CN113806966B - Construction method of nonlinear rock fatigue constitutive model based on rheological model application - Google Patents
Construction method of nonlinear rock fatigue constitutive model based on rheological model application Download PDFInfo
- Publication number
- CN113806966B CN113806966B CN202111191687.8A CN202111191687A CN113806966B CN 113806966 B CN113806966 B CN 113806966B CN 202111191687 A CN202111191687 A CN 202111191687A CN 113806966 B CN113806966 B CN 113806966B
- Authority
- CN
- China
- Prior art keywords
- fatigue
- rock
- constitutive model
- nonlinear
- epsilon
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
- 239000011435 rock Substances 0.000 title claims abstract description 182
- 238000010276 construction Methods 0.000 title claims abstract description 16
- 206010016256 fatigue Diseases 0.000 claims abstract description 301
- 230000006378 damage Effects 0.000 claims abstract description 164
- 238000000034 method Methods 0.000 claims abstract description 89
- 230000008569 process Effects 0.000 claims abstract description 29
- 230000008878 coupling Effects 0.000 claims abstract description 18
- 238000010168 coupling process Methods 0.000 claims abstract description 18
- 238000005859 coupling reaction Methods 0.000 claims abstract description 18
- 208000027418 Wounds and injury Diseases 0.000 claims abstract description 5
- 208000014674 injury Diseases 0.000 claims abstract description 5
- 238000012360 testing method Methods 0.000 claims description 57
- 125000004122 cyclic group Chemical group 0.000 claims description 49
- 230000006399 behavior Effects 0.000 claims description 16
- 230000006641 stabilisation Effects 0.000 claims description 15
- 238000011105 stabilization Methods 0.000 claims description 15
- 230000010354 integration Effects 0.000 claims description 6
- 230000006735 deficit Effects 0.000 claims description 5
- 230000000087 stabilizing effect Effects 0.000 claims description 5
- 238000006243 chemical reaction Methods 0.000 claims description 4
- 238000013016 damping Methods 0.000 claims description 4
- 230000007547 defect Effects 0.000 claims description 3
- 239000000463 material Substances 0.000 claims description 3
- 230000001052 transient effect Effects 0.000 claims description 3
- 241000284466 Antarctothoa delta Species 0.000 claims description 2
- 239000000126 substance Substances 0.000 claims 1
- 238000009661 fatigue test Methods 0.000 description 23
- 238000013001 point bending Methods 0.000 description 15
- 238000011160 research Methods 0.000 description 10
- 238000010586 diagram Methods 0.000 description 8
- 238000004364 calculation method Methods 0.000 description 6
- 238000010206 sensitivity analysis Methods 0.000 description 6
- 230000001133 acceleration Effects 0.000 description 5
- 238000003776 cleavage reaction Methods 0.000 description 5
- 238000006073 displacement reaction Methods 0.000 description 5
- 230000007017 scission Effects 0.000 description 5
- 238000004458 analytical method Methods 0.000 description 4
- 238000005056 compaction Methods 0.000 description 4
- 238000011161 development Methods 0.000 description 4
- 238000005065 mining Methods 0.000 description 4
- 230000000694 effects Effects 0.000 description 3
- 230000007246 mechanism Effects 0.000 description 3
- 238000005070 sampling Methods 0.000 description 3
- 235000002639 sodium chloride Nutrition 0.000 description 3
- FAPWRFPIFSIZLT-UHFFFAOYSA-M Sodium chloride Chemical compound [Na+].[Cl-] FAPWRFPIFSIZLT-UHFFFAOYSA-M 0.000 description 2
- 238000009825 accumulation Methods 0.000 description 2
- 230000015572 biosynthetic process Effects 0.000 description 2
- 238000012512 characterization method Methods 0.000 description 2
- 230000005489 elastic deformation Effects 0.000 description 2
- 230000000977 initiatory effect Effects 0.000 description 2
- 238000005259 measurement Methods 0.000 description 2
- 230000021715 photosynthesis, light harvesting Effects 0.000 description 2
- 238000012545 processing Methods 0.000 description 2
- 238000000518 rheometry Methods 0.000 description 2
- 230000035945 sensitivity Effects 0.000 description 2
- 239000011780 sodium chloride Substances 0.000 description 2
- 238000012795 verification Methods 0.000 description 2
- 101100283966 Pectobacterium carotovorum subsp. carotovorum outN gene Proteins 0.000 description 1
- 230000009286 beneficial effect Effects 0.000 description 1
- 230000008859 change Effects 0.000 description 1
- 230000006835 compression Effects 0.000 description 1
- 238000007906 compression Methods 0.000 description 1
- 230000001186 cumulative effect Effects 0.000 description 1
- 229910003460 diamond Inorganic materials 0.000 description 1
- 239000010432 diamond Substances 0.000 description 1
- 230000007613 environmental effect Effects 0.000 description 1
- 238000002474 experimental method Methods 0.000 description 1
- 238000011835 investigation Methods 0.000 description 1
- 230000003189 isokinetic effect Effects 0.000 description 1
- 230000003902 lesion Effects 0.000 description 1
- 238000012821 model calculation Methods 0.000 description 1
- 238000012986 modification Methods 0.000 description 1
- 230000004048 modification Effects 0.000 description 1
- 239000002245 particle Substances 0.000 description 1
- 230000002085 persistent effect Effects 0.000 description 1
- 238000002360 preparation method Methods 0.000 description 1
- 230000008458 response to injury Effects 0.000 description 1
- 150000003839 salts Chemical class 0.000 description 1
- 230000009528 severe injury Effects 0.000 description 1
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
- G06F17/13—Differential equations
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F2111/00—Details relating to CAD techniques
- G06F2111/10—Numerical modelling
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
- G06F2119/04—Ageing analysis or optimisation against ageing
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
- G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces
-
- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
- Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
- Y02T90/00—Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation
Landscapes
- Engineering & Computer Science (AREA)
- Physics & Mathematics (AREA)
- General Physics & Mathematics (AREA)
- Mathematical Physics (AREA)
- Theoretical Computer Science (AREA)
- Pure & Applied Mathematics (AREA)
- Mathematical Analysis (AREA)
- Mathematical Optimization (AREA)
- Computational Mathematics (AREA)
- Data Mining & Analysis (AREA)
- General Engineering & Computer Science (AREA)
- Geometry (AREA)
- Evolutionary Computation (AREA)
- Computer Hardware Design (AREA)
- Operations Research (AREA)
- Algebra (AREA)
- Databases & Information Systems (AREA)
- Software Systems (AREA)
- Investigating Strength Of Materials By Application Of Mechanical Stress (AREA)
Abstract
The invention discloses a construction method of a nonlinear rock fatigue constitutive model based on rheological model application, which comprises the following steps: constructing a rock fatigue constitutive model; obtaining the viscous element parameters of the fractional order description; obtaining rock fatigue constitutive model epsilon based on fractional derivative eve Is a strain expression of (2); obtaining a rock fatigue constitutive model epsilon based on fractional derivative after introducing Mittag-Leffler function eve (N); construction of nonlinear fatigue constitutive model epsilon reflecting mechanical characteristics of accelerated fatigue stage vp The method comprises the steps of carrying out a first treatment on the surface of the Acquisition of macroscopic initial injury D of rock mass ma And rock mass microscopic damage D mi A damage variable D after coupling; obtaining nonlinear fatigue constitutive model epsilon considering mechanical characteristics of initial damage in accelerated fatigue stage vp (N); a nonlinear fatigue constitutive model describing the overall process of fatigue of the rock mass is obtained taking into account the initial injury. According to the method, the nonlinear fatigue constitutive models describing different fatigue stages are combined to build the fatigue constitutive model describing the whole fatigue process of the rock mass, and the constitutive model is simple in parameters and clear in physical and mechanical significance.
