Method and apparatus for analyzing properties of viscoelastic materials

Abstract

The application relates to the technical field of performance analysis of viscoelastic materials, and discloses a performance analysis method and device of viscoelastic materials, wherein the method comprises the following steps: obtaining elastic operators of a specific viscoelastic material at multiple time points, and dividing the material into coarse element domains and fine element domains by using a multi-scale finite element method. A multi-scale base function is constructed for each coarse element domain, and an equivalent stiffness matrix set is calculated, and then a scale reduction calculation is performed to extract a micro information set. Based on the functional relationship of the micro information, viscoelastic analysis of the specified viscoelastic material is finally realized, and a basis is provided for performance evaluation of the material. The application has the advantages of high calculation efficiency, resource saving and large problem scale expansion compared with the prior art, reduces the historical data storage demand, significantly reduces the degree of freedom of the macro equation, is widely applicable to non-periodic structures, and has the potential to be extended to nonlinear viscoelastic problems.

Description

Technical Field

[0001] This invention relates to the field of performance analysis technology for viscoelastic materials, and more particularly to a method and apparatus for performance analysis of viscoelastic materials. Background Technology

[0002] Many engineering materials exhibit viscoelastic mechanical properties. Typical examples include rubber, plastics, textile fibers, and polymers. Their mechanical behavior under load is not only related to the load level but also a function of time. Additionally, some engineering materials, such as underground surrounding rock, asphalt, and concrete, exhibit stress-strain relationships independent of time under normal pressure or temperature, but show significant viscoelastic properties when pressure or temperature increases to a certain level. Therefore, research on the mechanical properties of viscoelastic materials is of practical significance.

[0003] It is widely believed that most engineering materials possess inherent nested multi-scale characteristics. Material properties and responses are not merely macroscopic quantities, but rather manifest across different scales, from microscopic to mesoscopic and even macroscopic. Numerous material and structural failure events have revealed that material deformation and failure originate at the mesoscopic and even microscopic scales. Therefore, to deeply understand the characteristics of material deformation and failure, one cannot remain at the superficial macroscopic scale but should appropriately conduct multi-scale analysis, combining microscopic, mesoscopic, and macroscopic analyses to identify the root causes of problems and the mechanisms of deformation and failure at a deeper level, thus avoiding erroneous judgments about the causes of macroscopic failure phenomena. This demonstrates the scientific and applied value of conducting multi-scale analysis. More specifically, under current computational conditions, conducting multi-scale analysis is the most realistic and feasible approach to revealing macroscopic structural failure mechanisms or developing new materials or new structural design processes.

[0004] Current multiscale analyses mostly focus on elastic or elastoplastic materials, with relatively few applications to viscoelastic materials. Current multiscale analyses of viscoelastic materials often utilize asymptotic homogenization methods. Based on elastic homogenization results, the elastic-viscoelastic correspondence principle is applied, replacing the corresponding elastic values ​​with viscoelastic values ​​after Laplace transformation. Constitutive relations containing mesostructural information are obtained in the Laplace domain, followed by an inverse Laplace transformation to derive the constitutive relations in the time domain. These are then combined with traditional analytical or finite element methods to complete macroscopic calculations. However, asymptotic homogenization methods require periodicity in the unit cells or representative volume elements of the mesostructural structure, which is not satisfied for many composite materials. Furthermore, handling viscoelastic problems using the elastic-viscoelastic correspondence principle involves an inverse Laplace transformation, which is challenging for some complex practical engineering problems. Summary of the Invention

[0005] Therefore, it is necessary to propose a method and apparatus for the performance analysis of viscoelastic materials to address the existing problems in the performance analysis of viscoelastic materials.

[0006] A method for performance analysis of viscoelastic materials, the method comprising: Obtain the elasticity operators of a specified viscoelastic material at multiple time points; The specified viscoelastic material is divided into multiple coarse element domains using a pre-defined multi-scale finite element method; wherein each coarse element domain includes multiple fine element domains; Based on the elastic operators at each time point, a multi-scale basis function is constructed for each of the coarse element domains at each time point, and the equivalent stiffness matrix set at each time point is obtained based on the multi-scale basis function; wherein, the equivalent stiffness matrix set includes the equivalent stiffness matrix corresponding to each of the coarse element domains at the same time point; Downscaling calculations are performed on the equivalent stiffness matrices in each of the aforementioned equivalent stiffness matrix sets to obtain the micro-information set corresponding to each time point; wherein, the micro-information set includes the micro-information corresponding to each of the aforementioned fine unit domains within the same time point; The functional relationship of the microscopic information set at adjacent time points is analyzed based on a preset model. Based on the aforementioned functional relationship, a viscoelastic analysis of the specified viscoelastic material is performed.

[0007] Furthermore, the step of dividing the specified viscoelastic material into multiple coarse element domains using a preset multi-scale finite element method includes: The basic coarse element domain and the basic fine element domain in each basic coarse element domain are pre-divided using a preset multi-scale finite element method. Obtain the material parameters of the specified elastic material, and input the material parameters of the specified elastic material into the basic fine element domain to obtain multiple coarse element domains.

[0008] Furthermore, the step of constructing the multi-scale basis function of each coarse unit domain at each time point based on the elastic operator at each time point includes: Apply unit displacements to each node degree of freedom in the coarse element domain, while keeping the other node degrees of freedom at 0, to set the equilibrium equations for each coarse element domain. The equilibrium equations are solved using the substructure stiffness matrix to obtain the basis functions for each degree of freedom in the coarse element domain. Based on the basis functions of each degree of freedom of the coarse unit domain, a multi-scale basis function for the coarse unit domain is established.

[0009] Further, the step of performing downscaling calculations on the equivalent stiffness matrices in each of the equivalent stiffness matrix sets to obtain the microscopic information set corresponding to each time point includes: The displacement vector of each node in each fine element domain is obtained based on the equivalent stiffness matrix in each set of equivalent stiffness matrices. The stress and strain corresponding to each fine element domain are obtained based on the displacement vector; wherein, the microscopic information includes the stress and strain of the fine element domain.

[0010] Furthermore, the step of analyzing the functional relationship of the microscopic information set at adjacent time points based on a preset model includes: The initial recursive formula for the stress is obtained using the preset model; The parameters in the initial recursive formula are solved based on the physical relationships of each fine element domain to obtain the recursive function relationship of stress changing with time.

[0011] Furthermore, the specified viscoelastic material is any one of rubber, plastic, textile fiber, and polymer.

[0012] Furthermore, in the step of analyzing the functional relationship of the micro-information set at adjacent time points based on a preset model, the preset model is a generalized Maxwell model.

[0013] A performance analysis device for viscoelastic materials, the device comprising: The elastic operator acquisition module is used to acquire the elastic operators of a specified viscoelastic material at multiple time points. The viscoelastic material partitioning module is used to divide the specified viscoelastic material into multiple coarse element domains using a preset multi-scale finite element method; wherein each coarse element domain includes multiple fine element domains; The equivalent stiffness matrix set acquisition module is used to construct multi-scale basis functions for each coarse element domain at each time point based on the elastic operators at each time point, and to acquire the equivalent stiffness matrix set for each time point based on the multi-scale basis functions; wherein, the equivalent stiffness matrix set includes the equivalent stiffness matrix corresponding to each coarse element domain at the same time point; The micro-information set acquisition module is used to perform downscaling calculations on the equivalent stiffness matrices in each of the equivalent stiffness matrix sets to obtain the micro-information set corresponding to each time point; wherein, the micro-information set includes the micro-information corresponding to each of the fine unit domains within the same time point; The micro-information set function relationship analysis module is used to analyze the micro-information set function relationship between adjacent time points based on a preset model; The viscoelasticity analysis module is used to perform viscoelasticity analysis on the specified viscoelastic material based on the functional relationship.

