A numerical simulation method and system for predicting the residual strength and failure mode of perforated circular plates

Abstract

The present invention discloses a numerical simulation method and system for predicting the residual strength and failure mode of a perforated circular plate, belonging to the fields of computational mechanics and engineering simulation. The method includes: according to the strength and fracture characteristics of a flat plate with singular and non-singular defects, a peridynamic hybrid failure model considering both stress and energy conditions is proposed. The method sequentially predicts the residual strength and failure mode of the perforated circular plate by establishing a perforated circular plate model, analyzing the material properties of the plate, determining the geometric parameters and the type parameters of the near field, determining the mixed crack length, and implementing the peridynamic method.

Description

Technical Field

[0001] The present invention belongs to the field of computational mechanics and engineering simulation research, and particularly relates to a numerical simulation method and system for predicting the residual strength and failure mode of a perforated circular plate. Background Art

[0002] Due to reasons such as part assembly and processing technology in engineering structures, stress concentration areas are inevitable, such as circular holes and notches. Under a certain load, cracks will initiate in the stress concentration areas, thereby causing the failure of materials and structures. Classical fracture mechanics can effectively analyze problems with initial pre-cracks (singular stress concentration), but for problems with elliptical holes (non-singular stress concentration), classical fracture mechanics is not applicable. In engineering, the residual strength of perforated flat plates is generally predicted using stress-based point stress and average stress empirical models, but empirical parameters obtained from experimental results need to be introduced in such models. The finite fracture mechanics (FFM) model takes into account both the strength and fracture parameters of materials and can successfully predict the residual strength of flat plates with non-singular stress concentration. However, for the analysis of the process from crack initiation to crack propagation, the FFM model is difficult to analyze. For problems with non-singular stress concentration, such as perforated flat plate problems, the peridynamic bond-breaking criterion based on fracture energy equivalence is no longer applicable. Therefore, a peridynamic failure model for analyzing non-singular stress concentration problems is needed. Summary of the Invention

[0003] Aiming at the deficiencies of the prior art, the present invention proposes a numerical simulation method and system for predicting the residual strength and failure mode of a perforated circular plate. The present invention proposes a peridynamic hybrid failure model that simultaneously considers stress and energy, and quantitatively analyzes the strength and fracture problems of flat plates with singular and non-singular defects.

[0004] To achieve the above object, the present invention provides the following solutions:

[0005] A numerical simulation method for predicting the residual strength and failure mode of a perforated circular plate, comprising the following steps:

[0006] Based on the strength and fracture characteristics of flat plates with singular and non-singular defects, establish a peridynamic hybrid failure model that simultaneously considers stress and energy conditions;

[0007] Solve the non-local stress field and energy release rate of the peridynamic hybrid failure model to obtain the numerical mixed crack length;

[0008] Input the numerical mixed crack length into the initialized peridynamic hybrid failure model for stress analysis;

[0009] If the result of the stress analysis meets the preset requirements, enter the phase transition point and apply symmetric initial cracks to the flat plate with an elliptical hole.

[0010] Based on the mixed crack length and critical load that simultaneously satisfy the stress and energy conditions, use the bond-breaking criterion based on the critical elongation rate or critical bond energy density to perform fracture analysis on the flat plate with an elliptical hole, and obtain the structural critical load and crack propagation energy.

[0011] Preferably, based on material, geometry, and peridynamic parameters, the method for obtaining the numerical mixed crack length using the stress and energy mixed failure theory includes: solving the peridynamic nonlocal stress field and solving the peridynamic energy release rate.

[0012] Preferably, the method for performing stress analysis by inputting the numerical mixed crack length into the initialized peridynamic mixed failure model includes:

[0013] Apply a linearly increasing displacement load, and numerically solve peridynamics to obtain F, ε, and σ;

[0014] Determine the relationship between the stress and the tensile strength. If the stress in the y direction is greater than the tensile strength, enter the phase transition point and apply symmetric initial cracks to the flat plate with an elliptical hole. If the stress in the y direction is less than the tensile strength, continue the stress analysis;

[0015] Among them, the method for applying a linearly increasing displacement load and numerically solving peridynamics to obtain F, ε, and σ includes:

[0016] Discretize the nodes x of the peridynamic numerical model i Nonlocal deformation gradient F i It is expressed as:

[0017]

[0018] In the formula, M is the node x i The total number of nodes x within the peridynamic range j Y is the deformation vector state, is the product of two vectors, ω is the influence function, B(x i ) is the shape tensor, V j is the volume;

[0019] Using the nonlocal deformation gradient F and considering the constitutive equation of continuum mechanics, the nonlocal strain and stress tensors of an isotropic material are:

[0020]

[0021] In the formula, I is the identity matrix and D is the elastic stiffness matrix.

