A method and device for predicting the mechanical response of liquid crystal elastomers

Abstract

The present invention discloses a method and device for predicting the mechanical response of liquid crystal elastomers, belonging to the technical field of material property prediction. The method comprises the following steps: constructing a free energy density function of the liquid crystal elastomer; calculating thermal spontaneous deformation and mechanical deformation; determining an order parameter affected by temperature and external load; constructing a rotational evolution equation for the director of the liquid crystal elastomer to predict the dynamic response of the microscopic liquid crystal molecules in the liquid crystal elastomer; and constructing a constitutive equation for the liquid crystal elastomer to conduct a uniaxial tensile test to predict its mechanical response. A prediction device is also provided based on the prediction method. The present invention is suitable for predicting the macroscopic mechanical properties and microscopic director evolution of monodomain liquid crystal elastomers under different temperatures and loading conditions. It can accurately and efficiently obtain the mechanical response of liquid crystal elastomers, enabling efficient utilization and design of liquid crystal elastomers.

Description

Technical Field

[0001] The present invention relates to the technical field of material property prediction, and in particular to a method and device for predicting the mechanical response of a liquid crystal elastomer. Background Art

[0002] Liquid crystal elastomers (LCEs) are a new type of intelligent soft material that combines the flexibility of traditional elastomers with the nematic order of liquid crystal molecules, resulting in unique and excellent properties. Under external stimuli, the liquid crystal molecules in LCEs can undergo a phase transition between the nematic phase and the isotropic phase, resulting in significant and reversible shape changes. At the same time, the rearrangement and distribution of LC molecules alter their microstructure and mechanical response. For example, LCEs exhibit soft elastic properties, with an elastic modulus close to zero during stretching and excellent energy dissipation capabilities. Monodomain LCEs exhibit significant differences in mechanical response and material stiffness under different LC molecule orientations, which is completely different from the isotropic mechanical properties of traditional elastomers.

[0003] Liquid crystal elastomers (LCEs) can deform in response to external stimuli such as heat, light, and electricity, but thermal actuation is currently the most effective method. Under the coupled effects of heat and force, LCEs undergo macroscopic deformation and microscopic rotational evolution of the LC director, resulting in a mechanical response characterized by strong nonlinear characteristics. Currently, the prediction of the macroscopic and microscopic mechanical properties of single-domain LCEs remains difficult, and there is a lack of efficient and reliable computational tools to predict their behavior under different conditions, which poses challenges to their application.

[0004] In response to the current shortcomings of single-domain liquid crystal elastomers in the prediction of macro- and micro-mechanical properties, as well as the lack of efficient and reliable tools to predict the performance of liquid crystal elastomers under different conditions, the present invention proposes a new multi-field coupling prediction method. Summary of the Invention

[0005] The purpose of the present invention is to provide a method and device for predicting the mechanical response of liquid crystal elastomers to solve the problems in the background technology.

[0006] To achieve the above object, the present invention provides a method for predicting the mechanical response of a liquid crystal elastomer, comprising the following steps:

[0007] S1. constructing a free energy density function of a liquid crystal elastomer, wherein the free energy density function includes the elastic energy of the cross-linked polymer network in the liquid crystal elastomer, the Landau free energy of liquid crystal molecule rotation and phase transition, and the anisotropic free energy caused by the difference in liquid crystal orientation;

[0008] S2. Determine the deformation of the liquid crystal elastomer under the influence of temperature and calculate the corresponding thermal spontaneous deformation and mechanical deformation;

[0009] S3. Calculate the Landau free energy of the liquid crystal elastomer affected by temperature and external load, and obtain the order parameter of the liquid crystal elastomer under the corresponding state;

[0010] S4. Average the orientation of the liquid crystal molecules in the liquid crystal elastomer into the director orientation, construct the rotation evolution equation of the director, and predict the dynamic response of the microscopic liquid crystal molecules in the liquid crystal elastomer;

[0011] S5. Based on the free energy function, director and order parameter of liquid crystal elastomer, the constitutive relation of liquid crystal elastomer under thermal-mechanical coupling conditions is constructed;

[0012] S6. Expand the Jaumann derivative of stress in the constitutive relation to obtain the tangent stiffness matrix and implement iterative calculation of stress;

[0013] S7. Perform uniaxial tensile tests on the liquid crystal elastomer at different temperatures and different director orientations, measure relevant material parameters, and calculate the stress state of the liquid crystal elastomer under different load and temperature conditions.

[0014] Preferably, in S1, the liquid crystal elastomer is composed of a cross-linked polymer network and liquid crystal molecules, wherein the rotation of the liquid crystal molecules will generate energy and even cause a phase change, introducing the Landau free energy f nem The energy generated by the rotation of liquid crystal molecules is also called phase change energy. The traditional Landau free energy only considers the effect of temperature on the order parameter. In fact, external force also has an impact on the order parameter. The force-order coupling term is introduced to correct the Landau free energy. The force-order coupling term is:

[0015] The formula for calculating Landau free energy is:

[0016]

[0017] Among them, Q is the order parameter; A0, B and C q is the material parameter related to Q, T * is the assumed second-order phase transition temperature; σ m is the Mises stress, μ is the shear modulus, and γ represents the coupling parameter between the order parameter and the Mises stress;

[0018] The cross-linked polymer network of a monodomain liquid crystal elastomer has high elasticity. At the same time, due to the anisotropic characteristics of liquid crystal elastomers, in order to express the anisotropy affected by orientation, the step length tensor is introduced to represent the influence of liquid crystal orientation on deformation:

[0019]

[0020]

[0021] Among them, l0 and l are the step length tensors in the initial configuration and the current configuration, respectively; d0 and d are the director vectors in the initial configuration and the current configuration, respectively; l ⊥ 、l ∥ are the effective step lengths perpendicular to and parallel to the director, respectively; related to the order parameter, for nematic single-domain liquid crystal elastomers, it is generally l ⊥ =1-Q,l ∥ =1+2Q; I is the unit tensor;

[0022] On the basis of the traditional hyperelastic neo-Hookean elastic energy, the effect of the director rotation on the deformation of the cross-linked polymer network is considered. At the same time, the semi-soft elastic energy is introduced to represent the mutual coupling between the cross-linked polymer network and the liquid crystal molecules in the liquid crystal elastomer. The elastic energy of the liquid crystal elastomer is obtained as follows:

[0023]

[0024] Where α is the semi-soft elastic parameter, J = det(F) represents the volume change during deformation, K is the bulk modulus, and F is the mechanical deformation gradient.

