CN109283215B - Method for measuring and calculating frost heaving force of tunnel surrounding rock in cold region after repeated freeze-thaw damage - Google Patents
Method for measuring and calculating frost heaving force of tunnel surrounding rock in cold region after repeated freeze-thaw damage Download PDFInfo
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Abstract
The invention discloses a method for measuring and calculating frost heaving force sigma of tunnel surrounding rock in a cold region after repeated freeze-thaw damagef(N)The calculation is performed according to the following formula:
(ii) a Wherein E is2(N)The elastic modulus of the rock in the freezing area after the Nth freezing-thawing cycle; n is0And nNThe method comprises the steps of firstly calculating the surrounding rock frost heaving force based on an elasticity theory, then considering damage of the freeze-thaw cycle to the rock, obtaining the deterioration rule of the rock elasticity modulus and the porosity along with the freeze-thaw cycle, finally obtaining a tunnel surrounding rock frost heaving force calculation method considering the damage of the rock freeze-thaw cycle, and providing practical guidance for the construction direction and the method of the rock tunnel maintenance engineering.
Description
Technical Field
The invention relates to the technical field related to cold region tunnel mechanics measurement and calculation, in particular to a surrounding rock frost heaving force damage mechanics calculation method considering rock elastic modulus and porosity change under freeze-thaw cycle.
Background
Rock engineering failure caused by freeze thawing is an important problem commonly encountered in cold region engineering construction at present, and tunnel lining cracking caused by surrounding rock frost heaving is considered as one of main forms of cold region tunnel damage. In the construction of tunnels in cold regions, the calculation of the frost heaving force of surrounding rocks of the tunnels is the premise and the foundation for designing lining structures. As the rock mass contains a plurality of pores and cracks, when water enters the rock mass and is frozen at low temperature, huge frost heaving force is generated[2]Further causing the rock mass to be further damaged and evolved, then the water after melting enters the newly formed pores again, and the process is repeated in a circulating way. Multiple freeze-thaw cycles lead the rock mass to generate a series of physical and mechanical changes, and finally the tunnel is damaged. Therefore, scholars at home and abroad carry out more intensive research on the frost heaving force of the tunnel surrounding rock respectively from a plurality of aspects such as experiments, theories, numerical calculation and the like.
First from a test point of view, the channel flies, etc[1]The method is characterized in that the low-temperature condition of the tunnel is simulated by adopting a cooling mode in the cavity of the tunnel through an indoor model test, the frost heaving generated by the migration of water in a rock crack is researched, and the result shows that the lining frost heaving strain is smaller at an inverted arch and an inverted arch foot, larger at an arch crown, an arch foot and a side wall, and the maximum frost heaving occurs at the side wall. To estimate the safety of the tunnel lining under the action of the frost-heave force, Hu, etc[2]The distribution characteristics of the frost heaving force acting on the lining are studied by an indoor model test. Qiu et al[3]The model test is adopted to research the magnitude and the distribution rule of the frost heaving force when the tunnel is excavated in the broken rock mass in the cold region, the influence of different constraint conditions and freezing depths on the frost heaving force is discussed, and the corresponding frost heaving force is found to be larger when the freezing depth is deeper and the top constraint is larger. Secondly in terms of theoretical studies, Lai et al[4]Providing a nonlinear analysis method considering the coupling of a cold region temperature field, a seepage field and a stress field, deriving a corresponding finite element calculation formula based on a Galerkin method, and finally showing that the frost heaving force is applied to the tunnel liningThe force effect is significant. Lai, etc[5-6]A viscoelastic theory calculation method of the frost heaving force of the tunnel in the cold region is provided by utilizing Laplace transformation, and the frost heaving force is found to be an important factor causing the plastic region of the surrounding rock to expand obviously. Feng et al[7]A new elastic-plastic model of tunnel surrounding rock in a cold region is provided, and the tunnel is divided into 4 regions, namely an unfrozen elastic region, a frozen plastic region and a supporting region, and the frozen plastic surrounding rock is considered to meet the ideal elastic-plastic model and the Moore-Coulomb yield criterion. Zhang quan Sheng, etc[8]A theoretical calculation model of the frost heaving force of the round tunnel surrounding rock is established based on an elasticity theory.
Although many scholars have conducted intensive research on the frost heaving force of rock mass tunnel surrounding rocks from the aspects of theory, experiment and the like, further improvement is still needed. Theoretical calculation model as described above[4-8]The influence of freeze-thaw cycles on the frost heaving force of the surrounding rock is not considered, and the frost heaving force under a single frost heaving condition can be calculated. In practical rock mass projects such as tunnels and the like, freeze-thaw cycles are performed for multiple times within service life, and freeze-thaw damage to rocks is inevitably caused, so how to consider the influence of the freeze-thaw cycle damage to the surrounding rock frost heaving force is an important problem to be solved urgently at present.
The relevant references are as follows.
[1] Indoor model test research on frost heaving force of tunnel lining in cold regions [ J ] report on rock mechanics and engineering, 2015, 34 (9): 1894-1900.