[0022] Preferably, the method for obtaining the mixed crack length and critical load that simultaneously satisfy the stress and energy conditions includes:

[0023] Simultaneously considering the stress field and energy release rate of the flat plate with an elliptical hole, that is, obtaining the mixed crack length a that simultaneously satisfies the stress and energy conditions through the peridynamic numerical model * and the critical load σ * :

[0024]

[0025] wherein, the stress σ yy (l) and the incremental energy release rate are linearly and quadratically related to the applied stress load σ0 respectively. At the same time, the critical load σ * and the mixed crack length a * obtained by numerical solution are not affected by the applied initial stress load σ0, and σ c and G c are the material tensile strength and critical energy release rate respectively. Preferably, based on the mixed crack length and critical load that simultaneously satisfy the stress and energy conditions, the method for performing fracture analysis on the flat plate with an elliptical hole and obtaining the structural critical load and crack propagation energy by using the bond-breaking criterion based on the critical elongation rate or critical bond energy density includes:

[0026] For the flat plate structure with a hole, as the tensile load increases continuously, the stress at the hole opening increases continuously and reaches:

[0027] σ yy (a + a * , y = 0) = σ c

[0028] wherein, a * is the numerical mixed crack length that simultaneously satisfies the stress and energy conditions obtained from the equation:

[0029]

[0030] wherein, σ yy (x) is the stress in the y direction along the straight line on the x-axis, G inc (l) is the average energy release rate considering the crack elongation as a finite length l, and σ c and G c are the material tensile strength and critical energy release rate respectively. Therefore, when σ yy (a + a * , y = 0) = σ c holds, the stress and energy simultaneously satisfy:

[0031]

[0032] At this time, the corresponding tensile load σ N is the critical tensile load σ of the perforated structure * .

[0033] The present invention also provides a numerical simulation system for predicting the residual strength and failure mode of a perforated circular plate, including: a construction module, an initialization module, a stress analysis module, a stage conversion module, and a fracture analysis module;

[0034] The construction module is used to establish a peridynamic hybrid failure model that simultaneously considers stress and energy conditions based on the strength and fracture characteristics of a flat plate with singular and non-singular defects;

[0035] The initialization module is used to solve the non-local stress field and energy release rate of the peridynamic hybrid failure model to obtain the numerical hybrid crack length;

[0036] The stress analysis module is used to input the numerical hybrid crack length into the initialized peridynamic hybrid failure model for stress analysis;

[0037] The stage conversion module is used to enter the stage conversion point if the result of the stress analysis meets the preset requirements, and apply symmetric initial cracks to the flat plate with an elliptical hole;

[0038] The fracture analysis module is used to perform fracture analysis on the flat plate with an elliptical hole based on the hybrid crack length and critical load that simultaneously satisfy stress and energy conditions, and adopt a bond-breaking criterion based on critical elongation or critical bond energy density to obtain the structural critical load and crack propagation energy.

[0039] Preferably, in the initialization module, the process of obtaining the hybrid crack length by peridynamic solution based on material, geometry, and peridynamic parameters includes: solving the peridynamic non-local stress field and solving the peridynamic energy release rate.

[0040] Preferably, in the stress analysis module, the process of inputting the hybrid crack length into the initialized peridynamic hybrid failure model for stress analysis includes:

[0041] Applying a linearly increasing displacement load, and numerically solving by peridynamics to obtain F, ε, and σ;

[0042] Determining the relationship between the stress and the tensile strength. If the stress in the y direction is greater than the tensile strength, enter the stage conversion point and apply symmetric initial cracks to the flat plate with an elliptical hole. If the stress in the y direction is less than the tensile strength, continue the stress analysis;

[0043] Among them, the method of applying a linearly increasing displacement load and numerically solving by peridynamics to obtain F, ε, and σ includes:

[0044] Discretized peridynamic numerical model node x i Nonlocal deformation gradient F i It is expressed as:

[0045]

[0046] Wherein, M is the node x i The total number of nodes x within the peridynamic range j Y is the deformation vector state, is the product of two vectors, ω is the influence function, B(x i ) is the shape tensor, V j is the volume;

[0047] Using the nonlocal deformation gradient F and considering the constitutive equation of continuum mechanics, the nonlocal strain and stress tensors of an isotropic material are:

[0048]

[0049] Wherein, I is the identity matrix and D is the elastic stiffness matrix.

[0050] Preferably, in the phase transition module, the process of obtaining the mixed crack length and critical load that simultaneously satisfy the stress and energy conditions includes:

[0051] Simultaneously considering the stress field and energy release rate of the flat plate with an elliptical hole, that is, obtaining the mixed crack length a that simultaneously satisfies the stress and energy conditions through the peridynamic numerical model * and the critical load σ * :

[0052]

[0053] Wherein, the stress σ yy (l) and the incremental energy release rate are linearly and quadratically related to the applied stress load σ0 respectively. At the same time, the critical load σ * and the mixed crack length a * values obtained by numerical solution are not affected by the applied initial stress load σ0, and σ c and G c are the material tensile strength and critical energy release rate respectively.

[0054] Preferably, in the fracture analysis module, based on the mixed crack length and critical load that simultaneously satisfy the stress and energy conditions, the process of performing fracture analysis on the flat plate with an elliptical hole to obtain the structural critical load and crack propagation energy includes:

[0055] For the flat plate structure with a hole, as the tensile load increases continuously, the stress at the hole opening increases continuously and reaches:

[0056] σ yy (a + a * , y = 0) = σ c

[0057] where a * is the numerical mixed crack length that simultaneously satisfies the stress and energy conditions obtained from the equations:

[0058]

[0059] where σ yy (x) is the stress in the y - direction along the straight line on the x - axis, G inc (l) is the average energy release rate considering the crack elongation as a finite length l, σ c and G c are the material tensile strength and the critical energy release rate respectively. Therefore, when σ yy (a + a * , y = 0) = σ c holds, the stress and energy simultaneously satisfy:

[0060]

[0061] At this time, the corresponding tensile load σ N is the critical tensile load σ * of the perforated structure.