Description
Technical Field
The invention relates to the technical field of constitutive models of rock structural surfaces, in particular to a construction method of a nonlinear rock fatigue constitutive model based on rheological model application.
Background
During the construction of geotechnical engineering and the exploitation of underground ores, rock masses are often subjected to cyclic fatigue loads. The macro-micro evolution of the displacement field and the stress field in the rock mass is an important reason for the final damage of the rock mass engineering. Starting from the constitutive model, the mechanical mechanism of the rock mass can be effectively revealed, for example, research on the damage evolution rule of the rock salt under different stress amplitude values, loading frequencies and loading rates is developed in documents "He, M., et al, experimental investigation and damage modeling of salt rock subjected to fatigue loading.International Journal of Rock Mechanics and Mining Sciences, 2019.114:p.17-23", and a fatigue life prediction model based on stress, frequency and loading rates is provided; as another example, in the literature "Li, T., et al, nonlinear behavior and damage model for fractured rock under cyclic loading based on energy dissipation principle.engineering Fracture Mechanics,2019.206:p.330-341," on the basis of an indoor test, a damage model of a fractured rock mass is established based on the energy dissipation principle; literature "Liu, y.and f.dai, adamage constitutive model for intermittent jointed rocks under cyclic uniaxial expression.international Journal of Rock Mechanics and Mining Sciences,2018.103:p.289-301," based on the principle of lematre strain equivalence, coupled fatigue constitutive models for discontinuous joint rock masses were derived; in literature "Meng, Q., et al, research on non-linear characteristics of rock energy evolution under uniaxial cyclic loading and unloading conditions, environmental Earth Sciences,2019.78 (23)", a nonlinear energy evolution model is proposed based on the impact of loading rate and lithology on rock energy evolution rules under cyclic loading conditions.
The constitutive model well researches the mechanical characteristics of the rock mass under cyclic fatigue loading, but has the problems of undefined physical and mechanical significance of parameters, insufficient description of important evolution rules of a displacement field and the like, and in the constitutive model research, the definition of the physical and mechanical significance of the model parameters is an important factor for determining the application value of model engineering. Therefore, it is necessary to build a rock fatigue constitutive model with a definite physical and mechanical meaning.
Disclosure of Invention
Based on the method, the invention aims to provide a construction method of a nonlinear rock fatigue constitutive model based on rheological model application, and the constitutive model describing the whole rock fatigue process is established, and has simple parameters and definite physical and mechanical significance.
In order to solve the technical problems, the invention adopts the following technical scheme:
the invention provides a construction method of a nonlinear rock fatigue constitutive model based on rheological model application, which comprises the following steps:
constructing a rock fatigue constitutive model epsilon corresponding to mechanical behaviors describing attenuation and stable fatigue stages in cyclic loading process of rock mass eve Describing rock fatigue constitutive model epsilon corresponding to mechanical behaviors of attenuation and stable fatigue stage in cyclic loading process of rock mass eve The constitutive equation of (2) is epsilon eve ＝ε e +ε ve The method comprises the steps of carrying out a first treatment on the surface of the Wherein ε e Characterized by strain, ε, of the transient loading phase ve Characterized by strain corresponding to the decay and stabilization fatigue phase;
introduction of Riemann-Liouville fractional integrationScore integration +.>Substituted into rock fatigue constitutive model epsilon eve Constitutive equation ε of (2) eve ＝ε e +ε ve In the method, a rock fatigue constitutive model epsilon based on fractional derivative is obtained eve Is a strain expression of (2);
introducing a Mittag-Leffler function to obtain a rock fatigue constitutive model epsilon based on fractional derivative after introducing the Mittag-Leffler function eve (N)；
Construction of nonlinear fatigue constitutive model epsilon reflecting mechanical characteristics of accelerated fatigue stage vp ；
Based on strain equivalence theory, obtaining macroscopic initial damage D of rock mass ma And rock mass microscopic damage D mi A damage variable D after coupling;
introducing the coupled damage variable D into a nonlinear fatigue constitutive model epsilon reflecting the mechanical characteristics of the accelerated fatigue stage vp In the method, a nonlinear fatigue constitutive model epsilon considering mechanical characteristics of initial damage in an accelerated fatigue stage is obtained vp (N)；
Rock fatigue constitutive model epsilon based on fractional derivative after Mittag-Leffler function is introduced eve (N) nonlinear fatigue constitutive model epsilon with mechanical properties of accelerated fatigue phase considering initial damage vp (N) series connection, obtaining a nonlinear fatigue constitutive model describing the whole process of rock body fatigue in consideration of initial damage.
In summary, the method for constructing the nonlinear rock fatigue constitutive model based on rheological model application provided by the invention combines nonlinear fatigue constitutive models describing different fatigue stages, and establishes a fatigue constitutive model describing the whole fatigue process of a rock mass, wherein parameters of the constitutive model are simple, and physical and mechanical meanings are clear.
Drawings
FIG. 1 is a graph showing similarity of rock mass deformation and damage mechanisms under rheological and cyclic loading and unloading conditions: (a) creep conditions, (b) cyclic loading and unloading conditions; stage I, II, III: damping, stabilizing, accelerating phases.
FIG. 2 is a nonlinear fatigue constitutive model based on fractional derivatives.
FIG. 3 is a constitutive model of nonlinear fatigue during acceleration phase.
Fig. 4 is a schematic diagram of initial damage to a rock mass.
FIG. 5 is a nonlinear fatigue constitutive model reflecting mechanical behavior of a rock mass in three stages under cyclic loading conditions.
FIG. 6 is a different E K Strain-cycle number curve under conditions.
Fig. 7 is a plot of strain versus cycle number for different gamma conditions.
FIG. 8 is a graph of the D-cycle number of the damage variable at various delta.
Fig. 9 is a rock sampling and cutting instrument.
FIG. 10 is a schematic diagram of a cleavage test specimen geometry and a room-side view.
Fig. 11 is a schematic diagram of a three-point bending test specimen geometry and a room physical diagram.
Fig. 12 is a cyclic fatigue test apparatus.
FIG. 13 is a schematic illustration of cyclic fatigue test loading and unloading.
FIG. 14 is a load-strain graph of a conventional cleavage test.
Fig. 15 is a three-point bending specimen loading process diagram.
Fig. 16 is a load-displacement graph of a conventional three-point bending test.
FIG. 17 is a graph of tensile stress versus strain for cyclic loading and unloading tests.
FIG. 18 is a graph of peak strain versus cycle number for a split fatigue test.
Fig. 19 is a comparison of the calculated model values presented herein with the split fatigue test values.
FIG. 20 is a graph of strain versus cycle number for a three-point bending fatigue test.
FIG. 21 is a comparison of three-point bending fatigue test values with the model calculations set forth herein.
Fig. 22 is a flow chart of a method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to an embodiment of the present invention.
Detailed Description
The invention will be further described in detail with reference to the drawings and the detailed description below, in order to further understand the features and technical means of the invention and the specific objects and functions achieved.
Referring to fig. 1, deformation and damage of a rock mass under cyclic loading and rheological test also increases nonlinearly with an independent variable such as time, and both are caused by crack evolution, which has three identical characteristic stages: the damping stage (stage I), the stabilizing stage (stage II) and the accelerating stage (stage III) show that the deformation and damage breaking mechanism of the rock mass under cyclic loading and rheological test, namely the evolution and deformation characteristics of cracks in space and time, have obvious similar characteristics.
Based on similarity of rock fatigue and rheology, the element model combination mode in rock rheology is applied to rock fatigue characteristic research, so that construction of a rock fatigue constitutive model is conveniently realized.
