Three-dimensional micropolar elastomer unit construction method based on reduced integral

Abstract

The invention relates to a three-dimensional micro-elastomer unit construction method based on reduced integral. The method comprises the following steps: S1, constructing a kinetic equation (formula 1) for a three-dimensional micro-elastomer; s2, building a constitutive equation (formula 2) of the three-dimensional microelectrode body; s3, based on the virtual work principle, establishing a weak form (formula 3) of the differential equation; s4, the dynamic component interpolation function is inserted into the formula 1, so that an interpolation micropole strain vector and an interpolation micropole curvature vector are obtained; s5, substituting (formula 2), the interpolation micropole strain vector and the interpolation micropole curvature vector into (formula 3) to obtain (formula 4) and (formula 5); and S6, solving (formula 4) and (formula 5) by adopting a single-point reduction integral method, introducing a characteristic hourglass shape vector in the solving process, and constructing to obtain a finite element model of the three-dimensional micro-extreme elastomer. According to the method, the finite element form can be simplified, so that the calculation time is shortened, the overall model efficiency is improved, and the shear locking problem in bending deformation can be effectively avoided.

Description

Technical Field

[0001] The present invention relates to the technical field of finite element simulation analysis, and in particular to a method for constructing a three-dimensional micropolar elastic body element based on reduced integration. Background Art

[0002] As an extension of classical elasticity theory, micropolar elasticity theory provides a theoretical basis for the mechanical modeling of microstructure-sensitive materials such as fiber-reinforced composites, biological bones, and artificial metamaterials by introducing micro-rotation degrees of freedom and coupled stress tensors. At the engineering application level, this theory has promoted the improvement of the prediction accuracy of biomechanics and has become a key basis for the strength design of composite materials.

[0003] When integrating micropolar theory into finite element analysis, traditional three-dimensional FEM (finite element) models are based on the assumption of homogeneous continuous media and only support translational displacement fields, unable to characterize the microstructural rotation-deformation coupling effect. For typical micropolar systems such as granular media and biomimetic materials, traditional FEM elements cannot capture these characteristics. Without corresponding modifications, FEM cannot consider the interactions between microstructures, resulting in inaccurate predictions. To solve this problem, finite elements specifically for micropolar materials have been developed, which integrate micro-rotation and coupled stress tensors, enabling more accurate simulation of micropolar materials under various loading conditions. The emergence of these new elements has significantly improved the prediction ability of FEM and narrowed the gap between micropolar elasticity theory and practical engineering applications, especially in the fields of nanotechnology, biomechanics, and advanced materials design.

[0004] In the prior art, a full integration scheme is usually adopted to construct micropolar finite elements. However, the full integration scheme has a complex form and a long calculation time, reducing the efficiency of the overall model; at the same time, the finite element model constructed based on the full integration scheme has a shear locking problem in bending deformation; therefore, there is an urgent need to provide a more efficient and accurate method for constructing finite element elements for micropolar elastic body finite element analysis. Summary of the Invention

[0005] Based on this, it is necessary to provide a method for constructing a three-dimensional micropolar elastic body element based on reduced integration to address the problems of complex form, long calculation time, low efficiency of the overall model, and shear locking in bending deformation existing in the construction of micropolar finite elements in the prior art.

[0006] The technical solution adopted by the present invention is as follows:

[0007] A method for constructing a three-dimensional micropolar elastic body element based on reduced integration, comprising the following steps:

[0008] S1. In the three-dimensional micropolar elastic body, each material point respectively includes three displacement degrees of freedom u i and three rotation degrees of freedom φi , construct the kinetic equation (Equation 1) containing displacement degrees of freedom u i and rotational degrees of freedom φ i for each material point;

[0009] S2. Construct the constitutive equation (Equation 2) of the three-dimensional micropolar body based on the stress equilibrium equation and strain equilibrium equation of the three-dimensional micropolar elastic body;

[0010] S3. Based on the principle of virtual work, establish the weak form of the differential equation (Equation 3);

[0011] S4. Insert the interpolation function of the kinetic component into (Equation 1) to obtain the interpolated micropolar strain vector ε and the interpolated micropolar curvature vector κ;

[0012] S5. Substitute (Equation 2), the interpolated micropolar strain vector ε and the interpolated micropolar curvature vector κ into (Equation 3) to obtain the expression of the element stiffness matrix (Equation 4) and the expression of the element mass matrix (Equation 5);

[0013] S6. Solve (Equation 4) and (Equation 5) using the single-point reduced integration method. During the solution process, introduce the characteristic hourglass shape vector to construct the finite element model of the three-dimensional micropolar elastic body;

[0014] The expression of the characteristic hourglass shape vector is:

[0015]

[0016] In the above formula, represents the hourglass base vector of the Lagrangian interpolation function of the hexahedron element;

