Prediction model for equivalent electro-mechanical coupling performance of flexible nanoparticle reinforced composite material deformation sensor and application of prediction model

Abstract

The invention discloses a prediction model for equivalent electro-mechanical coupling performance of a flexible nanoparticle reinforced composite material deformation sensor and application of the prediction model. According to the model, a large-deformation hyperelastic mechanical constitutive relation and a linear electrical constitutive relation under direct current are adopted, a function interface effect related to deformation under large deformation is considered, pseudo-modulus and conductivity are selected as tensile sensing analysis homogenization parameters, and a prediction model of equivalent electro-mechanical coupling performance is established; the model is simple and easy to implement, is convenient to operate, and is used for solving the limitation that the numerical simulation of the flexible nano composite material deformation sensor under large deformation consumes a large amount of computing resources. The model can be used for predicting the non-linear tensile sensing capacity of a flexible nanoparticle reinforced composite material sensor, and tests show that the relative errors of resistance change ratios predicted by the equivalent electro-mechanical coupling performance prediction model are 5.15%, 5.12% and 7.10% respectively when the volume fractions of nanoparticles are 0.062%, 0.077% and 0.093% respectively.

Description

Technical Field

[0001] The invention relates to a prediction model for the performance of a flexible deformation sensor, in particular to a prediction model for the equivalent electrical coupling performance of a flexible nanoparticle-reinforced composite material deformation sensor and its application, and belongs to the technical field of flexible deformation sensors. Background Art

[0002] Flexible deformation sensors have been widely used in human health monitoring, aerospace, wearable electronics and other fields. Compared with traditional strain sensors made of piezoelectric materials, flexible nanocomposite deformation sensors can accurately sense a wide range of tensile deformations. Flexible nanocomposite deformation sensors are made of silver nanoparticle-reinforced polymer composites. The advantage of adding silver nanoparticles to the polymer matrix is ​​that the electrical properties of the flexible nanocomposite deformation sensor can be improved while maintaining sufficient flexibility and antibacterial properties. Flexible nanoparticle-reinforced composites are considered to be candidate materials for flexible deformation sensors.

[0003] However, when a large number of nanoparticles are included in the flexible nanocomposite deformation sensor, the numerical simulation will consume high computing resources. Although theoretical research on flexible nanocomposite deformation sensors can overcome the disadvantage of high computational cost in numerical simulation, the existing homogenization method of strain sensors under small deformation cannot explain the highly nonlinear sensing behavior of deformation sensors under large deformation. At the same time, under large deformation conditions, the microscopic characteristics of the flexible nanocomposite deformation sensor have an important influence on the macroscopic tensile sensing performance of the flexible nanocomposite sensor. In this case, the tunnel interface effect and loading-related microstructural characteristics must be considered, including the stretch-related nanoparticle aspect ratio, nanoparticle volume fraction, tunnel distance, etc.

[0004] The inventor has previously submitted a patent application for "A method for predicting the sensitivity of a tensile and compressive carbon nanotube / polymer composite strain sensor", and in a published paper (Xia, X., et al. "A unified investigation into the tensile and compressive sensing performance in highly sensitive MWCNT / epoxy nanocomposite strain sensor through loading-dependent tunneling distance." Composites Science and Technology (2022).), the equivalent resistance change ratio and strain sensitivity coefficient and mechanical properties of the carbon nanotube / polymer composite strain sensor under axial tension and compression under small deformation were respectively given. However, the above patent application and paper did not consider the influence of large deformation conditions on the equivalent force electrical coupling performance of the flexible nanocomposite deformation sensor, and there is no large deformation force electrical coupling homogenization theory based on the microstructure characteristics of the material in the prior art that can link the microscopic characteristics of the flexible nanocomposite deformation sensor with the macroscopic nonlinear tensile sensing performance. Therefore, there is an urgent need for a method for predicting the equivalent force electrical coupling performance of the sensor based on the microscopic characteristics of the deformation sensor. Summary of the invention

[0005] In view of the problems existing in the prior art, the first purpose of the present invention is to provide a prediction model for the equivalent force-electric coupling performance of a flexible nanoparticle-reinforced composite material deformation sensor. The model adopts the large deformation hyperelastic mechanical constitutive relationship and the linear electrical constitutive relationship under direct current, considers the functional interface effect related to deformation under large deformation, selects pseudo modulus and conductivity as homogenization parameters for tensile sensing analysis, and establishes a prediction model for the equivalent force-electric coupling performance; the model is simple and easy to operate, and is used to solve the limitation that the numerical simulation of the flexible nanocomposite deformation sensor under large deformation consumes a large amount of computing resources.

[0006] The second object of the present invention is to provide an application of a prediction model of equivalent effective electrical coupling performance of a flexible nanoparticle reinforced composite material deformation sensor, which is used to predict the nonlinear tensile sensing ability of the flexible nanoparticle reinforced composite material sensor. The equivalent effective electrical coupling performance prediction model provided by the present invention obtains pseudo modulus and conductivity by step-by-step loading, and predicts the nonlinear tensile sensing performance of the flexible nanoparticle reinforced composite material deformation sensor by two specific indicators, namely resistance change ratio and tensile sensitivity coefficient. After testing, the resistance change ratio predicted by the equivalent effective electrical coupling performance prediction model of the present invention has relative errors of 5.15%, 5.12% and 7.10% when the volume fraction is 0.062%, 0.077% and 0.093%, respectively.

[0007] In order to achieve the above technical objectives, the present invention provides a prediction model for the equivalent electrical coupling performance of a flexible nanoparticle reinforced composite material deformation sensor, comprising:

[0008] Step S1. Obtaining characteristic parameters of a prediction model for equivalent electrical coupling performance of a flexible deformation sensor under large deformation;

[0009] Step S2. establishing a mechanical constitutive model of the flexible nanoparticle reinforced composite material component according to the characteristic parameters;

[0010] Step S3. establishing an electrical constitutive model of the flexible nanoparticle reinforced composite material component according to the characteristic parameters;

[0011] Step S4. Establish an equivalent force electrical coupling performance prediction model of the flexible deformation sensor under large deformation based on the mechanical and electrical constitutive models.

[0012] The equivalent force electrical coupling performance prediction model provided by the present invention is based on the extended Mori-Tanaka method and the equivalent medium approximation theory under large deformation, and selects pseudo modulus and conductivity as homogenization parameters for tensile sensing analysis; adopts nonlinear hyperelastic mechanical constitutive and linear electrical constitutive relations to strengthen the weights of two functional interface effects related to stretching, namely, incomplete interface connection and electron tunneling effect; and predicts the nonlinear tensile sensing performance of the flexible nanoparticle reinforced composite deformation sensor through two specific indicators, namely, resistance change ratio and tensile sensitivity coefficient.

[0013] As a preferred solution, the characteristic parameters include: the aspect ratio α of the nanoparticles p , the slenderness ratio of the tunnel area α t , the radius of the nanoparticle r p , the mass density of the nanoparticles ρ p , mass density of polydimethylsiloxane polymer ρ m , the conductivity of the nanoparticles p , the conductivity of polydimethylsiloxane χm , the conductivity of the interface layer χ int , the nanofiller-related tunnel distance parameter m tunnel 、n tunnel , the volume ratio of the tunnel area to the nanoparticle q tunnel , material parameters of polydimethylsiloxane related to distortion Material parameters of polydimethylsiloxane related to volume deformation Material parameters related to distortion of nanoparticles Material parameters of nanoparticles related to volume deformation Material parameters related to interface layer distortion Material parameters related to interface layer and volume deformation Interface layer thickness d int , tunnel area a, barrier height γ between adjacent nanoparticles, Planck constant h, electron charge e, electron mass m e , the cutoff distance d of the tunnel effect cutoff .

[0014] As a preferred solution, the process of establishing the mechanical constitutive model of the flexible nanoparticle reinforced composite material component is:

[0015] Step S2-1. Establish a nonlinear hyperelastic constitutive relationship, then establish a hyperelastic mechanical constitutive relationship of nanoparticles under large deformation based on the relationship, and then construct a hyperelastic mechanical constitutive relationship of the coating nanoparticles in the flexible deformation sensor under large deformation;

[0016] Step S2-2. Establish the nonlinear mechanical constitutive relationship of the polymer matrix in the flexible nanoparticle reinforced composite material under large deformation.

