CN109885980B - Joint shearing overall process damage constitutive model for determining yield point based on stress difference - Google Patents
Joint shearing overall process damage constitutive model for determining yield point based on stress difference Download PDFInfo
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Abstract
The invention discloses a joint shearing overall process damage constitutive model for determining a yield point based on stress difference, S1, setting a joint thin-layer mesoscopic unit body to be loaded, namely entering a linear elastic stage, wherein the joint thin-layer mesoscopic unit body is an isotropic continuous medium, the conversion of the mesoscopic unit body from a lossless state to a lossy state is instantly completed, and the process is irreversible; s2, defining an external load threshold F based on Weibull distribution function*Obtaining a damage probability density function of the microscopic unit body; s3, simulating the load condition of the mesoscopic unit body by adopting the combination of a spring and a friction plate based on the mechanical element of the rheological model to obtain a statistical damage constitutive model of joints under the shearing action; s4, determining an external load threshold value F according to the fact that the rock-soil material becomes damaged or undamaged after being subjected to load damage*Obtaining a damage evolution model of joint shearing deformation in the shearing deformation process and a joint shearing deformation damage constitutive model; s5, determining the joint shearing deformation damage constitutive model parameters m and u0、us(ii) a And S6, verifying the correctness of the constitutive model.
Description
Technical Field
The invention belongs to the technical field of rock shear surface characteristics, and particularly relates to a joint shearing overall process damage constitutive model for determining a yield point based on stress difference.
Background
The natural rock body is subjected to complex geological action in the forming process, various joints with different properties exist in the natural rock body, the failure mechanism of the engineering rock body is controlled by the joints to a great extent, the mechanical properties of the joints are mainly controlled by the shear strength of the joints, and the shear characteristic research of the joints is a key scientific problem and a hotspot problem in the rock mechanical research. In the past, the key point of the research of the joint shearing theory is to establish a constitutive model which can accurately reflect the joint mechanical characteristics and the whole shearing process. Scholars at home and abroad carry out a great deal of experimental analysis and theoretical research aiming at the joint shearing characteristic and provide some classical constitutive models for describing the shearing mechanical deformation characteristic of the joint. For example, Bandis[1]According to the direct shear test result, a shear stress-displacement curve in the front region of the peak value is fitted by adopting a hyperbolic function. s.Saeb and B.Amadei[2]Analyzing the whole process of the joint shear stress-displacement, providing a pure linear shear constitutive model, and describing the stress-displacement relation by adopting a piecewise linear function. Grasselli[3,4]The results of 37 sets of joint direct shear tests were analyzed, and it was considered that the front part of the peak was fitted with a linear function and the rear part of the peak was fitted with a hyperbolic function. In fact, a large amount of direct shear test data[4]Indicating that the shear stress-displacement relationship before and after the peak is not a simple linear relationship. Since the 70 s of the 20 th century, the rise of injury mechanics provides a new idea for researching joint shearing characteristics. Damage mechanics consider the natural microfractures present in large numbers in rock as a type of damage[5,6]Numerous scholars[7-13]The research on the rock constitutive model is carried out. C.s.desai[14]Starting from random distribution of defects, a joint DSC shearing constitutive model is established based on a damage theory. Wang Z L[15]It was found that microcracking from damage to fracture is a continuous process, comparing the Deluker-Lager criterion with Moire-After coulomb's criterion, a statistical constitutive model of rock damage softening that follows Weibull distribution based on unit intensities is proposed. Simon[16]And (3) finding that the nonlinear softening phenomenon can occur after the joint shear deformation reaches a peak value, and describing by using an exponential function curve.
The research on the joint shear damage constitutive model strongly promotes the development of the damage theory in rock mechanics. However, the existing constitutive model mainly aims at fitting with the shape of a test curve, has excessive introduced parameters and ambiguous physical meaning, is relatively deficient in the study of the mechanical response law of the joint unit, and is a more rare report on the study of the deformation before and after the joint shear peak as a whole. The above references are as follows:
[1]Bandis SC,Lumsden AC,Barton NR.Fundamentals of rock joint deformation[J].International Journal of Rock Mechanics&Mining Sciences&Geomechanics Abstracts,1983,20(6):249-268.
[2]Saeb S.Modelling of rock joints under shear and normal loading:Saeb,S；Amadei,B Int J Rock Mech Min SciV29,N3,May 1992,P267–278[J].International Journal of Rock Mechanics&Mining Sciences&Geomechanics Abstracts,1992,29(6):344.
[3]Grasselli G,Egger P.Constitutive law for the shear strength of rock joints based on three-dimensional surface parameters[J].International Journal of Rock Mechanics&Mining Sciences,2003,40(1):25-40.
[4]GRASSELLI.Shear strength of rock joints based on quantified surface description[J].Rock Mechanics&Rock Engineering,2006,39(4):295.
[5]Lemaitre J.How to use damage mechanics☆[J].Nuclear Engineering&Design,1984,80(2):233-245.
[6] xie and Hei rock concrete damage mechanics [ M ] China mining university Press; 1990.
[7]Kachanov ML.A microcrack model of rock inelasticity part I:Frictional sliding on microcracks[J].Mechanics of Materials,1982,1(1):19-27.
[8]Kachanov ML,Kachanov ML.A microcrack model of rock inelasticity part II:Propagation of microcracks[J].Mechanics of Materials,1982,1(1):29-41.
[9] mechanical characteristics of multi-fracture rock mass under action of pressure-shear stress-fracture damage evolution equation and experimental verification [ J ] geotechnical, 1994, (2):1-12.
[10] Kudzuvine repairing and moistening, conception, pyaibin, et al, CT dynamic test of coal rock triaxial mesoscopic damage evolution law [ J ]. report on rock mechanics and engineering, 1999, (05):497-502.