[0025] The effect of the director on anisotropy is also reflected in the significant differences in mechanical response and modulus for different director orientations. To better describe the anisotropic behavior caused by director orientation, the liquid crystal molecular orientation is used as a reinforcing fiber to introduce the anisotropic free energy, and the anisotropic free energy of the liquid crystal elastomer is obtained:

[0026]

[0027] Where k1 and k2 are material parameters related to the director orientation, I4 = d0·F T F·d0.

[0028] Preferably, in said S2, the deformation gradient F of the liquid crystal elastomer during the thermal-mechanical coupling deformation process is total It is divided into mechanical deformation part and thermal spontaneous deformation part:

[0029] F total =FF s ;

[0030] Among them, F s It is the thermal spontaneous deformation part;

[0031] The thermal spontaneous deformation of liquid crystal elastomers is mainly caused by temperature changes. During the spontaneous deformation process, it is assumed that the director does not rotate, only the shape changes. However, under the action of external forces, the director will rotate and deform. Due to the anisotropic and incompressible characteristics of liquid crystal elastomers, the thermal spontaneous deformation is defined as:

[0032]

[0033] Where S is the spontaneous deformation in the local coordinate system, λ s is the thermal deformation in the direction parallel to the director, λ s =1+G(T), G(T) is the thermal strain related to temperature, i i (i=1,2,3) is the unit vector in the main direction, R s It is the rotation tensor related to the orientation angle of the director. When only the rotation of the director in the plane is considered, the rotation tensor R s It can be expressed as:

[0034]

[0035] Where θ is the angle between the director and the vertical direction of the loading.

[0036] Preferably, in S3, the order parameter Q represents the average state of the orientation of the liquid crystal molecules. When Q=0, it is in the isotropic state, and when Q=1, it is in the perfect nematic state. In practice, it does not exist. From formula (1), it can be seen that the order parameter of the liquid crystal elastomer is affected by the temperature and stress state. In order to satisfy the thermodynamic equilibrium of the phase transition at the local minimum of the free energy, it is necessary to have:

[0037]

[0038] According to the Landau free energy calculation formula, we can get:

[0039]

[0040] From this we obtain the order parameter Q under different temperature and stress states, 0<Q<1.

[0041] Preferably, in S4, the objectivity of the material requires that the constitutive equation of the director rotation should remain unchanged under the change of the reference, so the Jaumann derivative of the director is used. Ensure that the rotation of the director in different reference frames is physically consistent, where W is the spin tensor, which is an antisymmetric tensor. The rotation of the director is only related to the external force, so the objective rate of the director is It can be expressed as:

[0042]

[0043] Among them, η D is the director-viscous material parameter, η0, k S are material parameters related to the rotation of the director; σ ij is the true stress of each component;

[0044] The rotation evolution equation of the director is expressed as:

[0045]

[0046]

[0047] According to the right-hand rule, the coordinates of the Cartesian coordinate system are established. The angle between the director and the Z axis of the coordinate is ξ, and the angle between its projection on the XY plane and the X axis is θ. The director is defined as:

[0048] d=(sinξcosθ,sinξsinθ,cosξ) T ;

[0049] Taking the derivative of the above formula we can get:

[0050]

[0051] in,

[0052] t1=(coSξcosθ, coSξsinθ, -sinξ) T ;

[0053] t2=(-sinξsinθ, sinξcosθ, 0) T ;

[0054] Then the dynamic evolution equation of the discrete director is:

[0055]

[0056]

[0057] Preferably, in S5, the total free energy density of the liquid crystal elastomer is:

[0058] f=f el +f nem +f ani ;

[0059] The constitutive equation of stress is defined as:

[0060]

[0061] Where σ is the true stress, and sym() indicates symmetry of the variables in the brackets. Therefore, the constitutive equation of the liquid crystal elastomer is expressed as:

[0062]

[0063] in,

[0064] Preferably, in said S6, when predicting the mechanical properties of the liquid crystal elastomer, due to the continuous deformation of the material and the update of the boundary, it is necessary to update the parameters such as stress and director orientation in the liquid crystal elastomer, and it is necessary to iterate these parameters and introduce the tangent stiffness matrix C ijkl , the tangent stiffness matrix is ​​given by Kirchhoff stress τ ij The Jaumann derivative of is determined by:

[0065] δ(τ ij )+τ it δW tj -δW im τ mj =J(C ijkl δD kl );

[0066] Among them, τ ij =Jσ ij , δD ij is the virtual deformation rate tensor, δW ij is the virtual spin tensor;

[0067] Simplifying the above formula, we can get the tangent stiffness matrix expression:

[0068]

[0069] in, (.) RT Indicates transposing the lM in the subscript ijlM;

[0070] Substituting the constitutive equation into the tangent stiffness matrix expression, we can get the tangent stiffness matrix, which is divided into the elastic part (C I ) ijkl and the anisotropic part (C II ) ijkl :

[0071]

[0072]

[0073]

