Plastic yield surface evaluation method for three-dimensional gradient metamaterials

Abstract

The present invention discloses a method for evaluating the plastic yield surface of a three-dimensional gradient metamaterial, and relates to the technical field of research on the plastic yield performance of lightweight structures, and includes the following steps: making two three-dimensional gradient metamaterial structures; obtaining the macroscopic elastic stiffness tensor of the three-dimensional gradient metamaterial structure; obtaining the effective elastic modulus of the three-dimensional gradient metamaterial structure; obtaining the uniaxial yield strength of the three-dimensional gradient metamaterial; obtaining the yield condition of the three-dimensional gradient metamaterial; obtaining a plastic yield function model; and verifying the plastic yield function. The present invention adopts the above-mentioned method for evaluating the plastic yield surface of the three-dimensional gradient metamaterial, gives the macroscopic elastic stiffness tensor of the three-dimensional gradient metamaterial, and obtains the effective elastic modulus and uniaxial yield strength of the three-dimensional gradient metamaterial; obtains the plastic yield function model of the three-dimensional gradient metamaterial based on the proposed yield condition, and verifies it through the yield surface predicted by finite element simulation, which can accurately predict the yield behavior of the three-dimensional gradient metamaterial.

Description

Technical Field

[0001] The invention relates to the technical field of plastic yield performance research of lightweight structures, and in particular to a plastic yield surface evaluation method of a three-dimensional gradient metamaterial. Background Art

[0002] Lightweight structures, due to their large internal pores, can provide sufficient space for large structural deformations, and thus can present high load-bearing efficiency. Porous structures such as honeycombs, foams, and foam-filled tubes have been widely used in the field of energy absorption and shock absorption, thus replacing energy absorption structures and buffer devices made of traditional solid structures. The lattice structure coupled with multiple directional gradient topologies is a new type of metamaterial that has better mechanical properties, damage behavior, and energy absorption than traditional uniform metamaterial structures. More importantly, the gradient metamaterial structure has excellent plastic yield performance and can simultaneously meet the needs of different parts of the structure under complex working conditions. At present, a large number of studies have reported the plastic yield performance of uniform metamaterial structures or single-cell metamaterial structures; and for gradient metamaterial structures, only their energy absorption capacity has been studied, but the plastic yield performance of gradient metamaterial structures has not been studied. Therefore, this patent proposes a plastic yield surface evaluation method for three-dimensional gradient metamaterials to characterize and describe the plastic yield behavior of three-dimensional gradient metamaterials, providing a certain reference for engineering applications. Summary of the invention

[0003] The purpose of the present invention is to provide a method for evaluating the plastic yield surface of a three-dimensional gradient metamaterial, to solve the problems raised in the above-mentioned background technology, and to predict and reveal the plastic yield behavior of the three-dimensional gradient metamaterial.

[0004] To achieve the above object, the present invention provides a method for evaluating the plastic yield surface of a three-dimensional gradient metamaterial, comprising the following steps:

[0005] Step S1, obtaining two three-dimensional gradient metamaterial structures TD-GL-I and TD-GL-II;

[0006] Step S2, obtaining the macroscopic elastic stiffness tensor of the three-dimensional gradient metamaterial structure;

[0007] Step S3, obtaining the effective elastic modulus of the three-dimensional gradient metamaterial structure;

[0008] Step S4, obtaining the uniaxial yield strength of the three-dimensional gradient metamaterial;

[0009] Step S5, obtaining the yield condition of the three-dimensional gradient metamaterial;

[0010] Step S6, obtaining a plastic yield function model;

[0011] Step S7: verify the plastic yield function.

[0012] Preferably, the specific steps of step S2 are as follows:

[0013] Step S21, calculating the local deformation of the pillar by establishing an assumption on the three-dimensional gradient metamaterial structure;

[0014] Step S22, calculating the local internal force of the pillar by establishing an assumption on the three-dimensional gradient metamaterial structure;

[0015] Step S23, obtaining a local stress tensor through local deformation and local internal force;

[0016] Step S24, obtaining a macroscopic stress tensor through the local stress tensor;

[0017] Step S25, obtain the macro elastic stiffness tensor, the expression is as follows:

[0018]

[0019] Among them, f η represents the volume fraction of support η, E m represents the Young's modulus of the substrate, is the unit axial vector of pillar η.