[2]Hu,Q.J.,Shi,R.D.,Hu,Y.,Cai,Q.J.,Qu,M.F.,Zhao,W.,He,L.P.,2018.Method to evaluate the safety of tunnels through steeply inclined stratain cold regions based on the sidewall frostheavemodel.J.Perform.Constr.Fac.32(4),DOI:10.1061/(ASCE)CF.1943-5509.0001165。
[3]Qiu W.G.,Sun B.,2010.Model test study of frost heaving pressuresin tunnels excavated in fractured rock mass in coldregions.J.Glaciol.Geocryol.32(3),557-561。
[4]Lai,Y.M.,Wu,Z.W.,Zhu,Y.L.,Zhu,L.N.,1998.Nonlinear analysis for thecoupled problem of temperature,seepage and stress fields in cold-regiontunnels.Tunn.Undergr.Space Technol.13(4):435–440。
[5]Lai,Y.M.,Wu,H.,Wu,Z.W.,Liu,S.Y.,Den,X.J.,2000a.Analyticalviscoelastic solution for frost force in cold-region tunnels.ColdReg.Sci.Technol.31(3),227–234。
[6]Lai,Y.M.,Wu,Z.W.,Zhu,Y.L.,et al.,2000b.Elastic visco-plasticanalysis for earthquake response of tunnels in cold regions.ColdReg.Sci.Technol.31,175–188。
[7]Feng,Q.,Jiang,B.S.,Zhang,Q.,WangL.P.,2014.Analytical elasto-plastic solution for stress and deformation of surrounding rock in coldregion tunnels.Cold Reg.Sci.Technol.108,59–68。
[8] Zhangheng, Pogger, Wanglihua, etc. computational analysis of soft rock tunnel frozen-swelling force under freeze-thaw conditions [ J ]. proceedings of the Siam institute of science and technology, 2003, 23 (1): 1-6.
[9] Xu inherits the trade, liu xin yu applied elasto-plastic mechanics [ M ]. beijing: qinghua university Press, 1995.
[10]MATSUOKAN.The rate of bedrock weathering by frost action：fieldmeasurements and a predictive model[J].Earth Surface Processes and Landforms,1990,15(1):73–90。
[11]MATSUOKAN.Mechanisms of rock breakdown by frost action：anexperimental approach[J].Cold Regions Science and Technology,1990,17(3):253–270。
[12] First summer, yellow souring, korea, formation of tangzhi, evaluation method of dynamic expansion rate of cold region tunnel rock and dynamic load sensitivity classification [ J ]. report on rock mechanics and engineering, 2013, 32 (9): 1876-1885.
[13] Li shi yu, and tai name, yixiang foundation rock fracture mechanics theory of rupture [ M ]. fertilizer: china university of science and technology press, 2010.
[14] Research on the change law of the elastic modulus of rock under the freeze-thaw cycle conditions [ J ] geotechnical mechanics, 2015, 36 (8): 2315-2322.
[15] Xi jia rice, populus, pluviality, and the like. 1262-1268.
[16] The book of the Erythium, von Wen bridge, Damage mechanics [ M ]. Beijing, Qinghua university Press, 1997.
[17]
F S,WEIΒR,WIEDEMANNG.Changes of pore structure of cementmortars due to temperature[J].Cement and Concrete Research,1980,10(2):157-164。
[18]TAN Xian-jun,CHEN Wei-zhong,TIAN Hong-ming,et al.Water flow andheat transport including ice/water phase change in porous media:Numericalsimulation and application[J].Cold Regions Science and Technology,2011,68:74-84。
[19] Analyzing the correlation of rock physical and mechanical indexes under the conditions of Tianyan hucho, Xushuan and freeze-thaw cycle [ J ]. mining safety and environmental protection, 2017, 44 (4): 24-27.
disclosure of Invention
The invention aims to provide a method for measuring and calculating the frost heaving force of tunnel surrounding rocks in a cold region after repeated freeze-thaw damage.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows.
Method for measuring and calculating frost heaving force of tunnel surrounding rock in cold region after repeated freeze-thaw damage, wherein the frost heaving force is recorded as sigmaf(N)The calculation is performed according to the following formula:
in the formula, a, b and c are respectively the inner diameter of the lining, the inner diameter of the freezing area/the outer diameter of the lining and the outer diameter of the freezing area; e2(N)The elastic modulus of the rock in the freezing area after the Nth freezing-thawing cycle; e1And upsilon1Respectively the elastic modulus and Poisson's ratio of the lining; e2And upsilon2Respectively the elastic modulus and the Poisson ratio of the surrounding rock at the freezing zone; e3And upsilon3Respectively the elastic modulus and Poisson's ratio of surrounding rock in the non-frozen region；n0And nNThe initial porosity of the rock and the porosity of the rock after N times of freeze-thaw cycles are respectively, s is a test fitting constant, η is a hydrothermal migration influence coefficient, the frost heaving sensitive rock is 1.58, and the non-frost heaving sensitive rock is 1.0.
As a preferred embodiment of the present invention, k is given by the following formula:
as a preferable aspect of the present invention, said E2(N)The elastic modulus of the rock in the freezing area after the Nth freezing-thawing cycle; the method comprises the following steps:
A. based on the fact that the microcracks are saturated all the time, the half-length of microcrack propagation in the Nth freeze-thaw cycle is recorded as delta lN(ii) a After N freeze-thaw cycles, microcrack propagation half-length lNComprises the following steps:
lN＝ΔlN+lN-1(32)；
B. the corresponding microcrack density parameter β is expressed as:
C. the number of microcracks activated to propagate per unit volume follows an exponential distribution, namely:
where ρ is the number of microcracks having a microcrack radius larger than l per unit area, ρ0Total number of microcracks,/cA microcrack distribution parameter;
D. under the two-dimensional condition, the interaction between the microcracks is considered, and an expression of the effective elastic modulus of the rock is obtained:
in the formula, E0Is the initial elastic modulus of the rock in MPa; eNThe equivalent modulus of elasticity in MPa for the rock after N times of freeze-thawing, β the microcrack density parameter, expressed as β ═ ρ (Δ l)2(ii) a Rho is the number of microcracks with half-length of delta l after freeze thawing in unit area, and the unit is strip/m2；
E. Finally, equations (32) to (34) are substituted for equation (30), for ENAnd solving to obtain the equivalent elastic modulus of the tunnel surrounding rock after N times of freeze thawing.
