Plastic return and implementation method suitable for epoxy resin parabola yield model

Abstract

The invention discloses a plasticity return and implementation method suitable for an epoxy resin parabola yield model, and relates to the field of mechanical property analysis of epoxy resin-based composites.The plasticity return and implementation method comprises the steps that according to the basic theory of plasticity mechanics, a parabola yield criterion, a non-correlation flow rule and a hardening rule suitable for epoxy resin are established; carrying out incremental updating on the plastic internal variable by adopting a self-adaptive substep return mapping algorithm; deriving a consistent tangent stiffness matrix under the implicit finite element framework; and a parabola yield criterion, a non-correlation flow rule, a hardening rule, a self-adaptive substep return mapping algorithm and a consistent tangent stiffness matrix are embedded into the damage model, mechanical response simulation of materials and structures in the elastic-plastic damage process is achieved, and plastic return and implementation suitable for the epoxy resin parabola yield model are completed. According to the method, the problems of poor stability of an iteration method and inaccurate stiffness matrix in the existing resin-based composite material elastic-plastic modeling are solved.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of mechanical property analysis of epoxy resin-based composites, and particularly relates to a plastic return and implementation method suitable for a parabolic yield model of epoxy resin. BACKGROUND

[0002] Resin-based composites, especially epoxy resin-based composites, have been widely used in aerospace, automobile manufacturing and high-performance structural parts due to their excellent forming processability, mechanical properties and interfacial bonding ability. Such materials naturally have significant multiscale structural characteristics, consisting of macroscopic reinforcements (such as fibers) and microscopic matrices (such as epoxy resin). When analyzing their mechanical properties, it is usually necessary to establish the constitutive model of the reinforcement and the matrix respectively, and to realize accurate prediction of the overall structural behavior by combining multiscale calculation methods.

[0003] As the main matrix material of epoxy resin, its constitutive response has obvious elastic-plastic behavior characteristics. Studies have shown that the parabolic yield criterion can well fit the yield and hardening behavior of such materials, with good applicability and physical consistency. However, in the implementation process, the existing models mostly use the traditional Newton-Raphson iteration method for return mapping solution. Such methods are highly sensitive to initial guess values, and when dealing with strongly nonlinear constitutive relationships, they often have slow convergence speed or even divergence problems, which seriously affect the efficiency and stability of finite element simulation. In order to improve the robustness of the solution, it is necessary to develop a nonlinear iterative method with adaptive control capability and a plastic correction strategy to enhance the numerical stability of the plastic iteration process under complex loading conditions.

[0004] In addition, in the implicit finite element framework, the consistent tangent stiffness matrix plays a key role in controlling the convergence of nonlinear equation systems. In traditional implementation, in order to simplify the derivation process, the dependence of material parameters on equivalent plastic strain is often ignored, resulting in the constructed tangent stiffness matrix being unable to accurately reflect the true stiffness evolution behavior of the material, affecting the global iteration convergence speed, and even causing simulation instability. Especially under complex service conditions such as heat and load of resin-based composites, this error is particularly evident. Therefore, it is necessary to derive a new expression of the consistent tangent stiffness matrix to improve the accuracy and convergence of finite element solution.

[0005] In summary, there are still key problems such as poor stability of iterative methods and inaccurate stiffness matrix in current elastoplastic modeling of resin-based composites, and technical breakthroughs are needed in nonlinear return mapping strategies and tangent stiffness derivation methods to build a stable, convergent and generalizable constitutive numerical solution framework to serve the multiscale mechanical analysis and engineering design of complex structures. SUMMARY

[0006] In view of the above problems in the prior art, the plastic return and implementation method suitable for the parabolic yield model of epoxy resin provided by the present application solves the problems of poor stability of the iterative method and inaccuracy of the stiffness matrix in the existing resin-based composite material elastic-plastic modeling.

