Construction method of mesoscopic random aggregate model of recycled concrete based on convex polygon

Abstract

The present invention discloses a method for constructing a convex polygon-based mesoscopic random aggregate model for recycled concrete, comprising the following steps: 1. determining basic parameters; 2. calculating the theoretically calculated placement area of aggregate for each particle size segment; 3. randomly generating different ellipses; 4. sequentially connecting the vertices of N randomly generated convex polygons on the ellipses to generate a convex polygonal model of common aggregate; 5. randomly generating a convex polygonal model of recycled aggregate composed of mortar; and 6. randomly placing common aggregate and recycled aggregate. The resulting mesoscopic model is closer to the mesoscopic structure of an actual recycled concrete cross-section, enabling simulation of the convex polygonal recycled concrete random aggregate model, more closely meeting the requirements of actual simulation test models.

Description

Technical Field

[0001] The present invention belongs to the technical field of material simulation, and in particular relates to a two-dimensional five-phase microscopic modeling method of recycled concrete based on a convex polygonal random aggregate model. Background Art

[0002] With the rapid development of my country's economy, the country's infrastructure consumption is enormous, depleting natural resources such as sand and gravel. At the same time, the demolition of old buildings generates an increasing amount of waste concrete, and the current method of landfilling has a significant impact on the environment. Waste concrete can be processed into recycled coarse aggregate, which can be used to replace natural aggregate in the production of recycled concrete. This not only addresses the shortage of natural aggregate resources but also enables the recycling of waste concrete.

[0003] In recent years, many scholars have conducted extensive research on the various properties of recycled concrete. For example, by changing the formulation of recycled concrete, they studied the effects of factors such as the replacement rate of recycled aggregate and the mortar adhesion rate on the mechanical properties and durability of recycled concrete. By changing the strength of the attached mortar aggregate, their influence on the basic mechanical properties of recycled concrete was studied. Chen Long used the sand ratio as a variable to prepare recycled coarse aggregates with different mortar adhesion rates in the laboratory. The study found that the greater the mortar adhesion rate of the recycled coarse aggregate, the greater the drying shrinkage of the recycled concrete and the lower its compressive strength. Lan Yue's study found that the lower the strength of the attached aggregate, the lower the compressive strength of the recycled concrete and the greater the drying shrinkage. However, most of these studies are based on macroscopic experimental research, ignoring the differences in the elastic modulus and strength properties of the various phase components.

[0004] Domestic researchers have conducted extensive research on the compressive and tensile mechanical properties of recycled concrete, focusing on factors such as water-cement ratio and recycled aggregate replacement rate. Feng Shuai et al. used microwave heating to control the mortar coverage of recycled aggregate to study its effect on the durability and mechanical properties of recycled concrete. Hu Xin, Cui Zhenglong et al. used hydrochloric acid titration to measure the mortar adhesion of different recycled aggregates to investigate their impact on the properties of recycled concrete. These studies are largely based on macroscopic experiments, ignoring the differences in elastic modulus and strength properties between the various phases of the materials.

[0005] Since recycled concrete has more significant heterogeneity and complex internal structure than ordinary concrete, it is necessary to study the influence of each phase composition on its performance based on the complex microstructure of recycled concrete materials, which can provide a basis for further improving the performance of recycled concrete.

[0006] With the development of computer information technology, establishing a computer model of the microstructure of recycled concrete has become an important research hotspot. Since the random aggregate model can more accurately reflect the failure mechanism of concrete, the establishment of a random aggregate model of recycled concrete has attracted the interest of more and more scholars. Jiang Baoku et al. established a random aggregate model of recycled concrete based on a circle, studied the tensile mechanical properties of recycled concrete, and simulated the two-dimensional diffusion process of chloride ions in recycled concrete; Xu Mengchan established an elliptical two-dimensional recycled concrete microstructure and studied the influence of the micro-component interface thickness on the mechanical properties of recycled concrete; CN114239345A constructed a recycled concrete micromodel based on concentric recycled aggregates; CN112464523A constructed a recycled concrete micromodel based on elliptical recycled aggregates. However, whether it is an ellipse or a circle, these models are too idealized and there are large differences with the actual aggregate shape. CN113591195A constructs a recycled concrete micromodel with concentric convex polygons, but this method simply regards the attached mortar as a layer wrapped in the outer circle of the convex polygon, which is also different from the actual recycled concrete.

[0007] To date, there is no method to construct a more realistic mesoscopic random aggregate model of recycled concrete. Summary of the Invention

[0008] In order to address the shortcomings of the above-mentioned prior art, the present invention proposes a method for constructing a mesoscopic random aggregate model of recycled concrete based on convex polygons, in order to construct a more realistic recycled concrete model, thereby facilitating the promotion and efficient application of recycled concrete.

