A method of calculating the viscoelastic deformation of drying shrinkage of a cement-based material

Abstract

The application discloses a kind of calculation methods for predicting cement-based material dry shrinkage viscoelastic deformation, comprising the following steps: establishing cement-based material structure model;Using Darcy's law, Fick's law and phase change law to describe the moisture transport behavior occurred in the drying process of cement-based material;Based on the basic law of thermodynamics, the constitutive equation of cement-based material in the drying process is established;Establish humidity-mechanical coupling equation, determine the instantaneous elastic deformation of cement-based material dry shrinkage;Based on Boltzmann superposition principle, determine the time-dependent viscous deformation of cement-based material dry shrinkage, obtain the total deformation of cement-based material dry shrinkage;Determine the calculation parameters, predict the dry shrinkage of target cement-based material under specific conditions.The present application solves the problem that the current cement-based material dry shrinkage viscoelastic deformation prediction model is often an empirical model fitted under specific experimental conditions, with low accuracy, no universality and scalability.

Description

Technical Field

[0001] This invention relates to the field of cement-based materials technology, and in particular to a calculation method for predicting the viscoelastic deformation of cement-based materials during drying shrinkage. Background Technology

[0002] Cement often undergoes volume deformation during molding and use, with shrinkage deformation being the most common. Cement shrinkage mainly includes settlement shrinkage, chemical shrinkage, autogenous shrinkage, temperature shrinkage, carbonation shrinkage, and drying shrinkage. Drying shrinkage accounts for over 80% of shrinkage deformation and is a long-term phenomenon that severely affects the durability of cement-based material structures. Therefore, predicting the viscoelastic deformation of cement-based materials during drying shrinkage is a significant and important issue in order to improve the durability of cement-based material structures and extend their service life.

[0003] Currently, models for predicting the drying shrinkage deformation of cement-based materials are mainly divided into two categories: statistical empirical models and thermodynamic theoretical models. Statistical empirical models, through regression analysis of large amounts of material shrinkage test data, have proposed various engineering-oriented empirical or semi-empirical formulas, including the CEB / FIP method, the ACI method, the method suggested by Bazant and Panula, the Parrott method, and the method of the Japan Society of Civil Engineers. However, this method lacks an understanding of the microscopic mechanisms and requires extensive experimentation to obtain corrections under new historical conditions. Thermodynamic theoretical models, starting from the thermodynamic expressions of various behaviors occurring in the system, obtain the deformation expression under quasi-static thermodynamic equilibrium by solving the coupled field equations of these behaviors. Examples include the thermodynamic model based on diffusion behavior proposed by Bazant and the thermodynamic model of unsaturated porous materials proposed by Coussy et al. This method is only applicable to thermodynamic equilibrium states and cannot be applied to unequilibrium kinetic processes. However, because the main component of cement-based materials, CSH gel, has viscoelastic properties, in addition to the instantaneous elastic deformation obtained under quasi-static thermodynamic equilibrium, there is also viscous deformation that increases over time. This deformation often requires a sufficiently long time to reach thermodynamic equilibrium. Therefore, calculating only the instantaneous elastic deformation of cement-based materials under thermodynamic equilibrium ignores the viscous deformation that increases over time, which leads to an underestimation of the structural deformation of cement-based materials in long-term service, especially in high-temperature and low-humidity environments. Summary of the Invention

[0004] Purpose of the invention: The purpose of this invention is to provide a calculation method that can accurately predict the viscoelastic deformation of the drying shrinkage of a target cement-based material under given experimental conditions.

[0005] Technical Solution: To achieve the above objective, the present invention provides a calculation method for predicting the viscoelastic deformation of cement-based materials during drying shrinkage, comprising the following steps:

[0006] Step S1: Establish a structural model of cement-based materials;

[0007] Step S2: Use Darcy's law, Fick's law, and the law of phase transition to describe the moisture transport behavior of cement-based materials during the drying process;

[0008] Step S3: Based on the fundamental laws of thermodynamics, establish the constitutive equation of cement-based materials during the drying process;

[0009] Step S4: Establish the humidity-mechanical coupling equation to determine the instantaneous elastic deformation of the drying shrinkage of cement-based materials;

[0010] Step S5: Based on the Boltzmann superposition principle, determine the viscous deformation of the drying shrinkage of cement-based materials as time increases, and obtain the total deformation of the drying shrinkage of cement-based materials;

[0011] Step S6: Determine the calculation parameters to predict the drying shrinkage of the target cement-based material under specific conditions.

[0012] Specifically, step S1 involves establishing a cement-based material structure model by considering the cement-based material as a combination of two interacting continuums, including a compressible skeleton and an incompressible fluid, and taking the basic unit dΩ of the cement-based material structure model.

[0013] In step S2, Darcy's law, Fick's law, and the law of phase transition are used to describe the moisture transport behavior of cement-based materials during the drying process. Specifically, the seepage of liquid water and mixed gas is as follows:

[0014]

[0015] In the formula, i is a subscript representing liquid water l or mixed gas m; w is the mass flux per unit time of the basic unit dΩ of the cement-based material structural model; ρ is the density; k i The permeability coefficient of liquid water or mixed gas; P i The pressure of the liquid water or the mixed gas is given by grad(); grad() is the gradient function.

