An analytical method for material damage problems based on the base surface force element method

Abstract

This invention provides an analytical method for material damage problems based on the basal surface force element method, specifically including: 1) establishing a basal surface force element model for static damage problems based on the potential energy principle; 2) constructing dynamic basal surface force element equations for dynamic material damage problems; 3) solving the dynamic basal surface force element equations in step 2) using the Newmark-β method to obtain the damage evolution process of the material under dynamic loads. This invention is logically conceived, significantly improves adaptability to mesh distortion, has a simpler mathematical model, and significantly enhances the efficiency of obtaining material damage information. It can be applied to finite element damage analysis of materials and has broad application prospects.

Description

Technical Field

[0001] This invention relates to the field of materials mechanics, and more specifically to an analytical method for material damage problems based on the basal surface force element method. Background Technology

[0002] Mechanical damage analysis of materials has always been an important research direction in engineering, especially in industries such as construction, aerospace, and civil engineering, where the performance requirements for materials are extremely high. Reliable prediction of mechanical properties is crucial for engineering safety. To gain a deeper understanding of the damage evolution process of materials under various external loads, researchers have proposed a variety of numerical analysis methods, among which the finite element method (FEM) is the most widely used. In recent years, with the development of computing technology, the FEM has demonstrated powerful capabilities in simulating complex material structures and nonlinear behaviors. However, the traditional FEM still faces many challenges in handling large deformations, mesh distortions, and complex material damage behaviors, especially in the study of complex boundary conditions and the microstructure of materials.

[0003] Current material damage analysis mainly relies on the traditional finite element method and some improved discontinuous medium models. However, these methods face the following technical challenges when dealing with complex geometries and large deformations:

[0004] (1) Mesh dependency and mesh distortion problems

[0005] When dealing with large deformation problems, the traditional finite element method suffers from decreased computational accuracy due to mesh distortion, which can even prevent the calculation from converging. This poses a great challenge to the accurate simulation of material failure processes, especially in the crack initiation and propagation stages, where mesh re-division is time-consuming and complex.

[0006] (2) Low computational efficiency

[0007] Existing finite element method damage analysis often requires mesh refinement to improve computational accuracy, which leads to a significant increase in computational load, long computation time, and low efficiency, making it difficult to meet real-time or high-efficiency engineering requirements.

[0008] In conclusion, it is necessary to further innovate existing technologies. Summary of the Invention

[0009] To address the technical problems existing in the background art, this invention proposes an analysis method for material damage problems based on the basal force element method. Its concept is reasonable, it can significantly improve the adaptability to mesh distortion, the mathematical model is more concise, and it significantly improves the efficiency of obtaining material damage problems. It can be applied to the finite element damage analysis of materials and has a relatively broad application prospect.

[0010] To address the aforementioned technical problems, this invention provides an analytical method for material damage problems based on the basal force element method, which specifically includes the following steps:

[0011] 1) Based on the principle of potential energy, a base force element model for static damage problems is established.

[0012] 2) Construct the dynamic datum force element equations for the material dynamic damage problem;

[0013] 3) Use the Newmark-β method to solve the dynamic datum force element equation in step 2) above to obtain the damage evolution process of the material under dynamic load.

[0014] The method for analyzing material damage problems based on the basal force element method, wherein the process of constructing the strain and stiffness matrices of the basal force elements for the static damage problem in step 1) is as follows:

[0015] Consider a three-dimensional elastic body region, where Q represents the radial vector after deformation, and x i (representing Lagrange coordinates, i = 1, 2, 3, then the deformed coordinate frame is)

[0016] To describe the stress state near point Q, the current configuration Q is constructed as a parallelepiped element with three edges dx and dx. 1 Q1, dx 2 Q2, dx 3 Q3, the force on its corresponding positive side is denoted as dT. 1 dT 2 dT 3 The base force is expressed as follows:

[0017]

[0018] In the above formula (1), it is agreed that 3+1=1 and 1-1=3; T i Let x be the coordinate system i The base surface forces at point Q, i = 1, 2, 3;

[0019] The relationship between the base force and the Cauchy stress tensor σ is:

[0020]

[0021] In equation (2) above, V Q For x i The system's basic capacity, Q i These are the basis vectors after deformation;

[0022] According to the basic force theory, the stiffness matrix for both two-dimensional and three-dimensional problems is expressed as:

[0023]

[0024] In equation (3), E and ν are the elastic modulus and Poisson's ratio, respectively; A is the area or volume of the base force element; U is the unit tensor; m I and m J It is a constant related to the shape of the base force element, m IJ =m I ·m J ;

[0025] When the strain deviation of the base force element is negligible, the average strain is used instead of the true strain.

