A numerical simulation method of a statistical-based nonlinear dynamic gel constitutive model
By establishing a statistically based nonlinear dynamic gel constitutive model, the problem of inaccurate analysis of active gel networks under large deformation conditions is solved, enabling more accurate structural design, which is suitable for applications such as tissue engineering, catalyst loading, and soft actuators.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- BEIHANG UNIV
- Filing Date
- 2023-11-21
- Publication Date
- 2026-06-19
Smart Images
Figure CN117574718B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of mechanical properties and nonlinear dynamics analysis of active gels, and particularly to a numerical simulation method for a nonlinear dynamic gel constitutive model based on statistics. This method considers the material exchange with the external solution and the internal chain reaction of the active gel network. It is a numerical simulation method for a nonlinear active gel constitutive model based on statistical chain theory that considers microscopic chain reactions. This provides a foundation for the design of future biomimetic structures using active gels as the main material. Background Technology
[0002] Natural materials in nature possess the ability to remain solid and provide mechanical strength while retaining fluidity, recombination, and self-repair capabilities. From an application perspective, the goal of materials science is to develop materials with sufficient dynamism to perform certain functions as effectively as in biological systems. This gap can be bridged at the chemical, physical, and biological levels, developing dynamic materials with a wide range of novel functions and turning dreams into reality. Almost all soft tissues in nature, including axons, muscles, blood vessels, ligaments, and nerves, exhibit the desired dynamic responses. For example, the repeating squid ring tooth (SRT) protein from cephalopods exhibits properties such as strength, self-healing, and biocompatibility. Through non-covalent hydrogen bonding, and through chain diffusion and physical cross-linking of protein matrices and folded nanostructures, the protein network rapidly repairs itself after damage and can be assembled into various morphologies and molecular structures. Recently, several synthetic strategies have been developed to learn from nature, developing dynamic polymers and polymer gels with biologically similar "life" properties. For example, a novel paradigm—covalent adaptive networks—has been explored in cross-linked polymers, where covalent cross-linked networks are formed through reversible bond or bond exchange reactions. This allows for the persistent presence of triggerable, reversible chemical structures throughout the network, enabling chain separation and reattachment. In addition to forming individual networks, monomers can also be inserted into the original network through specific polymerization processes, such as specific intermolecular hydrogen bonds between nucleic acid base pairs, free radical polymerization induced by various media, and ring-opening metathesis polymerization. These chemical reactions can be used to produce polymers with growth capabilities. Active gel network materials offer possibilities for the artificial development of dynamic materials.
[0003] Early hyperelastic models of hydrogel materials were mostly phenomenological models, such as the Mooney-Rivlin model and the Ogden model, which characterized strain energy using principal elongation. Hydrogels, as soft materials containing numerous polymer fiber chains, exhibit elastic characteristics different from hard materials like metals and ceramics. Changes in internal energy dominate the deformation process of hard materials, hence the elasticity of hard materials is termed "energy elasticity." However, for soft materials like hydrogels, the decisive factor affecting deformation is the change in the number of internal microstructures, i.e., the change in the entropy term. Therefore, the elasticity of hydrogels is also called "entropy elasticity." Consequently, researchers have developed a series of theoretical models based on statistical thermodynamics by studying the microstructure of hydrogel materials and statistically analyzing the length, orientation, and structure of polymer molecular chains. Based on the statistical mechanics of individual polymer chains, when a single chain is stretched, its configurational entropy decreases, providing the energy driving force for elastic recovery. This entropy elasticity can be captured by many single-chain models, such as freely connected chains with Gaussian statistics. In recent years, researchers have proposed defining a chain distribution tensor to describe the average deformation of all chains. Even though the chain distribution tensor is derived by integrating all chains, the macroscopic stress state is described by the stress distribution tensor. Building upon this theory, Haohui Zhang et al. further considered the variation in the number of chain segments during chemical reactions, realizing other reactions such as chain insertion dissociation and new chain formation. They also considered the influence of different diffusion rates on the reaction.
[0004] However, to date, no nonlinear mechanical properties of polymer chains under large deformation have been considered, nor have numerical simulation methods applicable to structural design been developed. Summary of the Invention
[0005] To overcome the shortcomings of existing technologies, this invention provides a numerical simulation method for a statistically based nonlinear dynamic gel constitutive model. This method fully considers the large deformations and chain reactions commonly encountered in practical engineering applications, providing feasible design methods for tissue engineering, catalyst loading, and soft actuators using active gels as materials. This results in design outcomes that better reflect real-world conditions and have stronger engineering applicability. This invention establishes a theoretical model to analyze the influence of introducing single-chain nonlinearity and considering microscopic chain reactions on the dynamic properties of active gel networks. The method proposed in this invention provides a new approach to accurately describing the dynamic properties of gel materials under large deformations and will provide a theoretical foundation for the further development and customization of high-performance gel materials.
