A numerical simulation method for flexible fiber bundles in a cement matrix

Abstract

The present invention relates to a numerical simulation method for flexible fiber bundles in a cement matrix, which is used to generate a random distribution model of flexible fibers in the cement matrix. This method first generates a fiber cluster containing multiple curved fibers. The fibers in the fiber cluster do not intersect with each other. Only by calculating the distances between the fibers at the centers of each fiber cluster can it be determined whether the fibers intersect. Therefore, the operation rate is greatly improved, the program running time is reduced, and it conforms to the existing state of flexible fibers in the cement matrix.

Description

Technical Field

[0001] The present invention relates to a numerical simulation method for flexible fiber bundles in a cement matrix. Background Art

[0002] Concrete is one of the main building materials and has been widely used in practical engineering. However, the disadvantages of brittle concrete matrix such as low tensile strength and poor toughness limit its wide application. Therefore, in the past few decades, fibers have been introduced into concrete to form fiber-reinforced cement-based composites (FRC). Rapid and accurate analysis of the mechanical properties of composites is the basis for material application. In view of the disadvantages of the traditional experimental-based mechanical property analysis method, such as high cost and long cycle, in recent years, meso-mechanical analysis of composites based on simulation has become an important direction for analyzing the mechanical properties and structural design of fiber-reinforced concrete (FRC).

[0003] The closer the model is to the actual structure, the more accurate the analysis result. The distribution of fibers in fiber concrete in the matrix is actually a random and disordered state, with a huge number of fibers and no intersection between them. In order to be able to truly simulate the random distribution of fibers in fiber-reinforced cement-based composites and establish a more realistic finite element model, it is necessary to study the algorithm for the random distribution of fibers in the concrete matrix.

[0004] The existing fiber placement techniques are mainly as follows:

[0005] (1) "Finite Element Simulation Study of Strain-Hardening Fiber-Reinforced Cementitious Composites"

[0006] This document simulates the existence state of PVA fibers in the cement matrix with a single straight-line segment model, randomly generates a straight fiber in space, determines whether the fiber is in the matrix, and then determines whether the fiber intersects with the already generated fibers. Repeat the above process until the required number of fibers is generated.

[0007] (2) "Research on the Early Plastic Crack Limiting Mechanism and Numerical Simulation of Fiber Reinforced Concrete"

[0008] This document approximately simulates the geometric shape characteristics of polypropylene fibers with arcs on a spatial circle. First, spherical coarse aggregates are generated in space, and it is determined whether the coarse aggregates exceed the matrix boundary and intersect. After all the aggregates are placed, arc-shaped polypropylene fibers are placed, and it is determined whether the fibers exceed the boundary and intersect with the coarse aggregates. Repeat the above process until the required number of fibers is generated. The specific flow chart is as Figure 1 shown.

[0009] Existing Technology 1: The technical disadvantages of the document "Finite Element Simulation Study of Strain-Hardening Fiber-Reinforced Cementitious Composites" are specifically as follows:

[0010] Disadvantage 1: It cannot accurately reflect the curved shape of flexible fibers in concrete, let alone the true state of "clumping, aggregation, and bunching" of the fibers.

[0011] Due to the characteristics of the material, structure, and surface hydrophobicity of flexible fibers themselves, they will bend in the concrete system and cannot be evenly dispersed, and are prone to clumping, aggregation, and bunching. The existing technology only generates single straight fibers in sequence, which cannot accurately reflect the curved shape of flexible fibers in concrete and the true state of "clumping, aggregation, and bunching".

[0012] Disadvantage 2: The fiber placement time generation model takes a long time.

[0013] Currently, the efficiency of spatial fiber placement is generally very low. The main reason is that it takes a huge amount of time to judge whether the placed spatial fibers intersect. Since the intersection judgment of two spatial fibers requires calculating the spatial distance between the fibers, the calculation amount is large. When generating the nth spatial fiber, it is necessary to judge whether it intersects with all the previously generated n - 1 fibers. When the number of placed fibers is very large, the cumulative calculation amount of the spatial distance is extremely large.

[0014] Disadvantage 3: It is impossible to simulate spatial fibers with a high fiber volume ratio.

[0015] Flexible fibers are randomly and disorderly distributed in the cement matrix. Using the single-fiber placement method, when the fiber volume content in the cement matrix is high, the randomly distributed placed fibers occupy too much matrix space, and it is difficult for the later placed fibers to find a suitable position in the matrix space and intersect with the placed fibers, resulting in ineffective placement, so the target fiber volume content cannot be achieved.

[0016] Existing technology 2: The technical disadvantages of the literature "Research on the Early Plastic Crack Limiting Mechanism and Numerical Simulation of Fiber Reinforced Concrete" are as follows:

[0017] Disadvantage 1: The placed fibers may have spatial overlap, which does not conform to the actual situation.

[0018] In the method of this literature, "it does not judge whether the fibers intersect". The model generated by this method not only cannot correctly simulate the true distribution of the fibers, but also does not conform to the true stress state of the fibers during the meso-mechanical analysis of fiber reinforced concrete.

[0019] Disadvantage 2: It cannot accurately reflect the true state of "clumping, aggregation, and bunching" of flexible fibers in concrete.

[0020] Although this literature uses arcs on a spatial circle to approximately simulate the geometric shape characteristics of flexible fibers, it does not present the true state of "clumping, aggregation, and bunching" of flexible fibers in concrete. Summary of the Invention

[0021] The purpose of the present invention is to provide a numerical simulation method for flexible fiber bundles in a cement matrix, which can truly simulate the curved shape presented by the flexible fibers in the cement matrix and the distribution state of bundling and agglomeration. Moreover, the present invention solves the bottleneck problems of slow operation speed of the existing three-dimensional curved fiber placement model and difficulty in realizing the placement of fibers with a high volume content during simulation.