[0062] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0063] (1) The peridynamics used in the present invention is based on a non - local theory, which transforms the traditional differential equation of motion into an integral equation, thus avoiding the stress singularity of continuum mechanics in discontinuous regions and having more advantages in analyzing problems such as fracture damage;

[0064] (2) The peridynamics mixed failure model used in the present invention can effectively reflect the stress situation of the structure, thus avoiding the problem that in the stress concentration region (circular hole or crack boundary), due to the surface effect problem of the non - local theory, the non - local stress expression is no longer applicable to the nodes within the near - field range of the circular hole boundary;

[0065] (3) The peridynamics numerical model used in the present invention solves the value of the mixed crack length a * , optimizing the problem that for most tension - loaded plates with complex elliptical orifices, it is difficult to obtain the theoretical solution of the corresponding mixed crack length a * ;

[0066] (4) The innovative method of the peridynamic hybrid failure model that simultaneously considers stress and energy conditions provided by the present invention can be applied to solve the strength and fracture problems of any flat plate with singular and non-singular defects. Brief Description of the Drawings

[0067] In order to more clearly illustrate the technical solutions of the present invention, the drawings required in the embodiments will be briefly introduced below. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0068] Figure 1 Flow chart for analyzing the failure of a flat plate with a hole using the peridynamic hybrid model in the embodiment of the present invention;

[0069] Figure 2 For the stress and energy release rate solution flow chart under the action of stress σ N = σ0 in the embodiment of the present invention;

[0070] Figure 3 For the embodiment of the present invention, applying a symmetric crack da (blue dashed line) to a flat plate with an elliptical hole: break all the bonds crossing the crack da (such as the red line connecting nodes x i and x j );

[0071] Figure 4 For the values of the mixed crack length a * and the critical load σ * when the crack occurs in the embodiment of the present invention, where the stress and energy release rates under a fixed load σ0 are used;

[0072] Figure 5 Tensile flat plate with an elliptical hole in the embodiment of the present invention;

[0073] Figure 6 For the embodiment of the present invention, under the tensile action in the y direction, the stresses (a) σ x , (b) σ y and (c) τ xy obtained by using (1) finite element and (2) peridynamics;

[0074] Figure 7 For the embodiment of the present invention, the stress in the y direction at a distance l from the hole opening along the x-axis corresponding to different peridynamic ranges δ;

[0075] Figure 8 For the embodiment of the present invention, the curve of the numerical solution of the energy release rate varying with the finite crack length l from the hole opening corresponding to different peridynamic ranges δ;

[0076] Figure 9In the embodiment of the present invention, considering both stress and energy release rate simultaneously, the mixed crack length a when the crack occurs in the flat plate with an elliptical hole is obtained * and the critical load value σ * (a * = 2.66 mm, σ * = 332.79 MPa);

[0077] Figure 10 It is the stress-strain curve during the failure process of the perforated flat plate under the action of linear displacement load in the embodiment of the present invention;

[0078] Figure 11 It is the curve of the total fracture energy of the system changing with the total crack length 2l during the crack propagation process in the embodiment of the present invention;

[0079] Figure 12 It is the final failure form of the flat plate with an elliptical hole under the action of tensile load in the embodiment of the present invention;

[0080] Figure 13 It is the mixed crack length a corresponding to different ellipticity a / b in the embodiment of the present invention * and the critical load σ * / σ c The curve of the change with the semi-major axis a of the ellipse. Specific embodiments

[0081] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0082] To make the above objects, features, and advantages of the present invention more obvious and understandable, the present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0083] Embodiment 1

[0084] The present invention provides a numerical simulation method for predicting the residual strength and failure form of a perforated circular plate, including the following steps:

[0085] Based on the strength and fracture characteristics of flat plates with singular and non-singular defects, a peridynamic hybrid failure model considering both stress and energy conditions is established;

[0086] Solve the non-local stress field and energy release rate of the peridynamic hybrid failure model to obtain the numerical mixed crack length;

[0087] Input the numerical mixed crack length into the initialized peridynamic mixed failure model for stress analysis;

[0088] If the result of the stress analysis meets the preset requirements, enter the stage conversion point and apply symmetric initial cracks to the flat plate with an elliptical hole;

[0089] Based on the mixed crack length and critical load that simultaneously satisfy the stress and energy conditions, use the bond-breaking criterion based on the critical elongation rate or critical bond energy density to perform fracture analysis on the flat plate with an elliptical hole, and obtain the structural critical load and crack propagation energy.

[0090] In this embodiment, based on the material, geometry, and peridynamic parameters, the method for obtaining the mixed crack length using the stress and energy mixed failure theory includes: obtaining the peridynamic nonlocal stress field and the energy release rate.

[0091] First, initialize the model, determine the material, geometry, and peridynamic parameters, and then obtain the mixed crack length according to peridynamics. In the peridynamic theory, when the structure fails, the bond-breaking criterion based on the critical elongation rate or critical bond energy density needs to be used.