[0020] Preferably, the expression of the effective elastic modulus in step S3 is as follows:

[0021]

[0022] Where ρ represents the relative density of the three-dimensional gradient metamaterial structure, E m The Young's modulus G representing the base material represents the effective shear modulus.

[0023] Preferably, the expression of the uniaxial yield strength in step S4 is as follows:

[0024]

[0025] Where ρ represents the relative density of the three-dimensional gradient metamaterial structure, σ y Indicates the yield strength of the base material.

[0026] Preferably, the expression of the yield condition in step S5 is as follows:

[0027]

[0028] Among them, d1-d7 are structural parameters representing the degree of anisotropy of TD-GLs, σ xx represents the normal stress in the x direction, σ yy represents the normal stress in the y direction, σ zzrepresents the normal stress in the z direction, τ xy represents the shear stress in the xy plane, τ xz represents the shear stress in the xz plane, τ yz represents the shear stress in the yz plane, σ h Represents hydrostatic pressure.

[0029] Preferably, the plastic yield function model in step S6 includes:

[0030] The plastic yield function expression of TD-GL-Ⅰ is as follows:

[0031]

[0032] Among them, σ xx represents the normal stress in the x direction, σ yy represents the normal stress in the y direction, σ zz represents the normal stress in the z direction, τ xy represents the shear stress in the xy plane, τ xz represents the shear stress in the xz plane, τ yz represents the shear stress in the yz plane, σ h represents the hydrostatic pressure, σ s represents the uniaxial yield strength;

[0033] The plastic yield function expression of TD-GL-Ⅱ is as follows:

[0034]

[0035] Among them, σ xx represents the normal stress in the x direction, σ yy represents the normal stress in the y direction, σ zz represents the normal stress in the z direction, τ xy represents the shear stress in the xy plane, τ xz represents the shear stress in the xz plane, τ yz represents the shear stress in the yz plane, σ h represents the hydrostatic pressure, σ s Represents the uniaxial yield strength.

[0036] Therefore, the present invention adopts the plastic yield surface evaluation method of the above-mentioned three-dimensional gradient metamaterial. First, the macroscopic elastic stiffness tensor of the three-dimensional gradient metamaterial is given, and the effective elastic modulus and uniaxial yield strength of the three-dimensional gradient metamaterial are obtained; finally, based on the proposed yield condition, the plastic yield function model of the three-dimensional gradient metamaterial is obtained, and verified by the yield surface predicted by finite element simulation, which can accurately predict the yield behavior of the three-dimensional gradient metamaterial.

[0037] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 It is a flow chart of an embodiment of a method for evaluating the plastic yield surface of a three-dimensional gradient metamaterial of the present invention;

[0039] Figure 2 A schematic diagram of the structure of a three-dimensional gradient metamaterial structure TD-GL-I for the plastic yield surface evaluation method of the three-dimensional gradient metamaterial of the present invention;

[0040] Figure 3 A schematic diagram of the structure of a three-dimensional gradient metamaterial structure TD-GL-Ⅱ for the plastic yield surface evaluation method of the three-dimensional gradient metamaterial of the present invention;

[0041] Figure 4 It is the normalized yield surface under biaxial loading conditions of the plastic yield surface evaluation method of the three-dimensional gradient metamaterial of the present invention, wherein (a) is the normalized yield surface of TD-GL-I under biaxial loading conditions, and (b) is the normalized yield surface of TD-GL-II under biaxial loading conditions;

[0042] Figure 5 It is the normalized yield surface under double shear loading condition of the plastic yield surface evaluation method of the three-dimensional gradient metamaterial of the present invention, wherein (a) is the normalized yield surface of TD-GL-I under double shear loading condition, and (b) is the normalized yield surface of TD-GL-II under double shear loading condition;

