A method for predicting the sensitivity of a tensile and compressive carbon nanotube / polymer composite material strain sensor

Abstract

The application discloses a kind of axial tensile and compression carbon nanotube / polymer composite material strain sensor sensitivity prediction method.The method is based on the electromechanical coupling microstructure theory in micromechanics, while the influence of carbon nanotube volume concentration, axial tensile, compression strain and the tunneling distance between adjacent carbon nanotubes on the mechanical properties and electrical properties of carbon nanotube / polymer composite material is investigated, and the resistance change ratio and strain sensitivity coefficient of carbon nanotube / polymer composite material strain sensor are quantitatively calculated, which has high fitting degree with actual test results.The method designs the corresponding calculation process from the microcosmic angle, fully embodies the nonlinear relationship between resistance change ratio and strain in the axial tension and compression range, greatly simplifies the design process of high-sensitivity strain sensor, and has important guiding significance for large-scale production of high-sensitivity strain sensor.

Description

Technical Field

[0001] This invention designs a method for predicting the sensitivity of a strain sensor, specifically a method for predicting the sensitivity of a strain sensor made of tensile and compressive carbon nanotube / polymer composite material, belonging to the field of strain sensors. Background Technology

[0002] With the rapid development of intelligent technologies in the information age, the demand for precise perception of the surrounding environment in daily life is increasing. As a crucial component of monitoring systems, intelligent sensors establish a close connection between the physical world and digital systems by converting monitoring information into electronic or digital signals. Specifically, strain sensors can quantitatively monitor the mechanical strain at a material point by measuring electrical signals. Currently, strain sensors are widely used in practical applications such as human-machine interfaces, structural health monitoring, and electronic skin.

[0003] Traditional strain sensors are mainly divided into silicon-based sensors and piezoelectric sensors. Silicon-based strain sensors, due to the strong correlation between their band gap and interatomic spacing, possess advantages such as high sensitivity and a large piezoresistive coefficient. However, due to the inherent brittleness and stiffness of semiconductor materials, these strain sensors are prone to fracture and exhibit low fracture toughness. As an alternative, piezoelectric strain sensors are made from functionalized ceramics or polymers with high-performance piezoelectric properties. However, they only operate within a narrow measurement range and are extremely expensive to manufacture. Unlike the previous two types of strain sensors, carbon nanotube / polymer composite strain sensors are fabricated by adding carbon nanotube nanofillers to a polymer matrix. In recent decades, research on carbon nanotube / polymer composite strain sensors has mainly focused on the DC strain sensing performance under axial tensile strain. Through experimental studies and theoretical modeling, researchers have reached similar conclusions: the resistance change ratio and strain sensitivity coefficient both increase with increasing axial tensile strain, while both decrease with increasing carbon nanotube volume concentration. However, some researchers have discovered that the DC strain sensing performance of carbon nanotube / polymer composite strain sensors differs under axial tensile and compressive conditions. Therefore, the DC strain sensing performance under axial tensile and compressive strain has attracted increasing attention. To date, research on DC sensing performance under axial tensile and compressive strain has mainly focused on experimental studies. Through these studies, researchers have discovered several interesting experimental phenomena: (1) the strain sensitivity coefficient under axial tensile strain loading is higher than that under axial compressive strain loading; (2) at lower carbon nanotube volume concentrations, the resistance change ratio under both axial tensile and compressive strain loading exhibits a significant nonlinear trend; (3) at lower carbon nanotube volume concentrations, the resistance change ratio under axial tensile strain loading increases more sharply with increasing axial tensile strain, while the resistance change ratio under axial compressive strain loading increases more gradually with increasing axial compressive strain. Current computational simulations and theoretical modeling cannot perfectly describe this nonlinear phenomenon, resulting in significant discrepancies with experimental data. Therefore, it is urgent to establish a unified theoretical model suitable for axial tensile and compressive strain loading to explain the DC sensing performance of carbon nanotube / polymer composite strain sensors related to axial strain. Summary of the Invention

[0004] To address the problems existing in the prior art, the present invention aims to provide a sensitivity prediction method for strain sensors made of tensile and compressive carbon nanotube / polymer composite materials. This method is based on micromechanics, considers the influence of tunneling distance on the sensitivity and resistance change ratio of the strain sensor, and establishes a unified prediction model. The prediction results obtained by this method have a high degree of agreement with the actual test results, greatly simplifying the design process of high-sensitivity strain sensors and providing important guidance for the large-scale production of high-sensitivity strain sensors.

[0005] To achieve the above technical objectives, this invention provides a method for predicting the sensitivity of a strain sensor for a carbon nanotube / polymer composite material under tension and compression, comprising the following steps: 1) determining the geometric, electrical, and mechanical parameters of the carbon nanotubes; 2) obtaining the mechanical and electrical characteristic parameters of the carbon nanotube / polymer composite material strain sensor under different carbon nanotube volume concentrations; 3) setting the axial strain. 4) Establish mechanical homogenization models for coated carbon nanotubes and carbon nanotube / polymer composite strain sensors; 5) Establish elastic constitutive models for the raw materials required for carbon nanotube / polymer composite strain sensors and models for the volume concentration of carbon nanotubes related to axial strain; 6) Establish models for conductivity, resistance change ratio, and strain sensitivity coefficient related to axial strain of carbon nanotube / polymer composite strain sensors; 7) Establish DC sensitivity prediction models related to axial strain of carbon nanotube / polymer composite strain sensors; 8) Plot curves for each of the above models and perform model verification and correction.

[0006] This invention, based on micromechanics, considers the impact of tunneling effects on the sensitivity of carbon nanotube / polymer composite strain sensors and establishes a unified prediction model. This model employs a seven-step method, which can rapidly calibrate the sensing performance of strain sensors under axial tensile and compressive strain by selecting specific carbon nanotube volume concentrations, offering significant advantages such as saving experimental time and economic costs. It can accurately predict the sensing performance across the entire axial tensile and compressive range by adjusting only a few parameters. Furthermore, the model is established within the elastic deformation range, reflecting the reusability of strain sensors and better reflecting real-world conditions. It can accurately predict experimental data on the resistance change ratio and strain sensitivity coefficient of strain sensors; and the prediction curves clearly demonstrate the nonlinear relationship between the resistance change ratio and axial strain, as well as the asymmetry of the resistance change ratio under tensile and compressive strain.