Fig. 22 is a flow chart of a method for constructing a nonlinear rock fatigue constitutive model based on rheological model application, which is provided in an embodiment of the present invention, as shown in fig. 22, and specifically includes the following steps:
s1, constructing a rock fatigue constitutive model epsilon corresponding to mechanical behaviors describing attenuation and stable fatigue stages in the cyclic loading process of the rock mass eve Describing rock fatigue constitutive model epsilon corresponding to mechanical behaviors of attenuation and stable fatigue stage in cyclic loading process of rock mass eve The constitutive equation of (2) is epsilon eve ＝ε e +ε ve 。
Wherein,,ε e characterized by strain, ε, of the transient loading phase ve Characterized by strain corresponding to the decay and stabilization fatigue phase; sigma is characterized by the upper stress limit of the load, E M The deformation modulus is characterized by the deformation modulus corresponding to the instantaneous strain generated under cyclic loading; e (E) K Characterised by attenuation and attenuation under cyclic loadingStabilizing deformation modulus corresponding to strain in fatigue stage; η (eta) K Is a viscous element parameter characterized by a strain rate at cyclic loading; epsilon Ek For modulus of deformation E K Corresponding strain, ε ηk Is the parameter eta of the viscous element K Corresponding strain.
S2, introducing Riemann-Liouville fraction integralScore integration +.>Substituted into rock fatigue constitutive model epsilon eve Constitutive equation ε of (2) eve ＝ε e +ε ve In the method, a rock fatigue constitutive model epsilon based on fractional derivative is obtained eve Is a strain expression of (2); wherein Riemann-Liouville fractional integration is introduced>Describing the parameter eta of the viscous element K 。
Wherein, step S2 is to introduce Riemann-Liouville fraction integralIntegrating the scoreSubstituted into rock fatigue constitutive model epsilon eve Constitutive equation ε of (2) eve ＝ε e +ε ve In the method, a rock fatigue constitutive model epsilon based on fractional derivative is obtained eve The specific operations include:
step S21, introducing Riemann-Liouville fraction integralWherein (1)>Xi, N isIntegrating the parameters of equation f (N); gamma (Gamma) is Gamma function, gamma E (0, 1), and->
S22, calculating gamma-order calculus of f (N); wherein the gamma-order calculus of f (N) satisfies
Step S23, riemann-Liouville fraction integrationSubstituted into rock fatigue constitutive model epsilon eve Constitutive equation ε of (2) eve ＝ε e +ε ve The following formula is obtained:
step S24, obtaining epsilon ve Expression of (2)Wherein epsilon is based on the initial condition n=0 ve =0 and fractional differential theory to obtain ε ve Expression +.>Fractional differential theory is described in literature data "kilbs AA, srivastatin HM, trujillo jj. Area and applications of fractional differential equivalents, amsterdam: elsevier;2006. "in;
step S25, combining the formulas in step S24Rock fatigue constitutive model epsilon eve Constitutive equation ε of (2) eve ＝ε e +ε ve Obtaining rock fatigue based on fractional derivativesConstitutive model epsilon eve Strain expression +.>
S3, introducing a Mittag-Leffler function to obtain a rock fatigue constitutive model epsilon based on fractional derivative after introducing the Mittag-Leffler function eve (N) the fatigue constitutive model epsilon based on fractional derivative after Mittag-Leffler function is introduced eve (N) effectively describes the mechanical behavior of the rock mass during the decay and stabilization fatigue phase.
The specific operation of the method in step S3 includes:
step S31, introducing Mittag-Leffer functionFor formula-> Conversion is carried out to obtain:
due to the formulaMiddle accumulation item->The calculation amount is heavy, when the number of times N of circulation is large, the calculation and the parameter inversion are difficult, and after the Mittag-Leffler function is introduced, the calculation amount is effectively reduced, and the parameter inversion cost is reduced.
Step S32, according to the formulaIt can be seen that->Thereby obtaining the formulaThe process of representation of gamma Γ (γp) is as follows:
thereby obtaining the fatigue constitutive model epsilon based on the fractional derivative after introducing Mittag-Leffler function eve Constitutive equation of (N):
s4, constructing a nonlinear fatigue constitutive model epsilon reflecting mechanical characteristics of an accelerated fatigue stage vp The method comprises the steps of carrying out a first treatment on the surface of the The nonlinear fatigue constitutive model epsilon reflecting the mechanical characteristics of the accelerated fatigue stage vp The constitutive equation of (2) satisfies the formula
Wherein sigma is the upper stress of the load; η (eta) vp Is a viscous element coefficient; d is a damage variable; />Nonlinear fatigue constitutive model epsilon for reflecting mechanical characteristics of accelerated fatigue stage vp The first derivative of the medium strain and the cycle number N reflects the increasing speed of the strain along with the cycle number; sigma (sigma) S For a threshold stress value at which the rock mass enters an accelerated fatigue phase, σ in this embodiment S The peak stress was 0.75 times, which is described in literature data "Geranmayeh Vaneghi, r., et al, fatigue damage response of typical crystalline and granular rocks to uniaxial cyclic compression.international Journal of Fatigue,2020.138," and "Ma, l., et al, mechanical properties of rock salt under combined creep and fatigue. International Journal of Rock Mechanics and Mining Sciences,2021.141.
Constitutive equation of fatigue constitutive model based on fractional derivative after Mittag-Leffler function is introducedIt can be seen that the fatigue constitutive model based on fractional derivative after Mittag-Leffler function is introduced can effectively describe the mechanical behavior of the fatigue stage, but lacks the reflection of the accelerated fatigue stage, so that the invention constructs a nonlinear fatigue constitutive model epsilon capable of reflecting the mechanical characteristics of the accelerated fatigue stage on the basis vp . As can be seen from fig. 1, the nonlinear characteristics of the rock mass are most remarkable when entering the accelerated fatigue stage, so that the nonlinear evolution of the mechanical characteristics is an important consideration when constructing a corresponding nonlinear fatigue constitutive model reflecting the mechanical characteristics of the accelerated fatigue stage.
The nonlinear fatigue constitutive model for reflecting the mechanical characteristics of the accelerated fatigue stage constructed by the invention is shown in the figure 3, and the nonlinear fatigue constitutive model reflecting the mechanical characteristics of the accelerated fatigue stage is formed by a characterization threshold stress value sigma S Is formed by connecting a plastic element representing the first derivative of the strain and the cycle number in parallel; when the loaded stress and the cycle number reach threshold values, the nonlinear fatigue constitutive model reflecting the mechanical characteristics of the accelerated fatigue stage is touched and starts to play a role.
S5, acquiring macroscopic initial damage D of the rock mass based on strain equivalence theory ma And rock mass microscopic damage D mi Damage variable D after coupling.
The specific operation of the method in step S5 includes:
step S51, based on strain equivalence theory, rock mass macroscopic initial damage D ma And rock mass microscopic damage D mi The expression of the impairment variable D after coupling satisfies d=d ma +D mi -D ma D mi The method comprises the steps of carrying out a first treatment on the surface of the Wherein D is ma Macroscopic initial damage to the rock mass; d (D) mi Is microscopic damage to the rock mass. From the expression d=d ma +D mi -D ma D mi It can be seen that when only microscopic damage exists in the rock mass, i.e. macroscopic damage D ma =0, the damage variable D after coupling is equal to the microscopic damage D mi The method comprises the steps of carrying out a first treatment on the surface of the When microscopic damage D mi =0, i.e. when only macroscopic damage of the rock mass occurs, the damage variable after coupling d=d ma The expression of the damage variable D after the coupling can be well applied to the research of mechanical properties under macro-micro damage coupling conditions.