[11] Conception, Kudzuvine modification, uniaxial compression rock damage evolution fine mechanism and constitutive model research [ J ] in the report of rock mechanics and engineering, 2001,20(4): 425-.
[12] Research on constitutive model of rock three-dimensional elastoplastic damage, Duyuli, Huangqiqi, Jinliu, et al, J, report of geotechnical engineering, 2017,39(6): 978-.
[13] Lingjian, Jiangguang, Damage mechanics analysis of mechanical properties of non-through fractured rock mass [ J ]. Proc on rock mechanics and engineering, 1992,11(4): 373-.
[14]Desai CS,Ma Y.Modeling of joints and interfaces using the disturbed state concept[J].International Journal for Numerical&Analytical Methods in Geomechanics,2010,16(9):623-653.
[15]Wang ZL,Li YC,Wang JG.A damage-softening statistical constitutive model considering rock residual strength[J].Computers&Geosciences,2007,33(1):1-9.
[16]Simon R,Aubertin M,Mitri HS.A non-linear constitutive model for rock joints to evaluate unstable slip[J].1999.
Disclosure of Invention
The present invention is directed to solve or improve the above-mentioned problems by providing a joint shear overall damage constitutive model for determining a yield point based on a stress difference.
In order to achieve the purpose, the invention adopts the technical scheme that:
a joint shear full-process damage constitutive model for determining yield point based on stress difference comprises the following steps:
s1, setting the joint thin-layer mesoscopic unit bodies to enter a linear elasticity stage when being loaded, wherein the joint thin-layer mesoscopic unit bodies are isotropic continuous media, the conversion of the mesoscopic unit bodies from a lossless state to a lossy state is instantly completed, and the process is irreversible;
s2, defining an external load threshold F based on Weibull distribution function*Obtaining a damage probability density function of the microscopic unit body;
s3, simulating the loading condition of the joint thin-layer mesoscopic unit body by adopting the combination of a spring and a friction plate based on mechanical elements in the rheological model, and obtaining a statistical damage constitutive model of the joint under the shearing action;
s4, determining an external load threshold value F according to the fact that the rock-soil material becomes damaged or undamaged after being subjected to load damage*Obtaining a damage evolution model of joint shearing deformation in the shearing deformation process and a joint shearing deformation damage constitutive model;
s5, determining shape parameter m and scale parameter u of the joint shear deformation damage constitutive model0And a yield point displacement parameter us；
S6, substituting the joint shearing test data into the joint shearing deformation damage constitutive model, verifying that the obtained joint shearing deformation damage constitutive model can accurately simulate the joint shearing full deformation process, and reflecting the stage characteristics of the shearing process.
Preferably, the method for obtaining the damage probability density function in step S2 is:
when the load on the infinitesimal body at a certain stage is larger than the threshold value F*When the rock material is damaged, the damage probability density function P (F) of the mesoscopic unit bodies is as follows:
wherein, m, F0Respectively, Weibull distribution shape parameters and scale parameters.
Preferably, the statistical damage constitutive model processed in step S3 is:
s3.1, simulating the loading condition of the mesoscopic unit body by adopting the combination of the spring and the friction plate, and considering the mechanical response of the mesoscopic unit body of the joint thin layer in the shearing process as the stiffness coefficient k1iAnd a spring and a stiffness coefficient of k2iThe two parts of elements of the spring friction plate are connected in parallel; since the damaged and undamaged portions are intermingled, i.e. the displacements are in harmony and equal to the macroscopic shear displacement during deformation, it is assumed that P1si、P2siThe two parts are respectively subjected to shearing force, and the sum of the two parts is equal to the shearing force P applied to the microscopic unit bodysiBecause the two parts of elements are connected in parallel, namely the displacements are equal, according to the stress relation, when the displacement u of the microscopic unit body is smaller than the critical value usWhen is, Psi＝P1si+P2si＝k1iu+k2iu; the displacement u is greater than a critical value usWhen the mesoscopic unit body enters the damaged state, Psi＝P2si＝Pnitan phi, number Pnitanφ＝k2ius(ii) a The displacement critical value u of the unit body is equal to the macroscopic shearing displacement because the damaged part and the undamaged part are mixed together and the displacement of the two parts is equal to the macroscopic shearing displacement according to the deformation coordination principlesNamely the displacement of the joint yield point; when the joint is subjected to a shearing force PsWhen the method is used, the total number of the joint units is set to be N, and the number of the damaged units is set to be NfThe material damage is caused by the continuous destruction of these mesoscopic cells, and the shear force can be expressed as:
wherein u is shear displacement and u issFor yield point displacement, N represents the total number of microscopic microelements, NfIs used to represent the number of destroyed microscopic micro-elements in a certain stage;
s3.2, obtaining the shearing force P on the shearing surface according to the fact that the stiffness coefficient after the two springs are connected in parallel is the slope of the elastic stage of the Psi-u image linesComprises the following steps:
Ps＝(N-Nf)ksu+Nfk2ius
wherein k iss＝k1i+k2iI.e. shear stiffness;
s3.3 according to the area A of the shearing surfaceNFrom the damaged area ANfAnd a yet undamaged portion a, yielding a damage variable D of:
s3.4, dividing the shear surface shear force expression in the step S3.1 by A at the same timeNAnd substituting into step S3.2 to obtain apparent shear stress tau:
τ＝ksu(1-D)+k2iusD
wherein the apparent shear stress τ comprises τ borne by the undamaged portion*＝ksu and τ partially undertaken by injuryr＝k2iusComposition, then the expression in step S3.4 becomes:
τ＝τ*(1-D)+τrD
s3.5, the Damage variable D under Weibull distribution is
Obtaining a statistical damage constitutive model of the joint by combining the apparent shear stress tau:
preferably, the method for obtaining the damage evolution model of the joint in the shear deformation process and the joint shear deformation damage constitutive model in step S4 includes:
s4.1, adopting the shear stress tau corresponding to the effective stress and yield point in a certain stress statesDifference τ of*-τsAnd (3) measuring whether the strength of the microscopic micro-element reaches a damage state:
F-F*＝τ*-τs＝ksu-τs
s4.2, when F-F*When the content is more than or equal to 0,