[0017] represents the volume average of the partial derivatives of the shape function of node I with respect to the three coordinate axes x, y, and z;

[0018] represents the coordinate components of node J on the three coordinate axes x, y, and z;

[0019] represents the hourglass base vector of the Lagrangian interpolation function of the hexahedron element;

[0020] Take the characteristic hourglass shape vector as the displacement hourglass control norm to obtain the displacement mode expression related to the displacement degree of freedom u i ;

[0021] Take the characteristic hourglass shape vector as the rotation hourglass control norm to obtain the rotation mode expression related to the rotational degree of freedom φ i ;

[0022] As a further improvement of the above technical solution:

[0023] In S1, the expression of the i-th kinetic equation is:

[0024]

[0025] In (Equation 1), ε ij represents the strain tensor;

[0026] u j,i represents the partial derivative tensor of the displacement degree of freedom;

[0027] e kij represents the Levi-Civita tensor;

[0028] φ k represents the rotational degree of freedom of the micropolar material point;

[0029] κ ij represents the curvature tensor;

[0030] φ j,i represents the partial derivative tensor of the rotational degree of freedom.

[0031] In S2, the constitutive equation of the three-dimensional micropolar elastic body is expressed by the following formula:

[0032]

[0033] In (Equation 2), σ ij represents the stress;

[0034] A ijkl , B ijkl and C ijkl all represent fourth-order elastic tensors and all have symmetry, that is, A ijkl = A klij , B ijkl = B klij , C ijkl = C klij .

[0035] In S3, the expression of the weak form of the differential equation is:

[0036]

[0037] In (Equation 3), σ ij represents the stress;

[0038] δ is the symbol of the first variation of the physical quantity in the basic theory of finite elements;

[0039] ε ij represents the strain tensor;

[0040] m ij represents the micropolar couple stress tensor;

[0041] κ ij represents the micropolar curvature tensor;

[0042] ρ represents the material density;

[0043] represents the acceleration component of the micropolar material point;

[0044] u i represents the displacement degree of freedom of the micropolar material point;

[0045] represents the angular acceleration component of the micropolar material point;

[0046] φ i represents the rotational degree of freedom of the micropolar material point.

[0047] f Si represents the surface force;

[0048] m Si represents the surface couple;

[0049] A represents the surface area on which the surface force and surface couple act;

[0050] f Vi represents the force of the three-dimensional micropolar elastic body;

[0051] m Vi represents the moment of the three-dimensional micropolar elastic body.

[0052] In S4., the expression of the interpolated micropolar strain vector ε is:

[0053]

[0054] where,

[0055] 0 represents a 9×3 zero matrix;

[0056] The expression of the sub-matrix is:

[0057]

[0058] In the above formula, represents the partial derivative of the shape function of node I in the x direction; represents the partial derivative of the shape function of node I in the y direction; represents the partial derivative of the shape function of node I in the Z direction; N I represents the shape function of node I.

[0059] In S4, the expression of the interpolated micropolar curvature vector κ is:

[0060]

[0061] In (Equation 4),

[0062] 0 represents a 9×3 zero matrix;

[0063] The expression of the sub - matrix is:

[0064]

[0065] In the above formula, represents the partial derivative of the shape function of node I along the x - direction; represents the partial derivative of the shape function of node I along the y - direction; represents the partial derivative of the shape function of node I along the z - direction.

[0066] In S5, the expression of the element stiffness matrix is:

[0067]

[0068] In (Equation 4), represents the transpose of the matrix;

[0069] represents the transpose of the matrix;

[0070] B u is a matrix;

[0071] Q φ is a matrix;

[0072] B represents the material constitutive tensor in (Equation 2);

[0073] represents the transpose of the matrix;

[0074] C represents the material constitutive tensor in (Equation 2);

[0075] B φ is a matrix.

[0076] In S5, the expression of the element mass matrix is:

[0077]

[0078] In (Equation 5), M1 represents the mass matrix corresponding to the displacement degrees of freedom;

[0079] $M_2$ represents the mass matrix corresponding to the rotational degree of freedom;

[0080] $\rho$ represents the density of the micropolar material;

[0081] $N$ T represents the transpose of the shape function matrix of each node of the micropolar finite element;

[0082] $N$ is the shape function matrix of 8 nodes of the micropolar finite element $N = [N$ 1 $N$ 2 ... $N$ 8 .

