High arch dam deformation state fractional order numerical analysis method and module

Abstract

This invention discloses a fractional-order numerical analysis method and module for the deformation behavior of high arch dams, relating to the fields of service behavior analysis and computational mechanics technology for hydraulic structures. By constructing fractional-order characterization methods for decaying rheology, steady-state rheology, and accelerated rheology under three-dimensional stress states, fractional-order rheological mechanical element models of the dam body and foundation of high arch dams are established; using the incremental finite element method, based on t n The stress, viscoelastic strain, viscoplastic strain, and deformation at time t are calculated using the predifference formula. n The stress increment, viscoelastic strain increment, viscoplastic strain increment, and deformation increment within the time period are used to derive t. n+1 By repeatedly calculating the stress, strain, and deformation at specific moments, the stress, strain, and deformation of the high arch dam within the analysis period are obtained. This invention not only objectively characterizes the nonlinear rheological effects of the dam body and foundation of the high arch dam, but also achieves scientific numerical quantitative analysis of deformation behavior, providing technical support for safety monitoring and early warning of potential hazards in in-service high arch dams, and has broad application prospects and market potential.

Description

Technical Field

[0001] This invention relates to the field of service behavior analysis and computational mechanics technology for hydraulic structures, and more specifically to a fractional-order numerical analysis method and module for the deformation behavior of high arch dams. Background Technology

[0002] my country has abundant water resources, totaling approximately 2.8124 trillion cubic meters. 3 China's rivers and lakes account for approximately 6% of the world's total runoff resources, with a theoretical hydropower potential of about 680 million kilowatts, ranking first in the world in both potential and development capacity. Hydropower has advantages such as stable output, low pollution, and renewability, and has become an important component of my country's energy structure. Since the 1950s, to meet the needs of flood control, power generation, irrigation, water supply, and navigation, my country has built more than 98,000 dams. With the deepening of hydropower development, the construction of high arch dams has achieved remarkable success. Currently, a number of high arch dams have been completed and put into operation, such as Jinping I, Xiaowan, Xiluodu, and Dagangshan. These mega-projects have played a vital role in my country's economic and social development.

[0003] The safe operation of dams has always been a major concern for governments and the international community, especially for high arch dams. A failure in such dams would cause enormous casualties and property damage downstream. The failure of an arch dam is a gradual process, progressing from localized structural failure to overall instability. However, if scientific and effective methods are used to monitor the dam's operation in real time, promptly detect any abnormal signs, and implement targeted engineering and non-engineering measures, accidents can potentially be avoided. Deformation changes are a comprehensive reflection of the structural behavior of high arch dams and are an important indicator of structural health. Numerous historical engineering cases have demonstrated the importance of analyzing the deformation behavior of arch dams. The Malpasse Arch Dam in France is 66 meters high, 222.7 meters long at the crest, and 6.78 meters thick at the base, with a thickness-to-height ratio of 0.103 and a reservoir capacity of 51 million cubic meters. 3In February 1959, the dam suddenly collapsed, resulting in over 500 deaths or disappearances and property losses of 30 billion francs. Comparison of deformation simulation results with monitoring data revealed that the deformation monitoring values ​​four months before the collapse were greater than calculated values. The radial deformation monitoring values ​​at the arch crown beam, 8-10 meters from the dam foundation, reached 12-16 mm, 2-2.8 times the calculated values. The deformation near the dam foundation was even greater, far exceeding the warning values. A typical case of avoiding accidents through effective analysis of deformation behavior is the Zeuzier hyperbolic arch dam in Switzerland. Under normal water levels, vertical deformation monitoring data showed abnormal upstream deformation of the dam. The reservoir was emptied and the dam inspected, revealing multiple cracks on the upstream dam face. Analysis of slope and reservoir settlement monitoring data revealed large-scale subsidence in the valley where the dam was located, and the valley width narrowing. After repairs, the dam returned to normal operation. Therefore, establishing scientific theories, methods, and technologies for analyzing the deformation behavior of high arch dams will help engineering technicians to detect potential hazards early and prevent dam failures.

[0004] The spatiotemporal characteristics of the deformation behavior of high arch dams are closely related to their structural physical and mechanical properties. When the external load changes, the high arch dam first undergoes instantaneous deformation. Under continuous load, due to the creep effect of the dam body and the creep effect of the dam foundation (both collectively referred to as rheological effects), the high arch dam undergoes time-dependent deformation. Scientific characterization of the physical and mechanical properties of high arch dams is the prerequisite and foundation for objectively and quantitatively analyzing their deformation behavior. The physical and mechanical properties of high arch dams include elasticity, plasticity, viscoelasticity, viscoplasticity, and viscoelastic-plasticity. Currently, models describing these mechanical properties can be categorized into empirical models, component models, and theoretical models. Among these, component models are widely used in engineering due to their simplicity and relatively clear physical meaning. Traditional component models are constructed based on integer-order calculus theory, using series and parallel combinations of basic mechanical components such as springs, sticky pots, and sliders to characterize the aforementioned physical and mechanical properties. While classic integer-order rheological element models are convenient to use, they have certain limitations. For example, when using integer-order standard linear bodies (Hooke and Kelvin bodies in series) to simulate rheological experiments, although instantaneous and delayed elasticity can be characterized, the calculated results for relaxation modulus and other parameters do not fit the experimental data well. Integer-order rheological element models alone are insufficient to adequately characterize the nonlinear rheological effects of high arch dams. The methods for characterizing physical and mechanical properties are a crucial factor limiting the objectivity of numerical analysis of the deformation behavior of high arch dams.

[0005] Fractional calculus is the theory of calculus that studies operations of fractional order. In the early 19th century, mathematicians began researching fractional calculus, with the work of scientists such as Laplace and Fourier laying its theoretical foundation. However, for a considerable period, due to the unclear physical meaning of fractional calculus and the lack of support from applied fields, related research stagnated in the theoretical exploration stage. The international conference "Fractional Calculus and Its Application" held at the University of New Haven in 1974 spurred the development of fractional calculus in applied fields. Currently, it has been successfully applied in geotechnical engineering, electromagnetism, fractals, and other fields, and its advantages over classical integer calculus are gradually being discovered. To date, scholars have rigorously defined fractional calculus from different perspectives, with commonly used definitions including the Riemann-Liouville definition, the Caputo definition, and the Grunwald-Letnikov definition. Essentially, fractional calculus is a generalization of integer calculus to fractional calculus. When the order of fractional calculus is an integer, the two are consistent. Furthermore, due to the inherent characteristics of fractional calculus, it is related to historical information; that is, fractional calculus possesses a memory function, while integer calculus does not. Combining fractional calculus theory with integer-order representation methods for rheological effects can clarify the physical meaning of component models, providing an effective approach to describing the nonlinear rheological behavior of high arch dams.