at this point for the yield point there is:
τs＝ksus
S4.3、usfor the corresponding shear displacement, i.e. yield point displacement, of the specimen just as it enters the strain hardening phase, Weibull parameter F0Expressed as:
F0＝ksu0
s4.4, substituting the expressions in the steps S4.1, S4.2 and S4.3 into the equation
And (3) obtaining a damage evolution model of the shear deformation process joint:
s4.5, substituting the steps S4.1, S4.2, S4.3 and S4.4 into the damage probability density function of the mesoscopic unit bodies in the step S2 to obtain a joint shear deformation damage constitutive model:
preferably, joint shear deformation damage constitutive model parameters m and u are determined0The method comprises the following steps:
according to the fact that the derivative of the joint shear deformation damage curve at the peak intensity point function is zero, the following can be obtained:
at the peak point, u is equal to uf，τ＝τfIf the structural model of joint shear deformation damage is still established, the step S5.1 is substituted into the structural model of joint shear deformation damageModel to obtain the parameters m, u of the determined constitutive model0The calculation formula of (2):
preferably, determining the joint shear deformation damage constitutive model parameter usThe method comprises the following steps:
selecting data of the test data in the linear elasticity stage in the front area of the peak value, and fitting to obtain the slope of the linear elasticity stage, namely the shear stiffness ks；
Drawing a reference line τ ═ k in a coordinate systemsu, calculating the shear stress value tau of the measured data valueMeasured in factτ on the reference line corresponding to the corresponding shear displacement uReference toIs calculated as the difference of (d), the difference is recorded as tau*；
Plotting tau in a coordinate system*Image u and reference line τ*Directly reading out the recurved point from the image as 0, wherein the recurved point is the required yield point, and u corresponding to the yield point coordinate is the model parameter us。
The joint shearing overall process damage constitutive model for determining the yield point based on the stress difference has the following beneficial effects:
1. the model of the invention describes all deformation characteristics before and after the joint peak and the damage evolution rule thereof, the model is simple in form and clear in parameter physical significance, the model prediction result and the test result have quite high goodness of fit, and the established model is proved to be reasonable; the problems and the defects existing in the existing joint shearing deformation damage constitutive model can be effectively solved.
2. The shear stress-displacement curve of the joint is generally composed of a hardening stage before the peak and a softening stage after the peak, the conventional model is generally described by adopting a piecewise empirical function, most of the conventional models only consider the hardening behavior before the peak, the shear stress-displacement full-shear constitutive model capable of reflecting the softening characteristic after the peak is provided by the invention, the conventional direct shear test data is analyzed by utilizing the shear stress-displacement full-shear constitutive model, and the high fitting precision indicates the rationality of the novel model.
3. The model of the invention takes the shear stress and the shear displacement as basic parameters, the model expression only contains the conventional rock mechanical parameters, the physical significance is clear, the fitting parameters needing to be determined are less, and the precision is better than that of the former model.
4. The damage evolution model provided by the invention is based on the micro-element hypothesis, can uniformly describe the damage rule of the whole process of joint shearing deformation, and has simple damage equation form and definite parameter physical significance.
5. On the basis of analyzing the advantages and the disadvantages of the existing yield strength determination method, a new method for determining the yield strength based on the damage evolution model is provided, the method not only theoretically illustrates the position of the yield point, but also avoids the interference of human factors as much as possible, and has the characteristic of easy operability, and the yield point determined by the method is substituted into the established constitutive model, so that the rationality and the feasibility of the method are verified.
Drawings
FIG. 1 shows a joint shear stress-shear relative displacement relationship of a joint shear overall process damage constitutive model based on stress difference determination of yield point.
FIG. 2 shows an ideal joint thin layer damage microscopic unit body of a joint shear overall process damage constitutive model based on stress difference determination of a yield point.
FIG. 3 is a mesoscopic microelement model of a joint shear overall process damage constitutive model for determining yield point based on stress differences.
FIG. 4 is an element simulation model of a joint shear overall process damage constitutive model for determining yield point based on stress difference.
FIG. 5 shows different u of a joint shear overall process damage constitutive model for determining yield point based on stress difference0Influence on the shear deformation curve.
FIG. 6 is a graph showing the effect of different m on the shear deformation curve of the joint shear overall process damage constitutive model for determining the yield point based on the stress difference.
FIG. 7 is a schematic diagram of a us method for determining a damage constitutive model of the joint shearing overall process for determining a yield point based on a stress difference.
FIG. 8 is a comparison of measured values of a joint shear overall process damage constitutive model test and a model calculation curve for determining a yield point based on a stress difference.
FIG. 9 is a comparison of measured values of a joint shear overall process damage constitutive model test and a model calculation curve for determining a yield point based on a stress difference.
FIG. 10 is a comparison of measured values of a joint shear overall process damage constitutive model and a model calculation curve for determining a yield point based on a stress difference.
FIG. 11 is a comparison of measured values of joint shear test model JM1 of the joint shear overall process damage constitutive model based on stress difference determination yield point and a model calculation curve.
FIG. 12 is a comparison of measured values of joint shear test model JM2 of joint shear overall process damage constitutive model based on stress difference determination yield point and a model calculation curve.
Fig. 13 is a comparison of joint shear test measured values of joint shear overall process damage constitutive model JM3 and model calculation curves for determining yield point based on stress difference.