[0083] In $S_6$, the displacement mode expression includes the hourglass displacement expression and the hourglass force expression, and the hourglass displacement is conjugate to the hourglass force;

[0084] The hourglass displacement expression is:

[0085] $q$ α $ = [q$ α1 $q$ α2 $q$ α3 $ T $ = x$ α $F - X$ α

[0086] In the above formula, represents the deformed hourglass mode;

[0087] represents the undeformed hourglass mode;

[0088] represents the deformation gradient;

[0089] The hourglass force expression is:

[0090] $Q$ α $ = [Q$ α1 $Q$ α2 $Q$ α3 $ T $ = K$ uγ $q$ α

[0091] In the above formula, $K$ uγ represents the deformed hourglass mode;

[0092] $K$ uγ The expression of is

[0093] $q$ α represents the hourglass displacement;

[0094] $G$ represents the shear modulus of the material;

[0095] g u represents the displacement hourglass stiffness factor.

[0096] In S6, the rotation mode expression includes an hourglass rotation expression and an hourglass moment expression;

[0097] The hourglass rotation expression is:

[0098]

[0099] In the above formula, represents the characteristic hourglass shape vector;

[0100] represents the rotation component of node I;

[0101] The hourglass moment expression is:

[0102] R α =[R α1 R α2 R α3 T =K φγ r α

[0103] In the above formula, r α represents the hourglass rotation mode related to the rotational degree of freedom;

[0104] K φγ represents the rotational hourglass stiffness;

[0105] K φγ The expression of

[0106] g φ represents the rotational hourglass stiffness factor;

[0107] G represents the material shear modulus;

[0108] V el represents;

[0109] represents the partial derivative of the shape function of node I in the x direction;

[0110] represents the partial derivative of the shape function of node I in the y direction;

[0111] represents the partial derivative of the shape function of node I in the z direction.

[0112] The beneficial effects of the present invention are as follows:

[0113] ​The structure of the present invention is compact and reasonable, and the operation is convenient. By adopting the kinetic component interpolation function and the single-point reduced integration solution method, the finite element form can be simplified, and the shear locking problem in bending deformation can be effectively avoided, thereby shortening the calculation time and improving the overall model efficiency. At the same time, by introducing the characteristic hourglass shape vector the hourglass instability phenomenon in displacement and rotation can be suppressed, and the modeling accuracy can be improved.

[0114] The present invention is verified through the beam structure bending and free vibration tests and the force and displacement patch test analysis, thereby proving the advantages of the three-dimensional micropolar elastic body finite element constructed based on the construction method provided by the present invention in terms of accuracy and calculation efficiency, which can significantly improve the simulation calculation efficiency of the three-dimensional micropolar elastic body, and provide a practical and efficient solution for the structure material design based on simulation, effectively promoting the efficient design and engineering application of materials with complex microstructures. BRIEF DESCRIPTION OF THE DRAWINGS

[0115] Figure 1 is a flowchart of the present invention.

[0116] Figure 2 is a schematic diagram of the finite element analysis under the axial load of the beam structure in Embodiment 1.

[0117] Figure 3 is a schematic diagram of the finite element model of the displacement patch test in Embodiment 1.

[0118] Figure 4 is a stress contour map of the displacement patch test in Embodiment 1.

[0119] Figure 5 is a schematic diagram of the finite element analysis under the pure bending load of the beam structure in Embodiment 1.

[0120] Figure 6 is a stress and displacement contour map of the cantilever beam under the pure bending load in Embodiment 1. DETAILED DESCRIPTION OF THE INVENTION

[0121] The following will describe the specific embodiments of the present invention with reference to the drawings.

[0122] As Figure 1 shown, a construction method of a three-dimensional micropolar elastic body element based on reduced integration includes the following steps:

[0123] S1. In a three-dimensional micropolar elastic body, each material point respectively includes three displacement degrees of freedom u i and three rotation degrees of freedom φ i , and a displacement degree of freedom u i and a rotation degree of freedom φ iThe kinetic equation (Equation 1);

[0124] Among them, the expression of the i-th kinetic equation is:

[0125]

[0126] In (Equation 1), ε ij represents the strain tensor;

[0127] u j,i represents the partial derivative tensor of the displacement degree of freedom;

[0128] e kij represents the Levi-Civita tensor;

[0129] φ k represents the rotational degree of freedom of the micropolar material point, where the subscript k represents the Einstein summation with e kij ;

[0130] κ ij represents the curvature tensor;

[0131] φ j,i represents the partial derivative tensor of the rotational degree of freedom.

