Transverse shear stiffness estimation method, homogenization simulation method, system and medium

Abstract

The present invention provides a transverse shear stiffness estimation method, a homogenization simulation method, a system and a medium. The method is applicable to a periodic shell. By introducing a coarse-scale coordinate system and a fine-scale coordinate system, and utilizing the progressive homogenization theory of the shell, a method for calculating the transverse shear stiffness of the periodic shell is obtained, and the coefficients of the transverse shear stiffness matrix E of the periodic shell that satisfies the displacement form are determined through a classical homogenization process. The method of the present invention is applicable to periodic shells containing arbitrary inclusions or boundaries, such as hollow floor slabs, sandwich structures, waffle structures, and plate-lattice structures, etc.; and is also applicable to the calculation of the transverse shear stiffness of homogeneous thick shells and laminated shells. On this basis, the obtained transverse shear stiffness is further substituted into the finite element model of the homogenized shell, and the periodic shell can be homogenized and simulated to obtain accurate displacement and internal force, and accurate displacement and internal force are obtained with extremely small calculation cost and calculation accuracy loss, meeting the needs of engineering design.

Description

Technical Field

[0001] The present invention relates to the technical field of performance detection of composite shells, and in particular to a transverse shear stiffness estimation method, a homogenization simulation method, a system and a medium, which are applicable to periodic shells. Background Art

[0002] Composite shell structure refers to a new type of material formed by combining multiple layers of materials through lamination and other methods. It has significant advantages in terms of light weight, high strength and rigidity, temperature resistance, corrosion resistance, and flexible design. It has broad application prospects in aerospace engineering, automotive industry, shipbuilding and marine engineering, chemical engineering, wind power generation, modern building engineering and construction, and other fields.

[0003] The performance of composite shell structures is affected by the component layer level, including component materials, material interfaces, geometric shapes, axial force effects, and adjacent supports. The shell structure performance such as stress, strain, failure and damage at the layer can be measured and estimated through experimental, numerical simulation or theoretical derivation methods. In engineering practice, the transverse shear stiffness of the shell has a significant impact on the mechanical properties and behavior of the shell. It represents the ratio of the shear stress caused by the relative displacement of the shell structure in the normal direction under the action of external forces to the displacement, indicating the ability of the shell structure to resist deformation when subjected to lateral forces. Whether it is the design of uniform thick shells, laminated shells, or periodic shell structures, understanding and determining the transverse shear stiffness of the shell is of vital importance for the design of lightweight structural parts, prefabricated components of buildings and other shell structures.

[0004] At present, there are theories and methods for calculating the transverse shear stiffness for homogeneous thick shells and laminated shells. The representative volume element method (RVE) and the asymptotic homogenization method (AH) are two numerical methods for predicting the equivalent performance of periodic materials. The representative volume element method is a numerical homogenization algorithm based on the mean field theory. By selecting representative volume elements and using micromechanical analysis methods to calculate the equivalent mechanical parameters of the representative volume elements, it is used for macroscopic solution of the problem, and the average mechanical properties of the representative volume elements represent the performance of the overall structure. The representative volume element method has a clear mechanical concept and is simple to operate. It is suitable for homogenized equivalent estimation of structures similar to honeycomb materials, but is not suitable for numerical calculations of complex periodic shell structures. The asymptotic homogenization method has a rigorous mathematical basis, and there are mature theories and algorithms for calculating the material equivalent properties of three-dimensional and two-dimensional periodic microstructures. However, for periodic shell structures, such as hollow floors, sandwich structures, waffle structures, and plate-lattice structures, there is no theory and method for calculating the lateral shear stiffness. Although AA Kalamkarov et al. proposed the asymptotic homogenization mathematical theory of periodic plate and shell structures, it still cannot realize the finite element realization and macroscopic homogenization simulation of periodic shell structures. Summary of the invention

[0005] In view of the defects and shortcomings of the prior art, according to the first aspect of the present invention, a method for estimating the lateral shear stiffness is provided, which is applicable to periodic shells. The method utilizes the asymptotic homogenization theory of shells to realize the estimation of the lateral shear stiffness of periodic shells. The method is applicable to periodic shells containing arbitrary inclusions or boundaries, such as hollow floors, sandwich structures, waffle structures and plate-lattice structures, and is also applicable to the estimation of the lateral shear stiffness of homogeneous thick shells and laminated shells.