FIG. 14 is a comparison of measured values of joint shear test model JM4 of the joint shear overall process damage constitutive model based on stress difference determination yield point and a model calculation curve.
FIG. 15 is a normal stress σ of the joint shear overall process damage constitutive model for determining yield point based on stress differencenAnd parameter mu0And (4) relationship.
FIG. 16 is a normal stress sigma of a joint shear overall process damage constitutive model for determining yield point based on stress differencenIn relation to the parameter m.
FIG. 17 is a graph of the change trend of the damage factor of the joint shear overall process damage constitutive model JM1 for determining the yield point based on the stress difference.
FIG. 18 is a graph of the change trend of the damage factor of the joint shear overall process damage constitutive model JM2 for determining the yield point based on the stress difference.
FIG. 19 is a graph of the change trend of the damage factor of the joint shear overall process damage constitutive model JM3 for determining the yield point based on the stress difference.
FIG. 20 is a graph of the change trend of the damage factor of the joint shear overall process damage constitutive model JM4 for determining the yield point based on the stress difference.
FIG. 21 is a typical joint direct shear damage evolution process of a joint shear overall process damage constitutive model based on stress difference determination of yield point.
Detailed Description
The following description of the embodiments of the present invention is provided to facilitate the understanding of the present invention by those skilled in the art, but it should be understood that the present invention is not limited to the scope of the embodiments, and it will be apparent to those skilled in the art that various changes may be made without departing from the spirit and scope of the invention as defined and defined in the appended claims, and all matters produced by the invention using the inventive concept are protected.
According to an embodiment of the application, the joint shear overall process damage constitutive model for determining yield point based on stress difference according to the scheme with reference to fig. 1 comprises the following steps:
s1, establishing an ideal joint thin-layer unit body as shown in figure 2, setting the joint thin-layer unit body to be loaded, namely, entering a linear elasticity stage, wherein the joint thin-layer fine-viewing unit body is an isotropic continuous medium, the conversion of the joint thin-layer fine-viewing unit body from a lossless state to a lossy state is instantly completed, and the process is irreversible;
s2, defining an external load threshold F based on Weibull distribution function*Obtaining a damage probability density function of the microscopic unit body;
s3, simulating the loading condition of the joint thin-layer mesoscopic unit body by adopting the combination of a spring and a friction plate based on a rheological model mechanical element to obtain a statistical damage constitutive model of the joint under the shearing action;
s4, determining an external load threshold value F according to the fact that the rock-soil material becomes damaged or undamaged after being subjected to load damage*Obtaining a damage evolution model of joint shearing deformation in the shearing deformation process and a joint shearing deformation damage constitutive model;
s5, determining shape parameter m and scale parameter u of the joint shear deformation damage constitutive model0And a yield point displacement parameter us；
S6, substituting the joint shearing test data into the joint shearing deformation damage constitutive model, verifying that the obtained joint shearing deformation damage constitutive model can accurately simulate the joint shearing full deformation process, and reflecting the stage characteristics of the shearing process.
The above steps are described in detail below
S1, establishing an ideal joint thin-layer unit body as shown in figure 2, setting the joint thin-layer unit body to be loaded, namely, entering a linear elasticity stage, wherein the joint thin-layer fine-viewing unit body is an isotropic continuous medium, the conversion of the joint thin-layer fine-viewing unit body from a lossless state to a lossy state is instantly completed, and the process is irreversible;
referring to fig. 1, a typical joint shear process may be divided into 4 stages, namely a linear elastic stage (OA), a strain hardening stage (AB), a strain softening stage (BC), a residual strength stage (CD), where u iss、τsRespectively the corresponding shear displacement and shear stress when entering the yield stagef、τfShear displacement, shear stress, tau, respectively, corresponding to the peak pointrIs the residual strength.
In order to establish a damage model reflecting the shear deformation characteristics of the jointed rock mass, the following settings are made:
1. loading the material and then entering a linear elastic stage;
2. the mesoscopic unit bodies of the joint thin layer are isotropic continuous media, the size of the mesoscopic unit bodies is small enough macroscopically, the mesoscopic unit bodies can be regarded as material particles, the size of the mesoscopic unit bodies is large enough from the aspect of mesoscopic view, the fundamental information of material damage is contained, and countless mesoscopic unit bodies form all the damaged bodies and reflect the statistical average property of the material;
3. the transformation of the mesoscopic unit bodies from the lossless state to the lossy state is accomplished instantaneously, and the process is irreversible.
S2, defining an external load threshold F based on Weibull distribution function*Obtaining a damage probability density function of the microscopic unit body;
the rock material has a large number of randomly distributed microcracks, gaps and other defects, the existence of the defects causes the shapes and the strengths of a plurality of microelements to be different, the damage of the microelements under the action of external load is random, and the defects are based on Weibull distribution[19]Function(s)The infinitesimal destruction probability density function of (a) is:
wherein, F is the material infinitesimal strength measurement mode: when the infinitesimal strength F is less than 0, no damage occurs at the moment, the infinitesimal damage variable is constantly zero, and the infinitesimal is in a linear elastic state;
when the infinitesimal strength is greater than 0, the infinitesimal body is in damage states of different degrees, the damage variable D value is in the interval [0,1], and the damage variable is directly related to the stress strain state of the infinitesimal body at the moment, so that F ═ 0 can be understood as the damage criterion of the infinitesimal body and a reasonable initial point of infinitesimal body damage.
At this point, there is an external load threshold F*，F*Is equal to the load of the micro-element when the micro-element is damaged in value, and when the load of the micro-element at a certain stage is less than a threshold value F*When the material is not damaged, the external load material is removed, and the original state can be recovered; when the load on the infinitesimal body at a certain stage is larger than the threshold value F*When the material is damaged, the damage probability density function of the mesoscopic unit bodies is as follows:
wherein, m, F0Weibull shape and scale parameters, respectively, where m reflects the shape parameter, F0Represents the macro-average strength of the material.