[0132] S2. Construct the constitutive equation (Equation 2) of the three-dimensional micropolar body based on the stress equilibrium equation and strain equilibrium equation of the three-dimensional micropolar elastic body;

[0133] For the three-dimensional micropolar elastic body, the stress equilibrium equation for the stress σ ij and the strain equilibrium equation for the couple stress m ij are:

[0134]

[0135] In the above formula, f Vi represents the force of the three-dimensional micropolar elastic body;

[0136] m Vi represents the moment of the three-dimensional micropolar elastic body;

[0137] ρ represents the material density;

[0138] represents the acceleration component of the micropolar material point;

[0139] e kij represents the Levi-Civita tensor;

[0140] σ kl represents the micropolar stress tensor;

[0141] represents the angular acceleration component of the micropolar material point;

[0142] The above equation adopts the Einstein summation convention for expressions with subscripts. For σ ji,j and m ji,j , the comma therein represents the partial derivative with respect to the corresponding variable;

[0143] Both i and j represent the Latin index range, which ranges from 1 to 3 and represents the x, y, and z coordinate axes in the spatial coordinate system respectively;

[0144] Then the constitutive equation of the three-dimensional micropolar elastic body is expressed by the following equation:

[0145]

[0146] (In Equation (2)), σ ij represents stress;

[0147] A ijkl , B ijkl and C ijkl all represent fourth-order elastic tensors and all have symmetry, that is, A ijkl = A klij , B ijkl = B klij , C ijkl = C klij ;

[0148] S3. In order to form the boundary value problem of the numerical solution, based on the principle of virtual work, the weak form (Equation (3)) of the differential equation is established;

[0149] Based on the principle of virtual work, the following equation can be obtained:

[0150]

[0151] In the above equation, σ ij represents stress;

[0152] δ is the symbol of the first-order variation of physical quantities in the basic theory of finite elements;

[0153] ε ij represents the strain tensor;

[0154] m ij represents the micropolar couple stress tensor;

[0155] κ ij represents the micropolar curvature tensor;

[0156] ρ represents the material density;

[0157] represents the acceleration component of the micropolar material point;

[0158] u i represents the displacement degree of freedom of the micropolar material point;

[0159] Denotes the angular acceleration component of a micropolar material point;

[0160] φ i Denotes the rotational degree of freedom of a micropolar material point

[0161] f Si Denotes the surface force;

[0162] m Si Denotes the surface couple;

[0163] A denotes the surface area on which the surface force and the surface couple act;

[0164] f Vi Denotes the force of a three-dimensional micropolar elastic body;

[0165] m Vi Denotes the moment of a three-dimensional micropolar elastic body;

[0166] Performing a variation on the above equation yields the following expression:

[0167]

[0168] (In Equation (3)), σ ij Denotes the stress;

[0169] δ is the symbol for the first variation of a physical quantity in the basic theory of finite elements;

[0170] ε ij Denotes the strain tensor;

[0171] m ij Denotes the micropolar couple stress tensor;

[0172] κ ij Denotes the micropolar curvature tensor;

[0173] ρ denotes the material density;

[0174] Denotes the acceleration component of a micropolar material point;

[0175] u i Denotes the displacement degree of freedom of a micropolar material point;

[0176] Denotes the angular acceleration component of a micropolar material point;

[0177] φ i Denotes the rotational degree of freedom of a micropolar material point.

[0178] f Si Denotes the surface force;

[0179] mSi Denotes the surface couple;

[0180] A represents the surface area on which the surface force and surface couple are applied;

[0181] f Vi Denotes the force of the three-dimensional micropolar elastic body;

[0182] m Vi Denotes the moment of the three-dimensional micropolar elastic body;

[0183] S4. Insert the dynamic component interpolation function into (Equation 1) to obtain the interpolated micropolar strain vector ε and the interpolated micropolar curvature vector κ;

[0184] Among them, the interpolation function for the motion component can be expressed as: and where I represents the number of nodes of the element; N I Denotes the Lagrangian interpolation function of the nodal displacement and rotation, and its functional form is related to the shape and order of the element;

[0185] In the local coordinate system (g, h, r), the three-dimensional isoparametric shape function N I The expression is:

[0186]

[0187] Among them,

[0188]

[0189] In the above formula, Is a vector representing the number of hourglass basis vectors;

[0190] Insert the three-dimensional isoparametric shape function N I Into (Equation 1), we can get:

[0191] The expression of the interpolated micropolar strain vector ε is:

[0192]

[0193] Among them,

[0194] 0 represents a 9×3 zero matrix;

[0195] The expression of the submatrix is:

[0196]

[0197] In the above formula, Represents the partial derivative of the shape function of node I in the x direction; Represents the partial derivative of the shape function of node I in the y direction; Denotes the partial derivative of the shape function of node I in the Z direction; N I Denotes the shape function of node I;

[0198] The expression of the interpolated micropolar curvature vector κ is:

[0199]

[0200] Among them,

[0201] 0 represents a 9×3 zero matrix;

[0202] The expression of the submatrix is:

[0203]

[0204] In the above formula, Denotes the partial derivative of the shape function of node I in the x direction; Denotes the partial derivative of the shape function of node I in the y direction; Denotes the partial derivative of the shape function of node I in the Z direction.

