obtained by the existing model is inaccurate. Summary of the invention
[0004] In view of the technical problem of the divergence of the crack band in the local gradient damage model, the present invention provides an optimization method for suppressing the divergence of the crack band in the local gradient damage model. The present invention modifies the damage parameter representing the residual strength in the damage evolution equation from a constant to an exponential function. This method can effectively avoid the divergence of the crack band. The local gradient damage model based on the new damage evolution equation can effectively avoid the divergence of the crack band in the process of analyzing the damage and failure of quasi-brittle materials, greatly improving the prediction ability of the crack trajectory.
[0005] The technical solution of the present invention is as follows:
[0006] An optimization method for suppressing the divergence of crack bands in a local gradient damage model, characterized by comprising the following steps:
[0007] In step 1), the damage parameter α representing the residual strength in the damage evolution equation is set to an exponential function, which is:
[0008] α=1-γexp[-λ(κ-κ 0 )]
[0009] In the formula, α is the damage parameter representing the residual strength in the damage evolution equation, κ is the historical parameter, and 0 is the damage initiation threshold, γ and λ are damage parameters.
[0010] In step 2), the damage evolution equation based on the exponential function to represent the residual strength is incorporated into the local gradient damage model to obtain the optimized damage evolution control equation.
[0011] Furthermore, the specific steps in step (2) are: the damage evolution equation of the residual strength expressed by an exponential function is:
[0012]
[0013] Where β is the damage parameter representing the softening rate;
[0014] The derivative of the damage evolution equation of residual strength expressed by exponential function with respect to the historical parameter κ is:
[0015]
[0016] The optimized damage evolution control equation includes the following two parts, one of which is the standard constitutive relation:
[0017] σ=(1-D)C:ε
[0018] Among them, σ is the stress tensor, ε is the strain tensor, and C is the fourth-order elastic tensor;
[0019] The other part is the microscopic force balance equation that characterizes the local gradient enhancement:
[0020]
[0021] in, and are the morphological variables describing the interaction between macro and micro levels and between micro and micro levels, e is the equivalent strain, g is the interaction relationship, and l is the length scale parameter.
[0022] Furthermore, it also includes step 3): introducing the optimized control equation obtained in step 2) into the finite element numerical framework or MATLAB software to obtain a local gradient damage model based on the optimized damage evolution equation.
[0023] Furthermore, the optimized control equations are introduced into the finite element numerical framework by discretizing the interpolation function. Specifically, the basic variables u and Discretize using interpolation functions,
[0024] u=N u a u
[0025]
[0026] Among them, a u and a e is the node freedom matrix, N u is the quadratic interpolation function, N e is a linear interpolation function;
[0027] The weak form of the differential governing equation is obtained as follows:
[0028]
[0029] Among them, B u and B e is the gradient operator matrix, and are the stress conjugates of their respective motion fields, dν and da are the area elements, and t is the surface traction of the element;
[0030] The final simplified numerical framework of plane strain is:
[0031]
[0032] Among them, K uu , K ue , K eu and K ee are the components of the element stiffness matrix, and the components of the force vector are:
[0033]
[0034] The specific form of the element stiffness matrix is:
[0035]
[0036] Furthermore, the damage initiation threshold κ 0 =0.00017, softening parameters β=800, γ=0.4, λ=30.
[0037] The local gradient damage model cannot control the damage energy well within a narrow crack band when simulating the damage and failure process of quasi-brittle materials, causing it to diffuse in large quantities to adjacent areas and causing false damage. In order to avoid the crack band divergence phenomenon that occurs when the local gradient damage model simulates the damage and failure process of quasi-brittle materials, the present invention modifies the damage parameter representing the residual strength in the damage evolution equation from a constant to an exponential function. The damage evolution equation based on the exponential function to represent the residual strength is incorporated into the local gradient damage model to obtain a new control equation. The control equation is discretized by an interpolation function and introduced into the finite element numerical framework, and is developed and implemented using the user unit subroutine UEL (user-defined element) in the finite element software ABAQUS to improve the computational efficiency and application scope of the local gradient damage model. The present invention avoids the divergence of the crack band and greatly improves the accuracy of the local gradient damage model in predicting the damage and failure process of quasi-brittle materials.
[0038] Compared with the prior art, the present invention has the following advantages:
[0039] The present invention solves the shortcomings of the original technology, that is, the existing local gradient damage model cannot control the damage energy well within a narrow crack zone when simulating the damage and destruction process of quasi-brittle materials, causing it to diffuse to adjacent areas in large quantities and cause false expansion, that is, the problem of crack zone divergence mentioned in the present invention. The present invention modifies the damage parameter representing the residual strength in the damage evolution equation from a constant to an exponential function. This method can increase the value of the softening parameter while ensuring the softening curve of the quasi-brittle material. A larger softening parameter can effectively avoid the divergence of the crack zone, greatly improving the accuracy of the local gradient damage model in predicting the damage and destruction process of quasi-brittle materials. BRIEF DESCRIPTION OF THE DRAWINGS
[0040] Figure 1 Schematic diagram of the geometric dimensions and boundary conditions of the three-point bending beam.