As a preferred embodiment of the present invention, in step a, when N is 1, Δ l is written1Δ l, calculated according to the following formula:
in the formula, p is the frost heaving force of a single microcrack, l is the approximate plane ellipse major axis radius of the rock microcrack, b is the inner diameter of the frost region of the tunnel surrounding rock/the outer diameter of the tunnel lining, Y is the energy release rate of the microcrack, and delta ViIs the single wide expansion volume of ice.
As a preferred technical scheme of the invention, Y is the Griffith energy release rate of the microcrack, and the calculation formula is as follows:
wherein E isrIs the low temperature rock modulus of elasticity, Er TFor the modulus of elasticity of the rock at T temperature, the modulus of elasticity of the rock increases as the temperature decreases, i.e. taking Er T＝mEr，Er TAnd ErThe elastic modulus of the rock at normal temperature and low temperature T respectively, m is an elastic modulus amplification coefficient caused by temperature reduction, the value of m is related to the temperature T, and the value is 1-2.
As a preferable embodiment of the present invention, the Δ ViThe method comprises the following steps:
without taking into account microcracking surfacesThe ice body in the microcrack is freely expanded when being constrained, in practical situation, the wall of the microcrack exerts a reaction force with the magnitude of p on the ice body, the ice body generates elastic strain, and the body strain epsilon of the ice under the condition of plane strainvComprises the following steps:
in the formula Ei、υi、KiRespectively, the elastic modulus, poisson's ratio and volume modulus of ice; the ice volume expansion rate at free expansion was 9%, and the microcrack volume change at an expansion pressure p was:
ΔVi＝πlb(1.09-εv)＝πlb(1.09-pKi) (26)。
as a preferred embodiment of the present invention, the single microcrack frost-swell force p is obtained by changing the size of the microcracks after expansion while maintaining the shape thereof:
meanwhile, according to the Griffith energy release rate theory, when water in the microcracks is frozen into ice, the volume is expanded, but due to the constraint of the microcrack surfaces, the ice body generates expansion pressure on the ice body, so that elastic strain energy is generated in a medium around the microcracks; when the stress intensity factor K of the microcrack tipΙGreater than fracture toughness value KΙCAt this time, the micro-cracks begin to expand and the elastic strain energy is released, so that there is:
W＝Z-U (16)；
wherein, W represents the work done by the frost heaving force, Z represents the elastic strain energy stored around the microcrack, and U represents the total potential energy of the whole system;
assuming that the elastic strain energy is fully released when the microcracks propagate, then:
W＝-U (17)；
when the stress intensity factor is greater than or equal to the fracture toughness, the microcrack begins to expand, and the expanded frost heaving force works along the normal direction of the wall surface of the microcrack, so the expression on the left side of the expression (17) can be expressed as:
W＝4pl×Δb (18)；
the potential energy reduction value of the system on the right side of equation (17) can be expressed as the expansion force work:
U＝-2Y×Δl (19)；
by substituting formulae (18) to (20) for formula (17):
the combination of formulas (21) and (27) is as follows:
as a preferred embodiment of the present invention, Δ l is solved by substituting equations (20), (26), and (28) into equation (24):
obtaining the expansion length value of the microcracks of the surrounding rocks of the tunnel in the cold region under the frost heaving force;
then Δ lNThe calculation is performed according to the following formula:
as a preferred embodiment of the present invention, E is obtained by substituting formulae (31) to (34) for formula (30)NAnd (3) solving:
and obtaining the equivalent elastic modulus of the tunnel surrounding rock after N times of freeze thawing.
As a preferred embodiment of the present invention, the following steps: substituting formula (35) into formula (40), neglecting the influence of freeze-thaw cycle on poisson ratio of rock in frozen region, and applying to sigmaf(N)Solving to obtainAnd (4) freezing and thawing the damaged tunnel surrounding rock repeatedly to obtain the frost heaving force value of the tunnel surrounding rock in the cold region.
Adopt the produced beneficial effect of above-mentioned technical scheme to lie in: according to the method, firstly, the frost heaving force of the surrounding rock based on the elasticity theory is calculated, then the damage of the freeze-thaw cycle to the rock is considered, the deterioration rule of the elastic modulus and the porosity of the rock along with the freeze-thaw cycle is obtained, finally, the tunnel surrounding rock frost heaving force calculation method considering the damage of the freeze-thaw cycle to the rock is provided, and practical guidance is provided for the construction direction and method of the rock tunnel maintenance engineering.
Drawings
FIG. 1 is a schematic diagram of a tunnel frost heaving force calculation model.
FIG. 2 is a schematic view of a microcrack model under the action of a frost-heave force.
FIG. 3 is a schematic view of a microcrack frost-heave expansion model.
FIG. 4 is a schematic diagram of the change rule of the rock elastic modulus along with the number of freeze-thaw cycles.
FIG. 5 is a schematic diagram showing the variation of frost heaving force with the number of freeze-thaw cycles.
FIG. 6 is a schematic diagram of the change rule of the frost heaving force and the rock elastic modulus with the freeze-thaw times without considering the change of the rock frost heaving rate.
FIG. 7 is a schematic diagram showing the variation law of the frost heaving force with the number of freeze-thaw cycles at different s.
FIG. 8 is a schematic diagram of the change rule of the rock porosity with the number of freeze-thaw cycles at different s.
Detailed Description
Example 1 calculation of the wall rock frost heave force based on the elasticity theory.
The calculation model is as shown in figure 1, in order to simplify the problem, firstly, the freezing area and the non-freezing area in the tunnel lining and the surrounding rock are regarded as a stress system formed by three axisymmetric elastomers which are in mutual complete contact, wherein the tunnel is a circular hole in an infinite mountain, and a, b and c are respectively the inner diameter of the lining, the inner diameter of the freezing area (also the outer diameter of the lining) and the outer diameter of the freezing area, and the following assumptions are made: (1) the surrounding rock is a homogeneous and isotropic continuous medium; (2) the stress of the tunnel lining and surrounding rock belongs to the plane strain problem of elastic mechanics; (3) the freezing area is always in a saturated state; (4) the dead weight of surrounding rocks and tunnel linings is not considered.