[0205] S5. Substitute (Equation 2), the interpolated micropolar strain vector ε, and the interpolated micropolar curvature vector κ into (Equation 3) to obtain the expression of the element stiffness matrix (Equation 4) and the expression of the element mass matrix (Equation 5);

[0206] Assume that there is no initial stress and coupling stress, that is, σ = 0, m = 0;

[0207] The expression of the element stiffness matrix is:

[0208]

[0209] (In Equation 4), Denotes The transpose of the matrix;

[0210] Denotes The transpose of the matrix;

[0211] B u Is A matrix;

[0212] Q φ Is A matrix;

[0213] B represents the material constitutive tensor in (Equation 2);

[0214] Denotes The transpose of the matrix;

[0215] C represents the material constitutive tensor in (Equation 2);

[0216] B φ is a matrix;

[0217] The expression of the element mass matrix is:

[0218]

[0219] In (Equation 5), M1 represents the mass matrix corresponding to the displacement degrees of freedom;

[0220] M2 represents the mass matrix corresponding to the rotational degrees of freedom;

[0221] ρ represents the micropolar material density;

[0222] N T represents the transpose of the shape function matrix of each node of the micropolar finite element;

[0223] N is the shape function matrix of 8 nodes of the micropolar finite element N = [N 1 N 2 ...N 8 , where, N I (I = 1, 2, 3, 4, 5, 6, 7, 8) is a submatrix, and its expression is:

[0224]

[0225] S6. The single-point reduced integration method is used to solve (Equation 4) and (Equation 5). During the solution process, the characteristic hourglass shape vector is introduced to construct the finite element model of the three-dimensional micropolar elastic body;

[0226] The characteristic hourglass shape vector has the following expression:

[0227]

[0228] In the above formula, represents the hourglass basis vector of the Lagrangian interpolation function of the hexahedron element, and its specific form is:

[0229]

[0230] represents the volume average of the partial derivatives of the shape function of node I with respect to the x, y, and z coordinate axes, and its specific form is:

[0231]

[0232] represents the coordinate components of node J on the x, y, and z coordinate axes;

[0233] Denote the hourglass base vectors of the Lagrange interpolation function for hexahedral elements, where the superscript J represents the Einstein summation with ;

[0234] Take the characteristic hourglass shape vector as the displacement hourglass control paradigm, so as to obtain the displacement mode expression related to the displacement degree of freedom u i ;

[0235] Take the characteristic hourglass shape vector as the rotation hourglass control paradigm, so as to obtain the rotation mode expression related to the rotation degree of freedom φ i ;

[0236] The single-point reduced integration scheme analyzes the first-order differential rotational solid elements based on the uniform strain formula and the uniform curvature formula, and obtains the average strain and average curvature of the three-dimensional micropolar elastic body within the entire element volume through analysis and calculation, so as to ensure that the first-order reduced integration elements can pass the patch test and maintain the calculation accuracy in distorted elements;

[0237] The single-point reduced integration method uses a scheme with one order lower than the full integration to calculate the internal forces and stiffness of the elements, which can alleviate the element locking problem and reduce the calculation cost under conditions such as incompressibility or Kirchhoff shear constraint. Especially in three-dimensional problems, it can bring significant time savings and usually provide faster convergence for nonlinear problems;

[0238] However, simply using reduced integration may introduce zero energy mode or hourglass mode problems. Introducing the characteristic hourglass shape vector can effectively suppress the hourglass instability phenomenon in displacement and rotation and improve the modeling accuracy;

[0239] The displacement mode expression includes the hourglass displacement expression and the hourglass force expression, and the hourglass displacement is conjugate to the hourglass force;

[0240] The hourglass displacement expression is:

[0241] q α = [q α1 q α2 q α3 T = x α F-X α

[0242] In the above formula, represents the deformed hourglass mode;

[0243] represents the undeformed hourglass mode; ​

[0244] Denote the deformation gradient. In this embodiment, only the linear elastic deformation behavior is considered, and the large geometric deformation effect is ignored. Therefore, F = I;

[0245] The expression of the hourglass force is:

[0246] Q α =[Q α1 Q α2 Q α3 T =K uγ q α

[0247] In the above formula, K uγ represents the hourglass mode of deformation;

[0248] K uγ The expression of is

[0249] q α represents the hourglass displacement;

[0250] G represents the shear modulus of the material;

[0251] g u represents the displacement hourglass stiffness factor.

[0252] The rotation mode expression includes the hourglass rotation expression and the hourglass moment expression;

[0253] The hourglass rotation expression is:

[0254]

[0255] In the above formula, represents the characteristic hourglass shape vector;

[0256] represents the rotation component of node I;

[0257] The hourglass moment expression is:

[0258] R α =[R α1 R α2 R α3 T =K φγ r α

[0259] In the above formula, r α represents the hourglass rotation mode related to the rotational degree of freedom;

[0260] K φγ represents the rotational hourglass stiffness; ​​

[0261] K φγ is expressed as

[0262] g φ represents the rotational hourglass stiffness factor;

[0263] G represents the shear modulus of the material;

[0264] V el represents;

[0265] represents the partial derivative of the shape function of node I in the x - direction;

[0266] represents the partial derivative of the shape function of node I in the y - direction;

[0267] represents the partial derivative of the shape function of node I in the z - direction.