[0074] Preferably, in S7, many material parameters in the constitutive equation formula need to be determined through experiments in order to accurately predict the mechanical response of the liquid crystal elastomer. The uniaxial tensile test is the simplest way to confirm the mechanical properties of the material. It is necessary to carry out uniaxial tensile tests of liquid crystal elastomers in different states to confirm the material parameters: the shear modulus μ is determined by the stress-strain relationship in the small deformation stage, and the calculation formula is:

[0075]

[0076] Among them, F T is the tensile load, A is the cross-sectional area, and λ is the stretching ratio;

[0077] Liquid crystal elastomers are incompressible and have a Poisson's ratio of 0.5, so the bulk modulus K can be converted using parameters such as the Poisson's ratio;

[0078] Since the director rotation and semi-soft elasticity are temperature-dependent, η0,k S , α can be calibrated through uniaxial tensile tests at different temperatures with the director perpendicular to the loading direction;

[0079] k1 and k2 are anisotropic parameters related to the orientation of the director, which can be calibrated through uniaxial tensile tests with the director perpendicular or parallel to the loading direction.

[0080] The present invention also provides a prediction device for the multi-field coupled mechanical response of a single-domain liquid crystal elastomer, comprising an energy calculation unit, a thermal spontaneous deformation calculation unit, an order parameter calculation unit, a director rotation evolution unit, a constitutive relationship calculation unit, and a parameter calibration unit;

[0081] The energy calculation unit is used to construct a free energy density function of the liquid crystal elastomer;

[0082] The thermal spontaneous deformation calculation unit is used to calculate the corresponding thermal spontaneous deformation and mechanical deformation of the liquid crystal elastomer under the influence of temperature;

[0083] The order parameter calculation unit is used to solve the Landau free energy of the liquid crystal elastomer affected by temperature and external load, and obtain the order parameter of the liquid crystal elastomer in the corresponding state;

[0084] The director rotation evolution unit is used to average the liquid crystal molecular orientation of the liquid crystal elastomer into the director orientation, construct the director rotation evolution equation, and predict the dynamic response of the microscopic liquid crystal molecules of the liquid crystal elastomer;

[0085] The constitutive relationship calculation unit is used to construct the constitutive relationship of the liquid crystal elastomer under thermal-mechanical coupling conditions based on the free energy function, director and order parameter of the liquid crystal elastomer;

[0086] The iterative unit is used to expand the Jaumann derivative of stress in the constitutive relationship, obtain the tangent stiffness matrix, and realize the iterative calculation of stress;

[0087] The parameter calibration unit is used to perform uniaxial stretching experiments on the liquid crystal elastomer at different temperatures and different director orientations, measure relevant material parameters, and calculate the stress state of the liquid crystal elastomer under different load and temperature conditions.

[0088] The present invention also provides a storage medium storing a program, which, when executed by a processor, implements a method for predicting the mechanical response of a monodomain liquid crystal elastomer under multi-field coupling.

[0089] Therefore, the present invention provides a method and device for predicting the mechanical response of liquid crystal elastomers. By separating the energy functions of the cross-linked polymer network and the liquid crystal molecules in the liquid crystal elastomer, a solid foundation is provided for establishing the constitutive equation. The influence of the thermal-mechanical coupling on the macroscopic and microscopic properties of the liquid crystal elastomer is decomposed. Specifically, the thermal field only affects the macroscopic deformation of the liquid crystal elastomer and does not affect the rotation of the microscopic liquid crystal molecules. The external force affects both the macroscopic deformation and the rotation of the microscopic liquid crystal molecules, thereby constructing a model of thermal spontaneous deformation and mechanical deformation. By solving the Landau free energy, the order parameter of the liquid crystal elastomer under different temperature and stress states is obtained. Characterize the degree of consistency in the orientation distribution of liquid crystal molecules and use it to evaluate the phase transition state of liquid crystal elastomers; average the orientation of liquid crystal molecules into director orientations, and establish a rotational evolution equation of the director based on the energy function to predict the dynamic response of the microscopic liquid crystal molecules of the liquid crystal elastomer under external excitation; then, through the free energy function, director evolution equation, and order parameter of the liquid crystal elastomer, construct the constitutive relationship of the liquid crystal elastomer under thermal-mechanical coupling to solve the stress state of the liquid crystal elastomer under different conditions; at the same time, to update parameters such as stress and director that vary with boundary conditions and loading conditions, a tangent stiffness matrix is ​​established to achieve iterative calculation of stress. The present invention is suitable for predicting the macroscopic mechanical properties and microscopic director evolution of single-domain liquid crystal elastomers under different temperatures and loading conditions. Through this method, the mechanical response of liquid crystal elastomers can be accurately and efficiently obtained, thereby achieving efficient utilization and design of liquid crystal elastomers.

[0090] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0091] Figure 1 Schematic diagram of the structure of the rectangular sheet sample of liquid crystal elastomer in Example 1 of the present invention, wherein a is a front view and b is a side view;

[0092] Figure 2 This is a flow chart of the prediction method according to Example 1 of the present invention;

[0093] Figure 3The figures show the experimental and predicted results of the uniaxial tensile mechanical response of the liquid crystal elastomer according to Example 1 of the present invention; wherein, a is the experimental value of the shear modulus for different director orientations at room temperature, and the inset is the deformation curve required to calculate the shear modulus; b is a comparison of the experimental and predicted uniaxial tensile stress-stretch ratio curves of the liquid crystal elastomer for different director orientations; c is the experimental uniaxial tensile stress-stretch ratio curve of the liquid crystal elastomer at different temperatures when θ = 0° (the director is perpendicular to the loading direction); d is the predicted uniaxial tensile stress-stretch ratio curve of the liquid crystal elastomer at different temperatures when θ = 0° (the director is perpendicular to the loading direction);