The frost heaving force of the tunnel surrounding rock is caused by the ice formed by the water in the tunnel surrounding rock, due to the effect of the frost heaving force, the frost heaving of the surrounding rock at the frost zone expands towards the inner direction and the outer direction, as shown in figure 1, assuming that the expansion displacement of the surrounding rock at the frost zone to the lining direction and the surrounding rock at the non-frost zone is respectively delta h1And Δ h2. Let sigmafAnd σhThe pressures of the surrounding rocks in the freezing zone acting on the lining and the surrounding rocks in the non-freezing zone respectively are important in the research, and the frost heaving pressure sigma is concernedf. As can be seen from FIG. 1, when the lining is subjected to frost heaving pressure σfWhen acting, it can be regarded as a thick-walled cylinder under external pressure, and thus is an axisymmetric problem. For the convenience of solution, polar coordinates are used for calculation. According to the theory of elasticity [9]The frost heaving pressure sigma of the lining can be obtainedfThe displacement under action is:
wherein E is1And υ 1 is the elastic modulus and poisson's ratio, respectively, of the lining.
At the outer diameter of the tunnel lining, i.e. r ═ b, its displacement δ1Comprises the following steps:
the inner and outer walls of the surrounding rock in the freezing zone are respectively subjected to frost heaving pressure sigmafActing force sigma of surrounding rock in unfrozen areahThe surrounding rock at the freezing zone can be regarded as an axisymmetric problem simultaneously acted by internal pressure and external pressure, and the displacement of the surrounding rock at the freezing zone can be obtained according to the elasticity theory:
wherein E is2And upsilon2Elastic modulus and Poisson of surrounding rock in frozen zoneAnd (4) the ratio.
At the inner wall of the surrounding rock in the freezing zone, namely r ═ b, the displacement delta isf1Comprises the following steps:
at the outer wall of the surrounding rock in the freezing zone, namely r ═ c, the displacement delta isf2Comprises the following steps:
as shown in figure 1, the surrounding rock of the non-frozen area is only subjected to sigma on the inner wallhThe displacement delta of the surrounding rock of the non-frozen zone at the inner wall can be obtained according to the elasticity theory2Comprises the following steps:
wherein E is3And upsilon3Respectively the elastic modulus and the Poisson ratio of surrounding rock at an unfrozen area.
According to the condition of interface displacement continuity between the lining and the frozen surrounding rock, the method comprises the following steps:
-δ1+δf1＝Δh1(7)
similarly, according to the condition of continuity of the interface displacement between the frozen surrounding rock and the unfrozen surrounding rock, the following steps are carried out:
δ2-δf2＝Δh2(8)
wherein Δ h1And Δ h2Can be respectively obtained by the following formula:
wherein α is the frost heaving ratio of surrounding rock, Delta V1And Δ V2Surrounding rock in the freezing zone at the inner diameter and the outer diameterThe frost heaviness at the diameter.
Finally, the frost heaving pressure σ can be obtained from the expressions (2), (4) to (10)fComprises the following steps:
wherein,
from the equation (11), it can be seen that the wall rock frost heaving force is not only related to the geometric dimensions of the model, such as the lining inner diameter a, the freezing zone inner diameter (also called lining outer diameter) b and the freezing zone outer diameter c, but also related to the elastic constants of the model, such as the lining elastic modulus and Poisson's ratio (E)1、v1) Modulus of elasticity and Poisson's ratio (E) of frozen rock2、v2) And modulus of elasticity to Poisson's ratio (E) of unfrozen rock3、v3) The two former parameters are easy to determine and are not repeated here, the following method for determining the frost heaviness is discussed, the frost heaviness is a quantitative description of the frost heaviness of the rock, N.Matsuoka [10 ]]The freezing test of saturated rock shows that the rock frost heaving rate is affected by the volume expansion of frozen pore water, hydrothermal migration, rock frost heaving restraining effect and other factors, and the frost heaving rate of rock is 0.1-0.5% [11 ]]. Considering the influence of hydrothermal migration on rock frost heaving rate, early summer (12)]A calculation formula of the frost heaving rate of the saturated rock under the open condition is provided:
α＝2.17％ηn (12)
wherein η is the influence coefficient of hydrothermal migration, frost heaving sensitivity rock is 1.58, non-frost heaving sensitivity rock is 1.0, and n is the porosity of rock.
Example 2 crack initiation criteria and propagation direction of single microcracks under the action of frost heave force.
The nature of freeze-thaw damage of the rock is that water in the microcracks is frozen into ice at low temperature to generate volume expansion, and then the microcracks expand under the action of expansion force to cause the deterioration of rock properties. Then when the temperature rises, the ice melts into water, which will flow along the expanded micro-cracks. If the water supply is sufficient and the melting time is long enough, the microcracks are considered to be always in a saturated state, and when the temperature is reduced, the water is frozen into ice, so that the microcracks are further expanded, and the process is repeated. Each freeze-thaw cycle causes some damage to the rock, which is microscopically manifested as an increase in microcrack length and macroscopically as a decrease in elastic modulus. Therefore, from the viewpoint of mesomechanics, the influence of freeze-thaw cycles on the elastic modulus of the rock is studied, and then a method for calculating the frost heaving force of the tunnel surrounding rock by considering damage is proposed on the basis of section 2.
The microcracks in the rock are considered to be flat elliptical fractures in a planar state (as in FIG. 2), with the major axis 2l, the minor axis 2b, and b < < l of the ellipse.