[0016] In step S2, Darcy's law, Fick's law, and the law of phase transition are used to describe the moisture transport behavior of cement-based materials during the drying process. Specifically, the diffusion of vapor water in the mixed gas is as follows:

[0017]

[0018] In the formula, v is a subscript representing gaseous water; f is the diffusion coefficient; w is the mass flux per unit time of the basic unit dΩ of the cement-based material structural model; ρ is the density; and P is the pressure.

[0019] In step S2, Darcy's law, Fick's law, and the law of phase transition are used to describe the moisture transport behavior of cement-based materials during the drying process. Specifically, the liquid-gas phase transition of water is as follows:

[0020] g l (P l ,T)=g v (P v ,T) (3),

[0021] In the formula, g l The Gibbs potential per unit mass of liquid water, g v v is the Gibbs potential per unit mass of vapor water; v is a subscript representing vapor water; P is the pressure.

[0022] Meanwhile, during the drying process, a meniscus forms in the capillaries inside the cement-based material, generating capillary stress, which in turn compresses the framework phase, specifically:

[0023]

[0024] In the formula, RH represents the relative humidity of the cement-based material; M v ρ is the molar volume of water vapor; R is the gas constant; T is the Kelvin temperature; ρ l P is the density of the liquid water phase. c This refers to the capillary stress generated within the cement-based material.

[0025] In step S3, the constitutive equation for cement-based materials during the drying process is established based on the fundamental laws of thermodynamics. Specifically, the first law of thermodynamics is introduced to express the energy changes in the structural model of the cement-based material.

[0026]

[0027] In the formula, E is the internal energy of the basic unit dΩ of the cement-based material structural model; t is time; i is a subscript representing liquid water l or mixed gas m; w is the mass flux per unit time of the basic unit dΩ of the cement-based material structural model; e i σ is the internal energy of phase i; q is the heat conduction flux per unit time; σ is the external stress acting on dΩ; ε is the strain generated by dΩ; P is the pressure; ρ is the density; F is the body force generated inside dΩ; div() is the divergence function.

[0028] In step S3, the constitutive equation for cement-based materials during the drying process is established based on the fundamental laws of thermodynamics. Specifically, the second law of thermodynamics is introduced to express the changes in the complexity of the structural model of the cement-based material.

[0029]

[0030] In the formula, S is the entropy of the basic unit dΩ of the cement-based material structure model; t is time; i is a subscript representing liquid water l or mixed gas m; w is the mass flux per unit time of the basic unit dΩ of the cement-based material structure model; q is the heat conduction flux per unit time; T is the Kelvin temperature; div() is the divergence function; φ is the heat conduction tensor spontaneously generated due to the energy dissipation of the model.

[0031] The energy dissipation of the model includes dissipation caused by irreversible deformation of the model skeleton, dissipation caused by phase transition, dissipation caused by heat conduction, and dissipation caused by mass transport.

[0032] φ=φ1+φ → +φ2+φ3 (8),

[0033] In the formula, φ1 represents the dissipation caused by the irreversible deformation of the model skeleton, and φ → φ1 represents the dissipation caused by phase change, φ2 represents the dissipation caused by heat conduction, and φ3 represents the dissipation caused by mass transport.

[0034] The dissipation caused by the irreversible deformation of the model skeleton is as follows:

[0035]

[0036] In the formula, σ is the external stress acting on dΩ; g i Ψ is the thermodynamic potential of phase t; Ψ is the Gibbs free energy of dΩ.

[0037] Under isothermal and isobaric conditions, the Gibbs free energy Ψ of the basic unit dΩ in the model represents the external variables ε and m. i and internal variable χ:

[0038] Ψ=Ψ(ε,m i ,χ) (10),

[0039] When only considering the instantaneous elastic deformation of the cement-based material structure model during moisture transport, the dissipation φ1 caused by the irreversible deformation of the model skeleton is 0.

[0040] Substituting equation (10) into equation (9) and combining it with φ1=0, we obtain the constitutive equations of the model, as shown in equations 12 and 13:

[0041]

[0042]

[0043]

[0044] Specifically, step S4, which involves establishing the humidity-mechanical coupling equation to determine the instantaneous elastic deformation of the drying shrinkage of cement-based materials, includes:

[0045]

[0046] In the formula, P is pressure; ρ is density; i is a subscript representing liquid water (l), mixed gas (m), or gaseous water (v); Ψ is the Gibbs free energy of dΩ; ε and m i Indicates an external variable; j is a subscript, representing liquid water (l), mixed gas (m), or gaseous water (v);

[0047] Define the parameters for the physical quantities in equation (14):

[0048]

[0049] In the formula, K is the bulk modulus of the cement-based material in the saturated state;

[0050]

[0051] In the formula, M ij The Biot modulus of cement-based materials;

[0052]