[0009] In order to achieve the above-mentioned object, the present invention adopts the following technical solutions:

[0010] The method for constructing a convex polygon-based recycled concrete mesoscopic random aggregate model of the present invention is characterized in that it comprises the following steps:

[0011] Step 1: Obtain basic parameters, including: rectangular cross-sectional dimensions of recycled concrete specimens, i.e., width W and height H of specimen cross-sectional dimensions, aggregate volume ratio P k , hole size [D min ,D2…D i-1 ,D i ,D i+1 … D max ], recycled aggregate replacement rate R, mortar adhesion content level; among them, D min Indicates the minimum aggregate particle size, D max Indicates the maximum particle size of aggregate; D i represents the size of the i-th hole sieve;

[0012] Step 2: Calculate the theoretical calculation area of aggregates in each size range according to the Walraven formula of the aggregate size curve in the two-dimensional plane of the Fuller size curve. S i | i=2,3,…,max}, where S i Indicates that the particle size range is (D i-1 , D i ) Theoretical calculation placement area of aggregate;

[0013] Step 3: With the lower left corner of the rectangular cross section of the specimen as the coordinate origin, the horizontal direction of the rectangular cross section as the x-axis, and the vertical direction of the rectangular cross section as the y-axis, a plane rectangular coordinate system is established, and a plane rectangular domain with a width of W and a height of H is generated;

[0014] Step 4: Initialize i=max;

[0015] Step 5: Generate a random number r∈(0,1) based on the Monte Carlo method. If r≤R, the generated particle size range is (D i-1 ,D i ) of the ordinary aggregate convex polygon model, if > R, then the generated particle size range is (D i-1 , D i ) of the recycled aggregate convex polygonal model; where R represents the recycled aggregate replacement rate;

[0016] Step 6: Determine the particle size range (D i-1 , D i ) to determine whether the aggregate convex polygon model generated by the method satisfies both the boundary conditions and the aggregate interference conditions. If both conditions are satisfied, the particle size range is set within (D i-1 , D i ) is put into the plane rectangular domain, and the particle size range is calculated in (D i-1 , D i ) the area a of the sth aggregate placement is , s=1,2,…,L; otherwise, return to step 5; until the particle size range is (D i-1 , D i ) of the L aggregate placement areas Until formula (1) is satisfied;

[0017]

[0018] In formula (1), Represents parameters between (0,1);

[0019] Step 7: After assigning i-1 to i, return to step 4 and execute sequentially until i<2, thereby obtaining the cumulative placement area of aggregates in each particle size range { A i | i=2,3,…,max};

[0020] Step 8: Calculate the total aggregate placement area ratio S A and the proportion of attached old mortar area S M , which is used to construct a mesoscopic random aggregate model of recycled concrete.

[0021] The method for constructing a convex polygon-based recycled concrete mesoscopic random aggregate model according to the present invention is also characterized in that the boundary function of the plane rectangular domain in step 3 is obtained using formula (2):

[0022]

[0023] In formula (2), 、 、 、 are the independent variables of the four boundary line equations.

[0024] The step of generating a common aggregate convex polygon model in step 5 includes:

[0025] Step 5.1a: Generate an ellipse E randomly, and let the major axis a∈ 0.5× (D i-1 ,D i+1 );

[0026] Step 5.2a: Randomly generate N angles {φ j | j=1,2,…,N}; Starting from the center of the ellipse E, generate N rays at N angles, and obtain N points after the N rays intersect with the ellipse E respectively; where φ j represents the angle between the jth ray and the positive direction of the x-axis of the plane rectangular coordinate system;

[0027] Step 5.3a: Sort the N angles in ascending order and calculate the difference between two adjacent angles. If any difference is less than the set angle, , then return to step 5.2a and execute sequentially; otherwise, execute step 5.4a;

[0028] Step 5.4a: Based on the N angles and the general parametric equation of the ellipse, calculate the coordinates of the N points on the ellipse E and connect the vertices in sequence to obtain a common aggregate convex polygon model.

[0029] The process of generating the recycled aggregate convex polygon model in step 5 includes:

[0030] On the basis of generating the ordinary aggregate convex polygon model, N1 consecutive vertices are randomly selected from the N vertices of the ordinary aggregate convex polygon model according to the level, and 2 points are randomly selected inside the ordinary aggregate convex polygon model, so as to sequentially connect the N1+2 points to obtain the attached mortar polygon model. The polygon part formed by the attached mortar polygon model and the remaining N-N1 vertices together constitute the recycled aggregate convex polygon model.

[0031] The method for determining whether the convex polygon model meets the boundary conditions in step 6 is as follows:

[0032] If the general parametric equations of the circumscribed ellipse of the convex polygonal aggregate model and the four boundary line equations in formula (2) all have no solutions, it means that the convex polygonal model meets the boundary conditions of the delivery area; otherwise, it means that the convex polygonal model does not meet the boundary conditions.