The inner wall of the microcrack is acted by uniform normal frost heaving force p, and after N times of freeze-thaw cycles, the half length of the microcrack is changed into lNIt is assumed that the micro-cracks propagate in the direction in which the strain energy density factor is smallest, i.e., the destabilizing propagation of the micro-cracks is due to the smallest strain energy density factor SminReach the corresponding critical value S of the materialcThe strain energy density field strength of the tip region of the pure I-type micro-crack under the plane strain condition is [13]]：
Wherein
GrIs the rock shear modulus; v isrAnd theta is Poisson's ratio and theta is the wing crack initiation angle.
From equation (14), it can be determined that the wing crack initiation angle θ is 0, i.e., the microcracks self-similar propagate.
The criterion of microcrack cracking is as follows:
wherein: kΙCIs the fracture toughness of the rock.
Example 3, single microcrack expansion length under the effect of the frost heave force.
When the micro-cracks have frost heaving effect, the inner wall of the micro-cracks has uniformly distributed frost heaving force, and correspondingly the micro-cracks expand towards the x direction and the y direction, as shown in figure 3, when the frost heaving expansion of the micro-cracks is researched, the following assumptions are made as [14] ① that the micro-cracks are plane ellipses before and after the frost heaving, namely that the shape is unchanged, the central position is unchanged, and only the size is changed, ② ignores the water migration and the deformation of a rock framework, ③ that the micro-cracks are always in a saturated state, and ④ that the micro-cracks stably expand and conform to the theory of linear.
According to Griffith energy release rate theory, when water in the microcracks freezes into ice, the volume expands, but due to the constraint of the microcrack surfaces, the ice body generates expansion pressure on the water, and thus elastic strain energy is generated in the medium around the microcracks. When the stress intensity factor K of the microcrack tipΙGreater than fracture toughness value KΙCAt this time, the micro-cracks begin to expand and the elastic strain energy is released, so that there is:
W＝Z-U (16)
wherein W represents the work done by the frost heaving force, Z represents the elastic strain energy stored around the microcrack, and U represents the total potential energy of the whole system descending.
Assuming that the elastic strain energy is fully released when the microcracks propagate, then:
W＝-U (17)
when the stress intensity factor is equal to or higher than the fracture toughness, the microcracks start to propagate, and the form of the propagated microcracks is shown by a broken line in fig. 3. At this time, the frost heaving force works along the normal direction of the microcrack wall surface, so the expression on the left side of the formula (17) can be expressed as:
W＝4pl×Δb (18)
the potential energy reduction value of the system on the right side of equation (17) can be expressed as the expansion force work:
U＝-2Y×Δl (19)
wherein: y is the Griffith energy release rate of the microcrack, and the calculation formula is as follows:
wherein: er TIs the elastic modulus of the rock at temperature T, according to xi's meter, etc. [15]When the temperature is lowered, the modulus of elasticity of the rock increases, i.e. E is takenr T＝mEr，Er TAnd ErThe elastic modulus of the rock at normal temperature and low temperature T respectively, m is the elastic modulus amplification coefficient caused by temperature reduction, and the value of m is related to the temperature T and is usually 1-2.
By substituting formulae (18) to (20) for formula (17):
then, depending on the volume change of the microcracks before and after the phase change of the water in the microcracks, it is possible to obtain:
πlb+ΔVi＝π(l+Δl)(b+Δb) (22)
the equations for the propagation length of the microcracks are given by equations (21) to (22):
A(Δl)2+B(Δl)+C＝0 (23)
wherein: a ═ pi Y, B ═ pi (2plb + lG), C ═ 2pl Δ Vi.
Solving the formula (23) to obtain a relational expression between the crack propagation length and the frost heaving stress under the plane strain condition:
it can be seen that when solving using equation (24)Determination of the single-width expansion volume Δ V of the ice massiAnd the expansion pressure p of the ice against the microcrack walls.
First, the Δ V will be discussediThe calculation method of (1) is that the ice body in the microcrack expands freely without considering the constraint of the microcrack surface, but in practical situation, the wall of the microcrack applies a reaction force with the size of p to the ice body, the ice body generates elastic strain, and according to the elastic theory, the ice body strain epsilon under the plane strain conditionvComprises the following steps:
wherein: ei、υi、KiRespectively, the elastic modulus, poisson's ratio and bulk modulus of ice.
Assuming that the ice has a volume expansion ratio of 9% when it is expanded freely, the amount of volume change of the microcracks at an expansion pressure p is:
ΔVi＝πlb(1.09-εv)＝πlb(1.09-pKi) (26)；
next, discussing the calculation of the single microcrack frost-swell force p, as can be seen from fig. 3, since the size of the microcracks will change after expansion, but the shape will remain unchanged, it can be assumed that:
the following binding formulas (21) and (27) can be obtained:
by substituting formulae (20), (26), and (28) for formula (24):
example 4 rock modulus of elasticity versus number of freeze-thaw cycles.
Based on the microscopic damage theory, an averaging method is adopted to reflect the microscopic damage mechanics research result to the macroscopic mechanical property of the material, a Mori-Tanaka method [16] is adopted, and under the two-dimensional condition, the interaction among microcracks is considered, so that the expression of the effective elastic modulus of the rock is obtained as follows:
wherein E is0Is the initial elastic modulus of rock, MPa; eNEquivalent modulus of elasticity, MPa, of the rock after N times of freeze-thawing, β the microcrack density parameter, expressed as β ═ ρ (Δ l)2Rho is the number of microcracks with half-length of delta l expanded after freeze thawing in unit area, bars/m2。
First, since the microcracks are assumed to be saturated at all times, at the Nth freeze-thaw cycle, the half-length of microcrack propagation is:
correspondingly, the microcracks propagate half-length l after N freeze-thaw cyclesNThen it should be:
lN＝ΔlN+lN-1(32)
the corresponding microcrack density parameter β at this time may be expressed as:
secondly, the first step is to carry out the first,
etc. [17 ]]Research suggests that after the rock is subjected to multiple freeze-thaw cycles, the total number of microcracks is not increased basically, but the microcracks with longer lengths are expanded, and the microcracks with smaller lengths are closed due to the extrusion of other microcracks. That is, as the freeze-thaw cycle progresses, the number of microcracks that can continue to propagate will be less. According to Griffith micro-crack propagation theoryIn theory, the number of microcracks activated to propagate per unit volume follows an exponential distribution, namely:
wherein: rho is the number of microcracks with a microcrack radius greater than l per unit area, rho0Total number of microcracks,/cIs a microcrack distribution parameter.