[0007] In order to achieve the above-mentioned purposes, the technical scheme adopted by the present application is as follows: a plastic return and implementation method suitable for the parabolic yield model of epoxy resin, comprising the following steps: S1: according to the basic theory of plastic mechanics, establishing a parabolic yield criterion, a non-associated flow rule and a hardening rule suitable for epoxy resin; S2: using a self-adaptive sub-step return mapping algorithm to incrementally update the plastic internal variable; S3: deriving a consistent tangent stiffness matrix under an implicit finite element framework; S4: embedding the parabolic yield criterion, the non-associated flow rule, the hardening rule, the self-adaptive sub-step return mapping algorithm and the consistent tangent stiffness matrix into a damage model to realize the simulation of the mechanical response of materials and structures in the elastic-plastic damage process, and to complete the plastic return and implementation of the parabolic yield model suitable for epoxy resin.

[0008] Further, the parabolic yield criterion in S1 is:

[0009] wherein, is a yield function, and is a first stress invariant and a second deviatoric stress invariant, and is the tensile and compressive yield strength of the material.

[0010] Further, the non-associated flow rule in S1 is:

[0011] wherein, is a plastic strain increment, is a plastic multiplier increment, is a flow tensor, is a flow rule, is a deviatoric stress tensor, is a second-order unit tensor, is a plastic parameter, is a plastic Poisson's ratio, is a stress tensor.

[0012] Further, the hardening rule in S1 is a function of the tensile and compressive strength, and the hardening depends on the equivalent plastic strain, and the formula is:

[0013]

[0014] in, and It is a function of tensile and compressive strength. For equivalent plastic strain, For the increment of plastic strain, It is an intermediate variable.

[0015] Furthermore, step S2 includes the following sub-steps: S21: Select an initial approximation of the plastic multiplier increment. and adaptive factor ; S22: The plastic multiplier increment is updated using an improved Newton-Raphson iteration method, with the following formula:

[0016] in, For the first Approximate value of the next iteration For the first The approximate value of the next iteration; S23: Calculation and compare and The size, specifically: a) If < hour: Then when At that time, take The calculation process is now complete. when At that time, take And return to S22 to continue the iteration; b) If > hour: Then when and At that time, take The calculation process is now complete. when and ,Pick Then return to S22 to continue the iteration; when and Take the adaptive factor Then return to S22 to continue the iteration; in, , , For adaptive threshold, It is a positive number; S24: Update stress based on plastic multiplier increment update results. .

[0017] Furthermore, the stress is updated in S24. The formula is:

[0018] in, For elastic testing stress, Shear modulus For the deviatoric stress tensor, Bulk modulus This is a plasticity parameter.

[0019] Furthermore, the uniform tangent stiffness matrix in S3 is:

[0020] in, To elastically test strain, For hydrostatic pressure, For unit skew tensor, As an intermediate variable related to the deviatoric stress tensor, As an intermediate variable related to hydrostatic pressure, To test the deviatoric stress elastically, As an intermediate variable related to the yield function, To flexibly test the purified water pressure, For intermediate variables related to hardening parameters, This is the derivative of the equivalent plastic strain with respect to the elastic test strain. This is the symbol for tensor product.

[0021] Furthermore, the damage model in S4 includes damage initiation criteria and damage evolution criteria; The damage initiation criterion is as follows:

[0022] in, As a criterion for damage initiation, As the first effective stress invariant, As the second effective stress invariant, and The ultimate strength under compression and tension; The damage evolution criterion is as follows:

[0023] in, As a damage variable, As a damage index factor, This is the damage threshold function.

[0024] The beneficial effects of this invention are: This invention proposes an adaptive substep return mapping algorithm and plasticity correction strategy for the parabolic yield model of epoxy resin, and derives a novel implicit finite element uniform tangent stiffness matrix. This method embeds the plasticity model into a damage constitutive framework, enabling nonlinear finite element analysis of materials and structures. This invention effectively overcomes the problems of existing plasticity return mapping algorithms being sensitive to initial values ​​and prone to divergence or slow convergence when solving highly nonlinear equations; it also corrects the insufficient convergence caused by neglecting the equivalent plastic strain dependence in the traditional uniform tangent stiffness matrix, thereby significantly improving the overall computational efficiency and stability.