[0006] Existing research on fractional-order component models focuses on rheological experimental analysis, with numerous one-dimensional stress state models. However, in actual operation, high arch dams operate under three-dimensional stress, yet research on three-dimensional models is scarce, especially fractional-order component models characterizing accelerated rheological effects under three-dimensional high stress conditions. Therefore, based on fractional-order calculus theory, researching methods for constructing fractional-order rheological component models of the dam body and foundation under three-dimensional stress conditions, and subsequently achieving fractional-order numerical analysis of the deformation behavior of high arch dams, is a crucial problem urgently needing to be solved by technicians in the fields of hydraulic structure service behavior analysis and computational mechanics. Summary of the Invention

[0007] In view of this, the present invention provides a fractional-order numerical analysis method and module for the deformation behavior of high arch dams to solve the bottleneck problems existing in the background technology.

[0008] To achieve the above objectives, the present invention adopts the following technical solution.

[0009] On the one hand, a fractional-order numerical analysis method for the deformation behavior of high arch dams is provided, including:

[0010] Based on fractional calculus theory, a fractional-order characterization method for each stage of rheological effect under three-dimensional stress is constructed, including decay rheology, steady-state rheology and accelerated rheology. On this basis, a fractional-order rheological mechanical element model of the high arch dam body and dam foundation is established.

[0011] Based on the fractional-order rheological element model of the dam body and foundation, fractional-order numerical analysis is performed using the incremental finite element method, according to t n The stress, viscoelastic strain, viscoplastic strain, and deformation at time t are calculated using the predifference formula. n The stress increment, viscoelastic strain increment, viscoplastic strain increment, and deformation increment within the time period are used to derive t. n+1 The stress, strain, and deformation at each moment are repeatedly calculated to obtain the stress, strain, and deformation of the high arch dam within the analysis period.

[0012] On the other hand, a fractional-order numerical analysis module for the deformation behavior of high arch dams is provided, including:

[0013] Fractional-order characterization units, based on a parallel structure design, include decaying rheological characterization subunits, steady-state rheological characterization subunits, and accelerated rheological characterization subunits;

[0014] The model building unit, based on parallel structure design, includes a dam body fractional-order rheological mechanical element model sub-unit and a dam foundation fractional-order rheological mechanical element model sub-unit;

[0015] The incremental calculation unit, based on a series-parallel structure design, includes a viscoelastic strain incremental calculation sub-unit, a viscoplastic strain incremental calculation sub-unit, a deformation incremental calculation sub-unit, and a stress incremental calculation sub-unit.

[0016] The visualization display unit outputs results including but not limited to instantaneous deformation, aged deformation, stress, and strain, presented to the operator in the form of cloud maps, contour maps, distribution maps, and process lines.

[0017] As can be seen from the above technical solution, compared with the prior art, the present invention discloses a fractional-order numerical analysis method and module for the deformation behavior of high arch dams, which has the following beneficial effects:

[0018] (1) By constructing fractional-order characterization methods for decaying rheology, steady-state rheology and accelerated rheology under three-dimensional stress state, fractional-order rheological mechanical element models of high arch dam body and dam foundation were established;

[0019] (2) Using the predifference formula, recursive formulas for viscoelastic strain increment and viscoplastic strain increment were constructed. Based on this, calculation methods for stress, strain and deformation were proposed according to the incremental finite element method.

[0020] (3) The order of the derivative characterizes the relationship between the hardening and recovery effects in the rheological process of soft matter. The higher the order, the more obvious the recovery effect; the lower the order, the more significant the hardening effect. Compared with the classical integer-order rheological element model, the fractional-order rheological element model has a clearer physical meaning.

[0021] (4) The fractional-order rheological element model describes the properties of Hooke bodies and Newton bodies, as well as the nonlinear viscous mechanical relationship between the two, and has a wider characterization range.

[0022] (5) Since the order of fractional calculus is continuously changing, it is more conducive to simulating accelerated rheology. The fractional rheological mechanical element model can achieve a better characterization of the entire rheological process under the premise of fewer parameters and simpler configuration.

[0023] (6) This invention aims to improve the accuracy and objectivity of deformation behavior analysis of high arch dams, and provide scientific basis and technical support for safety monitoring and early warning of dangers of high arch dams in service. It has a wide range of application prospects and market. Attached Figure Description

[0024] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0025] Figure 1 The present invention provides a fractional-order numerical analysis method and module architecture for the deformation behavior of high arch dams;

[0026] Figure 2 This invention provides a fractional-order rheological component model for a high arch dam body.

[0027] Figure 3 This invention provides a fractional-order rheological element model for the foundation of a high arch dam.

[0028] Figure 4 The calculation flow of the fractional-order numerical analysis program for the deformation behavior of high arch dams provided by this invention;

[0029] Figure 5 The finite element model of the Jia Gao arch dam provided by this invention;

[0030] Figure 6 (a) is a zoning diagram of the concrete material of the Jiagao arch dam provided by the present invention; (b) is a typical node distribution diagram of the Jiagao arch dam provided by the present invention.

[0031] Figure 7The distribution law of radial instantaneous deformation of cantilever beam of A-high arch dam under 6 working conditions;

[0032] Figure 8 The distribution law of radial instantaneous deformation of the arch ring at the top of the A-level arch dam under six working conditions;

[0033] Figure 9 This invention provides the radial time-dependent deformation variation law of nodes at different elevations of the Jia Gao arch dam under working condition 1.

[0034] Figure 10 The present invention provides the radial time-dependent deformation distribution law of the cantilever beam of the A-type arch dam under working condition 1.

[0035] Figure 11 The present invention provides the radial time-dependent deformation distribution law of the arch ring at the crest of the A-type high arch dam under working condition 1.

[0036] Figure 12 The present invention provides the variation law of radial instantaneous deformation and radial time-dependent deformation of characteristic points of the A-high arch dam under working condition 2.