[0017] As a preferred solution, the process of establishing the nonlinear hyperelastic constitutive relation is as follows:

[0018] Formula 1:

[0019] Formula 2: J = detF;

[0020] Formula 3: C=F T F;

[0021] Formula 4:

[0022] Formula 5:

[0023] Formula 6:

[0024] Formula 7:

[0025] Formula 8:

[0026] Formula 9:

[0027] In formulas 1 to 9: F in formula 1 e is the equivalent deformation gradient of the flexible nanoparticle reinforced composite deformation sensor, dimensionless, and the subscript e represents the overall equivalent performance of the flexible nanoparticle reinforced composite deformation sensor; in formula 2, J is the volume ratio defined by the equivalent deformation gradient, dimensionless, and F is obtained by formula 1; in formula 3, C is the uncorrected right Cauchy Green tensor, dimensionless, and F is obtained by formula 1, F T is the transpose of F; in formula 4 is the modified right Cauchy Green tensor, dimensionless, where J is obtained from Equation 2 and C is obtained from Equation 3; β = p is the nanoparticle phase, dimensionless; β = m is the polymer matrix phase, dimensionless; Ψ in Equation 5 β (F β ) is the Helmholtz free energy, with the dimension (J / m 3 ); is the isotropic part of the Helmholtz free energy, with dimensions (J / m 3 ); is the volume part of the Helmholtz free energy, with the dimension (J / m 3 );S in Formula 6 β is the second-kind Piola-Kirchhoff stress tensor, with the dimension of MPa, where Ψ β (F β ) is obtained from formula 5, J β From formula 2, C β From formula 3, Obtained from formula 4; in formula 7 is the nominal stress, dimension is MPa, where S β From formula 6, Obtained from formula 1; in formula 8, σ β is the Cauchy stress, with the dimension of MPa, where From formula 7, we can get β From formula 1, J β Obtained from formula 2; L in formula 9 β is the equivalent pseudo modulus of the β phase, with the dimension of MPa, where Ψ β (F β ) is obtained by formula 5, From formula 1, Obtained by Equation 7.

[0028] As a preferred solution, the nonlinear mechanical constitutive relation of the nanoparticle phase using the Neo-Hookean hyperelastic model is calculated as follows:

[0029] Formula 10:

[0030] Formula 11:

[0031] Formula 12:

[0032] Formula 13:

[0033] Formula 14:

[0034] Formula 15:

[0035]

[0036] In formulas 10 to 15: In formula 10 is the first invariant of the right Cauchy-Green tensor, dimensionless, where C β Obtained from formula 3; in formula 11 is the first invariant of the modified right Cauchy-Green tensor, dimensionless, where From formula 10, J β Obtained from formula 2; in formula 12, Ψ p (F p ) is the Helmholtz free energy of the nanoparticle phase using the Neo-Hookean hyperelastic model, with the dimension of (J / m 3 ), where From formula 11, J β From formula 2, and are the material parameters of the nanoparticles related to distortion and volume deformation, both in TPa. is the nominal stress of the nanoparticle phase, with the dimension of TPa, where J β From formula 2, we can get p From formula 1, From formula 10, and are the material parameters related to the distortion and volume deformation of the nanoparticles measured, both in TPa. p is the Cauchy stress of the nanoparticle phase, with the dimension of TPa, where J β From formula 2, we can get p From formula 1, Obtained from formula 10, I2 is the second-order invariant tensor, and are the material parameters related to the distortion and volume deformation of the nanoparticles, both in TPa. pis the equivalent pseudo modulus of the nanoparticles, with the dimension of TPa, where J β From formula 2, we can get p From formula 1, From formula 10, I4 is a fourth-order invariant tensor, and The material parameters related to the distortion and volume deformation of the nanoparticles were measured respectively, and the dimensions are both TPa.

[0037] As a preferred solution, the calculation process of the hyperelastic constitutive relation of the coating nanoparticles under large deformation is:

[0038] Formula 16:

[0039] Formula 17:

[0040] Formula 18:

[0041] Formula 19:

[0042] Formula 20:

[0043] Formula 21:

[0044] Formula 22:

[0045] L c (F c )=[c int L int (F int )+(1-c int )L p (F p ):A p (F int ,F p )]:

[0046] [c int I4+(1-c int )A p (F int ,F p )] -1 ;

[0047] In formulas 16 to 22, in formula 16, c int is the volume fraction of the interface layer in the coating nanoparticles, dimensionless, where d int and r p are the thickness of the ultra-thin interface layer and the radius of the nanoparticle, both in nm; in Eq. is a unit vector, dimensionless; in formula 18 is the parameter in the Hill tensor, dimension is (MPa), where From formula 17, L AiBj is the stiffness matrix; in Eq. 19 is the parameter in the Hill tensor, with the dimension (MPa -1 ), where Obtained by formula 18; P in formula 20 iAjB is the extended Hill tensor, dimensionless, where for The unit circle in space, Obtained by formula 17; A in formula 21 p (F int ,F p ) is the extended mechanical concentration tensor of nanoparticles embedded in the ultrathin interface layer, dimensionless, where From formula 20, we can get p (F p ) and L int (F int ) are obtained from formula 15, I4 is a fourth-order invariant tensor; L in formula 22 c (F c ) is the equivalent pseudo modulus of the coated nanoparticles, with the dimension of TPa, where c int From formula 16, we can get p (F p ) and L int (F int ) is obtained by formula 15, A p (F int ,F p ) is obtained by formula 21, and I4 is a fourth-order invariant tensor.

[0048] As a preferred solution, the process of establishing the nonlinear mechanical constitutive relationship of the polymer matrix under large deformation is:

[0049] Formula 23:

[0050] Formula 24:

[0051] Formula 25:

[0052] Formula 26:

[0053] Formula 27:

[0054]

[0055] Formula 28:

[0056]

[0057] Formula 29:

[0058] In formulas 23 to 29, in formula 23 is the second invariant of the right Cauchy Green tensor, dimensionless, where C β Obtained by formula 3; in formula 24 is the second invariant of the modified right Cauchy Green tensor, dimensionless, where From formula 23, J β Obtained from formula 2; in formula 25, Ψ m (F m ) is the Helmholtz free energy of the polymer matrix using the Mooney-Rivlin hyperelastic model, with dimensions (J / m 3 ), where From formula 11, From formula 24, J m From formula 2, and are the material parameters related to the distortion and volume deformation of the polymer matrix, both in MPa. m is the left Cauchy Green tensor of the polymer matrix phase, dimensionless, where F m Obtained from formula 1; in formula 27 is the nominal stress of the polymer matrix, with the dimension of MPa, where J m From formula 2, we can get m From formula 1, From formula 10, we can get m From formula 26, we can get and are the material parameters related to the distortion and volume deformation of the polymer matrix, both in MPa. m is the Cauchy stress of the polymer matrix, with the dimension of MPa, where J m From formula 2, From formula 10, we can get m From formula 26, From formula 24, we can obtain and are the material parameters related to the distortion and volume deformation of the polymer matrix, both in MPa. m is the equivalent pseudo modulus of the polymer matrix, with the dimension of MPa, where From formula 27, we can get m Obtained by formula 1.

[0059] As a preferred solution, the process of establishing the electrical constitutive model of the flexible nanoparticle-reinforced composite material component is as follows:

[0060] Step S3-1. Establishing the electrical constitutive relationship of the coating nanoparticles under large deformation;

[0061] Step S3-2. Establish a nonlinear electrical constitutive relation of the polymer matrix under large deformation considering the stretch-related tunneling effect.