[0022] To achieve the above object, the technical solution of the present invention is: a numerical simulation method for flexible fiber bundles in a cement matrix, which is used to generate a random distribution model of flexible fibers in the cement matrix. This method first generates a fiber cluster containing multiple curved fibers, and the fibers in the fiber cluster do not intersect each other. It only needs to calculate the distances between the fibers at the centers of each fiber cluster to determine whether the fibers intersect. As Figure 2 shown, the specific steps are as follows:

[0023] Step S1: Determine the relevant parameters of the target to be placed

[0024] Determine the length L, width B, and height H of the matrix; the fiber diameter d f , the fiber length l f , the target fiber volume fraction V f . Create an empty matrix MA and an empty matrix MB.

[0025] The MA matrix is used to store the spatial coordinates of all the fibers that have been placed. The MB matrix is used to store the spatial coordinates of the central fibers of the fiber clusters that have been placed and the radii of the fiber clusters.

[0026] Step S2: Calculate the number of fibers N to be placed

[0027] The number of fibers N to be placed is the total number of single curved fibers to be placed in a model to reach the target fiber volume fraction V f . The number of fibers N can be calculated by Equation (1):

[0028]

[0029] In the formula, d f is the fiber diameter, l f is the fiber length, V f is the target fiber volume fraction, and L, B, and H are the dimensions of the length, width, and height of the matrix. When the calculated number of fibers is a decimal, it is rounded to the nearest integer.

[0030] Step S3: Generate the nth (n≥1) fiber cluster

[0031] Step S3.1: Determine the number of single fibers i n in the nth fiber cluster and the radius R n

[0032] The fiber cluster is presented in the form that one fiber is the center and other fibers are tightly packed around it, as Figure 3 shown.

[0033] Randomly determine the number of fibers i in the nth fiber cluster n , where 1 ≤ i n ≤ i Max , and i Max is the maximum number of single fibers in the fiber cluster required in the model. e is the number of single-curve fibers that have been placed. When N - e ≤ i Max , i n = N - e.

[0034] The radius R of the fiber cluster n is the distance from the center fiber center point of the nth fiber cluster to the outermost fiber edge, as Figure 4 shown.

[0035] Since the fiber radii are the same and the fiber cross-sections are equal circles, at most 6 fibers can surround one fiber on the outside of the first layer. At most 12 fibers can surround it on the outside of the second layer, and so on. At most 6*s fibers can surround it on the s-th layer. According to the number of layers s n surrounding the central fiber of the nth fiber cluster, the radius R of its fiber cluster can be calculated n . The specific calculation of the radius R of the fiber cluster n is shown in Equation (2).

[0036] R n = s n *d f + 0.5*d f (2)

[0037] Step S3.2, generate the central fiber of the nth planar curve fiber cluster

[0038] Generate the first curve fiber starting from the origin in the X-Y plane. The curve fiber model consists of an arc part and nl (0 ≤ nl ≤ 2) straight lines, and the straight lines are tangent to the arc part. To simplify the parameters, the arc part is discretized into w straight line segments of equal length connected end to end. w is the number of divisions of the arc segment.

[0039] In the present invention, the single-curve model only takes 2 straight lines and an arc part as an example, and the arc part is discretized into 3 straight line segments of equal length connected end to end. That is, nl = 2, w = 3, then the u-th curve fiber in the nth fiber cluster consists of L nu1 (P nu1 , P nu2 ), L nu2 (P nu2 , P nu3 ), L nu3 (Pnu3 , P nu4 )、L nu4 (P nu4 , P nu5 )、L nu5 (P nu5 , P nu6 ) consists of five straight line segments, such as Figure 5 shown.

[0040] The steps for generating cluster center curve fibers are as follows:

[0041] (1) Select the origin (0, 0, 0) as the fiber starting point P n11 ;

[0042] (2) Within a reasonable range ((l n ) 11 +(l n ) 12 <2 / 3*l f ) for line segment L n11 and line segment L n12 Randomly select a value and then randomly define the horizontal orientation angle in the range of (-π, π) Determine the straight line segment L n11 direction;

[0043] (3) According to line segment L n11 The starting point coordinates P n11 , length and horizontal orientation angle To calculate point P n12 The coordinates of

[0044] (4) where δ n r n = l f -(l n ) 11 -(l n ) 12 To avoid the intersection of the straight line segments tangent to the two ends of the arc in the model, ensure that the central angle δ of the arc segment n The value is randomly defined in the range (0, π);

[0045] (5) Since the arc segments of the fiber are respectively n11 and the straight line segment L n12 If they are tangent, the horizontal orientation angle of each straight line segment can be calculated by formula (3);

[0046]

[0047]

[0048] Where w is the number of straight line segments of the discretized arc.

[0049] Calculate the end coordinates of each straight-line segment based on the starting point coordinates, length, and horizontal orientation angle of each straight-line segment until the calculation of all coordinate points is completed.

[0050] (7) Connect the points P n11 (x 11 , y 11 , z 11 ), P n12 (x 12 , y 12 , z 12 ), P n13 (x 13 , y 13 , z 13 ), P n14 (x 14 , y 14 , z 14 ), P n15 (x 15 , y 15 , z 15 ), P n16 (x 16 , y 16 , z 16 ), thus generating the central fiber of the nth planar curve cluster.

[0051] Step S3.3: Generate the nth spatial curve fiber cluster

[0052] Let the initial coordinates of the central fiber of the nth planar fiber cluster be (x a , y a , z a ). The fiber cluster rotates by angles α n , β n , γ n around the X, Y, and Z axes in three directions respectively; then, it translates by Δx n , Δy n , Δz n in the X, Y, and Z directions respectively. Finally, the spatial coordinates (x b , y b , z b ) of the central fiber of the three-dimensional spatial fiber cluster to be deployed are obtained, as shown in the following formula:

[0053]

[0054] where (x a , y a , z a ) is a certain coordinate point before rotation and translation, and (x b , y b , zb ) is the coordinate point after rotation and translation.

[0055] Step S4: Determine whether the nth spatial fiber cluster exceeds the matrix boundary

[0056] Let the maximum values of the X-axis, Y-axis, and Z-axis of the central fiber coordinate point of the nth spatial curve fiber cluster be (X n ) max , (Y n ) max , (Z n ) max , and the minimum values be (X n ) min , (Y n ) min , (Z n ) min .