[0053] In the formula, B is the Biot coefficient of cement-based materials;

[0054] Substituting the parameters into equation (14), we obtain the expression for the external stress:

[0055]

[0056] In the formula, K0 is the bulk modulus of the cement-based material in a dry state;

[0057] Substituting the values ​​of each phase, we finally obtain the expression for the external stress:

[0058] dσ=K0dε-B l dP c -dP m (19);

[0059] In the formula, P c B refers to the capillary stress generated within the cement-based material. l P is the Biot coefficient of the liquid phase water in cement-based materials. m The pressure of the mixed gas;

[0060] Since the cement-based material is under isothermal and isobaric conditions and the applied stress is zero during the drying process, we obtain:

[0061] K0dε=(S+1)dP c (20),

[0062] Substituting equation (4) into equation (20), we get:

[0063]

[0064] In the formula, ε e M represents the instantaneous elastic deformation that occurs in cement-based materials during the drying process. v This represents the molar volume of water vapor.

[0065] Specifically, step S5, based on the Boltzmann superposition principle, determines the viscous deformation of the cement-based material during drying shrinkage, thus obtaining the total deformation of the cement-based material during drying shrinkage. This is because, since the capillary stress acting on the cement gel is less than 50% of the strength of the cement-based material, the viscous deformation, i.e., creep, of the cement-based material is considered to have a linear relationship with the stress.

[0066]

[0067]

[0068]

[0069] In the formula, J(t, τ0) is the creep function, representing the magnitude of creep from time τ0 to time t under a unit stress; τ i For characteristic time; C M ε is the creep modulus; c ε represents the viscous deformation of the cement-based material during the drying process, which increases with time; ε represents the total strain of the cement-based material during the drying process; P c The capillary stress generated within the cement-based material; RH is the relative humidity of the cement-based material; K0 is the bulk modulus of the cement-based material in its dry state; S is the entropy of the basic element dΩ in the structural model of the cement-based material; T is the Kelvin temperature; ρ is the density of liquid water; M v This represents the molar volume of water vapor.

[0070] Specifically, step S6 involves determining the calculation parameters to predict the drying shrinkage of the target cement-based material under specific conditions. This includes determining the intrinsic properties of the target cement-based material, inputting the humidity field and saturation field, and thus determining the strain field of the cement-based material. The strain field refers to the relationship between the drying shrinkage strain inside the cement-based material and the changes in time and location. In addition, the mapping relationship between strain and humidity, and between strain and saturation can also be obtained.

[0071] Beneficial effects: The present invention has the following advantages: 1. Under given experimental conditions, by inputting the intrinsic performance parameters of cement-based materials as well as the humidity field and saturation field, the present invention can calculate the strain field of cement-based materials and obtain the mapping relationship between strain and humidity, and strain and saturation, thereby providing assistance for the study of the drying shrinkage mechanism of cement-based materials;

[0072] 2. This invention, from a thermodynamic perspective, first considers the instantaneous elastic deformation of cement-based materials during the drying process; secondly, it uses the Boltzmann superposition principle to express the deformation dynamics under non-equilibrium conditions, calculating the time-dependent viscous deformation of the cement-based material during drying, and finally obtaining the expression for the total deformation of the sample. The calculation results have universality and scalability, laying the foundation for real-world applications of multi-field coupling modeling. Attached Figure Description

[0073] Figure 1 This is a flowchart of the method of the present invention;

[0074] Figure 2 This is a schematic diagram of a cement-based material structure model;

[0075] Figure 3 This is a strain-time curve at different locations inside the hardened cement paste in 50% RH environment in Example 1;

[0076] Figure 4 This is a strain-humidity curve at different locations inside the hardened cement paste in 50% RH environment in Example 1;

[0077] Figure 5 This is a strain-saturation curve at different locations inside the hardened cement paste in 50% RH environment in Example 1;

[0078] Figure 6 This is a strain-time curve at different locations inside the hardened cement paste in 30% RH environment in Example 2;

[0079] Figure 7 This is a strain-humidity curve at different locations inside the hardened cement paste in an environment of 30% RH in Example 2;

[0080] Figure 8 This is a strain-saturation curve at different locations inside the hardened cement paste in 30% RH environment in Example 2. Detailed Implementation

[0081] The technical solution of the present invention will be described in detail below with reference to specific embodiments and accompanying drawings.

[0082] Example 1: Prediction of strain field of hardened cement paste under single-sided drying in 50% RH environment.

[0083] This invention provides a calculation method for predicting the viscoelastic deformation of cement-based materials during drying shrinkage, such as... Figure 1 The diagram shown is a flowchart of the method of the present invention, which includes the following steps:

[0084] Step S1: Establish a structural model of the cement-based material. Specifically, the cement-based material is considered as a combination of two interacting continuums, including a compressible skeleton and an incompressible fluid. The basic unit dΩ of the cement-based material structural model is taken as follows: Figure 2 As shown.