[0025] Furthermore, the method established in this invention is not only applicable to material models with nonlinear yielding behavior, such as epoxy resin, but can also be extended to more general elastoplastic constitutive frameworks, and can be further applied to complex material models such as viscoelastic-viscoplastic materials, demonstrating good versatility and engineering applicability. This method can provide reliable theoretical support and efficient numerical tools for material constitutive modeling, structural strength assessment, and high-precision finite element simulation, possessing broad prospects for scientific research and engineering applications. Attached Figure Description

[0026] Figure 1 This invention provides a method and flow chart for plastic return and implementation of a parabolic yield model for epoxy resin.

[0027] Figure 2 The diagram shows the evolution process of unit equivalent plastic strain and damage variables, as well as the load-displacement curve of the "dog bone" specimen in the embodiment of the present invention.

[0028] Figure 3 The diagram shows the equivalent plastic strain and damage variables of the "dog bone" specimen in the embodiments of the present invention.

[0029] Figure 4 This is a comparison chart of the computational efficiency of the new method and the original method in the embodiments of the present invention. Detailed Implementation

[0030] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0031] like Figure 1 As shown, a method for plastic return and implementation of a parabolic yield model for epoxy resin includes the following steps: S1: Based on the basic theory of plasticity, establish the parabolic yield criterion, non-associated flow rule, and hardening rule applicable to epoxy resin; The yield criterion for a parabola is:

[0032] in, Let be the yield function. and For the first stress invariant and the second deviatoric stress invariant, and It represents the tensile and compressive yield strength of the material.

[0033] If the associated flow rule is used, it will lead to positive volumetric plastic strain under hydrostatic pressure, which does not conform to actual physical phenomena. Therefore, the non-associated flow rule is used. The non-associated flow rule is as follows:

[0034] in, For the increment of plastic strain, For the plastic multiplier increment, For the flow tensor, For the law of flow, For the deviatoric stress tensor, It is a second-order unit tensor. For plasticity parameters, For plastic Poisson's ratio, Let be the stress tensor.

[0035] The hardening law is a function of tensile and compressive strength, and hardening depends on the equivalent plastic strain, as shown in the formula:

[0036]

[0037] in, and It is a function of tensile and compressive strength. For equivalent plastic strain, For the increment of plastic strain, It is an intermediate variable.

[0038] S2: The adaptive substep return mapping algorithm is used to incrementally update the plastic internal variables; This invention employs computational plasticity mechanics to achieve this process, thus deriving the return mapping algorithm for the epoxy resin matrix, i.e., given a time... Given the stress state, we can completely determine the time based on the definition of strain increment. The stress tensor. In other words, the stress at the end of the strain increment is calculated based on the stress, strain, and hardening variables at the beginning of the increment.

[0039] The general return mapping update formula for the stress tensor is given as follows:

[0040] in, For stress, To test the stress, Here is the elastic stiffness matrix; The last term corresponds to the plastic correction part. Substituting the plastic strain increment defined in the non-associated flow rule into the above equation, we obtain the following stress update formula:

[0041] The above is broken down into partial components and volume components (for clarity, from now on, the subscripts corresponding to the end of the increment will be removed). n +1”):

[0042] in, To test the deviatoric stress tensor, For hydrostatic pressure, To test the pressure of the purified water.

[0043] It should be noted that the deviatoric stress and volumetric stress at the end of the increment are both obtained by simply adjusting according to... The factor is obtained by proportionally reducing the trial deviatoric stress and volumetric stress. To simplify the writing, we redefine the denominator in the above formula:

[0044] The consistency condition defined by the yield surface equation is given by the following equation:

[0045] The two yield stresses in this equation are functions of the equivalent plastic strain defined above. Applying the non-associated flow rule and the equivalent plastic strain formula, the equivalent plastic strain of this model is defined as:

[0046] in, To test the first stress invariant, To explore the second stress invariant.

[0047] For ease of explanation, parameters are defined. A It replaces the dividend under the second square root; it is the plastic multiplier. An incremental function.