[0094] Figure 4 The uniaxial tensile stress and director orientation prediction simulation diagram and experimental deformation diagram of Example 1 of the present invention at room temperature with different director orientations; wherein a is θ = 0°; b is θ = 45°; c is θ = 90°;

[0095] Figure 5 This is the stress-strain ratio curve of the mechanical metamaterial based on liquid crystal elastomer design according to Example 2 of the present invention, where a is the mechanical response obtained by configuring different director orientations in regions I and II; b is the mechanical response obtained by configuring the temperature in regions I and II;

[0096] Figure 6 Schematic diagram of the structure of the prediction device in Example 3 of the present invention;

[0097] Figure 7 This is an architecture of a computer device according to embodiment 4 of the present invention;

[0098] Reference numerals:

[0099] 1. Device body; 11. Energy calculation unit; 12. Thermal spontaneous deformation calculation unit; 13. Order parameter calculation unit; 14. Director rotation evolution unit; 15. Constitutive relationship calculation unit; 16. Iteration unit; 17. Parameter calibration unit; 21. Computer system; 22. Input / output; 23. System bus; 24. One or more CPUs; 25. Memory. DETAILED DESCRIPTION

[0100] The technical solution of the present invention is further described below with reference to the accompanying drawings and embodiments.

[0101] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments.

[0102] Example 1

[0103] The sample used in this embodiment 1 is a rectangular sheet sample synthesized by a two-step method, and its shape is as follows Figure 1 As shown, L is the length, W is the width, and t is the thickness. The thickness is much smaller than the length and width of the specimen, and the director is assumed to be distributed along the plane. The unit of length is mm. In other embodiments, other types of specimens may also be used, and the present invention is not limited thereto.

[0104] With one corner of the rectangle as the origin of the Cartesian coordinate system, the long side as the Y axis, and the short side as the X axis, a Cartesian coordinate system is established according to the right-hand rule, and the angle between the director and the X axis is θ.

[0105] like Figure 2 As shown, a method for predicting the mechanical response of a single-domain liquid crystal elastomer under uniaxial tension with different director orientations at different temperatures using multi-field coupling is provided, which specifically includes the following steps:

[0106] S1. Construct a free energy density function of a liquid crystal elastomer, wherein the free energy density function includes the elastic energy of the cross-linked polymer network in the liquid crystal elastomer, the Landau free energy of liquid crystal molecule rotation and phase transition, and the anisotropic free energy caused by the difference in liquid crystal orientation. The specific operation is as follows:

[0107] Liquid crystal elastomers are composed of cross-linked polymer networks and liquid crystal molecules, in which the rotation of liquid crystal molecules will generate energy and even cause phase change, introducing the Landau free energy f nem The energy generated by the rotation of liquid crystal molecules is also called phase change energy. The traditional Landau free energy only considers the effect of temperature on the order parameter. In fact, external force also has an impact on the order parameter. The force-order coupling term is introduced to correct the Landau free energy. The force-order coupling term is:

[0108] The formula for calculating Landau free energy is:

[0109]

[0110] Among them, Q is the order parameter; A0, B and C q is the material parameter related to Q, T * is the assumed second-order phase transition temperature; σ m is the Mises stress, μ is the shear modulus, and γ represents the coupling parameter between the order parameter and the Mises stress;

[0111] The cross-linked polymer network of a monodomain liquid crystal elastomer has high elasticity. At the same time, due to the anisotropic characteristics of liquid crystal elastomers, in order to express the anisotropy affected by orientation, the step length tensor is introduced to represent the influence of liquid crystal orientation on deformation:

[0112]

[0113]

[0114] Among them, l0 and l are the step length tensors in the initial configuration and the current configuration, respectively; d0 and d are the director vectors in the initial configuration and the current configuration, respectively; l ⊥ 、l || are the effective step lengths perpendicular to and parallel to the director, respectively; related to the order parameter, for nematic single-domain liquid crystal elastomers, it is generally l ⊥ =1-Q,l || =1+2Q; I is the unit tensor;

[0115] On the basis of the traditional hyperelastic neo-Hookean elastic energy, the effect of the director rotation on the deformation of the cross-linked polymer network is considered. At the same time, the semi-soft elastic energy is introduced to represent the mutual coupling between the cross-linked polymer network and the liquid crystal molecules in the liquid crystal elastomer. The elastic energy of the liquid crystal elastomer is obtained as follows:

[0116]

[0117] Where α is the semi-soft elastic parameter, J = det(F) represents the volume change during deformation, K is the bulk modulus, and F is the mechanical deformation gradient.

[0118] The effect of the director on anisotropy is also reflected in the significant differences in mechanical response and modulus for different director orientations. To better describe the anisotropic behavior caused by director orientation, the liquid crystal molecular orientation is used as a reinforcing fiber to introduce the anisotropic free energy, and the anisotropic free energy of the liquid crystal elastomer is obtained:

[0119]

[0120] Where k1 and k2 are material parameters related to the director orientation, I4 = d0·F T F·d0.