[0014] An electronic device includes a memory and a processor, the memory storing a computer program that, when executed by the processor, causes the processor to perform the following steps: Obtain the elasticity operators of a specified viscoelastic material at multiple time points; The specified viscoelastic material is divided into multiple coarse element domains using a pre-defined multi-scale finite element method; wherein each coarse element domain includes multiple fine element domains; Based on the elastic operators at each time point, a multi-scale basis function is constructed for each of the coarse element domains at each time point, and the equivalent stiffness matrix set at each time point is obtained based on the multi-scale basis function; wherein, the equivalent stiffness matrix set includes the equivalent stiffness matrix corresponding to each of the coarse element domains at the same time point; Downscaling calculations are performed on the equivalent stiffness matrices in each of the aforementioned equivalent stiffness matrix sets to obtain the micro-information set corresponding to each time point; wherein, the micro-information set includes the micro-information corresponding to each of the aforementioned fine unit domains within the same time point; The functional relationship of the microscopic information set at adjacent time points is analyzed based on a preset model. Based on the aforementioned functional relationship, a viscoelastic analysis of the specified viscoelastic material is performed.

[0015] A computer-readable storage medium storing a computer program, which, when executed by a processor, causes the processor to perform the following steps: Obtain the elasticity operators of a specified viscoelastic material at multiple time points; The specified viscoelastic material is divided into multiple coarse element domains using a pre-defined multi-scale finite element method; wherein each coarse element domain includes multiple fine element domains; Based on the elastic operators at each time point, a multi-scale basis function is constructed for each of the coarse element domains at each time point, and the equivalent stiffness matrix set at each time point is obtained based on the multi-scale basis function; wherein, the equivalent stiffness matrix set includes the equivalent stiffness matrix corresponding to each of the coarse element domains at the same time point; Downscaling calculations are performed on the equivalent stiffness matrices in each of the aforementioned equivalent stiffness matrix sets to obtain the micro-information set corresponding to each time point; wherein, the micro-information set includes the micro-information corresponding to each of the aforementioned fine unit domains within the same time point; The functional relationship of the microscopic information set at adjacent time points is analyzed based on a preset model. Based on the aforementioned functional relationship, a viscoelastic analysis of the specified viscoelastic material is performed.

[0016] The beneficial effects of this invention are as follows: It adopts a preset model to reflect the viscoelastic properties of the microstructure, utilizes the idea of ​​the initial stress method, and combines it with the multi-scale finite element theory of elasticity problems to realize cross-scale analysis of viscoelastic materials, which greatly improves the computational efficiency and greatly expands the application scope of the multi-scale method; at the same time, due to the above characteristics, the proposed method has the potential to be extended to nonlinear viscoelastic problems. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] in: Figure 1 This is a diagram illustrating the application environment of a performance analysis method for viscoelastic materials in one embodiment. Figure 2 This is a flowchart of a performance analysis method for viscoelastic materials in one embodiment; Figure 3 This is a multi-scale finite element diagram of a viscoelastic material in one embodiment; Figure 4 A schematic diagram of the viscoelastic material of a three-section viscoelastic shaft tie rod in one embodiment; Figure 5 This is a schematic diagram of the multi-scale finite element calculation results of a viscoelastic shaft tension rod with three equal cross sections in one embodiment, based on the initial stress method. Figure 6 This is a schematic diagram illustrating the relative error of the multi-scale finite element calculation results based on the initial stress method for the viscoelastic material of the three groups of equal cross-section viscoelastic shaft tie rods in one embodiment. Figure 7 This is a schematic diagram of a homogeneous cantilever beam with a concentrated force applied to the end of a viscoelastic material in one embodiment; Figure 8 This is a schematic diagram of the viscoelastic displacement at the midpoint of the free end of a non-homogeneous cantilever beam made of viscoelastic material in one embodiment. Figure 9 This is a structural block diagram of a performance analysis device for viscoelastic materials in one embodiment; Figure 10 This is a structural block diagram of an electronic device in one embodiment; Figure 11 This is a viscoelastic multi-scale finite element solution for a viscoelastic material under different mesh divisions in one embodiment; Figure 12This represents the relative error of the viscoelastic multi-scale finite element solution for a viscoelastic material under different mesh divisions in one embodiment. Detailed Implementation

[0019] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. 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.

[0020] Figure 1 This is a diagram illustrating the application environment for performance analysis of a viscoelastic material in one embodiment. (Refer to...) Figure 1 This method for analyzing the performance of viscoelastic materials is applied to a performance analysis system for viscoelastic materials. The system includes a terminal 110 and a server 120. The terminal 110 and server 120 are connected via a network. The terminal 110 can be a desktop terminal or a mobile terminal; a mobile terminal can be at least one of a mobile phone, tablet, or laptop. The server 120 can be a standalone server or a server cluster consisting of multiple servers. The terminal 110 is used to acquire the elasticity operators of a specified viscoelastic material at multiple time points, and the server 120 is used to analyze the viscoelasticity of the specified viscoelastic material.

[0021] like Figure 2 As shown, in one embodiment, a method for performance analysis of viscoelastic materials is provided. This method can be applied to both terminals and servers; this embodiment uses terminal application as an example. The specific steps of this method for performance analysis of viscoelastic materials include: S1: Obtain the elasticity operator of a specified viscoelastic material at multiple time points; S2: The specified viscoelastic material is divided into multiple coarse element domains using a preset multi-scale finite element method; wherein each coarse element domain includes multiple fine element domains; S3: Construct multi-scale basis functions for each coarse element domain at each time point based on the elastic operators at each time point, and obtain the equivalent stiffness matrix set for each time point based on the multi-scale basis functions; wherein, the equivalent stiffness matrix set includes the equivalent stiffness matrix corresponding to each coarse element domain at the same time point; S4: Perform downscaling calculations on the equivalent stiffness matrices in each of the equivalent stiffness matrix sets to obtain the micro-information set corresponding to each time point; wherein, the micro-information set includes the micro-information corresponding to each of the fine unit domains at the same time point; S5: Analyze the functional relationship of the microscopic information set at adjacent time points based on a preset model; S6: Perform viscoelastic analysis on the specified viscoelastic material based on the functional relationship.

[0022] As described in steps S1-S2 above, divide the macroscopic finite element mesh and the fine element domain located within the coarse element domain; read in the viscoelastic material parameters: , , , ( Here, m is the number of Maxwell elements included in the generalized Maxwell model. Let r be the elastic modulus of the r-th Maxwell element. Let be the viscosity coefficient of the r-th Maxwell element. The elastic coefficient at infinity is the value of the elastic modulus. Let the time be Poisson's ratio. Discretize the time to be determined: Let the time interval be , , ( ); Read in load time history Based on the discrete time series above, write down the load values ​​at each time point. abbreviated as In practice, for loads that do not change with time, .

[0023] In one embodiment, step S2, which involves dividing the specified viscoelastic material into multiple coarse element domains using a preset multi-scale finite element method, includes: S201: Pre-divide the basic coarse element domain and the basic fine element domain in each basic coarse element domain using a preset multi-scale finite element method; S202: Obtain the material parameters of the specified elastic material, and input the material parameters of the specified elastic material into the basic fine element domain to obtain multiple coarse element domains.