[0041] Figure 2 Schematic diagram of mesh division for three-point bending beam.
[0042] Figure 3 The load and opening displacement curve of the three-point bending beam.
[0043] Figure 4 The original model and the fracture zones calculated by the present invention, where (a) is the experimental result, (b) is the original model, and (c) is the optimized model of the present invention. DETAILED DESCRIPTION
[0044] The present invention will be further described below in conjunction with the accompanying drawings.
[0045] The present invention modifies the damage parameter representing the residual strength in the damage evolution equation from a constant to an exponential function. This method can increase the value of the softening parameter while ensuring the softening curve of the quasi-brittle material. A larger softening parameter can effectively avoid the divergence of the crack zone, greatly improving the accuracy of the local gradient damage model in predicting the damage and failure process of quasi-brittle materials. The specific process includes the following steps:
[0046] 1) The damage parameter representing the residual strength in the damage evolution equation is changed from a constant to an exponential function. As the damage parameter representing the residual strength changes from a constant to a variable, the damage parameter representing the softening rate can be increased, so that the damage parameter representing the softening rate no longer needs to be kept at a small value to ensure the quasi-brittle softening curve. The increase in the damage parameter representing the softening rate can effectively avoid the false expansion of the damage area caused by the divergence of the crack zone;
[0047] 2) The damage evolution equation based on the exponential function to represent the residual strength is incorporated into the local gradient damage model to derive a new control equation;
[0048] 3) The local gradient damage model based on the new damage evolution equation is introduced into the finite element numerical framework by discretizing the control equation obtained in step 2) through the interpolation function, and the application of the local gradient damage model in the finite element software ABAQUS is realized through the user unit subroutine UEL (user-definedelement) to improve the computational efficiency and application scope of the local gradient damage model.
[0049] Specifically, in step 1), the damage parameter representing the residual strength in the damage evolution equation is modified from a constant to an exponential function, and the residual strength represented by the exponential function is specifically as follows:
[0050] α=1-γexp[-λ(κ-κ 0 )]
[0051] In the formula, α is the damage parameter representing the residual strength in the damage evolution equation, κ is the historical parameter, and 0 is the damage initiation threshold, γ and λ are damage parameters.
[0052] In step 2), the damage evolution equation based on the exponential function to represent the residual strength is incorporated into the local gradient damage model. The original damage evolution equation of the existing local gradient damage model is:
[0053]
[0054] The damage evolution equation of residual strength expressed by exponential function is:
[0055]
[0056] Where β is the damage parameter representing the softening rate.
[0057] The derivative of the damage evolution equation of residual strength expressed by exponential function with respect to the historical parameter κ is:
[0058]
[0059] The present invention uses the standard free energy density:
[0060]
[0061] Among them, ε is the strain tensor, C is the fourth-order elastic tensor, h is the coupling modulus, and are the morphological variables describing the interaction between macro and micro and between micro and micro levels, e is the equivalent strain, and l is the length scale parameter. The interaction equation is in the following form:
[0062]
[0063] Among them, η and R are interaction parameters.
[0064] The present invention mainly adopts the modified von Mises equivalent strain to characterize the macroscopic deformation, which is in the form of:
[0065]
[0066] Where k is the ratio of the compressive strength to the tensile strength of the material, ν is the Poisson’s ratio, and I 1 and J 2 are the first and second invariants of the deviatoric strain tensor, respectively.
[0067] The final control equation consists of the following two parts, one of which is the standard constitutive relationship:
[0068] σ=(1-D)C:ε
[0069] Here, σ is the stress tensor.
[0070] The other part is the microscopic force balance equation that characterizes the local gradient enhancement:
[0071]
[0072] In step 3), the local gradient damage model based on the new damage evolution equation is introduced into the finite element numerical framework by discretizing the control equation obtained in step 2) through the interpolation function. The basic variables u and Discretize using interpolation functions
[0073] u=N u a u
[0074]
[0075] Among them, a u and a e is the node freedom matrix, N u is the quadratic interpolation function, N e is the linear interpolation function.