[0092] The stress and energy conditions in Equation 6 respectively give the lower and upper limits of the finite crack length l corresponding to the constant tensile load σ N As shown, the notch critical strength σ Figure 5 corresponding to the occurrence of failure and the mixed crack length a * can be obtained. That is to say, the value of the mixed crack length a * simultaneously satisfies the strength and energy conditions. *

[0093] In this embodiment, based on the initialized model, perform stress analysis (in the non-destructive stress analysis stage of the flat plate with a hole, at this time, no cracks are applied, and only stress analysis is performed for this length region). Apply a linearly increasing displacement load, and the peridynamic numerical solution obtains F, ε, and σ (non-destructive stress analysis stage).

[0094] The mixed crack length a * needs to simultaneously satisfy the stress and energy conditions. Therefore, it is necessary to first calculate the nonlocal stress value based on peridynamics. In the discretized peridynamic numerical model, the nonlocal deformation gradient F i of node x i can be expressed as:

[0095]

[0096] where M is the total number of nodes x i within the peridynamic range of node x j , ω is the influence function,​Y is the deformed vector state, is the product of two vectors, B(x i ) is the shape tensor, V j is the volume. Using the nonlocal deformation gradient F and considering the constitutive equations of continuum mechanics, the nonlocal strain and stress tensors of an isotropic material are:

[0097]

[0098] where I is the identity matrix and D is the elastic stiffness matrix. Therefore, the nonlocal stress and strain values of conventional-state peridynamics can be obtained using Equation (2). It should be noted that the nonlocal stress obtained from Equation (2) has a different meaning from the stress in unconventional peridynamics. It is only the averaged calculation output of the stable displacement field and does not participate in the solution of the interaction forces between nodes.

[0099] Then, the relationship between the stress and the tensile strength is determined. If the stress in the y direction is greater than the tensile strength, it enters the stage conversion point, that is, a symmetric initial crack is applied. If the stress in the y direction is less than the tensile strength, it returns to the stress analysis stage;

[0100] In this embodiment, a symmetric initial crack a * (here it enters the conversion stage, that is, the fracture analysis stage, and a symmetric initial crack is applied). To analyze the non-singular stress concentration problem, the stress field and the energy release rate are considered simultaneously. Based on the peridynamics numerical model, the solution process of the stress field and the incremental energy release rate of a flat plate with an elliptical hole under tensile stress is given by Figure 2 where a constant load σ0 is applied to the structure, the quasi-static solution is carried out using the ADR method, and the nonlocal stress distribution near the hole is calculated using Equations (1) and (2). At the same time, the peridynamics finite crack elongation (PD_FCE) model is considered to calculate the incremental energy release rate G inc (l).

[0101] In the hybrid model, the peridynamics finite crack elongation (PD_FCE) model is used to calculate the energy release rate of the flat plate with a hole, where a finite crack is considered to replace the infinitesimal crack in the classical Griffith theory. As Figure 3 shown, under the action of a uniform load on the flat plate with an elliptical hole, after the system converges to a stable state, finite-length cracks l are symmetrically applied on both sides of the hole. Among them, the macroscopic crack is applied at the hole opening in the form of microscopic oriented bond breaking, that is, all the bonds crossing the finite crack l are broken (such as Figure 3 the red line connecting nodes x i and x j ).

[0102] Based on the PD_FCE theory, the total energy of the system before and after crack elongation is obtained, and then the corresponding numerical energy release rate is obtained by using the method of numerical difference. Considering the two-step difference scheme, the incremental energy release rate G inc (l) can be expressed as:

[0103]

[0104] where l is the finite crack length, U e (0) and U e (l) are the total strain energies of the perforated structure before and after crack application respectively, Δx is the unit finite crack elongation, N and M are the current and maximum finite crack application times respectively. Therefore, after the finite crack l = N*Δx is applied step by step to the perforated plate, the function of the total strain energy of the system with respect to the finite crack length l can be obtained, and then the incremental energy release rate G inc (l) with respect to the finite crack length l can be calculated.

[0105] As Figure 4 , by considering both the stress field and the energy release rate of the perforated plate with an elliptical hole, the mixed crack length a that satisfies both the stress and energy conditions can be obtained through the peridynamic numerical model * and the critical load σ * (here, the numerical mixed crack length is directly solved by peridynamics):

[0106]

[0107] where the stress σ yy (l) and the incremental energy release rate G inc (l) are linearly and quadratically related to the applied stress load σ0 respectively. At the same time, the critical load σ * and the mixed crack length a * obtained by numerical solution are not affected by the applied initial stress load σ0.

[0108] In this embodiment, the bond state is judged by the bond-breaking criterion. After the numerical solution obtains the mixed crack length σ * , the whole process failure of the perforated tensile plate can be analyzed by using the hybrid model based on peridynamics. As Figure 1 shown, the failure process of the perforated tensile plate can be divided into two stages: the non-destructive stress analysis stage and the fracture analysis stage. As the linear load increases continuously, when the structure stress meets certain conditions, the initial non-destructive stress stage will be transformed into the fracture stage. Specifically, for the perforated plate structure, as the tensile load increases continuously, the stress at the hole opening increases continuously and reaches:

[0109] σ yy (a + a* σ(y = 0)=σ c (5)

[0110] In the formula, a * is the numerical mixed crack length that simultaneously satisfies the stress and energy conditions obtained from Equation (8). Therefore, when Equation (13) holds, the stress and energy are simultaneously satisfied:

[0111]

[0112] The corresponding tensile load σ N is the critical tensile load σ * of the perforated structure.

