Method for analyzing material damage problem based on base surface force unit method

Abstract

The invention provides a method for analyzing a material damage problem based on a base surface force unit method, which specifically comprises the following steps of: 1) establishing a base surface force unit model of a static force damage problem based on a base surface force unit method of a potential energy principle; 2) constructing a power base surface force element equation of the material power damage problem; and (3) solving the dynamic base surface force element equation in the step (2) by adopting a Newmark-beta method, and obtaining the damage evolution process of the material under the action of the dynamic load. The method is reasonable in conception, the adaptability to grid distortion can be remarkably improved, the mathematical model is simpler, the obtaining efficiency of the material damage problem is remarkably improved, and the method can be applied to finite element damage analysis of materials and has wide application prospects.

Description

Technical Field

[0001] The invention relates to the technical field of material mechanics, and in particular to an analysis method for material damage problems based on a base force unit method. Background Art

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

[0003] Existing material damage problem analysis is mainly based on the traditional finite element method and some improved discontinuous medium models. However, these solutions face the following technical difficulties when dealing with complex geometries and large deformation problems:

[0004] (1) Grid dependency and grid distortion issues

[0005] When dealing with large deformation problems, the traditional finite element method will reduce the calculation accuracy due to mesh distortion, and even make the calculation unable to converge. This poses a great challenge to the accurate simulation of the material failure process, especially in the crack initiation and propagation stages, and the re-division of the mesh is time-consuming and complicated.

[0006] (2) Low computational efficiency

[0007] Existing finite element method damage analysis often requires mesh refinement to improve calculation accuracy, which leads to a significant increase in the amount of calculation, long calculation time and low efficiency, making it difficult to meet real-time or efficient engineering needs.

[0008] To sum up, it is necessary to make further innovations on existing technologies. Summary of the invention

[0009] In response to the technical problems existing in the above-mentioned background technology, the present invention proposes an analysis method for material damage problems based on the base force unit method. The method has a reasonable concept, can significantly improve the 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.

[0010] In order to solve the above technical problems, the present invention provides an analysis method for material damage problems based on the base force unit method, which specifically includes the following steps:

[0011] 1) The base force element method based on the potential energy principle is used to establish the base force element model for static damage problems;

[0012] 2) Construct the dynamic base force element equation for the material dynamic damage problem;

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

[0014] The analysis method of material damage problem based on the base force unit method, wherein the construction process of the strain and stiffness matrix of the base force unit of the static damage problem in step 1) is as follows:

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

[0016] In order to describe the stress state near point Q, the current configuration Q is made into a parallelepiped microelement, whose three edges are dx 1 Q1, dx 2 Q2, dx 3 Q3, the force on the corresponding positive side is recorded 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, 1-1=3; T i is the coordinate system x i Base force at point Q, i = 1, 2, 3;

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

[0020]

[0021] In the above formula (2), V Q For x i The base capacity of the system, Q i is the basis vector after deformation;

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

[0023]

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

[0025] When the strain deviator of the base force unit is negligible, the average strain is used instead of the true strain;

[0026] For two-dimensional problems, the three components of the average strain of the base force element are:

[0027]

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

[0029] For three-dimensional problems, the six components of the average strain of the base force element are:

[0030]

[0031]

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

[0033] The analysis method of material damage problem based on base force element method, wherein the solution of the base force element model of static damage problem in the material damage and destruction process in step 1) is to adopt direct iteration method, and the specific steps are as follows:

[0034] 1.2.1) Give the initial values ​​used to solve the node displacement field, stress field and damage factor field in the iterative process, for the Ith load step, that is, 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 needs to continue iterating. Initialize n = 0;

[0036] Form the effective stiffness matrix:

[0037]

[0038] In the above formula (13), K is the effective stiffness matrix of the entire structure; is the damage factor of the e-th base force unit at the n-th iteration; is the initial stiffness matrix of the e-th base force element;

[0039] For displacement loading, the effective load array is formed by multiplying large numbers:

[0040]

[0041] In the above formula (14), P I is the overall effective load array; is the external load array; k b is the penalty function coefficient corresponding to the boundary condition; u b is the known displacement vector of the displacement boundary condition;

[0042] 1.2.3) Perform the nth iteration and iteratively update the node displacement;

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

[0044]

[0045] 1.2.3.2) Calculate node displacement:

[0046]