S3, simulating the load condition of the mesoscopic unit body by adopting the combination of a spring and a friction plate based on the mechanical element of the rheological model to obtain a statistical damage constitutive model of joints under the shearing action;
the damage of the rock material is caused by the continuous destruction of microscopic micro-elements, N represents the total number of microscopic micro-elements, NfRepresenting the number of microscopically small elements destroyed at a certain stage, and defining a damage variable D as the ratio of the number of destroyed microsomes to the total number of microsomes at a certain time, i.e. DComprises the following steps:
external load is formed by F*Total number of destroyed microscopic micro-elements N in the process when loaded into FfComprises the following steps:
substituting equation (4) into equation (3) has:
referring to fig. 3, the combination of the spring and the friction plate is used to simulate the loading condition of the joint thin-layer mesoscopic unit body based on the method of reflecting the rheological constitutive model by the common basic elements and the combination thereof in the rheological model:
the mechanical response of the mesoscopic unit bodies in the shearing process is considered to be formed by the rigidity coefficient k1iAnd a spring and a stiffness coefficient of k2iThe two parts of elements of the spring friction plate are connected in parallel; since the damaged and undamaged portions are intermingled, i.e. the displacements are in harmony and equal to the macroscopic shear displacement during deformation, it is assumed that P1si、P2siThe two parts are respectively subjected to shearing force, and the sum of the two parts is equal to the shearing force P applied to the microscopic unit bodysiBecause the two parts of elements are connected in parallel, namely the displacements are equal, according to the stress relation, when the displacement u of the microscopic unit body is smaller than the critical value usWhen is, Psi＝P1si+P2si＝k1iu+k2iu; the displacement u is greater than a critical value usWhen the unit body enters the damage state Psi＝P2si＝Pnitan phi, number Pnitanφ＝k2ius(ii) a The displacement critical value u of the unit body is equal to the macroscopic shearing displacement because the damaged part and the undamaged part are mixed together and the displacement of the two parts is equal to the macroscopic shearing displacement according to the deformation coordination principlesI.e. the yield point displacement of the joint. When the joint is subjected to a shearing force PsWhen the method is used, the total number of the joint units is set to be N, and the number of the damaged units is set to be Nf(the hatched portion in fig. 2 is a damaged cell), the damage of the material is caused by the continuous damage of the microscopic cells, and the shearing force can be expressed as:
wherein u is shear displacement and u issFor yield point displacement, N represents the total number of microscopic microelements, NfTo indicate the number of microscopic micro-elements that have been destroyed at a certain stage.
Referring to FIG. 4, the stiffness coefficient of two springs connected in parallel is the slope of the elastic phase of the Psi-u image line, i.e., ks＝k1i+k2iTherefore, equation (7) becomes:
Ps＝(N-Nf)ksu+Nfk2ius (8)
according to the area A of the shear planeNFrom the damaged (destroyed) area ANfAnd a yet undamaged portion a, the damage variable D being:
for both sides of formula (7) simultaneously divided by ANSubstituting equation (8) can obtain:
τ＝ksu(1-D)+k2iusD (10)
τ apparent shear stress τ borne by undamaged portion*＝ksu and τ partially undertaken by injuryr＝k2iusComposition, then (10) becomes:
τ＝τ*(1-D)+τrD (11)
simultaneous (6), (11) statistical damage constitutive model of joints under shearing action:
however, to reflect the overall process of joint shear failure, F in the constitutive model of statistical damage must be determined*It will be analytically determined below.
S4, determining an external load threshold value F according to the fact that the rock-soil material becomes damaged or undamaged after being subjected to load damage*Obtaining a damage evolution model of joint shearing deformation in the shearing deformation process and a joint shearing deformation damage constitutive model;
in the prior art, the geotechnical material is changed into two parts of damaged part and undamaged part after being subjected to load damage, from the consideration of two extreme conditions of a damage factor D (0) and a damage factor 1, when a infinitesimal enters a yielding stage, the initial stage of the damage is considered, and when D (0), the effective stress of a linear elastic stage is smaller than tau at the momentsNo damage occurs, when the stress reaches the threshold value tausSince D is 1, one of them is used
Effective stress in the stressed state and shear stress τ corresponding to strain hardeningsDifference τ of*-τsAnd (3) measuring whether the strength of the microscopic micro-element reaches a damage state, namely:
F-F*＝τ*-τs＝ksu-τs (13)
when F-F*When the value is more than or equal to 0, then
For the demarcation point between the linear elasticity and the hardening phase, the yield point, there are at this point:
τs＝ksus (14)
usis the corresponding shear displacement of the sample just after the strain hardening phase, for Weibull parameter F0The following can also be expressed:
F0＝ksu0 (15)
substituting the expressions (13), (14) and (15) into the expression (6) can obtain a damage evolution model of the shear deformation process joint:
substituting expressions (13), (14), (15) and (16) into expression (10) to obtain the joint shear deformation damage constitutive equation as follows:
s5, determining shape parameter m and scale parameter u of the joint shear deformation damage constitutive model0And a yield point displacement parameter us；
Weibull random distribution parameter u0And m will influence the geometrical dimension and morphology of the joint shear deformation curve, for equation (17), other conditions are unchanged, and u is changed respectively0M, see FIGS. 5 and 6, Weibull distribution parameters m, u0The following rules exist in relation to the change of the shear deformation curve:
1. in the elastic stage before the peak of the shear stress-displacement curve, the curve shape is basically not influenced by parameters; after the joint reaches the yield strength, u follows0And the value of m is increased, and the curve is slightly shifted upwards; in the strain softening phase, parameter u0And m has a large influence on the curve.