[0113] Meanwhile, when the stress satisfies Equation (6), such as Figure 5 , pre-cracks with a length of the mixed crack length a * symmetrically attached along the x-axis on both sides of the elliptical hole. After the pre-cracks are attached, the non-destructive stress analysis of the structure is transformed into fracture analysis. In the fracture analysis stage, a bond-breaking criterion based on the critical elongation rate or critical bond energy density is adopted, and the crack starts to expand naturally along the pre-cracks.

[0114] Generally speaking, the critical stress condition (6) divides the analysis of the perforated tensile plate into two stages: non-destructive stress and fracture analysis. Among them, in the crack occurrence stage, the mixed crack length σ * can simultaneously satisfy the stress and energy conditions; in the fracture propagation stage, the bond-breaking model also naturally satisfies the fracture energy balance condition. Therefore, in the analysis of the entire failure process, the fracture energy conservation is always satisfied. Therefore, the mixed model based on peridynamics can predict the entire failure process of the perforated structure from crack occurrence to crack propagation, and can quantitatively obtain the critical load and crack propagation energy of the structure.

[0115] Figure 1 The following is a flow chart of a new numerical model provided by an embodiment of the present invention that can predict the residual strength and failure mode of a perforated circular plate, and the specific implementation operations are as follows:

[0116] Step 1: First, initialize the model, determine the material, geometry, and peridynamics parameters, and then obtain the mixed crack length a * .

[0117] The geometric dimensions and force conditions of the tensile test of the flat plate with an elliptical hole are determined by Figure 5Given that the geometric dimensions are \(W = 40\mathrm{mm}\), \(L = 80\mathrm{mm}\), the thickness of the plate is \(1\mathrm{mm}\), and the plane stress condition is considered. The material is an isotropic elastic material, and the material parameters are shown in Table 1. In the peridynamic numerical model, the ADR numerical integration method is used for quasi-static solution, and the peridynamic finite crack elongation (PD_FCE) model is adopted to calculate the incremental energy release rate of the structure.

[0118] Table 1 Material parameters of 1CrMoV steel

[0119]

[0120] Step 2: Based on the initialization of the model in Step 1, stress analysis is carried out. A linearly increasing displacement load is applied, and the peridynamic numerical solution gives \(F\), \(\varepsilon\), and \(\sigma\).

[0121] Figure 5 As shown, first consider the case where the elliptical hole size is \(a = 10\mathrm{mm}\), \(b = 5\mathrm{mm}\), and the uniform load on both sides of the plate is \(\sigma_0 = 10\mathrm{MPa}\). In the peridynamic model, the \(\delta -\)convergence analysis is carried out considering the continuously decreasing peridynamic ranges \(\delta = 4\mathrm{mm}\), \(2\mathrm{mm}\), and \(1\mathrm{mm}\) and the fixed \(m = 4\). In this case, the Figure 2 procedure is used to solve the peridynamic stress field and energy release rate. Figure 6 and Figure 7 respectively show the stress distribution of the plate with an elliptical hole under tensile stress, where the peridynamic and finite element models are adopted respectively. Generally speaking, the non - local stress results obtained from the peridynamic model are in good agreement with the finite element solution, thus verifying the effectiveness of the non - local stress formula (3). In particular, the stress in the \(y\) - direction at a distance \(l\) from the hole along the \(x\) - axis is given by Figure 7 considering different peridynamic range \(\delta\) values. As shown in the figure, except near the hole boundary, the peridynamic non - local stress coincides with the finite element solution; the non - local stress at the hole mouth approaches the finite element solution as the peridynamic range \(\delta\) decreases. However, the non - local stress at the hole boundary is unreliable. This is because in the non - local peridynamic theory, the nodes within a distance less than the peridynamic range \(\delta\) from the hole boundary do not have a complete peridynamic range, and their corresponding effective elastic response is different from the elastic properties inside the structure. This phenomenon is called the surface effect. At the same time, the surface effect phenomenon is enhanced at the stress concentration of the hole mouth; however, with the decrease of the peridynamic range, the error caused by the surface effect can be weakened to a certain extent.

[0122] Step 3: Apply a symmetric initial crack \(\sigma\) * . Figure 8 gives the numerical solution of the energy release rate obtained by PD_FCE The curve of the change of the finite crack length l with respect to the distance from the hole mouth, considering different values of the near-field range. As Figure 8 shown, the values obtained by PD_FCE approach the finite element values continuously as the near-field range δ decreases. Additionally, the values near the hole mouth are also affected by the surface effect of the hole mouth, and this effect also weakens as δ decreases.

[0123] As Figure 9 shown, considering both the peridynamic stress distribution ( Figure 7 ) and the energy release rate ( Figure 8 ), and using Equation (5), the mixed crack length a * at crack initiation and the critical load value σ * for the flat plate with an elliptical hole under tensile stress can be obtained. Among them, the curves of the critical load obtained from the stress condition (blue line) and the energy condition (red line) with respect to the length l intersect at the mixed point (2.66 mm, 332.79 MPa), that is, the mixed crack length a * = 2.66 mm and the critical tensile load σ * = 332.79 MPa that satisfy both the stress and energy conditions.