[0057] Judge whether it exceeds the matrix boundary through the following formula (5):

[0058] min{(X n ) min , (Y n ) min , (Z n ) min}>0 and (X n ) max <L and (Y n ) max <B and (Z n ) max <H(5)

[0059] When formula (5) holds, proceed to the next step, i.e., step S5

[0060] When formula (5) does not hold, remove this fiber cluster and return to step S2

[0061] Step S5: Determine whether the nth spatial fiber cluster intersects with the previously placed (n - 1) spatial fiber clusters (when placing the first fiber cluster, i.e., n = 1, no intersection judgment is required, and directly proceed to step S6)

[0062] Step S5.1: Extract the radius R of the previously placed n - 1 spatial fiber clusters and the spatial information of the central fiber of the cluster, i.e., matrix MB.

[0063] Step S5.2: Calculate the minimum distance (d n ) jm. In the above model, the curved fibers are all composed of straight line segments. Then, the minimum distance (d n ) jm = min{d nj1 , d nj2 , d nj3 , d nj4 , d nj5 ,..., d nj(nl+w)}, where the minimum distance d njk between the j-th straight line segment of the central fiber of the n-th fiber cluster and the k-th (1 ≤ k ≤ (nl + w)) straight line segment of the central fiber of the m-th fiber cluster that has been placed is calculated as follows:

[0064] Let the j-th straight line segment of the central fiber of the n-th spatial fiber cluster be the straight line l1, and the endpoint coordinates be denoted as P j1 (x1, y1, z1), P j2 (x2, y2, z2), and the k-th straight line segment of the central fiber of the m-th cluster of the placed fiber cluster be the straight line l2, and the endpoint coordinates be denoted as Q k1 (x1, y1, z1), Q k2 (x2, y2, z2). The direction vectors of the straight lines l1 and l2 are respectively Then, the coordinates of any point on the straight lines l1 and l2 can be expressed by the following formula:

[0065]

[0066] where 0 ≤ λ1 ≤ 1, 0 ≤ λ2 ≤ 1.

[0067] To find the shortest distance (d njk ) min between the two line segments, it can be transformed into finding the minimum value of the square of the distance between the two line segments (d njk ) 2 min .

[0068] The square of the distance f between the two line segments is as shown in Equation (7):

[0069]

[0070] When , the square of the distance f reaches a minimum value. Expanding and simplifying the formula gives:

[0071]

[0072] When the obtained parameters λ1 and λ2 satisfy 0 ≤ λ1, λ2 ≤ 1, then the shortest distance at this time When at least one of the parameters λ1 and λ2 is not within the range of [0, 1], the shortest distances d1 from point P j1 to the straight line segment l2, the shortest distances d2 from point P j2 to the straight line segment l2, the shortest distances d3 from point Q k1 to the straight line segment l1, the shortest distances d4 from point Q k2 to the straight line segment l1 are solved respectively. At this time

[0073] Then, the shortest distance (d n ) from the j-th straight line segment of the central fiber of the n-th fiber cluster to be placed to the central fiber of the m-th fiber cluster that has been placed jm = min{d nj1 , d nj2 , d nj3 , d nj4 , d nj5 ,..., d nj(nl+w)}

[0074] Step S5.3: Based on the following formula, determine whether the n-th spatial fiber cluster intersects with the fiber clusters that have been placed:

[0075] (d n ) jm > (R n + R m ) (9)

[0076] When formula (9) holds, j = j + 1. Return to step S5.2.

[0077] When formula (9) does not hold, remove this fiber cluster and return to step S2 again.

[0078] When j = nl + w, it means that the n-th spatial fiber cluster does not intersect with the m-th spatial fiber cluster that has been placed, and m = m + 1.

[0079] Step S5.4: When m = n, it means that the n-th spatial fiber cluster does not intersect with the n - 1 spatial fiber clusters that have been placed, and proceed to the next step, that is, step S6.

[0080] Step S6: Determine the positions of the remaining i n - 1 fibers in the n-th spatial fiber cluster (when i n = 1, directly enter step S7)

[0081] Step S6.1: Generate all the fibers within the s n layers of the n-th spatial fiber cluster

[0082] First, obtain the s nAll the fiber coordinates within the layer, and then according to the rotation angles α n , β n , γ n and the translation distances Δx n , Δy n , Δz n of the nth planar cluster center fiber obtained in step S3.3, perform rotation and translation to obtain the spatial coordinates of all the fibers in the nth three-dimensional space cluster fiber to be placed. n All the fiber coordinates within the layer

[0083] The algorithm for all the fiber coordinates within the layer of the nth planar cluster fiber is as follows: n As can be seen from the above model, the curved fibers are all composed of straight line segments. Determining the fiber position is to find the endpoint coordinates of each straight line segment. There are mainly the following two relative positions of two fibers in the planar fiber cluster, as

[0084] shown. In the figure, the minimum distance rd between the two fibers is the fiber diameter d Figure 6 , that is, the two fibers are adjacent. If the endpoint coordinates of each straight line segment of one fiber are known, the distance rd and the angle ε between the two fibers can be calculated to obtain the coordinates of each straight line segment of the other fiber. Let the endpoints of the straight line segments on the 1st fiber be {Z f , Z 11 , Z 12 ,..., Z 13 ,..., Z 1(nl+w+1)}, and the endpoints of the straight line segments on the 2nd fiber be {Z 21 , Z 22 , Z 23 ,..., Z 2(nl+w+1)}.