[0007] In terms of mechanics, this invention uses bulk modulus and shear modulus as average variables, and obtains the mechanical modulus of the composite material strain sensor using the Mori-Tanaka homogenization method. In terms of electrical properties, it uses electrical conductivity as an average variable, and obtains the electrical conductivity of the coated carbon nanotubes and composite material strain sensor using the Mori-Tanaka and effective medium homogenization methods. Considering the influence of axial strain and tunneling distance on the electrical and mechanical properties of the carbon nanotube / polymer composite material strain sensor and the tunneling effect, the invention finally calculates a sensitivity prediction method related to axial tensile and compressive strain of the carbon nanotube / polymer composite material strain sensor, thereby obtaining continuous variation curves of its resistance change ratio and strain sensitivity coefficient with respect to axial strain and carbon nanotube volume concentration.

[0008] As a preferred embodiment, the geometric, electrical, and mechanical parameters of the carbon nanotubes include: aspect ratio. Dimensionless; radius The dimensionless conductivity is (m); the in-plane and normal conductivity are (S / m); the axial Young's modulus is... Dimensions are (Pa); transverse bulk modulus Dimensions are (Pa); Transverse shear modulus Dimensions are (Pa); Plane shear modulus The dimensionless value is (Pa); plane Poisson's ratio Dimensionless; the electrical and mechanical parameters of the polymer include: electrical conductivity The dimensionless value is (S / m); Poisson's ratio Dimensionless; Young's modulus Its dimension is (Pa).

[0009] As a preferred embodiment, the geometric, mechanical, and electrical characteristic parameters of the carbon nanotube / polymer composite strain sensor include: interface layer thickness. The dimensions are (m) and Poisson's ratio Dimensionless; strain-free DC conductivity The dimension is (S / m); the strain resistance change ratio Dimensionless; strain sensitivity coefficient , dimensionless.

[0010] As a preferred embodiment, the mechanical and electrical characteristic parameters of the carbon nanotube / polymer composite strain sensor can be obtained experimentally. The specific process is as follows: 1) Prepare M carbon nanotube / polymer composite strain sensor samples with different carbon nanotube volume concentrations, and measure and obtain the interface layer thickness and Poisson's ratio of these samples by SEM; 2) Measure the DC conductivity of N samples with different carbon nanotube volume concentrations under strain-free conditions; and then measure the resistance change ratio and strain sensitivity coefficient of the N samples under different strains, wherein 4≤N≤M.

[0011] As a preferred embodiment, the mechanical homogenization model of the coated carbon nanotube and carbon nanotube / polymer composite strain sensor is established using the Mori–Tanaka method. This model includes: the elastic model of the coated carbon nanotube, the effective bulk modulus of the carbon nanotube / polymer composite strain sensor, and the relationship between the shear model and the strain.

[0012] As a preferred embodiment, the process of establishing the mechanical homogenization model for the coated carbon nanotube and carbon nanotube / polymer composite strain sensor includes:

[0013] 1) An ultrathin interlayer was introduced between carbon nanotubes and a polymer matrix, and the mechanical properties of the carbon nanotubes were calculated. The calculation process is as follows:

[0014] Formula 1:

[0015] Formula 2:

[0016] in:

[0017] ; ;

[0018] 2) Establish the relationship between the effective bulk modulus and shear modulus of the carbon nanotube / polymer composite strain sensor and the strain. The calculation process is as follows:

[0019] Formula 3:

[0020] Formula 4:

[0021]

[0022] Formula 5:

[0023]

[0024] Formula 6:

[0025]

[0026] Formula 7:

[0027] Formula 8: ;

[0028] In equations 1-8: The aspect ratio of carbon nanotubes is dimensionless. Let be the radius, with dimensions in meters (m). The thickness of the interlayer is expressed in meters (m). , where is the interlayer volume concentration in the coated carbon nanotubes, dimensionless; The elastic stiffness tensor of the coated carbon nanotubes is expressed in Pa. The elastic stiffness tensor of the interlayer is expressed in Pa. It is a unit tensor, dimensionless; , , , and These represent the planar strain modulus, cross-sectional modulus, transverse shear modulus, axial modulus under uniaxial strain, and axial shear modulus of carbon nanotubes, respectively, with dimensions in Pa; (function) This is a user-defined function; The Poisson's ratio is a polymer, dimensionless. , , , , , , , , , , and The component of the Eshelby tensor representing carbon nanotube inclusions embedded in a polymer matrix is ​​dimensionless. Let Eshelby be the dimensionless Eshelby tensor of the encapsulated carbon nanotubes in Hill-Walpole notation. , , , , , and These represent transverse isotropic parameters, which are dimensionless; , , ,and These are the parameters of the homogenized mechanical model, and are dimensionless. and These are the bulk modulus and shear modulus of the polymer matrix, respectively, with dimensions of (Pa); , Let represent the effective bulk modulus and shear modulus of the carbon nanotube / polymer composite material after a given axial strain, respectively, with dimensions of (Pa); Given an axial strain, the Young's modulus of the composite material strain sensor is given, with dimensions of (Pa). The Poisson's ratio of the composite material strain sensor after a given axial strain is dimensionless.

[0029] As a preferred embodiment, the process for establishing the elastic constitutive model of the raw materials required for the carbon nanotube / polymer composite strain sensor and the volume concentration model related to the axial strain of the carbon nanotube is as follows:

[0030] 1) The transverse isotropic elastic constitutive relation of carbon nanotubes is calculated as follows:

[0031] Formula 9: ;

[0032] 2) The mechanical constitutive relation of the isotropic polymer is calculated as follows:

[0033] Formula 10: ;

[0034] 3) The volume concentration of carbon nanotubes in the composite strain sensor after deformation is calculated as follows:

[0035] Formula 11: ;

[0036] In equations 9-11: The stress tensor of carbon nanotubes is expressed in Pa. The elastic stiffness tensor of carbon nanotubes is expressed in Pa. The strain tensor of carbon nanotubes is dimensionless. , , , and These are the planar strain modulus, cross-sectional modulus, transverse shear modulus, axial modulus under uniaxial strain, and axial shear modulus of carbon nanotubes, respectively, with dimensions of (Pa). The stress tensor of the polymer is expressed in Pa. The elastic stiffness tensor of the polymer is expressed in Pa. The strain tensor of the polymer is dimensionless. and The bulk modulus and shear modulus of a polymer are expressed in Pa. This represents the updated carbon nanotube volume concentration, dimensionless. This represents the updated polymer volume concentration, dimensionless. The Poisson's ratio of the composite material strain sensor after a given axial strain is dimensionless.