[0033] The particle size range in step 6 is (D i-1 , D i ) satisfies the aggregate interference condition as follows:

[0034] Step 6.1: Determine the particle size range (D i-1 , D i ) is inside any of the placed aggregate convex polygon models. If so, it means that the aggregate interference condition is not met. Otherwise, execute step 6.2;

[0035] Step 6.2: Calculate the particle size range (D i-1 , D i ) and calculate the generalized characteristic polynomial of the quadratic matrix A of the circumscribed ellipse of the aggregate convex polygon model, and the quadratic matrix B of the circumscribed ellipse of any one of the placed aggregate convex polygon models; determine whether the generalized characteristic polynomial has two different positive real roots; if so, it means that the circumscribed ellipses corresponding to the quadratic matrices A and B are separated, and execute step 6.3; otherwise, execute step 6.4;

[0036] Step 6.3: Continue to determine whether the quadratic matrix A and the circumscribed ellipses of other aggregate convex polygonal models are separated. Only when they are separated, it means that the particle size range is within (D i-1 , D i )’s aggregate convex polygon model satisfies the aggregate interference condition and exits the judgment process;

[0037] Step 6.4: Determine the particle size range in sequence (D i-1 , D i) are all the vertices of the aggregate convex polygon model inside all the aggregate convex polygon models that have been put in. If so, it means that the particle size range is (D i-1 , D i ) does not satisfy the aggregate interference condition, otherwise, execute step 6.5;

[0038] Step 6.5: Determine the particle size range (D i-1 , D i ) whether any line segment between all vertices of the aggregate convex polygon model intersects with any line segment between the vertices of all the placed aggregate convex polygon models; if so, it means that the aggregate interference condition is not met; otherwise, it means that the aggregate interference condition is met.

[0039] Compared with the prior art, the present invention has the following beneficial effects:

[0040] 1. Compared with the existing recycled concrete model, the present invention randomly generates an attached old mortar model around the ordinary aggregate model, so that the recycled concrete model based on convex polygons is more consistent with the microstructure of recycled concrete in real conditions, the shape and position of the attached mortar are also more realistic, and the simulation results are more accurate.

[0041] 2. Compared with the existing recycled concrete microscopic model, the present invention can control the replacement rate of recycled aggregate and the attachment content of old mortar by changing the input parameters, and can be used to simulate and study the effects of the mortar attachment amount and replacement rate of recycled aggregate on various properties of recycled concrete.

[0042] 3. Compared with the existing recycled concrete microscopic model, the boundary condition judgment and aggregate interference judgment in the present invention are more rigorous and accurate, and the aggregate filling rate is improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 is a flow chart of the method of the present invention;

[0044] Figure 2 Schematic diagram of the recycled aggregate model of the present invention;

[0045] Figure 3 This is the boundary condition determination diagram of the present invention;

[0046] Figure 4 This is a diagram showing the intersection of convex polygons according to the present invention;

[0047] Figure 5a This is a judgment diagram of the line segment intersection method of the present invention;

[0048] Figure 5b This is a schematic diagram of the intersection of line segments of the present invention;

[0049] Figure 6aThis is a microscopic random aggregate model diagram of recycled concrete with a replacement rate of 30% according to the present invention;

[0050] Figure 6b This is a microscopic random aggregate model diagram of recycled concrete with a replacement rate of 50% according to the present invention;

[0051] Figure 6c This is a microscopic random aggregate model diagram of recycled concrete with a replacement rate of 100% according to the present invention. DETAILED DESCRIPTION

[0052] In this embodiment, Figure 1 As shown in the figure, a method for constructing a recycled concrete mesoscopic random aggregate model based on convex polygons is to simultaneously generate a polygonal old mortar model based on the convex polygonal random aggregate, so that it is randomly attached. By controlling the number of polygonal vertices, the old mortar content can be controlled, which is more consistent with the real mesoscopic structure of recycled concrete. Specifically, the following steps are included:

[0053] Step 1: Obtain basic parameters, including: rectangular cross-sectional dimensions of recycled concrete specimens, i.e., width W and height H of specimen cross-sectional dimensions, aggregate volume ratio P k (Aggregate volume as a percentage of the total volume of recycled concrete), hole size [D min ,D2…D i-1 ,D i ,D i+1 … D max ], recycled aggregate replacement rate R∈(0,1), mortar adhesion content level; where D min Indicates the minimum aggregate particle size, D max Indicates the maximum particle size of aggregate; D i represents the size of the i-th hole sieve;

[0054] Step 2: According to the aggregate size curve of the Fuller size curve in the two-dimensional plane, that is, the Walraven formula shown in formula (1) to determine the area proportion of aggregate of each size, and calculate the theoretical calculation area of aggregate of each size range according to formula (2) { S i | i=2,3,…,max}, where S i Indicates that the particle size range is (D i-1 , D i ) Theoretical calculation placement area of aggregate;

[0055] (1)

[0056] (2)

[0057] In formulas (1) and (2), P(D < D0) is the probability that the aggregate particle size D within the cross-section is less than the sieve hole D0, and P k is the percentage of the aggregate volume in the total volume of recycled concrete, D max is the maximum aggregate particle size, S i represents the theoretically calculated aggregate placement area within the particle size range (D i-1 , D i ), P i (D < D i ) is the probability that the aggregate particle size D within the cross-section is less than the sieve hole D i range, W is the cross-sectional width of the specimen, and H is the cross-sectional height of the specimen.