[0085] For the first case, the 1st fiber and the 2nd fiber are perpendicular to each other in the Z-axis direction. If the coordinates of the 1st fiber are known, the coordinates of the 2nd fiber can be calculated by the following formulas (10), (11), and (12)

[0086] x 2g = x 1g (10)

[0087] y 2g = y 1g (11)

[0088] z 2g = z 1g + rd (12)

[0089] If the coordinates of the 2nd fiber are known, the coordinates of the 1st fiber can be calculated by the following formulas (13), (14), and (15)

[0090] x 1g = x2g (13)

[0091] y 1g = y 2g (14)

[0092] z 1g = z 2g -rd (15)

[0093] For the second case, if the coordinates of Fiber 1 are known, the Z-axis coordinate of Fiber 2 can be calculated by the following formula (16)

[0094] z 2g = z 1g + rd * sin(ε) (16)

[0095] To obtain the X-axis and Y-axis coordinates of Fiber 2, Fiber 2 needs to be vertically projected onto the X-Y plane where Fiber 1 is located along the Z-axis direction, as Figure 7 shown (the line below is the vertical projection line of Fiber 2 and the line above is Fiber 1). Since Fiber 2 is vertically projected along the Z-axis direction, the X-axis and Y-axis coordinates of the endpoints of the straight line segment of Fiber 2 are equal to those of the endpoints of the straight line segment of the projection line {Z’ 21 、Z’ 22 、Z’ 23 、...、Z’ 2(3+w)}.

[0096] x 2g = x’ 2g (17)

[0097] y 2g = y’ 2g (18)

[0098] Among them, the coordinates of the endpoints of the straight line segment of the projection line can be obtained according to the broken line parallel line algorithm. The broken line parallel line algorithm is as follows:

[0099] As Figure 8 shown, The vector is equal to the sum of the vectors of two adjacent sides of the parallelogram and vector . The directions of vector and vector can be represented by the coordinate differences between the vertex Ki and the endpoints K1, K2. The lengths of vector and vector are the same, equal to the division of the parallel line spacing M by the sine value of the included angle θ between the two line segments. The coordinates of point Q i are calculated as shown in formula (19)

[0100]

[0101] According to the above algorithm and equations (20), (21), (22), and (23), the projection line Z’ of Fiber No. 2 can be obtained. 22 , Z’ 23 ,..., Z’ 2(2+w) coordinates.

[0102]

[0103]

[0104]

[0105]

[0106] where the value of g ranges from 2 to (nl + w); M is the distance between the projection lines of Fiber No. 1 and Fiber No. 2, i.e., rd * cos(ε); θ g(g+1) represents the angle between the straight line segment L 1(g-1) of Fiber No. 1 and the straight line segment L 1g .

[0107] The Z’ 21 , Z’ 2(nl+w+1) of the projection line of Fiber No. 2 can be obtained respectively according to the lengths and unit vectors of the points Z’ 21 , Z’ 2(3+w) and the straight line segments L 11 , L 1(nl+w+1) of Fiber No. 1, and the specific calculation is as shown in equations (24) and (25) below.

[0108] Z' 21 = Z' 22 + l 11 * ((Z 11 - Z 12 ) / / Z 11 - Z 12 / ) (24)

[0109] Z' 2(nl+w+1) = Z' 2(nl+w) + l 15 * ((Z 1(nl+w+1) - Z 1(nl+w) ) / / Z 1(nl+w+1) - Z 1(nl+w) / ) (25)

[0110] For the second case, if the coordinates of Fiber No. 2 are known, the Z-axis coordinate of Fiber No. 1 can be calculated through the following equation (26).

[0111] z 1g = z 2g - rd * sin(ε) (26)

[0112] Obtain the X-axis and Y-axis coordinates of Fiber No. 1. Similarly, project Fiber No. 1 vertically along the Z-axis onto the X-Y plane where Fiber No. 2 is located. Then, according to the broken-line parallel-line algorithm, obtain the projection line Z’ of Fiber No. 1 through equations (27), (28), (29), and (30). 12 、Z’ 13 、...、Z’ 1(2+w) coordinates.

[0113]

[0114]

[0115]

[0116]

[0117] Where the value of g ranges from 2 to (nl + w); M is the distance between the projection lines of Fiber No. 1 and Fiber No. 2, that is, rd*cos(ε); θ g(g+1) represents the angle between the straight-line segment L 2(g-1) of Fiber No. 2 and the straight-line segment L 2g of Fiber No. 2.

[0118] The Z’ 11 、Z’ 1(nl+w+1) of the projection line of Fiber No. 1 can be obtained respectively according to the lengths and unit vectors of the points Z’ 11 、Z’ 1(nl+w+1) and the straight-line segments L 21 、L 2(nl+w+1) of Fiber No. 2. The specific calculation is as follows in equations (31) and (32).

[0119] Z' 11 =Z' 12 +l 21 *((Z 21 -Z 22 ) / / Z 21 -Z 22 / ) (31)

[0120] Z' 1(nl+w+1) =Z' 1(nl+w) +l 25 *((Z 2(nl+w+1) -Z 2(nl+w) ) / / Z 2(nl+w+1) -Z 2(nl+w) / ) (32)

[0121] Step S6.2: Randomly determine the coordinates of each fiber in the nth spatial fiber cluster

[0122] Select the fiber s in the nth three-dimensional spatial cluster nAll the fibers in layer -1, s n There are 3*s in layer -1 n *(s n -1)+1 fiber, and then randomly select (i - 1)-3*s n fibers from the s n *(s n -1) fibers.

[0123] Step S7. Storage of the spatial information of the nth spatial fiber cluster

[0124] Step S7.1. Update the information of the coordinate matrix MA

[0125] Successfully place the nth spatial fiber cluster, and store the coordinates of each single fiber in the nth spatial fiber cluster into the matrix MA.

[0126] Step S7.2. Update the information of the coordinate matrix MB

[0127] Store the coordinates of the central fiber of the nth spatial curve fiber cluster and the fiber cluster radius R n into the matrix MB.

[0128] Step S7.3. Update the value of the number of placed fibers e

[0129] At this time, the number of placed single - curve fibers e = e + i n .

[0130] Step S8. Determine whether the placement of the curve fiber is completed

[0131] When e ≠ N, update the value of n according to n = n + 1, and repeat the operations according to steps S3 - S7 of this method.

[0132] When e = N, the placement of all N single - spatial fibers is completed. Output the coordinates of all N spatial fiber nodes, that is, the model matrix MA.