2. Post-peak softening phase parameter u0The larger the peak intensity and residual intensity of the joint, the larger the shear displacement when the peak intensity is reached, the upward and rightward movement of the curve as a whole, u0When the intensity of the peak is increased from 1.02 to 3.02, the peak intensity is increased by 56.21 percent, the corresponding shearing displacement value is increased by 31.86 percent, and different parameters u0In this case, the post-peak curve groups are arranged approximately in parallel as a whole. Wherein u is0Mainly reflects the size of the macroscopic statistical average strength of the rock.
3. With the increase of the parameter m, the peak intensity tends to increase, and the whole slope of the curve after the peak increases. Under the condition of different parameters m, a common intersection point exists between the peak intensity and the residual intensity in the post-peak curve group, and the change trends of the curves before and after the intersection point are just opposite. When the parameter m is increased from 2.53 to 4.53, the peak intensity is increased by 10.42 percent, and m mainly reflects the brittleness characteristic of the rock and the distribution concentration degree of the infinitesimal intensity in the material.
Determining joint shear deformation damage constitutive model parameters m and u0Which comprises the following steps:
the derivative of the shear deformation failure curve at the peak intensity point function is zero according to the joint surface, namely:
and at the peak point, u is equal to uf，τ＝τfWhen the formula (17) is still satisfied, the formula (18) is substituted into the formula (17) to obtain the parameters m, u of the constitutive model0The calculation formula of (2):
from the formula (19), m, u are determined0The key to determining the demarcation point u between the linear elasticity and the hardening phase, and even for the entire shear constitutive models。
Determining joint shear deformation damage constitutive model parameter us：
During joint shear, there is an important stress threshold, yield strength, in the linear elastic phase and the nonlinear phase (peak frontal area). The yield strength point is a boundary point of joint macroscopic deformation from linearity to nonlinearity, and the determination of the yield limit is an important link for researching the joint shearing process.
There are two methods for determining the yield strength point in the shearing process:
(1) by directly establishing the turning point of the shearing stress-displacement curve from linearity to nonlinearity.
(2) Directly determining yield stress and then carrying out linear fitting by Mohr-Coulomb criterion to obtain yield strength index (c)s、ψs)。
Both of the above methods have their drawbacks: the determination of the turning point strongly depends on the subjective judgment of a tester and is greatly influenced by human factors, the determination of the yield stress also depends on the human judgment, and the key point for accurately determining the yield strength is how to objectively determine the conversion of the shear stress-displacement curve from linear to nonlinear.
The invention is based on the shear stress difference, and determines the specific numerical value of the yield point through the shear stress difference between a test curve and a reference straight line, and the principle is as follows:
at the data points in the linear elastic phase, the difference between the corresponding actual shear stress and the calculated theoretical shear stress fluctuates around 0 at the baseline τ, which is a function of u and is denoted as τ (u), while at the data points after the yield point, the difference τ increases as u increases, because the yield point has the following characteristics:
(1) the position of the reference line fluctuates up and down, and the reference line is intersected or almost intersected with the reference line;
(2) after the point, all data points are farther from the reference line, so that after the stress difference between each measured data point and the reference straight line is obtained, the yield point is almost on the reference line by taking τ as 0, the yield point τ (u) after the yield point is reduced along with the increase of u (negative number, the larger the absolute value is, the smaller the negative number is), and the corresponding yield point can be found intuitively from a coordinate system.
The specific operation steps are as follows:
referring to FIG. 7, the data in the linear elastic phase of the experimental data in the front of the peak is selected and fitted to obtain the slope of the linear elastic phase, i.e. the shear stiffness ks。
Drawing a reference line τ ═ k in a coordinate systemsu, calculating the shear stress value tau of the measured data valueMeasured in factτ on the reference line corresponding to the corresponding shear displacement uReference toIs calculated as the difference of (d), the difference is recorded as tau*。
Plotting tau in a coordinate system*Image u and reference line τ*Determining a reverse bending point, wherein the shearing displacement corresponding to the point is the required yield displacement us。
To this end, m, u0、usHas been completely determined and the modelThe parameter has definite physical significance and is suitable for the conditions under different normal stress states, however, the feasibility of the parameter is verified by test examples.
S6, substituting a plurality of joint shearing test data into the joint shearing deformation damage constitutive model, verifying that the obtained joint shearing deformation damage constitutive model can accurately simulate the joint shearing full deformation process and reflect the stage characteristics of the shearing process, wherein the method comprises the following specific steps:
direct shear test data analysis based on T.Papaliangas
According to direct shear test data (test results are shown in table 1) of the T.Papaliangas jointed rock mass under normal stresses of 0.10 and 0.05MPa, the constitutive relation model of the invention is adopted for fitting,
table 1 t. papaliangas direct shear test results
From experimental data, when σn0.05MPa, ks is 0.024, taurWhen u is 0.047, u is obtained by the method in step S5S1.637; when sigma isn0.1MPa, ks 0.026,. taurWhen u is 0.077, u can be obtained3＝3.045。
The two groups of parameters are respectively substituted into the formula (17), the effect of fitting the curve is shown in fig. 8, and as can be seen from fig. 8, the model can accurately simulate the joint shearing total deformation process, so that the stage characteristics of the shearing process can be better reflected.
Bandis based on s[26]Direct shear test data analysis of
Under normal stress conditions, a series of direct shear tests are carried out on natural joints by S.Bandis and the like, the joint size is 9cm multiplied by 5cm, the applied shear displacement is 6-7 mm, the uniaxial compressive strength of the joint surface wall is 2MPa, and the results of the direct shear tests are shown in Table 2. The analysis is carried out on the direct shear test data with the normal stress of 0.09MPa, and the fitting result is shown in FIG. 9:
table 2 s. bandis direct shear test results
From experimental data, when σnWhen k is 0.09MPa, ks＝0.131，τrWhen u is 0.077, u is obtainedsWhen the parameters are substituted into equation (17) at 0.547, the fitted curve is shown in fig. 9, and the goodness of fit R is obtained2The value of (A) is very close to 1, which shows that the constitutive model established in the method not only can well represent the whole shearing deformation process, but also can highlight the deformation characteristics of joint shearing stage.