[0037] Carbon nanotubes have high overall stiffness and only undergo elastic deformation when subjected to stress. Therefore, Hill notation can be used to describe the transverse isotropic elastic constitutive relation of carbon nanotubes. In contrast, polymer matrices are considered isotropic, so Hill notation is used to describe the isotropic mechanical constitutive relation of polymer matrices.

[0038] As a preferred embodiment, the model establishment process for the axial strain-related conductivity, resistance change ratio, and strain sensitivity coefficient of the carbon nanotube / polymer composite strain sensor is as follows:

[0039] 1) Establish the geometric settings related to the axial strain of the carbon nanotube / polymer composite strain sensor;

[0040] 2) Based on the tunneling effect related to axial strain at the interface, the relationship between the axial strain and the conductivity of the interlayer is established. The calculation process is as follows:

[0041] Formula 12: , in, , ;

[0042] Formula 13: ;

[0043] Formula 14: ;

[0044] Formula 15: ;

[0045] Formula 16: ;

[0046] Equation 17:

[0047] ;

[0048] Formula 18: ;

[0049] Formula 19: ;

[0050] 3) The electrical conductivity of the coated carbon nanotubes was calculated based on the tunneling effect. The calculation process is as follows:

[0051] Formula 20: ;

[0052] 4) Homogenization calculation of the axial strain-related electrical properties of the carbon nanotube / polymer composite strain sensor. The calculation process is as follows:

[0053] Equation 21:

[0054] Equation 22: ;

[0055] In equations 12-22: The distance between the updated axes of adjacent carbon nanotubes is in the dimension (m); This represents the original distance between the axes of two adjacent carbon nanotubes, with dimensions (m). This represents the initial tunneling distance between two adjacent carbon nanotubes, with dimensions in meters (m). , , , , , Let x, y, and z represent the x, y, and z coordinates of points P and Q after deformation, respectively, with dimensions in meters (m). for and The initial angle between them is dimensionless; This is a dimensionless correction parameter for the distance between the central axes of adjacent coated carbon nanotubes. The distance between adjacent coated carbon nanotube axes, which is related to axial strain, is expressed in units of (m). The denoted distance represents the average distance between adjacent coated carbon nanotube axes related to axial strain, with dimensions in meters (m). The average tunneling distance between adjacent coated carbon nanotubes, which is related to the axial strain, is expressed in units of (m). The tunneling effect cutoff distance is in units of (m); The contact area between adjacent carbon nanotubes, with dimensions (m²). 2 ); The potential barrier height between adjacent coated carbon nanotubes is expressed in eV. The initial interfacial phase resistance between adjacent coated nanotubes is given by the dimensionless ( ); , and These represent Planck's constant, with dimensions ( ). ), electron charge, dimensionless ( ), electron mass, dimensionless ); This represents the interface thickness, measured in meters (m). This represents the resistance related to the interface change, with dimensions ( ). ); This represents the interlayer volume concentration in coated carbon nanotubes, and is dimensionless. This represents the dimensionless component of the depolarization tensor of the electric field in electrostatics. The conductivity of the sandwiched carbon nanotubes is expressed in units of (S / m). The conductivity of the coated carbon nanotubes is expressed in units of (S / m). This represents the conductivity vector of carbon nanotubes. Indicates the axial direction. It indicates the horizontal direction and has the dimension (S / m); The electrical conductivity of the polymer matrix is ​​expressed in units of (S / m). and The axial or transverse electrical conductivity of the coated carbon nanotubes, respectively, is expressed in units of (S / m). The conductivity of the composite material strain sensor after deformation is expressed in units of (S / m). The ratio of resistance change is dimensionless. and These represent the original resistances of the undeformed composite material strain sensor, with dimensions ( ). ) and conductivity, with dimensions (S / m); The Poisson's ratio of the composite material strain sensor after a given axial strain is given; it is dimensionless. It represents the change of internal quantities during deformation and is an operator; is the DC strain sensitivity coefficient, which is dimensionless. Wherein, It depends on the volume concentration of carbon nanotubes and the tunneling distance between two adjacent carbon nanotubes.

[0056] This invention employs four steps to establish models for the conductivity, resistance change ratio, and strain sensitivity coefficient of a carbon nanotube / polymer composite strain sensor related to axial strain. Since the distance between adjacent carbon nanotubes is less than the cutoff distance, a tunneling effect occurs, and this tunneling distance changes with the application of axial strain. In the microscopic realm, the tunneling effect occurs when the distance between adjacent carbon nanotubes is less than the cutoff distance. However, from a macroscopic perspective, we cannot quantitatively measure the distance between each carbon nanotube and its neighbors. The greater the number of carbon nanotubes per unit volume, the smaller the average distance between them. Therefore, this invention reflects the average distance between adjacent carbon nanotubes through the volume concentration of carbon nanotubes. As the volume concentration of carbon nanotubes increases, the average distance between adjacent carbon nanotubes decreases. When it is less than the cutoff distance, additional electrons will pass through the interface due to the electron tunneling effect. Furthermore, since external strain has a significant impact on the average tunneling distance between adjacent carbon nanotubes, the electron tunneling effect will be related to axial strain.

[0057] As a preferred embodiment, the process for establishing the prediction model for the axial strain-related DC sensitivity of the carbon nanotube / polymer composite strain sensor is as follows:

[0058] 1) Based on the effective conductivity at different carbon nanotube volume concentrations, the tunneling effect parameters and interface material parameters are determined by fitting, including: This represents the initial tunneling distance parameter under high carbon nanotube loading conditions; it is dimensionless. A dimensionless characteristic index representing the initial distance between adjacent carbon nanotubes; This represents the contact area between adjacent carbon nanotubes, with dimensions (m²). 2 )and This represents the initial interfacial phase resistance, with dimensions ( ). );

[0059] 2) Based on the resistivity change ratio under different axial strains at different carbon nanotube volume concentrations, the remaining parameters are determined by fitting, including: This represents the interfacial phase elastic modulus, with dimensions in Pa. This represents the barrier height between adjacent coated carbon nanotubes, with dimensions (eV).