[0085] Step S2: Darcy's law, Fick's law, and the law of phase transition are used to describe the moisture transport behavior of cement-based materials during the drying process, including the seepage of liquid water and mixed gas, the diffusion of gaseous water in the mixed gas, and the liquid-gas phase transition of water. Specifically, the seepage of liquid water and mixed gas is as follows:

[0086]

[0087] In the formula, t is a subscript that can be used to represent liquid water (l) and mixed gas (m); w is the mass flux per unit time of dΩ; ρ is the density; k i P is the permeability coefficient of liquid water or mixed gas. i The pressure of the liquid water or the mixed gas is given by grad(); grad() is the gradient function.

[0088] The diffusion of vaporous water in the mixed gas is specifically as follows:

[0089]

[0090] In the formula, v is a subscript used to represent gaseous water; f is the diffusion coefficient; and P is the pressure.

[0091] The liquid-gas phase transition of water specifically includes:

[0092] g l (P l ,T)=g v (P v ,T) (3),

[0093] In the formula, g l The Gibbs potential per unit mass of liquid water, g v The Gibbs potential is given by the mass of vapor phase water.

[0094] Meanwhile, during the drying process, a meniscus forms in the capillaries inside the cement-based material, generating capillary stress, which in turn compresses the framework phase, specifically:

[0095]

[0096] In the formula, RH is the relative humidity; M v ρ is the molar volume of water vapor; R is the gas constant; T is the Kelvin temperature; ρ ε ρ is the density of the liquid water phase; l P is the density of the liquid water phase. c This refers to the capillary stress generated within the cement-based material.

[0097] Step S3: Based on the fundamental laws of thermodynamics, establish the constitutive equation for cement-based materials during the drying process, specifically:

[0098] The first law of thermodynamics is introduced to express the energy changes in the structural model of cement-based materials:

[0099]

[0100] In the formula, E is the internal energy of dΩ; t is time; e i ε is the internal energy of phase i; q is the heat conduction flux per unit time; ε is the strain generated by dΩ; F is the body force generated inside dΩ; div() is the divergence function.

[0101] The second law of thermodynamics is introduced to express the changes in complexity of cement-based material structural models:

[0102]

[0103] In the formula, S is the entropy of dΩ; φ is the spontaneous heat conduction tensor due to the energy dissipation of the model.

[0104] The energy dissipation of the model includes dissipation caused by irreversible deformation of the model skeleton, dissipation caused by phase transition, dissipation caused by heat conduction, and dissipation caused by mass transport.

[0105] φ=φ1+φ → +φ2+φ3 (8),

[0106] In the formula, φ1 represents the dissipation caused by the irreversible deformation of the model skeleton, and φ → φ1 represents the dissipation caused by phase change, φ2 represents the dissipation caused by heat conduction, and φ3 represents the dissipation caused by mass transport.

[0107] The dissipation caused by the irreversible deformation of the model skeleton is as follows:

[0108]

[0109] In the formula, σ is the external stress acting on dΩ; g i Ψ is the thermodynamic potential of phase i; Ψ is the Gibbs free energy of dΩ.

[0110] Based on the above discussion and under isothermal and isobaric conditions, the Gibbs free energy Ψ of the model's basic element dΩ can represent the external variables (ε, m). i A function of the model's energy dissipation and the internal variable (χ, representing the effect of model energy dissipation on model free energy):

[0111] Ψ=Ψ(ε,m i ,χ) (10),

[0112] When only considering the instantaneous elastic deformation of the cement-based material structure model during moisture transport, the dissipation φ1 caused by the irreversible deformation of the model skeleton can be assumed to be 0. Substituting Equation 10 into Equation 9 and combining it with φ1=0, the constitutive equations of the model are obtained, as shown in Equations 12 and 13:

[0113]

[0114]

[0115]

[0116] Step S4: Establish the humidity-mechanical coupling equation to determine the instantaneous elastic deformation of the drying shrinkage of cement-based materials, specifically:

[0117]

[0118] Define the parameters for the physical quantities in equation (14):

[0119]

[0120] In the formula, K is the bulk modulus of the cement-based material in the saturated state;

[0121]

[0122] In the formula, M ij The Biot modulus of cement-based materials;

[0123]

[0124] In the formula, B is the Biot coefficient of cement-based materials;

[0125] Substituting the parameters into equation (14), we obtain the expression for the external stress:

[0126]

[0127] In the formula, K0 is the bulk modulus of the cement-based material in a dry state;

[0128] Substituting the values ​​of each phase, we finally obtain the expression for the external stress:

[0129] dσ=K0dε-B l dP c -dP m (19);

[0130] In the formula, j is a subscript representing liquid water (l), mixed gas (m), or gaseous water (v); K is the bulk modulus of the cement-based material under saturated conditions; M ij B is the Biot modulus of the cement-based material; B is the Biot coefficient of the cement-based material; K0 is the bulk modulus of the cement-based material in the dry state; B l P is the Biot coefficient of the liquid phase water in cement-based materials. m This represents the pressure of the mixed gas.