[0121] S2. Measure the deformation of the liquid crystal elastomer affected by temperature and calculate the corresponding thermal spontaneous deformation and mechanical deformation. The specific operation is as follows:

[0122] Deformation gradient F of liquid crystal elastomer during thermal-mechanical coupled deformation total It is divided into mechanical deformation part F and thermal spontaneous deformation part F s :

[0123] F total =FF s (5)

[0124] Among them, F s It is the thermal spontaneous deformation part;

[0125] The thermal spontaneous deformation of liquid crystal elastomers is mainly caused by temperature changes. During the spontaneous deformation process, it is assumed that the director does not rotate, only the shape changes. However, under the action of external forces, the director will rotate and deform. Due to the anisotropic and incompressible characteristics of liquid crystal elastomers, the thermal spontaneous deformation is defined as:

[0126]

[0127] Where S is the spontaneous deformation in the local coordinate system, λ s is the thermal deformation in the direction parallel to the director, λ s =1+G(T), G(T) is the thermal strain related to temperature, i i (i=1,2,3) is the unit vector in the main direction, R s It is the rotation tensor related to the orientation angle of the director. When only the rotation of the director in the plane is considered, the rotation tensor R s It can be expressed as:

[0128]

[0129] Here, θ is the angle between the director and the direction perpendicular to the load. In this example, the liquid crystal elastomer is heated to the corresponding temperature before the uniaxial tensile test. Therefore, the thermal spontaneous deformation can be ignored, and the overall deformation of the material is entirely mechanical.

[0130] S3. Solve the Landau free energy affected by temperature and external load to obtain the order parameter of the liquid crystal elastomer under the corresponding state. The specific operation is as follows:

[0131] The order parameter Q represents the average state of the orientation of liquid crystal molecules. When Q = 0, it is in the isotropic state, and when Q = 1, it is in the perfect nematic state. In reality, it does not exist. From formula (1), it can be seen that the order parameter of liquid crystal elastomer is affected by temperature and stress state. In order to satisfy the thermodynamic equilibrium of phase transition at the local minimum of free energy, it must be:

[0132]

[0133] From formulas (1) and (8), we can get:

[0134]

[0135] From this we obtain the order parameter Q under different temperature and stress states, 0<Q<1.

[0136] S4. Average the liquid crystal molecular orientation of the liquid crystal elastomer to the director orientation, and create a rotational evolution equation of the director to predict the dynamic response of the microscopic liquid crystal molecules of the liquid crystal elastomer. The specific operations are as follows:

[0137] The objectivity of the material requires that the constitutive equation of the director rotation should remain unchanged under the change of the reference, so the Jaumann derivative of the director is used Ensure that the rotation of the pointing vector in different reference frames is physically consistent, where W is the rotation tensor, which is an antisymmetric tensor. The rotation of the pointing vector is only related to the external force, so the objective rate of the pointing vector is It can be expressed as:

[0138]

[0139] Among them, η D is the director-viscous material parameter, k S are material parameters related to the rotation of the director; σ ij is the true stress of each component;

[0140] Therefore, the rotation evolution equation of the director is expressed as:

[0141]

[0142] According to the right-hand rule, the coordinates of the Cartesian coordinate system are established. The angle between the director and the Z axis of the coordinate is ξ, and the angle between its projection on the XY plane and the X axis is θ. The director is defined as:

[0143] d=(sinξcosθ,sinξsinθ,cosξ) T (12)

[0144] By taking the derivative of formula (12), we can get:

[0145]

[0146] in,

[0147] t1=(cosξcosθ,cosξsinθ,-sinξ) T ,

[0148] t2=(-sinξsinθ, sinξcosθ, 0) T .

[0149] According to formulas (11) and (13), the dynamic evolution equation of the discretized director can be obtained as:

[0150]

[0151] In this example, since the dimension along the thickness direction is much smaller than the other two dimensions, it can be considered that the director is distributed in the plane, so ξ=90°, so formulas (14) and (15) can be simplified to:

[0152]

[0153] where t i =(-sinθ, cosθ, 0) T , Formula (16) can be further discretized to obtain the plane director evolution equation:

[0154]

[0155] Among them, M ij =(Fl0F T ) ij ,

[0156] S5. The free energy function, director, and order parameter of the liquid crystal elastomer are introduced to obtain the constitutive relation of the liquid crystal elastomer under thermal-mechanical coupling. The specific operations are as follows:

[0157] From formulas (1), (3), and (4), the total free energy density of the liquid crystal elastomer can be obtained as:

[0158] f=f el +f nem +f ani (18)

[0159] The constitutive equation of stress is defined as:

[0160]

[0161] Where σ is the true stress, and sym() indicates symmetry of the variables in the brackets. Therefore, the constitutive equation of the liquid crystal elastomer is expressed as:

[0162]

[0163] in,

[0164] S6. Expand the Jaumann derivative of stress in the constitutive relation to obtain the tangent stiffness matrix to implement stress iteration. The specific operations are as follows:

[0165] When predicting the mechanical properties of liquid crystal elastomers, due to the continuous deformation of the material and the update of the boundary, it is necessary to update the parameters such as stress and director orientation in the liquid crystal elastomer. These parameters need to be iterated and the tangent stiffness matrix C is introduced. ijkl , the tangent stiffness matrix is ​​given by Kirchhoff stress τ ij The Jaumann derivative of is determined by:

[0166] δ(τ ij )+τ it δWtj -δW im τ mj =J(C ijkl δD kl ) (twenty one)

[0167] Among them, τ ij =Jσ ij , δD ij is the virtual deformation rate tensor, δW ij is the virtual spin tensor;

[0168] Simplifying formula (21) to obtain the tangent stiffness matrix expression:

[0169]

[0170] in, (.) RT Indicates transposing the lM in the subscript ijlM;

[0171] Substituting formula (20) into (22) yields the tangent stiffness matrix, which we divide into the elastic part (C I ) ijkl and the anisotropic part (C II ) ijkl :

[0172]

[0173]

[0174]

[0175] S7. Perform uniaxial tensile tests on the liquid crystal elastomer at different temperatures and different director orientations to measure relevant material parameters to calculate the stress state of the liquid crystal elastomer under different loads and temperatures. The specific operations are as follows:

[0176] Many material parameters in the constitutive equations (10) and (13) need to be determined through experiments to accurately predict the mechanical response of liquid crystal elastomers. Uniaxial tensile tests are the simplest way to confirm the mechanical properties of materials. It is necessary to carry out uniaxial tensile tests of liquid crystal elastomers in different states to confirm the material parameters: the shear modulus μ is determined by the stress-strain relationship in the small deformation stage, and the calculation formula is:

[0177]

[0178] Among them, F Tis the tensile load, A is the cross-sectional area, and λ is the stretch ratio. Since the shear modulus is affected by temperature and director orientation, in this example, the shear modulus is measured at T = 25°C, 40°C, 60°C, 80°C and director orientation θ = 0°, 30°, 45°, 60°, 90°. Figure 3 As shown in a; Due to the incompressibility of liquid crystal elastomers, the bulk modulus K can be converted by parameters such as Poisson's ratio. Assuming that the Poisson's ratio is 0.499, the corresponding bulk modulus is 492 MPa;

[0179] Since the director rotation and semi-soft elasticity are temperature-dependent, η0,k S , α can be calibrated through uniaxial tensile tests at different temperatures (T = 25 °C, 40 °C, 60 °C, 80 °C) with the director perpendicular to the loading direction (θ = 0°);

[0180] k1 and k2 are anisotropic parameters related to the orientation of the director, which can be calibrated through uniaxial tensile tests with the director perpendicular or parallel to the loading direction (θ = 0° and 90°).

[0181] S8. Calculate the mechanical response of uniaxial stretching at different temperatures and different director orientations. In this example, the mechanical response of liquid crystal elastomer under uniaxial stretching at θ = 0° at T = 25°C, 40°C, 60°C, 80°C and at θ = 0°, 30°, 45°, 60°, 90° at T = 25°C is calculated. The proposed method can accurately predict the macroscopic mechanical response of liquid crystal elastomer under uniaxial stretching at θ = 0°, 30°, 45°, 60°, 90° at T = 25°C. Figure 3 As shown in b, the macroscopic mechanical response of the liquid crystal elastomer at T = 25 ° C, 40 ° C, 60 ° C, and 80 ° C can also be accurately predicted when θ = 0 °, as shown in Figure 3 In addition, the provided method can also predict the rotation evolution of the microscopic director, and realize the prediction of the macroscopic and microscopic mechanical properties of liquid crystal elastomers, such as Figure 4 As shown in the figure, the proposed method can accurately predict the deformation of the material, the rotation fireworks of the stress vector, and is consistent with the deformation, material necking and other phenomena observed in the experiment.

[0182] Example 2

[0183] Since liquid crystal elastomers are widely used in engineering applications such as soft actuators and soft robots, the prediction method in Example 2 is applied to engineering practice for verification.

[0184] In this embodiment 2, a mechanical metamaterial based on liquid crystal elastomer is designed. The metamaterial is a rectangular sheet of liquid crystal elastomer with periodic circular holes dug out. The metamaterial is then partitioned and programmed. In this embodiment 2, it is divided into two zones, I and II. The mechanical properties of the metamaterial are controlled by configuring different director orientations or temperatures. Figure 5 As shown in a, by controlling the different director orientations of regions I and II at room temperature, the stress platform of the metamaterial can be adjusted, and the bearing capacity of the material can be changed to achieve on-demand response of the metamaterial. Figure 5 In b, the mechanical properties of the metamaterial are controlled by setting different temperatures in regions I and II. The temperature changes the material's load-bearing capacity and nonlinear response. For θ = 0°, the increase in temperature in region II reduces the soft elastic mechanics-related stress associated with the rotation of the director. For θ = 45°, the increase in temperature in region II significantly reduces the slope of the stress-stretch ratio curve.

[0185] Example 3

[0186] like Figure 6 , a schematic diagram of a multi-field coupled mechanical response prediction device for a single-domain liquid crystal elastomer, comprising: a device body 1;

[0187] The device body 1 includes an energy calculation unit 11 for constructing a free energy density function, which includes the elastic energy of the cross-linked polymer network in the liquid crystal elastomer, the Landau free energy of liquid crystal molecule rotation and phase transition, and the anisotropic free energy caused by the difference in liquid crystal orientation;

[0188] The thermal spontaneous deformation calculation unit 12 is used to measure the deformation of the liquid crystal elastomer caused by the influence of temperature and calculate the corresponding thermal spontaneous deformation and mechanical deformation;

[0189] An order parameter calculation unit 13 is used to solve the Landau free energy affected by temperature and external load to obtain the order parameter of the liquid crystal elastomer in the corresponding state;

[0190] a director rotation evolution unit 14 , which averages the liquid crystal molecular orientations of the liquid crystal elastomer into director orientations and creates a director rotation evolution equation to predict the dynamic response of the microscopic liquid crystal molecules of the liquid crystal elastomer;

[0191] A constitutive relationship calculation unit 15 is used to introduce the free energy function, director and order parameter of the liquid crystal elastomer to obtain the constitutive relationship of the liquid crystal elastomer under thermal-mechanical coupling;

[0192] An iterative unit 16 is used to expand the Jaumann derivative of stress in the constitutive relationship to obtain a tangent stiffness matrix to implement stress iteration;

[0193] The parameter calibration unit 17 is used to perform uniaxial stretching experiments on the liquid crystal elastomer at different temperatures and different director orientations, and measure relevant material parameters to calculate the stress state of the liquid crystal elastomer under different load and temperature conditions.

[0194] Example 4

[0195] The device according to embodiment 4 of the present invention can also be used by Figure 7 The architecture of the computing device shown is implemented. Figure 7 As shown, the computer system 21, system bus 23, one or more CPUs 24, input / output 22, memory 25, etc. The memory 25 can store various data or files used for computer processing and / or communication and program instructions including the method of embodiment 1 executed by the CPU.