[0076] The weak form of the differential governing equation is obtained as follows:
[0077]
[0078]
[0079] Among them, B u and B e is the gradient operator matrix, and are the stress conjugates of their respective motion fields, dν and da are the area elements, and t is the surface traction of the element;
[0080] The final simplified numerical framework of plane strain is:
[0081]
[0082] Among them, K uu , K ue , K eu and K ee are the components of the element stiffness matrix, and the components of the force vector are:
[0083]
[0084] The element stiffness matrix is:
[0085]
[0086] The superior performance of the new damage evolution equation obtained in the present invention is verified by a three-point bending beam composite fracture test. The geometric dimensions and boundary conditions of the three-point bending beam are consistent with those of Gálvez et al. (Gálvez J, Elices M, Guinea G, Planas J. Mixed mode fracture of concrete under proportional and nonproportional loading [J]. International Journal of Fracture, 1998, 94 (3): 267-284.), such as Figure 1 shown. Figure 1In the above equation, a represents the eccentricity of the loading point, which is 150 mm; b represents the width of the prefabricated opening, which is 2 mm; c represents the height of the prefabricated opening, which is 75 mm; d represents the height of the beam, which is 150 mm; j and f represent the distances from the left and right support points to the midpoint of the beam, which are 75 mm and 300 mm, respectively; the total length of the beam is 675 mm; the thickness is 50 mm; and u represents the displacement load. The eccentric load and the eccentric prefabricated opening will lead to the formation of a bending crack trajectory that originates from the top of the prefabricated opening and gradually deviates toward the loading point. The material parameters used in the calculation are: elastic modulus E = 38 GPa, Poisson's ratio ν = 0.2. The model parameters required for the original damage evolution equation are: damage initiation threshold κ 0 =0.0001, softening parameter β = 50. The model parameters required for the optimized damage evolution equation are: damage initiation threshold κ 0 = 0.00017, softening parameters β = 800, γ = 0.4, λ = 30. The length scale used in both damage evolution equations is l = 2 mm. The grid division method used in the calculation is as follows: Figure 2 As shown. In order to improve the calculation accuracy, the grid is finely divided in the area where cracks may appear, with a size of 0.5mm. In order to improve the calculation efficiency, a larger size of 10mm is selected in other areas.
[0087] The calculated structural response is as follows: Figure 3 As shown in the figure, it can be seen that the optimized damage evolution equation can obtain a structural response similar to the original damage evolution equation and consistent with the test results. This shows that the optimized damage evolution equation is more accurate. Figure 4 The crack zone of the three-point bending composite fracture obtained by experiment and calculation. As can be seen from the figure, the crack trajectories obtained by the two damage evolution equations are consistent with the experimental results, but the new damage evolution equation obviously overcomes the divergence of the crack zone in the original damage evolution equation, and obtains a sharp crack that is closer to the experimental results. Especially at the bend of the crack trajectory where the shear effect is strong, the divergence of the crack zone is well controlled. This result fully demonstrates the superiority of the new damage evolution equation proposed in the present invention, which can well overcome the divergence of the crack zone in the original model, so that the local gradient damage model can more accurately analyze the damage and failure process of quasi-brittle materials.
Claims
1. An optimization method for suppressing the divergence of crack bands in a local gradient damage model, characterized in that: The following steps are involved: In step 1), the damage parameter α representing the residual strength in the damage evolution equation is set to an exponential function, which is: α=1-γexp[-λ(κ-κ0)] In the formula, α is the damage parameter representing the residual strength in the damage evolution equation, κ is the historical parameter, κ0 is the damage initiation threshold, γ and λ are damage parameters; In step 2), the damage evolution equation based on the exponential function to represent the residual strength is incorporated into the local gradient damage model to obtain the optimized damage evolution control equation.
2. The optimization method for suppressing the divergence of crack bands in a local gradient damage model according to claim 1, characterized in that: The specific steps in step (2) are: the damage evolution equation of residual strength expressed by exponential function is: Where β is the damage parameter representing the softening rate; The derivative of the damage evolution equation of residual strength expressed by exponential function with respect to the historical parameter κ is: The optimized damage evolution control equation includes the following two parts, one of which is the standard constitutive relation: σ=(1-D)C:ε Among them, σ is the stress tensor, ε is the strain tensor, and C is the fourth-order elastic tensor; The other part is the microscopic force balance equation that characterizes the local gradient enhancement: in, and are the morphological variables describing the interaction between macro and micro levels and between micro and micro levels, e is the equivalent strain, g is the interaction relationship, and l is the length scale parameter.
3. The optimization method for suppressing the divergence of crack bands in a local gradient damage model according to any one of claims 1 to 2, characterized in that: The method also includes step 3): introducing the optimized control equation obtained in step 2) into finite element software ABAQUS or MATLAB software to obtain a local gradient damage model based on the optimized damage evolution equation.
4. The optimization method for suppressing the divergence of crack bands in a local gradient damage model according to claim 3 is characterized in that: The optimized control equations are introduced into the finite element software ABAQUS through the discretization of interpolation functions. Specifically, the basic variables u and Discretize using interpolation functions, u=N u a u Among them, a u and a e is the node degree of freedom matrix, N u is the quadratic interpolation function, N e is the linear interpolation function; The weak form of the differential governing equation is obtained as follows: Among them, B u and B e is the gradient operator matrix, and are the stress conjugates of their respective motion fields, dν and da are the area elements, and t is the surface traction of the element; The final simplified numerical framework of plane strain is: Among them, K uu , K ue , K eu and K ee are the components of the element stiffness matrix, and the components of the force vector are: The specific form of the element stiffness matrix is:
5. The optimization method for suppressing the divergence of crack bands in a local gradient damage model according to claim 4, characterized in that: The method is developed and implemented using the user element subroutine UEL in the finite element software ABAQUS.
6. The optimization method for suppressing the divergence of crack bands in a local gradient damage model according to claim 5, characterized in that: The damage initiation threshold κ0=0.00017, the softening parameters β=800, γ=0.4, and λ=30.