[0006] The method for estimating the transverse shear stiffness according to the first aspect is applicable to a periodic shell, and the method comprises the following steps:

[0007] Step 1: Establish a coarse-scale coordinate system for characterizing the mathematical model of periodic shells ( α 1, α 2, c ) and fine-scale coordinate system ( y 1, y 2, z ), the coarse-scale coordinate system ( α 1, α 2, c ) is an orthogonal curvilinear coordinate system, the coarse-scale coordinate system ( α 1, α 2, c ) and fine-scale coordinate system ( y1, y 2, z ) based on the coarse-scale coordinate system ( α 1, α 2, c )The thickness of the lower shell and the fine-scale coordinate system ( y 1, y 2, z ) The relationship is established between the lateral dimensions of the unit cell of the lower shell;

[0008] Step 2: In the coarse-scale coordinate system ( α 1, α 2, c ) and the fine-scale coordinate system ( y 1, y 2, z ) and the relative displacement function of the shell is constructed by using the perturbation method;

[0009] Step 3: Based on the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, α 1, α 2, c The relative displacement in the direction is used to obtain the strain function based on the relative displacement of the shell, and the strain tensor corresponding to the perturbation coefficient is determined;

[0010] Step 4, according to the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, obtain the stress function based on the relative displacement of the shell;

[0011] Step 5, inputting the strain tensor corresponding to the perturbation coefficient in step 3 into the generalized Hooke's law of three-dimensional elasticity to obtain the stress tensor corresponding to the perturbation coefficient;

[0012] Step 6: According to the stress function based on the relative displacement of the shell in step 4, the equilibrium equation of the shell is input to obtain the equilibrium equation corresponding to the perturbation expansion coefficient;

[0013] Step 7, based on the stress tensor corresponding to the perturbation coefficient in step 5, the equilibrium equation corresponding to the perturbation expansion coefficient in step 6 is input to obtain the equilibrium equation considering the transverse shear deformation expressed in relative displacement;

[0014] Step 8, input the displacement form of the shell into the equilibrium equation considering the transverse shear deformation in step 7 to obtain the control equation in the domain Ω;

[0015] Step 9: Based on the control equation in the domain Ω, determine the control equation in the domain Θ;

[0016] Step 10: Convert the control equations in the domain Θ into finite element form and solve them numerically, and use the classical homogenization process to obtain the coefficients of the transverse shear stiffness matrix E of the periodic shell that satisfies the displacement form.

[0017] According to a second aspect of the present invention, a homogenization simulation method taking into account transverse shear stiffness is provided, which is applicable to periodic shells, has extremely low computational cost and loss of computational accuracy, and is applicable to periodic shells with arbitrary hollow inclusions or complex boundaries.

[0018] The homogenization simulation method considering transverse shear stiffness according to the second aspect is applicable to periodic shells, and the homogenization simulation method comprises the following steps:

[0019] Establish a finite element model of a unit cell of a periodic shell and input the material parameters of the unit cell, apply a periodic boundary to the unit cell, and apply an initial stress load to the unit cell;

[0020] The finite element results of the unit cell are solved according to the above method to obtain the coefficients of the transverse shear stiffness matrix E of the periodic shell;

[0021] A homogenized finite element model of a periodic shell is established, and the transverse shear stiffness matrix E is input. After applying boundary conditions and loads, the central deflection of the periodic shell is calculated.

[0022] According to a third aspect of the present invention, there is provided a computer system comprising one or more processors and a memory, wherein the memory stores operable instructions, wherein when the instructions are executed by the one or more processors, the one or more processors perform operations, wherein the operations include the process of executing the aforementioned method.

[0023] According to a fourth aspect of the present invention, there is provided a computer-readable medium storing a computer program, wherein the computer program comprises instructions / instruction sets that can be executed by one or more processors, and the instructions / instruction sets, when executed by the one or more processors, implement the process of the aforementioned method.

[0024] According to a fifth aspect of the present invention, there is provided a computer program product, comprising a computer program, wherein the computer program implements the steps of the aforementioned method when executed by a processor.

[0025] It should be understood that all combinations of the aforementioned concepts and the additional concepts described in more detail below can be considered as part of the inventive subject matter of the present disclosure as long as such concepts are not mutually inconsistent. In addition, all combinations of the claimed subject matter are considered as part of the inventive subject matter of the present disclosure.