[0026] For a two-dimensional problem, the three components of the average strain of the base force element are as follows:

[0027]

[0028] In equations (4)-(6) above, I is the element node number, n is the number of element nodes, A is the area of ​​the base force element, and u Ix and u Iy These represent the x and y displacements of the nodes of the base force element, respectively. and It is a constant related to the shape of the base force element.

[0029] For a three-dimensional problem, the six components of the average strain of the base force element are as follows:

[0030]

[0031]

[0032] In equations (7)-(12) above, I is the element node number, n is the number of element nodes, and V is the volume of the base force element; u Ix u Iy and u Iz These represent the displacements in the x, y, and z directions of the nodes of the base force element, respectively; and It is a constant related to the shape of the base force element.

[0033] The method for analyzing material damage problems based on the basal force element method, wherein the solution of the basal force element model for the static damage problem during the material damage and failure process in step 1) is performed using a direct iteration method, the specific steps of which are as follows:

[0034] 1.2.1) Provide the initial values ​​for solving the nodal displacement field, stress field and damage factor field during the iteration process, for the I-th load step, i.e. at time t+Δt;

[0035] 1.2.2) The iteration step counter is defined as n, which is used to record the number of iterations experienced in the solution process under the current load step, so as to determine whether the iteration has converged or whether it is necessary to continue iterating. Initialize n = 0;

[0036] Forming the effective stiffness matrix:

[0037]

[0038] In equation (13) above, K is the effective stiffness matrix of the entire structure; Let be the damage factor of the e-th base surface force element in the nth iteration; Let be the initial stiffness matrix of the e-th base surface force element;

[0039] For displacement loading, the effective load matrix is ​​formed using the multiplication method:

[0040]

[0041] In equation (14) above, P I For the overall effective load array; For external load array; k b The coefficients of the penalty function corresponding to the boundary conditions; u b The known displacement vector is the boundary condition for the displacement.

[0042] 1.2.3) Perform the nth iteration to iteratively update the node displacements;

[0043] 1.2.3.1) Calculate the effective stiffness matrix terms:

[0044]

[0045] 1.2.3.2) Calculate nodal displacements:

[0046]

[0047] 1.2.3.3) Calculate the stress-strain state of each basal force element in the basal force element analysis of step 2) above, and determine the damage state of the element;

[0048] 1.2.3.4) Calculate the damage factor of the base surface force element:

[0049]

[0050] In equation (17), the damage factor V refers to the damage measure corresponding to the damaged area that the element cannot withstand under damaged conditions. n V is the effective volume of the undamaged part that can actually bear the load. d For the damage volume; after calculating the updated damage factor Then, the overall effective stiffness matrix K is updated again according to the above formula (15) for the next iteration to solve the nodal displacement field, so as to ensure that the stiffness matrix matches the current damage state.

[0051] 1.2.3.5) Calculate the nominal stress and nominal strain of the material specimen under load.

[0052] The nominal stress is obtained by the following formula:

[0053]

[0054] In equation (18), F is the loading force on the specimen, and A is the cross-sectional area of ​​the specimen.

[0055] The nominal strain is obtained by the following formula:

[0056]

[0057] In equation (19), ΔL is the axial deformation of the specimen, and L is the initial length of the specimen.

[0058] 1.2.3.6) Verify whether the iteration converges, that is, check whether the node displacements satisfy the convergence criterion:

[0059]

[0060] In equation (20) above, u (n) and u (n-1) These are the node displacement vectors for the current and previous iterations, respectively;

[0061] If the requirements are not met and the number of iterations is less than the maximum allowable number of iterations, proceed to the next iteration. If the number of iterations is greater than the maximum allowable number of iterations or the accuracy requirements are met, proceed to step 1.2.4 below.

[0062] 1.2.4) Determine whether the specimen has become unstable and failed or reached the given load step. If so, stop the calculation. Otherwise, let I = I + 1 and proceed to the next loading step, that is, increase the load to the next predetermined level and return to step 1.2.1) above to restart the iterative calculation process to simulate the damage evolution process of the material under gradually increasing load.