[0024] Specifically, refer to Figure 3 The following section uses the analysis of a planar continuum problem as an example to introduce the implementation process and main formulas of the multi-scale finite element method. The subscripts 's' and 'c' represent the fine-grained element domain and the coarse-grained element domain, respectively. Figure 3 The internal domain of a coarse mesh element in the two-dimensional solution domain shown is... , The multi-scale basis functions of the coarse element domain are solved numerically, that is, the equilibrium equations inside the coarse element domain are solved under specific boundary conditions: exist Inside, (2-1); In the formula: For elastic operators, satisfying ; The fourth-order stiffness tensor representing material properties; It is the displacement vector; For coarse cell domain nodes The basis functions satisfy , ( ), Kronecker symbol; The number of nodes in the coarse cell domain is taken here. =4, For the gradient operator, div represents the divergence. This represents the x-direction displacement of the j-th coarse element node. This represents the y-direction displacement of the j-th coarse element node.

[0025] In one embodiment, step S3, which constructs the multi-scale basis functions of each coarse unit domain at each time point based on the elastic operator at each time point, includes: S301: Apply a unit displacement to each node degree of freedom in the coarse element domain, while keeping the other node degrees of freedom at 0, to set the equilibrium equations for each coarse element domain. S302: Solve the equilibrium equations using the substructure stiffness matrix to obtain the basis functions for each degree of freedom in the corresponding coarse element domain; S303: Based on the basis functions of each degree of freedom of the coarse element domain, establish the multi-scale basis functions of the coarse element domain. As described in steps S301-S303 above, calculate the material parameters at the initial time: elastic modulus. Poisson's ratio Load .

[0026] Using the initial material parameters of each detailed element, the stiffness matrix of each sub-element is calculated for the structure of the fine element domain within each coarse element domain, and the stiffness matrices of the substructures within the coarse element domain are integrated. By applying unit displacements to each nodal degree of freedom in the coarse element domain while keeping the other nodal degrees of freedom zero, the homogeneous equations are solved using the substructure stiffness matrix to obtain the basis functions for each degree of freedom in the corresponding coarse element domain. Then, the overall basis function matrix is ​​established. For example, for a coarse element domain with a planar 4-node quadrilateral shape, its basis function matrix takes the following form, where... Indicates the 4 node numbers of the macroscopic coarse cell field: ; ; ; .

[0027] here These represent the four node numbers of a specific microarray e within the fine element domain; This represents the displacement vector of the mesoscopic nodes within the fine element domain; This represents the nodal displacement vector of the coarse element domain.

[0028] Then, based on the macro node number, the coarse unit domains are... Integrated overall stiffness matrix .here Let be the stiffness matrix of element e in the fine element domain; This is the equivalent stiffness matrix of the coarse element domain.

[0029] Solve the equations to find the macroscopic nodal displacements at the initial time. ; From the obtained The displacements of each coarse element domain node are obtained. .

[0030] Find the nodal displacements of the substructure Then, the nodal displacements on each micro-element are calculated. Calculate the strain in each sub-unit at the initial time. At each moment Multiscale finite element analysis.

[0031] The relaxation stress on each microstructure element during this time period is calculated using a recursive formula: Using the above relaxation stress, the additional nodal forces of each element within the substructure are calculated. .

[0032] Seeking macroscopic additional force: ; Solve the equation: Find Time macroscopic nodal displacement Furthermore, the displacements of each coarse element domain node are obtained. ; To find the nodal displacements of the substructure, use the transformation formula. Find the nodal displacements on each sub-unit. ; Calculate the strain of each mesoscopic element at that moment. , which serves as the strain for calculating the relaxation stress in the next time period.

[0033] Determine if it satisfies If the conditions are met, the calculation exits, and the viscoelastic displacement at each time step is output. If the conditions are not met, the calculation continues for the next time step, thus calculating the viscoelastic displacement at each time step.

[0034] Specifically, for two-dimensional plane problems, Depend on and constitute, These are the four nodes of the coarse element domain. Thus, the displacement of any node in the fine mesh (fine element domain) within this element can be expressed as... (2-2); (2-3); In the formula: For the coupling additional terms of the basis functions, their physical meaning refers to the coupling at the nodes of the coarse element domain. occur When the displacement is equal to the unit displacement in the direction, each micro-node within the element generates Displacement value in the direction, This represents the horizontal displacement of the i-th macroscopic node. .

[0035] Equations (2-2) and (2-3) can be written as follows: (2-4); In the formula: It represents the displacement vector of all nodes in the fine element domain within the macroscopic coarse element domain; Let be the displacement vectors of the nodes in the macroscopic coarse element domain. They can be written as: (2-5); (2-6); (2-7); in: , (2-8).

[0036] In the formula: This represents the total number of nodes in the fine mesh within the coarse cell domain.

[0037] for Figure 1 middle For a specific sub-unit e within the sub-unit domain, the following transformation relation can also be established: , (2-9); In the formula: Let be the transformation matrix of element e in the fine element domain, representing the mapping relationship between the nodal displacement of a certain element e in the fine mesh and the nodal displacement of the coarse element domain in which it is located. Let be the nodal displacement vector of element e in the fine element domain, which can be expressed as: (2-10); Here This represents the four node numbers of element e in the fine element domain.

[0038] For scalar field problems, when constructing basis functions by solving equation (2-1), the following boundary conditions must be satisfied: exist On the boundary (2-11); In the formula: To construct basis functions The boundary conditions applied at that time.

[0039] For vector field problems, basis functions need to be constructed separately in different coordinate directions. Studies have shown that the applied force during the construction of basis functions... The differences will affect the accuracy of the calculation results. The following section will focus on solving the basis functions. Let's take an example to illustrate how to apply linear boundary conditions.

[0040] Finding basis functions under linear boundary conditions That is, add to edges 1-4 and 1-2 respectively. The linear boundary of the direction, i.e., by linearly change to and On sides 2-3 and 3-4 The directional displacement is set to 0, and at the same time, the displacement of each node on the boundary is... The directions are all fixed; see the constraint details below. Figure 2 As shown. Under the above boundary conditions, traditional numerical methods in the fine-cell domain... Solving the equilibrium equation (2-1) yields the displacements of each degree of freedom in the fine element domain, forming... By analogy, all the basis functions of the coarse unit domain can be obtained. .

[0041] It can be proven that the basis functions constructed using the above method satisfy the condition that their sum at any point within the element equals 1, i.e. (2-12); This ensures that the rigid body displacement of the coarse element domain is coordinated with the displacement between coarse element domains.

[0042] When constructing basis functions using linear boundary conditions, the boundaries of the coarse element domain are forced to undergo linear deformation. This artificial constraint generates a strong boundary layer effect at the boundaries of heterogeneous elements, resulting in an overestimation of stiffness in multi-scale calculations. To reduce the error caused by this boundary layer effect, the supersample technique approximates the oscillations at the original coarse element domain boundaries by expanding the mesh range when solving the basis functions, thereby improving the accuracy of the solution.

[0043] In the fine element domain, the conventional finite element discretization equations for linear elastic problems can be written in the following matrix form: (2-13); in It is the stiffness matrix of the integrated microstructures in the fine element domain within a certain coarse element domain; It is the nodal external force vector of the fine element domain. and The dimensions are respectively and ,here This represents the total degrees of freedom of the fine element domain within the coarse element domain, generally speaking. .