[0026] The foregoing and other aspects, embodiments and features of the present invention can be more fully understood from the following description in conjunction with the accompanying drawings. Other additional aspects of the present invention, such as the features and / or beneficial effects of the exemplary embodiments, will be apparent from the following description or learned from the practice of the specific embodiments according to the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] The accompanying drawings are not intended to be drawn to scale. In the accompanying drawings, each identical or nearly identical component shown in each figure may be represented by the same reference numeral. For clarity, not every component is labeled in each figure.

[0028] Figure 1 is a flow chart of a method for estimating the transverse shear stiffness of a periodic shell according to an embodiment of the present invention.

[0029] Figure 2 is a flow chart of a periodic shell homogenization simulation according to an embodiment of the present invention.

[0030] Figure 3 is a schematic diagram of a uniform thickness shell having periodically distributed square inclusions within a plate according to an embodiment of the present invention.

[0031] Figure 4 is a schematic diagram of a homogenized shell according to an embodiment of the present invention.

[0032] Figure 5 It is a schematic diagram of the simulation calculation results of DNS and HNS on the ANSYS platform according to an embodiment of the present invention.

[0033] Figure 6 It is a schematic diagram of the simulation calculation results by DNS and HNS on the ABAQUS platform according to an embodiment of the present invention. DETAILED DESCRIPTION

[0034] In order to better understand the technical content of the present invention, specific embodiments are described below in conjunction with the accompanying drawings.

[0035] Various aspects of the present invention are described in this disclosure with reference to the accompanying drawings, in which many illustrative embodiments are shown. The embodiments of the present disclosure are not necessarily intended to include all aspects of the present invention. It should be understood that the various concepts and embodiments introduced above, as well as those described in more detail below, can be implemented in any of many ways, because the concepts and embodiments disclosed by the present invention are not limited to any implementation. In addition, some aspects disclosed by the present invention can be used alone or in any appropriate combination with other aspects disclosed by the present invention.

[0036] The method for estimating the transverse shear stiffness according to the disclosed embodiment of the present invention is applicable to periodic shells. The method uses the theory of progressive homogenization of shells to obtain the theory and method for calculating the transverse shear stiffness of periodic shells. The method is applicable to periodic shells containing arbitrary inclusions or boundaries, such as hollow floor slabs, sandwich structures, waffle structures, and plate-lattice structures. At the same time, the method of the present invention can also be applied to the estimation of the transverse shear stiffness of homogeneous thick shells and laminated shells to obtain the transverse shear stiffness. On this basis, the obtained transverse shear stiffness is further substituted into the finite element model of the homogenized shell, and the homogenization simulation of the periodic shell considering the transverse shear stiffness can be realized, and accurate displacement and internal force can be obtained with extremely small calculation cost and calculation accuracy loss, meeting the needs of engineering design.

[0037] {Example 1}

[0038] The method for estimating the transverse shear stiffness according to the disclosed embodiment of the present invention is applicable to a periodic shell, that is, a method for estimating the transverse shear stiffness of a periodic shell, and can be executed by a computer system. The computer system can be implemented through a terminal, a server, a cloud server, or an interactive system between a terminal and a server.

[0039] In this embodiment, the computer system in the terminal is used as an example. Figure 1 The illustrated example of the method for estimating the transverse shear stiffness of a periodic shell includes the following steps:

[0040] Step 1: Establish a coarse-scale coordinate system for characterizing the mathematical model of periodic shells ( α 1, α 2, c ) and fine-scale coordinate system ( y 1, y 2, z ), the coarse-scale coordinate system ( α 1, α 2, c ) is an orthogonal curvilinear coordinate system, the coarse-scale coordinate system ( α 1, α 2, c ) and fine-scale coordinate system ( y 1, y 2, z ) based on the coarse-scale coordinate system ( α 1, α 2, c )The thickness of the lower shell and the fine-scale coordinate system ( y 1, y 2, z ) is related to the lateral dimensions of the unit cell of the lower shell;

[0041] Step 2: In the coarse-scale coordinate system ( α1, α 2, c ) and the fine-scale coordinate system ( y 1, y 2, z ) and the relative displacement function of the shell is constructed by using the perturbation method;

[0042] Step 3: Based on the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, α 1, α 2, c The relative displacement in the direction is used to obtain the strain function based on the relative displacement of the shell, and the strain tensor corresponding to the perturbation coefficient is determined;

[0043] Step 4, according to the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, obtain the stress function based on the relative displacement of the shell;

[0044] Step 5, inputting the strain tensor corresponding to the perturbation coefficient in step 3 into the generalized Hooke's law of three-dimensional elasticity to obtain the stress tensor corresponding to the perturbation coefficient;