[0006] The technical solution adopted by the present invention to solve the above-mentioned technical problems is as follows:
[0007] A numerical simulation method based on a statistical nonlinear dynamic gel constitutive model is used in the structural design of active gels as materials, comprising the following steps:
[0008] Step 1: Synthesize the evolution equations of chain distribution function, total chain concentration, and total chain segment concentration under various chain reactions;
[0009] Step 2: Based on the chain distribution function, define the free energy function of the macroscopic active gel network according to the energy function of a single chain;
[0010] Step 3: Based on the thermodynamic framework, derive the first and second thermodynamic laws of dynamic active gel behavior to obtain the general kinetic equilibrium equation;
[0011] Step 4: Considering the energy of the external solution and the mixing energy between the polymer, monomer and solvent inside the active gel network, as well as the mixing energy between the cross-linking agent and other molecules, the dynamic equilibrium relationship of the chemical potential of the active gel network immersed in the external solution is obtained.
[0012] Step 5: Combine the kinetic equilibrium equations from Step 3 and the dynamic equilibrium relationship of chemical potentials from Step 4 to form the general constitutive relation of the dynamic active gel, which is used to calculate and realize the numerical simulation of the active dynamic gel network.
[0013] Active gel materials exhibit significant nonlinearity under large deformation, and the active gel network undergoes a chain reaction when immersed in a specific solution. The significant nonlinearity and chain reaction have a considerable impact on structural properties. This invention proposes a solution and prediction method for the large deformation nonlinearity and the constant chain reaction of active gel networks, which can be used in the design of active gels as tissue engineering, catalyst loading, soft actuators, etc.
[0014] Beneficial effects:
[0015] This invention provides a novel approach to analyzing the nonlinear dynamics of active gel networks under chain reactions, including changes in the number of chain segments and chain insertion / dissociation and new chain formation. It overcomes and improves upon the shortcomings of traditional linear models of active gel networks based on ideal chains, which cannot accurately describe the characteristics under large deformations. The constructed dynamic and nonlinear dynamic model of the active gel network considers both the influence of microscopic chain reactions on its macroscopic properties and the strong nonlinear effects during actual chain stretching. Based on the aforementioned thermodynamic framework, high-dimensional finite element simulations are achieved, which can be used to simulate the mechanical analysis of hydrogels under complex multi-field coupling conditions. It also provides feasible design methods for tissue engineering, catalyst loading, and soft actuators using active gels as materials, making the design results more realistic and more applicable to engineering. Attached Figure Description
[0016] Figure 1 This is a flowchart of the numerical simulation of active dynamic gel networks according to the present invention;
[0017] Figure 2 It is the mesh generation model of the two-dimensional plate model of the active gel network;
[0018] Figure 3 These are deformation contour maps of a two-dimensional flat plate at different diffusion rates at various times.
[0019] Figure 4 These are stress-strain curves under different diffusion coefficients;
[0020] Figure 5(a) shows the diffusion rate. Results of internal chain concentration in a two-dimensional flat plate at the end of stretching;
[0021] Figure 5(b) shows the diffusion rate. Results of internal chain concentration in a two-dimensional flat plate at the end of stretching;
[0022] Figure 5(c) shows the diffusion rate. Results of internal chain concentration in a two-dimensional flat plate at the end of stretching;
[0023] Figure 6(a) shows the diffusion rate. Monomer concentration inside the two-dimensional plate at the end of the stretching process;
[0024] Figure 6(b) shows the diffusion rate. Monomer concentration inside the two-dimensional plate at the end of the stretching process;
[0025] Figure 6(c) shows the diffusion rate. The monomer concentration inside the two-dimensional plate at the end of the stretching process. Detailed Implementation
[0026] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the protection scope of the present invention.
[0027] This invention proposes a numerical simulation method based on a statistical nonlinear dynamic gel constitutive model, such as... Figure 1 As shown, it includes the following steps:
[0028] Step (1) synthesizes the evolution equations of chain distribution function, total chain concentration, and total chain segment concentration under various chain reactions. In statistical chain theory, at time t, the number of chains that satisfy the condition that the end-to-end vector of the chain is r and the number of chain segments is n is denoted as Φ(r,n,t), and is defined as the chain distribution function.
[0029] According to the definition of the chain distribution function:
[0030]
[0031]
[0032] In the above formula, C(t) and C n (t) represents the total chain concentration and the total number of chain segments concentration, respectively. The three-dimensional phase space corresponding to the end-to-end vector r, This corresponds to the set of all positive real numbers for the number of chain segments *n* (in practice, *n* can only be a positive integer, but considering that most chains have a large number of segments, the range of *n* is extended from the discrete space of positive integers to the field of positive real numbers starting from 1). For ease of subsequent discussion, we use <> to denote the integral in the four-dimensional chain space, i.e.