[0062] As a preferred solution, the process of establishing the electrical constitutive relationship of the coating nanoparticles under large deformation is:

[0063] Formula 30: K β =diag{ β ,x β ,x β},β=p or m;

[0064] Formula 31: K int =diag{ int ,x int ,x int};

[0065] Formula 32: J β =K β E β ;

[0066] Formula 33:

[0067] Formula 34:

[0068] Formula 35:

[0069] Formula 36:

[0070] In formulas 30 to 36, K in formula 30 β is the conductivity tensor of the β phase, with the dimension (S / m), where χ β is the isotropic conductivity of the β phase; K in Eq. int is the conductivity tensor of the interface layer, with the dimension (S / m), where χ int is the isotropic conductivity of the interface layer; in Eq. 32, J is the current density tensor, with the dimension (A / m 2 ), where K is obtained in formula 30, E is the electric field strength, dimension is (V / m); in formula 33 is the dimensionless correction for the aspect ratio of nanoparticles, where and is the diagonal component of the nanoparticle deformation gradient tensor, dimensionless, α pis the aspect ratio of the measured nanoparticles, dimensionless; K in Eq. c (F e ) is the conductivity tensor of the coated nanoparticles related to stretching, with the dimension of (S / m), where K m , K p , K int are the measured conductivity of the polymer matrix, nanoparticles and interface layer, c int From formula 16, is the dimensionless depolarization tensor determined by Eq. 53; K in Eq. 35 c (F e ) is the conductivity tensor of the coated nanoparticles related to stretching, with the dimension (S / m), where are the axial and transverse conductivity components of the coating nanoparticles, respectively; in Eq. is the conductivity component of the coating nanoparticles related to stretching, with the dimension (S / m), where c int From formula 16, we can obtain p , χ int are the conductivity components of the nanoparticles and the interface layer, is the dimensionless depolarization tensor component.

[0071] As a preferred solution, the process of establishing the nonlinear electrical constitutive relation of the polymer matrix under large deformation considering the stretch-related tunneling effect is as follows:

[0072] Formula 37:

[0073] Formula 38:

[0074] Formula 39:

[0075] Formula 40:

[0076] Formula 41:

[0077] Formula 42:

[0078] Formula 43: dx tunnel =F e dX tunnel ;

[0079] Formula 44:

[0080] Formula 45:

[0081] Formula 46:

[0082] In formulas 37 to 46, in formula 37 is the initial tunnel distance associated with the nanoparticle, with the dimension of nm, where c p is the volume fraction of nanoparticles, m tunnel is the corresponding parameter of the tunnel distance associated with the nanofiller, with dimensions of nm, n tunnel is the corresponding parameter of the tunnel distance related to the nanofiller, dimensionless; in Eq. 38, d0(c p ) is the initial distance between adjacent nanoparticles, with the dimension of nm, where r p To measure the radius of nanoparticles, the dimension is nm, Obtained from formula 37; in formulas 39 to 41 is the tunnel vector component, dimension is nm, where d0(c p ) is obtained by formula 38, θ, is the orientation angle of the tunnel distance between adjacent nanoparticles; dX in Formula 42 tunnel is the tunnel vector, dimension is nm, where Obtained from equations 39 to 41; in equation 43, dx tunnel is the modified tunnel vector, with dimension nm, where F e From formula 1, dX tunnel Obtained by formula 4; in formula 44 <dx tunnel > is the corrected tunnel vector dx tunnel The integral of , dimensionless, where dx tunnel From formula 43, we can get θ, is the orientation angle of the tunnel distance between adjacent nanoparticles; in Eq. is the average tunnel distance after orientation averaging and stretching, with the dimension of nm, where <dx tunnel From formula 44, we can get r p is the measured radius of the nanoparticle; R in Formula 46 tunnel (F e ) is the tensile resistance of the tunnel region, with the dimension of Ω, where From Equation 45, we can obtain: e is the electron charge, with the dimension of C; h is Planck's constant, with the dimension of (J·s); m e is the mass of the electron, with the dimension of kg, a is the tunnel distance area, with the dimension of (m 2 ), γ is the potential barrier height, dimension is eV, d cutoff is the cutoff distance of the tunnel effect, with the dimension of nm.

[0083] As a preferred solution, the process of establishing the prediction model of the equivalent force electrical coupling performance of the large deformation sensor under large deformation is as follows:

[0084] Formula 47:

[0085] Formula 48: L e =[(1-c p )L m (F m )+c p L c (F c ):A c (F c ,F m )]:[(1-c p )I4+c p A c (F c ,F m )] -1 ;

[0086] Formula 49: c t =q tunnel ·c p ;

[0087] Formula 50:

[0088] Formula 51: K t (F e ) = diag{ t (F e ),x t (F e ),x t (F e )};

[0089] Formula 52:

[0090] Formula 53:

[0091] Formula 54:

[0092]

[0093] Formula 55:

[0094]

[0095] Formula 56:

[0096]

[0097] Formula 57:

[0098]

[0099] Formula 58:

[0100] Formula 59:

[0101] Formula 60:

[0102] In formulas 47 to 60, A in formula 47 c (F c ,F m ) is the extended mechanical concentration tensor of the coating nanoparticles embedded in the polymer matrix, dimensionless, where I4 is the fourth-order invariant tensor, It has been obtained by formula 20, L c (F c ) has been obtained by formula 22, L m (F c ) has been obtained by equation 29; in equation 48, L e is the equivalent pseudo modulus of the flexible nanoparticle reinforced composite deformation sensor, with the dimension of (MPa), where c p is the volume fraction of nanoparticles, L c (F c ) has been obtained by formula 22, L m (F c ) has been obtained by formula 29, A c (F c ,F m ) is obtained by formula 47, I4 is a fourth-order invariant tensor; in formula 49, c t is the volume fraction of the tunnel region in the polymer matrix, dimensionless, where q tunnel is the volume ratio between the tunnel area and the nanoparticles to be fitted, c p is the volume fraction of nanoparticles; in formula 50, t (F e ) is the isotropic conductivity component of the tunnel region, with the dimension of (S / m), It has been obtained in formula 45, R tunnel (F e ) has been obtained in Eq. 46, a is the area of ​​the tunnel region, dimension is (m 2 ); K in Formula 51 t (F e ) is the conductivity tensor of the tunnel region, with the dimension (S / m), where χ t (F e ) has been obtained in Equation 50; in Equation 52 is the equivalent conductivity tensor of the polymer matrix with stretch-related tunneling effect, which is determined by Eq. 54 and has the dimension (S / m), where is the isotropic conductivity component of the polymer matrix with stretch-dependent tunnel distance; in Eq. is the depolarization tensor component, dimensionless, where αt is the slenderness ratio of the tunnel area; in Eq. 54 is the equivalent conductivity tensor of the polymer matrix with stretch-related tunneling effect to be calculated, with the dimension (S / m), where K t (F e ) has been determined by formula 51, c t From formula 49, we can get K m The conductivity of the polymer matrix is ​​measured. Obtained from formula 53; in formula 55 is the isotropic conductivity component of the polymer matrix with stretch-dependent tunneling distance, dimension (S / m), where c t From formula 49, we can obtain t (F e ) has been obtained in Eq. 50, From formula 53, we can obtain m (F e ) is the conductivity of the polymer matrix obtained by measurement; K in Formula 56 e (F e ) is the equivalent conductivity tensor of the overall flexible nanoparticle reinforced composite deformation sensor, with the dimension of (S / m), where c p is the volume fraction of nanoparticles, K c (F e ) has been obtained by formula 34, It has been obtained by formula 54. It has been obtained from Equation 53, I2 is a second-order invariant tensor; in Equation 57, χ e (F e ) is the isotropic component of the equivalent conductivity of the flexible nanoparticle reinforced composite deformation sensor, with the dimension of (S / m), where c p is the volume fraction of nanoparticles, and From formula 36, From formula 55, Calculated by formula 53; in formula 58 is the resistance change ratio of the deformation sensor of the flexible nanoparticle reinforced composite material, dimensionless, where R e (0), R e (F e ) are the equivalent resistances of the flexible nanoparticle reinforced composite deformation sensor in the initial reference configuration and the current deformation configuration, respectively, with the dimension of Ω; in Formula 59, λ x , z are axial tension and transverse tension, dimensionless, respectively, where l is the initial length of the flexible nanocomposite deformation sensor, dimension m, and A is the initial cross-sectional area of ​​the flexible nanocomposite deformation sensor, dimension (m 2 ), χe (F e ) is obtained by formula 57; in formula 60, SF(F e ) is the tensile sensitivity coefficient of the flexible nanoparticle reinforced composite deformation sensor, dimensionless, where λ x , z are axial tension and transverse tension, respectively. e (F e ) is obtained by formula 57, Calculated by formula 58.