[0048]

[0049] At this point, the system of equations is finally simplified to The yield function is a function, but this is an extremely complex nonlinear equation that requires Newton's iteration method to solve. Therefore, it is necessary to solve for the yield function with respect to... Partial derivative:

[0050] Subsequent yield strength and The two derivatives can be determined by applying the chain rule:

[0051] Where parameters and This represents the hardening modulus for the two yield strengths (compressive and tensile) considered here. To solve the above equations, the derivative of the equivalent plastic strain must be determined:

[0052] Now, we apply Newton's iteration method to the increment of the plastic multiplier. The solution is then performed. While the traditional Newton-Raphson iteration method has the advantage of fast convergence, it is sensitive to initial values ​​and prone to divergence or slow convergence when solving highly nonlinear equations. Therefore, this invention introduces an adaptive substep method, which can automatically increase or decrease the iteration step size based on the residual. Introducing an adaptive factor λ, the improved iteration format is as follows:

[0053] The parameters satisfy the following: ,and , The adaptive threshold is set to 1e-6.

[0054] The calculation steps are as follows: S2 includes the following sub-steps: S21: Select an initial approximation of the plastic multiplier increment. and adaptive factor ; S22: The plastic multiplier increment is updated using an improved Newton-Raphson iteration method, with the following formula:

[0055] in, For the first Approximate value of the next iteration For the first The approximate value of the next iteration; S23: Calculation and compare and The size, specifically: a) If < hour: Then when At that time, take The calculation process is now complete. when At that time, take And return to S22 to continue the iteration; b) If > hour: Then when and At that time, take The calculation process is now complete. when and ,Pick Then return to S22 to continue the iteration; when and Take the adaptive factor Then return to S22 to continue the iteration; in, , , For adaptive thresholding, it is generally set to 1e. -5 It can be fine-tuned according to the actual model. It is a positive number; S24: Update stress based on plastic multiplier increment update results. :

[0056] in, For elastic testing stress, Shear modulus For the deviatoric stress tensor, Bulk modulus This is a plasticity parameter.

[0057] In solving for the plastic multipliers, the method employed will approximate the convergent solution as closely as possible through iteration, with an upper limit of 20 iterations. After completing the iterations, the validity of the results needs to be checked. This is because the consistency condition function may exhibit irregular behavior in certain domain regions, causing the plastic multipliers to converge incorrectly to non-physical solutions, such as diverging to infinity or exhibiting negative values.

[0058] If the convergence result meets the validity condition, the mapping process ends. If the adaptive Newton iteration method fails to converge after 20 iterations or the result fails the validity judgment, the auxiliary correction function is called to improve the initial guess value, and the adaptive Newton iteration process is restarted accordingly, thereby improving the stability and accuracy of plastic multiplier calculation.

[0059] S3: Derive the uniform tangent stiffness matrix under the implicit finite element framework; This step involves conducting simulation analysis of the mechanical and thermophysical properties of ceramic matrix composites based on the analysis requirements.

[0060] The computational plasticity method requires determining a consistent tangent stiffness matrix.

[0061] The elastic-plastic uniform tangent stiffness matrix associated with a specific return mapping algorithm is defined as the derivative:

[0062] in, This is the result obtained from the previously described return mapping algorithm. Since the input to the elastoplastic integral procedure is the elastic test strain... rather than total strain , Direct differentiation yields:

[0063] To determine the uniform tangent stiffness matrix corresponding to the current parabolic yield criterion, the derivatives of the two stress update formulas in the partial component and volume component equations must be determined.

[0064]

[0065] Differentiating the yield function yields:

[0066] in, To explore the second stress invariant, To test the first stress invariant, For equivalent plastic strain, here This is to emphasize the numerical concept of "the elastoplastic consistent tangent stiffness matrix associated with a specific return mapping algorithm." Numerical solutions involve the concept of "incremental steps," which is indicated by the subscript "...". n +1”.

[0067] It is worth noting that this invention re-derives this formula. The original method did not consider the dependence of the subsequent yield strength on the equivalent plastic strain, which would lead to an incorrect uniform tangent stiffness matrix, resulting in a decrease in the convergence speed of the finite element calculation. This invention focuses on this term. First, a formal analysis is performed on the right-hand side of the equation. The terms inside the square brackets are constants for a given time increment and have been determined in the return mapping algorithm. From now on, the characters... This will be used to represent the term. Simplifying the left side of the equation, the terms inside the square brackets can be written as:

[0068] It's worth noting that this is a scalar, making it very convenient to organize the equations. Rearrange the equations as follows:

[0069] The derivatives of the two stress invariants are relatively easy to solve, and are given directly here:

[0070] in, To test the elastic deviatoric strain, To test the strain of the elastic body.