[0033] Following the assumptions of continuous medium theory, the polymer network is represented as a continuous set of points, and each point is labeled with coordinate X in this reference configuration. After the network deforms, the coordinates of the labeled points at time t move to a new position x. This process can be described by a function x = Γ(X,t). The deformation gradient tensor F is defined as:
[0034]
[0035] Therefore, the change in the chain distribution function is caused by the interaction between the following five physical processes:
[0036] 1. Under the assumption of affine deformation, the end-to-end vectors before and after deformation satisfy r = F·r0, where r0 is the end-to-end vector before deformation and r is the end-to-end vector after deformation. Under the action of the deformation gradient tensor F, the end-to-end vector r of the network chain undergoes a distorted change, while the number of chain segments n remains unchanged. Let C be the velocity gradient tensor; the chain distribution function changes as shown in equation (4) during this process, and the chain concentration C and the chain segment number concentration C are in the reference configuration of the X coordinate system. n The value remains unchanged, r in the equation i and r j L is the component form of the end-to-end vector r. ij and L ii Let L be the component form of the velocity gradient tensor, where i and j are dummy indices. Following Einstein's summation convention, the indexed terms of these indices are summed:
[0037]
[0038] 2. Monomers are inserted into or extracted from existing chains in a polymer network. The rate of change in the number of chain segments is determined by the insertion and extraction reaction rates. A decision can be expressed as Where ξ in (n) and ξ ex(n) represents the number of individuals inserted and removed in each of the n segments. Simultaneously, the end-to-end vector r increases or decreases with the number of segments, satisfying the following condition: The chain distribution function changes during this process as shown in equation (5), and the chain concentration C and chain segment number concentration C under the reference configuration are... n constant.
[0039]
[0040]
[0041] in and This represents the total insertion and extraction rate.
[0042] 3. Monomers directly transform into polymer network chains: The rate of change in the total number of chain segments is determined by the rate of new chain formation, expressed as... Let n be the average number of chain segments in the newly generated chain. ave ,have The new chain distribution satisfies the initial distribution, denoted as p. 0 (r,n), assuming the initial distribution satisfies In the above formula, b is the Kuhn length, n is the total number of chain segments, β is the number of chain segments in a Kuhn segment, r is the total chain length, and the initial chain segment distribution is defined as follows: satisfy
[0043]
[0044]
[0045]
[0046] 4. The new chain has a probability density function in a stress-free state. The associative chains attach to the polymer network at a certain correlation rate, and the number of associative chain segments and the distribution of chain end vectors conform to the initial configuration.
[0047]
[0048]
[0049]
[0050] in This represents the total number of associated chains per unit time and per reference volume.
[0051] 5. The separation of the attached chains in its stretched configuration, with a dissociation rate of Ξ d (r,n) highlights that the number of chain segments and the distribution of chain end vectors in the dissociated chain conform to the current configuration.
[0052]
[0053]
[0054]
[0055] in This represents the total number of chains that are dissociated per unit time and per unit reference volume.
[0056] The evolution equation of the chain distribution function satisfies:
[0057]
[0058] The evolution equations for total chain concentration and total chain segment concentration satisfy:
[0059]
[0060]
[0061] Step (2) Based on the chain distribution function, the free energy function of the macroscopic active gel network is defined according to the energy function of a single chain. According to the Gaussian statistical theory of freely connected chains, under the assumption of large deformation, the Gibbs free energy G(T,f,N) of a single chain is expressed as:
[0062]
[0063] Where k is the Boltzmann constant and T is the temperature. It is the tensile force, N is the number of Kuhn segments, and b is the Kuhn length, satisfying...
[0064] Based on the above equation, the elastic energy of a single chain in the second-order Taylor expansion can be obtained as follows:
[0065]
[0066] For simplicity, consider an incompressible polymer and assume that the stored elastic energy in the material volume is the sum of the energy stored in the configuration space of each active chain. Define ψ c Given the elastic energy of a single chain with end-to-end vector r and n chain segments, the free energy function Ψ of the active gel network is... e It can be written as:
[0067] Ψ e =<(Φ-Φ0)ψ c > (21)
[0068] Where Φ0 is the chain distribution function under the initial stress-free state.
[0069] Step (3) derives the first and second thermodynamic laws governing the dynamic active gel behavior based on the thermodynamic framework, obtaining the general kinetic equilibrium equation. Due to the slow motion of the polymer and solvent, the inertial effect is neglected. Therefore, the linear momentum equilibrium gives...
[0070] DivP T +B=0 (22)
[0071] Where P(X,t) (i.e., P) is the first Piola-Kirchhoff stress tensor, and B(X,t) (i.e., B) represents the external surface load per unit volume in the reference coordinate system. The Cauchy stress σ satisfies σ=J -1 P·F T The superscript T indicates the transpose of the matrix. Div is the divergence symbol.