[0063] The method for analyzing material damage problems based on the basal force element method, wherein step 2) is specifically as follows:

[0064] The dynamic equilibrium equations of each node of the base surface force element are obtained by applying d'Alembert's principle. Based on the dynamic equilibrium equations, the dynamic base surface force element equations are constructed as follows:

[0065]

[0066] In equation (21), ü t , and u t Let M, C, and K represent the acceleration, velocity, and displacement vectors of each node in the structure, respectively; M, C, and K represent the overall mass matrix, damping matrix, and stiffness matrix of the structure, respectively; P t This represents the array of dynamic loads on the structure.

[0067] The method for analyzing material damage based on the base-plane force element method, wherein the mass matrix adopts the lumped mass matrix method, and for planar elements, the mass of the element is evenly distributed to the nodes according to the static equivalence principle. The lumped mass matrix of the planar element can be expressed as:

[0068]

[0069] In the above formula (22), ρ is the element density, A is the element area, T is the element thickness, and n is the number of element nodes.

[0070] The method for analyzing material damage problems based on the basal force element method, wherein the damping matrix adopts the Rayleigh damping model, which can be expressed as a linear combination of the mass matrix and the stiffness matrix:

[0071] C = αM + βK (23);

[0072] In equation (23), α and β are frequency-independent proportionality coefficients.

[0073] The method for analyzing material damage problems based on the basal force element method, wherein in step 3), for a linear system (i.e., before material damage occurs, the elastic modulus of the material remains constant during loading, and the stiffness matrix of the structure does not change with displacement or time), the specific steps for calculating the dynamic basal force element equations of the material using the Newmark-β method are as follows:

[0074] 3.1) Initial Calculation

[0075] 3.1.1) Determine the initial motion values ​​ü0 of the system. And u0, ü0 are the nodal acceleration vectors at the initial time (t=0). Let u0 be the node velocity vector at the initial moment, and u0 be the node displacement vector at the initial moment.

[0076] 3.1.2) The entire structure is discretized using the base surface force element method, and the overall stiffness matrix K and mass matrix M are obtained by assembling according to the element stiffness and mass characteristics; the damping matrix C adopts the Rayleigh damping model and is expressed as a linear combination of the stiffness matrix K and the mass matrix M: C = αM + βK, where α and β are damping coefficients;

[0077] 3.1.3) Select the time step Δt, control parameters β and γ (β = 0.25(0.5 + γ) 2 , γ≥0.5), calculate the integration constant;

[0078]

[0079] 3.1.4) Forming the effective stiffness matrix

[0080]

[0081] 3.2) Calculation for each time step

[0082] 3.2.1) Calculate the effective load vector of the system at time t+Δt:

[0083]

[0084] 3.2.2) Solve for the displacements of each node of the base force element at time t+Δt:

[0085]

[0086] 3.2.3) Calculate the acceleration and velocity of each node of the base force element at time t+Δt:

[0087]

[0088] The method for analyzing material damage problems based on the basal force element method, wherein in step 3), for nonlinear systems (i.e., when material damage occurs, the elastic modulus of the material is related to the degree of material damage during loading), the steps for directly iteratively solving the dynamic basal force element equations of the material damage problem based on the Newmark-β method are as follows:

[0089] 4.1) Initial assignment

[0090] Set the time step Δt, and the Newmark-β method parameters β and γ;

[0091] Set initial displacement u0 and velocity and acceleration

[0092] Initialize damage factors Indicates no damage;

[0093] 4.2) Time step loop, from n=0 to the total number of time steps:

[0094] 4.2.1) Iterative Loop

[0095] 4.2.1.1) Initialize the iteration counter k = 0, and set the convergence tolerance ∈;

[0096] Repeat steps (a) to (g) until the convergence condition is met or the maximum number of iterations is reached:

[0097] (a) Calculate the stiffness matrix after damage;

[0098] (b) Form an effective stiffness matrix;

[0099] (c) Calculate the effective load vector;

[0100] (d) Solve for nodal displacements;

[0101] (e) Update acceleration and velocity;

[0102] (f) Calculate the new stress and strain, and update the damage factor.

[0103] (g) Check the convergence condition:

[0104] like Then the iteration converges;

[0105] Otherwise, let k = k + 1 and return to step (a) above;

[0106] 4.2.2) Time step progression

[0107] a. Use the converged displacement, velocity, and acceleration as the initial values ​​for the next time step;

[0108] Update time t n =t n +Δt;

[0109] 4.3) Termination Conditions

[0110] The calculation ends when the total calculation time is reached or the specimen fails, i.e., when the damage factor reaches the critical value.