[0060] As a preferred option, the model verification and calibration process is as follows:

[0061] 1) Compare the predicted curve of effective DC conductivity with respect to carbon nanotube volume concentration under no-strain loading with the actual measured data;

[0062] 2) Compare the predicted curves of conductivity, resistance change ratio and strain sensitivity coefficient related to axial strain of the obtained carbon nanotube / polymer composite strain sensor with the actual measurement data.

[0063] Furthermore, the specific steps of the model verification and calibration process are as follows:

[0064] 1) By substituting different carbon nanotube volume concentrations and zero strain into the homogenization calculation of the axial strain-related electrical properties of the carbon nanotube / polymer composite strain sensor, a complete prediction curve of the effective DC conductivity under no-strain loading with respect to the carbon nanotube volume concentration is obtained and verified with experimental data.

[0065] 2) Substitute the determined carbon nanotube volume concentration and different axial tensile or compressive strains into the homogenization calculation of the axial strain-related electrical properties of the carbon nanotube / polymer composite strain sensor established in step five to obtain the complete prediction curves of the resistance change ratio and strain sensitivity coefficient with respect to axial strain under different carbon nanotube volume concentrations, and verify them with experimental data.

[0066] The model provided by this invention also includes a self-checking process. When the predicted curve differs significantly from the actual quantity, the model can gradually adjust parameters such as the interfacial phase elastic modulus. The dimensionless value is (Pa), representing the barrier height between adjacent coated carbon nanotubes. The dimensionless parameter is (eV), representing the initial tunneling distance under high carbon nanotube loading conditions. Dimensionless; a characteristic index of the initial distance between adjacent carbon nanotubes. Dimensionless; contact area between adjacent carbon nanotubes, Dimensions are (m) 2 and the initial interface phase resistance, , dimensionless Then, the comparison is repeated until the relative error between the predicted curve and the actual quantity is less than 22%.

[0067] Compared with the prior art, the beneficial effects of the technical solution of the present invention are:

[0068] 1. The present invention provides a sensitivity prediction method for a strain sensor of carbon nanotube / polymer composite material under axial tension and compression. This method links the microscopic parameters of the strain sensor with the macroscopic DC sensing performance. By experimental measurement or literature search and based on limited experimental data, the parameters in the prediction model are quickly determined to obtain a complete prediction model. This model is then used to predict the DC sensing performance of the strain sensor of carbon nanotube / polymer composite material under continuous axial tension and compression strain ranges and different carbon nanotube volume concentrations.

[0069] 2. The technical solution provided by this invention can quickly calibrate the sensing performance of a strain sensor under axial tensile and compressive strain by selecting a specific carbon nanotube volume concentration. By adjusting a small number of parameters, the sensing performance can be accurately predicted within the entire axial tensile and compressive range. Furthermore, the model is established within the elastic deformation range, which reflects the characteristic that the strain sensor can be used multiple times. It is more in line with actual conditions and has obvious advantages such as saving test time and saving economic costs.

[0070] 3. The technical solution provided by this invention not only considers the influence of axial tensile and compressive strain loading on the volume change of composite material strain sensor, but also considers the influence of axial tensile and compressive strain loading on the tunneling distance between adjacent carbon nanotubes, and quantitatively reflects this through formula derivation. After verification, the prediction model provided by this invention can accurately predict the experimental data of the resistance change ratio and strain sensitivity coefficient of the strain sensor, and the prediction curve can clearly show the nonlinear relationship between the resistance change ratio and axial strain, as well as the asymmetric phenomenon of the resistance change ratio under tensile and compressive strain. Its prediction speed and accuracy are superior to the existing technical solutions. Attached Figure Description

[0071] Figure 1 This is a schematic diagram of a strain sensor.

[0072] Figure 2This is a schematic diagram of the geometric configuration of a carbon nanotube / polymer composite strain sensor subjected to axial tensile and compressive strain and DC loading.

[0073] Figure 3 (a) Calculation of the distance between the two types of coated carbon nanotubes, (b) Strain-induced distance variation, and (c) Relationship between the central axis distance of the two types of coated carbon nanotubes and the tunneling distance and interlayer thickness.

[0074] Figure 4 SEM image of a multi-walled carbon nanotube / epoxy resin composite strain sensor.

[0075] Figure 5 This figure shows a theoretical and experimental comparison of the effective conductivity of a multi-walled carbon nanotube / epoxy resin composite strain sensor under strain-free loading.

[0076] Figure 6 This is a comparison of theoretical and experimental results regarding the resistance change ratio of a multi-walled carbon nanotube / epoxy resin composite strain sensor with respect to axial strain.

[0077] Figure 7 This is a theoretical and experimental comparison of the strain sensitivity coefficient of a multi-walled carbon nanotube / epoxy resin composite strain sensor with respect to axial strain.

[0078] Figure 8 This is a comparison of the resistance change ratio of the multi-walled carbon nanotube / epoxy resin composite strain sensor with respect to the axial strain change, as predicted by the present invention, and the model prediction curve in Comparative Example 1.

[0079] Figure 9 The graph shows a comparison between the strain sensitivity coefficient prediction curve of the multi-walled carbon nanotube / epoxy resin composite strain sensor with respect to axial strain, as predicted by the present invention, and the prediction curve based on the model in Comparative Example 1.

[0080] Figure 10 This is a comparison of the curve showing the relative error between the strain sensitivity coefficient and experimental data of the multi-walled carbon nanotube / epoxy resin composite strain sensor predicted by this invention and the curve showing the relative error between the strain sensitivity coefficient and experimental data and the axial strain based on the model in Comparative Example 1. Detailed Implementation

[0081] The following specific embodiments are intended to further illustrate the content of the present invention, rather than to limit the scope of protection of the claims.