[0044] Substitute equation (2-4) into equation (2-13), and then multiply both sides of the equation by the left side. We can obtain: (2-14); here and These are the macroscopic equivalent stiffness matrix and macroscopic equivalent external force vector of the coarse element domain, respectively. For a planar 4-node coarse element domain, and The dimensions are respectively and Their expressions are as follows: (2-15); (2-16); The above is the stiffness matrix after integrating all micro-elements within a known fine-element domain. When solving for the equivalent stiffness matrix of the coarse element domain, the formula is used. If the stiffness matrix of a specific fine element in the fine element domain is known... ,For example Figure 1 middle For a specific fine element *e* within the fine element domain, the equivalent stiffness matrix of the coarse element domain can also be calculated using the following formula, in conjunction with formula (2-9): , (2-17); In the formula: This represents the number of elements in the fine-cell domain within a single coarse-cell domain.

[0045] Regardless of the method used, after obtaining the stiffness matrix and external force vector of a single coarse element domain, a method similar to conventional finite element analysis can be used to integrate the overall macroscopic equivalent stiffness matrix and external force vector of the entire structure, i.e.: , (2-18); here and These represent the overall macroscopic stiffness matrix and the external force vector, respectively; A is the matrix integration operator in the conventional finite element method. The number of coarse cell domains.

[0046] After obtaining the overall macroscopic stiffness matrix and external force vectors, the following equations can be solved at the macroscopic scale to obtain the displacement vectors at the macroscopic scale. To complete macroscopic calculations.

[0047] (2-19).

[0048] In one embodiment, step S4, which involves downscaling the equivalent stiffness matrices in each of the equivalent stiffness matrix sets to obtain the microscopic information set corresponding to each time point, includes: S401: Obtain the displacement vector of each node in each fine element domain based on the equivalent stiffness matrix in each set of equivalent stiffness matrices. S402: Calculate the stress and strain corresponding to each fine element domain based on the displacement vector; wherein the microscopic information includes the stress and strain of the fine element domain.

[0049] After the macro analysis is completed, from The nodal displacement vectors of the coarse element domain c can be obtained. Then, using the transformation relationship (2-4) or (2-9) between the macroscopic variables of the coarse element domain and the microscopic variables of the fine element domain, the displacement vectors of each node in the fine element domain are obtained. Furthermore, the stress of each detailed element e in the fine element domain can be calculated. and strain The process of calculating micro quantities from macro quantities, such as microscopic information, is called downscaling.

[0050] In one embodiment, step S5, which analyzes the functional relationship of the micro-information set at adjacent time points based on a preset model, includes: S501: Obtain the initial recursive formula for the stress using the preset model; S502: Solve the parameters in the initial recursive formula according to the physical relationship of each fine element domain to obtain the recursive function relationship of stress changing with time.

[0051] As described in steps S501-S502 above, the study of the mechanical properties of viscoelastic materials can generally be divided into two methods: analytical methods and numerical methods. Analytical methods mainly utilize the elastic-viscoelastic correspondence principle, obtaining the solution in the Laplace domain elastic form through Laplace transform. Then, the elastic parameters in the formula are replaced with the transformed viscoelastic parameters, and an inverse Laplace transform is performed to obtain the solution to the original viscoelastic problem. Solving using the correspondence principle requires a prerequisite: finding an analytical solution corresponding to the elastic problem under the same boundary conditions. Therefore, it is greatly limited in practical applications. Numerical methods for analyzing the mechanical properties of viscoelastic materials mainly employ the finite element method. Based on the form of the constitutive relation used, there are two different implementation approaches. One is the variable stiffness method using integral constitutive relations. This method is widely used in commercial programs, with advantages such as strong versatility and high computational accuracy. However, its disadvantages are also obvious: large computational load and low computational efficiency. The stiffness matrix must be recalculated based on the relaxation modulus for each time period. To improve computational efficiency, another approach is to use the constant stiffness method based on differential constitutive relations. This divides the entire solution time into a series of time intervals, assuming that the stress or strain remains constant within each interval. The creep strain or relaxation stress caused by material viscosity is treated as the initial strain or initial stress, thus forming the initial strain method or initial stress method. This type of method maintains a constant stiffness matrix throughout the calculation process and uses a recursive approach to solve for the initial strain or initial stress in each interval, thereby greatly improving the computational efficiency of viscoelastic problems. Considering the widespread application of the generalized Maxwell model, the basic principles and related formulas of the matching initial stress method for solving viscoelastic problems are introduced below.

[0052] Using the generalized Maxwell model, its relaxation modulus is: (3-1); In the formula, For the relaxation modulus of the generalized Maxwell model; is the elasticity coefficient of the model when time approaches infinity; m is the number of Maxwell elements included in the generalized Maxwell model; Let r be the elastic modulus of the r-th Maxwell element; Let be the relaxation time of the r-th Maxwell element, where Let be the viscosity coefficient of the r-th Maxwell element. , , The coefficients, including m, were obtained by fitting the relaxation test data using the least squares method.

[0053] When using integral constitutive relations, the constitutive relations can be expressed as: (3-2); The first term in the above formula represents the initial strain. The stress response at time t is caused by the stress, while the integral term represents the stress response over the entire subsequent time interval. The stress at time t caused by the change.

[0054] Decompose the total stress at any given time into elastic stress and viscous relaxation stress. From equation (3-2), for time... ; (3-3); Integrating by parts on the second term and simplifying, we get: (3-4); In the above formula This represents the elastic modulus at the initial moment. Indicates time Adapt to change in time, Indicates time Stress during time, express The total stress corresponds to the elastic stress at the initial system elastic constant. express The relaxation stress caused by material viscosity is included in the total stress.

[0055] As shown in formula (3-4), the total stress at any given time can be decomposed into an elastic component that is independent of past loading history and a viscous component that reflects past loading history. If the viscous relaxation stress is regarded as the initial stress, the initial stress method in finite element theory can be used for calculation.

[0056] From formula (3-4), we know that (3-5); According to the principle of virtual displacement, the internal forces of the element caused by the above stress are: (3-6); In the above formula for The internal forces within element e caused by the stress at that moment are It is an elasticity matrix that depends only on Poisson's ratio. Let e ​​be the strain matrix of element e. for The displacement vector of the element node at any given time is equal to the strain of the Maxwell body at any given time, since the generalized Maxwell model uses a spring element and a series of Maxwell bodies connected in parallel. Therefore, the strain on the spring element and the strain of the Maxwell body are the same at any given time, and are equal to the total strain of the system at that time.

[0057] The internal force caused by virtual displacement is equal to the external force at the node; (3-7); so: (3-8); Further results were obtained: (3-9); In short: (3-10); In the formula, , is the initial stiffness matrix of the element; for External load vector at the node of the time element; This refers to the additional load caused by stress relaxation, which is the additional load caused by the initial stress.

[0058] The element characteristic matrices in equation (3-10) are assembled into a global characteristic matrix by matching the elements to their corresponding values, as follows: (3-11); The above equation is the overall equation form obtained by the initial stress method. This is a system of linear equations, and the stiffness matrix on the left side... Determined by the material constants at the initial moment, it remains constant throughout the analysis. Right-hand external load vector. This indicates that the load at this moment can remain constant throughout the analysis or change over time. This represents the additional force caused by the relaxation stress of a viscous material. Solving this system of equations yields... Displacement response at time t.

[0059] In summary, the key to solving equation (3-11) is calculation. It is made by It is assembled. So how do we calculate viscous relaxation stress? This is the key to viscoelastic finite element analysis using the initial stress method.

[0060] For ease of derivation, the calculation of viscous stress is first considered in one dimension, and the conclusions obtained can be easily generalized to three dimensions.