[0058] Step 3: Taking the lower left corner point of the rectangular cross-section of the specimen as the coordinate origin, the horizontal direction of the rectangular cross-section as the x-axis, and the vertical direction of the rectangular cross-section as the y-axis, establish a plane rectangular coordinate system, and generate a plane rectangular domain with a width of W and a height of H. The boundary function of the rectangle is formula (3);

[0059] (3)

[0060] In formula (3), , , , are the independent variables of the four boundary straight-line equations.

[0061] Step 4: Initialize i = max, and place the aggregates in sequence from the largest particle size to the smallest;

[0062] Step 5: Randomly generate a number r ∈ (0, 1) based on the Monte Carlo method. If r ≤ the recycled aggregate replacement rate R, then generate a convex polygon model of normal aggregates within the particle size range (D i-1 , D i ), which is divided into the following steps:

[0063] Step 5.1a: Randomly generate an ellipse E, and let the long semi-axis a of the ellipse E ∈ 0.5×(D i-1 , D i+1 ), randomly generate the ratio of the short semi-axis b to the long semi-axis a (a > b), calculate the value of the short semi-axis b as a, randomly generate the ellipse center coordinates (m, n) that satisfy formula (4), randomly generate the angle α between the long axis direction of the ellipse and the positive direction of the x-axis of the plane rectangular coordinate system within (0, ), and generate random positions based on the general parametric equation of the ellipse in formula (5) for an ellipse with any inclination angle;

[0064] (4)

[0065] In formula (4), b is the minor axis of the ellipse E, W is the width of the specimen section, H is the height of the specimen section, m and n are the horizontal and vertical coordinates of the center of the ellipse E, respectively.

[0066] The ellipse E generated by the above formula intersects with the boundary in only one case: Figure 3 shown.

[0067] (5)

[0068] In formula (5), x and y are the horizontal and vertical coordinates of any point on the ellipse E, a and b are the major and minor axes of the ellipse E, α is the angle between the major axis of the ellipse E and the positive direction of x, and m and n are the horizontal and vertical coordinates of the center of the ellipse E, respectively.

[0069] Step 5.2a: Randomly generate N angles {φ j | j=1,2,…,N}, N∈(7,11); Starting from the center of ellipse E, generate N rays at N angles, and obtain N points after the N rays intersect with ellipse E respectively; where φ j represents the angle between the jth ray and the positive direction of the x-axis of the plane rectangular coordinate system;

[0070] Step 5.3a: Sort the N angles in ascending order and calculate the difference between two adjacent angles. If any difference is less than the set angle, , then return to step 5.2a and execute it in sequence, re-randomly generate N angles to avoid the occurrence of long, thin and deformed aggregates with greatly different side lengths; otherwise, execute step 5.4a;

[0071] Step 5.4a: Based on the N angles and the general parametric equation of the ellipse (5), calculate the coordinates of the N points on the ellipse E, and connect the vertices in sequence to obtain the general aggregate convex polygon model.

[0072] If r>recycled aggregate replacement rate R, the generated particle size range is (D i-1 , D i The specific steps of the recycled aggregate convex polygon model are as follows: on the basis of generating the ordinary aggregate convex polygon model, randomly select N1 consecutive vertices from the N vertices of the ordinary aggregate convex polygon model according to the level as part of the vertices of the old mortar polygon model. Figure 2As shown in the figure, when the level is 'small', N1 is 3; when the level is 'median', N1 is 4; and when the level is 'big', N1 is 5. As N1 increases, the mortar adhesion content increases. Two points are randomly selected from the interior of the ordinary aggregate convex polygonal model. The N1 + 2 points are sequentially connected to form the adhesion mortar polygonal model. The recycled aggregate convex polygonal model is composed of the adhesion mortar polygonal model and the polygonal portion formed by connecting the remaining N - N1 vertices.

[0073] Step 6: Determine the particle size range (D i-1 , D i ) to determine whether the aggregate convex polygon model generated by the method satisfies both the boundary conditions and the aggregate interference conditions. If both conditions are satisfied, the particle size range is set within (D i-1 , D i ) is put into the plane rectangular domain, and the particle size range is calculated in (D i-1 , D i ) the area a of the sth aggregate placement is , s=1,2,…,L (when the aggregate model is a recycled aggregate convex polygon model, calculate the area F of the attached mortar polygon model in the kth recycled aggregate convex polygon model k , k=1,2,…,L1); otherwise, return to step 5; when the particle size (D i-1 , D i ) Cumulative aggregate placement area A i Smaller than the theoretical calculation area S of this particle size i When the aggregate is 100%, return to step 5 and continue to add the aggregate of the particle size segment; A i Greater than S i When the last aggregate is abandoned, a smaller aggregate is generated in the same particle size range and the particles are continuously added until the particle size range is (D i-1 , D i ) of the L aggregate placement areas Until formula (6) is satisfied;

[0074] (6)

[0075] In formula (6), Represents a parameter between (0,1); in order to make the cumulative area of each particle size A i As close as possible to the theoretical calculation area S of the particle size i , 0.2 can be taken.

[0076] The aggregate convex polygon model first determines the boundary conditions: if the general parameter equation (5) of the circumscribed ellipse of the convex polygon aggregate model and the four boundary line equations in equation (3) all have no solution, it means that the convex polygon model meets the boundary conditions of the placement area; otherwise, it means that the convex polygon model does not meet the boundary conditions. The aggregate convex polygon model that meets the boundary conditions further determines the aggregate interference conditions and determines the particle size range in (D i-1 , D i ) satisfies the aggregate interference condition:

[0077] Step 6.1: Determine the particle size range (D i-1 , D i ) is inside any of the placed aggregate convex polygon models. If so, it means that the aggregate interference condition is not met. Otherwise, execute step 6.2;

[0078] Step 6.2: To improve the efficiency of aggregate interference judgment, first use the interference judgment between ellipses to determine whether the convex polygons inside them interfere. If the circumscribed ellipse of the convex polygon does not interfere with the circumscribed ellipse of the placed convex polygon, then the convex polygon must not interfere with other convex polygons and can be directly placed in the plane rectangular domain generated in step 3. Otherwise, further strict interference judgment of convex polygons is required. Interference judgment between ellipses is based on algebraic conditions: if the generalized eigenvalues of the quadratic form matrices of two ellipses A and B (generalized characteristic equation f( )=0) has two different positive real roots, then the ellipses are separated, otherwise they are not separated. The particle size range is calculated by formula (7) in (D i-1 , D i ) and calculate the generalized characteristic polynomial of the quadratic matrix A of the circumscribed ellipse of the aggregate convex polygon model, and calculate the generalized characteristic polynomial of the quadratic matrix B of the circumscribed ellipse of any of the aggregate convex polygon models that have been placed; determine whether the generalized characteristic polynomial (Formula (8)) has two different positive real roots. If so, it means that the circumscribed ellipses corresponding to the quadratic matrices A and B are separated, and execute step 6.3; otherwise, it can be determined that the external ellipse of the aggregate convex polygon model interferes with the circumscribed ellipse of the aggregate convex polygon model that has been placed, and further convex polygon interference judgment is required, and execute step 6.4;

[0079] (7)

[0080] In formula (8), A represents the quadratic matrix of the circumscribed ellipse. are the horizontal and vertical coordinates of the center of the ellipse, a and b are the major and minor axes of the ellipse respectively, and α is the angle between the major axis of the ellipse and the positive x direction.

[0081] (8)

[0082] In formula (3), A and B are the quadratic matrices of any two circumscribed ellipses, f( ) are the generalized characteristic polynomials of ellipses A and B.

[0083] Step 6.3: Continue to determine whether the quadratic matrix A and the circumscribed ellipses of other aggregate convex polygonal models are separated. Only when they are separated, it means that the particle size range is within (D i-1 , D i )’s aggregate convex polygon model satisfies the aggregate interference condition and exits the judgment process;

[0084] Step 6.4: The intersection of convex polygons includes three cases: Figure 4 As shown, the particle size range is determined in sequence (D i-1 ,D i ) are all the vertices of the aggregate convex polygon model inside all the aggregate convex polygon models that have been put in. If so, it means that the particle size range is (D i-1 , D i ) does not meet the aggregate interference condition. Otherwise, it can be determined that the aggregate convex polygon model does not have the interference type shown in Cases 1 and 2. Further determine whether it is Case 3 where the points do not contain intersections, and execute step 6.5;

[0085] Step 6.5: Determine the particle size range (D i-1 , D i ) whether any line segment between all vertices of the aggregate convex polygon model intersects with any line segment between all vertices of the aggregate convex polygon model that has been put in. If so, it means that the aggregate interference condition is not met. Otherwise, it means that the aggregate interference condition is met. Further methods for judging whether two line segments intersect (such as Figure 5a As shown in the figure, first perform a "quick repulsion experiment": use the coordinates of the endpoints of the two line segments to determine whether the rectangles with these two line segments as diagonals intersect. If they do not intersect, then it is obvious that the two line segments do not intersect. If they do intersect, proceed to the next step of the "straddling experiment": if the two line segments AB and CD intersect, then two conditions must be met:

[0086] (1) Points C and D are on either side of line segment AB;

[0087] (2) Points A and B are on both sides of line segment CD;

[0088] like Figure 5bAs shown, based on the knowledge of vector cross products and vector products, if c < 0 in equation (9), then points C and D are on either side of line segment AB. Similarly, using equation (10), we can calculate that f < 0, then points A and B are on either side of line segment CD. When c < 0 and f < 0, line segments AB and CD intersect; otherwise, they do not intersect.

[0089] (9)

[0090] (10)

[0091] Step 7: After assigning i-1 to i, return to step 4 and execute in sequence, placing aggregate models of each size in the order of large to small, until i<2, and calculate the cumulative placement area of aggregates in each size range { A i | i=2,3,…,max};

[0092] Step 8: Calculate the total aggregate placement area ratio S according to formula (11) A and the proportion of attached old mortar area S M , which is used to construct a mesoscopic random aggregate model of recycled concrete.

[0093] (11)

[0094] In formula (11), S A and S M are the total aggregate area ratio and the area ratio of attached old mortar, A i (D i-1 ,D i ) is the actual area of aggregate placed in the particle size segment, max is the number of holes in the sieve, W and H are the width and height of the specimen section, L1 is the number of particles of recycled aggregate placed, F k is the area of the attached mortar polygonal model in the kth recycled aggregate convex polygonal model.

[0095] Example: The following steps are used to construct a stochastic mesoscopic model of recycled concrete based on convex polygons:

[0096] Step 1: Determine basic parameters:

[0097] The cross-sectional width of the recycled concrete specimen is W = 100 mm, the height is H = 100 mm, and the percentage of aggregate volume to the total volume of recycled concrete is P k =0.75, minimum aggregate size D min =4.75 mm, maximum aggregate size D max=25 mm, the sieve size is 4.75 mm, 9.5 mm, 16 mm, 19 mm, 25 mm, and the replacement rate of recycled aggregate is 30%, 50%, and 100%, respectively, that is, R=0.3, 0.5, and 1;

[0098] Step 2: Calculate the area proportion of each size aggregate in the aggregate placement area:

[0099] According to the Walraven formula, the area of aggregate placement with a particle size of 19 to 25 mm is calculated to be 674 mm. 2 The aggregate with a particle size of 16 to 19 mm is placed on an area of 494 mm. 2 The aggregate with a particle size of 9.5 to 16 mm is placed on an area of 1401 mm. 2 The aggregate with a particle size of 4.75 to 9.5 mm is placed on an area of 1434 mm. 2 , the total aggregate input area accounts for 40%.

[0100] Step 3: With the lower left corner of the rectangular cross section of the specimen as the coordinate origin, the horizontal direction of the rectangular cross section as the x-axis, and the vertical direction of the rectangular cross section as the y-axis, a plane rectangular coordinate system is established, and a plane rectangular domain with a width of W and a height of H is generated;

[0101] Step 4: Initialize i=5;

[0102] Step 5: Randomly generate a number r=rand(1) in the range of (0,1). When r≤R, generate a convex polygon model of ordinary aggregate with a particle size range of (D4, D5): randomly generate the number of vertices N=6+randi(3,1) of the convex polygon model of ordinary aggregate, and randomly generate N angles {φ j | j = 1, 2, …, N}, =rand(1)×2 , use Matlab built-in function sort function to sort φ j Sort in ascending order and calculate the difference between two adjacent angles in turn. If it is less than the set angle, = / 6, then randomly generate N angles φ j According to the ellipse parametric equation, calculate the coordinates of N vertices and use the built-in function fill in Matlab to connect the N vertices in sequence and fill them with colors to obtain a convex polygon model of ordinary aggregate; if r>R, the generated particle size range is (D i-1 , D i) convex polygon model of recycled aggregate: Based on the generated convex polygon model of ordinary aggregate, determine the number of vertices N1 of the old mortar polygon: when level = 's', N1 = 3; when level = 'm', N1 = 4; when level = 'b', N1 = 5; and randomly select 2 points inside the convex polygon, use the fill function to sequentially connect (N1 + 2) points and fill them with color to obtain a random polygon model of recycled aggregate with old mortar attached;