[0082] Example 1

[0083] To facilitate understanding of the present invention, a comprehensive description of the invention will be provided below with reference to embodiments. Multi-walled carbon nanotubes (MWCNT-7) with a purity exceeding 99.5% (Japan NCTC product) were prepared via chemical vapor deposition. The epoxy resin was an insulating bis-F epoxy resin (JER806, Japan Epoxy Resin Co., Ltd.). The prediction method includes the following four steps:

[0084] 1. The geometric parameters, electrical and mechanical properties, and tunneling effect parameters of multi-walled carbon nanotubes and epoxy resin were measured under strain-free loading. The results are as follows:

[0085] Aspect ratio of multi-walled carbon nanotubes ,radius ; Transverse conductivity Axial conductivity Axial Young's modulus Lateral bulk modulus Transverse shear modulus Planar shear modulus plane Poisson ratio ; Conductivity of epoxy resin polymer Poisson's ratio Young's modulus Furthermore, Planck's constant was obtained by looking up a table. Electronic quality The charge of an electron Tunneling effect cutoff distance The above material properties and physical parameters will be used in the parameter fitting process during the implementation steps.

[0086] 2. Six strain sensor samples of multi-walled carbon nanotube / epoxy resin composite materials with different multi-walled carbon nanotube volume concentrations were prepared. .

[0087] (1) First, obtain SEM images of the sample, such as Figure 4 As shown. The thickness of the interface layer was measured. Poisson ratio of interface layer .

[0088] (2) Then, the volume concentrations of four types of multi-walled carbon nanotubes were measured under strain-free loading, preferably... The effective DC conductivity of the multi-walled carbon nanotube / polymer composite strain sensor sample is shown in the experimental data as follows: Figure 5 The diamond icon is shown in the image.

[0089] (3) Secondly, within the axial tensile and compressive strain range of -0.6% to 0.6%, the volume concentrations of multi-walled carbon nanotubes were measured as follows: The resistance change ratio and strain sensitivity coefficient of the multi-walled carbon nanotube / polymer composite strain sensor sample are shown in the experimental data as follows: Figure 6 and Figure 7 The rhombus in the middle ( ),triangle( ) and circles ( As shown in the figure.

[0090] (4) Substitute the above material properties and experimental data into the present invention, and determine the material parameters related to the conductivity of the prediction model through data fitting. Specifically, substitute the material properties and physical parameters obtained in steps 1 and 2 of the embodiment into equations 21 and 22; the equations after substitution are the final equations for solving the effective conductivity, resistance change ratio and strain sensitivity coefficient of the multi-walled carbon nanotube / polymer composite strain sensor.

[0091] Take the loading state with strain of 0, and Figure 5 Four points were selected from the effective conductivity data of different volume concentrations of multi-walled carbon nanotubes (specific values ​​are {0.03, 1.017}, {0.0429, 1.818}, {0.062, 1.978}, {0.095, ...).

[0092] Substitute 2.09 into equation 21 respectively; solve using Newton's method to obtain the parameters. , , and The specific values ​​are shown in Table 1.

[0093] The volume concentration of multi-walled carbon nanotubes is taken as a fixed value (preferably). ),Will Figure 6 middle Two points were randomly selected from the resistivity change ratio data under different axial tensile and compressive strains (specific values ​​are {0.291451, 4.241215} and {0.485373, 9.741366}), and substituted into Equation 22; the parameters were obtained by solving using Newton's method. and The specific values ​​are shown in Table 1.

[0094]

[0095] Thus, we have obtained a complete predictive model for the resistance change ratio and strain sensitivity coefficient of the multi-walled carbon nanotube / epoxy composite strain sensor, which includes all geometric and material physics parameters.

[0096] 3. Substitute equations 11, 19, 20, and 21, the material properties and physical parameters obtained in steps 1 and 2, and the material parameters obtained through data fitting in Table 1 into equation 22. Equation 22 is the final equation for solving the complete prediction model of this invention. In this embodiment, the tunneling effect needs to be considered.

[0097] (1) The accuracy of the prediction model established in this invention is verified by comparing the experimental data of the effective conductivity of different volume concentrations of multi-walled carbon nanotubes under strain-free loading with the prediction curve.

[0098] By setting the strain in the final equation 21 to 0, and substituting different multi-walled carbon nanotube volume concentrations into the final equation 21, the effective conductivity of the composite material strain sensor is numerically solved using Newton's method. A continuous prediction curve of the effective conductivity under strain-free loading conditions versus multi-walled carbon nanotube volume concentration is plotted, as follows: Figure 5 As shown by the curve in the middle.

[0099] Depend on Figure 5 As can be seen, the predicted curve of the effective conductivity of the multi-walled carbon nanotube / epoxy resin composite strain sensor is consistent with the experimental data points, verifying the accuracy of the prediction model in this invention at zero strain. The effective conductivity of the multi-walled carbon nanotube / epoxy resin composite strain sensor increases with increasing multi-walled carbon nanotube volume concentration. When the multi-walled carbon nanotube volume concentration reaches the percolation threshold, the effective conductivity increases significantly.

[0100] (2) The following will be conducted by measuring the volume concentration of multi-walled carbon nanotubes (preferred respectively). The experimental data and predicted curves of the resistance change ratio and strain sensitivity coefficient under different axial tensile and compressive strains were compared to verify the accuracy of the prediction model established in this invention.

[0101] The volume concentration of multi-walled carbon nanotubes in Equation 22 is taken as a constant volume concentration of multi-walled carbon nanotubes (preferred to be...). Substituting different strains into Equation 22, the resistance change ratio and strain sensitivity coefficient of the composite material strain sensor were numerically solved using Newton's method. Continuous prediction curves of the resistance change ratio and strain sensitivity coefficient with respect to axial tensile and compressive strain were plotted for three constant multi-walled carbon nanotube volume concentrations. Figure 6 and Figure 7 As shown by the curve in the middle.

[0102] Depend on Figure 6 and Figure 7 It can be seen that, within a continuous range of tensile and compressive strain, the resistance change ratio and strain sensitivity coefficient of the multi-walled carbon nanotube / epoxy resin composite strain sensor under different constant multi-walled carbon nanotube volume concentrations are consistent with the experimental data points, verifying the accuracy of the prediction model established in this invention. Figure 6 The results show that, under constant volume concentration of multi-walled carbon nanotubes, the absolute value of the resistance change ratio increases with increasing axial tensile and compressive strain, but the two are asymmetrical; the absolute value of the resistance change ratio becomes more drastic with increasing axial tensile strain, while the absolute value of the resistance change ratio becomes more gradual with increasing axial compressive strain; under constant strain, the absolute value of the resistance change ratio decreases with increasing volume concentration of multi-walled carbon nanotubes. Figure 7 The results show that at three different volume concentrations of multi-walled carbon nanotubes ( The strain sensitivity coefficient of the multi-walled carbon nanotube / epoxy resin composite strain sensor increases with increasing axial tensile strain and decreases with increasing axial compressive strain; its change with strain is nonlinear.