[0124] Step 4: Use the bond-breaking criterion to judge the state of the bond. Next, the peridynamic mixed failure model is used to analyze the entire process of tensile failure of the flat plate with a hole, considering the case where the elliptical dimensions are a = 10 mm and b = 5 mm, and using the mixed crack length a * = 2.66 mm obtained above. As Figure 5 shown, the flat plate is subjected to a symmetric displacement load that increases linearly, where the velocity of the displacement load is 4.0×10 -9 m per ADR iteration step. In this case, the process in Figure 1 is used for analysis.

[0125] Figure 10 shows the stress-strain curve during the failure process of the flat plate with a hole under a linear displacement load, where the stress and strain are the average values of the flat plate:

[0126]

[0127] As Figure 10 shown, the stress increases linearly with the displacement load first. When Equation (5) is satisfied, the strength of the flat plate fails and the stress immediately drops to 0. Table 2 gives the critical load values corresponding to different near-field ranges.

[0128] After the strength of the flat plate with a hole fails, the crack begins to propagate. Figure 11 gives the curve of the total fracture energy of the system with respect to the total crack length 2l during the crack propagation process, where the fracture energy is calculated by the formula

[0129]

[0130] Obtained as Figure 11 shown, the curve starts from the starting point (2a * , 2hG c a * ) and increases linearly in a stepped manner with the crack length; as the near-field range decreases, the stepped curve gradually approaches a straight line. Among them, Table 2 gives the numerical critical energy release rate and the total fracture energy value at the completion of crack propagation corresponding to different near-field ranges. It can be seen from Table 2 that the mixed model can accurately predict the tensile critical load of the perforated structure and the energy release during the crack propagation process after the structure fails. In addition, the final failure mode of the flat plate with an elliptical hole under tensile load is given by Figure 12 .

[0131] Table 2 Strength and fracture analysis results of flat plates with elliptical holes corresponding to different near-field ranges

[0132]

[0133] For Figure 5 the elliptical hole, three different ellipticities a / b = 0.5, 1, and 2 are considered; in each ellipticity a / b, the length a is continuously increased from 4 mm to 16 mm to analyze the influence of the ellipticity and the size of the ellipse on the critical load. In the peridynamic model, the discretization method δ = 2 mm, m = 4 is considered. For each ellipticity a / b and length a, the mixed crack length a * and the critical load value σ * are solved using the mixed model, and the solution results are compared with the analytical solution of the finite fracture mechanics (FFM) for a circular hole (a / b = 1).

[0134] Figure 13 Shows the curves of the mixed crack length a * and the critical load value σ * / σ c with the change of the semi-major axis a of the ellipse. As shown in the figure, the values of a * and σ * / σ c change with the change of the ellipticity a / b and the length a; generally speaking, the larger the ellipticity a / b, the smaller the mixed crack length a * . When a / b ≥ 1, the value of a * will increase slightly with a; but when a / b = 1 / 2, this trend no longer exists. At the same time, the critical load value σ * / σ c decreases significantly with a; when the value of a is relatively large, the ellipticity a / b has an impact on the critical tensile load σ * / σ c has little influence, which is in good agreement with the experimental conclusions in the literature. In addition, when a / b = 1, the values of a * and σ * / σ c obtained by the peridynamics numerical model are in very good agreement with the analytical solutions of the FFM model, with the maximum errors being 2.3% and 3.8% respectively.

[0135] Example Two

[0136] The present invention also provides a numerical simulation system for predicting the residual strength and failure mode of a perforated circular plate, including: a construction module, an initialization module, a stress analysis module, a stage conversion module, and a fracture analysis module;

[0137] The construction module is used to establish a peridynamics hybrid failure model that simultaneously considers stress and energy conditions based on the strength and fracture characteristics of flat plates with singular and non-singular defects;

[0138] The initialization module is used to initialize the peridynamics hybrid failure model, determine material, geometric, and peridynamics parameters, and obtain the hybrid crack length using peridynamics based on the material, geometric, and peridynamics parameters;

[0139] The stress analysis module is used to input the hybrid crack length into the initialized peridynamics hybrid failure model for stress analysis;

[0140] The stage conversion module is used to enter the stage conversion point if the result of the stress analysis meets the preset requirements, and apply symmetric initial cracks to the flat plate with an elliptical hole;

[0141] The fracture analysis module is used to perform fracture analysis on the flat plate with an elliptical hole based on the hybrid crack length and critical load that simultaneously satisfy stress and energy conditions, and obtain the structural critical load and crack propagation energy.

[0142] In this embodiment, in the initialization module, the process of obtaining the hybrid crack length using peridynamics based on the material, geometric, and peridynamics parameters includes:

[0143] Obtained according to the conditions that stress and energy need to satisfy when the structure fails in peridynamics theory; among them, the failure of the structure adopts a bond-breaking criterion based on the critical elongation rate or critical bond energy density.