[0037] Figure 13 The radial time-dependent deformation distribution law of the cantilever beam of the A-high arch dam under working condition 2 provided by the present invention. Detailed Implementation

[0038] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0039] This invention discloses a fractional-order numerical analysis method for the deformation behavior of high arch dams, such as... Figure 1 As shown, it includes:

[0040] Based on fractional calculus theory, a fractional-order characterization method for each stage of rheological effect under three-dimensional stress is constructed, including decay rheology, steady-state rheology and accelerated rheology. On this basis, a fractional-order rheological mechanical element model of the high arch dam body and dam foundation is established.

[0041] Based on the fractional-order rheological element model of the dam body and foundation, fractional-order numerical analysis is performed using the incremental finite element method, according to t n The stress, viscoelastic strain, viscoplastic strain, and deformation at time t are calculated using the predifference formula. n The stress increment, viscoelastic strain increment, viscoplastic strain increment, and deformation increment within the time period are used to derive t. n+1The stress, strain, and deformation at each moment are repeatedly calculated to obtain the stress, strain, and deformation of the high arch dam within the analysis period.

[0042] Specifically, the fractional-order characterization methods for decaying rheology, steady-state rheology, and accelerated rheology under three-dimensional stress states are described below.

[0043] a. Fractional-order characterization method of attenuated rheology under three-dimensional stress state

[0044] The elastic relationship between stress and strain in a high arch dam under three-dimensional stress state is as follows:

[0045]

[0046] In the formula: K is the bulk modulus characterizing elastic deformation; G is the shear modulus characterizing elastic deformation; σ m Let ε be the spherical stress tensor; m S is the spherical strain tensor; ij e is the deviatoric stress tensor; ij Let K and G be the deviatoric strain tensors. The relationships between K and G and the elastic modulus E and Poisson's ratio μ are as follows:

[0047]

[0048] Similar to the elastic case, the viscoelastic stress-strain relationship of a high arch dam under three-dimensional stress also includes two parts: the spherical tensor relationship and the deviatoric tensor relationship. Assumptions: ① The material is isotropic; ② The volumetric deformation of the dam is elastic deformation, caused by the spherical stress tensor, and completed instantaneously under stress; ③ The rheological deformation of the dam is caused only by the deviatoric stress tensor; ④ During the rheological process, Poisson's ratio does not change with time. Under one-dimensional stress, the general formula for the fractional-order element model characterizing the viscoelastic mechanical state of a high arch dam is:

[0049]

[0050] In the formula: t is time; a h b w H is a constant; h = 1, 2, ..., H and w = 1, 2, ..., W are non-negative integers; α h β w Let be a real number whose numerator and denominator are coprime; H and W are the number of strain and stress terms in the general formula, respectively; σ is stress; ε is strain. Let:

[0051]

[0052] Equation (4) can be expressed as:

[0053]

[0054] Under three-dimensional stress conditions, the viscoelastic stress-strain relationship of a high arch dam can be expressed as:

[0055]

[0056] Equation (7) first represents the relationship between spherical stress and spherical strain. Since spherical stress only causes elastic deformation, {Q2} = K, {P2} = 1. Equation (7) second represents the relationship between deviatoric stress and deviatoric strain. Since deviatoric stress causes viscoelastic deformation, {Q1} and {P1} correspond to {Q} and {P} respectively. However, it is necessary to convert all elastic moduli E in {Q} and {P} into shear elastic moduli G. The conversion formula is:

[0057]

[0058] In the formula: μ is Poisson's ratio.

[0059] Therefore, the axial stress σ in the fractional-order characterization model of decaying rheology under one-dimensional stress state is transformed into the deviatoric stress S. ij Furthermore, the retardation modulus is converted into the shear retardation modulus, thus obtaining the fractional-order characterization method of decay rheology under three-dimensional stress state.

[0060] Under three-dimensional stress, the viscoelastic strain of fractional-order Kelvin bodies and fractional-order mountain bodies can be expressed in the following form.

[0061] The fractional Kelvin body is:

[0062]

[0063] For hard rock, creep deformation generally occurs only after the stress reaches a stress threshold. For the deviatoric stress threshold tensor that exhibits decay rheology, when At that time, the mountain in the fractional-level village was:

[0064]

[0065] In equations (9) and (10): t is time; G1 is the viscoelastic strain tensor; G1 is the shear modulus characterizing the damped rheology; S ij Γ is the deviatoric stress tensor; k is a non-negative integer variable; η1 is the viscosity coefficient characterizing the decay rheology; γ1 is the order characterizing the decay rheology; and Γ is the Gamma function.

[0066] b. Steady-state rheological fractional-order characterization method under three-dimensional stress state

[0067] remember For the deviatoric stress threshold tensor that exhibits steady-state rheology, based on the P. Perzyna assumption, when At that time, the fractional-order Bingham viscoplastic strain is expressed as:

[0068]

[0069] In the formula: t is time; σ is the viscoplastic strain tensor; F is the yield function; Q is the plastic potential function; Φ is the flow function; σ is the strain tensor. ij Γ is the total stress tensor; η2 is the viscosity coefficient characterizing steady-state rheology and accelerated rheology; γ2 is the order characterizing steady-state rheology and accelerated rheology; Γ is the Gamma function.

[0070] The yield function F adopts the Drucker-Prager yield criterion, that is:

[0071]

[0072] In the formula: c is the cohesion; φ is the internal friction angle; I1 is the first invariant of the stress tensor; J2 is the second invariant of the deviatoric stress tensor.

[0073] Using the associated flow rule (F=Q) and the power-law flow function, when At that time, the expression for the fractional-order viscoplastic strain of a Bingham body under three-dimensional stress is:

[0074]

[0075] In the formula: t is time; σ is the viscoplastic strain tensor; F is the yield function; F0 is the yield function at the initial moment; σ ij η is the total stress tensor; η² is the viscosity coefficient characterizing steady-state rheology and accelerated rheology; γ² is the order characterizing steady-state rheology and accelerated rheology; m is the yield parameter; and Γ is the Gamma function.

[0076] c. Fractional-order characterization method of accelerated rheology under three-dimensional stress state

[0077] When the accumulated viscoplastic strain reaches a threshold, the fractional-order Bingham body... It changes in the form of an exponential function f(t), and its expression is:

[0078]

[0079] In the formula: t is time; η2 is the viscosity coefficient characterizing steady-state rheology and accelerated rheology; γ2 is the order characterizing steady-state rheology and accelerated rheology; α2 is the acceleration parameter. With... As the viscosity decreases, the viscoplastic strain rate increases, entering the accelerated rheological stage, and an equivalent viscoplastic strain is introduced. Its expression is:

[0080]

[0081] In the formula: It is the viscoplastic strain tensor.