[0103] 4. At this point, the predicted curves for the effective conductivity, resistance change ratio, and strain sensitivity coefficient of the multi-walled carbon nanotube / epoxy resin composite strain sensor with respect to axial tensile and compressive strain and multi-walled carbon nanotube volume concentration have been obtained, as follows: Figures 5 to 7 As shown, the axial strain-related sensitivity prediction method for strain sensors made of tensile and compressive carbon nanotube / polymer composite materials based on micromechanics proposed in this invention is feasible and accurate. By selecting specific carbon nanotube volume concentrations and axial strains, the macroscopic performance of the sensor can be quickly obtained, which can further simplify the design process of high-sensitivity strain sensors and improve design efficiency.

[0104] Comparative Example 1

[0105] The model building process for Comparative Example 1 is the same as that for Example 1, but the models used in the specific steps are different. The differences between the two models are as follows:

[0106] 1. The model in the comparative example considers the damage of the interlayer during strain loading. The damage process can be considered as an irreversible thermodynamic process. To characterize this phenomenon, a damage parameter is introduced. The concept of continuous damage mechanics. The specific damage parameter evolution equation can be obtained from the following equation, where the interface free energy density is defined as...

[0107]

[0108] in and The effective Mises stress and hydrostatic stress of the polymer, respectively, can be expressed by the field wave method as follows:

[0109]

[0110]

[0111] in The updated carbon nanotube volume concentration is expressed using the same formula as in Example 1. and It refers to the effective Mises stress and hydrostatic stress applied throughout the nanocomposite material. This indicates external stress. , , and Let represent the updated secant bulk modulus and secant shear modulus of the composite material, and the updated secant bulk modulus and secant shear modulus of the polymer, respectively, all in Pa. The updated secant modulus of the composite material can be expressed as:

[0112]

[0113] , , and The mechanical homogenization parameter is dimensionless and its expression is similar to that of Example 1. Note that the Poisson's ratio of the polymer in Example 1 needs to be changed. and polymer Young's modulus Change to secant modulus and .

[0114] Thermodynamic inequalities can be set as follows:

[0115]

[0116] The equation applies only to reversible processes. The thermodynamic driving force, given an external load, is expressed as...

[0117]

[0118] In order to characterize damage variables The evolutionary process requires the establishment of damage potential. In a specific state , and Damage potential It has the following forms

[0119]

[0120] in and This represents the damage index of the interlayer, and is dimensionless. It is the damage energy intensity, with dimensions of (Pa).

[0121] Finally, the evolution equation of the interface damage parameter D can be expressed by a Ginzburg-Landau-like equation.

[0122]

[0123] It is proportional to the thermodynamic driving force. According to this equation, the damage variable... As plastic strain increases, it gradually increases from 0 to 1.

[0124] 2. The tunneling effect in the proportional model is represented by the Cauchy cumulative probability distribution function, as shown below.

[0125]

[0126] in and Let represent the imperfect interface conductivity exponent considering strain-dependent electron tunneling effects and the initial imperfect interface conductivity exponent, respectively, with dimensions (m). 2 / S), where It will take into account how the damage parameter D changes as it increases; The updated carbon nanotube volume concentration is expressed using the same formula as in Example 1. It is the Cauchy cumulative probability distribution function, which can be expressed as:

[0127]

[0128] This represents the proportional parameter describing the rate of change of the Cauchy cumulative probability distribution function; it is dimensionless. This is expressed as the percolation threshold of the carbon nanotube / polymer nanocomposite material, which depends on the aspect ratio of the carbon nanotubes.

[0129]

[0130] The components of the Eshelby tensor are represented by the same expression as in Example 1.

[0131] 3. In the comparative model, the polymer constitutive model is an elastoplastic constitutive model, and the mechanical modulus needs to be expressed using the secant modulus method.

[0132]

[0133] In the formula, , , These are the yield stress, strength index, and strain hardening index of the polymer matrix, respectively. and These represent the effective Mises stress and effective plastic strain of the polymer matrix, respectively, and their expressions are related to the deviatoric stress and plastic strain of the polymer matrix. The secant Young's modulus after yielding of the polymer matrix phase is also given. Secant shear modulus It can be represented as

[0134]

[0135] These are the main differences between the theoretical model in Comparative Example 1 and the model of this invention.

[0136] Furthermore, the tensile and compressive sensing properties of the carbon nanotube / polymer composite strain sensor were predicted using the model in the comparative example, and the prediction results were compared with those in Example 1. The comparison results are as follows:

[0137] 1. Determine the volume concentration of multi-walled carbon nanotubes (preferably...) Substituting different tensile and compressive axial strain loads into the two prediction models, continuous prediction curves of the resistivity change ratio with respect to axial tensile and compressive strain under a given multi-walled carbon nanotube volume concentration were plotted, as follows: Figure 8 As shown. From Figure 8 It can be seen that the prediction model in Comparative Example 1 cannot well reflect the nonlinear relationship between the resistance change ratio and axial tensile and compressive strain. Moreover, the prediction model in Comparative Example 1 only considers the electrical damage of the strain sensor under axial tensile strain load, but not under axial compressive strain load, thus failing to effectively unify the study of axial tensile and compressive sensing performance. The prediction model in Example 1 can well illustrate this phenomenon, and the prediction results are closer to the experimental values.

[0138] 2. The determined volume concentration of multi-walled carbon nanotubes (preferably...) Substituting different axial strain loads into the two prediction models, continuous prediction curves of the strain sensitivity coefficient with respect to axial tensile and compressive strain were plotted for a given multi-walled carbon nanotube volume concentration, as shown below. Figure 9 As shown. From Figure 9 As can be seen, the model in Comparative Example 1 cannot accurately describe the variation trend of the strain sensitivity coefficient of the multi-walled carbon nanotube / epoxy resin composite strain sensor with respect to axial tensile and compressive strain. The prediction model in Example 1 can well illustrate this phenomenon, and its prediction results are closer to the experimental values.