[0268] The following is the part of specific embodiments.

[0269] Embodiment 1

[0270] In this embodiment, by setting force patch tests, displacement patch tests, and beam structure bending and free vibration tests, the convergence of regular finite elements can be evaluated, the correctness and convergence of irregular three - dimensional micropolar elements can be evaluated, so as to verify the accuracy of the three - dimensional micropolar elastic body finite element constructed based on the construction method provided by the present invention; meanwhile, by investigating the bending boundary value problem of a cantilever beam in the beam structure bending and free vibration tests, the numerical solution and the analytical solution are obtained and compared, thereby alleviating shear locking.

[0271] In this embodiment, the SAM (Structural Analysis of Marine) software is used to construct a full - integration hexahedral micropolar element (Hex8L) by directly generating finite element meshes and applying boundary conditions within the SAM platform;

[0272] Based on the above - mentioned three - dimensional micropolar elastic body element construction method within the SAM platform, a reduced - integration hexahedral micropolar element (Hex8R) is constructed;

[0273] In addition, an external model can also be imported through the INP file interface of ABAQUS software or HYPERMESH software, and it is required to convert the element type from C3D8R to a micropolar element.

[0274] In this embodiment, a large - scale eigenvalue problem is solved by a structural finite element solver integrated with the Hex8L / Hex8R element algorithm, and the numerical results are finally output in HDF5 format;

[0275] In this embodiment, a post-processing module integrated with the VTK visualization toolkit parses the HDF5 result file to generate multi-dimensional contour maps, characterizing the stress field, displacement distribution, and rotational response, thereby meeting the visualization requirements for the analysis of three-dimensional micropolar elastic body elements.

[0276] First, through the force patch test, it is checked whether the finite element elements maintain a theoretically constant stress field under any mesh discretization.

[0277] As Figure 2 shown, in this embodiment, the test object is a cantilever beam in a uniaxial tension configuration.

[0278] The geometric parameters of the cantilever beam are L = 5 m, h = 2 m, and b = 1 m, and the elastic tensor A ijkl is expressed as:

[0279]

[0280] The elastic tensor C replaces λ, μ, and ν with α, β, and γ respectively, and the material parameter values are defined as follows: μ = 1000 N / mm 2 , λ = 1000 N / mm 2 , ν = 500 N / mm 2 , α = 20 N, β = 20 N, and γ = 20 N; the left end of the beam is fully constrained, and an axial stress of F = 10 N / m 2 is applied at the right end to maintain a uniform stress distribution.

[0281] Figure 2 In

[0282] , (a) is a schematic diagram of the beam under axial load, which is used as a reference verification to check the numerical convergence characteristics; (b) is the finite element model of the beam under axial force; (c) is the stress distribution of a single Hex8R element (1×1×1); (d) is the stress distribution of a fine mesh Hex8R element (10×1×2). Figure 2 By comparing (c) and (d) in

[0283] it can be seen that even when only a single element is used, the calculated results show excellent agreement with the analytical solution, verifying the accuracy and reliability of the reduced integration hexahedral micropolar element. Figure 2 The geometric shape of the cantilever beam shown in (a) is still used for the displacement patch test, with dimensions of L = 0.24 m, h = 0.12 m, and b = 0.06 m.

[0284] The cantilever beam model is discretized into seven irregular block elements using eight nodes, and the material constitutive parameters are the same as those used in the force patch test, corresponding to the standard patch test in classical continuum theory.

[0285] By applying a linearly varying displacement field and a uniform rotation field, a symmetric stress and strain distribution can be obtained. The equations for the displacement and micro-rotation fields are expressed as:

[0286] u1 = 0.01(x + 0.5y + z), u2 = 0.01(x + y + 0.5z),

[0287] u z = 0.01(0.5x + y + z), φ1 = φ2 = φ3 = 0.00025,

[0288] The following analytical solutions are generated:

[0289] σ 11 = σ 22 = σ 33 = 5.0,

[0290] σ 12 = σ 21 = σ 23 = σ 32 = σ 13 = σ 31 = 1.5.

[0291] As Figures 3 - 4 shown, Figure 3 a finite element model of the initial displacement test is presented, Figure 4 showing the corresponding stress contour distribution. According to Figure 3 and Figure 4 it can be seen that the numerical results obtained using the Hex8R element are in complete agreement with the theoretical solution.