The rock mass is affected by various geological effects during the formation process and external factors such as stress, weathering and earth movement after the formation, so that natural macroscopic defects such as joints are generated in the rock mass, and as shown in fig. 4, the existence of the initial damages seriously affects the mechanical properties of the rock mass. Therefore, in the research of mechanical properties of the unfolded rock mass, microscopic damage caused by initial damage and cyclic load needs to be considered simultaneously. Under the assumption of strain equivalence theory, the references "Liu, h.y., et al, A dynamic damage constitutive model for a rock mass with persistent joints.international Journal of Rock Mechanics and Mining Sciences, 2015.75:p.132-139", "initial damage to rock mass D ma And rock mass microscopic damage D mi The impairment variable after coupling can be expressed as: d=d ma +D mi -D ma D mi 。
Step S52, through the formulaObtaining macroscopic initial damage D of rock mass ma Wherein E is C Modulus of elasticity, E, of rock mass containing natural macroscopic defects O Is the elastic modulus of the complete rock mass.
Step S53, determining microscopic damage D through Kachanov damage law mi Relationship with cycle number N, wherein the Kachanov's law of damage equation is:
A. delta is the material constant determined by the test; omega is the applied load;is a rock microscopic damage variable D mi A first derivative with respect to the number of cycles N; in this embodiment, as shown in fig. 1, the relationship between deformation and cycle times of rock mass under cyclic loading conditions and the relationship between deformation and time under creep conditions have similarity, and based on the above characteristics, the invention determines microscopic damage D by Kachanov damage law in creep mi Relationship to the number of cycles N.
When the number of cycles n=0, the rock mass sample is not subjected to cyclic loading, D mi =0, based on the initial conditions, and toIntegrating to obtain D mi The expression is
D when the rock mass is completely destroyed mi =1, corresponding complete damage cycle number N C The method meets the following conditions:
N C ＝[A(δ+1)ω δ ] -1 ；
bonding ofN C ＝[A(δ+1)ω δ ] -1 The rock mass damage evolution equation under the cyclic loading condition can be obtained to meet the condition:
Step S54, the nonlinear fatigue constitutive model epsilon reflecting the mechanical characteristics of the accelerated fatigue stage vp Described is the mechanical behaviour of a rock mass in the accelerated fatigue phase, n=n when damage begins to occur S Thus the formulaThe actual injury cycle number in the middle is +.>The actual complete damage cycle number is: />Under this condition, the formula ++in step S53 is given>The method is changed into that:
wherein Ns is the number of cycles of the rock mass into the accelerated fatigue phase; n (N) F The number of cycles at the moment of complete destruction of the rock mass, i.e. the fatigue life.
When n=ns, D mi ＝0，N＝N F At time D mi ＝1。
Step S55, the formula is calculatedAnd formula->The formula d=d substituted into step S51 ma +D mi -D ma D mi In the method, the initial macroscopic damage D of the rock mass can be considered ma And rock mass microscopic damage D mi The expression of the impairment variable D after coupling satisfies:
s6, introducing the coupled damage variable D into a nonlinear fatigue constitutive model epsilon reflecting the mechanical characteristics of the accelerated fatigue stage vp In the method, a nonlinear fatigue specimen considering the mechanical characteristics of the initial damage in the accelerated fatigue stage is obtainedConstruction model epsilon vp (N); wherein the damage variable D after coupling is the initial macroscopic damage D of the rock mass ma And rock mass microscopic damage D mi Damage variable D after coupling.
The specific operations of the method in step S6 include:
Step S61, formulaSubstituted into nonlinear fatigue constitutive model epsilon reflecting mechanical characteristics of accelerated fatigue stage vp Constitutive equation of->In the method, constitutive equation of the nonlinear fatigue constitutive model considering mechanical characteristics of initial damage in the accelerated fatigue stage can be obtained:
the conversion of the above formula can be obtained:
step S62, combining the initial conditions n=n S ，ε vp =0, pair of formulasSolving can be carried out to obtain a nonlinear fatigue constitutive model expression considering the mechanical characteristics of the initial damage in the accelerated fatigue stage, wherein the nonlinear fatigue constitutive model expression satisfies the following conditions:
s7, introducing a Mittag-Leffler function into the rock fatigue constitutive model epsilon based on fractional derivative eve (N) nonlinear fatigue with consideration of initial injury accelerated fatigue phase mechanical propertiesConstitutive model epsilon vp (N) series connection, obtaining a nonlinear fatigue constitutive model describing the whole process of rock body fatigue in consideration of initial damage.
Specifically, in the invention, as shown in fig. 2, 3 and 5, a rock fatigue constitutive model based on fractional derivative after the Mittag-Leffer function is introduced in fig. 2 and a nonlinear fatigue constitutive model considering mechanical characteristics of initial damage in an accelerated fatigue stage in fig. 3 are connected in series, so that the nonlinear fatigue constitutive model reflecting mechanical behaviors of the rock in three stages (an attenuation stage, a stabilization stage and an acceleration stage) under the cyclic loading condition of the rock as shown in fig. 5, namely, a nonlinear fatigue constitutive model describing the whole process of the rock fatigue in consideration of initial damage, can be obtained.
The specific operations of the method in step S7 include:
step S71, connecting a rock body fatigue constitutive model based on fractional derivative after introducing a Mittag-Leffler function and a nonlinear fatigue constitutive model considering mechanical characteristics of an initial damage in series, and obtaining a nonlinear fatigue constitutive model describing the whole rock body fatigue process by considering the initial damage, wherein a strain equation of the nonlinear fatigue constitutive model describing the whole rock body fatigue process by considering the initial damage is satisfied:
ε＝ε e +ε ve +ε vp ；
step S72, combining formulas
Sum formulaDetermining a specific strain equation epsilon (N) of a nonlinear fatigue constitutive model describing the whole process of rock mass fatigue taking initial damage into consideration:
the invention can obtain the concrete D (N) and epsilon (N) through the steps S1 to S7Expression formula, wherein the formula has parameters with unclear mechanical properties and E K Gamma and delta, the values of which vary with differences in rock mass properties, E K Rock fatigue characteristics characterized by gamma and delta values, different model parameters E can be calculated by a controlled variable method K D (N) and epsilon (N) values under gamma and delta to obtain E K The reflection conditions of gamma and delta values on the fatigue characteristics of the rock specifically comprise the following steps: (1) e is divided by epsilon (N) of a fatigue constitutive model by adopting a controlled variable method K Parameters other than gamma and delta are constants, and different E's are calculated respectively K Values of D (N), ε (N) under γ and δ conditions; (2) drawing corresponding D-N and epsilon-N curves to obtain E K And the gamma and delta values reflect the fatigue characteristics of the rock mass. Acquisition E described above K The method for reflecting the conditions of the gamma and delta values on the fatigue characteristics of the rock can obtain E through calculation and analysis K Fatigue characteristics reflected by gamma and delta values, so that the nonlinear fatigue constitutive model can be inverted from E after fitting test data K The gamma and delta values obtain the mechanical properties of the rock mass.
In addition, in the parameters of formula (A), E M 、η K And eta VP Meaning is relatively clear, representing respectively the instantaneous strain, the steady-phase base rate and the acceleration-phase initial rate at cyclic loading, while E K The mechanical properties of the characterization of γ and δ are not clear enough. The invention adopts a control variable method commonly used in sensitivity analysis to develop and analyze the sensitivity of the three parameters.
Let σ be<σ S Will E M ＝500MPa，η K =50gpa, γ=0.5 is substituted into formula (a), taking E K Parameter sensitivity analyses were performed for 1, 3, 5, 7, and 9GPa, and the results are shown in FIG. 6. As can be seen from the figure, E K Has obvious influence on strain, and is expressed as E K The larger the attenuation and the less strain stabilizing the fatigue phase, E K Inversely proportional to the strain at the decay and stabilization fatigue stage. On the other hand, different E K The difference of the strain nonlinear characteristics of the lower attenuation and stable fatigue stage is not obvious, E K The strain quantity of the decay and stable fatigue stage is mainly characterized, since in cyclic loading and unloading,the strain at the decay and stabilization fatigue stage represents to a large extent the degree of compaction of the plastic hysteresis loop and the damage characterized by the degree of compaction, thus E K The plastic hysteresis loop area and total amount of damage at the decay and isokinetic fatigue stage were also characterized.