[0133] Compared with the prior art, the present invention has the following beneficial effects:

[0134] Advantage 1: It accurately reflects the curved form of the flexible fiber in concrete, and the simulation result is more consistent with the real existence state of "clustering, aggregation, and bunching" presented by the flexible fiber;

[0135] Due to the characteristics of the flexible fiber's own material, structure, and the hydrophobicity of the fiber surface, it will bend and cannot be evenly dispersed in the concrete system, and is prone to clustering, aggregation, and bunching. The fiber model placed in the present invention is a curve fiber cluster model, which simulates the real existence state of the flexible fiber in the cement matrix.

[0136] Advantage 2: It significantly improves the generation efficiency of the flexible spatial curve fiber model;

[0137] Since the intersection judgment of two spatial fibers requires calculating the spatial distance between the fibers, the computational workload is large. When N fibers need to be placed in the model, if the existing technical solution is adopted, at least intersection judgments are required. In the fiber cluster of the present invention, there are multiple fibers and the fibers do not intersect with each other. Only the distance between the fibers at the center of each fiber cluster needs to be calculated to determine whether the fibers intersect, and on average only intersection judgments are required. Moreover, the flexible fibers are in a curved shape and randomly disordered distribution state in the cement matrix. When using the traditional algorithm, the randomly distributed fibers that have been placed occupy too much matrix space, and the fibers to be placed later are very likely to intersect with the fibers that have been placed and need to be continuously re-placed. The fibers in the fiber cluster of the present invention are closely packed together, which can leave a large number of suitable space positions for the fibers to be placed later, making it easy to place the fibers later. Therefore, the present invention greatly improves the operation rate and reduces the program running time.

[0138] Advantage Three: Completely solve the existing technical defect that when simulating the high-volume curve fiber content in the prior art, the simulation of the target volume fiber ratio cannot be achieved

[0139] The flexible fibers are in a curved shape and randomly disordered distribution state in the cement matrix. When the fiber volume content in the cement matrix is high, the randomly distributed fibers that have been placed occupy too much matrix space, and it is difficult for the fibers to be placed later to find suitable positions in the matrix space and intersect with the fibers that have been placed, resulting in ineffective placement. Therefore, the target fiber volume content cannot be achieved. The fibers in the fiber cluster of the present invention are closely packed together, and compared with the traditional algorithm, a large number of suitable space positions can be left for the fibers to be placed later. Brief Description of the Drawings

[0140] Figure 1 is a flowchart of the existing fiber placement technology.

[0141] Figure 2 is a flowchart of the method steps of the present invention.

[0142] Figure 3 is in the form of a fiber cluster.

[0143] Figure 4 is the radius of the fiber cluster.

[0144] Figure 5 is a single-curve fiber model.

[0145] Figure 6 is the relative position of two fibers.

[0146] Figure 7 is the projection line of Fiber No. 2.

[0147] Figure 8 It is a schematic diagram of the parallel line algorithm.

[0148] Figure 9 It is a fiber cluster model.

[0149] Figure 10 It is a simulation diagram of the placement of curved fibers. Specific implementation manners

[0150] The technical solution of the present invention will be specifically described below in conjunction with the accompanying drawings.

[0151] A numerical simulation method for flexible fiber bundles in a cement matrix according to the present invention is used to generate a random distribution model of flexible fibers in the cement matrix. In this method, a fiber cluster containing multiple curved fibers is first generated, and the fibers in the fiber cluster do not intersect each other. Only the distances between the fibers at the centers of the fiber clusters need to be calculated to determine whether the fibers intersect.

[0152] The following are specific implementation examples of the present invention.

[0153] Example description: The size of the specimen is a cube with dimensions of 40×40×40 mm, the fiber volume fraction is 2%, the fiber length is 12 mm, and the fiber diameter is 0.04 mm. The single-curved fiber model consists of only 2 straight lines and an arc part. The arc part is discretized into 3 straight line segments with equal lengths that are connected end to end. That is, nl = 2, w = 3. The maximum number of fibers in a single fiber cluster is 19.

[0154] Step S1, determining relevant parameters of the target to be placed

[0155] The length of the matrix L = 40 mm, the width B = 40 mm, and the height H = 40 mm; the fiber diameter d f = 0.04 mm, the fiber length l f = 12 mm, the target fiber volume fraction V f = 12 mm. Create an empty matrix MA and an empty matrix MB.

[0156] The MA matrix is used to store the spatial coordinates of all the fibers that have been placed. The MB matrix is used to store the spatial coordinates of the central fibers of the fiber clusters that have been placed and the radii of the fiber clusters.

[0157] Step S2, calculating the number of fibers N to be placed

[0158]

[0159] After rounding, the number of fibers to be placed is 84,883.

[0160] Step S3, generating the first fiber cluster

[0161] Step S3.1: Determine the number of single fibers i1 and the fiber cluster radius R1 in the first fiber cluster

[0162] The number of single fibers i1 in the first fiber cluster is randomly selected within the range of 1 ≤ i1 ≤ 19, and i1 = 5. The number of fibers already placed e = 0. N - e = 84883 - 0 = 84883 > 19.

[0163] When i n = 5, s1 = 0, so R1 = 0.5 * d f = 0.02 mm;

[0164] Step S6: Determine the positions of the remaining 4 single fibers in the first spatial fiber cluster

[0165] Step S6.1: Generate all the fibers within one layer of the first spatial fiber cluster.

[0166] Using the planar single fiber generated in Step S3.2 as the central fiber of the fiber cluster, generate the 2nd - 7th fibers at a position d f away from the central fiber of the fiber cluster, as shown in Figure 9 . Among them, fiber No. 1 is the central fiber of the fiber cluster. The positions of these fibers are controlled by the variable θ and the single fiber diameter d f . The 2nd - 7th fibers are respectively located in the directions of π / 6, π / 6 + π / 3, π / 6 + 2π / 3, π / 6 + π, π / 6 + 4π / 3, π / 6 + 5π / 3 of the central fiber of the fiber cluster. According to the coordinates of the central planar fiber of the fiber cluster, the distance between fiber centers, i.e., the diameter d f and the variable θ, the coordinates of all the fibers within the first layer of the fiber cluster can be calculated.