[0103] The flexible nanoparticle reinforced composite deformation sensor targeted by the present invention adopts the Neo-Hookean hyperelastic model and the linear electrical constitutive relationship for the nanoparticle phase, and adopts the Mooney-Rivlin hyperelastic model and the linear electrical constitutive relationship for the polymer matrix phase. On a microscopic scale, the nanoparticles are surrounded by an ultra-thin imperfect interface layer to form coating nanoparticles, which can be regarded as a two-phase composite material composed of nanoparticles and an interface layer; on a macroscopic scale, the hyperelastic polymer and the coating nanoparticles constitute an overall binary composite material; the mechanical and electrical constitutive model of the flexible component of the flexible nanoparticle reinforced composite material established by the present invention also fully considers the electron tunneling effect related to stretching, effectively unifies the microscopic and macroscopic scales, and has a higher degree of fit with the actual situation.

[0104] The present invention also provides an application of a prediction model of the equivalent effective electrical coupling performance of a flexible nanoparticle reinforced composite material deformation sensor, which is used to predict the nonlinear tensile sensing capability of the flexible nanoparticle reinforced composite material sensor, and the process is as follows:

[0105] 1) Calculate the initial Hill tensor in the initial undeformed configuration of the flexible nanocomposite deformation sensor and the depolarization tensor

[0106] 2) According to the initial Hill tensor and the depolarization tensor Calculate the initial equivalent pseudo modulus L e (0) and equivalent conductivity K e (0);

[0107] 3) Apply a large loading step to the flexible deformation sensor along the axial direction:

[0108] Equation 61: λ x =λ x +Δλ;

[0109] Equation 62: dσ z =0;

[0110] Equation 63: dFe =diag{dλ x ,dλ z ,dλ z};

[0111] Equation 64: dF m =[(1-c p )I4+c p A c (F c ,F m )] -1 :dF e ;

[0112] Formula 65: dF c =A c (F c ,F m ):dF m ;

[0113] Equation 66: dF int =[c int I4+(1-c int )A p (F c ,F m )] -1 :dF c ;

[0114] Equation 67: dF p =A p (F int ,F p ):dF int ;

[0115] 4) Calculate the Cauchy stress σ of the nanoparticles and polymer matrix in the deformed configuration p , σ m and the equivalent pseudo modulus L of the flexible nanoparticle reinforced composite deformation sensor e (F e );

[0116] 5) Calculate the corrected tunnel distance According to the corrected tunnel distance Update of the equivalent conductivity K of flexible nanoparticle-reinforced composite deformation sensors e (F e );

[0117] 6) Update resistance change ratio and the tensile sensitivity factor SF(F e );

[0118] 7) Calculate the next loading step and repeat step 3) until the final loading is completed. Plot the resistance change ratio and the tensile sensitivity factor SF(F e )

[0119] In equations 61 to 67, λ in equation 61 x is the axial stretch, dimensionless, where Δλ is the axial stretch increment, dimensionless; dσ in Eq. z is the stress in the transverse direction, with the dimension of (MPa); dF in Formula 63 e is the equivalent deformation gradient increment tensor of the overall deformation sensor, dimensionless, where dλ x is the axial tensile increment, dλ z is the lateral displacement increment caused by axial tension, represented by dλ x Determined by equation 62; dF in equation 64 m is the equivalent deformation gradient increment tensor of the polymer matrix, dimensionless, where c p is the volume fraction of nanoparticles, A c (F c ,F m ) has been calculated by formula 47, dF e Obtained from Equation 63; dF in Equation 65 c is the equivalent deformation gradient increment tensor of the coated nanoparticles, dimensionless, where A c (F c ,F m ) has been calculated by formula 47, dF m Calculated by equation 64; dF in equation 66 int is the equivalent deformation gradient increment tensor of the interface layer, dimensionless, dF c Obtained from Equation 65; dF in Equation 67 p is the equivalent deformation gradient increment tensor of the nanoparticle, dimensionless.

[0120] Compared with the prior art, the beneficial technical effects of the technical solution of the present invention are:

[0121] 1) The prediction model provided by the present invention selects pseudo modulus and conductivity as homogenization parameters for stretching sensing analysis, focusing on two functional interface effects related to stretching: incomplete interface connection and electron tunneling effect, and establishes an equivalent electrical coupling performance prediction model based on the extended Mori-Tanaka method and equivalent medium approximation theory under large deformation, which effectively solves the limitation of large computing resource consumption in the current numerical simulation of flexible nanocomposite deformation sensors under large deformation.

[0122] 2) In the technical solution provided by the present invention, the extended Mori-Tanaka method and equivalent medium approximation theory under large deformation are adopted to establish a theoretical prediction model for equivalent force electrical coupling performance under large deformation. By comparing with the calculation results of the small deformation model in the prior art, the current large deformation prediction model has obvious advantages over the small deformation prediction model in predicting the equivalent force electrical coupling performance under large deformation, and is more in line with the actual situation. It solves the limitation that the current small deformation prediction model cannot predict the nonlinear sensing performance of flexible nano-composite deformation sensors under large deformation.

[0123] 3) In the technical solution provided by the present invention, the established equivalent effective electrical coupling performance prediction model is applied to the flexible nanoparticle reinforced composite deformation sensor to effectively predict the nonlinear sensing ability of the flexible nanocomposite deformation sensor. After testing, the resistance change ratio obtained by the equivalent effective electrical coupling performance prediction model under large deformation has a relative error of 5.15% when the volume fraction of the nanoparticles is 0.062%; when the volume fraction is 0.077%, the relative error is 5.12%; when the volume fraction is 0.093%, the relative error is 7.10%.

[0124] 4) In the technical solution provided by the present invention, the main factors affecting the sensing performance of the flexible nanocomposite deformation sensor can be effectively explained by regulating the microscopic parameters. After testing, the key factors that determine the nonlinear sensing performance of the flexible nanocomposite deformation sensor under large deformation are the significant changes in tunnel distance and volume during the stretching process. At the same time, the superelastic properties of the polymer matrix and the increase in the barrier height have a positive effect on the stretching sensing performance of the flexible nanoparticle-reinforced composite deformation sensor. It can provide guidance for optimizing the flexible nanoparticle-reinforced composite sensor in the field of electronic skin. BRIEF DESCRIPTION OF THE DRAWINGS

[0125] Figure 1 Schematic diagram of the flexible deformation sensor fabricated for nanoparticle-reinforced composites under finite tensile load;

[0126] Figure 2 This is a scanning electron microscope image of a flexible silver nanoparticle / polydimethylsiloxane nanocomposite deformation sensor;

[0127] Figure 3 This is a calculation flow chart of the nonlinear sensing performance of the flexible nanoparticle reinforced composite material deformation sensor in an embodiment of the present invention;

[0128] Figure 4 A comparison diagram of the Cauchy stress calculation results and experimental data of the flexible nanoparticle reinforced composite material at different volume fractions in an embodiment of the present invention;

[0129] Figure 5This is a comparison chart of the resistance change ratio calculation results and experimental data of the flexible nanoparticle reinforced composite material at different volume fractions in an embodiment of the present invention;

[0130] Figure 6 This is a comparison chart of the calculated results and experimental data of the tensile sensitivity coefficient of the flexible nanoparticle-reinforced composite material at different volume fractions in an embodiment of the present invention;

[0131] Figure 7 A comparison diagram of the Cauchy stress calculation results and experimental data using a large deformation equivalent force electrical coupling performance prediction model and a small deformation equivalent force electrical coupling performance prediction model in an embodiment of the present invention;

[0132] Figure 8 This is a comparison chart of the resistance change ratio calculation results and experimental data using the large deformation equivalent effective electrical coupling performance prediction model and the small deformation equivalent effective electrical coupling performance prediction model in the embodiment of the present invention. DETAILED DESCRIPTION

[0133] The following specific embodiments are only for detailed explanation of the technical solutions of the present invention. The present invention is not limited to the following embodiments. Those skilled in the art should understand that any improvements and substitutions based on the above principles and spirits on the basis of the present invention should be within the protection scope of the present invention.