[0061] According to formula (3-4): (3-12); Integration interval Discretized into a series of small time intervals Then the above expression becomes: (3-13); Assuming that the strain remains constant within each small time interval and is equal to the strain at the beginning of that interval, the above equation simplifies to: (3-14); Combining formula (3-1), we have: ; ; ; Substituting the above results into formula (3-14) and simplifying, we get: (3-15); in represent The relaxation stress generated by the r-th Maxwell element at time r.

[0062] For the next moment Following the same logic as formulas (3-12) to (3-15), we can conclude that: (3-16); in for The relaxation stress generated by the r-th Maxwell element at time r is calculated by the following formula: (3-17); make ,but: ; ; ; ; .

[0063] Formula (3-17) is transformed into: (3-18); Formula (3-18) is the recursive formula for calculating the relaxation stress of the r-th Maxwell element. For the end of the first time period: (3-19); Once the relaxation stress of the r-th Maxwell element is obtained, it can be substituted into formula (3-16) to obtain the total relaxation stress of the generalized Maxwell model.

[0064] Extending the above conclusions to the three-dimensional case, the recursive formula for calculating relaxation stress is as follows: When n=1: , (3-20); Starting from the second period; , (n=1,2,...)(3-21); After determining the viscous relaxation stress in each time period, the additional load for that time period can be determined according to the previous introduction. Then, the equations are solved again to calculate the displacement at the end of that time period, and then the strain at the beginning of the next time period is determined. This process is repeated until the viscoelastic analysis for the entire time period is completed.

[0065] In some specific embodiments, please refer to Figure 4 It is composed of three different viscoelastic materials, each of which accounts for one-third of the total length, and the total length is... The left end is fixed and the right end is free. The free end bears a constant load P and has a cross-sectional area of ​​A. The viscoelastic material is assumed to be a generalized Maxwell model, which is assumed here to be a spring element and a Maxwell element connected in parallel.

[0066] The material parameters for each section are as follows: Section AB: The elastic modulus of the spring element is 3. The elastic modulus of a single Maxwell element is The viscosity coefficient of Maxwell components is The initial elastic modulus of the material is: The relaxation time is ; Section BC: The elastic modulus of the spring element is 2. The elastic modulus of a single Maxwell element is The viscosity coefficient of Maxwell components is The initial elastic modulus of the material is: The relaxation time is ; Section CD: The elastic modulus of the spring element is The elastic modulus of a single Maxwell element is The viscosity coefficient of Maxwell components is The initial elastic modulus of the material is: The relaxation time is ; In this example, it is assumed that , , , , , , , .

[0067] Find the displacement of the rod end over 36 seconds.

[0068] Based on the principles of mechanics of materials, the elastic displacements at different positions of the three sets of tie rods can be calculated as follows: , , ; According to the principle of elasticity-viscoelasticity correspondence, the displacements at various points of the three groups of viscoelastic materials can be obtained as follows: , , ; here, .

[0069] Based on the multi-scale finite element solution using the initial stress method, and according to the multi-scale finite element theory, the tie rod is divided into two mesh systems: macroscopic and mesoscopic. For this example, only a one-dimensional rod element is used to form a coarse mesh on the macroscopic level, containing two nodes. , Within this coarse element domain, three fine element domains are formed based on different material properties. The nodes of the fine element domains are numbered 1, 2, 3, and 4.

[0070] Using equal time intervals, , ; For the initial time The material parameters are: part: ; BC segment: ;CD section: .

[0071] Each sub-unit is rigid: , , Total stiffness over integrated fine-cell domain: The following method obtains multi-scale basis functions containing the microstructure material properties by solving homogeneous equations on fine element domains.

[0072] 1. Order , ,beg and ; (here In Indicates the macro node number. (Indicates the number of the microscopic node within the fine element domain).

[0073] .

[0074] From lines 2 and 3, we get: , ; so: ; make , ,beg and ; ; From lines 2 and 3, we get: , ; so: .

[0075] The basis function matrix is: ; , , , ; ; here This represents the nodal displacement within the fine element domain. This represents the node displacement of the macroscopic coarse element domain.

[0076] Find the equivalent stiffness matrix: , (e=1, 2, 3); Unit 1: , , ; Unit 2: , , ; Unit 3: , , ; = ; In this example, there is only one coarse element domain, so the equivalent stiffness matrix of the coarse element domain obtained above is the total stiffness of the entire structure.

[0077] Solve the equation: ; ; Introducing boundary conditions Afterwards, ; Therefore, the initial macroscopic displacement is ; Downscaling analysis: Using basis functions, the nodal displacements of the micro-element domain can be easily obtained from the macro-displacements: , , , ; The strain matrix of each element is ,so , , ; The subsequent calculations were completed using the initial stress method: for : , , ; ; ; , (e=1,2,3); , , ; Introduction ,but ; , , ; , , ; ; At the end of the first period, that is The total load at that time is ; Solve the equation: ; ,here ; Introducing boundary conditions Later, , ; Downscaling analysis: , , , ; , , ; : ; ; ; ,here ; Solve the equation: ; ; Introducing boundary conditions Later, , ; Downscaling analysis: , , , ; , , ; : ; ; ; ,here ; Solve the equation: ; ; Introducing boundary conditions Later, , ; Downscaling analysis: , , , ; , , ; This process can be repeated to complete the calculations for subsequent time points.

[0078] The results of the multi-scale finite element analysis based on the initial stress method are shown below. Figure 5 The figure also shows the analytical solution obtained according to the principle of elasticity-viscoelasticity correspondence. Figure 6 The relative errors of the multi-scale finite element calculation results are presented.

[0079] It can be seen that, when using only a single coarse element domain, the proposed method exhibits the same trend as the analytical solution, reflecting the displacement creep characteristics of viscoelastic materials under constant load. Figure 6 The error curve shows that the error is small initially, increases over time, reaches its maximum of about 5% around 10 seconds, and then gradually decreases. Throughout the process, the error remains negative, indicating that the numerical solution is smaller than the analytical solution. The initial increase followed by decrease in error is because, for the generalized Maxwell model, the relaxation modulus shows a rapid change in the curve slope initially (steep curve), followed by a flattening curve after the relaxation time, with minimal change in the relaxation modulus thereafter. In this example, to facilitate faster convergence, a relaxation time of [value missing] is assumed. mean The stress at that time is the initial stress. This means the stress decreased by 63.2%, which is considered a very rapid decrease. Therefore, material parameters change rapidly near the relaxation time (or within a time range of the same order of magnitude). When using a uniform time interval, the error will naturally increase in this range, while the error will decrease outside this range. To improve calculation accuracy, the time interval can be further reduced near the relaxation time, while the time interval can be appropriately increased further away from the relaxation time.

[0080] The calculation process of the example shows that for this type of tie rod composed of three different materials, if the traditional finite element method is used, at least three elements are needed, and the total stiffness matrix is ​​[missing information] when solving the equations in each time period. The stiffness matrix is ​​obtained by using the proposed multi-scale finite element method based on the initial stress method, where only a coarse element domain needs to be defined. The stiffness matrix for solving the equations at each time step is... It is foreseeable that for problems of considerable scale, the size of the equations obtained using the multi-scale finite element method will be much smaller than that obtained using the traditional finite element method. Furthermore, the advantage of the initial stress method is that the stiffness matrix remains unchanged throughout the analysis process, the basis functions only need to be solved once, and when calculating the relaxation stress in each time period, only the relaxation stress and strain of the previous time period need to be known; it is not necessary to store the strain or relaxation stress at all times. This can greatly improve computational efficiency and reduce memory requirements.