[0043] Figure 6 It is the normalized yield surface under the combined uniaxial-shear loading condition of the plastic yield surface evaluation method of the three-dimensional gradient metamaterial of the present invention, wherein (a) is the normalized yield surface of TD-GL-Ⅰ under the combined uniaxial-shear loading condition, and (b) is the normalized yield surface of TD-GL-Ⅱ under the combined uniaxial-shear loading condition;

[0044] Figure 7 It is the normalized yield surface under the combined hydrostatic-shear loading condition of the plastic yield surface evaluation method of the three-dimensional gradient metamaterial of the present invention, wherein (a) is the normalized yield surface of TD-GL-Ⅰ under the combined hydrostatic-shear loading condition, and (b) is the normalized yield surface of TD-GL-Ⅱ under the combined hydrostatic-shear loading condition. DETAILED DESCRIPTION

[0045] The technical solution of the present invention is further described below through the accompanying drawings and embodiments.

[0046] Unless otherwise defined, the technical terms or scientific terms used in the present invention should be understood by people with ordinary skills in the field to which the present invention belongs. The words "first", "second" and similar words used in the present invention do not indicate any order, quantity or importance, but are only used to distinguish different components. "Include" or "comprise" and similar words mean that the elements or objects appearing before the word include the elements or objects listed after the word and their equivalents, without excluding other elements or objects. "Connect" or "connected" and similar words are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect. "Up", "down", "left", "right" and the like are only used to indicate relative positional relationships. When the absolute position of the described object changes, the relative positional relationship may also change accordingly.

[0047] Example

[0048] See also Figure 1-7 The present invention provides a method for evaluating the plastic yield surface of a three-dimensional gradient metamaterial, comprising the following steps:

[0049] Step S1: First, two new three-dimensional gradient metamaterial structures TD-GL-I and TD-GL-II are obtained, such as Figure 2 and Figure 3 shown.

[0050] Step S2: Obtaining the macroscopic elastic stiffness tensor of the three-dimensional gradient metamaterial structure. Including:

[0051] Step S21, calculating the local deformation of the pillar by establishing an assumption on the three-dimensional gradient metamaterial structure;

[0052] Step S22, calculating the local internal force of the pillar by establishing an assumption on the three-dimensional gradient metamaterial structure;

[0053] Step S23, obtaining a local stress tensor through local deformation and local internal force;

[0054] Step S24, obtaining a macroscopic stress tensor through the local stress tensor;

[0055] Step S25, obtaining the macroscopic elastic stiffness tensor.

[0056] The specific steps are as follows:

[0057] The octet lattice structure is a lightweight structure that is mainly tensile deformed. According to the deformation characteristics of the octet lattice, under small deformation conditions, the three-dimensional gradient metamaterial structure should meet the following assumptions:

[0058] Each component support is a straight rod and remains a straight rod within the range of linear elasticity.

[0059] Each component pillar is affinely deformed within the linear elastic range, that is:

[0060] x=F·X (1)

[0061] The infinitesimal line segments of the initial configuration and the current configuration are represented by X and x respectively, and the deformation gradient tensor

[0062] Within the linear elastic range, the internal force of each support rod is distributed along its axial direction.

[0063] The response of each support substrate under small deformation is isotropic linear elastic, as follows

[0064]

[0065] Among them, E m and v m are Young’s modulus and Poisson’s ratio of the substrate, respectively. m is the fourth-order stiffness tensor of the substrate, J vol Represents the volume 4th-order unit tensor, J dev Represents a partial rank-4 unit tensor.