[0045] Step 6: According to the stress function based on the relative displacement of the shell in step 4, the equilibrium equation of the shell is input to obtain the equilibrium equation corresponding to the perturbation expansion coefficient;

[0046] Step 7, based on the stress tensor corresponding to the perturbation coefficient in step 5, the equilibrium equation corresponding to the perturbation expansion coefficient in step 6 is input to obtain the equilibrium equation considering the transverse shear deformation expressed in relative displacement;

[0047] Step 8, input the displacement form of the shell into the equilibrium equation considering the transverse shear deformation in step 7 to obtain the control equation in the domain Ω;

[0048] Step 9: Based on the control equation in the domain Ω, determine the control equation in the domain Θ;

[0049] Step 10: Convert the control equations in the domain Θ into finite element form and solve them, and use the classical homogenization process to obtain the coefficients of the transverse shear stiffness matrix E of the periodic shell that satisfies the displacement form.

[0050] In the embodiment of the present invention, a coarse-scale coordinate system ( α 1, α 2, c ) and fine-scale coordinate system ( y 1, y 2, z ), where the coarse-scale coordinate system is an orthogonal curvilinear coordinate system, let α = [ α 1, α 2],y= [ y 1,y 2].

[0051] In the embodiment of the present invention, for ease of explanation, we make the following rule definition: the subscript of a variable is represented by Greek letters and takes values ​​in the range of {1, 2}, and the subscript of a variable is represented by Latin letters and takes values ​​in the range of {1, 2, 3}.

[0052] As an optional implementation, in step 1, based on the coarse-scale coordinate system ( α 1, α 2, c )The thickness of the lower shell and the fine-scale coordinate system ( y 1, y 2, z ) The relationship between the lateral dimensions of the unit cell of the lower shell is established as follows:

[0053] ;

[0054] in, h 1, h 2 respectively represent the unit cells of the shell α 1 and α 2. Transverse dimension in the direction; d represents the perturbation coefficient and represents the thickness of the shell in the coarse-scale coordinate system.

[0055] It should be understood that in the embodiments of the present invention, the definition of a unit cell is the same as that of AA Kalamkarov et al. in the composite and reinforced elements of construction document, which refers to a microstructure that is periodically repeated in a periodic shell structure.

[0056] As an optional implementation, in step 2, in the coarse-scale coordinate system ( α 1, α 2, c ) and fine-scale coordinate system ( y 1, y 2, z ), the displacement function of the shell is obtained by the perturbation method, which is expressed as follows:

[0057] ;

[0058] In the formula, u 1, u 2 and u 3 respectively represent α 1, α 2 and c Relative displacement in direction. refers to higher-order infinitesimals, The variables are in .

[0059] As an optional implementation, in step 3, according to the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, based on α 1, α 2, c The relative displacement in the direction is used to obtain the strain function based on the relative displacement of the shell and determine the strain tensor corresponding to the perturbation coefficient, including:

[0060] Substituting the relative displacement function of the shell into the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, the strain function based on the relative displacement of the shell is obtained, which is expressed as follows:

[0061] ;

[0062] in, represents the strain tensor; Representation and perturbation expansion coefficient d l The corresponding strain tensor, l ∈{0,1, 2, …};

[0063] And, according to the strain function based on the relative displacement of the shell, we obtain d 2 The corresponding variable tensor , expressed as follows:

[0064] ;

[0065] in, u 1, u 2, u 3 respectively represent α 1, α 2 and c The relative displacement in the direction, Representation and perturbation expansion coefficient d l The corresponding displacement, i ∈{1, 2, 3}; l ∈{0, 1, 2, 3, …}; , Orthogonal curvilinear coordinate system ( α 1, α 2, c )The coefficients of the first quadratic form.

[0066] As an optional implementation, in step 4, according to the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, a stress function based on the relative displacement of the shell is obtained, which is expressed as follows:

[0067] ;

[0068] in, represents the stress tensor; Representation and perturbation expansion coefficient d l The corresponding stress tensor, l ∈{0,1, 2, …}. refers to higher-order infinitesimals, The variables are in .

[0069] As an optional implementation, in step 5, the stress tensor corresponding to the perturbation coefficient obtained above is expressed as:

[0070] ;

[0071] In the formula, Representation and perturbation expansion coefficient d l The corresponding stress tensor, l ∈{0, 1, 2, …}; represents the fourth-order material stiffness tensor in the fine-scale coordinate system, y = [ y 1, y 2]; Representation and and The associated strain tensor; Representation and The associated strain tensor.