[0139] 3. To more clearly compare the advantages and disadvantages of the two models, given a multi-walled carbon nanotube volume concentration of... Under the given conditions, the relative error values ​​between the strain sensitivity coefficients predicted by the models in Example 1 and Comparative Example 1 and the experimental data were calculated for the entire axial strain range. The comparison curves of these two values ​​with respect to axial strain are shown below. Figure 10 As shown. From Figure 10It can be seen that the relative error between the model prediction results and experimental data in Comparative Example 1 is a maximum of 38.02% and a minimum of 0.08% across the entire axial strain load range, with an average relative error of 13.97% under axial tensile strain load and 19.86% under axial compressive strain load. In contrast, the relative error between the prediction model in Example 1 and experimental data is a maximum of 21.8% and a minimum of 0.036% across the entire axial strain load range, with an average relative error of 8.31% under axial tensile strain load and 3.09% under axial compressive strain load. Therefore, the average, maximum, and minimum relative errors of the prediction model in Example 1 are all smaller than those of the prediction model in Comparative Example 1. Thus, the prediction model in Example 1 has a significant advantage over the prediction model in Comparative Example 1.

Claims

1. A method for predicting the sensitivity of a tensile and compressive carbon nanotube / polymer composite strain sensor, characterized by: The method comprises the following steps: 1) determining the geometric, electrical and mechanical parameters of carbon nanotubes; 2) obtaining the mechanical and electrical characteristic parameters of the carbon nanotube / polymer composite strain sensor under different carbon nanotube volume concentrations; 3) Set axial strain and establish the mechanical homogenization model of the axial strain of the coated carbon nanotube and carbon nanotube / polymer composite strain sensor The process of establishing the mechanical homogenization model is: introducing an ultrathin interlayer between the carbon nanotube and the polymer matrix, and calculating the mechanical properties of the coated carbon nanotube; establishing the relationship between the effective volume modulus, shear modulus and strain of the carbon nanotube / polymer composite strain sensor; 4) establishing the elastic constitutive model of the required raw material of the carbon nanotube / polymer composite and the volume concentration model related to the axial strain of the carbon nanotube; The process of establishing the volume concentration model is: sequentially establishing the transversely isotropic elastic constitutive relationship of the carbon nanotube, the isotropic mechanical constitutive relationship of the polymer and the volume concentration of the carbon nanotube in the deformed composite strain sensor; 5) establishing the conductivity, resistance change ratio and strain sensitivity coefficient model related to the axial strain of the carbon nanotube / polymer composite strain sensor; The process of establishing the conductivity, resistance change ratio and strain sensitivity coefficient model related to the axial strain of the carbon nanotube / polymer composite strain sensor is: establishing the geometric setting related to the axial strain of the carbon nanotube / polymer composite strain sensor; based on the tunneling effect related to the axial strain on the interface, the relationship between the axial strain and the conductivity of the interlayer is established; the conductivity of the coated carbon nanotube is calculated based on the tunneling effect; the electrical properties related to the axial strain of the carbon nanotube / polymer composite strain sensor are homogenized calculated; 6) establishing a direct current sensitivity prediction model related to the axial strain of the carbon nanotube / polymer composite strain sensor; The process of establishing the direct current sensitivity prediction model is: based on the effective conductivity under different carbon nanotube volume concentrations, fitting and determining the tunneling effect parameters and the parameters of the interface material; based on the resistance change ratio under different axial strains under different carbon nanotube volume concentrations, fitting and determining the remaining parameters including the Young's modulus of the interface phase and the potential barrier height between adjacent coated carbon nanotubes; 7) drawing the curve graphs of the above models and performing model verification and correction; The process of model verification and correction is: comparing the predicted curve of the effective direct current conductivity without strain loading and the carbon nanotube volume concentration with the actual measurement data; comparing the predicted curve of the conductivity, resistance change ratio and strain sensitivity coefficient related to the axial strain of the carbon nanotube / polymer composite strain sensor with the actual measurement data.

2. The method of claim 1, wherein the method is characterized by: geometric, electrical and mechanical parameters of the carbon nanotubes include: aspect ratio , dimensionless; radius , dimension (m); in-plane and normal conductivities, dimension (S / m); axial Young's modulus , dimension (Pa); transverse bulk modulus , dimension (Pa); transverse shear modulus , dimension (Pa); in-plane shear modulus , dimension (Pa); in-plane Poisson's ratio , dimensionless; electrical and mechanical parameters of the polymer include: conductivity , dimension (S / m); Poisson's ratio , dimensionless; Young's modulus , dimension (Pa).

3. The method of claim 1, wherein the method is characterized by: Geometrical, mechanical and electrical characteristic parameters of the carbon nanotube / polymer composite strain sensor include: interface layer thickness , dimension (m) and Poisson's ratio , dimensionless; no-strain direct current conductivity , dimension (S / m); strain resistance change ratio , dimensionless; strain sensitivity coefficient , dimensionless.

4. The method of claim 1, wherein the method is characterized by: The mechanical homogenization model related to the axial strain of the coated carbon nanotube and the carbon nanotube / polymer composite strain sensor is established by the Mori-Tanaka method, and the model comprises: an elastic model of the coated carbon nanotube, a relationship between the effective volume modulus, shear modulus model and strain of the carbon nanotube / polymer composite strain sensor; The process of establishing the mechanical homogenization related to the axial strain of the coated carbon nanotube and the carbon nanotube / ploymer composite strain sensor comprises: 1) introducing an ultrathin interlayer between the carbon nanotube and the polymer matrix, and the mechanical properties of the coated carbon nanotube are calculated, and the calculation process is: Formula 1: ; Formula 2: ; wherein: ； ； ； 2) The relationship between the effective bulk modulus, shear modulus and strain of the carbon nanotube / polymer composite strain sensor is established, and the calculation process is as follows: Formula 3: ; Formula 4: ; Formula 5: ; Formula 6: ; Formula 7: ; Formula 8: ; in formulas 1-8: L is the aspect ratio of the carbon nanotube, dimensionless; R is the radius, dimension (m); t is the interlayer thickness, dimension (m); C is the interlayer volume concentration in the coated carbon nanotube, dimensionless; C denotes the elastic stiffness tensor of the coated carbon nanotube, dimension (Pa); C denotes the elastic stiffness tensor of the interlayer, dimension (Pa); I is the identity tensor, dimensionless; , , , and are the planar strain bulk modulus, cross-sectional modulus, transverse shear modulus, axial modulus under uniaxial strain, and axial shear modulus of the carbon nanotube, dimension (Pa); the function is a user-defined function; P is the polymer Poisson's ratio, dimensionless; , , , , , , , , , , and denote the components of the Eshelby tensor for carbon nanotube inclusions embedded in a polymer matrix, dimensionless; C is the Eshelby tensor for the wrapped carbon nanotube under Hill-Walpole representation, dimensionless; , , , , , and denote the transverse isotropic parameters, dimensionless; , , , and are the mechanical homogenization model parameters, dimensionless; and are the bulk and shear moduli of the polymer matrix, dimension (Pa); and denote the effective bulk and shear moduli of the carbon nanotube / polymer composite strain sensor after a given axial strain, dimension (Pa); E is the Young's modulus of the composite strain sensor after a given axial strain, dimension (Pa); Poisson's ratio for a given axial strain after composite strain sensor, dimensionless.