[0196] In this embodiment 4, Figure 7 The architecture shown is only exemplary and can be adjusted according to actual needs when implementing different devices. Figure 7 One or more components in. The memory 25, as a computer-readable storage medium, can be used to store software programs, computer executable programs and modules, such as program instructions / modules corresponding to the subject updating method in the embodiment of the present invention (for example, the energy calculation unit 11, thermal spontaneous deformation calculation unit 12, order parameter calculation unit 13, director rotation evolution unit 14, constitutive relationship calculation unit 15, iteration unit 16 and parameter calibration unit 17 in the residual stress prediction device 1). One or more CPUs 24 execute various functional applications and data processing of the device of the present invention by running the software programs, instructions and modules stored in the memory 25, that is, to implement the above-mentioned residual normal stress prediction method, which includes:

[0197] Construct a free energy density function that includes the elastic energy of the cross-linked polymer network in the liquid crystal elastomer, the Landau free energy of liquid crystal molecule rotation and phase transition, and the anisotropic free energy caused by the difference in liquid crystal orientation;

[0198] The deformation of liquid crystal elastomers affected by temperature is measured, and the corresponding thermal spontaneous deformation and mechanical deformation are calculated;

[0199] The Landau free energy affected by temperature and external load is solved to obtain the order parameters of the liquid crystal elastomer under the corresponding state;

[0200] The liquid crystal molecular orientation of the liquid crystal elastomer is averaged into the director orientation, and the rotation evolution equation of the director is established to predict the dynamic response of the microscopic liquid crystal molecules in the liquid crystal elastomer;

[0201] The free energy function, director and order parameter of liquid crystal elastomer are introduced to obtain the constitutive relation of liquid crystal elastomer under thermal-mechanical coupling.

[0202] Expand the Jaumann derivative of stress in the constitutive relation to obtain the tangent stiffness matrix to implement stress iteration;

[0203] The liquid crystal elastomer was subjected to uniaxial tensile tests at different temperatures and with different director orientations, and the relevant material parameters were measured to calculate the stress state of the liquid crystal elastomer under different load and temperature conditions;

[0204] Of course, the processor of the server provided in this embodiment 4 is not limited to executing the method operations described above, but can also execute related operations in the multi-field coupled mechanical response prediction method of monodomain liquid crystal elastomer provided in any embodiment of the present invention.

[0205] The memory 25 may primarily include a program storage area and a data storage area. The program storage area may store an operating system and at least one application required for a function; the data storage area may store data generated based on the use of the terminal. Furthermore, the memory 25 may include high-speed random access memory and non-volatile memory, such as at least one disk storage device, flash memory device, or other non-volatile solid-state memory device. In some instances, the memory 25 may further include memory remotely located relative to one or more CPUs 24, and such remote memory may be connected to the device via a network. Examples of such networks include, but are not limited to, the Internet, an intranet, a local area network, a mobile communication network, and combinations thereof.

[0206] The input / output 22 may be used to receive input digital or character information and generate key signal input related to user settings and function control of the device. The input / output 22 may also include a display device such as a display screen.

[0207] Therefore, the present invention provides a prediction method and device for the mechanical response of liquid crystal elastomers, which is suitable for predicting the macroscopic mechanical properties and microscopic director evolution of single-domain liquid crystal elastomers under different temperatures and loading conditions. By separating the energy functions of the cross-linked polymer network and the liquid crystal molecules in the liquid crystal elastomer, a solid foundation is provided for establishing the constitutive equation; the influence of the thermal-mechanical coupling on the macroscopic and microscopic properties of the liquid crystal elastomer is decomposed; by solving the Landau free energy, the order parameter of the liquid crystal elastomer under different temperatures and stress states is obtained, which characterizes the consistency of the orientation distribution of the liquid crystal molecules and is used to evaluate the phase transition state of the liquid crystal elastomer; the orientation of the liquid crystal molecules is determined. The orientation of the director is averaged, and the rotation evolution equation of the director is established based on the energy function to predict the dynamic response of the microscopic liquid crystal molecules of the liquid crystal elastomer under external excitation; then, the constitutive relationship of the liquid crystal elastomer under thermal-mechanical coupling is constructed through the free energy function, director evolution equation and order parameter of the liquid crystal elastomer, which is used to solve the stress state of the liquid crystal elastomer under different conditions; at the same time, in order to update the parameters such as stress and director that change with the boundary and loading conditions, the tangent stiffness matrix is ​​established to realize the iterative calculation of stress; through this method, the mechanical response of liquid crystal elastomer can be accurately and efficiently obtained, thereby realizing the efficient utilization and design of liquid crystal elastomer.