Let σ be<σ S Will E M ＝500MPa，E K ＝5GPa，η K The result of parameter sensitivity analysis of parameter γ, which was substituted with 50GPa in equation (a) and was 0.1, 0.3, 0.5, 0.7, and 0.9, is shown in fig. 7. It can be seen from the figure that the ultimate strain at different γ is almost uniform, and γ has no effect on the amount of strain in the decay and stable fatigue stage. But the nonlinear characteristic difference of the curve under different gamma is obvious, and the larger the gamma is, the higher the rate of increase of the strain with the cycle number is, and the more obvious the nonlinear characteristic of the curve is. Gamma mainly characterizes the nonlinear characteristics of the decay and stabilization fatigue phases. From E K The reflection of strain on the degree of plastic hysteresis loop density in the parameter sensitivity analysis of (a), gamma also characterizes the nonlinear characteristics of plastic hysteresis loop area expansion and damage variable evolution in the decay and stable fatigue stage in cyclic loading and unloading.
The parameter delta is an important parameter for representing the damage in the accelerated fatigue stage, and to understand the damage characteristic reflected by delta, the sigma is assumed<σ S Taking outN F ＝100，N S Let δ= -0.8, -0.5,0,0.5,1 be substituted into equation (21) = 90, and the result of parameter sensitivity analysis is shown in fig. 8. It can be seen that although the damage evolution end points under different delta are consistent, the difference of the damage evolution trend is very prominent, and the specific expression is as follows: when delta<At 0, the damage evolves with a trend of rate decay, and the smaller delta, the more obvious the trend of rate decay; when delta=0, the damage evolution trend is linear, and the damage-cycle number curve is evolved in a straight line form; when delta>At 0, the damage evolves with a continuously increasing rate, and the greater delta, the more pronounced the trend of rate increase. It can be known that delta is an important parameter for representing the damage trend of the accelerated fatigue stage, and the rock mass accelerated fatigue stage can be obtained through inversion numerical value of deltaAnd (5) damage evolution characteristics.
The construction method of the nonlinear rock fatigue constitutive model based on rheological model application is completely determined, the schematic diagram of the constitutive model is shown in fig. 5, the physical and mechanical significance of parameters of the constitutive model is clear, the method is suitable for describing the mechanical behavior of the whole fatigue process, and the feasibility of the construction method of the nonlinear rock fatigue constitutive model based on rheological model application is verified through a test example.
Substituting the indoor test data into a nonlinear rock fatigue constitutive model strain equation epsilon (N) and determining a threshold stress value sigma of an accelerated fatigue stage S And verifying the effectiveness and rationality of the model through experiments, and obtaining the fatigue characteristics of the rock mass through inverted parameters.
In order to verify the rationality and effectiveness of the fatigue constitutive model, an indoor cycle fatigue loading and unloading test is developed. The method adopts widely distributed red sandstone in geology as a research object, and samples are taken from red sandstone bodies in certain areas of Shandong China, belong to fine sandstone, are light brown in color, and have granulated sugar-like internal particles. After sampling by the drill coring method, professionals are entrusted to cut and process the sample according to the test requirement, and the sampling and cutting instrument is shown in fig. 9. In engineering practice, the initiation and the expansion of the pulling crack are important reasons for the damage of the rock mass, so the invention aims at the mechanical property of the pulling crack under the initiation and the expansion actions of the pulling crack under the cyclic loading condition, expands the cyclic fatigue loading and unloading test of which the two pulling cracks are main damage reasons and explores the mechanical property of the rock mass under the cyclic fatigue loading. The splitting test adopts a disc-shaped sample with prefabricated herringbone cracks, the diameter d of the disc is 100mm, the thickness t of the disc is 35mm, and a diamond slice is adopted to cut and prefabricate the diameter R S The geometric schematic diagram and the indoor physical diagram of the test sample are shown in figure 10, and the elastic modulus of the tensile stress-strain curve of the complete test sample is 572.64MPa. The three-point bending test uses a semicircular disk-shaped specimen containing a slit, the diameter d of the specimen is 100mm, the thickness t of the specimen is 35mm, the length of the slit is 20mm, and the width of the slit is 2mm, as shown in FIG. 11. The tensile stress-strain curve elastic modulus of the complete test specimen is 840.1MPa.
Test preparation
The test was developed on a WHY-200/100 microcomputer controlled universal tester as shown in FIG. 12. The instrument consists of a host, measurement and control software, a measurement and control system and the like, and the loading of the sample is realized through force or displacement control. The method has the advantages of strong stability, high precision, large measuring range and the like, and can meet the requirements of Brazilian splitting and cyclic loading and unloading tests of the red sandstone sample. In the development cycle fatigue test, the test piece was subjected to fatigue loading under constant temperature and constant humidity in a loading/unloading manner as shown in fig. 13.
Split fatigue test
To determine the cleavage fatigue test protocol, a conventional indoor cleavage test is first developed. And loading three identical herringbone fracture disc-shaped samples in a stress control mode, wherein the loading rate is 0.1KN/s. After the test, the data were processed, and as shown in fig. 14, it can be seen that as loading proceeds, the curve enters the elastic deformation phase from a distinct micro-crack compaction phase, and then brittle fracture occurs near the peak load point. And (3) substituting the average peak load of the three samples into a split test tensile strength calculation formula shown in the formula (B), so that the average tensile strength of the samples is 1.903MPa, and meanwhile, calculating according to a test curve, the elastic modulus of the samples is 753.51MPa.
Middle sigma t Is tensile strength; p is the peak load; d, t is the sample diameter and thickness.
Because the sample in the splitting test is destroyed mainly by stretching, the tensile stress is selected as the upper and lower limit stress of the cyclic loading and unloading test, and 3 groups of upper and lower limit stress amplitude values are unfolded to be 0.6sigma by taking the tensile stress as a variable t Is a cyclic fatigue loading and unloading test of (c). The test protocol is shown in table 1. The test is ended and the corresponding data recorded when the load break or cycle number reaches 100.
Table 1 split fatigue test protocol
Three-point bending fatigue test
Likewise, a conventional three-point bend test in the room was first developed and the test loading procedure is shown in FIG. 15. The loading was performed by displacement control, loading from the top center row at a loading rate of 0.1 mm/min. After the test was completed, data processing was performed, and the results are shown in fig. 16. Since the tensile strength calculation formula of the three-point bending test is consistent with that of the splitting test, the average peak load of the three samples of 2.62KN is substituted into (B) to calculate that the tensile strength of the sample is 0.953MPa, and a three-point bending fatigue test scheme is designed based on the tensile strength, as shown in Table 2. The elastic modulus of the test specimen is calculated by a test curve and is 381.39MPa.