[0072] Angular momentum equilibrium has the following:
[0073] P·F T =F·P T (twenty three)
[0074] Dynamically active gels consist of a polymer network cross-linked by dynamic bonds and adsorbed mobile molecules. The following derivation primarily considers three mobile substances: monomers, solvents, and cross-linking agents. Nominal concentrations are expressed in C1. m C s and C cr This is expressed as follows. The mass conservation law for each substance is given:
[0075]
[0076]
[0077]
[0078] J i Let i = m, s, and cr be the fluxes of monomer, solvent, and crosslinking agent, respectively. Equation (24) describes the mass conservation of monomer. The concentration change of monomer can be attributed to the formation of new chains and chain insertion and extraction reactions. Equation (25) describes the mass conservation of solvent, i.e., the solvent does not participate in the reaction. Equation (26) describes the mass conservation of free crosslinking agent, where f is the coordination number of the network, i.e., the number of chains connected to each crosslinking agent. Considering that each chain is connected to two crosslinking agents, the formation of a new chain consumes 2 / f free crosslinking agents.
[0079] The chain distribution function describes chains with different properties; a network is viewed as a set of chains. Define the chain distribution tensor:
[0080]
[0081]
[0082]
[0083] in
[0084] Further define the fourth-order chain distribution tensor:
[0085]
[0086]
[0087]
[0088] In the above formula, κ is a constant describing the initial chain segment number distribution, satisfying...
[0089] Define a second-order tensor θ and θ n Satisfying θ ij =τ ijkk θ nij =τ nijkk i and j are free indices, and all values of the index are traversed. k is a dummy index, and the index items of the index are summed.
[0090] The evolution equations for the chain distribution tensors γ and τ satisfy:
[0091]
[0092]
[0093] Where L is the velocity gradient tensor, L kj L lj L mj and L nj All of these are in their component form, τ jlmn τ kjmn τ kljn and τ klmj All are component forms of the fourth-order tensor τ, where I is a second-order unit vector. kl I mn I km I nl I kn and I lm ,τ0=I kl I mn +I km I nl +I kn I lm i, k, l, m, and n are free indices, and all values of the index are iterated over; j is a dummy index, and the index terms of the index are summed; I is a second-order unit vector.
[0094] The polymerization rate is related to the concentration of free radicals and monomers in the gel. Once stimulated, the nominal concentration C of free radicals within the gel increases. ra Keeping constant, the current free radical concentration is C ra / J. The rates of chain insertion and new chain formation can be expressed as:
[0095]
[0096]
[0097] in k represents the conversion rate from monomer to single chain. p This is the polymerization reaction constant. The average number of segments in the new chain depends on the ratio of monomer to crosslinking agent, i.e., 2C. m / (fC cr ).
[0098] The chain extraction reaction rate is related to the chain density, as follows:
[0099]
[0100] In the formula: k ex is the chain extraction reaction constant.
[0101] The insertion and removal of monomers from a chain changes the number of chain segments. If the number of monomers inserted or removed from a chain is proportional to the number of chain segments, then... and therefore,
[0102]
[0103] in is a constant, where k ex k is the chain extraction reaction constant. p The polymerization reaction constant is... C represents the rate of change of the number of chain segments. m Monomer concentration;
[0104] Chain dissociation and association can occur through various processes, such as dynamic bonding, mechanical damage, and chain exchange reactions. Only chain exchange reactions with equal rates of chain dissociation and association are evaluated. Furthermore, it is assumed that the reaction rate constant is independent of r and n. These assumptions imply that:
[0105] Ξ d (r,n)=k d Φ(r,n)C ra / J (39)
[0106]
[0107] In the formula: k d Let be the chain reaction rate constant. The total rate of dissociation and association is:
[0108]
[0109]
[0110] Substituting all reaction kinetics into the evolution equation, we get:
[0111]
[0112]
[0113] For any body B0 in the reference configuration, energy can flow across the boundary in the form of mechanical work and chemical species transport. In an isothermal system, it is assumed that each point of matter is connected to a heat source Q(X,t) to maintain a constant temperature T for the entire system. The first law of thermodynamics states:
[0114]
[0115] Where U represents the internal energy per unit reference volume. The left side represents the time rate of change of energy within volume B0. The terms on the right represent different mechanisms for adding power to the control volume; the first and second terms represent the mechanical power inputs from the surface traction force R = P·N and the body force B. The velocity of the material point is... Reference control volume boundary The unit normal is N. The third term is the energy carried by the chemical flux. The enthalpy carried by each particle in the i-th type of substance is h. i In the last term, Q(X,t) represents the rate at which heat is supplied to a material point per unit reference volume.
[0116] The second law of thermodynamics states that the increase in entropy within the control volume must be greater than or equal to the entropy flowing into the volume B0. Entropy can enter the control volume through chemical substances and the heat supplied to each point of matter in an isothermal system. The second law of thermodynamics is expressed as:
[0117]
[0118] Where S(X,t) (i.e., S) represents the entropy per unit reference volume. The left side of the equation represents the rate of change of entropy within volume B0 over time. The first term on the right side represents the entropy transferred by the chemical flux, where η is the entropy carried by each particle in the i-th type of substance. i The second term refers to the entropy released from the heat source.