[0131] Because cement-based materials are under isothermal and isobaric conditions throughout the drying process and the applied stress is 0 (σ=0, P m =P atm =constant), resulting in:

[0132] K0dε=(S+1)dP c (20),

[0133] Substituting equation 4 into equation 20, we get:

[0134]

[0135] In the formula, ε e This refers to the instantaneous elastic deformation that occurs in cement-based materials during the drying process.

[0136] Step S5: Based on the Boltzmann superposition principle, determine the viscous deformation of the cement-based material during drying shrinkage over time, and obtain the total deformation of the cement-based material during drying shrinkage. Specifically, since the capillary stress acting on the cement gel is less than 50% of the strength of the cement-based material, the viscous deformation, i.e., creep, of the cement-based material can be considered to satisfy a linear relationship with the stress.

[0137]

[0138]

[0139]

[0140] In the formula, J(t, τ0) is the creep function, representing the magnitude of creep from time τ0 to time t under a unit stress; τ i For characteristic time; C M ε is the creep modulus; c ε represents the viscous deformation of the cement-based material during the drying process, which increases over time; ε represents the total strain of the cement-based material during the drying process.

[0141] Step S6: Determine the calculation parameters and predict the drying shrinkage of the target cement-based material under specific conditions. Specifically, determine the intrinsic properties of the target cement-based material, input the humidity field and saturation field, and thus determine the strain field of the cement-based material. The strain field refers to the relationship between the drying shrinkage strain inside the cement-based material and the time and location. In addition, the mapping relationship between strain and humidity, and strain and saturation can also be obtained.

[0142] The intrinsic properties of the target cementitious material are: water-cement ratio: 0.5, bulk modulus: 7 GPa, characteristic time: 10 days, and creep modulus: 1 GPa. The experimental conditions were isothermal and isobaric, with room temperature (298.15 K), standard atmospheric pressure (101.325 kPa), and ambient humidity (50% RH). Furthermore, the Parrot humidity prediction model and empirical formula for the saturation field were used, as shown below:

[0143] RH=RH0+(100-RH0)f(t) (25),

[0144] f(t) = 1 / (1+t / b) (26),

[0145] b = d 1.35 (70-e)(w-0.19) / 8 (27),

[0146] In the formula, RH0 is the ambient relative humidity; f(t) is a function of time t; b is a parameter related to position d and material composition; d is the depth from the dry surface; e is the proportion of cement substitute; and w is the water-cement ratio.

[0147]

[0148] λ1=(2.9142w / c-2.5849)×10 -3 T-0.1994w / c+0.1647 (29)

[0149] λ2=(2.907w / c-1.1446×10 -3 T+1.5594×10- 5 T 3 +4.4465)×10 -3 (30),

[0150] λ3=(2.158w / c-3.2774)×10 -3 T-0.3272w / c+0.3154 (31),

[0151] In the formula, S is the saturation of the sample; λ1 is a parameter related to the water-cement ratio and temperature of the sample; λ2 is a parameter related to the water-cement ratio and temperature of the sample; λ3 is a parameter related to the water-cement ratio and temperature of the sample; w / c is the water-cement ratio of the sample; and T is the temperature of the sample.

[0152] By substituting relevant parameters and writing a Matlab program to predict the strain field of cement-based materials, the strain variation over time at different locations within the hardened cement paste under a 50% RH environment can be obtained. It can be observed that the strain at any location continuously increases with time. Figure 3 As shown. Furthermore, the mapping relationship between strain and relative humidity, as well as the mapping relationship between strain and saturation, at different locations within the hardened cement paste can be obtained. It can be found that the closer to the dry surface, the smaller the strain state when reaching the same humidity or saturation state, such as... Figure 4 , Figure 5 As shown.

[0153] Example 2: Prediction of strain field of hardened cement paste under single-sided drying in 30% RH environment.

[0154] This invention provides a calculation method for predicting the viscoelastic deformation of cement-based materials during drying shrinkage, such as... Figure 1 The diagram shown is a flowchart of the method of the present invention, which includes the following steps:

[0155] Step S1: Establish a structural model of the cement-based material. Specifically, the cement-based material is considered as a combination of two interacting continuums, including a compressible skeleton and an incompressible fluid. The basic unit of the model is dΩ, as shown in the figure. Figure 2 As shown.

[0156] Step S2: Darcy's law, Fick's law, and the law of phase transition are used to describe the moisture transport behavior of cement-based materials during the drying process, specifically the seepage of liquid water and mixed gas, the diffusion of gaseous water in the mixed gas, and the liquid-gas phase transition of water:

[0157]

[0158] In the formula, t is a subscript that can be used to represent liquid water (l) and mixed gas (m); w is the mass flux of dΩ per unit time; k i P is the permeability coefficient of liquid water or mixed gas. i It is the pressure of liquid water or a mixture of gases.

[0159]

[0160] In the formula, v is a subscript used to represent gaseous water; f is the diffusion coefficient.

[0161] g l (Pl ,T)=g v (P v ,T) (3),

[0162] In the formula, g l The Gibbs potential per unit mass of liquid water, g v The Gibbs potential is given by the mass of vapor phase water.

