Analytic model for predicting tensile rigidity of zero-Poisson-ratio corrugated lattice structure

Abstract

According to the performance of the material and the geometric parameters of the zero-Poisson-ratio corrugated lattice structure, carrying out stress analysis on the zero-Poisson-ratio corrugated lattice structure; the tensile deformation process of the zero-Poisson-ratio corrugated lattice structure is calculated according to the geometrical relationship, the geometrical nonlinear effect in the tensile process of the zero-Poisson-ratio corrugated lattice structure is considered, and further, the tensile load-displacement curve and the tensile rigidity of the zero-Poisson-ratio corrugated lattice structure are obtained according to the ideas of an iteration method and an energy method.

Description

Technical Field

[0001] The present invention provides an analytical model for predicting the tensile stiffness of a zero-Poisson's ratio corrugated lattice structure, belonging to the intersection of aviation and mechanics. Background Art

[0002] Zero Poisson's ratio corrugated lattice structures are widely used in lightweight and multifunctional designs in the aerospace field. Tensile stiffness is one of the important indicators for measuring the mechanical properties of zero Poisson's ratio corrugated lattice structures. Traditional empirical formulas do not take into account the changes in the geometric morphology of the zero Poisson's ratio corrugated lattice structure during the stretching process. However, the geometric nonlinear effect will significantly affect the prediction results of the tensile stiffness. Therefore, the present invention provides an analytical model for predicting the tensile stiffness of the zero Poisson's ratio corrugated lattice structure, which takes into account the geometric nonlinear effect of the zero Poisson's ratio corrugated lattice structure during the stretching process, and realizes the high-precision prediction of the tensile stiffness of the zero Poisson's ratio corrugated lattice structure. Summary of the Invention

[0003] The present invention establishes an analytical model for predicting the tensile stiffness of a zero Poisson's ratio corrugated lattice structure. This method has the advantages of simple calculation and high accuracy. The technical solution is as follows:

[0004] The present invention takes the zero Poisson's ratio corrugated lattice structure as the research object and establishes an analytical model. Figure 1 Schematic diagram of the geometric configuration of the zero Poisson's ratio corrugated lattice structure. Figure 2 The figure is a schematic diagram of the tensile deformation and stress analysis of the zero Poisson's ratio corrugated lattice structure. Based on the energy method, the present invention establishes a theoretical prediction model for the tensile properties of the zero Poisson's ratio corrugated lattice structure and selects one of the corrugated units as the research object for analysis.

[0005] The shape function of the corrugated element is expressed as

[0006]

[0007] Here, H represents the amplitude and L represents the period length.

[0008] Through the force balance condition and moment balance condition, the expressions of axial force and moment on any cross section can be derived as follows:

[0009]

[0010] Under the action of horizontal tensile load, the period length of the corrugated unit gradually increases, the amplitude gradually decreases (the thickness and width of the corrugation remain unchanged), and the line length gradually lengthens, that is, the geometric configuration of the corrugated unit is constantly changing during the stretching process. Based on the idea of iteration, the present invention divides the corrugated unit stretching deformation process into n steps, and the result of the calculation of the i-1 step is the input of the calculation of the i step. Calculate the ΔU of the displacement of the i step iWhen the input is ΔP i ,H i-1 ,L i-1 ,S i ,ΔS i-1 Where i = 1, 2, ..., n, ΔP i is the horizontal load applied in step i, H i-1 and L i-1 are the amplitude and cycle length of the ripple unit after the end of step i-1, S i is the line length of the corrugated unit after the end of step i-1, ΔS i-1 is the change in line length after the end of step i-1, ΔP i , H0, L0 are known quantities, ΔP1=ΔP2=···=ΔP i =···=ΔP n .

[0011] According to the energy method, the displacement of the corrugated unit under the tensile load can be expressed as

[0012]

[0013] in

[0014]

[0015] Where E is the elastic modulus of the material, W is the width of the corrugated unit, and t is the thickness of the corrugated unit.

[0016] L in formula (7) and formula (8) i-1 is the length of the cycle after the end of step i-1

[0017] L i-1 =L i-2 +ΔU i-1 (9)

[0018] H in formula (8) i-1 is the amplitude after the end of step i-1, and equations (10) to (14) are H i-1 The solution steps.

[0019] After step i-1, the length of the line changes by

[0020]

[0021] The line length input of the ripple at step i (the line length after step i-1) is

[0022] S i =S i-1 +ΔS i-1 (11)

[0023] The line length expression of the corrugated unit is

[0024]

[0025] Substituting equation (1) into equation (12) yields the line length expression of the i-th step corrugated unit:

[0026]

[0027] By Leibniz integration rule, S i About H i-1 The derivative of can be expressed as

[0028]

[0029] Use Newton's iteration method to solve, first select the initial value (H i-1 )0, then calculate the iterative formula Until|(H i-1 ) n+1 -(H i-1 ) n | is less than the set threshold 0.001, the amplitude input H of step i can be obtained i-1 .