[0103] Step 6: First determine the particle size range (D i-1 , D i ) Whether the aggregate convex polygon model generated by the ellipse satisfies the boundary conditions and aggregate interference conditions: Use the Matlab built-in function solve to determine whether the equation (4) for the parameters of any position of the ellipse and the equations (5) for the four boundary lines have solutions. If there are no solutions, it means that the convex polygon model meets the boundary conditions of the placement area and the next step of aggregate interference judgment is performed. Otherwise, return to step 5. The aggregate convex polygon model that meets the boundary conditions is further subjected to the determination of the aggregate interference conditions: Use the Matlab built-in function inpolygon to determine whether the center of the circumscribed ellipse of the newly generated convex polygon aggregate is inside any of the placed aggregate convex polygon models. If so, it means that the aggregate interference conditions are not met and return to step 5 to regenerate the aggregate convex polygon model. Otherwise, proceed to the next step of interference judgment between ellipses and determine the positional relationship between the circumscribed ellipse of the newly generated convex polygon aggregate and all the generated ellipses in turn. Use the built-in function eig(A,B) of Matlab to find the generalized eigenvalues of the quadratic form A and B of any two ellipses. If there are two different negative real roots, it can be determined that ellipse A and ellipse B are separated (when the built-in function eig(A,B) of Matlab is used to find the generalized eigenvalues, it is defined as f( )=det( AB) is a generalized characteristic polynomial. If the circumscribed ellipse of the newly generated convex polygon is completely separated from the circumscribed ellipse of the already generated convex polygon, it is placed directly; otherwise, the next step is determined; Matlab's built-in function inpolygon is used to determine whether all vertices of the newly generated convex polygon are outside all the already placed aggregate convex polygon models. If not, return to step 5. If all vertices are not inside, the next step is determined; Matlab's cross function and dot function are used to calculate the cross product and dot product between vectors, respectively, to determine whether all edges, i.e., line segments, of the newly generated convex polygon intersect with all edges of the already generated convex polygon. If all line segments do not intersect, placement is successful. Otherwise, return to step 5. For successfully placed aggregate convex polygon models, record their aggregate area. If they are recycled aggregate models, also record the area of their attached mortar polygons.

[0104] Step 7: After assigning i-1 to i, return to step 4 and execute sequentially until i<2, thereby obtaining the cumulative placement area of each particle size range; when the cumulative placement area of aggregate of a certain particle size is less than the theoretically calculated area of the particle size, continue to place aggregate of this particle size segment; when the cumulative placement area is greater than the theoretically calculated area, abandon the placement of the last aggregate and generate a smaller aggregate in the particle size segment and continue to place it until the cumulative placement area of the current particle size aggregate reaches 0.98~1.02 times the theoretically calculated area of the particle size aggregate, then place the next particle size aggregate until all particle size segments are placed;

[0105] Step 8: Calculate the total aggregate placement area ratio S A and the proportion of attached old mortar area S M .

[0106] Figure 6a The random aggregate model diagram of recycled concrete at the replacement rate of 30% and the levels of 's', 'm', and 'b' is shown. The S A is 41.25%, S M is 2.54%; when level is 'm', S A is 42.39%, S M is 8.29%; when level is 'b', S A is 41.85%, S M It is 12.65%. Figure 6b The random aggregate model diagram of recycled concrete at the level of 's', 'm' and 'b' when the replacement rate is 50% is obtained. A is 40.98%, S M is 4.20%; when level is 'm', S A is 41.33%, S M is 9.99%; when level is 'b', S A is 41.76%, S M It is 32.78%. Figure 6c The random aggregate model diagram of recycled concrete at the level of 's', 'm' and 'b' is shown in Figure 2. The S is calculated when the level is 's'. A is 41.69%, S M is 9.32%; when level is 'm', S A is 41.45%, S M is 25.02%; when level is 'b', S A is 42.01%, S M It is 55.04%.

Claims

1. A method for constructing a mesoscopic random aggregate model of recycled concrete based on convex polygons, characterized in that: The following steps are involved: Step 1: Obtain basic parameters, including: rectangular cross-sectional dimensions of recycled concrete specimens, i.e., width W and height H of specimen cross-sectional dimensions, aggregate volume ratio P k , hole size [D min ,D2…D i-1 ,D i ,D i+1 …D max ], recycled aggregate replacement rate R, mortar adhesion content level; among them, D min Indicates the minimum aggregate particle size, D max Indicates the maximum particle size of aggregate; D i represents the size of the i-th hole sieve; Step 2: Calculate the theoretical calculation area of aggregates in each size range according to the Walraven formula of the aggregate size curve in the two-dimensional plane of the Fuller size curve {S i |i=2,3,…,max}, where S i Indicates that the particle size range is (D i-1 ,D i ) Theoretical calculation placement area of aggregate; Step 3: With the lower left corner of the rectangular cross section of the specimen as the coordinate origin, the horizontal direction of the rectangular cross section as the x-axis, and the vertical direction of the rectangular cross section as the y-axis, a plane rectangular coordinate system is established, and a plane rectangular domain with a width of W and a height of H is generated; Step 4: Initialize i=max; Step 5: Generate a random number r∈(0,1) based on the Monte Carlo method. If r≤R, the generated particle size range is (D i-1 ,D i ) of the ordinary aggregate convex polygon model, if r>R, the generated particle size range is (D i-1 ,D i ) of the recycled aggregate convex polygonal model; where R represents the recycled aggregate replacement rate; Step 6: Determine the particle size range (D i-1 ,D i ) to determine whether the aggregate convex polygon model generated by the method satisfies both the boundary conditions and the aggregate interference conditions. If both conditions are satisfied, the particle size range is set within (D i-1 ,D i ) is put into the plane rectangular domain, and the particle size range is calculated in (D i-1 ,D i ) the area a of the sth aggregate placement is , s=1,2,…,L; otherwise, return to step 5; until the particle size range is (D i-1 ,D i ) of the cumulative placement area of L aggregates Until formula (1) is satisfied; A i ∈((1-σ)S i ,(1+σ)S i (1) In formula (1), σ represents the parameter between (0,1); Step 7: After assigning i-1 to i, return to step 4 and execute sequentially until i<2, thereby obtaining the cumulative placement area of aggregates in each particle size range {A i |i=2,3,…,max}; Step 8: Calculate the total aggregate placement area ratio S A and the proportion of attached old mortar area S M , which is used to construct a mesoscopic random aggregate model of recycled concrete.