[0292] Finally, bending and free vibration tests of the beam structure are carried out to evaluate the alleviation of shear locking of the three-dimensional micropolar elastic body element;

[0293] As Figure 5 shown, Figure 5 in (a) is the geometric configuration and boundary conditions of a rectangular beam, with one end fixed and a surface force f s and a surface bending moment m s applied at the other end; (b) is the corresponding finite element model, which contains two brick-shaped elements along the y direction. The corresponding material parameters are E = 1500 N / m 2 , n = 0.25 N / m 2 , λ = 600 N / m 2 and μ = 600 N / m 2 ;

[0294] The applied pure bending moment is defined as:

[0295]

[0296] Among them, b represents the width of the beam; h represents the height of the beam; x2 represents the dimension of the beam in the y direction;

[0297] The corresponding surface force f s and the surface moment m s can be expressed as follows:

[0298]

[0299] Among them, W = bh 2 / 6 represents a constant related to the cross-sectional dimension of the beam; δ = 24(l b / h) 2 represents a coefficient related to the slenderness ratio of the beam; represents the bending characteristic length value;

[0300] Then, the analytical solutions of the displacement and micro-rotation fields are:

[0301]

[0302] Among them, D = Eh 3 / (12 - 12n 2 ).

[0303] The pure bending boundary value problem is numerically studied by using the full integration Hex8L and reduced integration Hex8R elements; for comparative analysis, the C3D8R element in the ABAQUS software is used to simulate the scenario with only the surface force f s for (l b = 0.0); the calculation results include the displacement u2 and micro-rotation φ3 at the point P((L / 2, -h / 2, b / 2)), and the stress σ 11 at the integration point (7.88675, 0.211325, 0.788675); the calculation results are shown in Table 1 and Figure 6 as shown. In Table 1, A represents the analytical solution and N represents the numerical solution;

[0304] Table 1 Calculation results of Hex8L element and Hex8R element

[0305]

[0306] According to Table 1, the Hex8R element shows a high degree of consistency with the analytical solution in predicting the displacement and micro-rotation fields, demonstrating its reliability in capturing the motion response;

[0307] According to Figure 6 it can be seen that the stress contour diagram of the Hex8R element is highly consistent with the contour of the C3D8R element Figure 1 , indicating that the calculation accuracies of the two are very close, thus verifying the correctness of the Hex8R form.

[0308] The above description is an explanation of the present invention, not a limitation thereof. For the scope defined by the present invention, refer to the claims. Any form of modification may be made within the scope of protection of the present invention.

Claims

1. A method for constructing a three-dimensional micropolar elastic body unit based on reduced integration, characterized in that: The steps include: S1. In the three-dimensional micropolar elastic body, each material point includes three displacement degrees of freedom u i and three rotational degrees of freedom φ i , construct a displacement degree of freedom u for each material point i and the rotational freedom φ i The kinetic equation of (Eq. 1); S2. Construct the constitutive equation of the three-dimensional micropolar body based on the stress balance equation and strain balance equation of the three-dimensional micropolar elastic body (Formula 2); S3. Based on the principle of virtual work, the weak form of the differential equation is established (Equation 3); S4. inserting the dynamic component interpolation function into (Equation 1) to obtain the interpolated micropolar strain vector ε and the interpolated micropolar curvature vector κ; S5. Substitute (Equation 2), the interpolated micropolar strain vector ε and the interpolated micropolar curvature vector κ into (Equation 3) to obtain the expression of the unit stiffness matrix (Equation 4) and the expression of the unit mass matrix (Equation 5); S6. Use the single point reduced integration method to solve (Equation 4) and (Equation 5). In the solution process, the characteristic hourglass shape vector is introduced A finite element model of a three-dimensional micropolar elastic body is constructed; Characteristic hourglass shape vector The expression is: In the above formula, Hourglass basis vectors representing the Lagrangian difference function of the hexahedral element; Represents the volume average of the partial derivatives of the shape function of node I with respect to the three coordinate axes x, y and z; Represents the coordinate components of node J on the three coordinate axes of x, y and z; Hourglass basis vectors representing the Lagrangian difference function of the hexahedral element; The characteristic hourglass shape vector As the displacement hourglass control paradigm, we can get i Related displacement mode expressions; The characteristic hourglass shape vector As a rotation hourglass control paradigm, we can obtain i Related rotation mode expressions.

2. The method for constructing a three-dimensional micropolar elastic body unit based on reduced integration according to claim 1, characterized in that: In S1, the expression of the i-th kinetic equation is: In (Equation 1), ε ij represents the strain tensor; u j,i The partial derivative tensor representing the displacement degrees of freedom; e kij represents the Lévy–Civita tensor; φ k represents the rotational degrees of freedom of a micropolar material point; κ ij represents the curvature tensor; φ j,i A tensor representing the partial derivatives of the rotational degrees of freedom.