[0030] Equation (5)-Equation (9) and H i-1 Substituting into formula (4), we can get the displacement of step i, as follows:

[0031]

[0032] The tensile stiffness of the corrugated element at step i is

[0033]

[0034] According to formula (15), the tensile displacement ΔU of each corrugated unit can be obtained i (i=1,2,3,…,n). By accumulating ΔU at each step i and ΔP i , the total displacement and total tensile load of the kth step in the stretching process of the zero Poisson's ratio corrugated lattice structure can be obtained

[0035]

[0036]

[0037] Where s and s' represent the number of corrugated units in the zero Poisson's ratio corrugated lattice structure in the y direction and the x direction, respectively.

[0038] By combining equations (17) and (18), we can obtain the tensile load-displacement curve of the zero Poisson's ratio corrugated lattice structure.

[0039] The least squares linear fit [δ k , P k ](k=1,2,3,…,n), the tensile stiffness of the zero Poisson's ratio corrugated lattice structure is

[0040]

[0041] in, and are the average displacement and average tensile load of the zero Poisson's ratio corrugated lattice structure, respectively.

[0042]

[0043] The present invention is an analytical model for predicting the tensile stiffness of a zero Poisson's ratio corrugated lattice structure. The model is characterized in that the tensile stiffness of the zero Poisson's ratio corrugated lattice structure can be predicted quickly and easily based on the performance parameters and geometric parameters of the component materials of the zero Poisson's ratio corrugated lattice structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 Schematic diagram of the geometric configuration of the zero Poisson's ratio corrugated lattice structure.

[0045] Figure 2 Schematic diagram of tensile deformation and stress analysis of zero Poisson's ratio corrugated lattice structure.

[0046] The symbols in the figure are explained as follows:

[0047] Figure 2 Where F is the tensile load of the zero Poisson's ratio corrugated lattice structure, H is the amplitude of the corrugated unit, L is the period length of the corrugated unit, P is the tensile load of the corrugated unit, N(x) is the axial force of any section of the corrugated unit, M(x) is the moment of any section of the corrugated unit, P(x) is the tensile load at any position of the corrugated unit, and M1 is the moment of the fixed end of the corrugated unit. DETAILED DESCRIPTION

[0048] The tensile deformation and stress analysis process of the zero Poisson's ratio corrugated lattice structure is as follows Figure 2 As shown, the present invention establishes a theoretical prediction model for predicting the tensile properties of a zero Poisson's ratio corrugated lattice structure based on the energy method, and selects one of the corrugated units as a research object for analysis.

[0049] The shape function of the corrugated element is expressed as

[0050]

[0051] Here, H represents the amplitude and L represents the period length.

[0052] Through the force balance condition and moment balance condition, the expressions of axial force and moment on any cross section can be derived as follows:

[0053]

[0054] Under the action of horizontal tensile load, the period length of the corrugated unit gradually increases, the amplitude gradually decreases (the thickness and width of the corrugation remain unchanged), and the line length gradually lengthens, that is, the geometric configuration of the corrugated unit is constantly changing during the stretching process. Based on the idea of iteration, the present invention divides the corrugated unit stretching deformation process into n steps, and the result of the calculation of the i-1 step is the input of the calculation of the i step. Calculate the ΔU of the displacement of the i step i When the input is ΔP i ,H i-1 ,L i-1 ,S i ,ΔS i-1 Where i = 1, 2, ..., n, ΔP i is the horizontal load applied in step i, H i-1 and L i-1 are the amplitude and cycle length of the ripple unit after the end of step i-1, S i is the line length of the corrugated unit after the end of step i-1, ΔS i-1 is the change in line length after the end of step i-1, ΔP i , H0, L0 are known quantities, ΔP1=ΔP2=···=ΔP i =···=ΔP n .

[0055] According to the energy method, the displacement of the corrugated unit under the tensile load can be expressed as

[0056]

[0057] in

[0058]

[0059] Where E is the elastic modulus of the material, W is the width of the corrugated unit, and t is the thickness of the corrugated unit.

[0060] L in formula (7) and formula (8) i-1 is the length of the cycle after the end of step i-1

[0061] L i-1 =L i-2 +ΔU i-1 (9)

[0062] H in formula (8) i-1 is the amplitude after the end of step i-1, and equations (10) to (14) are Hi-1 The solution steps.

[0063] After step i-1, the length of the line changes by

[0064]

[0065] The line length input of the ripple at step i (the line length after step i-1) is

[0066] S i =S i-1 +ΔS i-1 (11)

[0067] The line length expression of the corrugated unit is

[0068]

[0069] Substituting equation (1) into equation (12) yields the line length expression of the i-th step corrugated unit:

[0070]

[0071] By Leibniz integration rule, S i About H i-1 The derivative of can be expressed as

[0072]

[0073] Use Newton's iteration method to solve, first select the initial value (H i-1 )0, then calculate the iterative formula Until|(H i-1 ) n+1 -(H i-1 ) n | is less than the set threshold 0.001, the amplitude input H of step i can be obtained i-1 .