[0111] Through the above steps 4.1)-4.3), the damage evolution process of materials under dynamic load can be accurately simulated, and the dynamic response of the specimen can be obtained.

[0112] By adopting the above technical solution, the present invention has the following beneficial effects:

[0113] The material damage analysis method based on the basal force element method of this invention is reasonably conceived. The mathematical model derivation of the basal force element method is novel, the formula is concise and clear, and the physical meaning is more intuitive and clear. This not only simplifies the calculation process and reduces the computational complexity, but also makes it more convenient and faster to implement in programming, and significantly improves the computational efficiency.

[0114] Furthermore, the material damage analysis method based on the basal force element method of this invention can effectively simulate the mechanical behavior of materials during damage and failure processes, and is applicable to the analysis of complex structures and nonlinear problems. This method has significant application value in the finite element damage analysis of materials and can be widely used in the design and safety assessment of engineering structures, providing a new research tool for the fields of materials science and engineering mechanics, and has broad application prospects. Attached Figure Description

[0115] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0116] Figure 1 This is a flowchart of the material damage analysis method based on the basal force element method of the present invention;

[0117] Figure 2 This is a schematic diagram of the basal forces involved in the material damage problem analysis method based on the basal force element method of this invention. Detailed Implementation

[0118] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0119] The present invention will be further explained below with reference to specific embodiments.

[0120] like Figure 1 As shown in the figure, this embodiment provides an analysis method for static and dynamic material damage problems based on the basal surface force element method, which can be applied to the finite element damage analysis of materials. The specific steps are as follows:

[0121] S100. Based on the principle of potential energy, a base surface force element model for static damage problems is established.

[0122] S110, Construction of strain and stiffness matrices for basic surface force elements in static damage problems.

[0123] The basal force element method, as a novel finite element method, possesses numerous unparalleled advantages over other methods, whether based on the complementary energy principle or the potential energy principle. In this invention, the strain and stiffness matrices of the basal force element are derived using the potential energy principle and expressed explicitly, without the need for Gaussian integrals.

[0124] Consider a three-dimensional elastic body region, where Q represents the radial vector after deformation, and x i (i = 1, 2, 3) represents Lagrange coordinates, then the transformed coordinate frame is:

[0125] To describe the stress state near point Q, the current configuration Q is constructed as a parallelepiped element with three edges dx and dx. 1 Q1, dx 2 Q2, dx 3 Q3, the force on its corresponding positive side is denoted as dT. 1 dT 2 dT 3 ,like Figure 2 As shown, the base surface forces are represented as follows:

[0126]

[0127] In equation (1), it is agreed that 3+1=1 and 1-1=3; T i (i = 1, 2, 3) represents the coordinate system x i The base force at point Q;

[0128] The relationship between the base force and the Cauchy stress tensor σ is:

[0129]

[0130] In equation (2), V Q For x i The system's basic capacity, Q i These are the basis vectors after deformation;

[0131] According to the basic force theory, the stiffness matrix for both two-dimensional and three-dimensional problems is expressed as:

[0132]

[0133] In equation (3), E and ν are the elastic modulus and Poisson's ratio, respectively; A is the area (two-dimensional) or volume (three-dimensional) of the base force element; U is the unit tensor; m I and m J It is a quantity related to the shape of the base force element, m IJ =m I ·m J ;

[0134] When the strain deviation of the base force element is negligible, the average strain is used instead of the true strain.

[0135] For a two-dimensional problem, the three components of the average strain of the base force element are as follows:

[0136]

[0137] In equations (4)-(6) above, I is the element node number, n is the number of element nodes, and A is the area of ​​the base force element; u Ix and u Iy These represent the x and y displacements of the nodes of the base force element, respectively. and It is a constant related to the shape of the base force element;

[0138] For a three-dimensional problem, the six components of the average strain of the base force element are as follows:

[0139]

[0140]

[0141] In equations (7)-(12) above, I is the element node number, n is the number of element nodes, V is the volume of the base force element, and u Ix u Iy and u Iz These represent the displacements in the x, y, and z directions of the nodes of the base force element, respectively. and It is a constant related to the shape of the base force element.

[0142] The stiffness matrix of the static damage problem during material damage and failure is solved using a direct iteration method, with the specific steps as follows:

[0143] S121. Give initial values ​​(initial values ​​refer to the initial values ​​used to solve the nodal displacement field, stress field and damage factor field during the iteration process; specifically, when the calculation of the I-th load step begins, that is, at the corresponding loading time, the initial value of the nodal displacement field is assigned to the nodal displacement value when the previous load step finally converges), for the I-th load step, that is, at time t+Δt.