Table 2 three point bending fatigue test protocol
Cleavage fatigue test results and model verification
The tensile stress-strain curve of the split fatigue test is shown in FIG. 17, and it can be seen from the graph that the upper limit tensile stress is 0.75σ t The test piece curve plastic hysteresis loop is in a tight state and is not destroyed after repeated cyclic loading, which indicates that the crack inside the test piece is insufficient in development and the deformation is mainly elastic deformation under the loading condition. And for an upper limit tensile stress of 0.85 sigma t The S-2 test sample is characterized in that the plastic hysteresis loop of the test sample curve is loose, and the test sample is destroyed after about 100 cycles, which indicates that the test sample crack development environment is good under the condition, but a certain cycle number is needed for the destruction. For an upper tensile stress of 0.95 sigma t The S-3 test sample has the most sparse curve plastic hysteresis loop, and the test sample is subjected to fatigue damage after being subjected to 6 times of cyclic load, so that the test sample is deformed mainly by plastic deformation under the load condition, cracks have sufficient development environment, and the damage only needs few cyclic times. It can be seen that the red sandstone has obvious nonlinear characteristics at each fatigue stage,and the influence of the cycle times and the stress is larger, and the research on the influence of the cycle times and the stress on the fatigue characteristics of the engineering rock mass has higher engineering value. Since the peak strain most intuitively reflects the fatigue characteristics of the rock mass, in the invention, the relationship between the peak strain and the peak stress at different times of the cycle is described by adopting the formula (A). Processing the data in fig. 17 results in a peak strain versus cycle number plot as shown in fig. 18.
The data shown in fig. 18 are fitted by using the formula (a), and the comparison result of the calculated value and the experimental value of the model proposed by the present invention is shown in fig. 19. From the graph, the fatigue constitutive model provided by the invention is well matched with the test curve, the fitting effect is obvious, and the rationality and the effectiveness of the model are effectively verified. The results of the parametric inversion are shown in table 3. The combination model parameter sensitivity analysis results show that: (1) Parameter E M ：S-1<S-2<S-3, showing that the deformation modulus corresponding to the instantaneous strain of the red sandstone is increased along with the increase of the upper limit stress; as can be seen from the stress-strain curve of fig. 14, this is a result of the longer compaction phase of the sandstone microcrack and no significant softening before peak stress: the larger the upper stress, the higher the linear phase of maximum slope in the curve is, and therefore the larger the corresponding instantaneous deformation modulus. (2) Parameter E K Obeying S-2<S-1<S-3, the strain and damage caused by cyclic fatigue load to the sample S-2 is greatest, S-1 times is least, and S-3 is least in the damping and stabilizing fatigue stage. This is because the upper limit stress of sample S-1 is minimal, the fatigue load causes limited fatigue damage to the sample, and the corresponding strain is less than sample S-2; and for an upper stress of 0.95 sigma t The S-3 sample of (2) has the attenuation and stable fatigue stage only existing in the early few cyclic loading stages, so that the damage caused by the fatigue load is small and the corresponding strain is minimum. (3) parameter γ: s-2<S-1<S-3, the plastic hysteresis loop area of the sample S-3 has the most obvious trend with the increase of the cycle number, S-1 times and the minimum S-2 in the attenuation and stable fatigue stage. The higher the upper limit stress is, the more obvious the sample decays and the damage evolution trend is in the stable fatigue stage, while the lower stress is in the non-accelerated fatigue stageUnder conditions, the evolution trend of damage is not always as pronounced as under high stress conditions. (4) parameter δ: s-3>S-2, the plastic hysteresis loop expansion degree of the S-3 sample is more obvious in the accelerated fatigue stage, and the damage evolution is more severe, so that the damage evolution characteristics in the accelerated fatigue stage are closely related to the stress level.
TABLE 3 parameter inversion results
Three-point bending fatigue test result and model verification
After the three-point bending fatigue test was completed, the test data were processed to obtain a strain-cycle number curve as shown in fig. 20. As can be seen from the figure, the curves at different stress levels have 3 morphologies as well, similar to the split fatigue test: upper limit stress of 0.7σ t The strain-cycle number curve only has the decay and stable fatigue stage, and the test curve of the T-1 sample is free from accelerated fatigue and damage. The upper limit stress is 0.8sigma exceeding the threshold stress value of the accelerated fatigue stage t The curve contains T-2 sample test curves for decay, stabilization and acceleration of the three fatigue phases. The upper limit stress is 0.9sigma of threshold stress value in the far super-acceleration fatigue stage t The cycle times of the decay and stabilization and acceleration fatigue stage are very small, and the test curve of the T-3 sample which rapidly enters the acceleration fatigue stage and is damaged is provided. Indicating that stress levels have a significant impact on curve morphology. The data shown in fig. 20 were fitted using formula (a), and the calculated values of the proposed model of the present invention were compared with the experimental values, as shown in fig. 21. The fatigue constitutive model provided by the invention can accurately describe a test curve, and effectively verify the rationality and effectiveness of the model. The results of the parametric inversion are shown in table 4. The combination model parameter sensitivity analysis results show that: (1) For parameter E K ：T-2<T-1<T-3, which illustrates that the decay of sample T-2 and the corresponding strain at the steady fatigue stage are maximum, T-1 times, and T-3 is minimum. Indicating that the cyclic fatigue loading was applied to sample T-2 during the decay and stabilization fatigue phase The damage is the largest, T-1 times, and T-3 is the smallest. This is because the upper stress limit of the T-1 specimen is minimal and the cumulative total amount of fatigue damage in the decay and stabilization fatigue phase is small; whereas the T-3 specimen has reached 0.9σ due to the upper stress limit t The initial sample is cyclically loaded to enter an unstable accelerated fatigue stage quickly, and the time for the decay and stable fatigue stage is very short, so that the corresponding damage total amount is minimum. (2) parameter γ: t-2<T-1<T-3, shows that in the decay and steady fatigue stage, the tendency of the specimen T-3 plastic hysteresis loop area to increase with cycle number is most pronounced, T-1 times, and T-2 is smallest. The most obvious trend of the area of the plastic hysteresis loop of the sample T-3 along with the increase of the cycle times shows that the damage characteristics of the decay and stable fatigue stage have obvious dependence on the stress level; while the plastic hysteresis loop of sample T-1 has a more pronounced trend than T-2, in combination with E K The analysis of the minimum total damage amount of the T-1 sample in the inversion of (a) shows that in the attenuation and stable fatigue stage, the total damage amount and the stress level are in positive correlation, but the damage evolution trend under the low stress condition without accelerated fatigue is not always less obvious than that under the high stress condition. (3) parameter δ: t-3 >T-2 shows that in the accelerated fatigue stage, the expansion degree of the plastic hysteresis loop of the T-3 sample is more obvious, the damage evolution is more severe, and the damage characteristics in the accelerated fatigue stage have higher dependence on stress. Comparing analysis results of parameter inversion of the split fatigue test, the change characteristics of each parameter of the three-point bending fatigue test along with the stress level are similar to the parameter inversion characteristics of the split fatigue test, so that the fatigue characteristics under two test conditions are common, and the fatigue constitutive model provided by the invention has wide applicability and good accuracy, and can be suitable for fatigue characteristic research under different boundary conditions.
TABLE 4 three-point bending fatigue test model parameter inversion results
Compared with the existing construction method of the constitutive model, the construction method has the following beneficial effects:
1. aiming at the rock mass mechanical behaviors in the attenuation and fatigue stages, a nonlinear fatigue constitutive model based on fractional derivatives is provided, wherein the model not only can describe the relationship between the fatigue deformation of the rock mass and the cycle times, but also can reflect nonlinear characteristics in the fatigue process; model parameters E K The plastic hysteresis loop area and the total damage amount in the fatigue stage can be reflected; gamma can characterize the nonlinear characteristics of plastic hysteresis loop area expansion trend and lesion variable evolution in the decay and stabilization fatigue stage.
2. For an accelerated fatigue stage with severe damage evolution, describing microscopic damage of a rock mass by adopting a Kachanov damage law, and obtaining a damage expression considering initial damage after combining a coupling expression of macroscopic damage and microscopic damage; applying the damage expression to the description of the damage evolution of the viscous element, and establishing a nonlinear fatigue constitutive model reflecting the mechanical characteristics of the accelerated fatigue stage, wherein the model parameters delta can reflect the damage evolution characteristics of the rock mass in the accelerated fatigue stage, and the larger the delta is, the more severe the rock mass damage is; the nonlinear fatigue constitutive model combinations of different fatigue stages are described, a fatigue constitutive model describing the whole fatigue process of the rock mass is established, and parameters of the constitutive model are simple, and the physical and mechanical significance is clear.
