A method for calculating a rock-concrete interface rate-dependent shear softening constitutive relationship

Abstract

The application relates to a method for calculating a rate-dependent shear softening constitutive relationship of a rock-concrete interface, and belongs to the field of rock-concrete interface fracture performance researches.The method is established from the perspective of energy conservation, based on the balance relationship between the energy consumed by rock-concrete interface crack propagation and the external force work, that is, the external force work for interface crack propagation is equal to the energy consumed for overcoming the shear cohesion in the fracture process zone in the numerical value, to establish the rate-dependent shear softening constitutive relationship.According to the proposed rate-dependent shear softening constitutive relationship equation, the shear cohesion stress of any point in the fracture process zone under a certain slip rate and slip displacement can be simply and conveniently calculated, and then the quantitative description of the energy consumption per unit area along the interface of each point is realized.The established constitutive model only takes the quasi-static average shear stress as a parameter, and comprehensively covers the influences of rock types, concrete strength and interface roughness, and can provide a reference for the safety evaluation of rock foundation concrete structures during the service period.

Description

Technical Field

[0001] This invention belongs to the field of rock-concrete interface fracture performance research, and relates to a method for calculating the rate-dependent constitutive relationship of rock-concrete interface shear softening. Background Technology

[0002] Rock-concrete interfaces are widely present in concrete structures based on rock masses, such as the contact surface between the concrete gravity dam body and bedrock, the lining-surrounding rock interface in tunnels and underground caverns, the concrete-rock contact surface in slope reinforcement projects, and the lining-surrounding rock interface in large underground powerhouses. These structures often withstand dynamic loads during service, including earthquakes, excavation blasting, and water hammer caused by rapid reservoir discharge. The strain rate corresponding to seismic loads is typically 10-1. -4 ~10 -1 The strain rate generated by the explosion can reach 10 on the order of magnitude. 1 The above are 10 quasi-static tests. -6 ~10 -5 Several orders of magnitude higher. At higher strain rates, the stress near the crack tip cannot be redistributed through creep and microcrack development in time, the rate of bond failure in the fracture process zone accelerates, and crack propagation needs to overcome greater resistance. Macroscopically, this is manifested as an increase in fracture toughness and fracture energy with increasing loading rate.

[0003] Fracture toughness and fracture energy only characterize the discrete states during the fracture process, such as crack initiation and instability, and cannot describe the continuous degradation of cohesive stress from crack initiation to propagation. In virtual crack models and cohesive zone models, this degradation process is described by softening constitutive relations. Under dynamic loads, cohesive stress is affected by the crack opening or slip rate, and its softening relation differs from the quasi-static form. If quasi-static fracture parameters or quasi-static softening relations are directly used in dynamic analysis, the true fracture resistance at the interface will be underestimated, leading to deviations in crack propagation and structural damage predictions, and reducing the reliability of safety assessments.

[0004] Currently, there are two main approaches to establishing rate-dependent softening constitutive relations. The first is the numerical inversion identification method, which involves fitting a softening function of assumed form (such as bilinear, exponential, or multilinear) to the measured load-displacement curve and correcting the quasi-static softening relation by introducing an amplification factor related to the loading rate or strain rate. This method is easy to implement in finite element analysis, but the rate variable used is the externally applied loading rate, rather than the local opening or slip rate at the crack surface. For specimens of different sizes, loading methods, or structural forms, the same external loading rate may correspond to different local rates. Therefore, the obtained softening relation is an empirical expression specific to the test conditions and may not be applicable to other specimen geometries or structural forms. The second approach is based on local stress equilibrium, which estimates the local shear stress and slip distribution along the interface using rock-side strain gauges in a single-sided push-out test. However, its rate-dependent relation is also handled by replacing the characteristic parameters in the quasi-static softening function with a function of the loading rate, thus retaining the quasi-static function form.

[0005] Establishing a softening constitutive relationship based on energy conservation, with relative displacement and rate of the crack surface as variables, is theoretically applicable to Type II shear fracture, but more difficult to achieve than Type I fracture. This is due to the characteristics of shear crack propagation: in existing Type II fracture tests, cracks often propagate simultaneously from both ends of the specimen, with unequal propagation lengths and rates on both sides, making it difficult to establish a clear correspondence between the work done by the external force and the work done to overcome shear cohesion. Based on this, the applicant's research group conducted relevant research, using a free-end push-out interface shear test to allow cracks to propagate unilaterally from the loaded end to the free end, and proposed a method for calculating the shear softening constitutive relationship related to the rock-concrete interface ratio. Summary of the Invention

[0006] Based on the limitations of current research, this invention, from the perspective of energy conservation, considers that the work done by external forces generates two parts of energy: one part is used to promote the propagation of interfacial cracks, and the other part is stored in the specimen as elastic strain energy. The former is numerically equal to the energy consumed to overcome the shear cohesion in the fracture process zone. The curve relating shear cohesion to interfacial slip displacement and slip rate is called the rock-concrete interface rate-dependent shear softening constitutive curve. In this invention, our research group proposes a method for calculating the rate-dependent shear softening constitutive relationship of the rock-concrete interface.