[0081] Example of a homogeneous cantilever beam: A cantilever beam, fixed at the left end and free at the right end, such as... Figure 7 As shown. Beam length Liang Gao ,thickness , The free end is subjected to a tangential force with a net force of P. Poisson's ratio The viscoelastic properties of the material are simulated using a generalized Maxwell model, assuming a single Maxwell element. The material parameters are: , , , The initial elastic modulus is .

[0082] Calculate the viscoelastic displacement of the cantilever beam over 36 seconds under the aforementioned load.

[0083] The cantilever beam can be considered as a plane stress problem. According to the principles of elasticity, the elastic solution for the displacement at its free end is: ; Viscoelastic analytical solution: Based on the principle of elastic-viscoelastic correspondence, the viscoelastic displacement at the free end of the cantilever beam can be calculated by the following formula: ; in ; when hour, , ; when hour, ,at this time: .

[0084] Viscoelastic multiscale finite element solution based on the initial stress method: This example is a plane stress problem, analyzed using four-node rectangular elements. Based on multi-scale finite element theory, the cantilever beam is divided into two sets of finite element meshes: coarse and fine. Macroscopically, 12 rows of elements are defined along the span direction, and two layers of elements are defined along the height direction, resulting in a total of 24 macroscopic coarse element domains with element sizes of [missing information]. The macroscopic node count is 39, and the total degrees of freedom is 78. Each coarse element domain is further divided into sub-elements. For comparison, each coarse element domain is further divided into sub-elements. and Two partitioning schemes are proposed: horizontally, divide the column into 2 or 4 columns; vertically, divide the column into 2 or 4 layers. For In terms of the partitioning scheme, each coarse element domain contains 4 fine elements, 9 internal nodes, and 18 internal degrees of freedom. The size of the fine element is... And for In terms of the partitioning scheme, each coarse element domain contains 16 fine elements, 25 internal nodes, and 50 internal degrees of freedom. The size of the fine element is... The geometric information for the two mesh divisions is shown in Table 1.

[0085] Table 1. Mesh Generation Information for Cantilever Beams Note: The unit of cell size in the table is cm; detailed meshing information refers to the detailed cell information within a typical coarse cell domain.

[0086] To analyze the cantilever beam using the algorithm proposed in this paper, a multi-scale finite element program was developed in Julia following the steps in Section 4. The vertical viscoelastic displacements of the mid-node at the free end under two different mesh settings were calculated as follows: Figure 11 As shown in the figure, the calculation results obtained according to the analytical solution are also listed. Figure 12 This represents the relative error between the results obtained under two different detailed meshing methods and the analytical solution. Since this example involves a periodic structure, the geometry and material properties of each coarse element domain are completely identical. Therefore, it is only necessary to solve for the basis functions and equivalent stiffness matrix for a typical coarse element domain. The integrated global stiffness matrix remains unchanged when solving for displacements at subsequent time steps, resulting in high computational efficiency. Furthermore, due to the characteristics of the recursive formula used to solve for relaxation stress at each time step in this algorithm, it is not necessary to retain all historical data; only the strain and relaxation stress information from the previous step is needed. Therefore, the memory requirements for computation are not high.

[0087] Figure 11 The calculation results show the vertical displacement of the node at the midpoint of the free end section of the cantilever beam as a function of time. Multi-scale finite element results under two different mesh divisions and analytical solutions obtained using the elastic-viscoelastic correspondence principle are presented. For the load conditions and material parameters in this example, the initial displacement of the cantilever beam's free end is approximately 1.7 mm. Due to the viscoelasticity of the material, under a fixed load, the displacement increases with time, reaching approximately 2.5 times the initial displacement at 36 seconds. The calculation time of 36 seconds is chosen to account for the relaxation time of the viscous elements in the generalized Maxwell model. This means that the displacement was calculated for 10 times the relaxation time, at which point the viscoelastic displacement had largely stabilized. Results from different mesh divisions show that, under the same macroscopic mesh division, the computational accuracy improves with increasing mesh size within the coarse element domain. Furthermore, from... Figure 12The relative error curve shows that the calculation error pattern of this planar problem is consistent with that of the one-dimensional shaft tension rod, both first increasing and then decreasing with time, reaching a peak error within approximately 10 seconds. The reason for this has been explained above, mainly due to the equal-time discretization scheme. To improve this phenomenon, a denser time discretization scheme can be used near the relaxation time in practical applications. It should also be noted that the viscoelastic displacement analytical solution here is obtained based on the elastic solution using the elastic-viscoelastic correspondence principle, while the elastic solution is obtained under the constraint condition... and The result is obtained under the given premise, when the constraints change. and When the elastic solution changes, the form of the two constraint conditions differs in that the former assumes that the differential line segment passing through the origin and parallel to the x-axis remains parallel to x after deformation, but the differential line segment parallel to the y-axis at that point will rotate clockwise by a small angle after deformation. The latter assumes that the differential line segment passing through the origin and parallel to the y-axis remains parallel to y after deformation, but the differential line segment parallel to the x-axis at that point will rotate counterclockwise by a small angle after deformation. Neither constraint method is strictly feasible in practice. Finite element analysis typically approximates the fixed constraint conditions by setting the horizontal and vertical displacements of each node at the fixed end section to zero. This does not completely match the constraint conditions assumed in the analytical solution, which can, to some extent, affect the accuracy of the solution. Figure 12 The relative error in the data contributes to this.

[0088] Example of a nonhomogeneous cantilever beam: The geometry, constraints, and loading conditions of the cantilever beam are the same as in the previous example, except that the material is no longer homogeneous; instead, its elastic modulus is assumed to vary randomly within a certain range. Multi-scale finite element analysis can still be performed using the algorithm proposed in this paper. However, since the material parameters are different in each coarse element domain, it is necessary to calculate the basis functions and equivalent stiffness matrix of each coarse element domain individually, and then integrate the overall stiffness for solution. For non-homogeneous cantilever beams, there is no analytical solution. For comparison, the beam is now meshed with a 48-8 fine mesh, i.e., 48 columns of elements horizontally and 8 layers of elements vertically. Thus, the size of each element is... It consists of 384 elements, 441 nodes, and 882 degrees of freedom. The generalized Maxwell model is still used to reflect the viscoelasticity of the material, and the elastic modulus of the material within different elements is assumed. The initial elastic modulus of the material within different units is randomly varied between 2000 MPa and 40000 MPa, while other viscoelastic parameters remain constant. MPa and The pressure varies randomly between MPa, resulting in an aperiodic structure. Based on this refined mesh, viscoelastic analysis is performed using the initial stress method, and the results are used as a comparison standard. When using the algorithm presented in this paper for multi-scale analysis, the macroscopic mesh generation is the same as in the above example, i.e., 12 horizontal columns and 2 vertical layers. The fine mesh generation within each coarse mesh adopts a 4-4 fine element domain partitioning scheme, i.e., the size of the sub-element is... The mesh generation information for multi-scale finite element method and traditional finite element method is shown in Table 2. The vertical viscoelastic displacement of the free end of the cantilever beam in the first 36 seconds under both algorithms is shown in Table 2. Figure 8 As shown, the average error is around 5% to 6%.