[0072] Further, we can get d 2 The corresponding stress tensor , expressed as follows:

[0073] ;

[0074] ;

[0075] In the formula, , They represent the principal curvature functions of the surface, and the independent variables are α 1, α 2. For surfaces of constant curvature, , is a constant.

[0076] As an optional implementation, in step 6, the stress function based on the relative displacement of the shell in step 4 is input into the equilibrium equation of the shell to obtain the equilibrium equation corresponding to the perturbation expansion coefficient, that is, d -1 , d 0 and d 1The equilibrium equation is as follows:

[0077] First equation: ;

[0078] Second equation: ;

[0079] The third equation: ;

[0080] In the formula, and Both represent differential operators.

[0081] It should be understood that the shell equilibrium equation here refers in particular to the shell equilibrium equation defined by AA Kalamkarov et al. in the literature on composite and reinforced elements of construction.

[0082] Further, in step 7, the stress tensor corresponding to the perturbation coefficient in step 5 is input into the equilibrium equation corresponding to the perturbation expansion coefficient in step 6 to obtain the equilibrium equation considering the transverse shear deformation expressed in relative displacement. As an optional implementation, specifically, d 2 The corresponding stress tensor is input into the aforementioned third equation to obtain the equilibrium equation considering the transverse shear deformation expressed in relative displacement, which is expressed as:

[0083] ;

[0084] In the formula, , , represent differential operators respectively.

[0085] As an optional implementation, in step 8, the displacement form of the aforementioned shell is expressed as:

[0086] ;

[0087] In the formula, , They represent the assumed functions of the displacement field respectively.

[0088] Furthermore, the displacement form of the shell is input into the equilibrium equation considering the transverse shear deformation in the aforementioned step 7 to obtain the control equation in the domain Ω, which is expressed as follows:

[0089] ;

[0090] In the formula, represents the upper and lower boundary normal vectors of the unit cell in the domain Ω of j The direction component. represents the differential operator, An assumed function representing the stress tensor, and is its component expression; , Ω refers to the coordinate system ( x 1, x 2, z ), ∈{1,2}, i ∈{1, 2, 3}, j ∈{1, 2, 3}.

[0091] As an optional implementation, in step 9, based on the control equation in the domain Ω, the control equation in the domain Θ is determined, and the control equation in the domain Θ is expressed as follows:

[0092] ;

[0093] In the formula, , ; Θ refers to the coordinate system ( or 1, or 2, or 3) The domain below; η = [ or 1, or 2, or 3]; d = [ d 1, d 2, 0], d 1 and d 2 respectively represent the domain Θ in or 1 and or Length in 2 directions; denotes the vertices of the domain Θ.

[0094] As an optional implementation, in step 10, the control equation in the domain Θ is converted into a finite element form, expressed as follows:

[0095] ;

[0096] In the formula, , They refer to the strain matrices of discrete elements in domain Θ respectively; represents the global stiffness matrix of domain Θ; represents the equivalent global load vector of domain Θ; In the domain Θ Numerical solution of ;

[0097] The coefficients of the transverse shear stiffness matrix E of the periodic shell satisfying the displacement form are obtained by using the classical homogenization process and are expressed as follows:

[0098] ;

[0099] represents the classical homogenization process.

[0100] {Example 2}

[0101] Combination Figure 2 As shown, according to the present disclosure, a homogenization simulation method considering transverse shear stiffness is also proposed, which is applicable to periodic shells, that is, a periodic shell homogenization simulation method considering transverse shear stiffness, wherein the periodic shell refers to a periodic shell (i.e., a homogenized shell) with equal thickness and a principal curvature equal to a constant assumption, and the periodic shell homogenization simulation method comprises the following steps:

[0102] Establish a finite element model of a unit cell of a periodic shell and input the material parameters of the unit cell, apply a periodic boundary to the unit cell, and apply an initial stress load to the unit cell;

[0103] The finite element results of the unit cell are solved according to the aforementioned numerical solution method to obtain the coefficients of the transverse shear stiffness matrix E of the periodic shell;

[0104] A homogenized finite element model of a periodic shell is established, and the transverse shear stiffness matrix E is input. After applying boundary conditions and loads, the central deflection of the periodic shell is calculated.

[0105] {Example 3}

[0106] Based on Examples 1 and 2, Figure 3~6 As shown, a shell of equal thickness with periodically distributed square inclusions in the plate is taken as an example for further explanation.