[0119] The Helmholtz free energy per unit reference volume is defined as W = U - TS(X,t), where U represents the internal energy per unit reference volume and S(X,t) represents the entropy per unit reference volume. Combining the first and second laws of thermodynamics, the inequalities governing the evolution of the Helmholtz free energy satisfy:
[0120]
[0121] Where μ i =h i -Tη i Represents the chemical potential carried by each particle in the i-th type of matter, e.g., μ m =h m -Tη m Represents the chemical potential carried by a single particle. This represents the change in the concentration of particles in the i-th type of substance;
[0122] According to the Flory–Huggin theory, the Helmholtz free energy W of the gel system is expressed as the sum of the elastic energy and mixing energy of the network.
[0123] W = Ψ e +W mix +W cross (48)
[0124] Among them Ψ e W represents the elastic properties of the gel network. mix W represents the mixing energy between the polymer, monomer, and solvent. cross Represents the mixing energy between the crosslinking agent and other molecules.
[0125] For simplicity, consider an incompressible polymer and assume that the stored elastic energy in the material volume is the sum of the energies of each active chain in its configuration space. Define ψ c For the elastic energy of a single chain with end-to-end vector r and n chain segments, the stored elastic energy Ψ is... e and the rate of change of elastic energy over time satisfy:
[0126] Ψ e =<(Φ-Φ0)ψ c > (49)
[0127]
[0128] Where tr is the trace symbol, k is the Boltzmann constant, and p n (n,t) is the probability density function of the chain segment number distribution. γ represents the rate of change of the deformation gradient tensor F. n and θ nTo define the second-order chain distribution tensor;
[0129] The mixing free energy generated by the interaction of the polymer network, monomers, and solvent, and the mixing energy of the crosslinking agent are respectively:
[0130]
[0131]
[0132] It is the chemical potential of the crosslinking agent under its standard conditions;
[0133] Solve the above equations simultaneously:
[0134]
[0135] Where Ω is the volume of a single molecule, and Π is an introduced Lagrange multiplier used to enforce the incompressibility condition;
[0136] Since equation (52) must hold for all possible thermodynamic processes, the following constitutive relation is obtained: (where Π represents the introduced Lagrange multiplier used to enforce the incompressibility condition.
[0137]
[0138]
[0139]
[0140]
[0141] Step (4) considers the energy of the external solution to obtain the chemical potential equilibrium equation for the active gel network immersed in the external solution. The free energy density of the external solution consists of the mixing energy between the monomer and the solvent, and the mixing energy between the solution and the free crosslinking agent:
[0142]
[0143] in and These represent the quantities of monomer, solvent, and crosslinking agent per unit volume of the external solution, respectively. For the external solution, the volume fraction of monomer is... volume fraction of solvent and the volume fraction of the crosslinking agent These are constants in external solutions; and These are the chemical potentials of the monomer, solvent, and crosslinking agent under their standard conditions; and the chemical potentials of these substances in the external solution. and It is also a constant:
[0144]
[0145]
[0146]
[0147] For the diffusion kinetics of monomers, crosslinking agents, and solvents within a polymer network, a simple model is adopted: the flow is proportional to its own chemical potential gradient. Diffusion is considered isotropic, and the diffusion coefficient is constant. The nominal flux J... m J s and J cr It can be represented as:
[0148]
[0149]
[0150]
[0151] Step (5) combines the kinetic equilibrium from step (3) and the dynamic chemical potential equilibrium from step (4) to form the general constitutive relation of the dynamic active gel, which is used to calculate and realize the numerical simulation of the active dynamic gel network, laying a theoretical foundation for the structural design of active gel materials. In this part, various variables are normalized, and all the formulas prior to this invention are summarized for subsequent numerical simulation. The normalization table is shown in Table 1:
[0152] Table 1
[0153]
[0154]
[0155] All variables and equations used to fully solve the problem have been summarized and normalized. In the following sections, all signed variables have been normalized using the normalized variables specified in Table 1. All derivatives are calculated relative to normalized coordinates and time.
[0156] The force balance equations are:
[0157]
[0158]
[0159]
[0160]
[0161] The chemical potential equilibrium equation is:
[0162]
[0163]
[0164]
[0165] The incompressibility constraint, expressed as a dimensionless variable, was applied.
[0166]
[0167] After considering diffusion effects, the mass conservation equation is:
[0168]
[0169]
[0170]
[0171]
[0172]
[0173]
[0174]
[0175]
[0176]
[0177] Based on the above constitutive equations, high-dimensional finite element simulations of active gel networks are performed, and mechanical analysis of active hydrogels under complex conditions is conducted, which can lay a theoretical foundation for the structural design of active gel materials.
[0178] Example:
[0179] To gain a fuller understanding of the features of this invention and its applicability to practical engineering, this invention validates the proposed nonlinear constitutive model of active hydrogels based on statistical chain theory, considering microscopic chain reactions, using numerical simulations of active dynamic hydrogel networks under solution immersion. Subsequently, to verify the proposed analysis method for active hydrogel networks considering diffusion effects, this invention performs finite element simulations of two-dimensional flat plates using COMSOL, considering diffusion effects, and investigates the influence of different diffusion efficiencies on the mechanical properties and chemical parameter concentrations of the active hydrogel network.