[0134] Example 1

[0135] The following is a complete description of the prediction model of the equivalent effective electrical coupling performance of the flexible nanoparticle reinforced composite material deformation sensor under large deformation in combination with this embodiment, which specifically includes the following steps:

[0136] 1. Obtain the microscopic parameters of each component material

[0137] Silver nanoparticles and polydimethylsiloxane were selected as component materials. The aspect ratio of the silver nanoparticles was measured to be α p =1, the radius of the silver nanoparticle is r p =1×10 -8 m, density of silver nanoparticles ρ p =3150kg / m 3 , the density of polydimethylsiloxane ρ m =970kg / m 3 , the conductivity of silver nanoparticles χ p =8.32×10 4 S / m, conductivity of polydimethylsiloxane χ m =1.0×10 -9 S / m; Material parameters of polydimethylsiloxane related to distortion Material parameters of polydimethylsiloxane related to volume deformation Material parameters of silver nanoparticles related to distortion Material parameters of silver nanoparticles related to volume deformation Looking up the table, we get Planck's constant h = 6.626×10 -34 J·s, electron charge e=1.602×10 - 19 C, electron mass m e =9.109×10 -31 kg; the barrier height between adjacent silver nanoparticles is obtained by searching the data, γ = 0.17 eV, and the cutoff distance d of the tunnel effect is cutoff =8nm.

[0138] 2. Preparation of silver nanoparticle / polydimethylsiloxane composite flexible deformation sensor samples

[0139] Silver nanoparticles and polydimethylsiloxane were used as raw materials and a drop casting method was used to prepare silver nanoparticle / polydimethylsiloxane composite flexible deformation sensors with silver nanoparticle volume fractions (vol%) of 0.062%, 0.077%, and 0.093%, respectively.

[0140] 3. Experimental data on the mechanical and electrical properties of deformation sensors made of silver nanoparticles / polydimethylsiloxane composite materials

[0141] The mechanical performance experimental data of the silver nanoparticle / polydimethylsiloxane composite deformation sensor were obtained by tensile testing with a universal mechanical testing machine. The electrical performance experimental data of the silver nanoparticle / polydimethylsiloxane composite deformation sensor were obtained by measuring the electrical performance of the silver nanoparticle / polydimethylsiloxane sensor with a PRS-812 resistor meter. The mechanical and electrical performance experimental data will be used in step 4 to determine the remaining parameters of the prediction model of the equivalent electrical coupling performance of the flexible nanoparticle reinforced composite deformation sensor under large deformation.

[0142] 4. Determine the remaining parameters of the prediction model of the equivalent electro-coupling performance of the flexible nanoparticle reinforced composite deformation sensor under large deformation

[0143] The experimental data of mechanical and electrical properties of some nanoparticle / polydimethylsiloxane composite flexible deformation sensors were introduced into the equivalent electrical coupling performance prediction model established in step 1, and the remaining model parameters were determined by data fitting, including the interface layer conductivity χ int =1.0×10 -3 S / m; the slenderness ratio of the tunnel area is α t = 50; parameter m of the tunnel distance associated with the nanofiller tunnel =2.07×10 -14 nm; parameter n for the tunnel distance associated with the nanofillertunnel =1.48; the volume ratio of the tunnel area to the silver nanoparticles q tunnel =17; Material parameters related to interface layer and distortion Material parameters related to interface layer and volume deformation Interface layer thickness d int =0.001nm; tunnel area a = 2×10 -3 m 2 .

[0144] 5. Experimental data on measuring nonlinear electromechanical sensing performance

[0145] The resistance of the silver nanoparticle / polydimethylsiloxane sensor sheet was measured by a PRS-812 resistor meter, and the tensile test was performed by a universal mechanical testing machine. The sensing characteristics of the sensor were measured by connecting the two ends of the sample to the two ends of the machine, and then applying a uniform stretching / releasing cycle at a loading speed of 10 mm / min at room temperature, and the resistance change ratio and stretching sensitivity coefficient during the stretching process were obtained. The Cauchy stress experimental data under different volume fractions are shown in Figure 1. Figure 4 The experimental data of resistance change ratio and stretch sensitivity coefficient at different volume fractions are shown in the circular data points and square data points. Figure 5 , 6 As shown in the figure, there are circle data points, square data points and triangle data points.

[0146] 6. Predicting the nonlinear tensile sensing performance of flexible nanoparticle-reinforced composite deformation sensors

[0147] 1) According to the prediction model of equivalent force electrical coupling performance under large deformation established in step 1, the determined prediction model parameters of equivalent force electrical coupling performance are introduced;

[0148] 2) In the initial undeformed configuration of the flexible nanocomposite deformation sensor, the initial Hill tensor is calculated by Eq. 20 and Eq. 53: and the depolarization tensor

[0149] 3) According to the initial Hill tensor and the depolarization tensor The initial equivalent pseudo modulus L is calculated by equations 48 to 56. e (0) and equivalent conductivity K e (0);

[0150] 4) According to equations 61 to 67, a loading step with a larger deformation is applied to the flexible deformation sensor along the axial direction;

[0151] 5) The Cauchy stress σ of the nanoparticles and the polymer matrix in the deformed configuration is calculated by using Eq. 14, Eq. 28 and Eq. 48: p , σm and the equivalent pseudo modulus L of the flexible nanoparticle reinforced composite deformation sensor e (F e );

[0152] 6) Calculate the corrected tunnel distance using equation 45 According to the corrected tunnel distance Update the equivalent conductivity K of the flexible nanoparticle reinforced composite deformation sensor by Equation 56 e (F e );

[0153] 7) Update the resistance change ratio according to equations 58 to 60 and the tensile sensitivity factor SF(F e );

[0154] 8) Calculate the next loading step, repeat step 3) until the final loading is completed, and plot the resistance change ratio and the tensile sensitivity factor SF(F e ) curve graph.

[0155] 7. Verify the correctness of the prediction method of the equivalent electrical coupling performance of flexible nanoparticle reinforced composite deformation sensors under large deformation

[0156] By applying the established prediction model of equivalent electrical coupling performance under large deformation, the Cauchy stress calculation results and experimental results of the flexible nanoparticle reinforced composite deformation sensor are compared under the conditions of 0% and 3.32% nanoparticle volume fraction. Figure 4 As shown, when the volume fraction is 0%, the relative error is 7.2%; when the volume fraction is 3.32%, the relative error is 5.2%.

[0157] By applying the established prediction model for equivalent electrical coupling performance under large deformation, the resistance change ratio of flexible nanoparticle reinforced composites is compared with the calculated results and experimental results under the conditions of volume fractions of 0.062%, 0.077% and 0.093%. Figure 5 As shown in Figure 2, the relative errors are 5.15%, 5.12%, and 7.10% respectively; the comparison between the tensile sensitivity coefficient and the experimental results is shown in Figure 2. Figure 6 As shown, the relative errors are 5.11%, 3.63% and 9.16% respectively.

[0158] The calculation results of the above equivalent electrokinetic coupling performance prediction model show that the nonlinear sensing performance of the flexible nanoparticle-reinforced composite deformation sensor can be predicted based on the model of the present invention, and the nonlinear sensing ability of the flexible composite deformation sensor can be accurately predicted; it effectively solves the limitation that the current numerical simulation of the flexible nanoparticle-reinforced composite deformation sensor under large deformation consumes a lot of computing resources, and can provide guidance for optimizing the flexible nanoparticle-reinforced composite sensor in the field of electronic skin.

[0159] Comparative Example 1

[0160] The following will describe the prediction model of the equivalent force electrical coupling performance of a flexible nanoparticle reinforced composite deformation sensor of the present invention in combination with this comparative example. The initial data and parameters of the deformation sensor used in this comparative example are exactly the same as those in Example 1, except that the model establishment process is carried out in accordance with the published paper Xia, X., et al. "Aunified investigation into the tensile and compressive sensing performancein highly sensitive MWCNT / epoxy nanocomposite strain sensor through loading-dependent tunneling distance." Composites Science and Technology (2022). The final prediction model of the equivalent force electrical coupling properties under small deformation is as follows: the Cauchy stress at a volume fraction of 3.32% is Figure 7 As shown, the resistance change when the volume fraction is 0.078% is as follows Figure 8 shown.