3. Comparing the calculated value of the fatigue constitutive model provided by the invention with the indoor split and three-point bending fatigue test value, the nonlinear fatigue mechanical behavior of the rock mass can be accurately reflected by the fatigue constitutive model provided by the invention, and the rationality and reliability of the model provided by the invention are verified. According to the model parameter inversion result, the stress and the cycle number are important factors influencing the fatigue damage evolution.
(1) Stage of fatigue, decay and stabilization: under different stress conditions, damage is accumulated continuously with the increase of the cycle times. However, under the condition of low stress, the sample is not subjected to fatigue damage, and the damage is slowly accumulated along with the increase of the cycle times and is converged to a certain characteristic value; with the increase of the stress level, the accumulation of damage in the decay and stabilization fatigue stage starts to increase, and in the case of accelerated fatigue, the higher the upper limit stress, the higher the damage evolution rate.
(2) Accelerated fatigue phase: the higher the stress level is, the more severe the damage evolves, and the fewer the number of cycles required for fatigue failure to occur, indicating that the characteristics of the accelerated fatigue stage damage evolution are closely related to the stress level.
The above examples merely represent a few embodiments of the present invention, which are described in more detail and are not to be construed as limiting the scope of the present invention. It should be noted that it will be apparent to those skilled in the art that several variations and modifications can be made without departing from the spirit of the invention, which are all within the scope of the invention. Accordingly, the scope of the invention should be assessed as that of the appended claims.
Claims (10)
1. The construction method of the nonlinear rock fatigue constitutive model based on rheological model application is characterized by comprising the following steps,
constructing a rock fatigue constitutive model epsilon corresponding to mechanical behaviors describing attenuation and stable fatigue stages in cyclic loading process of rock mass eve Describing rock fatigue constitutive model epsilon corresponding to mechanical behaviors of attenuation and stable fatigue stage in cyclic loading process of rock mass eve The constitutive equation of (2) is epsilon eve ＝ε e +ε ve The method comprises the steps of carrying out a first treatment on the surface of the Wherein ε e Characterized by strain, ε, of the transient loading phase ve Characterized by strain corresponding to the decay and stabilization fatigue phase;
introduction of Riemann-Liouville fractional integrationScore integration +.>Substituted into rock fatigue constitutive model epsilon eve Constitutive equation ε of (2) eve ＝ε e +ε ve In the method, a rock fatigue constitutive model epsilon based on fractional derivative is obtained eve Is a strain expression of (2);
introducing a Mittag-Leffler function to obtain a Mittag-Leffler functionRock fatigue constitutive model epsilon based on fractional derivative eve (N)；
Construction of nonlinear fatigue constitutive model epsilon reflecting mechanical characteristics of accelerated fatigue stage vp ；
Based on strain equivalence theory, obtaining macroscopic initial damage D of rock mass ma And rock mass microscopic damage D mi A damage variable D after coupling;
introducing the coupled damage variable D into a nonlinear fatigue constitutive model epsilon reflecting the mechanical characteristics of the accelerated fatigue stage vp In the method, a nonlinear fatigue constitutive model epsilon considering mechanical characteristics of initial damage in an accelerated fatigue stage is obtained vp (N)；
Rock fatigue constitutive model epsilon based on fractional derivative after Mittag-Leffler function is introduced eve (N) nonlinear fatigue constitutive model epsilon with mechanical properties of accelerated fatigue phase considering initial damage vp (N) connecting the two components in series to obtain a nonlinear fatigue constitutive model which is used for describing the whole fatigue process of the rock mass and considers initial damage; the strain equation of the nonlinear fatigue constitutive model satisfies the following conditions: epsilon=epsilon e +ε ve +ε vp 。
2. The method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 1, wherein the rock fatigue constitutive model epsilon corresponding to mechanical behaviors of damping and stabilizing fatigue stages in the cyclic loading process of the descriptive rock mass eve The constitutive equation of (2) is epsilon eve ＝ε e +ε ve In (a) Wherein σ is characterized by the upper stress limit of the load, E M The deformation modulus is characterized by the deformation modulus corresponding to the instantaneous strain generated under cyclic loading; e (E) K The deformation modulus is characterized by attenuation generated under cyclic loading and corresponding to strain in a stable fatigue stage; η (eta) K Is a viscous element parameter characterized by a strain rate at cyclic loading; epsilon Ek For modulus of deformation E K Corresponding strain, ε ηk Is the parameter eta of the viscous element K Corresponding strain.
3. The method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 1 or 2, wherein the steps introduce a Riemann-liooville fractional integralScore integration +.>Substituted into rock fatigue constitutive model epsilon eve Constitutive equation ε of (2) eve ＝ε e +ε ve In the method, a rock fatigue constitutive model epsilon based on fractional derivative is obtained eve The specific operations of the method of strain expression of (a) include,
introduction of Riemann-Liouville fractional integration Wherein (1)>ζ and N are parameters of an integral equation f (N); gamma (Gamma) is Gamma function, gamma E (0, 1), and->
Calculating gamma calculus of f (N); wherein the gamma-order calculus of f (N) satisfies
The Riemann-Liouville fraction was integratedSubstituted into rock fatigue constitutive model epsilon eve Constitutive equation ε of (2) eve ＝ε e +ε ve The following formula is obtained:
acquisition of epsilon ve Expression of (2)
Combination formulaConstitutive equation epsilon of rock fatigue constitutive model eve ＝ε e +ε ve Obtaining rock fatigue constitutive model epsilon based on fractional derivative eve Strain expression +.>
4. The method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 3, wherein the step of obtaining epsilon ve Expression of (2)The method comprises the following specific operations:
epsilon based on the initial condition n=0 ve =0 and fractional differential theory to obtain ε ve Expression of (2)
5. The method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 3, wherein the steps introduce a Mittag-Leffler function to obtain a rock fatigue constitutive model epsilon based on fractional derivative after introducing the Mittag-Leffler function eve The method of (N), the specific operations include,
step S31, introducing Mittag-Leffer function For formula-> Conversion is carried out to obtain:
step S32, according to the formulaIt can be seen that->Thereby obtaining the formulaThe process of representation of gamma Γ (γp) is as follows:
thereby obtaining the fatigue constitutive model epsilon based on the fractional derivative after introducing Mittag-Leffler function eve Constitutive equation of (N):
6. the method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 5, wherein the method comprises the following steps: the steps construct a nonlinear fatigue constitutive model epsilon reflecting the mechanical characteristics of the accelerated fatigue stage vp Nonlinear fatigue constitutive model epsilon reflecting mechanical characteristics of accelerated fatigue stage vp The constitutive equation of (2) satisfies the formulaWherein sigma is the upper stress of the load; η (eta) vp Is a viscous element coefficient; d is a damage variable; />Nonlinear fatigue constitutive model epsilon for reflecting mechanical characteristics of accelerated fatigue stage vp First derivative of medium strain and cycle number N; sigma (sigma) S The threshold stress value for the rock mass entering the accelerated fatigue phase.
7. The method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 6, wherein the method comprises the following steps: the sigma S The value was 0.75 times the peak stress.