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

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

[0049]

[0050] In formula (17), the damage factor Refers to the damage measure corresponding to the damaged area that the unit cannot bear in the damaged state, V n V is the undamaged equivalent volume that can actually bear the load. d is the damage volume; after calculating the updated damage factor After that, the overall effective stiffness matrix K is updated again according to the above formula (15) for solving the node displacement field in the next iteration to ensure that the stiffness matrix matches the current damage state;

[0051] 1.2.3.5) Calculate the nominal macroscopic stress and nominal strain of the material sample being analyzed

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

[0053]

[0054] In formula (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 formula (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 displacement meets the convergence criteria:

[0059]

[0060] In the above formula (20), u (n) and u (n-1) are the node displacement vectors of this and previous iterations respectively;

[0061] If the requirements are not met and the number of iterations is less than the maximum allowed number, proceed to the next iteration; if the number of iterations is greater than the maximum allowed number or the accuracy meets the requirements, proceed to the following step 1.2.4);

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

[0063] The analysis method of material damage problem based on the base force unit method, wherein the specific process of step 2) is:

[0064] The dynamic balance equation of each node of the base force unit is obtained by using the D'Alembert principle, and the dynamic base force element equation is constructed based on the dynamic balance equation, as shown below:

[0065]

[0066] In formula (21), ü t , and u t They represent the acceleration vector, velocity vector and displacement vector of each node of the structure respectively; M, C, K represent the overall mass matrix, damping matrix and stiffness matrix of the structure respectively; P t Represents the dynamic load array of the structure.

[0067] The analysis method of material damage problem based on the base force unit method, wherein the mass matrix adopts the method of concentrated mass matrix, and for the plane unit, the mass of the unit is evenly distributed to the nodes according to the static equivalence principle. The concentrated mass matrix of the plane unit can be expressed as:

[0068]

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

[0070] The analysis method of the material damage problem based on the base force unit 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 formula (23), α and β are proportionality coefficients that are independent of frequency.

[0073] The analysis method for material damage problem based on the base force element method, wherein in the step 3), for a linear system, that is, when the material has not been damaged, the elastic modulus of the material remains constant during the loading process, and the stiffness matrix of the structure does not change with displacement or time, the specific steps of using the Newmark-β method to calculate the dynamic base force element equation of the material are as follows:

[0074] 3.1) Initial calculation

[0075] 3.1.1) Determine the initial motion value ü0 of the system, and u0, ü0 are the node acceleration vectors at the initial time (t = 0), is the node velocity vector at the initial moment, u0 is the node displacement vector at the initial moment;

[0076] 3.1.2) The entire structure is discretized by the base force method, and the overall stiffness matrix K and mass matrix M are assembled according to the unit 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) For each time step, calculate

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

[0083]

[0084] 3.2.2) Solve the displacement of each node of the base force unit at time t+Δt:

[0085]

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

[0087]

[0088] The analysis method for material damage problem based on the base force element method, wherein in the step 3), for a nonlinear system, that is, when the material is damaged, the elastic modulus of the material is related to the degree of material damage during the loading process, and the steps of directly iterating the dynamic base force element equation 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, Newmark-β method parameters β and γ;

[0091] Set initial displacement u0, speed and acceleration

[0092] Initialization damage factor Indicates no damage;

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

[0094] 4.2.1) Iteration loop

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

[0096] Repeat the following 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) Forming the effective stiffness matrix;

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

[0100] (d) Solve the node displacement;

[0101] (e) Update acceleration and velocity;

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

[0103] (g) Check the convergence conditions:

[0104] like The iteration converges;

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

[0106] 4.2.2) Time step advancement

[0107] a. Use the converged displacement, velocity, and acceleration as the initial value of 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 is damaged, that is, the damage factor reaches the critical value;

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

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

[0113] The analysis method for material damage problems based on the base force unit method of the present invention is reasonably conceived, the mathematical model derivation idea of ​​the base force unit method adopted is novel, the formula expression is concise and clear, and the physical meaning is more intuitive and clear, which not only simplifies the calculation process and reduces the calculation complexity, but also makes programming implementation more convenient and quick, and significantly improves the calculation efficiency.