[0161] By comparing the calculation results of the model provided in Example 1 of the present invention with the calculation results of the model provided in Comparative Example 1, it can be concluded that the technical solution adopted by the present invention has the following advantages:

[0162] Comparison of the Cauchy stress calculation results and experimental data obtained by the prediction model of equivalent force electrical coupling properties under large deformation and the prediction model of equivalent force electrical coupling properties under small deformation. Figure 7 As shown, compared with the previous small deformation model, the relative error of the Cauchy stress predicted by the technical solution of the present invention is reduced from 191% under the small deformation equivalent effective electrical coupling performance prediction model in comparative example 1 to 5.7%, and the relative error of the small deformation equivalent effective electrical coupling performance prediction model in comparative example 1 continues to increase with the increase of stretching.

[0163] It can be seen that the Cauchy stress calculation results of the small deformation equivalent force electrical coupling performance prediction model increase linearly with axial stretching, which is significantly higher than the experimental data; the Cauchy stress calculation results of the large deformation equivalent force electrical coupling performance prediction model proposed in the present invention are consistent with the experiment. When the axial stretching growth rate decreases, the predicted Cauchy stress increases nonlinearly with axial stretching.

[0164] In addition, the resistance change ratio results predicted by the prediction model of equivalent electrical coupling performance under large deformation and the prediction model of equivalent electrical coupling performance under small deformation are compared with the experimental data. Figure 8 As shown, the relative error of the predicted resistance change ratio is reduced from 109% under the original prediction model of the equivalent effective electrical coupling properties under small deformation to 5.15% under the prediction model of the equivalent effective electrical coupling properties under large deformation, which can effectively predict the nonlinear sensing performance of flexible nanoparticle reinforced composite deformation sensors under large deformation.

[0165] It can be seen that in the prediction model of the equivalent force electrical coupling properties under small deformation, the resistance change ratio result is significantly higher than the experimental result; the resistance change ratio predicted by the prediction model of the equivalent force electrical coupling properties under large deformation described in this method is consistent with the experimental data. The equivalent resistance change increases nonlinearly with the continuous increase of axial stretching.

[0166] The prediction model of equivalent force electrical coupling performance under large deformation provided by the present invention has good convergence and robustness, and the predicted nonlinear sensing performance of the flexible nanoparticle reinforced composite material deformation sensor is more in line with the actual situation. Therefore, the prediction model of equivalent force electrical coupling performance under large deformation of the flexible nanoparticle reinforced composite material deformation sensor established by the present invention is superior to the previous prediction model of equivalent force electrical coupling performance under small deformation.

Claims

1. A prediction model for the equivalent electrical coupling performance of flexible nanoparticle reinforced composite deformation sensors, Features: Step S1. Obtaining characteristic parameters of a prediction model for equivalent electrical coupling performance of a flexible deformation sensor under large deformation; Step S2. establishing a mechanical constitutive model of the flexible nanoparticle reinforced composite material component according to the characteristic parameters; Step S3. establishing an electrical constitutive model of the flexible nanoparticle reinforced composite material component according to the characteristic parameters; Step S4. Establish an equivalent force electrical coupling performance prediction model of the flexible deformation sensor under large deformation based on the mechanical and electrical constitutive models.

2. The prediction model of the equivalent electrical coupling performance of the flexible nanoparticle reinforced composite deformation sensor according to claim 1, Features: The characteristic parameters include: the aspect ratio α of the nanoparticles p , the slenderness ratio of the tunnel area α t , the radius of the nanoparticle r p , the mass density of the nanoparticles ρ p , mass density of polydimethylsiloxane polymer ρ m , the conductivity of the nanoparticles p , the conductivity of polydimethylsiloxane χ m , the conductivity of the interface layer χ int , the nanofiller-related tunnel distance parameter m tunnel 、n tunnel , the volume ratio of the tunnel area to the nanoparticle q tunnel , material parameters of polydimethylsiloxane related to distortion Material parameters of polydimethylsiloxane related to volume deformation Material parameters related to distortion of nanoparticles Material parameters of nanoparticles related to volume deformation Material parameters related to interface layer and distortion Material parameters related to interface layer and volume deformation Interface layer thickness d int , tunnel area a, barrier height γ between adjacent nanoparticles, Planck constant h, electron charge e, electron mass m e , the cutoff distance d of the tunnel effect cutoff .

3. The prediction model for the equivalent electrical coupling performance of a flexible nanoparticle reinforced composite deformation sensor according to claim 2, Features: The process of establishing the mechanical constitutive model of the flexible nanoparticle reinforced composite material component is as follows: Step S2-1. Establish a nonlinear hyperelastic constitutive relationship, then establish a hyperelastic mechanical constitutive relationship of nanoparticles under large deformation based on the relationship, and then construct a hyperelastic mechanical constitutive relationship of the coating nanoparticles in the flexible deformation sensor under large deformation; Step S2-2. Establish the nonlinear mechanical constitutive relationship of the polymer matrix in the flexible nanoparticle reinforced composite material under large deformation.

4. The prediction model for the equivalent electrical coupling performance of a flexible nanoparticle reinforced composite deformation sensor according to claim 3, Features: The process of establishing the nonlinear hyperelastic constitutive relation is as follows: Formula 1: Formula 2: J = detF; Formula 3: C = F T F; Formula 4: Formula 5: Formula 6: Formula 7: Formula 8: Formula 9: In formulas 1 to 9: F in formula 1 e is the equivalent deformation gradient of the flexible nanoparticle reinforced composite deformation sensor, dimensionless, and the subscript e represents the overall equivalent performance of the flexible nanoparticle reinforced composite deformation sensor; in formula 2, J is the volume ratio defined by the equivalent deformation gradient, dimensionless, and F is obtained by formula 1; in formula 3, C is the uncorrected right Cauchy Green tensor, dimensionless, and F is obtained by formula 1, F T is the transpose of F; in formula 4 is the modified right Cauchy Green tensor, dimensionless, where J is obtained from Equation 2 and C is obtained from Equation 3; β = p is the nanoparticle phase, dimensionless; β = m is the polymer matrix phase, dimensionless; Ψ in Equation 5 β (F β ) is the Helmholtz free energy, with the dimension (J / m 3 ); is the isotropic part of the Helmholtz free energy, with dimensions (J / m 3 ); is the volume part of the Helmholtz free energy, with the dimension (J / m 3 );S in Formula 6 β is the second-kind Piola-Kirchhoff stress tensor, with the dimension of MPa, where Ψ β (F β ) is obtained from formula 5, J β From formula 2, C β From formula 3, Obtained from formula 4; in formula 7 is the nominal stress, dimension is MPa, where S β From formula 6, Obtained from formula 1; in formula 8, σ β is the Cauchy stress, with the dimension of MPa, In the formula From formula 7, we can get β From formula 1, J β Obtained from formula 2; L in formula 9 β is the equivalent pseudo modulus of the β phase, with the dimension of MPa, where Ψ β (F β ) is obtained by formula 5, From formula 1, Obtained by formula 7; The process of establishing the hyperelastic mechanical constitutive relation of the nanoparticles under large deformation is as follows: the nonlinear mechanical constitutive relation of the nanoparticle phase using the Neo-Hookean hyperelastic model is established through the nonlinear hyperelastic constitutive relation of the coating nanoparticles, and the calculation process is as follows: Formula 10: Formula 11: Formula 12: Formula 13: Formula 14: Formula 15: In formulas 10 to 15: In formula 10 is the first invariant of the right Cauchy-Green tensor, dimensionless, where C β Obtained from formula 3; in formula 11 is the first invariant of the modified right Cauchy-Green tensor, dimensionless, where From formula 10, J β Obtained from formula 2; in formula 12, Ψ p (F p ) is the Helmholtz free energy of the nanoparticle phase using the Neo-Hookean hyperelastic model, with the dimension of (J / m 3 ), where From formula 11, J β From formula 2, and are the material parameters of the nanoparticles related to distortion and volume deformation, both in TPa. is the nominal stress of the nanoparticle phase, with the dimension of TPa, where J β From formula 2, we can get p From formula 1, From formula 10, and are the material parameters related to the distortion and volume deformation of the nanoparticles measured, both in TPa. p is the Cauchy stress of the nanoparticle phase, with the dimension of TPa, where J β From formula 2, we can get p From formula 1, From formula 10, I 2 is a second-order invariant tensor, and are the material parameters related to the distortion and volume deformation of the nanoparticles, both in TPa. p is the equivalent pseudo modulus of the nanoparticles, with the dimension of TPa, where J β From formula 2, we can get p From formula 1, From formula 10, I 4 is a fourth-order invariant tensor, and They are the material parameters related to the distortion and volume deformation of the nanoparticles measured, both in TPa. The calculation process of the hyperelastic constitutive relation of the coating nanoparticles under large deformation is: Formula 16: Formula 17: Formula 18: Formula 19: Formula 20: Formula 21: Formula 22: L c (F c )=[c int L int (F int )+(1-c int )L p (F p ):A p (F int ,F p )]: [c int I 4 +(1-c int )A p (F int ,F p )] -1 ; In formulas 16 to 22, in formula 16, c int is the volume fraction of the interface layer in the coating nanoparticles, dimensionless, where d int and r p are the thickness of the ultra-thin interface layer and the radius of the nanoparticle, both in nm; in Eq. is a unit vector, dimensionless; in formula 18 is the parameter in the Hill tensor, dimension is (MPa), where From formula 17, L AiBj is the stiffness matrix; in Eq. 19 is the parameter in the Hill tensor, with the dimension (MPa -1 ), where Obtained by formula 18; P in formula 20 iAjB is the extended Hill tensor, dimensionless, where for The unit circle in space, Obtained by formula 17; A in formula 21 p (F int ,F p ) is the extended mechanical concentration tensor of nanoparticles embedded in the ultrathin interface layer, dimensionless, where From formula 20, we can get p (F p ) and L int (F int ) are obtained by formula 15, I 4 is a fourth-order invariant tensor; L in formula 22 c (F c ) is the equivalent pseudo modulus of the coated nanoparticles, with the dimension of TPa, where c int From formula 16, we can get p (F p ) and L int (F int ) is obtained by formula 15, A p (F int ,F p ) is obtained by formula 21, I 4 is a fourth-order invariant tensor.