By substituting formulae (31) to (34) for formula (30):
the elastic modulus E of the rock after N times of freeze-thaw cycles can be obtained by the formula (35)N. To verify the correctness of the above calculation method [18 ] with Tan et al]The test data of (A) are compared, and the adopted rock is granite taken from Galongla mountain area in Tibet, and a cylindrical test piece with the diameter of 50mm and the height of 100mm is manufactured. The physical and mechanical parameters are as follows: the dry mass and the saturated mass are 521.87g and 523.13g respectively, and the dry density is 2.77g/cm3Initial porosity 0.0067, uniaxial compression peak strength 135.73MPa, initial modulus of elasticity E037.64GPa and Poisson's ratio v00.25. The half length l of the microcrack is 9.0e-7m, the half width b is 1.7e-8m, and the distribution parameter l of the microcrackc5.5e-7m, microcrack density rho 1.7e13 bars/m2Elastic modulus E of Icei600MPa and Poisson's ratio upsiloniWhen m is 0.33, m is 1.42. The change rule of the elastic modulus of the rock along with the number N of the freeze-thaw cycles is shown in FIG. 4, and it can be seen that the elastic modulus of the rock gradually decreases along with the increase of the number of the freeze-thaw cycles, and the decrease speed gradually decreases along with the increase of the number of the freeze-thaw cycles, so that the rock elastic modulus is well matched with the test result.
Example 5, injury mechanics solution of wall rock frost heave force under freeze-thaw cycles.
Freeze-thaw cycles can cause varying degrees of damage to the rock, which in turn results in a reduction in the elastic modulus of the rock. In section 2, the calculation of the frost heaving force of the surrounding rock, i.e., equation (11), is based on the theory of elasticityAnd (3) calculating, namely not considering the reduction of the elastic modulus of the rock and the increase of the porosity after freeze-thaw cycling, so that the method is only suitable for solving the initial frost heaving force of the surrounding rock. The actual rock is usually subjected to multiple freeze-thaw cycles in the engineering service period, so that the damage of the freeze-thaw cycles to the surrounding rock is considered, and a damage mechanical solution of the frost heaving force of the surrounding rock under the freeze-thaw cycles is provided. It is assumed that the freeze-thaw cycle has an effect only on the mechanical properties of the rock in the frozen zone, but not on the mechanical properties of the lining concrete and rock in the non-frozen zone. The frost heaving force σ of the surrounding rock at the Nth freeze-thaw cycle can be obtained from the formulas (35) and (11)f(N)Comprises the following steps:
wherein:
the elastic modulus of the rock in the freezing area after the Nth freeze-thaw cycle is shown, and the other parameters are the same as the above. It is assumed here that the influence of the freeze-thaw cycle on the poisson's ratio of the freezing zone rock is small and negligible.
Meanwhile, along with the increase of the number of freeze-thaw cycles, the elastic modulus of the rock is reduced, and the corresponding physical process is the increase of the porosity of the rock, and the formula (12) shows that along with the increase of the porosity, the frost heaving rate of the rock is obviously increased, and finally, the frost heaving force of the surrounding rock is influenced. The effect of the freeze-thaw cycle on the porosity of the rock is therefore discussed below. Tianyan philosophy and Xueshi [19] have been studied to determine the following relationship between the elastic modulus of rock and the porosity of rock under freeze-thaw cycling conditions:
wherein E is0And ENRespectively the initial elastic modulus of the rock and the elastic modulus of the rock subjected to N times of freeze-thaw cycles; n is0And nNRespectively the initial porosity of the rock and after N freeze-thaw cycles(ii) rock porosity of; s is the trial fit constant.
Equation (37) can be written as:
substitution of formula (38) for formula (12) resulted in a rock frost heave rate of α after N freeze-thaw cyclesNComprises the following steps:
the surrounding rock frost heaving force sigma at the Nth freeze-thaw cycle can be obtained by substituting formula (39) for formula (36)f(N)Comprises the following steps:
example 6 law of wall rock frost heaving force variation with freeze-thaw cycle number N.