[0066] From the above assumptions, it can be seen that the macroscopic strain tensor E can be obtained by the strain localization tensor Mapped to the local deformation of pillar η, the local deformation λ of pillar η η for

[0067]

[0068] where n η is the unit axial vector of the pillar η. The local internal force γ of the pillar η η for

[0069]

[0070] Therefore, the local stress tensor σ can be expressed as:

[0071]

[0072] The local stress tensor σ of all the struts in the three-dimensional gradient metamaterial is averaged, and the expression of the macroscopic stress tensor Σ is obtained as follows:

[0073]

[0074] Among them, V Ф is the total volume of the space occupied by the three-dimensional gradient metamaterial, and Ф is the collection of all pillars. η represents the volume fraction of the support η. The simplest and clearest relationship between the macroscopic strain tensor E and the macroscopic stress tensor Σ is as follows:

[0075] Σ=L:E (7)

[0076] Among them, the fourth-order tensor L represents the macroscopic elastic stiffness of the three-dimensional gradient metamaterial. Combining formulas (6) and (7), the macroscopic strain tensor E and the strain localization tensor The symmetry of can be used to write the expression of the macroscopic elastic stiffness tensor L:

[0077]

[0078] Among them, f η represents the volume fraction of support η, E m represents the Young's modulus of the substrate, is the unit axial vector of pillar η.

[0079] Step S3: Obtaining the effective elastic modulus of the three-dimensional gradient metamaterial structure.

[0080] The unit vector n of any pillar in the three-dimensional gradient metamaterial structure η It has two inherent properties, namely

[0081]

[0082] From formulas (8)-(11), two useful relations of the macroscopic elastic stiffness tensor L can be obtained, namely:

[0083]

[0084] Among them, the relative density of the three-dimensional gradient metamaterial structure is represented by ρ,

[0085]

[0086] Among them, V and V s They represent the volume occupied by structural materials and the volume of space occupied by the structure respectively.

[0087] In order to facilitate the subsequent derivation of the effective elastic modulus, vector and matrix symbols are used to describe formula (7).

[0088]

[0089] The three-axis symmetric gradient topology strategy is applied to the structural design of the three-dimensional gradient metamaterial, so that the elastic stiffness matrix of the three-dimensional gradient metamaterial satisfies cubic symmetry. Therefore, the elastic stiffness matrix of the three-dimensional gradient metamaterial can be expressed as

[0090]

[0091] Among them, C1, C2 and C3 are three independent components, which fully represent the cubic symmetry of the three-dimensional gradient metamaterial. C1, C2 and C3 can be derived from the effective shear modulus G, the effective elastic modulus E and the effective Poisson's ratio v.

[0092]

[0093] Combining equations (12), (13), (19) and (20), the equation containing the effective elastic modulus E and the effective Poisson's ratio v can be expressed as:

[0094]

[0095] Therefore, the effective elastic modulus E of the three-dimensional gradient metamaterial is

[0096]

[0097] Where ρ represents the relative density of the three-dimensional gradient metamaterial structure, E m The Young's modulus G representing the base material represents the effective shear modulus.

[0098] It should be noted that in formula (20), the effective elastic modulus E is only related to C1 and C2, and C1, C2 and C3 are independent of each other. Therefore, the effective shear modulus G can be measured by experiment or finite element method to obtain the theoretical solution of the effective elastic modulus E of the three-dimensional gradient metamaterial.

[0099] Step S4: Obtaining the uniaxial yield strength of the three-dimensional gradient metamaterial.

[0100] Yield strength is the boundary between recoverable elastic deformation and permanent plastic deformation, and is a resistance of a structure or material to small plastic deformation. Therefore, the uniaxial yield strength of the three-dimensional gradient metamaterial is derived as a standard for measuring the plastic deformation or complete elastic deformation of the three-dimensional gradient metamaterial. The local internal force of any component pillar η is given by formula (4), which can be rewritten as the following formula using formula (7):

[0101]

[0102] Among them, L -1 represents the inverse tensor of the macroscopic elastic stiffness tensor L. -1 It can be broken down into:

[0103]

[0104] Where J res represents the cofourth-order unit tensor and is defined as follows:

[0105]

[0106] Substituting formula (25)-(26) into formula (24), we can obtain:

[0107]

[0108] where tri(Σ) and dev(Σ) represent the orbital and deviator parts of the tensor Σ. I is a second-order unit tensor. Consider the yield strength σ of the substrate y , the closed yield surface f of the three-dimensional gradient metamaterial is defined as:

[0109]

[0110] For static pressure loading P = tri (Σ) / 3 = σ s When , the uniaxial yield strength of the three-dimensional gradient metamaterial can be obtained:

[0111]

[0112] Where ρ represents the relative density of the three-dimensional gradient metamaterial structure, σ y Indicates the yield strength of the base material.