[0082] remember To accelerate the equivalent viscoplastic strain during rheological events, the corresponding strain is calculated using equation (15). Determined based on rheological tests. When Under three-dimensional stress, the viscoplastic strain of a variable-parameter fractional-order Bingham body is expressed as:

[0083]

[0084] In the formula: t is time; Γ is the viscoplastic strain tensor; F is the yield function; F0 is the yield function at the initial moment; η2 is the viscosity coefficient characterizing steady-state rheology and accelerated rheology; γ2 is the order characterizing steady-state rheology and accelerated rheology; k is a non-negative integer variable; m is the yield parameter; Γ is the Gamma function; σ ij α is the total stress tensor; α2 is the acceleration parameter; t s To accelerate the time it takes for rheology to emerge.

[0085] Specifically, based on the fractional-order characterization methods of decaying rheology, steady-state rheology, and accelerated rheology under three-dimensional stress state, the fractional-order rheological mechanical element models of the high arch dam body and dam foundation are given below.

[0086] The dam model adopts the following Figure 2 The configuration shown is expressed as:

[0087]

[0088] The dam foundation model adopts the following Figure 3 The configuration shown is expressed as:

[0089]

[0090] In equations (17) and (18): t is time; σ ij ε is the total stress tensor; ij σ is the total strain tensor; m S is the spherical stress tensor; ij For the deviatoric stress tensor; δ ij For Kronecker notation, when i = j, δ ij =1, otherwise δ ij =0; denoted as the deviatoric stress threshold tensor for decaying rheology; K is the bulk modulus characterizing elastic deformation; G is the shear modulus characterizing elastic deformation; G1 is the shear modulus characterizing decaying rheology; η1 is the viscosity coefficient characterizing decaying rheology; η2 is the viscosity coefficient characterizing steady-state and accelerated rheology; γ1 is the order characterizing decaying rheology; γ2 is the order characterizing steady-state and accelerated rheology; k is a non-negative integer variable; m is the yield parameter. Γ is the deviatoric stress threshold tensor for steady-state rheology; Γ is the Gamma function. Equivalent viscoplastic strain; The equivalent viscoplastic strain during the acceleration of rheological processes; F is the yield function; F0 is the yield function at the initial moment; α2 is the acceleration parameter; t s To accelerate the time it takes for rheology to emerge.

[0091] Specifically, based on the fractional-order rheological element model of the dam body and foundation, and with the help of the incremental finite element principle, the following is a method for fractional-order numerical analysis of the deformation behavior of high arch dams.

[0092] Note t n The load at time {R} n ;t n The total stress at time t is {σ} ij} n ;t n The viscoelastic strain at time t is The viscoplastic strain at time tn is The transformation at time tn is {U} n The total strain increment during the time period Δtn is {Δε} ij} n ;Δt n The instantaneous strain increment within the time period is Δt n The increase in viscoelastic strain over the time period is Δt n The increase in viscoplastic strain over the time period is Δt n The deformation increment during the time period is {ΔU} n ;Δt n The total stress increment during the time period is {Δσ} ij} n The following gives t n+1 Stress at time {σ ij} n+1 viscoelastic strain viscoplastic strain and deformation {U} n+1 The calculation method.

[0093] This invention employs a pre-difference method to construct viscous strain increments. With time increment Δt n The recursive relationship between them. In Δt n The increment of viscous strain over the time period is expressed as:

[0094]

[0095] In the formula: This represents the viscous strain increment. For t n Viscous strain rate at time t.

[0096] a. Calculate the viscoelastic strain increment

[0097] Taking the fractional-order Kelvin body as an example, the fractional-order village mountain body is similar. Differentiating the time t in the fractional-order expression of decay rheology under one-dimensional stress state, and combining it with equation (19), we obtain Δt. n The expression for the fractional-order Kelvin volume viscoelastic strain increment over a time interval is:

[0098]

[0099] In the formula: t n For a certain moment; Δt under one-dimensional stress state n The increase in viscoelastic strain over a given time period; E1 is the elastic modulus characterizing the decaying rheology; σ n For t n The axial stress at time t; k is a non-negative integer variable; η1 is the viscosity coefficient characterizing the decay rheology; γ1 is the order characterizing the decay rheology; Γ is the Gamma function.

[0100] Under three-dimensional stress, Δt n Viscoelastic strain increment of fractional-order Kelvin bodies over time period Represented as:

[0101]

[0102] In the formula: t n For a certain moment; For Δt n The increase in viscoelastic strain over a given time period; μ is Poisson's ratio; Let S be the Poisson ratio matrix; k is a non-negative integer variable; {S ij} n For t n The deviatoric stress at time t; η1 is the viscosity coefficient characterizing the decaying rheology; γ1 is the order characterizing the decaying rheology; G1 is the shear modulus characterizing the decaying rheology; Γ is the Gamma function.

[0103] b. Calculate the viscoplastic strain increment

[0104] Differentiating the time t in the fractional-order Bingham body expression under three-dimensional stress state, we obtain the expression for the viscoplastic strain rate as follows:

[0105]

[0106] Therefore Δt n The viscoplastic strain increment of the fractional-order Bingham body within the time period is:

[0107]

[0108] Differentiating the time t in the fractional-order Bingham viscoplastic strain expression with respect to varying parameters yields the expression for the viscoplastic strain rate:

[0109]

[0110] Then Δt n Fractional order Bingham viscoplastic strain increments over time period for:

[0111]

[0112] In equations (22) to (25): t n For a certain moment; For t n Viscoplastic strain rate at time t; For Δt n The viscoplastic strain increment over the time interval; F is the yield function; F0 is the yield function at the initial moment; η2 is the viscosity coefficient characterizing steady-state rheology and accelerated rheology; γ2 is the order characterizing steady-state rheology and accelerated rheology; k is a non-negative integer variable; m is the yield parameter; Γ is the Gamma function; {σ ij} n For t n Total stress at time t; α2 is the acceleration parameter; t s To accelerate the time it takes for rheology to emerge.

[0113] Solve The key is calculation Its decomposition is as follows:

[0114]

[0115] In equation (26):

[0116]

[0117]

[0118] In equations (26) to (29): I1 is the first invariant of the stress tensor; J2 and J3 are the second and third invariants of the deviatoric stress tensor, respectively; S xx S yy S zz S xy S yz and S zx The components of the deviatoric stress tensor; {σ ij} n Let be the total stress at time tn.