[0114] In addition, the material damage analysis method based on the base force unit method of the present invention can effectively simulate the mechanical behavior of materials during damage and destruction, and is suitable for the analysis of complex structures and nonlinear problems. This method has important application value in the finite element damage analysis of materials, can be widely used in the design and safety assessment of engineering structures, provides a new research tool for the fields of materials science and engineering mechanics, and has broad application prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0115] In order to more clearly illustrate the specific implementation methods of the present invention or the technical solutions in the prior art, the drawings required for use in the specific implementation methods or the description of the prior art will be briefly introduced below. Obviously, the drawings in the following description are some implementation methods of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0116] Figure 1 It is a flow chart of the method for analyzing material damage problems based on the base force unit method of the present invention;

[0117] Figure 2 It is a schematic diagram of the base force involved in the analysis method of material damage problem based on the base force unit method of the present invention. DETAILED DESCRIPTION

[0118] The technical solution of the present invention will be described clearly and completely below in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0119] The present invention is further explained below in conjunction with specific implementation modes.

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

[0121] S100, the base force element method based on the potential energy principle is used to establish the base force element model for static damage problems;

[0122] S110, Construction of strain and stiffness matrices of base force elements for static damage problems

[0123] As a new finite element method, the base force unit method has many incomparable advantages over other methods, whether based on the complementary energy principle or the potential energy principle. In the present invention, the strain and stiffness matrices of the base force unit are derived using the potential energy principle and expressed explicitly without using Gaussian integrals.

[0124] Considering a three-dimensional elastic body, Q represents the radial vector after deformation, x i (i=1,2,3) represents the Lagrange coordinates, then the deformed coordinate frame is

[0125] In order to describe the stress state near point Q, the current configuration Q is made into a parallelepiped microelement, whose three edges are dx 1 Q1, dx 2 Q2, dx 3 Q3, the force on the corresponding positive side is recorded as dT 1 , dT 2 , dT 3 ,like Figure 2 As shown, the base force is expressed as follows:

[0126]

[0127] In formula (1), it is agreed that 3+1=1, 1-1=3; T i (i=1,2,3) is the coordinate system x i Base force at the middle Q point;

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

[0129]

[0130] In formula (2), V Q For x i The base capacity of the system, Q i is the basis vector after deformation;

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

[0132]

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

[0134] When the strain deviator of the base force unit is negligible, the average strain is used instead of the true strain;

[0135] For two-dimensional problems, the three components of the average strain of the base force element are:

[0136]

[0137] In the above formulas (4)-(6), I is the unit node number, n is the number of unit nodes, A is the base force unit area; u Ix and u Iy They represent the x- and y-direction displacements of the base force unit nodes respectively; and is a constant related to the shape of the base force unit;

[0138] For three-dimensional problems, the six components of the average strain of the base force element are:

[0139]

[0140]

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

[0142] Among them, the solution of the stiffness matrix of the static damage problem in the process of material damage and destruction is a direct iteration method. The specific steps are as follows:

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

[0144] S122, initializing n=0 (n is an iteration step counter, 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 needs to continue to iterate);

[0145] Form the effective stiffness matrix:

[0146]

[0147] In the above formula (13), K is the effective stiffness matrix of the entire structure; is the damage factor of the e-th base force unit at the n-th iteration; is the initial stiffness matrix of the e-th base force unit (stiffness matrix when not damaged);

[0148] For displacement loading, the effective load array is formed by multiplying large numbers:

[0149]

[0150] In the above formula (14), P I is the overall effective load array; is the external load array; k b is the penalty function coefficient corresponding to the boundary condition (usually a large value); u b is the known displacement vector of the displacement boundary condition;

[0151] S123, perform the nth iteration and iteratively update the node displacement:

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

[0153]

[0154] S1232. Calculate node displacement:

[0155]

[0156] S1233, calculating the stress-strain state of each base force unit (a unit refers to a unit divided in the base force unit analysis, such as a plane triangle unit in a two-dimensional problem or a tetrahedron unit in a three-dimensional problem) in the base force unit analysis of the above step 2), and determining the damage state of the unit;

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

[0158] D=V d / V n (0≤D≤1) (17);

[0159] In formula (17), the damage factor Refers to the damage measure corresponding to the damaged area that the unit cannot bear in the damaged state, V n V is the undamaged equivalent volume that can actually bear the load; d For the damage volume, find the effective stiffness matrix; after calculating the updated damage factor After that, the overall effective stiffness matrix K is updated again according to the above formula (15) for solving the node displacement field in the next iteration to ensure that the stiffness matrix matches the current damage state;

[0160] S1235. Calculate the nominal macroscopic stress and nominal strain of the material sample being analyzed.