[0144] In this embodiment, in the stress analysis module, the process of inputting the hybrid crack length into the initialized peridynamics hybrid failure model for stress analysis includes:

[0145] Apply a linearly increasing displacement load, and numerically solve peridynamics to obtain F, ε, and σ;

[0146] Determine the relationship between the stress and the tensile strength. If the stress in the y direction is greater than the tensile strength, enter the stage conversion point. If the stress in the y direction is less than the tensile strength, continue the stress analysis;

[0147] Among them, the method of applying a linearly increasing displacement load and obtaining F, ε, and σ through peridynamic numerical solution includes:

[0148] Discretize the nodes x of the peridynamic numerical model i Nonlocal deformation gradient F i It is expressed as:

[0149]

[0150] In the formula, M is the node x i Nodes x within the peridynamic range j The total number of Y Is the deformation vector state, Is the product of two vectors, ω is the influence function, B(x i ) is the shape tensor, V j Is the volume;

[0151] Using the nonlocal deformation gradient F and considering the constitutive equation of continuum mechanics, the nonlocal strain and stress tensors of an isotropic material are:

[0152]

[0153] In the formula, I is the identity matrix and D is the elastic stiffness matrix.

[0154] In this embodiment, in the stage conversion module, the process of obtaining the mixed crack length and the critical load that simultaneously satisfy the stress and energy conditions includes:

[0155] Simultaneously consider the stress field and the energy release rate of the plate with an elliptical hole, that is, obtain the mixed crack length σ * And the critical load σ * :

[0156]

[0157] In the formula, the stress σ yy (l) and the incremental energy release rate Are linearly and quadratically related to the applied stress load σ0 respectively. At the same time, the critical load σ * And the mixed crack length a * Values are not affected by the applied initial stress load σ0, σ c And G c Are the material tensile strength and the critical energy release rate respectively.

[0158] In this embodiment, in the fracture analysis module, the process of performing fracture analysis on a flat plate with an elliptical hole based on the mixed crack length and critical load that simultaneously satisfy the stress and energy conditions, and obtaining the structural critical load and crack propagation energy includes:

[0159] For the flat plate structure with a hole, as the tensile load increases continuously, the stress at the hole opening increases continuously and reaches:

[0160] σ yy (a + a * , y = 0) = σ c

[0161] where a * is the numerical mixed crack length that simultaneously satisfies the stress and energy conditions obtained from the equation:

[0162]

[0163] where σ yy (x) is the stress in the y direction along the straight line on the x-axis, G inc (l) is the average energy release rate considering the crack elongation as a finite length l, σ c and G c are the material tensile strength and critical energy release rate respectively. Therefore, when σ yy (a + a * , y = 0) = σ c holds, the stress and energy simultaneously satisfy:

[0164]

[0165] At this time, the corresponding tensile load σ N is the critical tensile load σ * of the hole structure.

[0166] The above-described embodiments are only descriptions of the preferred embodiments of the present invention, and do not limit the scope of the present invention. Without departing from the design spirit of the present invention, various deformations and improvements made by those of ordinary skill in the art to the technical solutions of the present invention shall fall within the protection scope determined by the claims of the present invention.

Claims

1. A numerical simulation method for predicting the residual strength and failure form of a circular plate with a hole, characterized in that: The following steps are involved: Based on the strength and fracture characteristics of plates with singular and non-singular defects, a peridynamic hybrid failure model is established that considers both stress and energy conditions. The nonlocal stress field and energy release rate are solved for the peridynamic hybrid failure model to obtain the numerical hybrid crack length. The numerical hybrid crack length is input into the initialized peridynamic hybrid failure model for stress analysis; If the result of stress analysis meets the preset requirements, the stage transition point is entered, and a symmetrical initial crack is applied to the plate with elliptical hole to obtain the mixed crack length and critical load that simultaneously meets the stress and energy conditions; Based on the mixed crack length and critical load that simultaneously satisfies stress and energy conditions, the critical elongation or critical bond energy density bond breaking criterion is used to perform fracture analysis on the plate with elliptical holes, and the critical load and crack extension energy are obtained. Methods for inputting numerical hybrid crack length into the initialized peridynamic hybrid failure model for stress analysis include: Applying a linearly increasing displacement load, the peridynamic numerical solution is obtained , and ; Determine the relationship between stress and tensile strength. If the stress in the y direction is greater than the tensile strength, enter the stage transition point and apply a symmetrical initial crack to the plate with an elliptical hole. If the stress in the y direction is less than the tensile strength, continue the stress analysis. In which, a linearly increasing displacement load is applied and the peridynamic numerical solution is obtained: , and The methods include: Discretized Peridynamic Numerical Model Node Nonlocal deformation gradient It is expressed as: In the formula, Is a node Nodes within near field range The total number of is the deformation vector state, is the product of two vectors, is the influence function, is a tensor of shape, is the volume; Using non-local deformation gradient , considering the constitutive equation of continuum mechanics, the nonlocal strain and stress tensors of isotropic materials are: In the formula, is the identity matrix, is the elastic stiffness matrix.

2. The numerical simulation method for predicting residual strength and failure form of a circular plate with a hole according to claim 1 is characterized in that: Based on material, geometry and peridynamic parameters, using stress and energy mixed failure theory, the method of obtaining the numerical mixed crack length includes: solving the peridynamic non-local stress field and solving the peridynamic energy release rate.