[0180] In this embodiment, the material parameter is κ = 30. χPEG =0.55, χ NIPAM =0.36, k p C ra =9.45×10 -4 s -1 α = 0.64; dimensional parameters are L1 = L2 = L = 1. On the two sides of a two-dimensional plate, one end is completely fixed and a displacement load is applied to the other side.
[0181] Figure 2 It is a two-dimensional square flat plate grid. Figure 3 This demonstrates the effect of different diffusion efficiencies on the active gel network at a loading rate of [missing value]. The deformation at that time. The diffusion rate is among them. The values are 1, 0.25, and 0.05, respectively. At low diffusion efficiency, the active gel network experiences a central depression due to the Poisson effect, similar to the uniaxial tensile deformation of other metals or rubber materials. However, as the diffusion rate increases, the depression rate caused by the Poisson effect becomes less than the mass exchange rate in the external solution, and the Poisson ratio of the active gel material gradually becomes 0 and then negative. Figure 4 The variation of stress at the center point of the structure with tensile strength under uniaxial tension at different diffusion rates is presented. As the diffusion rate increases, more and more molecules enter through the boundary, leading to relaxation of the active gel network and a significant decrease in stress at the center point. Figures 5(a)-5(c) and Figures 6(a)-6(c) The results show the chain and monomer concentrations at different diffusion efficiencies, respectively. As diffusion efficiency increases, the overall volume increases rapidly. When the total chain volume remains relatively constant, the chain density decreases with increasing volume. Due to the faster volume growth at the two ends, the chain density gradually decreases from the center to the sides. The monomer concentration distribution in the gel is opposite to the chain concentration distribution, with the monomer concentration gradually increasing from the center to the sides. Monomers continuously enter the active gel network from the external solution via a chemical potential gradient, resulting in higher monomer concentrations at the two ends.
[0182] The parts of this invention not described in detail are well-known to those skilled in the art.
[0183] The above are merely specific steps of the present invention and do not constitute any limitation on the scope of protection of the present invention; it can be extended to the field of nonlinear dynamic analysis of chain reactions containing active dynamic hydrogels. All technical solutions formed by equivalent transformation or equivalent substitution fall within the scope of protection of the present invention.
Claims
1. A numerical simulation method of a statistical-based nonlinear dynamic gel constitutive model for use in structural design with active gels as materials, characterized by, Includes the following steps: Step 1: Synthesize the evolution equations of chain distribution function, total chain concentration, and total chain segment concentration under various chain reactions, including: In statistical chain theory, At time t, the end-to-end vector of the chain is Vector and the number of chain segments is The number of chains is denoted as , is defined as the chain distribution function; According to the definition of the chain distribution function: （1） （2） In the above formula and These are the total chain concentration and the total number of chain segments concentration, respectively. Corresponding to end-to-end vector The three-dimensional phase space, Corresponding to the number of chain segments The set of all positive real numbers, The range is the field of positive real numbers starting from 1; using The integral representing the four-dimensional chain space is... ; Following the assumptions of continuous medium theory, the polymer network is represented as a continuous set of points, using coordinates. Mark each point; after the network deforms, the marked points are at time [time missing]. Move the coordinates to the new position Using functions Description; The deformation gradient tensor F is defined as: （3） Therefore, the change in the chain distribution function is caused by the interaction between the following five physical processes: (1) Under the assumption of affine deformation, the end-to-end vectors before and after deformation satisfy the following conditions: , The end-to-end vector of the chain before deformation. Let the deformed chain end-to-end vector be the vector from which the deformation occurs; in the deformation gradient tensor... Under the influence of this effect, the end-to-end vector of the network chain Twisting changes, number of chain segments No change; among which Let be the velocity gradient tensor; the chain distribution function changes as shown in equation (4) during this process, and the chain concentration is in the X-coordinate reference configuration. and chain segment number concentration Unchanged, in the equation and End-to-end vector component form, and For velocity gradient tensor component form, and As a dummy index, according to Einstein's summation convention, the indexed terms of that index are summed: （4） (2) In a polymer network, the rate of change in the number of chain segments is determined by the insertion and extraction reaction rates of monomers into or from existing chains, and is expressed as follows: ,in and Each insertion and extraction is respectively The number of individual units in each chain segment; simultaneously, the end-to-end vector r of the chain increases or decreases with the number of chain segments, satisfying the following: The chain distribution function changes during this process as shown in equation (5), and the chain concentration under the reference configuration is... and chain segment number concentration Unchanged, in the equation End-to-end vector component form, Since it is a dummy index, the indexed terms of that index must be summed according to Einstein's summation convention: （5） （6） wherein and is the total insertion and extraction rate; (3) Monomers directly transform into polymer network chains: The rate of change in the total number of chain segments is determined by the rate of new chain formation reaction, expressed as: The average number of chain segments in the newly generated chain is denoted as . ,have The new chain distribution satisfies the initial distribution, denoted as . Assume the initial