[0113] Step S5: Obtaining the yield condition of the three-dimensional gradient metamaterial.

[0114] The Mises yield criterion generalized by Hill will be used to study the yield surface of the three-dimensional gradient metamaterial. Hill's generalization of the Mises yield condition has the following form:

[0115]

[0116] Among them, d1-d6 are material parameters that characterize the degree of anisotropy of the material. It is worth noting that equation (30) is proposed for orthogonal anisotropic materials, and the basis vectors of the stress tensor coincide with the orthogonal principal axes of the material. Obviously, the three-dimensional gradient metamaterial has orthogonal principal axes, and its coordinate axes (x, y and z) coincide with these principal axes. Therefore, it is reasonable to use the Hill generalized form of the Mises yield condition to describe the yield behavior of the three-dimensional gradient metamaterial. In addition, formula (30) does not consider the effect of hydrostatic pressure on plastic yield. If the influence of hydrostatic pressure is not considered, there will always be a stress state in which the stress component on one of the principal axes exceeds the uniaxial yield strength. This situation is extremely unreasonable for the three-dimensional gradient metamaterial. Therefore, the introduction of hydrostatic pressure (σ h =σ ii / 3) correction term is used to characterize the effect of hydrostatic pressure on structural plastic yield, and its form is:

[0117]

[0118] The yield condition of the three-dimensional gradient metamaterial can finally be expressed as

[0119]

[0120] Among them, d1-d7 are structural parameters representing the degree of anisotropy of TD-GLs, σ xx represents the normal stress in the x direction, σ yy represents the normal stress in the y direction, σ zz represents the normal stress in the z direction, τ xy represents the shear stress in the xy plane, τ xz represents the shear stress in the xz plane, τ yz represents the shear stress in the yz plane, σ h Represents hydrostatic pressure.

[0121] Step S6: Obtaining a plastic yield function model.

[0122] The plastic yield function of the three-dimensional gradient metamaterial is derived from the yield condition shown in formula (32), and the structural parameters d1-d7 are determined by the yield strength under various loading conditions. The yield strength is shown in Table 1. The closed function of the yield surface is normalized by the uniaxial yield strength and is expressed as:

[0123]

[0124] Finally, the plastic yield functions of TD-GL-Ⅰ and TD-GL-Ⅱ are as follows:

[0125] Plastic yield function of TD-GL-Ⅰ

[0126]

[0127] Among them, σ xx represents the normal stress in the x direction, σ yy represents the normal stress in the y direction, σ zz represents the normal stress in the z direction, τ xy represents the shear stress in the xy plane, τ xz represents the shear stress in the xz plane, τ yz represents the shear stress in the yz plane, σ h represents the hydrostatic pressure, σ s Represents the uniaxial yield strength.

[0128] Plastic yield function of TD-GL-Ⅱ

[0129]

[0130] Among them, σ xx represents the normal stress in the x direction, σ yy represents the normal stress in the y direction, σ zz represents the normal stress in the z direction, τxy represents the shear stress in the xy plane, τ xz represents the shear stress in the xz plane, τ yz represents the shear stress in the yz plane, σ h represents the hydrostatic pressure, σ s Represents the uniaxial yield strength.

[0131] Table 1 Normalized effective shear modulus G / E of three-dimensional gradient metamaterials m , shear yield strength τ s / σ y , uniaxial yield strength σ s / σ y and hydrostatic yield strength σ hs / σ y Functional expression of relative density

[0132]

[0133] Note: Variable y represents G / E m , τ s / σ y , σ s / σ y or hs / σ y The variable x represents the relative density ρ, and the parameter A represents the coefficient. The yield strength of the substrate is represented by σ y express.

[0134] Step S7: Verify the plastic yield function.