[0144] S122. Initialize n = 0 (n is an iteration step counter, used to record the number of iterations experienced in the solution process under the current load step, so as to determine whether the iteration has converged or whether it needs to continue iterating);

[0145] Forming the effective stiffness matrix:

[0146]

[0147] In equation (13) above, K is the effective stiffness matrix of the entire structure; Let be the damage factor of the e-th base surface force element in the nth iteration; Let be the initial stiffness matrix (stiffness matrix before damage) of the e-th base surface force element;

[0148] For displacement loading, the effective load matrix is ​​formed using the multiplication method:

[0149]

[0150] In equation (14) above, P I For the overall effective load array; For external load array; k b The coefficients of the penalty function corresponding to the boundary conditions (usually taken as a very large value); u b The known displacement vector is the boundary condition for the displacement.

[0151] S123. Perform the nth iteration and iteratively update the node displacements:

[0152] S1231. Calculate the effective stiffness matrix terms:

[0153]

[0154] S1232. Calculate nodal displacements:

[0155]

[0156] S1233. Calculate the stress-strain state of each basal force element (an element refers to a unit divided in the basal force element analysis, such as a planar triangular element in a two-dimensional problem or a tetrahedral element in a three-dimensional problem) in the basal force element analysis of step 2) above, and determine the damage state of the element.

[0157] S1234. Calculate the damage factor of the base surface force element:

[0158] D = V d / V n (0≤D≤1) (17);

[0159] In equation (17), the damage factor V refers to the damage measure corresponding to the damaged area that the element cannot withstand under damaged conditions. n V is the undamaged equivalent volume that can actually bear the load; d For the damage volume, calculate the effective stiffness matrix; then calculate the updated damage factor. Then, the overall effective stiffness matrix K is updated again according to the above formula (15) for the next iteration to solve the nodal displacement field, so as to ensure that the stiffness matrix matches the current damage state.

[0160] S1235. Calculate the macroscopic nominal stress and nominal strain of the material specimen under load.

[0161] The nominal stress is obtained by the following formula:

[0162]

[0163] In equation (18), F is the loading force on the specimen, and A is the cross-sectional area of ​​the specimen.

[0164] The nominal strain is obtained by the following formula:

[0165]

[0166] In equation (19), ΔL is the axial deformation of the specimen, and L is the initial length of the specimen.

[0167] S1236. Verify whether the iteration converges, that is, check whether the node displacements meet the convergence criteria:

[0168]

[0169] In equation (20) above, u (n) and u (n-1) These are the node displacement vectors for the current and previous iterations, respectively;

[0170] If the requirements are not met and the number of iterations is less than the maximum allowable number of iterations, proceed to the next iteration. If the number of iterations is greater than the maximum allowable number of iterations or the accuracy requirements are met, proceed to step S124 below.

[0171] S124. Determine whether the specimen has become unstable and failed or reached the given load step. If it does, stop the calculation. Otherwise, let I = I + 1 and proceed to the next loading step, that is, increase the load to the next predetermined level, and return to step S121 above to restart the iterative calculation process to simulate the damage evolution process of the material under gradually increasing load.

[0172] S200, Constructing the dynamic datum force element equations for material dynamic damage problems.

[0173] The difference between dynamic and static analysis is that dynamic analysis needs to consider inertial forces. Therefore, a mass matrix and a damping matrix are added to the equilibrium equations. In dynamic analysis, a time coordinate is introduced. Using d'Alembert's principle, the dynamic equilibrium equations for each node of the base surface force element can be obtained. In constructing the dynamic equations, the base surface force element method uses the same method as the conventional finite element method. Therefore, the dynamic base surface force element equations are as follows:

[0174]

[0175] In equation (21),

[0176] and u t Let M, C, and K represent the acceleration, velocity, and displacement vectors of each node in the structure, respectively; M, C, and K represent the overall mass matrix, damping matrix, and stiffness matrix of the structure, respectively; P t This represents the array of dynamic loads on the structure.

[0177] In this invention, the mass matrix adopts the lumped mass matrix method; for example, for a planar element, the mass of the element is evenly distributed to the nodes according to the static equivalence principle, and the lumped mass matrix of the planar element can be expressed as:

[0178]

[0179] In equation (22), ρ is the element density, A is the element area, T is the element thickness, and n is the number of element nodes.