[0074] Equation (5)-Equation (9) and H i-1 Substituting into formula (4), we can get the displacement of step i, as follows:

[0075]

[0076] The tensile stiffness of the corrugated element at step i is

[0077]

[0078] According to formula (15), the tensile displacement ΔU of each corrugated unit can be obtained i (i=1,2,3,…,n). By accumulating ΔU at each step i and ΔP i, the total displacement and total tensile load of the kth step in the stretching process of the zero Poisson's ratio corrugated lattice structure can be obtained

[0079]

[0080] Where s and s' represent the number of corrugated units in the zero Poisson's ratio corrugated lattice structure in the y direction and the x direction, respectively.

[0081] By combining equations (17) and (18), we can obtain the tensile load-displacement curve of the zero Poisson's ratio corrugated lattice structure.

[0082] The least squares linear fit [δ k , P k ](k=1,2,3,…,n), the tensile stiffness of the zero Poisson's ratio corrugated lattice structure is

[0083]

[0084] in, and are the average displacement and average tensile load of the zero Poisson's ratio corrugated lattice structure, respectively.

[0085]

[0086] The present invention is an analytical model for predicting the tensile stiffness of a zero Poisson's ratio corrugated lattice structure. The model is characterized in that the tensile stiffness of the zero Poisson's ratio corrugated lattice structure can be predicted quickly and easily based on the performance parameters and geometric parameters of the component materials of the zero Poisson's ratio corrugated lattice structure.

Claims

1. An analytical model for predicting the tensile stiffness of a zero Poisson's ratio corrugated lattice structure, characterized by: Based on the energy method, the present invention establishes a theoretical prediction model for the tensile properties of a zero Poisson's ratio corrugated lattice structure, and selects one of the corrugated units as the research object for analysis; The shape function of the corrugated element is expressed as Where H represents the amplitude and L represents the period length; Through the force balance condition and moment balance condition, the expressions of axial force and moment on any cross section can be derived as follows: Under the action of horizontal tensile load, the period length of the corrugated unit gradually increases, the amplitude gradually decreases (the thickness and width of the corrugation remain unchanged), and the line length gradually becomes longer, that is, the geometric configuration of the corrugated unit is constantly changing during the stretching process; Based on the idea of iteration, the present invention divides the tensile deformation process of the corrugated unit into n steps, and the result of the calculation of the i-1 step is the input of the calculation of the i-th step; the displacement ΔU of the i-th step is calculated. i When the input is ΔP i ,H i-1 ,L i-1 ,S i ,ΔS i-1 ; where i = 1, 2, ..., n, ΔP i is the horizontal load applied in step i, H i-1 and L i-1 are the amplitude and cycle length of the ripple unit after the end of step i-1, S i is the line length of the corrugated unit after the end of step i-1, ΔS i-1 is the change in line length after the end of step i-1, ΔP i , H0, L0 are known quantities, ΔP1=ΔP2=···=ΔP i =···=ΔP n ; According to the energy method, the displacement of the corrugated unit under the tensile load can be expressed as in Where E is the elastic modulus of the material, W is the width of the corrugated unit, and t is the thickness of the corrugated unit; L in formula (7) and formula (8) i-1 is the length of the cycle after the end of step i-1 L i-1 =L i-2 +ΔU i-1 (9) H in formula (8) i-1 is the amplitude after the end of step i-1, and equations (10) to (14) are H i-1 The solution steps: After step i-1, the length of the line changes by The line length input of the ripple at step i (the line length after step i-1) is S i =S i-1 +ΔS i-1 (11) The line length expression of the corrugated unit is Substituting equation (1) into equation (12) yields the line length expression of the i-th step corrugated unit: By Leibniz integration rule, S i About H i-1 The derivative of can be expressed as Use Newton's iteration method to solve, first select the initial value (H i-1 )0, then calculate the iterative formula Until|(H i-1 ) n+1 -(H i-1 ) n | is less than the set threshold 0.001, the amplitude input H of step i can be obtained i-1 ; Equation (5)-Equation (9) and H i-1 Substituting into formula (4), we can get the displacement of step i, as follows: The tensile stiffness of the corrugated element at step i is According to formula (15), the tensile displacement ΔU of each corrugated unit can be obtained i (i=1,2,3,…,n); by accumulating ΔU at each step i and ΔP i , the total displacement and total tensile load of the kth step in the stretching process of the zero Poisson's ratio corrugated lattice structure can be obtained Where s and s' represent the number of corrugated units in the zero Poisson's ratio corrugated lattice structure in the y direction and the x direction, respectively; Combining equations (17) and (18), we can obtain the tensile load-displacement curve of the zero Poisson's ratio corrugated lattice structure; The least squares linear fit [δ k , P k ](k=1,2,3,…,n), the tensile stiffness of the zero Poisson's ratio corrugated lattice structure is in, and are the average displacement and average tensile load of the zero Poisson's ratio corrugated lattice structure, respectively.

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