[0208] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit the same. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solutions of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for predicting the mechanical response of a liquid crystal elastomer, characterized in that: The following steps are involved: S1. constructing a free energy density function of a liquid crystal elastomer, wherein the free energy density function includes the elastic energy of the cross-linked polymer network in the liquid crystal elastomer, the Landau free energy of liquid crystal molecule rotation and phase transition, and the anisotropic free energy caused by the difference in liquid crystal orientation; The force-order coupling term is introduced to correct the Landau free energy. The force-order coupling term is: ; The formula for calculating Landau free energy is: ； in, is the order parameter; 、 and For Related material parameters, is the temperature, is the assumed second-order phase transition temperature; is the Mises stress, is the shear modulus, represents the coupling parameter between order parameter and Mises stress; The step length tensor is introduced to represent the effect of liquid crystal orientation on deformation: ； ； in, 、 are the step size tensors of the initial configuration and the current configuration respectively, and are the directors under the initial configuration and the current configuration respectively; 、 are the effective step lengths perpendicular and parallel to the director, respectively; is the unit tensor; The semi-soft elastic energy is introduced to represent the mutual coupling between the cross-linked polymer network and the liquid crystal molecules in the liquid crystal elastomer, and the elastic energy of the liquid crystal elastomer is obtained as: ； in, is the semi-soft elastic parameter, represents the volume change during the deformation process, is the bulk modulus; is the mechanical deformation gradient; The anisotropic free energy of liquid crystal elastomers is obtained by introducing the orientation of liquid crystal molecules as reinforcing fibers: ； in, and is a material parameter related to the director orientation, ; S2. Determine the deformation of the liquid crystal elastomer under the influence of temperature and calculate the corresponding thermal spontaneous deformation and mechanical deformation; S3. Calculate the Landau free energy of the liquid crystal elastomer affected by temperature and external load, and obtain the order parameter of the liquid crystal elastomer under the corresponding state; S4. Average the orientation of the liquid crystal molecules in the liquid crystal elastomer into the director orientation, construct the rotation evolution equation of the director, and predict the dynamic response of the microscopic liquid crystal molecules in the liquid crystal elastomer; S5. Based on the free energy function, director and order parameter of liquid crystal elastomer, the constitutive relation of liquid crystal elastomer under thermal-mechanical coupling conditions is constructed; S6. Expand the Jaumann derivative of stress in the constitutive relation to obtain the tangent stiffness matrix and implement iterative calculation of stress; S7. Perform uniaxial tensile tests on the liquid crystal elastomer at different temperatures and different director orientations, measure relevant material parameters, and calculate the stress state of the liquid crystal elastomer under different load and temperature conditions.

2. The method for predicting the mechanical response of a liquid crystal elastomer according to claim 1, wherein: In S2, the deformation gradient of the liquid crystal elastomer during the thermal-mechanical coupling deformation process It is divided into mechanical deformation part and thermal spontaneous deformation part: ； in, It is the thermal spontaneous deformation part; Thermal spontaneous deformation is: ； in, is the spontaneous deformation in the local coordinate system, is the thermal deformation in the direction parallel to the director, is the temperature-dependent thermal strain, is the rotation tensor related to the director orientation angle, (i=1,2,3) is the unit vector in the main direction.

3. The method for predicting the mechanical response of a liquid crystal elastomer according to claim 2, wherein: In S3, in order to satisfy the thermodynamic equilibrium of the phase transition and to achieve the local minimum of free energy, then: ； According to the Landau free energy calculation formula, we can get: ； Get the order parameters under different temperature and stress states , .

4. The method for predicting the mechanical response of a liquid crystal elastomer according to claim 3, wherein: In S4, the Jaumann derivative of the director is used. Ensure that the rotation of the director in different reference frames is physically consistent; Where W is the spin tensor, the objective rate of the director is for: ； in, is the director-viscous material parameter, , 、 are all material parameters related to the director rotation; is the true stress of each component; The rotation evolution equation of the director is: 。 5. The method for predicting the mechanical response of a liquid crystal elastomer according to claim 4, wherein: In S5, the total free energy density of the liquid crystal elastomer is: ； The constitutive equation is defined as: ； in, is the true stress, sym( ) indicates symmetry of the variables in the brackets; The constitutive equation of liquid crystal elastomer is expressed as: ； in, .

6. The method for predicting the mechanical response of a liquid crystal elastomer according to claim 5, wherein: In the above S6, the stress and director orientation parameters in the liquid crystal elastomer are iterated, and the tangent stiffness matrix is ​​introduced. , the tangent stiffness matrix is ​​given by Kirchhoff stress The Jaumann derivative of is determined by: ； in, , is the virtual deformation rate tensor, is the virtual spin tensor; Simplifying the above formula, we can get the tangent stiffness matrix expression: ； in, , Indicates subscript in Perform transposition; Substituting the constitutive equation into the tangent stiffness matrix expression, the tangent stiffness matrix can be obtained. The tangent stiffness matrix is ​​divided into the elastic part and anisotropic part : ； 。 7. The method for predicting the mechanical response of a liquid crystal elastomer according to claim 6, wherein: In S7, the material parameters, shear modulus, and The calculation formula is: ； in, is the tensile load, A is the cross-sectional area, is the stretch ratio in the stretching direction.

8. A device for predicting the mechanical response of a liquid crystal elastomer according to any one of claims 1 to 7, characterized in that: It includes energy calculation unit, thermal spontaneous deformation calculation unit, order parameter calculation unit, director rotation evolution unit, constitutive relationship calculation unit and parameter calibration unit; The energy calculation unit is used to construct a free energy density function of the liquid crystal elastomer; The thermal spontaneous deformation calculation unit is used to calculate the corresponding thermal spontaneous deformation and mechanical deformation of the liquid crystal elastomer under the influence of temperature; The order parameter calculation unit is used to solve the Landau free energy of the liquid crystal elastomer affected by temperature and external load, and obtain the order parameter of the liquid crystal elastomer in the corresponding state; The director rotation evolution unit is used to average the liquid crystal molecular orientation of the liquid crystal elastomer into the director orientation, construct the director rotation evolution equation, and predict the dynamic response of the microscopic liquid crystal molecules of the liquid crystal elastomer; The constitutive relationship calculation unit is used to construct the constitutive relationship of the liquid crystal elastomer under thermal-mechanical coupling conditions based on the free energy function, director and order parameter of the liquid crystal elastomer; The iterative unit is used to expand the Jaumann derivative of stress in the constitutive relationship, obtain the tangent stiffness matrix, and realize the iterative calculation of stress; The parameter calibration unit is used to perform uniaxial stretching experiments on the liquid crystal elastomer at different temperatures and different director orientations, measure relevant material parameters, and calculate the stress state of the liquid crystal elastomer under different load and temperature conditions.

9. A storage medium storing a program, characterized in that: When the program is executed by a processor, the method for predicting the mechanical response of a liquid crystal elastomer as described in any one of claims 1 to 7 is implemented.

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