[0180] The damping matrix adopts the Rayleigh damping model, which can be expressed as a linear combination of the mass matrix and the stiffness matrix:

[0181] C = αM + βK (23);

[0182] In equation (23), α and β are frequency-independent proportionality coefficients.

[0183] S300. Solve the dynamic datum force element equations of step S200 using the Newmark-β method to obtain the damage evolution process of the material under dynamic load.

[0184] For a linear system (i.e., the elastic modulus of the material remains constant during loading, and the stiffness matrix of the structure does not change with displacement or time), the Newmark-β method is used for calculation (the Newmark-β method is a commonly used direct integration method widely applied in structural dynamics to solve second-order ordinary differential equations and calculate the response of a structure under dynamic loads). The steps are as follows:

[0185] S310, Initial Calculation

[0186] S311. Determine the initial motion value ü0 of the system. And u0, ü0 are the nodal acceleration vectors at the initial time (t=0). Let u0 be the node velocity vector at the initial moment, and u0 be the node displacement vector at the initial moment.

[0187] S312. The entire structure is discretized using the base surface force element method, and the overall stiffness matrix K and mass matrix M are obtained by assembling according to the element stiffness and mass characteristics. The damping matrix C adopts the Rayleigh damping model and is expressed as a linear combination of the stiffness matrix K and the mass matrix M: C = αM + βK, where α and β are damping coefficients.

[0188] S313. Select the time step Δt, control parameters β and γ (β = 0.25(0.5 + γ)). 2 (γ≥0.5), and calculate the integration constant;

[0189]

[0190] S314. Forming an effective stiffness matrix

[0191]

[0192] S320, Calculate for each time step

[0193] S321. Calculate the effective load vector of the system at time t+Δt:

[0194]

[0195] S322. Solve for the displacements of each node of the base force element at time t+Δt:

[0196]

[0197] S323. Calculate the acceleration and velocity of each node of the base force element at time t+Δt:

[0198]

[0199] One simple but very useful method for selecting the time step is to first solve the problem with a seemingly reasonable time step, then repeat the solution with a slightly smaller time step, compare the results, and continue this process until two consecutive solutions are close enough.

[0200] For nonlinear systems, i.e., when material damage occurs, the elastic modulus of the material is related to the degree of material damage during loading. The steps for directly iteratively solving the dynamic base surface force element equations of the material damage problem based on the Newmark-β method are as follows:

[0201] S410, Initial Assignment

[0202] Set the time step Δt, and the Newmark-β method parameters β and γ;

[0203] Set initial displacement u0 and velocity and acceleration ü0;

[0204] Initialize damage factors Indicates no damage;

[0205] S420, Time Step Loop, from n=0 to the total number of time steps:

[0206] S421, Iterative Loop

[0207] S4211) Initialize the iteration counter k = 0 and set the convergence tolerance ε;

[0208] Repeat steps (a) to (g) until the convergence condition is met or the maximum number of iterations is reached:

[0209] (a) Calculate the stiffness matrix after damage;

[0210] (b) Form an effective stiffness matrix;

[0211] (c) Calculate the effective load vector;

[0212] (d) Solve for nodal displacements;

[0213] (e) Update acceleration and velocity;

[0214] (f) Calculate the new stress and strain, and update the damage factor.

[0215] (g) Check the convergence condition:

[0216] like Then the iteration converges;

[0217] Otherwise, let k = k + 1 and return to step (a) above;

[0218] S422, Time Step Advancement

[0219] a. Use the converged displacement, velocity, and acceleration as the initial values ​​for the next time step;

[0220] Update time t n =t n +Δt;

[0221] S430, Termination Conditions

[0222] The calculation ends when the total calculation time is reached or the specimen fails, i.e., when the damage factor reaches the critical value.

[0223] Through the above steps S410-S430, the damage evolution process of materials under dynamic load can be accurately simulated, and the dynamic response of the specimen can be obtained.

[0224] This invention has a reasonable concept, can significantly improve adaptability to mesh distortion, has a simpler mathematical model, and significantly improves the efficiency of obtaining material damage problems. It can be applied to finite element damage analysis of materials and has a broad application prospect.