In this example, the change law of the wall rock frost-heaving force with the number N of freeze-thaw cycles is described by example, and a circular tunnel in a cold region (as shown in fig. 1) is provided, where the inner diameter a of the concrete lining is 3.0m, the outer diameter b is 3.6m, the outer diameter c of the frozen layer is 5.0m, and the elastic constant of the concrete lining is E1＝10GPa、v1Initial modulus of elasticity E of frozen zone surrounding rock037.64GPa and Poisson's ratio v00.25, initial porosity n00.0067. The elastic constant of surrounding rock of the unfrozen layer is E3＝37.64GPa、v20.25. The half length l of the microcrack is 9.0e-7m, the half width b is 1.7e-8m, and the distribution parameter l of the microcrackc5.5e-7m, microcrack density rho 1.7e13 bars/m2Elastic modulus E of Icei600MPa and Poisson's ratio upsiloniAssuming that the rock is always saturated during the freeze-thaw cycle, the wall rock frost expansion force σ can be obtained after 150 freeze-thaw cyclesf(N)The relationship with the change in the number of freeze-thaw cycles N is shown in FIG. 5. It can be seen that when the rock begins to frost heave, the frost heave rate of the rock is 0.000189 according to the formula (39), and then the frost heave force of the rock is only calculatedIs 0.13 MPa. Then, as the number of freeze-thaw cycles is increased, the porosity of the rock is increased, and accordingly the frost heaving rate of the rock is increased, and finally the frost heaving force of the surrounding rock is increased to 1.16MPa, which is about 8.92 times of the original frost heaving force, so that the increase amplitude is larger. However, as the number of freeze-thaw cycles increases, the rate of increase gradually slows. This is because rock frost heaving is primarily caused by the volumetric expansion of the water in the pores freezing to ice, which is almost negligible in comparison to the volumetric expansion of rock particles caused by freezing. In the initial stage, the porosity of the rock is small, the frost heaving force generated by freezing is correspondingly small, and along with the progress of freeze-thaw cycle, the microcracks expand under the effect of the frost heaving force, so that the damage of the rock is increased, and the porosity is increased. Since the rock is assumed to be saturated at all times, the rock moisture content will also increase, which in turn will result in an increase in rock frost heave force. By circulating in this way, the frost heaving force of the surrounding rock is finally gradually increased along with the increase of the number of freeze-thaw cycles. However, as shown in fig. 4, as the number of freeze-thaw cycles increases, the elastic modulus of the freeze-thaw rock gradually tends to a constant value, and at this time, the rock damage also tends to a constant value, and accordingly, the porosity and the moisture content of the rock also tend to a constant value, which finally results in that the frost heaviness of the surrounding rock tends to a constant value as the number of freeze-thaw cycles increases. It can therefore be assumed from the above analysis that the magnitude of the rock frost heave force is primarily due to the volume expansion of the water in the rock pores due to freezing to ice. In order to prove the above view, the rock frost heaving rate is assumed to be constant, namely 0.000189, and the freeze-thaw cycle is considered to only cause the reduction of the rock elastic modulus, so that the change rule of the surrounding rock frost heaving force along with the number of freeze-thaw cycles can be obtained as shown in fig. 6. It can be seen that as the number N of freeze-thaw cycles is gradually increased from 0 to 150, the elastic modulus of the surrounding rock is gradually decreased from 37.64GPa to 13.93GPa initially, and the corresponding frost heaving force of the surrounding rock is gradually increased from 0.13MPa initially to 0.137MPa, the maximum increase is only 5.38%, which is very small compared with fig. 5, and this indicates that the change of the elastic modulus of the surrounding rock has a very limited effect on the frost heaving force. Therefore, it can be considered that rock frost heaving force is mainly caused by freezing expansion of water in rock pores, and from the engineering point of view, the rock frost heaving force is reducedThe damage of surrounding rocks caused by frost heaving is reduced, and the infiltration of water is mainly controlled, namely, corresponding water interception and drainage measures are adopted to prevent underground water and surface water from infiltrating into the surrounding rocks of the roadway. Meanwhile, as can be seen from fig. 7 to 8, the coefficient s in the formula (40) has a significant influence on the frost heaving pressure and the rock porosity of the surrounding rock, and the frost heaving pressure and the rock porosity are reduced along with the increase of s, so that an accurate s value is obtained as much as possible according to test data in the test.
In summary, the embodiments of the invention establish a change rule of the rock elastic modulus along with the number of freeze-thaw cycles based on the Griffith fracture theory and the microscopic damage theory, and it is considered that the rock elastic modulus gradually decreases along with the increase of the number of freeze-thaw cycles, but the decrease rate gradually slows down. Meanwhile, the invention provides a damage mechanics solution of the round tunnel surrounding rock frost heaving force considering rock damage under freeze-thaw cycles by considering the influence of the freeze-thaw cycles on the elastic modulus, porosity and frost heaving rate of the rock on the basis of a round tunnel surrounding rock frost heaving force calculation model based on an elasticity theory. The calculation example shows that the frost heaving force of the surrounding rock gradually increases and tends to a certain value along with the increase of the number of times of freeze-thaw cycles. Therefore, the rock frost heaving force is mainly caused by the freezing expansion of water in rock pores, so that the frost heaving damage of surrounding rocks is reduced by mainly adopting a water interception and drainage measure to control the infiltration of water.
The above description is only presented as an enabling solution for the present invention and should not be taken as a sole limitation on the solution itself.
Claims (9)
1. The method for measuring and calculating the frost heaving force of the tunnel surrounding rock in the cold region after repeated freeze-thaw damage is characterized by comprising the following steps of:
the frost heaviness force is recorded as sigmaf(N)The calculation is performed according to the following formula:
in the formula, a, b and c are respectively the inner diameter of the lining, the inner diameter of the freezing area/the outer diameter of the lining and the outer diameter of the freezing area; e2(N)For rock in the frozen area after the Nth freeze-thaw cycleThe modulus of elasticity of (a); e1And upsilon1Respectively the elastic modulus and Poisson's ratio of the lining; e2And upsilon2Respectively the elastic modulus and the Poisson ratio of the surrounding rock at the freezing zone; e3And upsilon3Respectively representing the elastic modulus and Poisson's ratio of surrounding rock at an unfrozen area; n is0And nNRespectively the initial porosity of the rock and the porosity of the rock after N times of freeze-thaw cycles, s is a test fitting constant, η is a hydrothermal migration influence coefficient, 1.58 is taken as frost heaving sensitive rock, and 1.0 is taken as non-frost heaving sensitive rock;
the k is given by the following formula:
2. the method for measuring and calculating the frost heaving force of the cold area tunnel surrounding rock after repeated freeze-thaw damage according to claim 1, wherein the method comprises the following steps: said E2(N)The elastic modulus of the rock in the freezing area after the Nth freezing-thawing cycle; the method comprises the following steps:
A. based on the fact that the microcracks are saturated all the time, the half-length of microcrack propagation in the Nth freeze-thaw cycle is recorded as delta lN(ii) a After N freeze-thaw cycles, microcrack propagation half-length lNComprises the following steps:
lN＝ΔlN+lN-1(32)；
B. the corresponding microcrack density parameter β is expressed as:
C. the number of microcracks activated to propagate per unit volume follows an exponential distribution, namely:
where ρ is the number of microcracks having a microcrack radius larger than l per unit area, ρ0To be the total number of micro-cracks,lca microcrack distribution parameter;
D. under the two-dimensional condition, the interaction between the microcracks is considered, and an expression of the effective elastic modulus of the rock is obtained:
in the formula, E0Is the initial elastic modulus of the rock in MPa; eNThe equivalent modulus of elasticity in MPa for the rock after N times of freeze-thawing, β the microcrack density parameter, expressed as β ═ ρ (Δ l)2(ii) a Rho is the number of microcracks with half-length of delta l after freeze thawing in unit area, and the unit is strip/m2；
E. Finally, equations (32) to (34) are substituted for equation (30), for ENAnd solving to obtain the equivalent elastic modulus of the tunnel surrounding rock after N times of freeze thawing.