[0167] After obtaining the coordinates of each point of the 2nd - 7th planar fibers, perform rotation and translation according to the rotation angles α, β, γ and translation distances Δx, Δy, Δz of fiber No. 1 on the X, Y, and Z axes obtained in Step S3.3, to obtain the spatial coordinates (x b , y b , z b ) of each fiber in the three - dimensional spatial fiber cluster to be placed.

[0168] Step S6.2: Randomly determine the coordinates of each fiber in the first spatial fiber cluster

[0169] Select the coordinates of fiber No. 1 in space, and then randomly select the coordinates of 4 spatial fibers from the 2nd - 7th fibers.

[0170] Step S7: Store the information of the first spatial fiber cluster

[0171] Step S7.1: Update the information of the coordinate matrix MA

[0172] Store the coordinates of 5 single fibers in the first spatial fiber cluster into matrix MA

[0173] Step S7.2, update the information of coordinate matrix MB

[0174] Store the coordinates of the central fiber of the first spatial fiber cluster and the cluster radius R1 = 0.02 mm into matrix MB.

[0175] Step S7.3, update the value of the number of fibers e that have been placed

[0176] At this time, the number of single-curve fibers e that have been placed = 0 + 5 = 5.

[0177] Step S8, determine whether the placement of the curve fibers is completed

[0178] e = 5 ≠ 84883, update the value of n, n = 1 + 1 = 2, and repeat steps S3 - S7 to place the second fiber cluster.

[0179] Keep repeating steps S3 - S8. When 9448 fiber clusters have been actually placed in the model, 84883 spatial curve fibers that are intended to be placed are reached in the matrix, the placement ends, and matrix MA is output.

[0180] The placement result is as Figure 10 shown. Only the central fibers of each spatial curve fiber cluster are shown in the figure.

[0181] The above are the preferred embodiments of the present invention. All changes made according to the technical solution of the present invention that do not exceed the scope of the technical solution of the present invention in terms of the functions and effects produced belong to the protection scope of the present invention.

Claims

1. A numerical simulation method for flexible fiber bundles in a cement matrix, characterized in that, A method for generating a random distribution model of flexible fibers in a cement matrix. First, a fiber cluster containing multiple curved fibers is generated, and the fibers in the fiber cluster do not intersect each other. Only the distances between the fibers at the centers of each fiber cluster need to be calculated to determine whether the fibers intersect. The implementation steps of this method are as follows: Step S1: Determine the relevant parameters of the target to be placed. Step S2: Calculate the number of fibers N to be placed. Step S3: Generate the nth fiber cluster, where n≥1. The specific implementation is as follows: Step S3.1: Determine the number i of single fibers in the nth fiber cluster n and the radius R of the fiber cluster n : The fiber clusters are presented in the form of other fibers tightly packed around a central fiber; randomly determine the number of fibers i in the nth fiber cluster n , where 1 ≤ i n ≤ i Max , and i Max is the maximum number of single fibers in a fiber cluster required in the model; e is the number of single-curve fibers that have been placed. When N - e ≤ i Max , then i n = N - e; the radius R of the fiber cluster n is the distance from the center point of the central fiber of the nth fiber cluster to the edge of the outermost fiber; since the fiber radii are the same and the fiber cross-sections are equal circles, at most 6 fibers can surround one fiber in the first layer; at most 12 fibers can surround it in the second layer, and so on, with 6*s fibers surrounding it in the s-th layer; according to the number of layers s of the fibers surrounding the central fiber in the nth fiber cluster n , calculate its fiber cluster radius R n ; the specific calculation of the fiber cluster radius R n is shown in Equation (2): R n = s n * d f + 0.5 * d f (2) d f is the fiber diameter; Step S3.2: Generate the central fiber of the nth planar curved fiber cluster: Generate the first curved fiber starting from the origin in the X-Y plane. The curved fiber model consists of an arc part and nl straight lines. The straight lines are tangent to the arc part, where 0≤nl≤2. For simplicity of parameters, the arc part is discretized into w straight line segments of equal length connected end to end, and w is the number of divisions of the arc segment. Step S3.3: Generate the nth spatial curved fiber cluster: Assume that the initial coordinates of the center fiber of the nth plane fiber cluster are (x a ,y a ,z a ), the fiber clusters rotate in three directions around the X, Y, and Z axes at an angle α n , β n , γ n Then, rotate Δx in the X, Y, and Z directions respectively. n , Δy n , Δz n Finally, the spatial coordinates (x b ,y b ,z b ), as follows: where (x a , y a , z a ) is a certain coordinate point before rotation and translation, and (x b , y b , z b ) is the coordinate point after rotation and translation; Step S4: Determine whether the nth spatial fiber cluster exceeds the matrix boundary. Step S5: Determine whether the nth spatial fiber cluster intersects with the previously placed n-1 spatial fiber clusters. When placing the first fiber cluster, i.e., when n = 1, no intersection judgment is required, and directly go to Step S6; Step S6. Determine the positions of the remaining i - 1 fibers in the nth spatial fiber cluster. When i = 1, directly proceed to Step S7; n - 1 root fiber positions, when i n = 1, directly enter step S7; Step S7: Store the spatial information of the nth spatial fiber cluster. Step S8: Determine whether the placement of the curved fibers is completed.

2. The numerical simulation method for flexible fiber bundles in a cement matrix according to claim 1, characterized in that, The specific implementation of Step S1 is as follows: Determine the length L, width B, and height H of the matrix; fiber diameter d f , fiber length l f , target fiber volume fraction V f ; Create an empty matrix MA and an empty matrix MB; The MA matrix is used to store the spatial coordinates of all the placed fibers; the MB matrix is used to store the spatial coordinates of the central fibers of the placed fiber clusters and the radii of the fiber clusters.