[0225] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. An analytical method for material damage problems based on the base surface force element method, characterized in that, Specifically, the following steps are included: 1) Based on the principle of potential energy, establish a base force element model for static damage problems; 2) Construct the dynamic datum force element equations for the material dynamic damage problem; The specific process is as follows: The dynamic equilibrium equations of each node of the base surface force element are obtained by applying d'Alembert's principle. Based on the dynamic equilibrium equations, the dynamic base surface force element equations are constructed as follows: （21）； In equation (21), , and These represent the acceleration vector, velocity vector, and displacement vector of each node in the structure, respectively. , , These represent the overall mass matrix, damping matrix, and stiffness matrix of the structure, respectively. This represents the dynamic load array of the structure; The mass matrix adopts the lumped mass matrix method. For planar elements, the mass of the element is evenly distributed to the nodes according to the static equivalence principle. The lumped mass matrix of the planar element can be expressed as: （22）； In the above formula (22), For unit density, For unit area, For unit thickness, This represents the number of unit nodes. The damping matrix adopts the Rayleigh damping model, which can be expressed as a linear combination of the mass matrix and the stiffness matrix: (23)； In equation (23), α and β are frequency-independent proportionality coefficients; 3) Use the Newmark-β method to solve the dynamic datum force element equation in step 2) above to obtain the damage evolution process of the material under dynamic load.

2. The method for analyzing material damage problems based on the basal surface force element method as described in claim 1, characterized in that, The process of constructing the strain and stiffness matrices of the base force elements in step 1) of the static damage problem is as follows: Considering the three-dimensional elastic body region, Represents the radial vector after deformation. Let i = 1, 2, 3 represent Lagrange coordinates. Then the deformed coordinate frame is: ; To describe the stress state near point Q, the current configuration Q is constructed as a parallelepiped element, with its three edges being... , , The force on its corresponding frontal side is denoted as , , The base force is expressed as follows: ， （1）； In the above formula (1), it is agreed that 3+1=1 and 1-1=3; coordinate system middle The base surface forces at points, i=1,2,3; Base surface force and Cauchy stress tensor The relationship is as follows: （2）； In equation (2) above, for The system's basic content, These are the basis vectors after deformation; According to the basic force theory, the stiffness matrix for both two-dimensional and three-dimensional problems is expressed as: （3）； In equation (3), E and ν are the elastic modulus and Poisson's ratio, respectively; A is the area or volume of the base force element; and U is the unit tensor. and It is a constant related to the shape of the base force element. ; When the strain deviation of the base force element is negligible, the average strain is used instead of the true strain. For a two-dimensional problem, the three components of the average strain of the base force element are as follows: （4）； （5）； （6）； In equations (4)-(6) above, I is the element node number, n is the number of element nodes, and A is the area of ​​the base force element. and These represent the x and y displacements of the nodes of the base force element, respectively. and It is a constant related to the shape of the base force element; For a three-dimensional problem, the six components of the average strain of the base force element are as follows: （7）； （8）； （9）； （10）； （11）； （12）； In the above equations (7)-(12), I is the element node number, n is the number of element nodes, and V is the volume of the base force element; , and These represent the displacements in the x, y, and z directions of the nodes of the base force element, respectively; , and It is a constant related to the shape of the base force element.

3. The method for analyzing material damage problems based on the basal surface force element method as described in claim 1, characterized in that, The solution of the basic force element model for the static damage problem during the material damage and failure process in step 1) is achieved using the direct iteration method, and the specific steps are as follows: 1.2.1) Provide initial values ​​for solving the nodal displacement field, stress field, and damage factor field during the iteration process. For the I-th load step, i.e., in time; 1.2.2) The iteration step counter is defined as n, which is used to record the number of iterations in the solution process under the current load step, so as to determine whether the iteration has converged or whether it is necessary to continue iterating. Initialize n=0; Forming the effective stiffness matrix: （13）； In the above formula (13), This is the effective stiffness matrix of the entire structure; Let be the damage factor of the e-th base surface force element in the nth iteration; Let be the initial stiffness matrix of the e-th base surface force element; For displacement loading, the effective load matrix is ​​formed using the multiplication method: （14）； In the above formula (14), For the overall effective load array; For external load array; These are the penalty function coefficients corresponding to the boundary conditions; The known displacement vector is the boundary condition for the displacement. 1.2.3) Perform the nth iteration to iteratively update the node displacements; 1.2.3.1) Calculate the effective stiffness matrix terms: （15）； 1.2.3.2) Calculate nodal displacements: （16）； 1.2.3.3) Calculate the stress-strain state of each basal force element in the basal force element analysis of step 2) above, and determine the damage state of the element; 1.2.3.4) Calculate the damage factor of the base surface force element: （17）； In equation (17), the damage factor It refers to the damage measure corresponding to the damaged area that the element cannot withstand under damaged conditions. The undamaged equivalent volume that can actually bear the load. For the damage volume; after calculating the updated damage factor Then, the overall effective stiffness matrix K is updated again according to the above formula (15) for the next iteration to solve the nodal displacement field, so as to ensure that the stiffness matrix matches the current damage state. 1.2.3.5) Calculate the nominal stress and nominal strain of the material specimen under load. The nominal stress is obtained by the following formula: (18)； In equation (18), The loading force applied to the specimen. The cross-sectional area of ​​the sample; The nominal strain is obtained by the following formula: (19)； In equation (19), This represents the axial deformation of the specimen. The initial length of the sample; 1.2.3.6) Verify whether the iteration converges, that is, check whether the node displacements satisfy the convergence criterion: (20)； In the above formula (20), and These are the node displacement vectors for the current and previous iterations, respectively; If the requirements are not met and the number of iterations is less than the maximum allowable number, proceed to the next iteration. If the number of iterations is greater than the maximum allowable number or the accuracy requirements are met, proceed to step 1.2.4 below. 1.2.4) Determine whether the specimen has become unstable and failed or reached the given load step. If so, stop the calculation; otherwise, let... Then proceed to the next loading step, which is to increase the load to the next predetermined level and return to step 1.2.1 above to restart the iterative calculation process to simulate the damage evolution of the material under gradually increasing load.