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

[0162]

[0163] In formula (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 formula (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 displacement meets the convergence criterion:

[0168]

[0169] In the above formula (20), u (n) and u (n-1) are the node displacement vectors of this and previous iterations respectively;

[0170] If the requirement is not met and the number of iterations is less than the maximum allowed number, the next iteration is entered; if the number of iterations is greater than the maximum allowed number or the accuracy meets the requirement, the process proceeds to the following step S124.

[0171] S124. Determine whether the specimen is unstable and damaged or reaches a given load step. If so, stop the calculation. Otherwise, set I=I+1 and enter the next loading step, that is, increase the load to the next predetermined level, return to the above step S121 and restart the iterative calculation process to simulate the damage evolution process of the material under gradually increasing loads.

[0172] S200, Constructing dynamic base force element equations for material dynamic damage problems

[0173] The difference between dynamic analysis and static analysis is that inertia force needs to be considered, so mass matrix and damping matrix are added to the equilibrium equation. In dynamic analysis, time coordinates are introduced. The dynamic equilibrium equation of each node of the base force unit can be obtained by using the D'Alembert principle. In the process of constructing the dynamic equation, the base force unit method adopts the same method as the conventional finite element method. Therefore, the dynamic base force element equation is as follows:

[0174]

[0175] In formula (21),

[0176] and u t They represent the acceleration vector, velocity vector and displacement vector of each node of the structure respectively; M, C, K represent the overall mass matrix, damping matrix and stiffness matrix of the structure respectively; P t Represents the dynamic load array of the structure.

[0177] In the present invention, the mass matrix adopts the method of concentrated mass matrix; for example, for a plane unit, the mass of the unit is evenly distributed to the nodes according to the static equivalence principle, and the concentrated mass matrix of the plane unit can be expressed as:

[0178]

[0179] In formula (22), ρ is the unit density, A is the unit area, T is the unit thickness, and n is the number of unit 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 formula (23), α and β are proportionality coefficients that are independent of frequency.

[0183] S300, using the Newmark-β method to solve the dynamic base force element equation of the above step S200, to obtain the damage evolution process of the material under the 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 (Newmark-β method is a commonly used direct integration method, widely used in structural dynamics, for solving second-order ordinary differential equations and calculating the response of the 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 node acceleration vectors at the initial time (t = 0), is the node velocity vector at the initial moment, u0 is the node displacement vector at the initial moment;

[0187] S312. The entire structure is discretized by the base force method, and the overall stiffness matrix K and mass matrix M are assembled according to the unit 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 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, for each time step calculation

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

[0194]

[0195] S322. Calculate the displacement of each node of the base force unit at time t+Δt:

[0196]

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

[0198]

[0199] Regarding the selection of the time step, a simple but very useful method for selecting the time step is to first solve the problem with a time step that seems reasonable, 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, that is, when the material is damaged, the elastic modulus of the material is related to the degree of material damage during loading. The steps for directly iterating and solving the dynamic base force element equation of the material damage problem based on the Newmark-β method are as follows:

[0201] S410, initial assignment

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

[0203] Set initial displacement u0, speed and acceleration ü0;

[0204] Initialization damage factor Indicates no damage;

[0205] S420, time step cycle, from n = 0 to the total number of time steps:

[0206] S421, Iteration loop

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

[0208] Repeat the following 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) Forming the effective stiffness matrix;

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

[0212] (d) Solve the node displacement;

[0213] (e) Update acceleration and velocity;

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

[0215] (g) Check the convergence conditions:

[0216] like The iteration converges;

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

[0218] S422, time step advancement

[0219] a. Use the converged displacement, velocity, and acceleration as the initial value of the next time step;

[0220] Update time t n =t n +Δt;

[0221] S430, Termination Condition

[0222] The calculation ends when the total calculation time is reached or the specimen is damaged, that is, the damage factor reaches the critical value;