[0135] Four loading conditions, namely biaxial loading, bi-shear loading, combined hydrostatic-shear loading and combined uniaxial-shear loading, were considered, the yield surfaces of TD-GL-Ⅰ and TD-GL-Ⅱ lattices were simulated, and the proposed plastic yield function was verified using these simulation results. The yield responses of TD-GL-Ⅰ and TD-GL-Ⅱ predicted by finite element simulation under these four loading conditions and the yield behavior predicted by the plastic yield function, such as Figure 3-6 As shown. Although there is a certain error between the plastic yield response predicted by the finite element simulation and the closed plastic yield surface defined by formulas (34)-(35), the shape of the plastic yield surface predicted by the finite element method is consistent with formulas (34)-(35). Therefore, the established plastic yield function can well describe the plastic yield behavior of TD-GL-Ⅰ and TD-GL-Ⅱ under various loading conditions.

[0136] Therefore, the present invention adopts the plastic yield surface evaluation method of the above-mentioned three-dimensional gradient metamaterial. First, the macroscopic elastic stiffness tensor of the three-dimensional gradient metamaterial is given, and the effective elastic modulus and uniaxial yield strength of the three-dimensional gradient metamaterial are obtained; finally, based on the proposed yield condition, the plastic yield function model of the three-dimensional gradient metamaterial is obtained, and verified by the yield surface predicted by finite element simulation, which can accurately predict the yield behavior of the three-dimensional gradient metamaterial.

[0137] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. A method for evaluating the plastic yield surface of a three-dimensional gradient metamaterial, characterized in that: The following steps are involved: Step S1, obtaining two three-dimensional gradient metamaterial structures TD-GL-I and TD-GL-II; Step S2, obtaining the macroscopic elastic stiffness tensor of the three-dimensional gradient metamaterial structure; The specific steps of step S2 are as follows: Step S21, calculating the local deformation of the pillar by establishing an assumption on the three-dimensional gradient metamaterial structure; Step S22, calculating the local internal force of the pillar by establishing an assumption on the three-dimensional gradient metamaterial structure; Step S23, obtaining a local stress tensor through local deformation and local internal force; Step S24, obtaining a macroscopic stress tensor through the local stress tensor; Step S25, obtain the macro elastic stiffness tensor, the expression is as follows: Among them, f η represents the volume fraction of support η, E m represents the Young's modulus of the substrate, is the unit axial vector of the strut η; Step S3, obtaining the effective elastic modulus of the three-dimensional gradient metamaterial structure; The expression of the effective elastic modulus in step S3 is as follows: Where ρ represents the relative density of the three-dimensional gradient metamaterial structure, E m represents the Young's modulus of the substrate, G represents the effective shear modulus; Step S4, obtaining the uniaxial yield strength of the three-dimensional gradient metamaterial; The expression of the uniaxial yield strength in step S4 is as follows: Where ρ represents the relative density of the three-dimensional gradient metamaterial structure, σ y Indicates the yield strength of the substrate; Step S5, obtaining the yield condition of the three-dimensional gradient metamaterial; The expression of the yield condition in step S5 is as follows: Among them, d1-d7 are structural parameters representing the degree of anisotropy of TD-GLs, σ xx represents the normal stress in the x-direction, σ yy represents the normal stress in the y direction, σ zz represents the normal stress in the z direction, τ xy represents the shear stress in the xy plane, τ xz represents the shear stress in the xz plane, τ yz represents the shear stress in the yz plane, σ h represents hydrostatic pressure; Step S6, obtaining a plastic yield function model; The plastic yield function model in step S6 includes: The plastic yield function expression of TD-GL-Ⅰ is as follows: The plastic yield function expression of TD-GL-Ⅱ is as follows: Among them, σ xx represents the normal stress in the x-direction, σ yy represents the normal stress in the y direction, σ zz represents the normal stress in the z direction, τ xy represents the shear stress in the xy plane, τ xz represents the shear stress in the xz plane, τ yz represents the shear stress in the yz plane, σ h represents the hydrostatic pressure, σ s represents the uniaxial yield strength; Step S7: verify the plastic yield function.

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