3. The method for measuring and calculating the frost heaving force of the cold area tunnel surrounding rock after repeated freeze-thaw damage according to claim 2, wherein the method comprises the following steps: in step A, when N is 1, the symbol Δ l1Δ l, calculated according to the following formula:
in the formula, p is the frost heaving force of a single microcrack, l is the approximate plane ellipse major axis radius of the rock microcrack, b is the inner diameter of the frost region of the tunnel surrounding rock/the outer diameter of the tunnel lining, Y is the energy release rate of the microcrack, and DeltaViIs the single wide expansion volume of ice.
4. The method for measuring and calculating the frost heaving force of the cold area tunnel surrounding rock after repeated freeze-thaw damage according to claim 3, wherein the method comprises the following steps: and Y is the Griffith energy release rate of the microcrack, and the calculation formula is as follows:
wherein E isrIs the low temperature rock modulus of elasticity, Er TFor the modulus of elasticity of the rock at T temperature, the modulus of elasticity of the rock increases as the temperature decreases, i.e. taking Er T＝mEr，Er TAnd ErThe elastic modulus of the rock at normal temperature and low temperature T respectively, m is an elastic modulus amplification coefficient caused by temperature reduction, the value of m is related to the temperature T, and the value is 1-2.
5. The method for measuring and calculating the frost heaving force of the cold area tunnel surrounding rock after repeated freeze-thaw damage according to claim 4, wherein the method comprises the following steps: the Δ ViThe method comprises the following steps:
the ice body in the microcrack expands freely without considering the constraint of the microcrack surface, and in practical situation, the reacting force with the size of p is applied to the ice body by the microcrack wall, the ice body generates elastic strain, and the ice body strain epsilon is generated under the condition of plane strainvComprises the following steps:
in the formula Ei、υi、KiRespectively, the elastic modulus, poisson's ratio and volume modulus of ice; the ice volume expansion rate at free expansion was 9%, and the microcrack volume change at an expansion pressure p was:
ΔVi＝πlb(1.09-εv)＝πlb(1.09-p/Ki) (26)。
6. the method for measuring and calculating the frost heaving force of the cold area tunnel surrounding rock after repeated freeze-thaw damage according to claim 5, wherein the method comprises the following steps: regarding the individual microcrack frost-swell force p, since the size of the microcracks will change after expansion, while the shape remains unchanged, we obtain:
at the same time, according to Griffith energy release rate theory, when water in the microcracks is frozen into ice, the volume expands, but due to the constraint of the microcrack surfaces, the ice body generates expansion pressure on the water, so that elastic strain energy is generated in a medium around the microcracks; when the stress intensity factor K of the microcrack tipΙGreater than fracture toughness value KΙCAt this time, the micro-cracks begin to expand and the elastic strain energy is released, so that there is:
W＝Z-U (16)；
wherein, W represents the work done by the frost heaving force, Z represents the elastic strain energy stored around the microcrack, and U represents the total potential energy of the whole system;
assuming that the elastic strain energy is fully released when the microcracks propagate, then:
W＝-U (17)；
when the stress intensity factor is greater than or equal to the fracture toughness, the microcrack begins to expand, and the expanded frost heaving force works along the normal direction of the wall surface of the microcrack, so the expression on the left side of the expression (17) can be expressed as:
W＝4pl×Δb (18)；
the potential energy reduction value of the system on the right side of equation (17) can be expressed as the expansion force work:
U＝-2Y×Δl (19)；
by substituting formulae (18) to (20) for formula (17):
the combination of formulas (21) and (27) is as follows:
7. the method for measuring and calculating the frost heaving force of the cold area tunnel surrounding rock after repeated freeze-thaw damage according to claim 6, wherein the method comprises the following steps: by substituting equations (20), (26), and (28) into equation (24), Δ l is solved:
obtaining the expansion length value of the microcracks of the surrounding rocks of the tunnel in the cold region under the frost heaving force;
then Δ lNThe calculation is performed according to the following formula:
8. the method for measuring and calculating the frost heaving force of the cold area tunnel surrounding rock after repeated freeze-thaw damage according to claim 7, wherein the method comprises the following steps: substitution of formulae (31) to (34) for formula (30), for ENAnd (3) solving:
and obtaining the equivalent elastic modulus of the tunnel surrounding rock after N times of freeze thawing.
9. The method for measuring and calculating the frost heaving force of the cold area tunnel surrounding rock after repeated freeze-thaw damage according to claim 8, wherein the method comprises the following steps: substituting formula (35) into formula (40), neglecting the influence of freeze-thaw cycle on poisson ratio of rock in frozen region, and applying to sigmaf(N)And solving to obtain the frost heaving force value of the tunnel surrounding rock in the cold region after repeated freeze-thaw damage.
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