[0163] Meanwhile, during the drying process, a meniscus forms in the capillaries inside the cement-based material, generating capillary stress, which in turn compresses the framework phase, specifically:

[0164]

[0165] In the formula, RH is the relative humidity; M v ρ is the molar volume of water vapor; R is the gas constant; T is the Kelvin temperature; ρ l This is the density of the liquid water.

[0166] Step S3: Based on the fundamental laws of thermodynamics, establish the constitutive equation for cement-based materials during the drying process, specifically:

[0167] The first law of thermodynamics is introduced to express the energy changes in the structural model of cement-based materials:

[0168]

[0169] In the formula, E is the internal energy of dΩ; t is time; q is the heat conduction flux per unit time; and ε is the strain generated by dΩ.

[0170] The second law of thermodynamics is introduced to express the changes in complexity of cement-based material structural models:

[0171]

[0172] In the formula, S is the entropy of dΩ; φ is the spontaneous heat conduction tensor due to the energy dissipation of the model.

[0173] The energy dissipation of the model includes dissipation caused by irreversible deformation of the model skeleton, dissipation caused by phase transition, dissipation caused by heat conduction, and dissipation caused by mass transport.

[0174] φ=φ1+φ → +φ2+φ3 (8),

[0175] In the formula, φ1 represents the dissipation caused by the irreversible deformation of the model skeleton, and φ → φ1 represents the dissipation caused by phase change, φ2 represents the dissipation caused by heat conduction, and φ3 represents the dissipation caused by mass transport.

[0176] The dissipation caused by the irreversible deformation of the model skeleton is as follows:

[0177]

[0178] In the formula, Ψ is the Gibbs free energy of dΩ.

[0179] Based on the above discussion and under isothermal and isobaric conditions, the Gibbs free energy Ψ of the model's basic element dΩ can represent the external variables (ε, m). i A function of the model's energy dissipation and the internal variable (χ, representing the effect of model energy dissipation on model free energy):

[0180] Ψ=Ψ(ε,m i ,χ) (10),

[0181] When only considering the instantaneous elastic deformation of the cement-based material structure model during moisture transport, the dissipation φ1 caused by the irreversible deformation of the model skeleton can be assumed to be 0. Substituting Equation 10 into Equation 9 and combining it with φ1=0, we obtain the constitutive equation of the model:

[0182]

[0183]

[0184]

[0185] Step S4: Establish the humidity-mechanical coupling equation to determine the instantaneous elastic deformation of the drying shrinkage of cement-based materials, specifically:

[0186]

[0187]

[0188]

[0189]

[0190]

[0191] dσ=K0dε-B l dP c -dP m (19),

[0192] In the formula, K is the bulk modulus of the cement-based material in the saturated state; M ij B represents the Biot modulus of cement-based materials. j is the Biot coefficient of cement-based materials; K0 is the bulk modulus of cement-based materials in the dry state.

[0193] Because cement-based materials are under isothermal and isobaric conditions throughout the drying process and the applied stress is 0 (σ=0, P m =P atm =constant), resulting in:

[0194] K0dε=(S+1)dP c (20),

[0195] Substituting equation 4 into equation 20, we get:

[0196]

[0197] In the formula, ε e This refers to the instantaneous elastic deformation that occurs in cement-based materials during the drying process.

[0198] Step S5: Based on the Boltzmann superposition principle, determine the viscous deformation of the cement-based material during drying shrinkage over time, and obtain the total deformation of the cement-based material during drying shrinkage. Specifically, since the capillary stress acting on the cement gel is less than 50% of the strength of the cement-based material, the viscous deformation, i.e., creep, of the cement-based material can be considered to satisfy a linear relationship with the stress.

[0199]

[0200]

[0201]

[0202] In the formula, J(t, τ0) is the creep function, representing the magnitude of creep from time τ0 to time t under a unit stress; τ i For characteristic time; C M ε is the creep modulus; c ε represents the viscous deformation of the cement-based material during the drying process, which increases over time; ε represents the total strain of the cement-based material during the drying process.

[0203] Step S6: The calculation process is the same as in Example 1, except that the calculation parameters are modified as follows:

[0204] The intrinsic properties of the target cementitious material are: water-cement ratio: 0.5, bulk modulus: 7 GPa, characteristic time: 10 days, and creep modulus: 1 GPa. The experimental conditions were isothermal and isobaric, with room temperature (298.15 K), standard atmospheric pressure (101.325 kPa), and ambient humidity (30% RH). Furthermore, the Parrot humidity prediction model and empirical formula for the saturation field were used, as shown below:

[0205] RH=RH0+(100-RH0)f(t) (25),

[0206] f(t) = 1 / (1+t / b) (26),

[0207] b = d 1.35 (70-e)(w-0.19) / 8 (27),

[0208] In the formula, RH0 is the ambient relative humidity; f(t) is a function of time t; b is a parameter related to position d and material composition; d is the depth from the dry surface; e is the proportion of cement substitute; and w is the water-cement ratio.