[0119] For the Drucker-Prager yield criterion, there exist C1 = α2, C2 = 1 and C3 = 0. Substituting equations (27) to (29) into equation (26) yields:

[0120]

[0121] Where: F is the yield function; α2 is the acceleration parameter; I1 is the first invariant of the stress tensor; J2 is the second invariant of the deviatoric stress tensor; {σ ij} n For t n The total stress at time t; {X} and {Y} are constant matrices.

[0122] {X} and {Y} are represented as follows:

[0123]

[0124]

[0125] c. Calculate the deformation increment {ΔU} within the time interval Δtn. n and tn +1 Transformation of time Un +1

[0126] According to elasticity theory, Δt n During the time period, the stress increment of the high arch dam {Δσ ij} n Expressed as:

[0127]

[0128] In the formula: {D} is the elasticity matrix; {Δσ} ij} n For Δt n Total stress increment over the period; For Δt n The increase in elastic strain over a given time period.

[0129] The total strain increment at a certain point in a high arch dam is {Δε} ij} nIncludes instantaneous strain increment Viscoelastic strain increment and viscoplastic strain increment satisfy:

[0130]

[0131] Substituting equation (34) into equation (33), we get:

[0132]

[0133] According to the principle of virtual work, the incremental balance equation after the structure is discretized is:

[0134]

[0135] In the formula: {B} n Let {ΔR} be the geometric matrix within the time interval Δtn. n The external load increment during the time period Δtn; {Δσ ij} n For Δt n The total stress increment over the time period. Substituting equation (35) into equation (36), the incremental equilibrium equation of the viscoelastic-plastic finite element method is expressed as:

[0136] {K}{ΔU} n ={ΔR} n +{ΔR ve} n +{ΔR vp} n (37)

[0137] In the formula: {K} is the global stiffness matrix; {ΔU} n For Δt n The deformation increment matrix within the time period; {ΔR ve} n For Δt n Increment of viscoelastic strain over time The corresponding equivalent nodal load increment; {ΔR vp} n The increase in viscoplastic strain during the time interval Δtn The corresponding equivalent nodal load increment; {ΔR} n This represents the increase in external load during the time interval Δtn.

[0138] {K}、{ΔR ve} n and {ΔR vp} n The expressions are as follows:

[0139]

[0140] In equations (38) to (40): {B} n For Δt n The geometric matrix within the time period; {D} is the elasticity matrix; For Δt n The increase in viscoelastic strain over a given period of time; For Δt n The increase in viscoplastic strain over a given time period. The integration objects of equations (38) to (40) are all elements, elements in a viscoelastic state, and elements in a viscoplastic state, respectively.

[0141] Calculate the deformation increment {ΔU} of the high arch dam during the time period Δtn. n ,Right now:

[0142] {ΔU}n={K}- 1 ({ΔR}n+{ΔRve}n+{ΔRvp}n)(41)

[0143] Therefore, tn is obtained. +1 The deformation value of the high arch dam at any given time {U} n+1 ,Right now:

[0144] {U} n+1 ={U} n +{ΔU} n (42)

[0145] In equations (41) and (42): {ΔR ve} n The viscoelastic strain increment during the time interval Δtn The corresponding equivalent nodal load increment; {ΔR vp} n The increase in viscoplastic strain during the time interval Δtn The corresponding equivalent nodal load increment; {ΔR} n For Δt n The external load increment within the time period; {K} is the global stiffness matrix; {U} n This is the transformation at time tn.

[0146] d. Calculate Δt n Total stress increment within the time period {Δσ ij} n and t n+1 The total stress at time {σ ij} n+1

[0147] Calculate Δt according to formula (33) n The total stress increment of high arch dams within a time period {Δσ ij} n , then t n+1 The total stress at time {σij} n+1 Expressed as:

[0148] {σ ij} n+1 ={σ ij} n +{Δσ ij} n (43)

[0149] In the formula: {σ ij} n Let be the total stress at time tn.

[0150] Through the above analysis steps, we can calculate {Δσ} ij} n , and {ΔU} n Combined with {σ ij} n , and{U} n , thus obtaining {σ ij} n+1 , and{U} n+1 By repeatedly calculating the stress, strain, and deformation of the high arch dam during the analysis period, a scientific quantitative analysis of the rheological effects and deformation behavior of the dam body and foundation can be achieved.

[0151] The calculation flow of the fractional-order numerical analysis program for the deformation behavior of high arch dams is as follows: Figure 4 As shown.

[0152] Specifically, based on the above method, a fractional-order numerical analysis module for the deformation behavior of high arch dams is provided below, such as... Figure 1 As shown, it includes:

[0153] Fractional-order characterization units, based on a parallel structure design, include decaying rheological characterization subunits, steady-state rheological characterization subunits, and accelerated rheological characterization subunits;

[0154] The model building unit, based on parallel structure design, includes a dam body fractional-order rheological mechanical element model sub-unit and a dam foundation fractional-order rheological mechanical element model sub-unit;

[0155] The incremental calculation unit, based on a series-parallel structure design, includes a viscoelastic strain incremental calculation sub-unit, a viscoplastic strain incremental calculation sub-unit, a deformation incremental calculation sub-unit, and a stress incremental calculation sub-unit.

[0156] The visualization display unit outputs results including but not limited to instantaneous deformation, aged deformation, stress, and strain, presented to the operator in the form of cloud maps, contour maps, distribution maps, and process lines.

[0157] The following embodiment further illustrates the method of the present invention.

[0158] The Jiagao Arch Dam is a concrete double-curvature arch dam, belonging to the large (1) type project. Its permanent main hydraulic structures are Class I structures. The dam crest elevation is 1885m, the maximum dam height is 305m, the dam crest width is 16m, the arc length of the dam crest centerline is 552.23m, the maximum span is 480m, the lowest foundation elevation is 1580m, and the dam base thickness is 63m. The valley width-to-height ratio is 1.57, and the valley is a typical deep-cut "V" shape. The normal water level is 1880m, the dead water level is 1800m, and the reservoir capacity below the normal water level is 7.76 billion m³. 3 Adjusting reservoir capacity by 4.91 billion cubic meters 3 On October 23, 2009, the first concrete pour for section 14 (elevation 1580-1581.5m) of the dam began, and the entire dam was sealed in December 2013. The completed dam's three-dimensional finite element model is shown below. Figure 5 As shown, the dam body concrete material is divided into zones as follows: Figure 6 As shown in (a), the typical node distribution of the cantilever beam and the dam crest arch ring is as follows: Figure 6 As shown in (b).