distribution satisfies In the above formula It is the Kuhn length. The total number of chain segments, It is the number of chain segments in a Kuhn segment. Given the total chain length, the initial chain segment distribution is defined as follows: satisfy ; （7） （8） （9） (4) The probability density function of the new chain in a stress-free state is The associative chains attach to the polymer network at a certain correlation rate, and the number of associating chain segments and the distribution of chain end vectors conform to the initial configuration: （10） （11） （12） wherein represents the total number of associated chains in a unit of time and per reference volume; (5) Separation of the attached chains in its stretched configuration, with a dissociation rate of The number of dissociated chain segments and the distribution of chain end vectors are consistent with the current configuration. （13） （14） （15） wherein represents the total number of strands dissociated per unit time and per unit reference volume; The evolution equation of the chain distribution function satisfies: （16） The evolution equations for total chain concentration and total chain segment concentration satisfy: （17） （18） Step 2: Based on the chain distribution function, define the free energy function of the macroscopic active gel network according to the energy function of a single chain; Step 3: Based on the thermodynamic framework, derive the first and second thermodynamic laws of dynamic active gel behavior to obtain the kinetic equilibrium equation; Step 4: Considering the energy of the external solution and the mixing energy between the polymer, monomer and solvent inside the active gel network, as well as the mixing energy between the cross-linking agent and other molecules, the dynamic equilibrium relationship of the chemical potential of the active gel network immersed in the external solution is obtained. Step 5: Combine the kinetic equilibrium equations from Step 3 and the dynamic equilibrium relationship of chemical potentials from Step 4 to form the constitutive relation of the dynamic active gel, which is used to calculate and realize the numerical simulation of the active dynamic gel network.
2. The numerical simulation method for a statistically based nonlinear dynamic gel constitutive model according to claim 1, characterized in that: The second step includes; According to the theory of free-joint chain with Gaussian statistics, the Gibbs free energy of a single chain is expressed as is expressed as: （19） wherein is the Boltzmann constant, is the temperature, is the stretching force, is the Kuhn segment number, is the Kuhn length, satisfying ; Based on the above equation, the elastic energy of a single chain obtained from the second-order Taylor expansion is: （20） Definitions to have an end-to-end vector , the number of segments is of the single strand, then the free energy function of the active gel network is written as: （21） wherein is the chain distribution function in the initial unstressed state.
3. The numerical simulation method for a statistically based nonlinear dynamic gel constitutive model according to claim 1, characterized in that: The third step includes: Due to the slow motion of the polymer and solvent, inertial effects are negligible; therefore, the linear momentum balance is given by: （22） in For the first Piola-Kirchhoff stress tensor, Represents the external surface load per unit volume in the reference coordinate system; Cauchy stress. satisfy , The divergence sign; Angular momentum equilibrium has the following: （23） Dynamically active gels consist of a polymer network cross-linked by dynamic bonds and adsorbed mobile molecules. Consider three mobile substances: monomer, solvent, and cross-linking agent, with nominal concentrations expressed as follows: , and This indicates that the mass conservation law for each substance is given: （24） （25） （26） in , These represent the fluxes of monomer, solvent, and crosslinking agent, respectively; Equation (24) describes the mass conservation of monomer, with the concentration change attributed to new chain formation and chain insertion and extraction reactions; Equation (25) describes the mass conservation of solvent, i.e., the solvent does not participate in the reaction; Equation (26) describes the mass conservation of free crosslinking agent, where... This is the coordination number of the network, i.e., the number of chains connecting each crosslinking agent; considering that each chain is connected to two crosslinking agents, the formation of a new chain consumes [a certain amount of time / efficiency]. A free crosslinking agent; The chain distribution function describes chains with different properties; a network is viewed as a set of chains. The chain distribution tensor is defined as follows: （27） （28） （29） wherein , ; Further define the fourth-order chain distribution tensor: （30） （31） （32） In the above formula A constant describing the initial chain segment number distribution, satisfying ; Define a second-order tensor and satisfy , , and Given a free index, iterate through all possible values for that index. This is a dummy index; it sums the indexes at that index. chain distribution tensor and the evolution equation for is given by （33） （34） in For the velocity gradient tensor, , , and All of them are in their component form. , , and All are fourth-order tensors component form, It is a second-order unit vector. , , , , and , , , , , and Given a free index, iterate through all possible values for that index. This is a dummy index; it sums the indexes at that index. The rate of polymerization is related to the free radical concentration and monomer concentration in the gel; once stimulated, the nominal concentration of free radicals inside the gel remains constant, the current free radical concentration is ; The rates of chain insertion and new chain formation are expressed as: （35） （36） in This represents the conversion rate from monomer to single chain. The polymerization reaction constant is given; the average number of segments in the new chain depends on the ratio of monomer to crosslinking agent, i.e. ; The chain extraction reaction rate is related to the chain density, as follows: （37） wherein: is the chain extraction reaction constant; If the number of monomers inserted or extracted from a chain is proportional to the number of chain segments, then we have and ,therefore, （38） wherein is a constant, wherein is a chain extraction