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

[0224] The invention has a reasonable concept, can significantly improve the adaptability to mesh distortion, has a simpler mathematical model, significantly improves the efficiency of obtaining material damage problems, can be applied to finite element damage analysis of materials, and has a relatively 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, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements 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 analysis method for material damage problems based on the base force unit method, characterized in that: The specific steps include: 1) The base force element method based on the potential energy principle is used to establish the base force element model for static damage problems; 2) Construct the dynamic base force element equation for the material dynamic damage problem; 3) The Newmark-β method is used to solve the dynamic base 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 base force unit method according to claim 1, characterized in that: The construction process of the strain and stiffness matrix of the base force unit of the static damage problem in step 1) is as follows: Considering a three-dimensional elastic body, Q represents the radial vector after deformation, x i (represents Lagrange coordinates, i = 1, 2, 3, then the deformed coordinate frame is In order to describe the stress state near point Q, the current configuration Q is made into a parallelepiped microelement, whose three edges are dx 1 Q1, dx 2 Q2, dx 3 Q3, the force on the corresponding positive side is recorded as dT 1 , dT 2 , dT 3 , the base force is expressed as follows: In the above formula (1), it is agreed that 3+1=1, 1-1=3; T i is the coordinate system x i Base force at point Q, i = 1, 2, 3; The relationship between the basal force and the Cauchy stress tensor σ is: In the above formula (2), V Q For x i The base capacity of the system, Q i is the basis vector after deformation; According to the base force theory, the stiffness matrix for both two-dimensional and three-dimensional problems is expressed as: In formula (3), E and ν are elastic modulus and Poisson's ratio respectively; A is the area or volume of the base force unit; U is the unit tensor; m I and m J is a constant related to the shape of the base force unit, m IJ =m I ·m J ; When the strain deviator of the base force unit is negligible, the average strain is used instead of the true strain; For two-dimensional problems, the three components of the average strain of the base force element are: In the above formulas (4)-(6), I is the unit node number, n is the number of unit nodes, A is the base force unit area, and u is Ix and u Iy represent the x- and y-direction displacements of the base force unit nodes, respectively. and is a constant related to the shape of the base force element. For three-dimensional problems, the six components of the average strain of the base force element are: In the above formulas (7)-(12), I is the unit node number, n is the number of unit nodes, V is the base force unit volume; u Ix 、u Iy and u Iz They represent the x, y and z displacements of the base force unit nodes respectively; and is a constant related to the shape of the base force element.

3. The method for analyzing material damage problems based on the base force unit method according to claim 1, characterized in that: In the step 1), the solution of the base force element model of the static damage problem in the material damage and destruction process is solved by using a direct iteration method, and the specific steps are as follows: 1.2.1) Give the initial values ​​used to solve the node displacement field, stress field and damage factor field in the iterative process, for the Ith load step, that is, at time t+Δt; 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 needs to continue iterating. Initialize n = 0; Form the effective stiffness matrix: In the above formula (13), K is the effective stiffness matrix of the entire structure; is the damage factor of the e-th base force unit at the n-th iteration; is the initial stiffness matrix of the e-th base force element; For displacement loading, the effective load array is formed by multiplying large numbers: In the above formula (14), P I is the overall effective load array; is the external load array; k b is the penalty function coefficient corresponding to the boundary condition; u b is the known displacement vector of the displacement boundary condition; 1.2.3) Perform the nth iteration and iteratively update the node displacement; 1.2.3.1) Calculate the effective stiffness matrix items: 1.2.3.2) Calculate node displacement: 1.2.3.3) Calculate the stress-strain state of each base force unit in the base force element analysis of step 2) above, and determine the damage state of the unit; 1.2.3.4) Calculate the damage factor of the base force unit: In formula (17), the damage factor Refers to the damage measure corresponding to the damaged area that the unit cannot bear in the damaged state, V n V is the undamaged equivalent volume that can actually bear the load. d is the damage volume; after calculating the updated damage factor After that, the overall effective stiffness matrix K is updated again according to the above formula (15) for solving the node displacement field in the next iteration to ensure that the stiffness matrix matches the current damage state; 1.2.3.5) Calculate the nominal macroscopic stress and nominal strain of the material sample being analyzed The nominal stress is obtained by the following formula: In formula (18), F is the loading force on the specimen, and A is the cross-sectional area of ​​the specimen; The nominal strain is obtained by the following formula: In formula (19), ΔL is the axial deformation of the specimen, and L is the initial length of the specimen; 1.2.3.6) Verify whether the iteration converges, that is, check whether the node displacement meets the convergence criteria: In the above formula (20), u (n) and u (n-1) are the node displacement vectors of this and previous iterations respectively; If the requirements are not met and the number of iterations is less than the maximum allowed number, proceed to the next iteration; if the number of iterations is greater than the maximum allowed number or the accuracy meets the requirements, proceed to the following step 1.2.4); 1.2.4) Determine whether the specimen is unstable or has reached a given load step. If so, stop the calculation. Otherwise, set I = I + 1 and enter the next loading step, that is, increase the load to the next predetermined level, return to the above step 1.2.1) and restart the iterative calculation process to simulate the damage evolution process of the material under gradually increasing loads.