[0107] Combination Figure 3 As shown in , (a), (b), (c), and (d) represent the plane view and cross-section view of a shell of equal thickness, and the plane view and cross-section view of a unit cell, respectively. Figure 3 As shown in (a) and (b), the periodic shell is an equal-thickness shell with periodically distributed square inclusions in a plate, with a length and width of 9.4m and a thickness of 0.3m.

[0108] The engineering constants of the base material of the plate are as follows: Young's modulus E m =25.5GPa, Poisson's ratio v m =0.2.

[0109] Combination Figure 3 (a) There are periodically distributed square inclusions in the plate.

[0110] Combination Figure 3In (c) and (d), each square inclusion is 0.814m long and wide, and 0.2m thick.

[0111] The engineering constants of the inclusion material are as follows: Young's modulus E i ∈{0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2,1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.9, 3.0, 3.2, 3.4, 3.6, 3.8, 4.0} E m , Poisson's ratio v m =0.2.

[0112] like Figure 3 As shown in (c) and (d), the volume of the inclusion in the unit cell of the shell accounts for 50%.

[0113] A uniformly distributed load of 10 kPa was applied to the top surface of the plate.

[0114] In this embodiment, we use traditional direct numerical simulation (DNS) and three homogenized numerical simulations (HNS) based on the method of the present invention for simulation calculations. The parameters and environment used in the simulation calculations are shown in Table 1. Figure 4 As shown, Figure 4 (a) and (b) respectively represent the plane and cross-section diagrams of the homogenized shell.

[0115]

[0116] Combined with Figure 5 , which shows the comparison of simulation results by DNS and HNS methods under ANSYS platform. The comparison of the center deflection of the shell is shown in Figure 5 (a); the comparison of the bending moments of the shell sections A and B under the four-side simply supported condition is shown in Figure 5 (b); the comparison of the bending moments of the shell sections A and B under the four-side clamped condition is shown in Figure 5 (c); Comparison of the bending moments of the shell sections A and B under the four-corner support condition is shown in Figure 5 (d) Figure 5 It can be seen from the comparison that the method of the present invention substitutes the transverse shear stiffness matrix E as a necessary condition into the equivalent homogenized shell and obtains the central deflection of the homogenized shell through simulation calculation. Compared with the central deflection of the direct numerical simulation of the periodic shell, the two maintain a high degree of consistency, and the error does not exceed 5%.

[0117] Combined with Figure 6 , which shows the comparison of simulation results of DNS and HNS methods under ABAQUS platform. The comparison of the central deflection of the shell under the condition of four-side simple support is shown in Figure 6 (a), where the green line is the analytical solution of the classical plate and shell theory (CPT) and the first-order shear deformation theory (FSDT); the comparison of the central deflection of the shell under the four-side clamped condition is shown in Figure 6 (b); the comparison of the central deflection of the shell under the four-corner support condition is shown in Figure 6 (c). Figure 6 From the comparison, it can be seen that the method of the present invention substitutes the transverse shear stiffness matrix E as a necessary condition into the equivalent homogenized shell and obtains the central deflection of the homogenized shell through simulation calculation. Compared with the central deflection of the periodic shell simulated by the traditional classical plate and shell theory (CPT) and the first-order shear deformation theory (FSDT), the two maintain a high degree of consistency and have a very low error.

[0118] At the same time, combined with the comparison of the above simulation environment and calculation time, it can be seen that the method of the present invention can greatly reduce the demand for computing memory and computing time, and obtain accurate homogenized shell displacement and internal force with extremely small computing cost and loss of computing accuracy, thus meeting the needs of engineering design.

[0119] {Example 4}

[0120] In an exemplary embodiment, a computer system is provided, which may be a server.

[0121] As an example, a computer device includes a processor, a memory, an input / output interface (I / O) and a communication interface. The processor, the memory and the input / output interface are connected via a system bus, and the communication interface is connected to the system bus via the input / output interface. The processor is used as a processing core for data and applications and is used to provide computing and control capabilities. The memory, especially the shell, adopts a combination of non-volatile memory and cache. The non-volatile memory, especially the shell, is used to store operating system programs, applications and databases executed by the processor. It should be understood that the cache is usually configured to be integrated with the processor.

[0122] It should be understood that the above example structures are merely typical examples of partial structures of computer devices, and do not constitute a limitation on the computer system to which the scheme of the present disclosure is applied. In one or more embodiments, the computer system may include more or fewer components, or combine certain components, or have different component arrangements.