reaction constant, is a polymerization reaction constant, denotes the rate of change of the number of segments, is the monomer concentration; Assuming that the reaction rate constant is independent of and then we have: （39） （40） In the formula: The chain reaction rate constant is: The total rate of dissociation and association is: （41） （42） Substituting all reaction kinetics into the evolution equation, we get: （43） （44） For any body in the reference configuration Energy can flow across the boundary in the form of mechanical work and transport of chemical species ; In an isothermal system, each material point is assumed to be connected to a heat source Heat is supplied to maintain the temperature of the entire system Constant; the first law of thermodynamics states that （45） in The left side represents the internal energy within a unit reference volume; the right side represents the volume. The time rate of change of internal energy; the terms on the right represent different mechanisms for adding power to the control volume, and the first and second terms represent the surface traction force. and physical strength The mechanical power input; the velocity of the matter point is Reference control volume boundary The unit normal is ; The third item is the energy carried by the flux of chemical substances; The enthalpy carried by each particle in this type of matter is In the last item, This represents the rate at which heat is supplied to a material point per unit reference volume. The second law of thermodynamics states that the control of the increase in entropy within a body is greater than or equal to the entropy flowing into the body The second law of thermodynamics is expressed as: （46） in The left side of the equation represents the entropy within a unit reference volume. The rate of change of internal entropy over time, where the first term on the right represents the entropy transferred by the chemical flux, and the second term... The entropy carried by each particle in matter-like substances The second term refers to the entropy released from the heat source. The Helmholtz free energy per unit reference volume is defined as , Internal energy within a unit reference volume The entropy within a unit reference volume; combining the first and second laws of thermodynamics, the inequality relationship of the Helmholtz free energy evolution satisfies: （47） in Representing the The chemical potential carried by each particle in a substance, for example Represents the chemical potential carried by a single particle. Representing the Changes in the concentration of particles in a substance; The Helmholtz free energy W of the gel system is expressed as the sum of the elastic energy and mixing energy of the network: （48） wherein represents the elastic energy of the gel network, represents the mixing energy between the polymer, monomer and solvent, represents the mixing energy between the crosslinker and other molecules; Consider an incompressible polymer and assume that the total elastic energy stored per unit volume of the material is the sum of the energies of the active chains; definition For having end-to-end vectors The number of chain segments is The elastic energy in a single chain, then the stored elastic energy satisfy: （49） （50） in To find the trace symbol, It is Boltzmann's constant. Let be the probability density function of the chain segment number distribution. Represents the deformation gradient tensor rate of change, and To define the second-order chain distribution tensor; The mixing free energy generated by the interaction of the polymer network, monomers, and solvent, and the mixing energy of the crosslinking agent are respectively: （51） in , and The Flory–Huggins parameters describe the interactions between polymers and monomers, polymers and solvents, and monomers and solvents, respectively. and These are the chemical potentials of the monomer and the solvent under standard conditions, respectively. （52） is the chemical potential of the crosslinking agent at its standard conditions; Solving the simultaneous equations (47)-(52), we get: （53） in It is the volume of a single molecule. These are introduced Lagrange multipliers used to enforce the incompressibility condition; Since equation (52) must hold for all possible thermodynamic processes, the following constitutive relation is obtained: （54） （55） （56） （57） wherein .
4. The numerical simulation method for a statistically based nonlinear dynamic gel constitutive model according to claim 1, characterized in that: The fourth step includes: The free energy density of the external solution consists of the mixing energy between the monomer and the solvent, and the mixing energy between the solution and the free crosslinking agent: （58） in , and These represent the quantities of monomer, solvent, and crosslinking agent per unit volume of the external solution, respectively. For the external solution, the volume fraction of the monomer is... volume fraction of solvent and the volume fraction of the crosslinking agent. These are constants in external solutions; , and These are the chemical potentials of the monomer, solvent, and crosslinking agent under their standard conditions; and the chemical potentials of these substances in the external solution. , and It is also a constant: （59） （60） （61） For the diffusion kinetics of monomer, crosslinker and solvent within the polymer network, a simple model was adopted: flow is proportional to its own chemical potential gradient; diffusion is considered to be isotropic and the diffusion coefficient is constant; the nominal flux , and is expressed as: （62） (63) (64)。 5. The numerical simulation method for a statistically based nonlinear dynamic gel constitutive model according to claim 1, characterized in that: The fifth step includes; All variables were normalized. All variables with the sign ̂ are normalized variables. All derivatives are calculated relative to normalized coordinates and time. The force balance equations are: （65） （66） （67） （68） The chemical potential equilibrium equation is: （69） （70） （71） In which, the incompressibility constraint is expressed in terms of dimensionless variables: （72） The conservation of mass equation is: （73） （74） （75） （76） （77） （78） （79） （80） （81） Based on the above constitutive equations, high-dimensional finite element simulation of active gel network is carried out, and mechanical analysis of active hydrogel under complex conditions is carried out.