[0071] The following needs to be done Perform analysis, based on

[0072] so

[0073] Due to tensor N Due to the symmetry, the above equation can be simplified to:

[0074] The item within the rightmost parenthesis can be written as:

[0075] in, It is a fourth-order unit symmetric tensor.

[0076] Based on the above formula, the corresponding indicator form is:

[0077] in, For the indicator symbol of strain, The indicator symbol for the flow law. To explore the index sign of deviatoric stress, The symbol for the Kronecker index. This is the index symbol for a four-stage bit-symmetric tensor. This is the symbol for the Kronecker index. and They represent the same thing, but are distinguished as different because of the different solution order. i, j, k, l .

[0078] For ease of subsequent derivation, it is written in tensor form:

[0079] according to

[0080] This yields the new uniform tangent stiffness matrix:

[0081] in, To elastically test strain, For hydrostatic pressure, For unit skew tensor, As an intermediate variable related to the deviatoric stress tensor, As an intermediate variable related to hydrostatic pressure, To test the deviatoric stress elastically, As an intermediate variable related to the yield function, To flexibly test the purified water pressure, For intermediate variables related to hardening parameters, This is the derivative of the equivalent plastic strain with respect to the elastic test strain. This is the symbol for tensor product.

[0082] S4: The parabolic yield criterion, non-associated flow rule, hardening rule, adaptive substep return mapping algorithm, and consistent tangent stiffness matrix are embedded into the damage model to realize the mechanical response simulation of materials and structures in the elastoplastic damage process, and to complete the plastic return and implementation of the parabolic yield model applicable to epoxy resin.

[0083] Considering the failure behavior requirements of materials under actual working conditions, the proposed plasticity algorithm is embedded into the damage model to simulate the mechanical response of materials and structures during the elastoplastic damage process, which is applicable to engineering problems such as fracture strength analysis. Since damage is not the focus of this invention, only the damage initiation and evolution criteria used are briefly introduced.

[0084] The damage model in S4 includes damage initiation criteria and damage evolution criteria; The damage initiation criterion is as follows:

[0085] in, As a criterion for damage initiation, As the first effective stress invariant, As the second effective stress invariant, and The ultimate strength under compression and tension; The damage evolution criterion is as follows:

[0086] in, As a damage variable, As a damage index factor, This is the damage threshold function.

[0087] Finally, based on the finite element software ABAQUS, the aforementioned plasticity return mapping algorithm was implemented by writing a user material subroutine (UMAT), and a numerical efficiency comparison analysis was conducted to verify its stability and computational performance under complex loading conditions. In this embodiment, a standard dog bone-shaped specimen was selected as the simulation example model, and the relevant calculation results are shown below. Figure 2 , Figure 3 and Figure 4 . Figure 2 In figure (a), the curves represent the equivalent plastic strain versus damage evolution. Figure 2 In diagram (b), the load-displacement curve is shown. Figure 2 It can be seen that the method of the present invention can accurately simulate the plastic evolution behavior of epoxy resin materials, exhibiting good stress-strain response characteristics; at the same time, as Figure 3 As shown, once damage occurs, the plastic evolution process of the material stops, which conforms to the mechanical behavior of actual materials. From Figure 4 The comparison results show that, compared with traditional methods, the present invention achieves a significant improvement in overall computational efficiency, effectively reduces the number of iterations in each incremental step and the total number of loading steps, demonstrating superior convergence performance and engineering application potential.

[0088] This invention addresses the problems of existing plasticity return mapping algorithms in nonlinear finite element analysis, such as high initial value sensitivity, slow convergence speed, and inaccurate uniform tangent stiffness matrix. It proposes and implements an adaptive substep return mapping algorithm and plasticity correction method suitable for the parabolic yield model of epoxy resin. This method significantly improves the stability and computational efficiency of plasticity iteration, and enhances the convergence performance of the finite element solver under complex loads by deriving a novel uniform tangent stiffness matrix. By embedding this plasticity model into a damage constitutive framework, high-precision elastoplastic damage simulation of materials and structures can be achieved, providing reliable support for strength assessment and life prediction of epoxy resin and similar materials in engineering. With this invention, researchers and engineers can more efficiently conduct complex material behavior modeling, providing a theoretical foundation and numerical tools for high-performance structural design and material safety analysis.