8. The method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 1, wherein the method comprises the following steps: the step is based on strain equivalence theory to obtain macroscopic initial damage D of rock mass ma And rock mass microscopic damage D mi The method of damaging the variable D after coupling, the specific operations include,
step S51, based on strain equivalence theory, rock mass macroscopic initial damage D ma And rock mass microscopic damage D mi The expression of the impairment variable D after coupling satisfies d=d ma +D mi -D ma D mi The method comprises the steps of carrying out a first treatment on the surface of the Wherein D is ma Macroscopic initial damage to the rock mass; d (D) mi Is microscopic damage to the rock mass;
step S52, through the formulaObtaining macroscopic initial damage D of rock mass ma Wherein E is C Modulus of elasticity, E, of rock mass containing natural macroscopic defects O The modulus of elasticity of the complete rock mass;
step S53, determining microscopic damage D through Kachanov damage law mi Relationship with cycle number N, wherein the Kachanov's law of damage equation is:
A. delta is the material constant determined by the test; omega is the applied load; />Is a rock microscopic damage variable D mi A first derivative with respect to the number of cycles N; wherein, pair-> Integrating to obtain D mi Expression->D when the rock mass is completely destroyed mi =1, corresponding complete damage cycle number N C Satisfy N C ＝[A(δ+1)ω δ ] -1 ；
Bonding ofN C ＝[A(δ+1)ω δ ] -1 Obtaining the rock mass damage performance under the cyclic loading conditionThe equation of the chemical formula satisfies the condition->
Step S54, the formula in step S53The method is changed into that:
wherein Ns is the number of cycles of the rock mass into the accelerated fatigue phase; n (N) F For the number of cycles at the moment of complete destruction of the rock mass, the actual number of cycles of damage +.>True complete injury cycle +.>When n=ns, D mi ＝0，N＝N F At time D mi ＝1；
Step S55, the formula is calculatedAnd formula->The formula d=d substituted into step S51 ma +D mi -D ma D mi In the method, the initial macroscopic damage D of the rock mass is considered ma And rock mass microscopic damage D mi The expression of the impairment variable D after coupling satisfies:
9. according to claimThe method for constructing the nonlinear rock fatigue constitutive model based on rheological model application is characterized by comprising the following steps of: the step introduces the damage variable D after coupling into a nonlinear fatigue constitutive model epsilon reflecting the mechanical characteristics of the accelerated fatigue stage vp In the method, a nonlinear fatigue constitutive model epsilon considering mechanical characteristics of initial damage in an accelerated fatigue stage is obtained vp The method of (N), the specific operations include,
step S61, formulaSubstituted into nonlinear fatigue constitutive model epsilon reflecting mechanical characteristics of accelerated fatigue stage vp Constitutive equation of->In the method, constitutive equation of a nonlinear fatigue constitutive model considering mechanical characteristics of initial damage in an accelerated fatigue stage is obtained:
the conversion is carried out on the above materials to obtain the following components:
step S62, combining the initial conditions n=n S ，ε vp =0, for the aboveSolving to obtain a nonlinear fatigue constitutive model expression considering the mechanical characteristics of the initial damage in the accelerated fatigue stage, wherein the nonlinear fatigue constitutive model expression satisfies the following conditions:
10. The method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 9, wherein the method comprises the following steps: the steps are to introduce a rock fatigue constitutive model epsilon based on fractional derivative after Mittag-Leffler function eve (N) nonlinear fatigue constitutive model epsilon with mechanical properties of accelerated fatigue phase considering initial damage vp (N) series connection, a method for obtaining a nonlinear fatigue constitutive model describing the whole process of rock mass fatigue by considering initial damage, the specific operation comprises,
step S71, connecting a nonlinear fatigue constitutive model based on fractional derivatives and a nonlinear fatigue constitutive model considering initial damage in series to obtain a nonlinear fatigue constitutive model describing the whole process of rock body fatigue considering initial damage, wherein a strain equation of the nonlinear fatigue constitutive model meets the following conditions:
ε＝ε e +ε ve +ε vp ；
step S72, combining formulas
Sum formulaDetermining a specific strain equation epsilon (N) of a nonlinear fatigue constitutive model describing the whole process of rock mass fatigue taking initial damage into consideration:
Applications Claiming Priority (2)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN2021108788709 | 2021-08-02 | ||
| CN202110878870 | 2021-08-02 |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN113806966A CN113806966A (en) | 2021-12-17 |
| CN113806966B true CN113806966B (en) | 2023-09-12 |
Family
ID=78897704
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202111191687.8A Active CN113806966B (en) | 2021-08-02 | 2021-10-13 | Construction method of nonlinear rock fatigue constitutive model based on rheological model application |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN113806966B (en) |
Families Citing this family (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN117409902B (en) * | 2023-06-05 | 2024-06-21 | 河海大学常州校区 | Mechanical model construction method for high polymer cyclic loading behavior |
Citations (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN110274835A (en) * | 2019-07-12 | 2019-09-24 | 中南大学 | A method of improving Burgers rock shearing creep model |
-
2021
- 2021-10-13 CN CN202111191687.8A patent/CN113806966B/en active Active
Patent Citations (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN110274835A (en) * | 2019-07-12 | 2019-09-24 | 中南大学 | A method of improving Burgers rock shearing creep model |
Non-Patent Citations (1)
| Title |
|---|
| 低动应力下岩石分数阶Burgers本构模型;浦少云;黄质宏;饶军应;穆锐;郑红超;王田龙;刘小浪;李磊;王义红;;长江科学院院报(第02期);113-119 * |
Also Published As
| Publication number | Publication date |
|---|---|
| CN113806966A (en) | 2021-12-17 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| Wu et al. | Experimental study and modeling of shear rheology in sandstone with non-persistent joints | |
| Hashiba et al. | Effect of water on the deformation and failure of rock in uniaxial tension | |
| Khandelwal | Correlating P-wave velocity with the physico-mechanical properties of different rocks | |
| CN107907409B (en) | Method, equipment and storage equipment for determining rock cracking stress | |
| Vogler et al. | Hugoniot and strength behavior of silicon carbide | |
| Shi et al. | Research on the fracture mode and damage evolution model of sandstone containing pre-existing crack under different stress paths | |
| CN113806966B (en) | Construction method of nonlinear rock fatigue constitutive model based on rheological model application | |
| Holt et al. | Static and dynamic moduli—so equal, and yet so different | |
| CN112986020A (en) | Method for representing progressive rock destruction based on combination of stress and sound wave change | |
| Wang et al. | Study on acoustic emission characteristics of sandstone under different fracture modes | |
| Abbass et al. | A review of methods, techniques and approaches on investigation of rock anisotropy | |
| Shu et al. | Parameter calibration of the tensile-shear interactive damage constitutive model for sandstone failure | |
| Quaglio et al. | Determination of Young’s modulus by specific vibration of basalt and diabase | |
| Song-tao et al. | Fatigue performance of aging asphalt mixtures | |
| Moomivand | Development of a new method for estimating the indirect uniaxial compressive strength of rock using Schmidt hammer | |
| Govindaraj | A new approach for estimation of properties of metamorphic rocks | |
| Kegang et al. | Effect of Drying-Wetting Cycles on Triaxial Compression Mechanical Properties of Sandstone. | |
| Hu et al. | Investigation and Control of the Blasting‐Induced Ground Vibration under Cold Condition | |
| Milašinović | Modeling of porous-hardened concrete by rheological-dynamical analogy | |
| Li et al. | Study on Granite Damage and Fracture Behavior and Theoretical Model Based on Pre-Peak Constant Amplitude Cyclic Loading Test Combined with Acoustic Emission Technology | |
| Wang et al. | Study on the creep damage characteristics of shale under the combined action of osmotic pressure and disturbance load | |
| Hasbani et al. | On the characterization of the viscoelastic response of the Vaca Muerta Formation. | |
| CN112800583A (en) | Medium and high strain rate dynamic damage constitutive model suitable for hard rock and application method thereof | |
| Mo et al. | A state parameter-based cavity expansion analysis for interpretation of CPT data in sands | |
| Bao et al. | Study on creep properties of weak layer rock of slope under damage coupling |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |