CN110737972A - Two-dimensional irregular inter-particle contact force calculation method - Google Patents

Two-dimensional irregular inter-particle contact force calculation method Download PDF

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CN110737972A
CN110737972A CN201910924902.7A CN201910924902A CN110737972A CN 110737972 A CN110737972 A CN 110737972A CN 201910924902 A CN201910924902 A CN 201910924902A CN 110737972 A CN110737972 A CN 110737972A
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contour
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contact force
circumscribed rectangle
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CN110737972B (en
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苏栋
王翔
陈湘生
江月琴
洪成雨
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Shenzhen University
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Abstract

The invention discloses an two-dimensional irregular inter-particle contact force calculation method which comprises the following steps of S10, obtaining a 0 profile corresponding to a th particle and obtaining a second profile corresponding to a second particle, S20, determining a mass center of a th profile and determining a second mass center of the second profile, S30, determining a th distance from a point on the th profile to an th mass center, S60, judging whether the th profile and the second profile are intersected, if the th profile and the second profile are intersected, calculating a normal direction of each point on the th profile in an intersected area, namely calculating the normal direction of each point on the second profile in the intersected area, S70, calculating the direction of each contact pair contact force, S80, calculating the contact force on each contact pair, S90, and calculating the contact force between the two particles.

Description

Two-dimensional irregular inter-particle contact force calculation method
Technical Field
The invention belongs to the technical field of contact force calculation, and particularly relates to a contact force calculation method for two-dimensional irregular particles.
Background
At present, the normal contact force of two-dimensional irregular particles needs to be calculated when the method is applied to discrete element simulation, and the conventional technology firstly calculates the embedding depth between binding circles of different particles, and then independently calculates the normal force of each contact point by assuming that the normal force of each contact point is a function of the embedding depth. There is a large error between the calculated contact force and the true contact force.
Therefore, the prior art is to be improved.
Disclosure of Invention
The invention mainly aims to provide methods for calculating the contact force between two-dimensional irregular particles, and aims to solve the technical problems mentioned in the background technology, reduce errors and improve the precision.
The two-dimensional irregular inter-particle contact force calculation method comprises the following steps of:
step S10, acquiring a th contour corresponding to the th particle, and acquiring a second contour corresponding to the second particle;
step S20, determining the th center of mass of the th contour, and determining the second center of mass of the second contour;
step S30, determining a distance from the point on the th contour to the th center of mass, determining a second distance from the point on the second contour to the second center of mass;
step S40, establishing a rectangular coordinate system, determining a th circumscribed rectangle corresponding to the th outline, and determining a second circumscribed rectangle corresponding to the second outline;
step S50, judging whether the circumscribed rectangle and the second circumscribed rectangle are intersected, if the circumscribed rectangle and the second circumscribed rectangle are intersected, executing step S60;
step S60, determining whether the th contour and the second contour intersect in the intersection portion of the th circumscribed rectangle and the second circumscribed rectangle, if the th contour and the second contour intersectAnd (3) scattering the th contour in the intersection region, and calculating the normal n of each scattered point1I.e. by
Figure BDA0002218629600000011
Calculating the normal n of points on the second contour associated with discrete points of the contour in the intersection region2I.e. by
Figure BDA0002218629600000021
Wherein x is1And y1Is the coordinate of each point of the th contour on a rectangular coordinate, x2And y2Coordinates of each point of the second contour on the rectangular coordinate;
step S70, calculating the direction of each contact pair contact force, i.e. the
Figure BDA0002218629600000022
Step S80, calculating the contact force F on each contact pairiI.e. Fi＝kSFinFiWherein S isFiDenotes the area of each contact pair, k being the particle contact stiffness;
in step S90, a contact force F between two particles is calculated, i.e. F ═ Fi。
Preferably, the contact pair includes a th point on the th contour and a second point on the second contour, the straight line connecting the th point and the second point passing through the center of mass of the th contour, the th point and the second point both being located in the intersection region.
The method for calculating the contact force between the two-dimensional irregular particles has the following beneficial effects: the invention directly calculates the real contact area between irregular particles, calculates the contact force of each division area by dividing the embedded area, and then obtains the total contact force by superposition, and the obtained result has higher precision compared with the prior art.
Drawings
FIG. 1 is a schematic diagram of the th contour, the second contour, the th bounding rectangle and the second bounding rectangle in the method for calculating the contact force between two-dimensional irregular particles according to the present invention;
FIG. 2 is a schematic view of th contour in the method for calculating contact force between two-dimensional irregular particles according to the present invention;
FIG. 3 is a schematic diagram of a second profile in the method for calculating contact force between two-dimensional irregular particles according to the present invention;