[0159] The design values ​​of the elastoplastic physical and mechanical parameters of the dam body and foundation of the Jiagao arch dam are shown in Table 1 for fractional-order numerical analysis. The retardation modulus of the dam body concrete is taken as E1 = 70 GPa, and the viscosity coefficient is taken as η1 = 2 × 10⁻⁶. 3 GPa·d, η2=1.5×10 4 GPa·d, with orders γ1 and γ2 set to 0.7 and 0.8 respectively, and the attenuation rheological stress threshold... Taken as 0.15 times the instantaneous strength of concrete, viscoplastic parameter It is taken as 0.8 times the instantaneous strength of concrete, and parameter α2 is taken as 0.003h. -1 The retardation modulus of the dam foundation rock mass is taken as E1 = 50 GPa, and the viscosity coefficient is taken as η1 = 3 × 10⁻⁶. 3 GPa·d, η2=1.5×10 4 GPa·d, with orders γ1 and γ2 set to 0.7 and 0.8 respectively, and the attenuation rheological stress threshold... The viscoplasticity parameter is taken as 0.15 times the instantaneous strength of the rock mass. The instantaneous strength of the rock mass is taken as 0.8 times, and the parameter α2 is taken as 0.003h. -1 .

[0160] Table 1 Design values ​​of elastic-plastic physical and mechanical parameters of the Jia Gao arch dam

[0161]

[0162] During construction, the main loads borne by the high arch dam include water pressure, temperature load, and self-weight. Before arch sealing and grouting, all loads are borne by the cantilever beams. After arch sealing and grouting, the loads are shared by the arch-beam system, while the loads of the unsealed and ungrouted sections are still borne by the cantilever beams. The phased pouring, phased arch sealing, and phased water impoundment process of the Jia high arch dam are relatively complex, and this example simplifies the process to some extent. Since the Jia high arch dam was fully sealed in December 2013, all relevant analyses in this invention start from January 1, 2014, when the reservoir water level was approximately 1840m. Assuming that the pouring elevation of the Jia high arch dam is the same as the arch sealing elevation, starting from the foundation elevation (approximately 1580m), the dam's self-weight and the reservoir water pressure before the 1840m water level are applied in five stages. The loading process is shown in Table 2, and the calculated stress field is used as the initial stress field.

[0163] Table 2 Elevations of the A-level arch dam during pouring, arch sealing, and water storage.

[0164]

[0165] a. Instantaneous deformation and change law

[0166] The following analysis uses six typical working conditions as examples to illustrate the instantaneous deformation variation law of the Jia Gao arch dam. The working conditions selected for calculation are: dead water level, dead water level + maximum temperature rise, dead water level + maximum temperature drop, normal water level, normal water level + maximum temperature rise, and normal water level + maximum temperature drop. Figure 7 The radial instantaneous deformation distribution of the cantilever beam of the A-high arch dam is shown under six calculation conditions. The location of the cantilever beam is as follows: Figure 6 As shown in (b) Figure 8 The following conclusions were drawn regarding the radial instantaneous deformation distribution of the arch ring at the crest of the A-high arch dam under six calculation conditions:

[0167] (1) The upper part of the Jiagao arch dam, especially the middle part of the dam crest, is less constrained by the dam foundation and bank slope, and is more sensitive to changes in load state than other parts. For example, during the process of the upstream water level changing from 1800m to 1880m, the calculated value of the radial instantaneous deformation at the top node of the cantilever beam (node ​​number 9561) changed from -7.63mm to 34.94mm, with the deformation increasing downstream by 42.57mm. Meanwhile, the calculated value of the radial instantaneous deformation at the node at elevation 1630m (node ​​number 9498) changed from 11.27mm to 21.65mm, with the deformation increasing downstream by 10.38mm. Similarly, during the same process, the calculated value of the radial instantaneous deformation at the middle node of the dam crest arch (node ​​number 10484) changed from -7.66mm to 39.47mm, with the deformation increasing downstream by 47.13mm. And the calculated value of the radial instantaneous deformation at the node near the left slope of the dam crest arch (node ​​number 9683) changed from -4.71mm to -0.87mm, with the deformation increasing downstream by 3.84mm.

[0168] (2) The effect of temperature load on the instantaneous deformation of the dam can be summarized as follows: when the temperature rises, the dam deforms relatively upstream; when the temperature falls, the dam deforms relatively downstream. For example, when the upstream water level is 1880m, the calculated value of the radial instantaneous deformation of the top node of the cantilever beam (node ​​number 9561) is 34.94mm. When the temperature load reaches the maximum temperature rise condition, the calculated value of the radial instantaneous deformation of this node is 17.04mm, which is 17.9mm relative to the upstream. When the temperature load reaches the maximum temperature drop condition, the calculated value of the radial instantaneous deformation of this node is 52.83mm, which is 17.89mm relative to the downstream.

[0169] (3) The effect of water pressure load on the instantaneous deformation of the dam can be summarized as follows: when the water level rises, the dam body deforms downstream; when the water level falls, the downstream deformation of the dam body decreases or it deforms upstream. For example, when the upstream water level is 1800m, the calculated value of the radial instantaneous deformation of the cantilever beam top node (node ​​number 9561) is -7.63mm. When the water level reaches 1880m, the calculated value of the radial instantaneous deformation of this node is 34.94mm. That is, during the process of the water level rising from 1800m to 1880m, the cantilever beam top deformed downstream by 42.57mm.

[0170] b. Laws of deformation over time

[0171] Taking the following two working conditions as examples, we will calculate and analyze the time-dependent deformation variation law of the A-level arch dam.

[0172] Operating Condition 1: The reservoir water level was increased from 1840m to the normal storage level of 1880m in 21 stages, with a time interval of 1 day (d) and a unit of 2m. The water level was then maintained at 1880m until day 300. The reservoir water load application method is shown in Table 3, and the reservoir water level change process is as follows: Figure 9 As shown by the double-dotted line.