3. The method for constructing a three-dimensional micropolar elastic body unit based on reduced integration according to claim 1, characterized in that: In S2, the constitutive equation of the three-dimensional micropolar elastic body is expressed as follows: In (Equation 2), σ ij Indicates stress; A ijkl , B ijkl and C ijkl Both represent fourth-order elastic tensors and have symmetry, namely, A ijkl =A klij , B ijkl =B klij , C ijkl =C klij .

4. The method for constructing a three-dimensional micropolar elastic body unit based on reduced integration according to claim 1, characterized in that: In S3, the weak form of the differential equation is expressed as: In (Equation 3), σ ij Indicates stress; δ is the symbol of the first-order variation of physical quantities in the basic finite element theory; ε ij represents the strain tensor; m ij represents the micropolar dipole stress tensor; κ ij represents the micropolar curvature tensor; ρ represents the material density; represents the acceleration component of the micropolar material point; u i represents the displacement degrees of freedom of a micropolar material point; represents the angular acceleration component of the micropolar material point; φ i Represents the rotational degree of freedom of a micropolar material point. f Si represents surface force; m Si represents the surface couple; A represents the surface area exerted by the surface forces and surface couples; f Vi Represents the forces of a three-dimensional micropolar elastic body; m Vi Represents the torque of a three-dimensional micropolar elastic body.

5. The method for constructing a three-dimensional micropolar elastic body unit based on reduced integration according to claim 1, characterized in that: In S4., the expression of the interpolated micropolar strain vector ε is: in, 0 represents a 9×3 zero matrix; The expression of the submatrix is: In the above formula, represents the partial derivative of the shape function of node I along the x direction; Represents the partial derivative of the shape function of node I along the y direction; represents the partial derivative of the shape function of node I along the Z direction; N I Represents the shape function of node I.

6. The method for constructing a three-dimensional micropolar elastic body unit based on reduced integration according to claim 1, characterized in that: In S4., the expression of the interpolated micropolar curvature vector κ is: In (Formula 4), 0 represents a 9×3 zero matrix; The expression of the submatrix is: In the above formula, represents the partial derivative of the shape function of node I along the x direction; Represents the partial derivative of the shape function of node I along the y direction; Represents the partial derivative of the shape function of node I along the Z direction.

7. The method for constructing a three-dimensional micropolar elastic body unit based on reduced integration according to claim 1, characterized in that: In S5, the expression of the element stiffness matrix is: In (Formula 4), express Transpose of a matrix; express Transpose of a matrix; B u for matrix; Q φ for matrix; B represents the material constitutive tensor in (Equation 2); B φ T express Transpose of a matrix; C represents the material constitutive tensor in (Equation 2); B φ for matrix.

8. The method for constructing a three-dimensional micropolar elastic body unit based on reduced integration according to claim 1, characterized in that: In S5, the expression of the unit mass matrix is: In (Equation 5), M1 represents the mass matrix corresponding to the displacement degree of freedom; M2 represents the mass matrix corresponding to the rotational degree of freedom; ρ represents the density of microelectrode material; N T Represents the transpose of the shape function matrix of each node of the micropolar finite element; N is the shape function matrix of the 8 nodes of the micropolar finite element N = [N 1 N 2 ...N 8 ].

9. The method for constructing a three-dimensional micropolar elastic body unit based on reduced integration according to claim 1, characterized in that: In S6, the displacement mode expression includes an hourglass displacement expression and an hourglass force expression, and the hourglass displacement is conjugate with the hourglass force; The hourglass displacement expression is: q α =[q α1 q α2 q α3 ] T =x α F-X α In the above formula, An hourglass pattern representing deformation; represents the undeformed hourglass pattern; represents the deformation gradient; The hourglass force expression is: Q α =[Q α1 Q α2 Q α3 ] T =K uγ q α In the above formula, K uγ An hourglass pattern representing deformation; K uγ The expression is q α represents the hourglass displacement; G represents the shear modulus of the material; g u represents the displacement hourglass stiffness factor.

10. The method for constructing a three-dimensional micropolar elastic body unit based on reduced integration according to claim 1, characterized in that: In S6, the rotation mode expression includes an hourglass rotation expression and an hourglass torque expression; The hourglass rotation expression is: In the above formula, represents the characteristic hourglass shape vector; represents the rotation component of node I; The hourglass moment expression is: R α =[R α1 R α2 R α3 ] T =K φγ r α In the above formula, r α represents the hourglass rotation mode associated with the rotational degree of freedom; K φγ represents the rotation hourglass stiffness; K φγ The expression is g φ represents the rotational hourglass stiffness factor; G represents the shear modulus of the material; V el express; represents the partial derivative of the shape function of node I along the x direction; Represents the partial derivative of the shape function of node I along the y direction; Represents the partial derivative of the shape function of node I along the z direction.

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