[0089] Table 2 Mesh Generation Information for Heterogeneous Cantilever Beams Compared to the asymptotic homogenization methods commonly used in multi-scale analysis, the multi-scale finite element method adopted in this paper relaxes the assumption that unit cells must be periodic. Therefore, it can be used to solve non-periodic structures with randomly varying material properties or irregular geometric shapes, such as those in this example. However, in this case, since the material or geometric properties within each coarse element domain are no longer the same, the basis functions and equivalent stiffness matrices must be solved separately for each coarse element domain, resulting in lower computational efficiency than for periodic structures under the same coarse element domain partitioning. Fortunately, the mesoscopic analysis for each coarse element domain is independent, making parallel programming highly suitable for solving large-scale practical engineering problems to improve computational efficiency. Due to the small overall computational scale of this example, the program was not optimized. Even so, Table 2 still shows the advantages of the proposed algorithm compared to traditional methods. Solving the linear equation system is a time-consuming step in finite element calculations. When using the traditional viscoelastic finite element analysis based on the initial stress method, this example has 882 degrees of freedom, meaning that at each moment, the basis functions must be solved separately. The total stiffness matrix is ​​used to solve the linear equations at each time step to obtain the displacement at that moment. However, by using the multi-scale method proposed in this paper, the total degrees of freedom are reduced to 78, and the size of the stiffness matrix used to solve the linear equations at each time step is reduced to [missing information]. The computational load is significantly reduced. Figure 8 The comparison of calculation results from different methods shows that, with a significant reduction in the solution scale, the calculation results of the algorithm proposed in this paper have an error of approximately 5% to 6% compared to the traditional finite element results with refined mesh generation, which basically meets engineering requirements. To further improve the calculation accuracy, the supersample technique from the elastic multi-scale finite element algorithm or the more accurate multi-node quadrilateral element can be considered. This is also a problem worthy of further research in this field.

[0090] This paper elucidates the basic principles and implementation process of a multi-scale finite element analysis method for viscoelastic materials based on the initial stress method. The generalized Maxwell model is used to reflect the viscoelastic properties of the microstructure. Utilizing the idea of ​​the initial stress method and combining it with the multi-scale finite element theory of elasticity problems, cross-scale analysis of viscoelastic materials is achieved. The paper presents the computational steps of the proposed method and demonstrates its implementation process and computational accuracy with three examples. Research shows that the proposed method has the following advantages compared to existing methods: (1) High computational efficiency. For linear viscoelastic problems with periodic structures, the multi-scale basis functions only need to be calculated once, and the total equivalent stiffness matrix only needs to be calculated once, which greatly improves computational efficiency; (2) Saves computational resources. When calculating the relaxation stress of each element within the fine element domain in each time period, a recursive formula is used. Only the relaxation stress and strain of the previous time period need to be known, and there is no need to save all historical data, which greatly saves computational memory; (3) Expanding the scale of solutions. Compared with the traditional finite element method, implementing multi-scale analysis greatly reduces the degrees of freedom of macroscopic equations, which can significantly improve computational efficiency or increase the scale of solutions; (4) Wide range of applications. Compared with the asymptotic homogenization method, the proposed method does not require the representative volume element to be periodic, nor does it require the macroscopic parameters to have separation characteristics at different scales, which greatly expands the application range of the multi-scale method; at the same time, due to the above characteristics, the proposed method has the potential to be extended to nonlinear viscoelastic problems.

[0091] Reference Figure 9 The present invention also provides a performance analysis device for viscoelastic materials, the device comprising: Elastic operator acquisition module 902 is used to acquire the elastic operators of a specified viscoelastic material at multiple time points; The viscoelastic material partitioning module 904 is used to partition the specified viscoelastic material into multiple coarse element domains using a preset multi-scale finite element method; wherein each coarse element domain includes multiple fine element domains; The equivalent stiffness matrix set acquisition module 906 is used to construct multi-scale basis functions for each coarse element domain at each time point based on the elastic operators at each time point, and to acquire the equivalent stiffness matrix set for each time point based on the multi-scale basis functions; wherein, the equivalent stiffness matrix set includes the equivalent stiffness matrix corresponding to each coarse element domain at the same time point; The micro information set acquisition module 908 is used to perform downscaling calculations on the equivalent stiffness matrices in each of the equivalent stiffness matrix sets to obtain the micro information set corresponding to each time point; wherein, the micro information set includes the micro information corresponding to each fine unit domain within the same time point. The micro-information set function relationship analysis module 910 is used to analyze the micro-information set function relationship between adjacent time points based on a preset model; The viscoelasticity analysis module 912 is used to perform viscoelasticity analysis on the specified viscoelastic material based on the functional relationship.

[0092] In one embodiment, the viscoelastic material segmentation module 904 includes: The basic coarse element domain partitioning submodule is used to pre-partition the basic coarse element domain and the basic fine element domain in each basic coarse element domain using a preset multi-scale finite element method. The material parameter input submodule is used to obtain the material parameters of the specified elastic material and input the material parameters of the specified elastic material into the basic fine element domain to obtain multiple coarse element domains.

[0093] In one embodiment, the equivalent stiffness matrix set acquisition module 906 includes: The unit displacement application submodule is used to apply unit displacements to each node degree of freedom in the coarse element domain while keeping the other node degrees of freedom at 0, so as to set the equilibrium equations for each coarse element domain. The equilibrium equation solving submodule is used to solve the equilibrium equations using the substructure stiffness matrix and to find the basis functions for each degree of freedom in the corresponding coarse element domain. The multi-scale basis function establishment submodule is used to establish multi-scale basis functions for the coarse unit domain based on the basis functions of each degree of freedom of the coarse unit domain.

[0094] In one embodiment, the microscopic information set acquisition module 908 includes: The displacement vector retrieval submodule is used to retrieve the displacement vector of each node in each fine element domain based on the equivalent stiffness matrix in each set of equivalent stiffness matrices. The stress calculation submodule is used to calculate the stress and strain corresponding to each fine element domain based on the displacement vector; wherein, the microscopic information includes the stress and strain of the fine element domain.

[0095] In one embodiment, the micro-information set function relationship analysis module 910 includes: The initial recursive formula acquisition submodule is used to obtain the initial recursive formula of the stress using the preset model; The recursive function relationship acquisition submodule is used to solve the parameters in the initial recursive formula based on the physical relationship of each fine element domain, so as to obtain the recursive function relationship of stress changing with time.

[0096] In one embodiment, the specified viscoelastic material is any one of rubber, plastic, textile fiber, and polymer.

[0097] In one embodiment, in the step of analyzing the functional relationship of the micro-information set at adjacent time points based on a preset model, the preset model is a generalized Maxwell model.

[0098] Figure 10 An internal structural diagram of an electronic device in one embodiment is shown. This electronic device can specifically be a terminal or a server, and more specifically, a computer device. Figure 10 As shown, the electronic device includes a processor, a memory, and a network interface connected via a system bus. The memory includes a non-volatile storage medium and internal memory. The non-volatile storage medium stores an operating system and may also store a computer program. When executed by the processor, this computer program enables the processor to implement a method for analyzing the performance of viscoelastic materials. The internal memory may also store a computer program, which, when executed by the processor, enables the processor to implement a method for analyzing the performance of viscoelastic materials. Those skilled in the art will understand that... Figure 10 The structure shown is merely a block diagram of a portion of the structure related to the present application and does not constitute a limitation on the electronic device to which the present application is applied. The specific electronic device may include more or fewer components than shown in the figure, or combine certain components, or have different component arrangements.