4. The method for analyzing material damage problems based on the base force unit method according to claim 1, characterized in that: The specific process of step 2) is as follows: The dynamic balance equation of each node of the base force unit is obtained by using the D'Alembert principle, and the dynamic base force element equation is constructed based on the dynamic balance equation, as shown below: In formula (21), and u t They represent the acceleration vector, velocity vector and displacement vector of each node of the structure respectively; M, C, K represent the overall mass matrix, damping matrix and stiffness matrix of the structure respectively; P t Represents the dynamic load array of the structure.

5. The method for analyzing material damage problems based on the base force unit method according to claim 4, characterized in that: The mass matrix adopts the method of concentrated mass matrix. For the plane unit, the mass of the unit is evenly distributed to the nodes according to the static equivalence principle. The concentrated mass matrix of the plane unit can be expressed as: In the above formula (22), ρ is the unit density, A is the unit area, T is the unit thickness, and n is the number of unit nodes.

6. The method for analyzing material damage problems based on the base force unit method according to claim 4, characterized in that: The damping matrix adopts the Rayleigh damping model, which can be expressed as a linear combination of the mass matrix and the stiffness matrix: C = αM + βK (23); In formula (23), α and β are proportionality coefficients that are independent of frequency.

7. The method for analyzing material damage problems based on the base force unit method according to claim 1, characterized in that: In step 3), for a linear system, that is, when the material has not been damaged, the elastic modulus of the material remains constant during the loading process, and the stiffness matrix of the structure does not change with displacement or time. The specific steps of calculating the dynamic base force element equation of the material using the Newmark-β method are as follows: 3.1) Initial calculation 3.1.1) Determine the initial motion value of the system and u0, is the node acceleration vector at the initial time (t=0), is the node velocity vector at the initial moment, u0 is the node displacement vector at the initial moment; 3.1.2) The entire structure is discretized by the base force method, and the overall stiffness matrix K and mass matrix M are assembled according to the unit 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; 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; 3.1.4) Forming the effective stiffness matrix 3.2) For each time step, calculate 3.2.1) Calculate the effective load vector of the system at time t+Δt: 3.2.2) Solve the displacement of each node of the base force unit at time t+Δt: 3.2.3) Calculate the acceleration and velocity of each node of the base force unit at time t+Δt:

8. The method for analyzing material damage problems based on the base force unit method according to claim 7, characterized in that: In step 3), for a nonlinear system, that is, when the material is damaged, the elastic modulus of the material is related to the degree of material damage during loading. The steps of directly iterating and solving the dynamic base force element equation of the material damage problem based on the Newmark-β method are as follows: 4.1) Initial assignment Set the time step Δt, Newmark-β method parameters β and γ; Set initial displacement u0, speed and acceleration Initialization damage factor Indicates no damage; 4.2) Time step cycle, from n = 0 to the total number of time steps: 4.2.1) Iteration loop 4.2.1.1) Initialize the iteration counter k=0 and set the convergence tolerance ∈; Repeat the following 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 the effective stiffness matrix; (c) Calculate the effective load vector; (d) Solve the node displacement; (e) Update acceleration and velocity; (f) Calculate new stress and strain and update damage factor (g) Check the convergence conditions: like The iteration converges; Otherwise, let k = k + 1 and return to the above step (a); 4.2.2) Time step advancement a. Use the converged displacement, velocity, and acceleration as the initial value of the next time step; Update time t n =t n +Δt; 4.3) Termination conditions The calculation ends when the total calculation time is reached or the specimen is damaged, that is, the damage factor reaches the critical value; Through the above steps 4.1)-4.3), the damage evolution process of the material under dynamic load can be accurately simulated to obtain the dynamic response of the specimen.

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