[0123] In one embodiment, the present disclosure further provides a computer system, including one or more processors and a memory. The memory can be used to store operable instructions, which, when executed by one or more processors, cause the one or more processors to perform operations, including the process of executing the transverse shear stiffness estimation method or the homogenization simulation method of the periodic shell of the aforementioned embodiment.

[0124] {Example 5}

[0125] In one embodiment, the present disclosure also provides a computer-readable medium storing software, wherein the software includes instructions that can be executed by one or more computers, and when the instructions are executed by the one or more computers, the process of the transverse shear stiffness estimation method or the homogenization simulation method of the periodic shell of the aforementioned embodiment is performed.

[0126] {Example 6}

[0127] In one embodiment, the present disclosure further provides a computer program product, including a computer program, which, when executed by a processor, implements the process of the transverse shear stiffness estimation method or the homogenization simulation method of the periodic shell of the aforementioned embodiment.

[0128] Although the present invention has been disclosed as above with preferred embodiments, it is not intended to limit the present invention. A person with ordinary knowledge in the technical field to which the present invention belongs may make various changes and modifications without departing from the spirit and scope of the present invention. Therefore, the protection scope of the present invention shall be determined by the definition of the claims.

Claims

1. A method for estimating transverse shear stiffness, applicable to periodic shells, characterized in that: The method comprises the following steps: Step 1: Establish a coarse-scale coordinate system for characterizing the mathematical model of periodic shells ( α 1, α 2, γ ) and fine-scale coordinate system ( y 1, y 2, z ), the coarse-scale coordinate system ( α 1, α 2, γ ) is an orthogonal curvilinear coordinate system, the coarse-scale coordinate system ( α 1, α 2, γ ) and the fine-scale coordinate system ( y 1, y 2, z ) based on the coarse-scale coordinate system ( α 1, α 2, γ )The thickness of the lower shell and the fine-scale coordinate system ( y 1, y 2, z ) The relationship is established between the lateral dimensions of the unit cell of the lower shell; Step 2: In the coarse-scale coordinate system ( α 1, α 2, γ ) and the fine-scale coordinate system ( y 1, y 2, z ) and the relative displacement function of the shell is constructed by using the perturbation method; Step 3: Based on the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, α 1, α 2, γ The relative displacement in the direction is used to obtain the strain function based on the relative displacement of the shell, and the strain tensor corresponding to the perturbation coefficient is determined; Step 4: According to the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, a stress function based on the relative displacement of the shell is obtained; Step 5, inputting the strain tensor corresponding to the perturbation coefficient in step 3 into the generalized Hooke's law of three-dimensional elasticity to obtain the stress tensor corresponding to the perturbation coefficient; Step 6: According to the stress function based on the relative displacement of the shell in step 4, the equilibrium equation of the shell is input to obtain the equilibrium equation corresponding to the perturbation expansion coefficient; Step 7: Based on the stress tensor corresponding to the perturbation coefficient in step 5, the equilibrium equation corresponding to the perturbation expansion coefficient in step 6 is input to obtain the equilibrium equation considering the transverse shear deformation expressed in relative displacement; Step 8, input the displacement form of the shell into the equilibrium equation considering the transverse shear deformation in step 7 to obtain the control equation in the domain Ω; Step 9: Based on the control equation in the domain Ω, determine the control equation in the domain Θ; Step 10: Convert the control equations in the domain Θ into finite element form and solve them, and use the classical homogenization process to obtain the coefficients of the transverse shear stiffness matrix E of the periodic shell that satisfies the displacement form.

2. The method for estimating transverse shear stiffness according to claim 1, characterized in that: In the step 1, the coarse-scale coordinate system ( α 1, α 2, γ )The thickness of the lower shell and the fine-scale coordinate system ( y 1, y 2, z ) The relationship between the lateral dimensions of the unit cell of the lower shell is established as follows: ； in, h 1, h 2 respectively represent the unit cells of the shell α 1 and α 2. Transverse dimension in the direction; δ represents the perturbation coefficient and represents the thickness of the shell in the coarse-scale coordinate system.