[0099] In one embodiment, an electronic device is provided, including a memory and a processor, the memory storing a computer program that, when executed by the processor, causes the processor to perform the following steps: Obtain the elasticity operators of a specified viscoelastic material at multiple time points; The specified viscoelastic material is divided into multiple coarse element domains using a pre-defined multi-scale finite element method; wherein each coarse element domain includes multiple fine element domains; Based on the elastic operators at each time point, a multi-scale basis function is constructed for each of the coarse element domains at each time point, and the equivalent stiffness matrix set at each time point is obtained based on the multi-scale basis function; wherein, the equivalent stiffness matrix set includes the equivalent stiffness matrix corresponding to each of the coarse element domains at the same time point; Downscaling calculations are performed on the equivalent stiffness matrices in each of the aforementioned equivalent stiffness matrix sets to obtain the micro-information set corresponding to each time point; wherein, the micro-information set includes the micro-information corresponding to each of the aforementioned fine unit domains within the same time point; The functional relationship of the microscopic information set at adjacent time points is analyzed based on a preset model. Based on the aforementioned functional relationship, a viscoelastic analysis of the specified viscoelastic material is performed.

[0100] Compared with existing technologies, it has the advantages of high computational efficiency, resource saving and expansion of problem-solving scale, reduces the need for historical data storage, significantly reduces the degrees of freedom of macroscopic equations, is widely applicable to non-periodic structures, and has the potential to be extended to nonlinear viscoelastic problems.

[0101] In one embodiment, a computer-readable storage medium is provided storing a computer program that, when executed by a processor, causes the processor to perform the following steps: Obtain the elasticity operators of a specified viscoelastic material at multiple time points; The specified viscoelastic material is divided into multiple coarse element domains using a pre-defined multi-scale finite element method; wherein each coarse element domain includes multiple fine element domains; Based on the elastic operators at each time point, a multi-scale basis function is constructed for each of the coarse element domains at each time point, and the equivalent stiffness matrix set at each time point is obtained based on the multi-scale basis function; wherein, the equivalent stiffness matrix set includes the equivalent stiffness matrix corresponding to each of the coarse element domains at the same time point; Downscaling calculations are performed on the equivalent stiffness matrices in each of the aforementioned equivalent stiffness matrix sets to obtain the micro-information set corresponding to each time point; wherein, the micro-information set includes the micro-information corresponding to each of the aforementioned fine unit domains within the same time point; The functional relationship of the microscopic information set at adjacent time points is analyzed based on a preset model. Based on the aforementioned functional relationship, a viscoelastic analysis of the specified viscoelastic material is performed.

[0102] Compared with existing technologies, it has the advantages of high computational efficiency, resource saving and expansion of problem-solving scale, reduces the need for historical data storage, significantly reduces the degrees of freedom of macroscopic equations, is widely applicable to non-periodic structures, and has the potential to be extended to nonlinear viscoelastic problems.

[0103] Those skilled in the art will understand that all or part of the processes in the above embodiments can be implemented by a computer program instructing related hardware. The program can be stored in a non-volatile computer-readable storage medium, and when executed, it can include the processes of the embodiments described above. Any references to memory, storage, databases, or other media used in the embodiments provided in this application can include non-volatile and / or volatile memory. Non-volatile memory can include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory can include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in various forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), dual data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), RAMbus direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and RAMbus dynamic RAM (RDRAM), etc.

[0104] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0105] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of this patent application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this patent application should be determined by the appended claims.

Claims

1. A method for analyzing the properties of viscoelastic materials, characterized in that, The method includes: Obtain the elasticity operators of a specified viscoelastic material at multiple time points; The specified viscoelastic material is divided into multiple coarse element domains using a pre-defined multi-scale finite element method; wherein each coarse element domain includes multiple fine element domains; Based on the elastic operators at each time point, a multi-scale basis function is constructed for each of the coarse element domains at each time point, and the equivalent stiffness matrix set at each time point is obtained based on the multi-scale basis function; wherein, the equivalent stiffness matrix set includes the equivalent stiffness matrix corresponding to each of the coarse element domains at the same time point; Downscaling calculations are performed on the equivalent stiffness matrices in each of the aforementioned equivalent stiffness matrix sets to obtain the micro-information set corresponding to each time point; wherein, the micro-information set includes the micro-information corresponding to each of the aforementioned fine unit domains within the same time point; The functional relationship of the microscopic information set at adjacent time points is analyzed based on a preset model. Based on the aforementioned functional relationship, a viscoelastic analysis of the specified viscoelastic material is performed.

2. The method for analyzing the performance of viscoelastic materials according to claim 1, characterized in that, The step of dividing the specified viscoelastic material into multiple coarse element domains using a preset multi-scale finite element method includes: The basic coarse element domain and the basic fine element domain in each basic coarse element domain are pre-divided using a preset multi-scale finite element method. Obtain the material parameters of the specified elastic material, and input the material parameters of the specified elastic material into the basic fine element domain to obtain multiple coarse element domains.

3. The method for analyzing the performance of viscoelastic materials according to claim 2, characterized in that, The step of constructing multi-scale basis functions for each coarse unit domain at each time point based on the elastic operator at each time point includes: Apply unit displacements to each node degree of freedom in the coarse element domain, while keeping the other node degrees of freedom at 0, to set the equilibrium equations for each coarse element domain. The equilibrium equations are solved using the substructure stiffness matrix to obtain the basis functions for each degree of freedom in the coarse element domain. Based on the basis functions of each degree of freedom of the coarse unit domain, a multi-scale basis function for the coarse unit domain is established.

4. The method for analyzing the performance of viscoelastic materials according to claim 1, characterized in that, The step of performing downscaling calculations on the equivalent stiffness matrices in each of the equivalent stiffness matrix sets to obtain the microscopic information set corresponding to each time point includes: The displacement vector of each node in each fine element domain is obtained based on the equivalent stiffness matrix in each set of equivalent stiffness matrices. The stress and strain corresponding to each fine element domain are obtained based on the displacement vector; wherein, the microscopic information includes the stress and strain of the fine element domain.

5. The method for analyzing the performance of viscoelastic materials according to claim 4, characterized in that, The step of analyzing the functional relationship of the micro-information set at adjacent time points based on a preset model includes: The initial recursive formula for the stress is obtained using the preset model; The parameters in the initial recursive formula are solved based on the physical relationships of each fine element domain to obtain the recursive function relationship of stress changing with time.

6. The method for analyzing the performance of viscoelastic materials according to claim 1, characterized in that, The specified viscoelastic material is any one of rubber, plastic, textile fiber, and polymer.

7. The method for analyzing the performance of viscoelastic materials according to claim 1, characterized in that, In the step of analyzing the functional relationship of the micro-information set at adjacent time points based on a preset model, the preset model is a generalized Maxwell model.

8. A performance analysis device for viscoelastic materials, characterized in that, The device includes: The elastic operator acquisition module is used to acquire the elastic operators of a specified viscoelastic material at multiple time points. The viscoelastic material partitioning module is used to divide the specified viscoelastic material into multiple coarse element domains using a preset multi-scale finite element method; wherein each coarse element domain includes multiple fine element domains; The equivalent stiffness matrix set acquisition module is used to construct multi-scale basis functions for each coarse element domain at each time point based on the elastic operators at each time point, and to acquire the equivalent stiffness matrix set for each time point based on the multi-scale basis functions; wherein, the equivalent stiffness matrix set includes the equivalent stiffness matrix corresponding to each coarse element domain at the same time point; The micro-information set acquisition module is used to perform downscaling calculations on the equivalent stiffness matrices in each of the equivalent stiffness matrix sets to obtain the micro-information set corresponding to each time point; wherein, the micro-information set includes the micro-information corresponding to each of the fine unit domains within the same time point; The micro-information set function relationship analysis module is used to analyze the micro-information set function relationship between adjacent time points based on a preset model; The viscoelasticity analysis module is used to perform viscoelasticity analysis on the specified viscoelastic material based on the functional relationship.

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