5. The prediction model for the equivalent electrical coupling performance of a flexible nanoparticle reinforced composite deformation sensor according to claim 4, Features: The process of establishing the nonlinear mechanical constitutive relationship of the polymer matrix under large deformation is as follows: Formula 23: Formula 24: Formula 25: Formula 26: Formula 27: Formula 28: Formula 29: In formulas 23 to 29, in formula 23 is the second invariant of the right Cauchy Green tensor, dimensionless, where C β Obtained by formula 3; in formula 24 is the second invariant of the modified right Cauchy Green tensor, dimensionless, where From formula 23, J β Obtained from formula 2; in formula 25, Ψ m (F m ) is the Helmholtz free energy of the polymer matrix using the Mooney-Rivlin hyperelastic model, with dimensions (J / m 3 ), where From formula 11, From formula 24, J m From formula 2, and are the material parameters related to the distortion and volume deformation of the polymer matrix, both in MPa. m is the left Cauchy Green tensor of the polymer matrix phase, dimensionless, where F m Obtained from formula 1; in formula 27 is the nominal stress of the polymer matrix, with the dimension of MPa, where J m From formula 2, we can get m From formula 1, From formula 10, we can get m From formula 26, we can get and are the material parameters related to the distortion and volume deformation of the polymer matrix, both in MPa. m is the Cauchy stress of the polymer matrix, with the dimension of MPa, where J m From formula 2, From formula 10, we can get m From formula 26, we can get From formula 24, we can obtain and are the material parameters related to the distortion and volume deformation of the polymer matrix, both in MPa. m is the equivalent pseudo modulus of the polymer matrix, with the dimension of MPa, where From formula 27, we can get m Obtained by formula 1.

6. The prediction model for the equivalent electrical coupling performance of a flexible nanoparticle reinforced composite deformation sensor according to claim 2, Features: The process of establishing the electrical constitutive model of the flexible nanoparticle reinforced composite material component is as follows: Step S3-1. Establishing the electrical constitutive relationship of the coating nanoparticles under large deformation; Step S3-2. Establishing a nonlinear electrical constitutive relation of the polymer matrix under large deformation considering the stretch-related tunneling effect.

7. The prediction model for the equivalent electrical coupling performance of a flexible nanoparticle reinforced composite deformation sensor according to claim 6, Features: The process of establishing the electrical constitutive relationship of the coating nanoparticles under large deformation is as follows: Equation 30: K β = diag{χ β , χ β , χ β}, β = p or m; Equation 31: K int = diag{χ int , χ int , χ int}; Formula 32: J β =K β E β ; Formula 33: Formula 34: Formula 35: Formula 36: In formulas 30 to 36, K in formula 30 β is the conductivity tensor of the β phase, with the dimension (S / m), where χ β is the isotropic conductivity of the β phase; K in Eq. int is the conductivity tensor of the interface layer, with the dimension (S / m), where χ int is the isotropic conductivity of the interface layer; in Eq. 32, J is the current density tensor, with the dimension (A / m 2 ), where K is obtained in formula 30, E is the electric field strength, dimension is (V / m); in formula 33 is the dimensionless correction for the aspect ratio of nanoparticles, where and is the diagonal component of the nanoparticle deformation gradient tensor, dimensionless, α p is the aspect ratio of the measured nanoparticles, dimensionless; K in Eq. c (F e ) is the conductivity tensor of the coated nanoparticles related to stretching, with the dimension of (S / m), where K m , K p , K int are the measured conductivity of the polymer matrix, nanoparticles and interface layer, c int From formula 16, is the dimensionless depolarization tensor determined by Eq. 53; K in Eq. 35 c (F e ) is the conductivity tensor of the coating nanoparticles related to stretching, with the dimension of (S / m), where are the axial and transverse conductivity components of the coating nanoparticles, respectively; in Eq. is the conductivity component of the coating nanoparticles related to stretching, with the dimension (S / m), where c int From formula 16, we can get p , χ int are the conductivity components of the nanoparticles and the interface layer, is the dimensionless depolarization tensor component.

8. The prediction model for the equivalent electrical coupling performance of a flexible nanoparticle reinforced composite deformation sensor according to claim 6, Features: The process of establishing the nonlinear electrical constitutive relation of the polymer matrix under large deformation considering the stretch-related tunneling effect is as follows: Formula 37: Formula 38: Formula 39: Formula 40: Formula 41: Formula 42: Equation 43: dx tunnel = F e dX tunnel ; Formula 44: Formula 45: Formula 46: In formulas 37 to 46, in formula 37 is the initial tunnel distance associated with the nanoparticle, with the dimension of nm, where c p is the volume fraction of nanoparticles, m tunnel is the corresponding parameter of the tunnel distance associated with the nanofiller, with dimensions of nm, n tunnel is the corresponding parameter of the tunnel distance associated with the nanofiller, dimensionless; d in Eq. 0 (c p ) is the initial distance between adjacent nanoparticles, with the dimension of nm, where r p To measure the radius of nanoparticles, the dimension is nm, Obtained from formula 37; in formulas 39 to 41 is the tunnel vector component, dimension is nm, where d 0 (c p ) is obtained by formula 38, θ, is the orientation angle of the tunnel distance between adjacent nanoparticles; dX in Formula 42 tunnel is the tunnel vector, dimension is nm, where Obtained from equations 39 to 41; in equation 43, dx tunnel is the modified tunnel vector, with dimension nm, where F e From formula 1, dX tunnel Obtained by formula 4; in formula 44 <dx tunnel > is the corrected tunnel vector dx tunnel The integral of , dimensionless, where dx tunnel From formula 43, we can get θ, is the orientation angle of the tunnel distance between adjacent nanoparticles; in Eq. is the average tunnel distance after orientation averaging and stretching, with the dimension of nm, where <dx tunnel >According to formula 44, r p is the measured radius of the nanoparticle; R in Formula 46 tunnel (F e ) is the tensile resistance of the tunnel region, with the dimension of Ω, where From Equation 45, we can obtain: e is the electron charge, with the dimension of C; h is Planck's constant, with the dimension of (J·s); m e is the mass of the electron, with the dimension of kg, a is the tunnel distance area, with the dimension of (m 2 ), γ is the potential barrier height, dimension is eV, d cutoff is the cutoff distance of the tunnel effect, with the dimension of nm.