[0173] Table 3 Reservoir water load application methods (condition 1)

[0174]

[0175] Figure 9 The radial deformation over time at characteristic points at different elevations of the Jia Gao arch dam is shown to be variable. Figure 10 The radial deformation distribution of the cantilever beam of the A-high arch dam is shown in the figure. The characteristic points and the locations of the cantilever beams are as follows: Figure 6 As shown in (b) Figure 11 The following conclusions are drawn regarding the radial deformation distribution of the arch ring at the crest of the Jiagao arch dam (for ease of description, the upstream nodes of the arch ring at the crest are numbered sequentially from left bank to right bank as 1 to 53):

[0176] (1) Under the continuous action of reservoir water load, the radial time-dependent deformation of the Jia Gao arch dam developed rapidly in the early stage of water impoundment and gradually converged in the later stage. By the 300th day of the calculation period, the radial time-dependent deformation convergence value of the node (node ​​number 9561) at the 1885m elevation of the cantilever beam was 4.77mm; the radial time-dependent deformation convergence value of the node (node ​​number 9533) at the 1800m elevation of the cantilever beam was 4.65mm; and the radial time-dependent deformation convergence value of the node at the 1750m elevation of the cantilever beam was 4.65mm. The radial aging deformation convergence value of node (node ​​number 9528) is 4.26 mm; the radial aging deformation convergence value of node (node ​​number 9519) at an elevation of 1720 m is 3.86 mm; the radial aging deformation convergence value of node (node ​​number 9507) at an elevation of 1660 m is 2.61 mm; and the radial aging deformation convergence value of node (node ​​number 9506) at an elevation of 1600 m is 0.98 mm.

[0177] (2) The radial time-dependent deformation distribution of the cantilever beam of the Jiagao arch dam is closely related to the reservoir water load. When the reservoir water level is low, the radial time-dependent deformation in the middle of the cantilever beam is large. By day 10, the maximum radial time-dependent deformation of the cantilever beam is 0.34 mm, which occurs at the 1720 m elevation of the cantilever beam (node ​​number 9519). When the reservoir water level is high, the radial time-dependent deformation in the upper part of the cantilever beam is large. By day 300, the maximum radial time-dependent deformation of the cantilever beam is 4.77 mm, which occurs at the 1880 m elevation of the top of the cantilever beam (node ​​number 9561).

[0178] (3) The radial time-dependent deformation of the middle part of the arch ring on the left bank of the Jia Gao arch dam is relatively large. By the 300th day of the calculation period, the radial time-dependent deformation of node 10484 of the arch ring on the dam is 5.49 mm. Due to the constraint of the mountains on both sides, the radial time-dependent deformation of the part near the bank slope is relatively small. The radial time-dependent deformation of node 9600 on the left slope is only 0.68 mm.

[0179] Operating Condition 2: The reservoir water level changes according to the water storage process from January 1, 2014 (Day 1) to March 31, 2015 (Day 455), with a time interval of 1 day. The reservoir water load application method is shown in Table 4, and the reservoir water level change process is as follows. Figure 12 As shown by the double-dotted line.

[0180] Table 4 Reservoir water load application methods (condition 2)

[0181]

[0182] Figure 12 The study examines the instantaneous radial deformation and time-dependent radial deformation variation patterns at a characteristic point of the Jiagao arch dam (elevation 1750m, node number 9528). Figure 13 The radial deformation distribution of the cantilever beam of the A-high arch dam is shown in the figure. The characteristic points and the locations of the cantilever beams are as follows: Figure 6As shown in (b), the following conclusion can be drawn:

[0183] (1) The radial instantaneous deformation of the cantilever beam at elevation 1750m (node ​​number 9528) of the Jiagao arch dam is significantly affected by the reservoir water level. When the reservoir water level rises, the node deforms downstream, and when the reservoir water level falls, the node deforms upstream. The radial time-dependent deformation of this node exhibits a nonlinear variation characteristic, developing rapidly in the early stage of water storage and gradually converging in the later stage. By the end of the calculation period, the radial time-dependent deformation of this node is 2.82mm. The radial time-dependent deformation during this period is fitted using c1θ+c2lnθ, and the fitting result is as follows. Figure 12 The fitted multiple correlation coefficient for the single-dot dashed line is 0.976, and the coefficients c1 and c2 are 0.2735 and 0.9346, respectively.

[0184] (2) The radial deformation distribution of the cantilever beam of the Jiagao arch dam is closely related to the reservoir water load. When the reservoir water level is low, the radial deformation in the middle of the cantilever beam is larger. For example, on the 150th day (February 19, 2014), the maximum radial deformation of the cantilever beam was 0.81 mm, which occurred at an elevation of 1690 m (node ​​number 9526). On the 200th day (July 19, 2014), the maximum radial deformation of the cantilever beam was 1.14 mm, which occurred at an elevation of 1720 m (node ​​number 9519). From day 250 to day 380 (October 27, 2014 to February 4, 2015), the reservoir water level was relatively high, fluctuating around the normal storage level of 1880m. The radial deformation of the upper part of the cantilever beam was relatively large. From the elevation of 1800m to the top of the dam, the radial deformation value was close. By day 380, the maximum radial deformation of the cantilever beam was 2.78mm, which occurred at the top elevation of the cantilever beam of 1885m (node ​​number 9561).

[0185] This example, based on the fractional-order numerical analysis method and module for the deformation behavior of high arch dams, analyzes the variation law of radial instantaneous deformation and radial time-dependent deformation of a high arch dam. The calculation results generally reflect the actual situation, thus verifying the feasibility of the present invention.

[0186] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0187] Therefore, the present invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A fractional-order numerical analysis method for the deformation behavior of high arch dams, characterized in that, include: Based on fractional calculus theory, a fractional-order characterization method for each stage of rheological effect under three-dimensional stress is constructed, including decay rheology, steady-state rheology and accelerated rheology. On this basis, a fractional-order rheological mechanical element model of the high arch dam body and dam foundation is established. The expression for the fractional-order rheological element model of the dam body is as follows: The expression for the fractional-order rheological element model of the dam foundation is as follows: In the formula: For time; This is the total stress tensor; This is the total strain tensor; For spherical stress tensor; It is the deviatoric stress tensor; For Kroneck symbol, when hour, ,otherwise ; The deviatoric stress threshold tensor for the decaying rheology; The bulk modulus characterizing elastic deformation; The shear modulus is used to characterize elastic deformation. The shear elastic modulus is used to characterize decay rheology; The viscosity coefficient characterizing decay rheology; The viscosity coefficient characterizes steady-state rheology and accelerated rheology; The order characterizing the decay rheology; The order is used to characterize steady-state rheology and accelerated rheology; It is a non-negative integer variable; The yield parameter; The deviatoric stress threshold tensor for steady-state rheology; It is the Gamma function; Equivalent viscoplastic strain; To accelerate the equivalent viscoplastic strain during rheological processes; Let be the yield function; Let be the yield function at the initial moment; For acceleration parameters; To accelerate the time it takes for rheology to emerge; Based on the fractional-order rheological element model of the dam body and foundation, fractional-order numerical analysis is performed using the incremental finite element method. The stress, viscoelastic strain, viscoplastic strain, and deformation at time points are calculated using the predifference formula. The stress increment, viscoelastic strain increment, viscoplastic strain increment, and deformation increment within the time period are then used to derive... The stress, strain, and deformation at each moment are repeatedly calculated to obtain the stress, strain, and deformation of the high arch dam within the analysis period.