[0209]

[0210] λ1=(2.9142w / c-2.5849)×10 -3 T-0.1994w / c+0.1647 (29)

[0211] λ2=(2.907w / c-1.1446×10 -3 T+1.5594×10 -5 T 3 +4.4465)×10 -3 (30),

[0212] λ3=(2.158w / c-3.2774)×10 -3 T-0.3272w / c+0.3154 (31),

[0213] In the formula, S is the saturation of the sample; λ1 is a parameter related to the water-cement ratio and temperature of the sample; λ2 is a parameter related to the water-cement ratio and temperature of the sample; λ3 is a parameter related to the water-cement ratio and temperature of the sample; w / c is the water-cement ratio of the sample; and T is the temperature of the sample.

[0214] By substituting relevant parameters and writing a Matlab program to predict the strain field of cement-based materials, the strain variation over time at different locations within the hardened cement paste under a 30% RH environment can be obtained. It can be observed that the strain at any location continuously increases with time. Figure 6 As shown. Furthermore, the mapping relationship between strain and relative humidity, as well as the mapping relationship between strain and saturation, at different locations within the hardened cement paste can be obtained. It can be found that the closer to the dry surface, the smaller the strain state when reaching the same humidity or saturation state, such as... Figure 7 , Figure 8 As shown.

Claims

1. A calculation method for predicting the viscoelastic deformation of cement-based materials during drying shrinkage, characterized in that, Includes the following steps: Step S1: Establish a structural model of cement-based materials. Specifically, cement-based materials are considered as a combination of two interacting continuums, including a compressible skeleton and an incompressible fluid. The basic unit of the cement-based material structural model is dΩ. Step S2: Based on the structural model of cement-based materials, the moisture transport behavior of cement-based materials during the drying process is described by Darcy's law, Fick's law, and the phase transition law. Among them, the seepage of liquid water and mixed gas is described by Darcy's law; the diffusion of gaseous water in mixed gas is described by Fick's law; the liquid-gas phase transition of water is described by the phase transition law, and the compression of the skeleton by capillary stress is considered. Step S3: Based on the fundamental laws of thermodynamics, establish the constitutive equation of cement-based materials during the drying process; Step S4: Establish the humidity-mechanical coupling equation to determine the instantaneous elastic deformation of the drying shrinkage of cement-based materials; Step S5: Based on the Boltzmann superposition principle, determine the viscous deformation of the cement-based material during drying shrinkage as time increases; based on the instantaneous elastic deformation and the viscous deformation as time increases, obtain the total deformation calculation formula for the drying shrinkage of the cement-based material; Step S6: Determine the intrinsic performance parameters of the target cement-based material and obtain its internal humidity field and saturation field. Based on the total deformation calculation formula, determine the strain field of the cement-based material and predict the drying shrinkage of the target cement-based material under specific conditions. The strain field refers to the relationship between the drying shrinkage strain inside the cement-based material and time and location. In addition, the mapping relationship between strain and humidity, and strain and saturation can also be obtained.

2. The calculation method for predicting the viscoelastic deformation of cement-based materials during drying shrinkage according to claim 1, characterized in that, Step S2 describes the moisture transport behavior of cement-based materials during the drying process using Darcy's law, Fick's law, and the law of phase transition. Specifically, the seepage of liquid water and mixed gas is as follows: （1）， In the formula, i is a subscript representing liquid water l or mixed gas m; w is the mass flux per unit time of the basic unit dΩ of the cement-based material structural model. ρ is density; k i The permeability coefficient of liquid water or mixed gas; P i The pressure of the liquid water or the mixed gas is given by grad(); grad() is the gradient function.

3. The calculation method for predicting the viscoelastic deformation of cement-based materials during drying shrinkage according to claim 1, characterized in that, Step S2 describes the moisture transport behavior of cement-based materials during the drying process using Darcy's law, Fick's law, and the law of phase transition. Specifically, the diffusion of vapor water in the mixed gas is as follows: （2）， In the formula, v is a subscript representing gaseous water; f is the diffusion coefficient; w is the mass flux per unit time of the basic unit dΩ of the cement-based material structural model; ρ is the density; and P is the pressure.

4. The calculation method for predicting the viscoelastic deformation of cement-based materials during drying shrinkage according to claim 1, characterized in that, Step S2 describes the moisture transport behavior of cement-based materials during the drying process using Darcy's law, Fick's law, and the law of phase transition. Specifically, the liquid-gas phase transition of water is as follows: （3）， In the formula, The Gibbs potential per unit mass of liquid water. The Gibbs potential per unit mass of vapor water; The subscript indicates vapor phase water; Pressure; Meanwhile, during the drying process, a meniscus forms in the capillaries inside the cement-based material, generating capillary stress, which in turn compresses the framework phase, specifically: （4）， In the formula, The relative humidity of cement-based materials; This represents the molar volume of water vapor. It is the gas constant; Kelvin temperature, Let be the density of the liquid water. This refers to the capillary stress generated within the cement-based material.