2. The method for constructing a convex polygon-based recycled concrete mesoscopic random aggregate model according to claim 1, characterized in that: Use formula (2) to get the boundary function of the plane rectangular domain in step 3: In formula (2), x1, x2, y1, and y2 are the independent variables of the four boundary line equations.

3. The method for constructing a convex polygon-based recycled concrete mesoscopic random aggregate model according to claim 1, characterized in that: The step of generating a common aggregate convex polygon model in step 5 includes: Step 5.1a: Generate an ellipse E randomly, and let the major axis a∈0.5×(D i-1 ,D i+1 ); Step 5.2a: Generate N random angles Starting from the center of ellipse E, N rays are generated at N angles. N points are obtained after the N rays intersect with ellipse E. represents the angle between the jth ray and the positive direction of the x-axis of the plane rectangular coordinate system; Step 5.3a: Sort the N angles in ascending order and calculate the difference between two adjacent angles. If any difference is less than the set angle δ, return to step 5.2a and execute the sequence; otherwise, execute step 5.4a. Step 5.4a: Based on the N angles and the general parametric equation of the ellipse, calculate the coordinates of the N points on the ellipse E and connect the vertices in sequence to obtain a common aggregate convex polygon model.

4. The method for constructing a convex polygon-based recycled concrete mesoscopic random aggregate model according to claim 1, characterized in that: The process of generating the recycled aggregate convex polygon model in step 5 includes: On the basis of generating the ordinary aggregate convex polygon model, N1 consecutive vertices are randomly selected from the N vertices of the ordinary aggregate convex polygon model according to the level, and 2 points are randomly selected inside the ordinary aggregate convex polygon model, so as to obtain the attached mortar polygon model after connecting the N1+2 points in sequence. The polygon part formed by the attached mortar polygon model and the remaining N-N1 vertices together constitute the recycled aggregate convex polygon model.

5. The method for constructing a convex polygon-based recycled concrete mesoscopic random aggregate model according to claim 2, characterized in that: The method for determining whether the convex polygon model meets the boundary conditions in step 6 is as follows: If the general parametric equations of the circumscribed ellipse of the convex polygonal aggregate model and the four boundary line equations in formula (2) all have no solutions, it means that the convex polygonal model meets the boundary conditions of the delivery area; otherwise, it means that the convex polygonal model does not meet the boundary conditions.

6. The method for constructing a convex polygon-based recycled concrete mesoscopic random aggregate model according to claim 1, characterized in that: The particle size range in step 6 is (D i-1 ,D i ) satisfies the aggregate interference condition as follows: Step 6.1: Determine the particle size range (D i-1 ,D i ) is inside any of the placed aggregate convex polygon models. If so, it means that the aggregate interference condition is not met. Otherwise, execute step 6.2; Step 6.2: Calculate the particle size range (D i-1 ,D i ) and calculate the generalized characteristic polynomial of the quadratic matrix A of the circumscribed ellipse of the aggregate convex polygon model, and the quadratic matrix B of the circumscribed ellipse of any one of the placed aggregate convex polygon models; determine whether the generalized characteristic polynomial has two different positive real roots; if so, it means that the circumscribed ellipses corresponding to the quadratic matrices A and B are separated, and execute step 6.3; otherwise, execute step 6.4; Step 6.3: Continue to determine whether the quadratic matrix A and the circumscribed ellipses of other aggregate convex polygonal models are separated. Only when they are separated, it means that the particle size range is within (D i-1 ,D i )’s aggregate convex polygon model satisfies the aggregate interference condition and exits the judgment process; Step 6.4: Determine the particle size range in sequence (D i-1 ,D i ) are all the vertices of the aggregate convex polygon model inside all the aggregate convex polygon models that have been put in. If so, it means that the particle size range is (D i-1 ,D i ) does not satisfy the aggregate interference condition, otherwise, execute step 6.5; Step 6.5: Determine the particle size range (D i-1 ,D i ) whether any line segment between all vertices of the aggregate convex polygon model intersects with any line segment between the vertices of all the placed aggregate convex polygon models; if so, it means that the aggregate interference condition is not met; otherwise, it means that the aggregate interference condition is met.

Images