2. The fractional-order numerical analysis method for the deformation behavior of high arch dams according to claim 1, characterized in that, The attenuation rheological effect was characterized using fractional-order Kelvin bodies and fractional-order Cunshan bodies; Under three-dimensional stress, the fractional-order Kelvin volume viscoelastic strain is expressed as: remember For the deviatoric stress threshold tensor that exhibits decay rheology, when Under three-dimensional stress, the fractional-order viscoelastic strain of the mountain mass is expressed as: In the formula: For time; It is the viscoelastic strain tensor; The shear elastic modulus is used to characterize decay rheology; It is the deviatoric stress tensor; It is a non-negative integer variable; The viscosity coefficient characterizing decay rheology; The order characterizing the decay rheology; This is the Gamma function.

3. The fractional-order numerical analysis method for the deformation behavior of high arch dams according to claim 1, characterized in that, Steady-state rheological processes were characterized using fractional-order Bingham volumes. remember For the deviatoric stress threshold tensor that exhibits steady-state rheology, when Under three-dimensional stress, the expression for the fractional-order viscoplastic strain of a Bingham body is: In the formula: For time; It is the viscoplastic strain tensor; Let be the yield function; Let be the yield function at the initial moment; This is the total stress tensor; The viscosity coefficient characterizes steady-state rheology and accelerated rheology; The order is used to characterize steady-state rheology and accelerated rheology; It is the Gamma function; This is the yield parameter.

4. The fractional-order numerical analysis method for the deformation behavior of high arch dams according to claim 1, characterized in that, Accelerated rheological processes were characterized using variable-parameter fractional-order Bingham volume representation. When the accumulated viscoplastic strain reaches a threshold, the fractional-order Bingham body... With exponential function The form change is expressed as: along with As the viscosity decreases, the viscoplastic strain rate increases, entering the accelerated rheological stage, and an equivalent viscoplastic strain is introduced. The expression is: remember To accelerate the equivalent viscoplastic strain during rheological events, when Under three-dimensional stress, the viscoplastic strain of a variable-parameter fractional-order Bingham body is expressed as: In the formula: For time; It is the viscoplastic strain tensor; Let be the yield function; Let be the yield function at the initial moment; This is the total stress tensor; The viscosity coefficient characterizes steady-state rheology and accelerated rheology; The order is used to characterize steady-state rheology and accelerated rheology; For acceleration parameters; To accelerate the time it takes for rheology to emerge; It is the Gamma function; It is a non-negative integer variable; This is the yield parameter.

5. A fractional-order numerical analysis method for the deformation behavior of high arch dams according to claim 3 or 4, characterized in that, Yield function Adopt the Drucker-Prager criterion; The yield function of the Drucker-Prager criterion The expression is: In the formula: It is cohesive force; It is the internal friction angle; It is the first invariant of the stress tensor; It is the second invariant of the deviatoric stress tensor.

6. The fractional-order numerical analysis method for the deformation behavior of high arch dams according to claim 1, characterized in that, Calculation using incremental finite element method Stress, strain, and deformation at any given moment; exist The viscoelastic strain increment of a fractional-order Kelvin body over a given time period is expressed as: In the formula: For a certain moment; for The increase in viscoelastic strain over a given period of time; Poisson's ratio; The Poisson ratio matrix; It is a non-negative integer variable; for Deviatoric stress at time; The viscosity coefficient characterizing decay rheology; The order characterizing the decay rheology; The shear elastic modulus is used to characterize decay rheology; It is the Gamma function; exist The viscoplastic strain increment of a fractional-order Bingham body over a given time period is expressed as: exist The viscoplastic strain increment of a variable-parameter fractional-order Bingham body over a given time period is expressed as: In the formula: For a certain moment; for The increase in viscoplastic strain over a given period of time; Let be the yield function; Let be the yield function at the initial moment; The viscosity coefficient characterizes steady-state rheology and accelerated rheology; The order is used to characterize steady-state rheology and accelerated rheology; It is a non-negative integer variable; The yield parameter; It is the Gamma function; for Total stress at time t; For acceleration parameters; To accelerate the time it takes for rheology to emerge; Deformation increment during the time period and Deformation value at time The calculation formula is: In the formula: The overall stiffness matrix; for Increment of viscoelastic strain over time The corresponding equivalent nodal load increment; for Increment of viscoplastic strain over time The corresponding equivalent nodal load increment; for The increase in external load within a time period; for The deformation value at time; Stress increment within a time period and Stress at any time The calculation formula is: In the formula: This represents the elastic strain increment. It is the elasticity matrix; for Total stress at time t.

7. The fractional-order numerical analysis method for the deformation behavior of high arch dams according to claim 6, characterized in that, In the expressions for the viscoplastic strain increments of fractional-order Bingham bodies and variable-parameter fractional-order Bingham bodies The calculation formula is: In the formula: Let be the yield function; for Total stress at time t; For acceleration parameters; It is the first invariant of the stress tensor; It is the second invariant of the deviatoric stress tensor; and It is a constant matrix.

8. A fractional-order numerical analysis module for the deformation behavior of high arch dams, characterized in that, The fractional-order numerical analysis method for the deformation behavior of high arch dams according to any one of claims 1-7 includes: Fractional-order characterization units, based on a parallel structure design, include decaying rheological characterization subunits, steady-state rheological characterization subunits, and accelerated rheological characterization subunits; The model building unit, based on parallel structure design, includes a dam body fractional-order rheological mechanical element model sub-unit and a dam foundation fractional-order rheological mechanical element model sub-unit; The incremental calculation unit, based on a series-parallel structure design, includes a viscoelastic strain incremental calculation sub-unit, a viscoplastic strain incremental calculation sub-unit, a deformation incremental calculation sub-unit, and a stress incremental calculation sub-unit. The visualization display unit outputs results including but not limited to instantaneous deformation, aged deformation, stress, and strain, presented to the operator in the form of cloud maps, contour maps, distribution maps, and process lines.

Images