3. The method for estimating the transverse shear stiffness according to claim 1, characterized in that: In step 3, based on the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, α 1, α 2, γ The relative displacement in the direction is used to obtain the strain function based on the relative displacement of the shell and determine the strain tensor corresponding to the perturbation coefficient, including: Substituting the relative displacement function of the shell into the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, the strain function based on the relative displacement of the shell is obtained, which is expressed as follows: ； in, represents the strain tensor; Representation and perturbation expansion coefficient δ l The corresponding strain tensor, l ∈{0, 1,2, …}; And, according to the strain function based on the relative displacement of the shell, we obtain δ 2 The corresponding strain tensor , expressed as follows: ； in, u 1, u 2, u 3 respectively represent α 1, α 2 and γ The relative displacement in the direction, Representation and perturbation expansion coefficient δ l The corresponding displacement, i ∈{1, 2, 3}; l ∈{0, 1, 2, 3, …}; , Orthogonal curvilinear coordinate system ( α 1, α 2, γ )The coefficients of the first quadratic form.

4. The method for estimating transverse shear stiffness according to claim 1, characterized in that: In step 5, the stress tensor corresponding to the perturbation coefficient is obtained and expressed as: ； In the formula, Representation and perturbation expansion coefficient δ l The corresponding stress tensor, l ∈{0, 1, 2, …}; represents the fourth-order material stiffness tensor in the fine-scale coordinate system, y = [ y 1, y 2]; Representation and and The associated strain tensor; Representation and The associated strain tensor; in, δ 2 The corresponding stress tensor , expressed as follows: ； ； In the formula, , They represent the principal curvature functions of the surface, and the independent variables are α 1, α 2.

5. The method for estimating transverse shear stiffness according to claim 4, characterized in that: In step 7, the equilibrium equation considering the lateral shear deformation expressed in terms of relative displacement is expressed as: ； In the formula, , , represent differential operators respectively.

6. The method for estimating transverse shear stiffness according to claim 5, characterized in that: In step 8, the displacement form of the shell is expressed as: ; In the formula, , They represent the assumed functions of displacement field respectively; The displacement form of the shell is input into the equilibrium equation considering the transverse shear deformation in step 7 to obtain the control equation in the domain Ω, which is expressed as follows: ; In the formula, represents the upper and lower boundary normal vectors of the unit cell in the domain Ω of j Directional component; represents the differential operator, An assumed function representing the stress tensor, and is its component expression; , Ω refers to the coordinate system ( ξ 1, ξ 2, z ), ∈{1,2}, i ∈{1, 2, 3}, j ∈{1, 2, 3}.

7. The method for estimating transverse shear stiffness according to claim 6, characterized in that: In step 9, based on the control equation in the domain Ω, the control equation in the domain Θ is determined, and the control equation in the domain Θ is expressed as follows: ; In the formula, , ; Θ refers to the coordinate system ( η 1, η 2, η 3) The domain below; η = [ η 1, η 2, η 3]; d = [ d 1, d 2, 0], d 1 and d 2 respectively represent the domain Θ in η 1 and η Length in 2 directions; denotes the vertices of the domain Θ.

8. The method for estimating transverse shear stiffness according to claim 7, characterized in that: In step 10, the control equation in the domain Θ is converted into a finite element form and expressed as follows: ; In the formula, , They refer to the strain matrices of discrete elements in domain Θ respectively; represents the global stiffness matrix of domain Θ; represents the equivalent global load vector of domain Θ; In the domain Θ Numerical solution of ; The coefficients of the transverse shear stiffness matrix E of the periodic shell satisfying the displacement form are obtained by using the classical homogenization process and are expressed as follows: ； in, represents the classical homogenization process.

9. A homogenization simulation method considering transverse shear stiffness, applicable to periodic shells, wherein the periodic shells are periodic shells of equal thickness and principal curvatures equal to constant assumptions, characterized in that: The homogenization simulation method comprises the following steps: Establish a finite element model of a unit cell of a periodic shell and input the material parameters of the unit cell, apply a periodic boundary to the unit cell, and apply an initial stress load to the unit cell; Solve the finite element results of the unit cell according to the method described in any one of claims 1 to 8 to obtain the coefficients of the transverse shear stiffness matrix E of the periodic shell; A homogenized finite element model of the periodic shell is established, and the transverse shear stiffness matrix E is input. After applying boundary conditions and loads, the central deflection of the periodic shell is calculated.

10. A computer system, characterized in that: include: one or more processors; A memory storing operable instructions, wherein when the instructions are executed by the one or more processors, the one or more processors are caused to perform operations, wherein the operations include the process of executing the method described in any one of claims 1 to 9.

11. A computer-readable medium storing software, characterized in that: The software includes instructions that can be executed by one or more computers, and when the instructions are executed by the one or more computers, the process of the method according to any one of claims 1 to 9 is performed.

12. A computer program product, comprising a computer program, characterized in that When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 9 are implemented.

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