9. The prediction model for the equivalent electrical coupling performance of a flexible nanoparticle reinforced composite deformation sensor according to claim 1, Features: The process of establishing the prediction model of the equivalent electrical coupling performance of the large deformation sensor under large deformation is as follows: Formula 47: Equation 48: L e = [(1 - c p )L m (F m ) + c p L c (F c ): A c (F c , F m )]: [(1 - c p )I 4 + c p A c (F c , F m )] -1 ; Formula 49: c t =q tunnel ·c p ; Formula 50: Equation 51: K t (F e ) = diag{χ t (F e ), χ t (F e ), χ t (F e )}; Formula 52: Formula 53: Formula 54: Formula 55: Formula 56: Formula 57: Formula 58: Formula 59: Formula 60: In formulas 47 to 60, A in formula 47 c (F c ,F m ) is the extended mechanical concentration tensor of the coating nanoparticles embedded in the polymer matrix, dimensionless, where I 4 is a fourth-order invariant tensor, It has been obtained by formula 20, L c (F c ) has been obtained by formula 22, L m (F c ) has been obtained by equation 29; in equation 48, L e is the equivalent pseudo modulus of the flexible nanoparticle reinforced composite deformation sensor, with the dimension of (MPa), where c p is the volume fraction of nanoparticles, L c (F c ) has been obtained by formula 22, L m (F c ) has been obtained by formula 29, A c (F c ,F m ) is obtained by formula 47, I 4 is a fourth-order invariant tensor; in formula 49, c t is the volume fraction of the tunnel region in the polymer matrix, dimensionless, where q tunnel is the volume ratio between the tunnel area and the nanoparticles to be fitted, c p is the volume fraction of nanoparticles; in formula 50, t (F e ) is the isotropic conductivity component of the tunnel region, with the dimension of (S / m), It has been obtained in formula 45, R tunnel (F e ) has been obtained in Eq. 46, a is the area of ​​the tunnel region, dimension is (m 2 ); K in Formula 51 t (F e ) is the conductivity tensor of the tunnel region, with the dimension (S / m), where χ t (F e ) has been obtained in Equation 50; in Equation 52 is the equivalent conductivity tensor of the polymer matrix with stretch-related tunneling effect, which is determined by Eq. 54 and has the dimension (S / m), where is the isotropic conductivity component of the polymer matrix with stretch-dependent tunnel distance; in Eq. is the depolarization tensor component, dimensionless, where α t is the slenderness ratio of the tunnel area; in Eq. 54 is the equivalent conductivity tensor of the polymer matrix with stretch-related tunneling effect to be calculated, with the dimension (S / m), where K t (F e ) has been determined by formula 51, c t From formula 49, we can get K m The conductivity of the polymer matrix is ​​measured. Obtained from formula 53; in formula 55 is the isotropic conductivity component of the polymer matrix with stretch-dependent tunneling distance, dimension (S / m), where c t From formula 49, we can obtain t (F e ) has been obtained in Eq. 50, From formula 53, we can obtain m (F e ) is the conductivity of the polymer matrix obtained by measurement; K in Formula 56 e (F e ) is the equivalent conductivity tensor of the overall flexible nanoparticle reinforced composite deformation sensor, with the dimension of (S / m), where c p is the volume fraction of nanoparticles, K c (F e ) has been obtained by formula 34, It has been obtained by formula 54, It has been obtained by formula 53, I 2 is a second-order invariant tensor; in formula 57, e (F e ) is the isotropic component of the equivalent conductivity of the flexible nanoparticle reinforced composite deformation sensor, with the dimension of (S / m), where c p is the volume fraction of nanoparticles, and From formula 36, From formula 55, Calculated by formula 53; in formula 58 is the resistance change ratio of the deformation sensor of the flexible nanoparticle reinforced composite material, dimensionless, where R e (0), R e (F e ) are the equivalent resistances of the flexible nanoparticle reinforced composite deformation sensor in the initial reference configuration and the current deformation configuration, respectively, with the dimension of Ω; in Formula 59, λ x , z are axial tension and transverse tension, dimensionless, respectively, where l is the initial length of the flexible nanocomposite deformation sensor, dimension m, and A is the initial cross-sectional area of ​​the flexible nanocomposite deformation sensor, dimension (m 2 ), χ e (F e ) is obtained by formula 57; in formula 60, SF(F e ) is the tensile sensitivity coefficient of the flexible nanoparticle reinforced composite deformation sensor, dimensionless, where λ x , z are axial tension and transverse tension, respectively. e (F e ) is obtained by formula 57, Calculated by formula 58.

10. Application of a prediction model for the equivalent electrical coupling performance of a flexible nanoparticle reinforced composite material deformation sensor according to any one of claims 1 to 9, Features: The process used to predict the nonlinear stretch sensing capability of flexible nanoparticle-reinforced composite sensors is as follows: 1) Calculate the initial Hill tensor in the initial undeformed configuration of the flexible nanocomposite deformation sensor and the depolarization tensor 2) According to the initial Hill tensor and the depolarization tensor Calculate the initial equivalent pseudo modulus L e (0) and equivalent conductivity K e (0); 3) Apply a large loading step to the flexible deformation sensor along the axial direction: Equation 61: λ x = λ x + Δλ; Equation 62: dσ z = 0; Equation 63: dF e = diag{dλ x , dλ z , dλ z}; Equation 64: dF m = [(1 - c p )I 4 + c p A c (F c , F m )] -1 : dF e ; Equation 65: dF c = A c (F c , F m ): dF m ; Equation 66: dF int = [c int I 4 + (1 - c int ) A p (F c , F m )] -1 : dF c ; Equation 67: dF p = A p (F int , F p ): dF int ; 4) Calculate the Cauchy stress σ of the nanoparticles and polymer matrix in the deformed configuration p , σ m and the equivalent pseudo modulus L of the flexible nanoparticle reinforced composite deformation sensor e (F e ); 5) Calculate the corrected tunnel distance According to the corrected tunnel distance Update of equivalent conductivity K of flexible nanoparticle-reinforced composite deformation sensors e (F e ); 6) Update resistance change ratio and the tensile sensitivity factor SF(F e ); 7) Calculate the next loading step and repeat step 3) until the final loading is completed. Plot the resistance change ratio and the tensile sensitivity factor SF(F e ) In equations 61 to 67, λ in equation 61 x is the axial stretch, dimensionless, where Δλ is the axial stretch increment, dimensionless; dσ in Eq. z is the stress in the transverse direction, with the dimension of (MPa); dF in Formula 63 e is the equivalent deformation gradient increment tensor of the overall deformation sensor, dimensionless, where dλ x is the axial tensile increment, dλ z is the lateral displacement increment caused by axial tension, represented by dλ x Determined by equation 62; dF in equation 64 m is the equivalent deformation gradient increment tensor of the polymer matrix, dimensionless, where c p is the volume fraction of nanoparticles, A c (F c ,F m ) has been calculated by formula 47, dF e Obtained from Equation 63; dF in Equation 65 c is the equivalent deformation gradient increment tensor of the coated nanoparticles, dimensionless, where A c (F c ,F m ) has been calculated by formula 47, dF m Calculated by equation 64; dF in equation 66 int is the equivalent deformation gradient increment tensor of the interface layer, dimensionless, dF c Obtained from Equation 65; dF in Equation 67 p is the equivalent deformation gradient increment tensor of the nanoparticle, dimensionless.

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