[0089] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of the invention.

Claims

1. A method for achieving plastic return in a parabolic yield model of epoxy resin, characterized in that, Includes the following steps: S1: Based on the basic theory of plasticity, establish the parabolic yield criterion, non-associated flow rule, and hardening rule applicable to epoxy resin; S2: The adaptive substep return mapping algorithm is used to incrementally update the plastic internal variables; S3: Derive the uniform tangent stiffness matrix under the implicit finite element framework; S4: The parabolic yield criterion, non-associated flow rule, hardening rule, adaptive substep return mapping algorithm, and consistent tangent stiffness matrix are embedded into the damage model to realize the mechanical response simulation of materials and structures in the elastoplastic damage process, and to complete the plastic return and implementation of the parabolic yield model applicable to epoxy resin.

2. The method for plastic return and implementation of the parabolic yield model of epoxy resin according to claim 1, characterized in that, The parabolic yield criterion in S1 is: in, Let be the yield function. and For the first stress invariant and the second deviatoric stress invariant, and It represents the tensile and compressive yield strength of the material.

3. The method for plastic return and implementation of the epoxy resin parabolic yield model according to claim 2, characterized in that, The non-associated flow rule in S1 is as follows: in, For the increment of plastic strain, For the plastic multiplier increment, For the flow tensor, For the law of flow, For the deviatoric stress tensor, It is a second-order unit tensor. For plasticity parameters, For plastic Poisson's ratio, Let be the stress tensor.

4. The method for plastic return and implementation of the epoxy resin parabolic yield model according to claim 3, characterized in that, The hardening law in S1 is a function of tensile and compressive strength, and hardening depends on the equivalent plastic strain, as shown in the formula: in, and It is a function of tensile and compressive strength. For equivalent plastic strain, For the increment of plastic strain, It is an intermediate variable.

5. The method for plastic return and implementation of the parabolic yield model of epoxy resin according to claim 1, characterized in that, S2 includes the following sub-steps: S21: Select an initial approximation of the plastic multiplier increment. and adaptive factor ; S22: The plastic multiplier increment is updated using an improved Newton-Raphson iteration method, with the following formula: in, For the first Approximate value of the next iteration For the first The approximate value of the next iteration; S23: Calculation and compare and The size, specifically: a) If < hour: Then when At that time, take The calculation process is now complete. when At that time, take And return to S22 to continue the iteration; b) If > hour: Then when and At that time, take The calculation process is now complete. when and ,Pick Then return to S22 to continue the iteration; when and Take the adaptive factor Then return to S22 to continue the iteration; in, , , For adaptive threshold, It is a positive number; S24: Update stress based on plastic multiplier increment update results. .

6. The method for plastic return and implementation of the epoxy resin parabolic yield model according to claim 5, characterized in that, Stress update in S24 The formula is: in, For elastic testing stress, Shear modulus For the deviatoric stress tensor, Bulk modulus This is a plasticity parameter.

7. The method for plastic return and implementation of the epoxy resin parabolic yield model according to claim 6, characterized in that, The uniform tangent stiffness matrix in S3 is: in, To elastically test strain, For hydrostatic pressure, For unit skew tensor, As an intermediate variable related to the deviatoric stress tensor, As an intermediate variable related to hydrostatic pressure, To test the deviatoric stress elastically, As an intermediate variable related to the yield function, To flexibly test the purified water pressure, For intermediate variables related to hardening parameters, This is the derivative of the equivalent plastic strain with respect to the elastic test strain. This is the symbol for tensor product.

8. The method for plastic return and implementation of the parabolic yield model of epoxy resin according to claim 1, characterized in that, The damage model in S4 includes damage initiation criteria and damage evolution criteria; The damage initiation criterion is as follows: in, As a criterion for damage initiation, As the first effective stress invariant, As the second effective stress invariant, and The ultimate strength under compression and tension; The damage evolution criterion is as follows: in, As a damage variable, As a damage index factor, This is the damage threshold function.

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