4. The method for analyzing material damage problems based on the basal surface force element method as described in claim 1, characterized in that, In step 3), for a linear system, i.e., when the material has not yet been damaged, the elastic modulus of the material remains constant during loading, and the stiffness matrix of the structure does not change with displacement or time, the specific steps for calculating the dynamic datum force element equations of the material using the Newmark-β method are as follows: 3.1) Initial Calculation 3.1.1) Determine the initial values ​​of the system's motion. , and , Let be the nodal acceleration vector at the initial time (t=0). Let be the nodal velocity vector at the initial moment. This represents the nodal displacement vector at the initial moment; 3.1.2) The entire structure is discretized using the basic force element method, and the overall stiffness matrix is ​​obtained by assembling the elements according to their stiffness and mass characteristics. and mass matrix Damping matrix The Rayleigh damping model is adopted, which is expressed as a stiffness matrix. and mass matrix Linear combination: ,in, α and β are damping coefficients; 3.1.3) Select the time step Control parameters β and ( ), calculate the integration constant; ， ， ， ， ， ， ， （24）； 3.1.4) Forming an effective stiffness matrix ; （25）； 3.2) Calculation for each time step 3.2.1) Calculation time The effective load vector of the system: （26）； 3.2.2) Solution time Displacement of each node of the base force element: （27）； 3.2.3) Calculation time The acceleration and velocity of each node of the base force element: （28）； （29）。 5. The method for analyzing material damage problems based on the basal surface force element method as described in claim 4, characterized in that, In step 3), for nonlinear systems, i.e., when material damage occurs, the elastic modulus of the material is related to the degree of material damage during loading. The steps for directly iteratively solving the dynamic basic force element equations of the material damage problem based on the Newmark-β method are as follows: 4.1) Initial assignment Set the time step Δt, and the Newmark-β method parameters β and γ; Set initial displacement ,speed and acceleration ; Initialize damage factors , indicates no damage; 4.2) Time step loop, from n=0 to the total number of time steps: 4.2.1) Iterative Loop 4.2.1.1) Initialize the iteration counter k=0 and set the convergence tolerance. ; Repeat steps (a) to (g) until the convergence condition is met or the maximum number of iterations is reached: (a) Calculate the stiffness matrix after damage; (b) Forming an effective stiffness matrix; (c) Calculate the effective load vector; (d) Solve for nodal displacements; (e) Update acceleration and velocity; (f) Calculate the new stress and strain, and update the damage factor. ; (g) Check convergence conditions: like If the iteration converges, then the iteration will be successful. Otherwise, let k = k + 1 and return to step (a) above; 4.2.2) Time step progression a. Use the converged displacement, velocity, and acceleration as the initial values ​​for the next time step; Update time ; 4.3) Termination Conditions The calculation ends when the total calculation time is reached or when the specimen fails, i.e., when the damage factor reaches the critical value. Through the above steps 4.1)-4.3), the damage evolution process of materials under dynamic load can be accurately simulated, and the dynamic response of the specimen can be obtained.

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