[0007] The technical solution of this invention: A method for calculating the shear softening constitutive relation of the rock-concrete interface ratio is presented. During the service life of rock-concrete foundation structures, the rock-concrete interface is often subjected to dynamic shear loads. Interface cracks propagate along the interface, with the fracture process zone as the leading factor. Shear cohesion is distributed within the fracture process zone, and its magnitude decreases with the increase of interface slip displacement. The curve reflecting the relationship between shear cohesion and slip displacement is called the shear softening constitutive curve. Under dynamic loads, the calculation method for shear cohesion differs from that under quasi-static conditions. The rate effect of shear cohesion cannot be ignored; that is, under dynamic load conditions, the interface shear cohesion is related not only to slip displacement but also to slip rate. From the perspective of energy conservation, based on the balance between the energy consumed by interface crack propagation and the work done by external forces, i.e., the work done by external forces for interface crack propagation... Energy consumed to overcome shear cohesion in the fracture process zone To establish rate-dependent shear softening constitutive relations, the values ​​are made equal. Step (1) is based on actual measurements P - δ Curve subtraction of elastic strain energy The external force work used for the propagation of the interface crack is obtained. ; Step (2) discretizes the crack propagation into N The step involves setting the slip displacement to be linearly distributed along the interface, which will consume the energy to overcome the shear cohesion. It is expressed as the integral of shear cohesive stress; Step (3) is based on the law of conservation of energy. and dimensionless quantity The shear cohesive stress is obtained by taking the second derivative, and a rate amplification factor is introduced. With critical slip correction factor Establish based on slip displacement and slip rate Rate-dependent shearing softening constitutive relation for variables.

[0008] The specific steps are as follows: (1) Calculate the external force work required for the propagation of the interface crack. A free-end push-out interface shear test was conducted on a rock-concrete composite specimen. The end of the specimen furthest from the loading point was unsupported (e.g., if the load was applied to the upper end of the specimen, then the lower end was unsupported), allowing the interface crack to propagate unilaterally from the loading end to the free end, forming a clear and quantifiable crack propagation path. The digital image correlation (DIC) method was used to acquire the full-field displacement of the rock-concrete composite specimen surface. The interface slip displacement was defined as the difference in vertical displacement between two points at the same height on the rock and concrete sides of the interface. The slip displacements at the crack tip and crack opening were pre-defined as follows: and CMSD Strain gauges were vertically attached in front of the pre-set crack tip to determine the crack initiation load. When a crack initiates and begins to propagate, the fracture energy is released, causing a sudden drop in the strain gauge readings. This drop is used to determine the initiation load. and its corresponding initial preset crack tip slip displacement The load during the entire loading process was collected using a servo-controlled testing machine. P and displacement of loading point δ Data, drawing P - δ curve; P - δ The area under the curve represents the total external work done, a portion of which is used for the propagation of interface cracks, i.e., the target computational amount. Part of it is used as elastic strain energy Stored in the specimen; Equal to a certain moment P - δ Area under the curve minus elastic strain energy The calculation formula is shown in equation (1): (1) In the formula, P For external loads, δ For the displacement of the loading point, It is the elastic strain energy. This represents the integral variable, a dummy variable representing the displacement of the loading point; (2) Calculate the energy consumed to overcome shear cohesion. The result is obtained by integrating the shear cohesion within the fracture process region. Calculating the shear cohesion at different times and locations is the accurate calculation A key step; rate-related shear cohesion is denoted as , τ Represents shear cohesive stress, Represents slip displacement. That is, d s / d t Represents slip rate, rate-dependent shear cohesion and and The relationship is called the rate-dependent shear softening constitutive expression; the integral of shear cohesion over the slip displacement is the energy dissipated per unit area. e Its expression is shown in equation (2): (2) Due to the rate-dependent shear cohesion at different locations in the interfacial fracture process zone The difference lies in the slip rate at each point. Since the values ​​are unequal and change continuously over time, they cannot be directly obtained by integrally integrating the entire crack propagation. e The total crack propagation is now discretized as follows: N Step, in the i Set the sliding rate within the step As a constant, the local stress-slip relationship is denoted as When the distance from the walk is not greater than the final amount of crack expansion. 5%, that is, the number of walks N When the value is not less than 20, the slip rate is considered to be equal on each micro-segment, i.e. τ - s The curves are the same; the energy dissipation per unit area on each micro-segment is proportional to the specimen thickness. b Integral and N The total energy consumed by the propagation of the interface cracks can be obtained by summing the steps. Its expression is shown in equation (3): (3) in, This represents the slip value at step j; Outer layer (spatial dimension): The interface is segmented along the crack propagation direction using the y-coordinate. The entire crack propagation process is discretized into N steps. This represents the cumulative crack propagation length at each step. Since the crack propagates from the loaded end to the free end, the segment that cracked earliest experiences the most slip steps. Just for "experienced" i The "Step-by-Step History" interface section: =N corresponds to the segment closest to the loading end, which starts sliding from step 1; i =1 corresponds to the segment that cracks only in the last step. Upper and lower limits of the outer integral. and That is, the positions at both ends of the section.

[0009] Inner layer (historical dimension): The slip history of a single point within the aforementioned segment. The slip at this point is... The step increments from 0 to the current value, the th... Step inside Increase to , that is, the lower and upper limits of the inner integral. Due to the different slip rates at each step, the relationship between shear cohesion stress and slip displacement gradually changes (denoted as ). The energy consumption at a single point must be accumulated step by step and cannot be accumulated all at once.

[0010] To transform the spatial integral along the crack length into an integral over the slip displacement, we assume that the slip displacement is linearly distributed along the crack propagation direction, i.e., the distance from the actual crack tip is... y slip displacement at point satisfy Based on this, we obtain d y With d s The differential relationship is shown in equation (4): (4) In the formula, a For the crack propagation length, b For the specimen thickness, For the final amount of crack propagation, Preset the slip displacement at the crack tip; (3) Calculate the rate-dependent shear softening constitutive model Substitute equation (4) into equation (3) and change the integration variable from y Change to s Multiply both sides of the equation Then, denote the dimensionless quantity on the left side of the equation as By the law of conservation of energy Therefore From actual measurement P - δ The curve is calculated as shown in equation (5): (5) right about Find the second derivative at the same normalized position. The stress at the point has a weight. The contribution signs of adjacent expansion steps at the same position are opposite, as shown in equation (6); Assume that the shear cohesive stress at the interface of each micro-segment is continuously distributed under any equilibrium state, i.e., the first... N - i +1 step and the N - i The stress at the same position is equal, as shown in equation (7); Substituting equation (7) into equation (6), adjacent terms at the same position cancel each other out, leaving only the first term, resulting in equation (8), which states that the rate-dependent shear cohesion stress is derived from... The second derivative is directly determined, while Based solely on actual measurements P - δ Curve calculation: (6) (7) (8) With the preset crack tip sliding displacement The x-axis is determined by the measured values ​​according to formula (5). P - δ Dimensionless quantities obtained from curve calculations Plot the measured values ​​as the ordinate. - s pct curve; For actual measurement - s pct For curve fitting, the fitting function must satisfy the following three constraints: ① coefficient of determination R 2 Greater than 0.9 to ensure consistency with the experimental curve; ② s =0 =0;③ right s First derivative characterization - s pct The fracture energy corresponding to the area under the curve. The fitting function determined accordingly is shown in equation (9), and its fitting parameters are... , : (9) Regarding equation (9) s Taking the second derivative yields the quasi-static shear softening constitutive model. As shown in equation (10): (10) Introducing a rate amplification factor into shear stress Introducing a correction factor in critical slip Based on this, a rate-related shear softening constitutive expression is established, as shown in equation (11); and The expressions are shown in Equations (12) and (13), respectively; the rate parameter is determined by fitting the test data under different concrete strengths and loading rates. , ; The reference slip rate is within the quasi-static slip rate range; when hour, and Both degenerate to 1, and equation (11) degenerates into the quasi-static shear softening constitutive form of equation (10): (11) (12) (13) Equation (11) only requires quasi-static parameters and It can be applied immediately; these two parameters can be obtained from the mean shear stress. estimate, Equals peak load divided by bonded area; As given by equation (14), Quasi-static critical slip The relationship is given by equation (15): (14) (15) In the formula, For the quasi-static maximum shear stress, The coefficients 2.7 and 0.01 in equations (14) and (15) are obtained by fitting quasi-static test data, representing the quasi-static critical slip displacement. , The rate parameter is determined by fitting experimental data.

[0011] Furthermore, the pre-set crack in step (1) is formed by pasting a polyvinyl chloride film on one end of the bonding surface near the loading point, with a film length of 50 mm; grid-like grooves are opened on the bonding surface of the rock block to control the interface roughness, with a groove width of 5 mm and a groove depth of 2 mm.

[0012] Furthermore, in step (1), multiple pairs of measuring points are arranged along the interface height in front of the preset crack tip. The sliding displacement of each measuring point is obtained by DIC and fitted into a linear distribution along the interface height, so that the sliding displacement reaches the initial preset crack tip sliding displacement. The location is taken as the crack tip, and the distance from this location to the predetermined crack tip is the crack propagation length. a .

[0013] Furthermore, in step (2), the total crack propagation is discretized at equal step sizes as follows: N Steps, distance from walking number N Not less than 20; if N If the relative change in shear cohesive stress calculated by equation (8) after doubling exceeds 5%, then increase... N Until the change does not exceed 5%.

[0014] Furthermore, in step (3), a pre-set crack tip slip displacement is implemented. Data is continuously collected by DIC at a fixed sampling frequency, with a preset crack tip slip rate. Depend on The derivative with respect to time is obtained.

[0015] Furthermore, the reference slip rate The value is 0.001 mm / s, which is within the range of quasi-static slip rate.

[0016] Furthermore, the rate-dependent shear softening constitutive model is based solely on the quasi-static average shear stress. As a parameter, it encompasses the combined effects of rock type, concrete strength, and interface roughness, and is applicable to quasi-static mean shear stress. Rock-concrete interface in the range of 0.86 to 1.87 MPa.

[0017] Furthermore, the rate-dependent shear softening constitutive model is embedded into finite element software via a user subroutine to numerically simulate the propagation of interface cracks; and the constitutive model is integrated along the slip displacement from 0 to the critical slip. s 0, Derivation rate-related Type II fracture energy The expression enables the direct calculation of fracture energy at different slip rates, as shown in equation (16): (16) The beneficial effects of this invention are as follows: Based on the rate-dependent shear softening constitutive equation proposed by this invention, the shear cohesion stress at any point in the fracture process zone under a certain slip rate and slip displacement can be calculated simply and conveniently, thereby achieving a quantitative description of the energy dissipation per unit area at each point along the interface; the established constitutive model uses only quasi-static average shear stress. As a parameter, it encompasses the combined effects of rock type, concrete strength, and interface roughness, and is applicable to various rock-concrete interfaces with quasi-static average shear stress in the range of 0.86–1.87 MPa. Furthermore, by integrating the constitutive model, a rate-dependent Type II fracture energy expression can be derived, enabling direct calculation of fracture energy at different slip rates, and providing a reference for the safety evaluation of rock-foundation concrete structures during service. Attached Figure Description

[0018] Figure 1 A schematic diagram of a rock-concrete composite specimen and a shear test setup with free end extension is shown; (a) shows the specimen geometry, and (b) shows the test configuration. The values ​​for each dimension symbol are as follows: L1 = 70 mm, L2 = 220 mm, L3 = 240 mm, L4 = 260 mm, L5 = 50 mm, L6 = 100 mm, L7 = 50 mm, L8 = 30 mm, L9 = 20 mm, L... 10 For 100 mm, L 11 It is 20 mm, L 12 For 200 mm, L 13 It is 160 mm, L 14 It is 390 mm, L 15 It is 250 mm, L 16 It is 170 mm, L 17 It is 70 mm, L 18The length is 100 mm. A grid-like groove is made on the bonding surface of the rock block, with a groove width of 5 mm and a groove depth of 2 mm; the total length of the polyvinyl chloride film is 50 mm, of which the 30 mm section overlapping with the bonding surface forms the initial crack.

[0019] Figure 2 A schematic diagram of the loading device for the shear test of the free end; where (a) is the front view and (b) is the rear view.

[0020] Figure 3 This diagram illustrates the determination of the initiation load, initial slip displacement, and crack propagation length.

[0021] Figure 4 The diagram shows the interface crack propagation process of specimen G-C35-2-2 at different loading stages.

[0022] Figure 5 The effect of loading rate and interface type on type II initiation fracture toughness and instability fracture toughness The influence curve; where (a) is (b) is .

[0023] Figure 6 For different loading rates and concrete strengths The curve showing the change in the ratio.

[0024] Figure 7 The average shear stress at four loading rates .

[0025] Figure 8 Load-displacement at loading point Curve and elastic strain energy External force work A schematic diagram for determining the [specific element].

[0026] Figure 9 This is a schematic diagram showing the linear distribution of slip displacement along the interface.

[0027] Figure 10 for curve and Curves; where (a) and (b) are the G-C25 interface curves respectively. curve and Curves (c) and (d) represent the G-C45 interface, respectively. curve and Curves (e) and (f) represent the G-C35 interface, respectively. curve and curve.

[0028] Figure 11For rate-related shear softening constitutive model A three-dimensional surface plot.

[0029] Figure 12 The curves showing the relationship between quasi-static constitutive parameters and mean shear stress; where (a) is... and The relationship curve, (b) is and The relationship curve.

[0030] Figure 13 For different slip rates Curves and different slip displacements Curves; where (a) and (b) represent the G-C25 interface, (c) and (d) represent the G-C35 interface, and (e) and (f) represent the G-C45 interface. The former in each group is... Curve, the latter is curve.

[0031] Figure 14 Flowchart for numerical simulation of crack expansion on complete interface.

[0032] Figure 15 This is a schematic diagram of the finite element model and boundary conditions.

[0033] Figure 16 For G-C45-3-3 specimens at 20%, 40%, 80%, and 100% concentrations... A comparison of DIC observations and numerical simulations of the crack propagation process.

[0034] Figure 17 G-C25 Interface Prediction A comparison graph of the curve and the test results; where (a) is the G-C25-5 specimen, (b) is the G-C25-4 specimen, (c) is the G-C25-3 specimen, and (d) is the G-C25-2 specimen.

[0035] Figure 18 Predicting the S-C35 interface A comparison graph of the curve and the test results; where (a) is the S-C35-5 specimen, (b) is the S-C35-4 specimen, (c) is the S-C35-3 specimen, and (d) is the S-C35-2 specimen.

[0036] Figure 19 For S-C35-2-3 specimens at 20%, 40%, 80%, and 100% concentrations... A comparison of DIC observations and numerical simulations of the crack propagation process.

[0037] Figure 20 Type II fracture energy related to different concrete strength reduction rates With slip rate A changing curve.

[0038] Figure 21 For different loading rates and concrete strengths , and The comparison chart; where (a) is (b) is (c) is . Detailed Implementation

[0039] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.

[0040] The following example uses a granite-concrete free-end ejection specimen to illustrate the process of determining the shear softening constitutive model of the rock-concrete interface under different loading rates, in conjunction with the accompanying drawings. From the derivation of equations (8) and (11) above, it can be seen that the following quantities are required for calculating and fitting the constitutive relationship: the external force work used for interface crack propagation. Preset crack tip slip displacement Specimen thickness Crack propagation length and preset crack tip slip rate ,in The calculation of the parameters needs to be determined through cyclic loading and unloading tests and free end ejection tests at different loading rates. The other parameters can be obtained by combining free end ejection tests with DIC measurements. The following section will detail the acquisition methods of each parameter, the experimental curves after data processing, and the entire derivation process of the rate-related shear softening constitutive relation.

[0041] (1) Free end pull-out interface shearing test The free-end ejection specimen described in this embodiment is as follows: Figure 1 As shown, the specimen was formed by bonding a 240 mm × 200 mm × 100 mm concrete block and a 220 mm × 100 mm × 20 mm rock block together for a bond length of 170 mm. Two 50 mm long polyvinyl chloride (PVC) films were adhered to the ends of the rock block to create pre-designed cracks, resulting in an interfacial bonding area of ​​170 mm × 100 mm. To investigate the fracture performance of the interface under different bonding conditions, concrete of strength grades C25, C35, and C45 were used. Grooves were artificially carved on the contact surface of the rock block, dividing the short side of the rock block into four equal segments, forming a 4×4 groove pattern with a groove width of 5 mm and a depth of 2 mm. The end of the specimen furthest from the loading point was unsupported, and the interfacial cracks propagated unilaterally from top to bottom along the interface, ensuring a clear and quantifiable crack propagation path.

[0042] (2) Determine the loading rate The test was conducted on a 250 kN closed-loop servo-controlled testing machine, with the loading device as follows: Figure 2 As shown, four loading rates are set. The velocity values ​​are 0.0022 mm / s, 0.022 mm / s, 0.22 mm / s, and 2.2 mm / s, denoted by 5, 4, 3, and 2 respectively. 5 corresponds to the lowest velocity, 2 to the highest velocity, and 0.0022 mm / s is close to the quasi-static condition. The full-field displacement of the specimen surface was obtained using DIC (Diverterless Interpolation). To determine the crack initiation load, strain gauges were vertically attached 5 mm in front of a pre-set crack. When the crack initiates and propagates, the strain gauge readings drop abruptly due to the release of fracture energy, and the crack initiation load is determined accordingly. ,like Figure 3 As shown. The specimens are numbered G-Cxx-AB, where G represents granite, Cxx represents concrete strength grade, A represents loading rate grade, and B represents specimen number.

[0043] (3) Determine the crack propagation length and slip rate Interface slip displacement is defined as the difference in vertical displacement between two points at the same height on the rock and concrete on either side of the interface. The slip displacements at the crack tip and crack opening are denoted as follows: and CMSD Cracking load corresponding Defined as the initial preset crack tip slip displacement Seven pairs of measuring points were arranged along the interface height on the rock and concrete to obtain seven slip values. Based on these values, a linear distribution of the slip displacement along the interface height was fitted, and the slip displacement was calculated to reach... The location is taken as the crack tip, and the distance from this location to the predetermined crack tip is the crack propagation length. After establishing a coordinate system at the new crack tip, the slip velocity at any position on the ligament increases with the increase of crack propagation. Based on this, the preset crack tip slip velocity can be calculated from the preset crack tip slip displacement. .

[0044] (4) Cyclic loading and unloading test External force used for the propagation of interface cracks Numerically equal to a certain moment Total area under the curve minus elastic strain energy ,like Figure 8 As shown. A three-point cyclic loading and unloading test was conducted. The recoverable deformation during unloading is the elastic displacement, and the area enclosed by the elastic displacement and the corresponding load is... By repeatedly loading the load, the relationship curve between the total displacement at the loading point and the residual displacement at the loading point is obtained, and then the displacement at each moment is calculated. .

[0045] (5) Rate-related Type II fracture parameters The crack initiation load at different loading rates was obtained from the free-end push-out test. Peak load Average shear stress Type II fracture toughness Critical crack propagation length Type II instability fracture toughness . , , and Both increase with increasing loading rate, and are related to... The relationship is approximately linear, such as Figure 5 As shown. Taking C25, C35, and C45 concrete strengths as examples, when the loading rate increases from 0.0022 mm / s to 2.2 mm / s, They increased by approximately 51.1%, 77.8%, and 69.7% respectively. They increased by approximately 57.3%, 34.8%, and 47.0% respectively. These represent increases of approximately 65.2%, 72.3%, and 80.6%, respectively. At the same loading rate, and All increase with increasing concrete strength. (Under different loading rates and concrete strengths) The ratio change is as follows Figure 6 As shown. Average shear stress The curve of change with loading rate is as follows Figure 7 As shown.

[0046] (6) Calculate the rate-dependent shear softening constitutive model Based on the data from the free-end ejection test and the cyclic loading and unloading test, the crack propagation can be determined. Energy required to form the current fracture process zone after the step Preset crack tip slip displacement Preset crack tip slip rate and current crack propagation length The dimensionless quantity can be obtained from equation (5). ,right about The second derivative can be obtained from equation (8) to obtain the rate-dependent shear cohesive stress. and Curves Figure 10 As shown. With slip displacement and slip rate As parameters, a two-parameter analysis was performed on the experimental results, introducing a rate amplification factor. and critical slip correction factor Establish the constitutive relation of rate-related shear softening As shown in equations (11) to (13), where =0.15、 =0.14、 =0.001 mm / s. The three-dimensional surface of the rate-dependent shear softening constitutive model is as follows: Figure 11 As shown: It increases with increasing slip velocity and decreases with increasing slip displacement; due to the quasi-static properties of C25, C35, and C45. and The three surfaces differ in that their curvatures rise sequentially, with C45 above C35 and C35 above C25. Only quasi-static parameters are needed for application. and Both can be determined by average shear stress. according to Figure 12 Estimation of the relationship in the middle.

[0047] (7) Numerical simulation and verification To verify the applicability of the established constitutive model, it was embedded in the finite element software ABAQUS for numerical simulation of interface crack propagation. A three-dimensional finite element model was established, with a total rock height of 220 mm. The upper 30 mm section of the interface was pre-cracked and no constitutive relation was assigned to this section. Hard contact was applied in the normal direction to prevent mutual intrusion, and no additional constraints were applied in the tangential direction. The rock and concrete were discretized using three-dimensional linear elastic solid elements C3D8R. The interface was simulated using contact surfaces, and a rate-dependent shear softening constitutive model was assigned to the interface through a user subroutine. The simulation process is as follows: Figure 14 As shown, the finite element model and boundary conditions are as follows: Figure 15 As shown. In each incremental step, the slip displacement is obtained from the finite element solution. The data is then passed to a subroutine, which calculates the slip rate by combining the displacement increment, time increment, and state variables. Then calculate the shear stress. Introducing initial slip ,when Stress develops along the elastic segment when The stress develops along the softening section, when hour Interface stiffness degradation is characterized by the damage variable CSDMG.

[0048] The established constitutive model was first applied to the granite-concrete specimens used in its derivation, simulating crack propagation downwards along the interface from the pre-crack tip, consistent with DIC observations, such as... Figure 16 As shown, simulation The curve agrees well with the experimental results, such as Figure 17As shown. In the constitutive derivation, rock type, concrete strength, and interface roughness are not introduced as independent variables, but are instead determined through quasi-static mean shear stress. Overall, the data used The range is 0.86–1.87 MPa, therefore the constitutive model is applicable. Any rock-concrete interface falling within this range. To further examine the applicability of the constitutive model to interfaces not involved in the derivation, we use... Independent verification was conducted on the sandstone-C35 interface, which also falls within the range of 0.86–1.87 MPa, and simulations were performed. The curves and crack propagation process agree well with the experimental results, as shown below. Figure 18 and Figure 19 As shown, the constitutive model using only the quasi-static mean shear stress as a parameter is applicable to different rock types and interface conditions within the specified mean shear stress range.