5. The calculation method for predicting the viscoelastic deformation of cement-based materials during drying shrinkage according to claim 1, characterized in that, Step S3 describes establishing the constitutive equation for cement-based materials during the drying process based on the fundamental laws of thermodynamics. Specifically, the first law of thermodynamics is introduced to express the energy changes in the structural model of the cement-based material. （5）， In the formula, dΩ represents the internal energy of the basic unit dΩ in the structural model of cement-based materials; t represents time. The subscript indicates liquid water. or mixed gas ; dΩ represents the mass flux per unit time of the basic unit dΩ in the cement-based material structural model. for The internal energy of a phase; The heat conduction flux per unit time; The external stress acting on dΩ; The strain generated by dΩ; Pressure; Density; The body force generated inside dΩ; Let be the divergence function.

6. The calculation method for predicting the viscoelastic deformation of cement-based materials during drying shrinkage according to claim 1, characterized in that, Step S3 describes establishing the constitutive equation for cement-based materials during the drying process based on the fundamental laws of thermodynamics. Specifically, the second law of thermodynamics is introduced to express the changes in the complexity of the cement-based material's structural model. （6）， In the formula, dΩ is the entropy of the basic unit dΩ in the structural model of cement-based materials; t is time. The subscript indicates liquid water. or mixed gas ; dΩ represents the mass flux per unit time of the basic unit dΩ in the cement-based material structural model. The heat conduction flux per unit time; Kelvin temperature; It is the divergence function; This is due to the spontaneous heat conduction tensor caused by the energy dissipation of the model; The energy dissipation of the model includes dissipation caused by irreversible deformation of the model skeleton, dissipation caused by phase transition, dissipation caused by heat conduction, and dissipation caused by mass transport. （8）， In the formula, This refers to the dissipation caused by the irreversible deformation of the model skeleton. Dissipation caused by phase transition Dissipation due to heat conduction, Dissipation resulting from the transport of matter; The dissipation caused by the irreversible deformation of the model skeleton is as follows: （9）， In the formula, The external stress acting on dΩ; for Thermodynamic potential of the phase; The Gibbs free energy of dΩ; Under isothermal and isobaric conditions, the Gibbs free energy Ψ of the model's basic element dΩ is represented as an external variable. , and internal variable χ: （10）， When only considering the instantaneous elastic deformation of the cement-based material structure model during moisture transport, the dissipation caused by the irreversible deformation of the model skeleton... =0; Substitute equation (10) into equation (9) and combine The constitutive equations of the model are obtained, as shown in Equations 12 and 13: （11）， （12）， （13）。 7. The calculation method for predicting the viscoelastic deformation of cement-based materials during drying shrinkage according to claim 1, characterized in that, Step S4, which establishes the humidity-mechanical coupling equation to determine the instantaneous elastic deformation of cement-based materials during drying shrinkage, specifically involves: （14）， In the formula, Pressure; Density; The subscript indicates liquid water. or mixed gas or vapor phase water ; The Gibbs free energy of dΩ; , Represents external variables; The subscript indicates liquid water. or mixed gas or vapor phase water ; Define the parameters for the physical quantities in equation (14): （15）， In the formula, K is the bulk modulus of the cement-based material in the saturated state; （16）， In the formula, The Biot modulus of cement-based materials; （17）， In the formula, B is the Biot coefficient of cement-based materials; Substituting the parameters into equation (14), we obtain the expression for the external stress: （18）， In the formula, K0 is the bulk modulus of the cement-based material in a dry state; Substituting the values ​​of each phase, we finally obtain the expression for the external stress: （19）； In the formula, This refers to the capillary stress generated within cement-based materials. The Biot coefficient for the liquid phase water in cement-based materials; The pressure of the mixed gas; Since the cement-based material is under isothermal and isobaric conditions and the applied stress is zero during the drying process, we obtain: （20）， Substituting equation (4) into equation (20), we get: （21）， In the formula, This refers to the instantaneous elastic deformation that occurs in cement-based materials during the drying process; The saturation level of the sample; RH is the molar volume of water vapor; R is the relative humidity of the cement-based material; T is the gas constant; and T is the Kelvin temperature.

8. The calculation method for predicting the viscoelastic deformation of cement-based materials during drying shrinkage according to claim 1, characterized in that, Step S5, based on the Boltzmann superposition principle, determines the viscous deformation of cement-based materials during drying shrinkage over time, yielding a formula for calculating the total deformation of cement-based materials during drying shrinkage. Specifically, since the capillary stress acting on the cement gel is less than 50% of the strength of the cement-based material, the viscous deformation, i.e., creep, of the cement-based material is considered to satisfy a linear relationship with the stress. （22）， （23）， （24）， In the formula, Let be the creep function, representing The unit stress applied at any time to The magnitude of creep over time; Characteristic time; For creep modulus; This refers to the viscous deformation of cement-based materials that occurs over time during the drying process; This represents the total strain that occurs during the drying process of cement-based materials. This refers to the capillary stress generated within cement-based materials. The relative humidity of cement-based materials; This is the bulk modulus of cement-based materials in their dry state. The saturation level of the sample; Kelvin temperature; liquid water density; R is the molar volume of water vapor; R is the gas constant.

Images