5. The method of claim 1 or 4, wherein the method is characterized by: The establishment process of the elastic constitutive model of the raw material required by the carbon nanotube / polymer composite strain sensor and the bulk concentration model related to the axial strain of the carbon nanotube is as follows: 1) Carbon nanotube transverse isotropic elastic constitutive relation, the calculation process is as follows: Formula 9: ; 2) The mechanical constitutive relation of the polymer is isotropic, and the calculation process is as follows: Formula 10: ; 3) The volume concentration of the carbon nanotube in the deformed composite strain sensor, the calculation process is as follows: Formula 11: ; in formulas 9-11: represents the stress tensor of the carbon nanotube, dimensionally (Pa); represents the elastic stiffness tensor of the carbon nanotube, dimensionally (Pa); represents the strain tensor of the carbon nanotube, dimensionless; and are the in-plane strain body modulus, cross-sectional modulus, transverse shear modulus, axial modulus under uniaxial strain, and axial shear modulus of the carbon nanotube, dimensionally (Pa); represents the stress tensor of the polymer, dimensionally (Pa); represents the elastic stiffness tensor of the polymer, dimensionally (Pa); represents the strain tensor of the polymer, dimensionless; represent the bulk modulus and shear modulus of the polymer, dimensionally (Pa); represents the updated carbon nanotube volume concentration, dimensionless; represents the updated polymer volume concentration, dimensionless; is the Poisson’s ratio of the composite strain sensor after a given axial strain, dimensionless.​​​​ 6. The method of claim 1, wherein the method is characterized by: The model establishment process of the axial strain related conductivity, resistance change ratio and strain sensitivity coefficient of the carbon nanotube / polymer composite strain sensor is as follows: The calculation process of the relationship between the axial strain and the conductivity of the sandwich is as follows: Formula 12: wherein, , ; Formula 13: ; Formula 14: ; Formula 15: ; Formula 16: ; Formula 17: ； Formula 18: ; Formula 19: ; The calculation process of the conductivity of the coated carbon nanotube based on the tunneling effect is as follows: Formula 20: ; 4) The homogenization calculation of the axial strain related electrical properties of the carbon nanotube / polymer composite strain sensor, the calculation process is as follows: Formula 21: Formula 22: ; In formulas 12~22: is the distance between two adjacent carbon nanotube axes after updating, with the dimension of (m); is the original distance between two adjacent carbon nanotube axes, with the dimension of (m); is the original tunneling distance between two adjacent carbon nanotubes, with the dimension of (m); , , , , , are respectively the x coordinate, y coordinate, z coordinate of the deformed P point and Q point, with the dimension of (m); is the is the is the initial angle between is the correction parameter of the distance between the central axes of two adjacent coated carbon nanotubes, dimensionless; is the distance related to the axial strain between the axes of two adjacent coated carbon nanotubes, with the dimension of (m); is the average distance related to the axial strain between the axes of two adjacent coated carbon nanotubes, with the dimension of (m); is the average tunneling distance related to the axial strain between two adjacent coated carbon nanotubes, with the dimension of (m); is the tunneling effect cutoff distance, with the dimension of (m); is the contact area of two adjacent carbon nanotubes, with the dimension of (m 2 ); is the potential barrier height between two adjacent coated carbon nanotubes, with the dimension of (eV); is the initial interfacial phase resistance between two adjacent coated carbon nanotubes, with the dimension of ; , and respectively represent the Planck constant, with the dimension of , the electronic charge, with the dimension of , and the electron mass, with the dimension of ; is the interfacial thickness, with the dimension of (m); is the interfacial phase strain related resistance, with the dimension of ; is the interlayer volume concentration in the coated carbon nanotube, dimensionless; is the component of the depolarization tensor of the electric field in electrostatics, dimensionless; is the interlayer carbon nanotube conductivity, with the dimension of (S / m); is the coated carbon nanotube conductivity, with the dimension of (S / m); is the carbon nanotube conductivity vector, Indicates the axial direction. It indicates the horizontal direction and has the dimension (S / m); The electrical conductivity of the polymer matrix is ​​expressed in units of (S / m). and The axial or transverse electrical conductivity of the coated carbon nanotubes, respectively, is expressed in units of (S / m). The conductivity of the composite material strain sensor after deformation is expressed in units of (S / m). The ratio of resistance change is dimensionless. and These represent the original resistances of the undeformed composite material strain sensor, with dimensions ( ). ) and conductivity, with dimensions (S / m); The Poisson's ratio of the composite material strain sensor after a given axial strain is given; it is dimensionless. It represents the change of internal quantities during deformation and is an operator; is the DC strain sensitivity coefficient, which is dimensionless.

7. The sensitivity prediction method for a strain sensor of a tensile and compressive carbon nanotube / polymer composite material according to claim 1, characterized in that: The tunneling effect parameters and parameters of the interface material include: represents the initial tunneling distance parameter under high carbon nanotube loading conditions, dimensionless; represents a characteristic index of the initial distance between adjacent carbon nanotubes, dimensionless; represents the contact area of adjacent carbon nanotubes, dimension (m 2 ) and represents the initial interface phase resistance, dimension (Ω ).

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