Based on the first summer-heat and the Ming Liu Yuan Ming[27]Direct shear test data analysis of
Liu Yuan Ming, summer just first carry out direct shear test to 3 kinds of non-through joint rock masses that contain the profile of tooth simulation material joint under different normal stress conditions, carry out the analysis to the experimental data that the normal stress is 0.5MPa, the test result is shown in Table 3:
TABLE 3 initial direct shear test results of Liu Yuan Ming and Xia Cheng
From experimental data, when σn0.5MPa, ks is 1.512,. taurWhen it is 1.278, u is obtainedsWhen the parameters are substituted into equation (17) at 1.04, the effect of fitting the curve is shown in fig. 10, and it can be seen from fig. 10 that the degree of matching between the established model and the test result is extremely high.
The invention is directed to the parameters m and u0Discussion of (1)
Parameters m, u0In connection with the joint itself and the external load, the invention combines the literature in order to study the form and extent of the influence of various factors on the parameters[28]The physical and mechanical parameters of the test piece are shown in table 4:
TABLE 4 test piece physical parameter table
The direct shear test results of 4 kinds of non-through joints are analyzed, 4 kinds of different relief angles are adopted in the test to simulate the joints, and the joint types are shown in table 5:
TABLE 5 Joint characteristics
Substituting equation (17) into the shear test results for verifying four types of joints, the fitted images are shown in fig. 11-14:
FIGS. 11-14 show that the constitutive model provided by the invention has good goodness of fit with the test, and model parameters under the same stress and the positive stress sigma of different methods are respectively considerednRelationships, data are shown in table 6:
TABLE 6 Weibull distribution parameters vs. normal stress
Fitting the above results, parameter u0M and sigmanThe fitted images of (2) are shown in fig. 15 and 16:
it can be seen that two parameters u0M and positive stress σnHas better linear relation and generalizes the following empirical formula:
u0＝aσn+b (17)
m＝cσn+d (18)
and a, b, c and d are test curve fitting coefficients, and the formula (17) and the formula (18) are substituted into the formula (14) to obtain a parameter determination method for the damage constitutive model of the joint shearing overall process under different normal stresses.
And (3) drawing a damage factor evolution curve of four types of rational test pieces according to the formula (16), as shown in FIGS. 17, 18, 19 and 20:
fig. 17-20 show the evolution curves of the damage variables of the joint shearing under different working conditions, and the damage evolution model obtained according to the model can correspond to the whole process of the joint shearing damage one by one. For reasons of space, only taking the direct shear data of the JM1 type joint under the positive stress of 0.5MPa as an example, the damage evolution rule of the joint in the direct shear test process is analyzed. The injury process can be divided into 3 stages as shown in fig. 21:
(1) a damage-maintenance phase in which almost no damage is generated and the value of the damage variable remains substantially constant at a level very close to 0. The joint is in the initial stage of shearing, the shearing deformation is small, the shearing stress is increased quickly, and the shearing stress and the displacement are increased in a linear relation;
(2) the stage of injury starting, which is a stage of generation, expansion and communication of joint fractures, wherein the injury variable approaches from 0 to 1 along with the continuous expansion and extension of micro fractures;
(3) and in the damage and damage stage, the shearing process enters the residual stage, and the damage variable tends to be gentle and stable at about 1 in the stage along with the displacement change.
Compared with the prior art, the joint shearing overall process damage constitutive model for determining the yield point based on the stress difference describes all deformation characteristics before and after the joint peak and the damage evolution rule thereof, the model is simple in form, the parameter physical significance is clear, the model prediction result and the test result have quite high goodness of fit, and the established model is proved to be reasonable; the problems and the defects existing in the existing joint shearing deformation damage constitutive model can be effectively solved.
While the embodiments of the invention have been described in detail in connection with the accompanying drawings, it is not intended to limit the scope of the invention. Various modifications and changes may be made by those skilled in the art without inventive step within the scope of the appended claims.
Claims (5)
1. A joint shear overall process damage constitutive model for determining yield point based on stress difference is characterized by comprising the following steps:
s1, setting the joint thin-layer mesoscopic unit bodies to enter a linear elasticity stage when being loaded, wherein the joint thin-layer mesoscopic unit bodies are isotropic continuous media, the conversion of the mesoscopic unit bodies from a lossless state to a lossy state is instantly completed, and the process is irreversible;
s2, defining an external load threshold F based on Weibull distribution function*Obtaining a damage probability density function of the microscopic unit body;
s3, simulating the load condition of the mesoscopic unit body by adopting the combination of a spring and a friction plate based on mechanical elements in the rheological model to obtain a statistical damage constitutive model of joints under the shearing action;
s4, determining an external load threshold value F according to the fact that the rock-soil material becomes damaged or undamaged after being subjected to load damage*Obtaining a damage evolution model of joint shearing deformation in the shearing deformation process and a joint shearing deformation damage constitutive model;
s5, determining shape parameter m and scale parameter u of the joint shear deformation damage constitutive model0And a yield point displacement parameter us；
Wherein the joint shear deformation damage constitutive model parameter u is determinedsThe method comprises the following steps:
selecting data of the test data in the linear elasticity stage in the front area of the peak value, and fitting to obtain the slope of the linear elasticity stage, namely the shear stiffness ks；
Drawing a reference line τ ═ k in a coordinate systemsu, calculating the shear stress value tau of the measured data valueMeasured in factτ on the reference line corresponding to the corresponding shear displacement uReference toThe difference is recorded as stress difference taudiff；
Plotting tau in a coordinate systemdiffImage u and reference line τdiffDirectly reading out the recurved point from the image as 0, wherein the recurved point is the required yield point, and u corresponding to the yield point coordinate is the model parameter us；
S6, substituting the joint shearing test data into the joint shearing deformation damage constitutive model, verifying that the obtained joint shearing deformation damage constitutive model can accurately simulate the joint shearing full deformation process, and reflecting the stage characteristics of the shearing process.