3. The numerical simulation method for predicting residual strength and failure form of a circular plate with a hole according to claim 1 is characterized in that: Methods to obtain mixed crack length and critical load that satisfy both stress and energy conditions include: The stress field and energy release rate of the plate with an elliptical hole are considered at the same time, that is, the mixed crack length that satisfies both stress and energy conditions is obtained through the peridynamic numerical model. and critical load : In the formula, stress and incremental energy release rate With the applied stress load They are linear and square correlations respectively. At the same time, the critical load obtained by numerical solution is and mixed crack length The value will not be affected by the initial stress load applied The impact of and are the material tensile strength and critical energy release rate respectively.

4. The numerical simulation method for predicting residual strength and failure form of a circular plate with a hole according to claim 1 is characterized in that: Based on the mixed crack length and critical load that simultaneously satisfies the stress and energy conditions, the fracture analysis of the plate with an elliptical hole is performed to obtain the critical load of the structure and the crack extension energy. The methods include: For a flat plate structure with a hole, as the tensile load continues to increase, the hole stress continues to increase and reaches: In the formula, It is given by the equation: The obtained numerical mixed crack length that satisfies both stress and energy conditions is: It is along Axis Directional stress, The crack extension is considered to be finite length The average energy release rate, and are the tensile strength and critical energy release rate of the material, so when When established, stress and energy simultaneously satisfy: The corresponding tensile load is is the critical tensile load of the hole structure .

5. A numerical simulation system capable of predicting the residual strength and failure form of a circular plate with a hole, characterized in that: include: Construction module, initialization module, stress analysis module, phase transition module and fracture analysis module; The building module is used to establish a peridynamic hybrid failure model that considers both stress and energy conditions based on the strength and fracture characteristics of a plate with singular and non-singular defects; The initialization module is used to solve the non-local stress field and energy release rate of the peridynamic hybrid failure model to obtain the numerical hybrid crack length; The stress analysis module is used to input the numerical hybrid crack length into the initialized peridynamic hybrid failure model to perform stress analysis; The stage transition module is used to enter the stage transition point if the result of the stress analysis meets the preset requirements, and to apply a symmetrical initial crack to the plate with the elliptical hole; The fracture analysis module is used to perform fracture analysis on the flat plate with elliptical holes based on the mixed crack length and critical load that simultaneously meet stress and energy conditions, and obtain the structural critical load and crack extension energy; In the stress analysis module, the process of inputting the hybrid crack length into the initialized peridynamic hybrid failure model for stress analysis includes: Applying a linearly increasing displacement load, the peridynamic numerical solution is obtained , and ; Determine the relationship between stress and tensile strength. If the stress in the y direction is greater than the tensile strength, enter the stage transition point and apply a symmetrical initial crack to the plate with an elliptical hole. If the stress in the y direction is less than the tensile strength, continue the stress analysis. In which, a linearly increasing displacement load is applied and the peridynamic numerical solution is obtained: , and The methods include: Discretized Peridynamic Numerical Model Node Nonlocal deformation gradient It is expressed as: In the formula, Is a node Nodes within near field range The total number of is the deformation vector state, is the product of two vectors, is the influence function, is a tensor of shape, is the volume; Using non-local deformation gradient , considering the constitutive equation of continuum mechanics, the nonlocal strain and stress tensors of isotropic materials are: In the formula, is the identity matrix, is the elastic stiffness matrix.

6. The numerical simulation system capable of predicting residual strength and failure form of a circular plate with a hole according to claim 5 is characterized in that: In the initialization module, based on material, geometry and peridynamic parameters, the process of obtaining the mixed crack length by using peridynamic solution includes: solving the peridynamic non-local stress field and solving the peridynamic energy release rate.

7. The numerical simulation system capable of predicting residual strength and failure form of a circular plate with a hole according to claim 5, characterized in that: In the stage conversion module, the process of obtaining the mixed crack length and critical load that simultaneously satisfies the stress and energy conditions includes: The stress field and energy release rate of the plate with an elliptical hole are considered at the same time, that is, the mixed crack length that satisfies both stress and energy conditions is obtained through the peridynamic numerical model. and critical load : In the formula, stress and incremental energy release rate With the applied stress load They are linear and square correlations respectively. At the same time, the critical load obtained by numerical solution is and mixed crack length The value will not be affected by the initial stress load applied The impact of and are the material tensile strength and critical energy release rate respectively.

8. The numerical simulation system capable of predicting residual strength and failure form of a circular plate with a hole according to claim 5, characterized in that: In the fracture analysis module, based on the mixed crack length and critical load that simultaneously satisfy the stress and energy conditions, a bond breaking criterion based on critical elongation or critical bond energy density is adopted to perform fracture analysis on the plate with an elliptical hole, and the process of obtaining the critical load of the structure and the crack extension energy includes: For a flat plate structure with a hole, as the tensile load continues to increase, the hole stress continues to increase and reaches: In the formula, It is given by the equation: The obtained numerical mixed crack length that satisfies both stress and energy conditions is: It is along Axis Directional stress, The crack extension is considered to be finite length The average energy release rate, and are the tensile strength and critical energy release rate of the material, so when When established, stress and energy simultaneously satisfy: The corresponding tensile load is is the critical tensile load of the hole structure .

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