3. The numerical simulation method for flexible fiber bundles in a cement matrix according to claim 2, characterized in that, The specific implementation of Step S2 is as follows: The number of fibers N to be placed is the total number of single-curve fibers to be placed in a model to achieve the target fiber volume fraction V f ; The number of fibers N is calculated by Equation (1): where l f is the fiber length, V f is the target fiber volume fraction, and L, B, and H are the dimensions of the length, width, and height of the matrix; when the calculated number of fibers is a decimal, round it to the nearest integer by rounding up or down.

4. The numerical simulation method for flexible fiber bundles in a cement matrix according to claim 1, characterized in that, The specific implementation of Step S4 is as follows: Let the maximum values of the X-axis, Y-axis, and Z-axis of the coordinate point of the central fiber of the nth spatial curve fiber cluster be (X n ) max , (Y n ) max , (Z n ) max , and the minimum values be (X n ) min , (Y n ) min , (Z n ) min ; Judge whether it exceeds the matrix boundary through Equation (5): min{(X n ) min ,(Y n ) min ,(Z n ) min}>0 and (X n ) max < L and (Y n ) max < B and (Z n ) max < H (5) When Equation (5) holds, go to Step S5; When Equation (5) does not hold, remove this fiber cluster and go back to Step S2.

5. A numerical simulation method for flexible fiber bundles in a cement matrix according to claim 4, wherein, The specific implementation of Step S5 is as follows: Step S5.1: Extract the radii R of the previously placed n-1 spatial fiber clusters and the spatial information of the central fibers of the clusters, i.e., the matrix MB; Step S5.

2. Calculate the minimum distance (d n ) jm between the j-th (1 ≤ j ≤ (nl + w)) straight-line segment of the central fiber of the n-th spatial fiber cluster and the central fiber of the m-th (1 ≤ m ≤ (n - 1)) spatial fiber cluster that has been placed in sequence; since the curved fibers are all composed of straight-line segments, the minimum distance (d n ) jm between the j-th straight-line segment of the central fiber of the n-th fiber cluster and the central fiber of the m-th fiber cluster that has been placed is d nj1 = min{d nj2 , d nj3 , d nj4 , d nj5 ,..., d nj(nl+w)}, where the algorithm for the minimum distance d njk between the j-th straight-line segment of the central fiber of the n-th fiber cluster and the k-th (1 ≤ k ≤ (nl + w)) straight-line segment of the central fiber of the m-th fiber cluster that has been placed is as follows: Let the j-th straight line segment of the n-th spatial fiber cluster's central fiber be the straight line segment l1, and the endpoint coordinates are respectively denoted as P j1 (x1, y1, z1), P j2 (x2, y2, z2), and the k-th straight line segment of the m-th cluster central fiber of the already placed fiber cluster be the straight line segment l2, and the endpoint coordinates are respectively denoted as Q k1 (x1, y1, z1), Q k2 (x2, y2, z2); the direction vectors of the straight line segment l1 and the straight line segment l2 are respectively Then the coordinates of any point on the straight line segment l1 and the straight line segment l2 are expressed by the following formula: where 0≤λ1≤1 and 0≤λ2≤1; Find the shortest distance (d njk ) min between two line segments, and transform it into finding the minimum value of the square of the distance between the two line segments (d njk ) 2 min ; The square of the distance f between two line segments is shown in Equation (7): When occurs, the square of the distance f obtains a minimum value; expanding and simplifying the formula gives: When the obtained parameters λ1 and λ2 satisfy 0 ≤ λ1, λ2 ≤ 1, the shortest distance at this time When at least one of the parameters λ1 and λ2 is not within the range [0, 1], the shortest distances d1 from point P j1 to line segment l2, the shortest distances d2 from point P j2 to line segment l2, the shortest distances d3 from point Q k1 to line segment l1, and the shortest distances d4 from point Q k2 to line segment l1 are solved respectively. At this time Then the shortest distance (d n ) jm from the j-th straight segment of the central fiber of the n-th fiber cluster put in to the central fiber of the m-th fiber cluster already put in is nj1 = min{d nj2 , d nj3 , d nj4 , d nj5 , d nj(nl+w)}; Step S5.3: Based on the following formula, determine whether the nth spatial fiber cluster intersects with the previously placed fiber clusters: (d n ) jm >(R n +R m ) (9) When Equation (9) holds, j = j + 1, and go back to Step S5.2; When Equation (9) does not hold, remove this fiber cluster and go back to Step S2; When j = nl + w, it means that the nth spatial fiber cluster does not intersect with the previously placed mth spatial fiber cluster, and m = m + 1; Step S5.4: When m = n, it means that the nth spatial fiber cluster does not intersect with the previously placed n-1 spatial fiber clusters, and go to Step S6.