2. The joint shear full-process damage constitutive model for determining yield point based on stress difference as claimed in claim 1, wherein the method for obtaining the failure probability density function in step S2 is as follows:
when the load on the infinitesimal body at a certain stage is larger than the threshold value F*When the rock material is damaged, the damage probability density function P (F) of the mesoscopic unit bodies is as follows:
wherein, m, F0Respectively, Weibull distribution shape parameters and scale parameters.
3. The joint shear full process damage constitutive model for determining yield point based on stress difference as claimed in claim 1, wherein the statistical damage constitutive model of the joint in step S3 is:
s3.1, simulating the loading condition of the mesoscopic unit body by adopting the combination of the spring and the friction plate, and considering the mechanical response of the mesoscopic unit body of the joint thin layer in the shearing process as the stiffness coefficient k1iAnd a spring and a stiffness coefficient of k2iThe two parts of elements of the spring friction plate are connected in parallel; since the damaged and undamaged portions are intermingled, i.e. the displacements are in harmony and equal to the macroscopic shear displacement during deformation, it is assumed that P1si、P2siThe two parts are respectively subjected to shearing force, and the sum of the two parts is equal to the shearing force P applied to the microscopic unit bodysiBecause the two parts of elements are connected in parallel, namely the displacements are equal, according to the stress relation, when the displacement u of the microscopic unit body is smaller than the critical value usWhen is, Psi＝P1si+P2si＝k1iu+k2iu; the displacement u is greater than a critical value usWhen the mesoscopic unit body enters the damaged state, Psi＝P2si＝Pnitan phi, number Pnitanφ＝k2ius(ii) a The displacement critical value u of the unit body is equal to the macroscopic shearing displacement because the damaged part and the undamaged part are mixed together and the displacement of the two parts is equal to the macroscopic shearing displacement according to the deformation coordination principlesIs just a sectionDisplacement of the theoretical yield point; when the joint is subjected to a shearing force PsWhen the method is used, the total number of the joint units is set to be N, and the number of the damaged units is set to be NfThe material damage is caused by the continuous destruction of these mesoscopic cells, and the shear force can be expressed as:
wherein u is shear displacement and u issFor yield point displacement, N represents the total number of microscopic microelements, NfIs used to represent the number of destroyed microscopic micro-elements in a certain stage;
s3.2, obtaining the shearing force P on the shearing surface according to the fact that the stiffness coefficient after the two springs are connected in parallel is the slope of the elastic stage of the Psi-u image linesComprises the following steps:
Ps＝(N-Nf)ksu+Nfk2ius
wherein k iss＝k1i+k2iI.e. shear stiffness;
s3.3 according to the area A of the shearing surfaceNFrom the damaged area ANfAnd a yet undamaged portion a, yielding a damage variable D of:
s3.4, dividing the shear surface shear force expression in the step S3.1 by A at the same timeNAnd substituting into step S3.2 to obtain apparent shear stress tau:
τ＝ksu(1-D)+k2iusD
wherein the apparent shear stress τ comprises the shear stress τ borne by the undamaged portion*＝ksu and the damaged part bear the shear stress taur＝k2iusComposition, then the expression in step S3.4 becomes:
τ＝τ*(1-D)+τrD
s3.5, Weibull distributionLower damage variable D is
Obtaining a statistical damage constitutive model of the joint by combining the apparent shear stress tau:
4. the joint shear overall process damage constitutive model for determining the yield point based on the stress difference as claimed in claim 3, wherein the method for obtaining the damage evolution model of the shear deformation process joint and the joint shear deformation damage constitutive model in the step S4 is as follows:
s4.1, adopting the shear stress born by the undamaged part in a certain stress state and the shear stress tau corresponding to the yield pointsDifference τ of*-τsAnd (3) measuring whether the strength of the microscopic micro-element reaches a damage state:
F-F*＝τ*-τs＝ksu-τs
s4.2, when F-F*When the content is more than or equal to 0,
at this point for the yield point there is:
τs＝ksus
S4.3、usfor the corresponding shear displacement, i.e. yield point displacement, of the specimen just as it enters the strain hardening phase, Weibull parameter F0Expressed as:
F0＝ksu0
s4.4, substituting the correlation expressions obtained in the steps S4.1, S4.2 and S4.3 into an equation
And (3) obtaining a damage evolution model of the shear deformation process joint:
s4.5 substituting the correlation expressions obtained in steps S4.1, S4.2, S4.3 and S4.4 into the formula τ ═ k obtained in step S3.4su(1-D)+k2iusIn the step D, obtaining a joint shear deformation damage constitutive model:
5. the joint shear full-process damage constitutive model for determining yield point based on stress difference as claimed in claim 1, characterized in that parameters m, u of the joint shear deformation damage constitutive model are determined0The method comprises the following steps:
according to the fact that the derivative of the shear deformation failure curve of the joint surface at the peak intensity point function is zero, the following can be obtained:
at the peak point, u is equal to uf，τ＝τfSubstituting the parameters into the concrete expression of the joint shearing deformation damage constitutive model established in the step S4 to obtain the parameters m, u of the determined constitutive model0The calculation formula of (2):
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