[0049] (8) Rate-dependent Type II fracture energy The law-dependent shear softening constitutive integral from 0 to critical slip along the slip displacement. That is, yield-related type II fracture energy As shown in equation (16), where Let from equation (11) The results show that, given the concrete strength, , and All three increase with increasing slip velocity; at a given slip velocity, all three increase with increasing concrete strength. When the slip velocity increases from 0.0022 mm / s to 2.2 mm / s, For C25, the increase was from 7.1 N / m to 19.3 N / m; for C35, from 11.2 N / m to 31.1 N / m; and for C45, from 22.9 N / m to 63.0 N / m, representing a relative increase of approximately 172% to 178%. Figure 20 and Figure 21 As shown.

Claims

1. A method for calculating the constitutive relation of rock-concrete interface ratio related to shear softening, characterized in that, From the perspective of energy conservation, based on the balance between the energy consumed in the propagation of interfacial cracks and the work done by external forces, that is, the work done by external forces for the propagation of interfacial cracks... Energy consumed to overcome shear cohesion in the fracture process zone To establish rate-dependent shear softening constitutive relations, the values ​​are made equal. Step (1) is based on actual measurements P - δ Curve subtraction of elastic strain energy The external force work used for the propagation of the interface crack is obtained. ; Step (2) discretizes the crack propagation into N The step involves setting the slip displacement to be linearly distributed along the interface, which will consume the energy to overcome the shear cohesion. It is expressed as the integral of shear cohesive stress; Step (3) is based on the law of conservation of energy. and dimensionless quantity The shear cohesive stress is obtained by taking the second derivative, and a rate amplification factor is introduced. With critical slip correction factor Establish based on slip displacement and slip rate Rate-dependent shearing softening constitutive relation for variables.

2. The method for calculating the constitutive relation of rock-concrete interface ratio-related shear softening according to claim 1, characterized in that, The specific steps (1) are as follows: A free-end push-out interface shear test was conducted on a rock-concrete composite specimen. The end of the specimen furthest from the loading point was unsupported, allowing the interface crack to propagate unilaterally from the loading end to the free end, forming a clear and quantifiable crack propagation path. The digital image correlation (DIC) method was used to acquire the full-field displacement of the rock-concrete composite specimen surface. The interface slip displacement was defined as the difference in vertical displacement between two points at the same height on the rock and concrete sides of the interface. The slip displacements at the crack tip and crack opening were pre-defined as follows: and CMSD Strain gauges were vertically attached in front of the pre-set crack tip to determine the crack initiation load. When a crack initiates and begins to propagate, the fracture energy is released, causing a sudden drop in the strain gauge readings. This drop is used to determine the initiation load. and its corresponding initial preset crack tip slip displacement ; Load data were collected throughout the loading process using a servo-controlled testing machine. P and displacement of loading point δ Data, drawing P - δ curve; P - δ The area under the curve represents the total external work done, a portion of which is used for the propagation of interface cracks, i.e., the target computational amount. Part of it is used as elastic strain energy Stored in the specimen; Equal to a certain moment P - δ Area under the curve minus elastic strain energy The calculation formula is shown in equation (1): （1） In the formula, P For external loads, δ For the displacement of the loading point, It is the elastic strain energy. This represents the integral variable, which is a dummy variable representing the displacement of the loading point.

3. The method for calculating the constitutive relation of rock-concrete interface ratio-related shear softening according to claim 2, characterized in that, Step (2) is specifically as follows: The result is obtained by integrating the shear cohesion within the fracture process region. The shear cohesion related to the rate is denoted as , τ Represents shear cohesive stress, Represents slip displacement. That is, d s / d t Represents slip rate, rate-dependent shear cohesion and and The relationship is called the rate-dependent shear softening constitutive expression; the integral of shear cohesion over the slip displacement is the energy dissipated per unit area. e Its expression is shown in equation (2): （2） Discretize the total crack propagation as N Step, in the i Set the sliding rate within the step As a constant, the local stress-slip relationship is denoted as When the distance from the walk is not greater than the final amount of crack expansion. 5%, that is, the number of walks N When the value is not less than 20, the slip rate is considered to be equal on each micro-segment, i.e. τ - s The curves are the same; the energy dissipation per unit area on each micro-segment is proportional to the specimen thickness. b Integral and N The total energy consumed by the propagation of the interface cracks can be obtained by summing the steps. Its expression is shown in equation (3): （3） in, This represents the slip value at step j; To transform the spatial integral along the crack length into an integral over the slip displacement, we assume that the slip displacement is linearly distributed along the crack propagation direction, i.e., the distance from the actual crack tip is... y slip displacement at point satisfy Based on this, we obtain d y With d s The differential relationship is shown in equation (4): （4） In the formula, a For the crack propagation length, b For the specimen thickness, For the final amount of crack propagation, This is a preset displacement for the crack tip sliding.