6. A numerical simulation method for flexible fiber bundles in a cement matrix according to claim 5, wherein, The specific implementation of Step S6 is as follows: Step S6.1: Generate all fibers within the s-th layer of the n-th spatial fiber cluster n of the s-th layer of the n-th spatial fiber cluster First, obtain s of the nth planar cluster fiber n coordinates of all fibers within the layer, and then, according to the rotation angles α n 、β n 、γ n in the three directions of the X, Y, and Z axes of the nth planar cluster center fiber obtained in step S3.3 n and translation distances Δx n 、Δy n 、Δz n , perform rotation and translation to obtain the spatial coordinates of all fibers within the s n layer of the nth three-dimensional space cluster fiber to be placed; s of the nth planar cluster fiber n The algorithm for the coordinates of all fibers within the layer is as follows: The curved fibers are all composed of straight line segments. Determining the fiber positions is to find the endpoint coordinates of each straight line segment. There are two cases for the relative positions of two fibers in a planar fiber cluster. One is that the two fibers are perpendicular to each other in the Z-axis direction, and the other is that the two fibers are not perpendicular to each other in the Z-axis direction. If the endpoint coordinates of each straight line segment of one fiber are known, the distance rd and the angle ε between the two fibers are used to calculate the coordinates of each straight line segment of the other fiber. Let the endpoints of the straight line segments on the No. 1 fiber be {Z 11 、Z 12 、Z 13 、...、Z 1(nl+w+1)}, and the endpoints of the straight line segments on the No. 2 fiber be {Z 21 、Z 22 、Z 23 、...、Z 2(nl+w+1)}; For the first case, the No. 1 fiber and the No. 2 fiber are perpendicular to each other in the Z-axis direction. If the coordinates of the No. 1 fiber are known, the coordinates of the No. 2 fiber can be calculated by the following Equations (10), (11), and (12) x 2g = x 1g (10) y 2g = y 1g (11) z 2g = z 1g + rd(12) If the coordinates of the No. 2 fiber are known, the coordinates of the No. 1 fiber can be calculated by the following Equations (13), (14), and (15) x 1g = x 2g (13) y 1g = y 2g (14) z 1g = z 2g -rd (15) For the second case, Fiber 1 and Fiber 2 form an angle along the X-axis direction. If the coordinates of Fiber 1 are known, the Z-axis coordinate of Fiber 2 can be calculated by the following formula (16). z 2g = z 1g + rd * sin(ε) (16) To obtain the X-axis and Y-axis coordinates of Fiber No. 2, Fiber No. 2 needs to be vertically projected onto the X-Y plane where Fiber No. 1 is located along the Z-axis direction. Since Fiber No. 2 is vertically projected along the Z-axis direction, the X-axis and Y-axis coordinates of the endpoints of the straight line segment of Fiber No. 2 are equal to the X-axis and Y-axis coordinates of the endpoints of the projection line straight line segment {Z’ 21 、Z’ 22 、Z’ 23 、...、Z’ 2(3+w)}; x 2g = x' 2g (17) y 2g = y' 2g (18) Among them, the endpoint coordinates of the straight line segment of the projection line can be obtained according to the broken line parallel line algorithm. The broken line parallel line algorithm is as follows: The vector is equal to the vectors of two adjacent sides of the parallelogram and the vector of the sum, the vector and the vector The direction of is represented by the coordinate differences between the vertex Ki and the endpoints K1, K2; the vector and the vector have the same length, which is equal to the division of the parallel line spacing M by the sine value of the angle θ between the two line segments; the coordinates of point Q i are calculated as shown in Equation (19): According to the above algorithm and equations (20), (21), (22), and (23), the projection line Z’ of Fiber No. 2 can be obtained. 22 , Z’ 23 ,..., Z’ 2(2+w) coordinates: where the value of g ranges from 2 to (nl + w); M is the distance between the projection lines of the first fiber and the second fiber, that is, rd*cos(ε); θ g(g+1) is represented as the straight line segment L of the first fiber 1(g-1) and the straight line segment L 1g the included angle between them; Z' of the projection line of Fiber No. 2 21 and Z' 2(nl+w+1) can be obtained respectively according to point Z' 21 and Z' 2(3+w) and the lengths and unit vectors of the straight line segments L of Fiber No. 1 11 and L 1(nl+w+1) as follows. The specific calculations are shown in the following equations (24) and (25): Z′ 21 = Z′ 22 + l 11 * ((Z 11 - Z 12 ) / |Z 11 - Z 12 |) (24) Z′ 2(nl+w+1) = Z′ 2(nl+w) + l 15 * ((Z 1(nl+w+1) - Z 1(nl+w) ) / |Z 1(nl+w+1) - Z 1(nl+w) |) (25) For the second case, if the coordinates of Fiber 2 are known, the Z-axis coordinate of Fiber 1 can be calculated by the following formula (26): z 1g = z 2g -rd*sin(ε) (26) Obtain the X-axis and Y-axis coordinates of Fiber No.

1. Similarly, Fiber No. 1 needs to be vertically projected along the Z-axis direction onto the X-Y plane where Fiber No. 2 is located; then, according to the broken-line parallel-line algorithm, the projection line Z’ of Fiber No. 1 is obtained through equations (27), (28), (29), and (30). 12 , Z’ 13 ,..., Z’ 1(2+w) coordinates: where the value of g ranges from 2 to (nl + w); M is the distance between the projection lines of the first fiber and the second fiber, that is, rd*cos(ε); θ g(g+1) is represented as the straight line segment L of the second fiber 2(g-1) and the straight line segment L 2g the included angle between them; The Z' of the No. 1 fiber projection line 11 , Z' 1(nl+w+1) can be respectively obtained according to the points Z' 11 , Z' 1(nl+w+1) and the length and unit vector of the No. 2 fiber straight line segment L 21 , L 2(nl+w+1) as follows. The specific calculations are shown in the following equations (31) and (32): Z′ 11 = Z′ 12 + l 21 * ((Z 21 - Z 22 ) / |Z 21 - Z 22 |) (31) Z′ 1(nl+w+1) = Z′ 1(nl+w) + l 25 * ((Z 2(nl+w+1) - Z 2(nl+w) ) / |Z 2(nl+w+1) - Z 2(nl+w) |) (32) Step S6.2: Randomly determine the coordinates of each fiber in the nth spatial fiber cluster Select the nth three-dimensional space cluster fiber s n - All the fibers within layer -1, s n - There are 3 * s n *(s n - 1) + 1 fibers within layer -1, and then randomly select (i - 1) - 3 * s n fibers from layer s n *(s n - 1).

7. A numerical simulation method for a flexible fiber bundle in a cement matrix according to claim 1, wherein, The specific implementation of the said Step S7 is as follows: Step S7.1: Update the information of the coordinate matrix MA Successfully place the nth spatial fiber cluster, and store the coordinates of each single fiber in the nth spatial fiber cluster into the matrix MA Step S7.2: Update the information of the coordinate matrix MB Store the coordinates of the central fiber of the nth spatial curve fiber cluster and the fiber cluster radius R n into the matrix MB; Step S7.3: Update the value of the number e of the fibers that have been placed At this time, the number of single-curve fibers already placed e = e + i n .

8. A numerical simulation method for a flexible fiber bundle in a cement matrix according to claim 1, wherein, The specific implementation of the said Step S8 is as follows: When e≠N, update the value of n by n=n + 1, and repeat the operations according to Steps S3 - S7 of this method When e = N, the placement of all N single spatial fibers is completed; output the node coordinates of all N spatial fibers, that is, the model matrix MA

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