4. The method for calculating the constitutive relation of rock-concrete interface ratio-related shear softening according to claim 3, characterized in that, Step (3) is specifically as follows: Substitute equation (4) into equation (3) and change the integration variable from y Change to s Multiply both sides of the equation Then, denote the dimensionless quantity on the left side of the equation as By the law of conservation of energy Therefore From actual measurement P - δ The curve is calculated as shown in equation (5): （5） right about Find the second derivative at the same normalized position. The stress at the point has a weight. The contribution signs of adjacent expansion steps at the same position are opposite, as shown in equation (6); Assume that the shear cohesive stress at the interface of each micro-segment is continuously distributed under any equilibrium state, i.e., the first... N - i +1 step and the N - i The stress at the same position is equal, as shown in equation (7); Substituting equation (7) into equation (6), adjacent terms at the same position cancel each other out, leaving only the first term, resulting in equation (8), which states that the rate-dependent shear cohesion stress is derived from... The second derivative is directly determined, while Based solely on actual measurements P - δ Curve calculation: （6） （7） （8） With preset crack tip slip displacement The x-axis is determined by the measured values ​​according to formula (5). P - δ Dimensionless quantities obtained from curve calculations Plot the measured values ​​as the ordinate. - s pct curve; For actual measurement - s pct For curve fitting, the fitting function must satisfy the following three constraints: ① coefficient of determination R 2 Greater than 0.9 to ensure consistency with the experimental curve; ② s =0 =0;③ right s First derivative characterization - s pct The fracture energy corresponding to the area under the curve; the fitting function determined accordingly is shown in equation (9), and its fitting parameters are: , : （9） Regarding equation (9) s Taking the second derivative yields the quasi-static shear softening constitutive model. As shown in equation (10): （10） Introducing a rate amplification factor into shear stress Introducing a correction factor in critical slip Based on this, a rate-related shear softening constitutive expression is established, as shown in equation (11); and The expressions are shown in Equations (12) and (13), respectively; the rate parameter is determined by fitting the test data under different concrete strengths and loading rates. , ; The reference slip rate is within the quasi-static slip rate range; when hour, and Both degenerate to 1, and equation (11) degenerates into the quasi-static shear softening constitutive form of equation (10): （11） （12） （13） Equation (11) only requires quasi-static parameters and It can be applied immediately; these two parameters can be obtained from the mean shear stress. estimate, Equals peak load divided by bonded area; As given by equation (14), Quasi-static critical slip The relationship is given by equation (15): （14） （15） In the formula, For the quasi-static maximum shear stress, For quasi-static critical slip displacement, , The relational function is obtained by fitting quasi-static experimental data. , The rate parameter is determined by fitting experimental data.

5. The method according to claim 2, characterized in that, The pre-set crack in step (1) is formed by attaching a polyvinyl chloride film to one end of the bonding surface near the loading point. The film is 50 mm long. A grid-like groove is opened on the bonding surface of the rock block to control the interface roughness. The groove is 5 mm wide and 2 mm deep.

6. The method according to claim 2, characterized in that, In step (1), multiple pairs of measuring points are arranged along the interface height in front of the preset crack tip. The slip displacement of each measuring point is obtained by DIC and fitted into a linear distribution along the interface height. The slip displacement is then adjusted to reach the initial preset crack tip slip displacement. s The location of ini pct is taken as the crack tip, and the distance from this location to the preset crack tip is the crack propagation length. a .

7. The method according to claim 4, characterized in that, In step (2), the total crack propagation is discretized at equal step sizes as follows: N Steps, distance from walking number N Not less than 20; if N If the relative change in shear cohesive stress calculated by equation (8) after doubling exceeds 5%, then increase... N Until the change does not exceed 5%.

8. The method according to claim 4, characterized in that, In step (3), the pre-set crack tip slip displacement is implemented. Data is continuously collected by DIC at a fixed sampling frequency, with a preset crack tip slip rate. Depend on The reference slip rate is obtained by differentiating it with respect to time. The value is 0.001 mm / s, which is within the range of quasi-static slip rate.

9. The method according to claim 4, characterized in that, The rate-dependent shear softening constitutive model is based solely on the quasi-static mean shear stress. As a parameter, it encompasses the combined effects of rock type, concrete strength, and interface roughness, and is applicable to quasi-static mean shear stress. Rock-concrete interface in the range of 0.86 to 1.87 MPa.

10. The method according to claim 4, characterized in that, The rate-dependent shear softening constitutive model was embedded into finite element software via a user subroutine to numerically simulate the propagation of interface cracks; and the constitutive model was integrated along the slip displacement from 0 to the critical slip. s 0, Derivation rate-related Type II fracture energy The expression enables the direct calculation of fracture energy at different slip rates, as shown in equation (16): （16）。

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