FIG. 4 is a schematic diagram of the th contour and the second contour intersection judgment in the two-dimensional irregular interparticle contact force calculation method of the present invention;
FIG. 5 is a schematic diagram of an intersection region in the method for calculating contact force between two-dimensional irregular particles according to the present invention;
FIG. 6 is a schematic flow chart of the method for calculating the contact force between two-dimensional irregular particles according to the present invention.
The objects, features, and advantages of the present invention are further described in with reference to the accompanying drawings.
Detailed Description
It should be understood that the specific embodiments described herein are merely illustrative of the invention and not limiting, it being noted that related terms such as "," "second," etc. may be used to describe various components but these terms are not limiting.
Referring to fig. 1, 2, 3 and 6, fig. 1 is a schematic diagram of an th outline and a second outline and a th circumscribed rectangle and a second circumscribed rectangle in the method for calculating the contact force between two-dimensional irregular particles, fig. 2 is a schematic diagram of a th outline in the method for calculating the contact force between two-dimensional irregular particles, fig. 3 is a schematic diagram of the second outline in the method for calculating the contact force between two-dimensional irregular particles, and fig. 6 is a schematic flow chart of the method for calculating the contact force between two-dimensional irregular particles.
The two-dimensional irregular inter-particle contact force calculation method comprises the following steps of:
acquiring a th contour 10 corresponding to the th particle and a second contour 20 corresponding to the second particle in step S10, and acquiring contours corresponding to the particles, such as the th contour 10 and the second contour 20 shown in fig. 1, by photographing or scanning with an electron microscope (SEM) in step S10;
step S20, determining the mass center P1 of the th contour and the second mass center P2 of the second contour, specifically, determining the mass center P1 of the th contour by principal component analysis and determining the second mass center P2 of the second contour by principal component analysis.
Step S30, determining a th distance from a point on the th contour to the th center of mass, determining a second distance from a point on the second contour to the second center of mass, expressing the distance r as a function of the polar angle θ, i.e., r ═ r (θ), fitting r (θ), i.e., using a Fourier series expansion
Figure BDA0002218629600000031
Figure BDA0002218629600000032
In the formula, a0，an,bnIs the Fourier coefficient, and N is the harmonic number of the Fourier series. a is0，an,bnAs determined by least squares fitting r (θ), N may be 25 as shown in FIG. 2, the radius of the th contour 10 is denoted as r1＝r1(theta) its Fourier series expansion is
Figure BDA0002218629600000041
In the formula, a01，an1,bn1Is the Fourier coefficient of the th particle the radius of the second profile 20 is denoted as r2＝r2(theta), Fourier series expansion of
Figure BDA0002218629600000042
In the formula, a02，an2,bn2Is the fourier coefficient of the second particle.
Step S40, establishing a rectangular coordinate system XY (as shown in FIG. 1), determining the circumscribed rectangle corresponding to the th contour, and determining the second contour corresponding to the second contourA second external rectangle, in step S40, a rectangular coordinate system XY is established, and the minimum coordinate value X in the X-axis direction on the th contour is obtainedl1The maximum coordinate value X in the X-axis directionu1The minimum coordinate value Y in the Y-axis directionl1And the maximum coordinate value Y in the Y-axis directionu1Determining four coordinate values to calculate th length and th width to determine th circumscribed rectangle corresponding to th contour 10, and obtaining the minimum coordinate value X in X-axis direction on the second contour 20l2The maximum coordinate value X in the X-axis directionu2The minimum coordinate value Y in the Y-axis directionl2And the maximum coordinate value Y in the Y-axis directionu2Determining four coordinate values to calculate a second length and a second width to determine a second circumscribed rectangle corresponding to the second contour 20, i.e., a th circumscribed rectangle, xl1≤x≤xu1，yl1≤y≤yu1(ii) a A second circumscribed rectangle: x is the number ofl2≤x≤xu2，yl2≤y≤yu2。
Step S50, judging whether the circumscribed rectangle and the second circumscribed rectangle are intersected, if the circumscribed rectangle and the second circumscribed rectangle are intersected, executing step S60;
the step S50 specifically includes a step S51 of determining whether the distance between the center point of the th circumscribed rectangle and the center point of the second circumscribed rectangle in the x-axis direction is less than half of the sum of the sides of the th circumscribed rectangle and the second circumscribed rectangle in the x-axis direction, and whether the distance between the center point of the th circumscribed rectangle and the center point of the second circumscribed rectangle in the y-axis direction is less than half of the sum of the sides of the th circumscribed rectangle and the second circumscribed rectangle in the y-axis direction, that is, the step S51 of determining whether the distance
When it is satisfied with
Figure BDA0002218629600000043
And satisfy
Figure BDA0002218629600000051
(satisfies the above equations 2-1 and 2-2) indicates that the th bounding rectangle intersects the second bounding rectangle, then step S60 is performed;
step S60, judging whether the th contour and the second contour in the intersection region formed by the th circumscribed rectangle and the second circumscribed rectangle are intersected, if so, performing discrete processing on the th contour in the intersection region to obtain each discrete point, and calculating the normal n of each discrete point1I.e. by
Figure BDA0002218629600000052
Calculating the normal n of each point on the second contour in the intersection region2I.e. byWherein x is1And y1Is the coordinate of each point of the th contour on a rectangular coordinate, x2And y2The coordinates of each point of the second contour on the rectangular coordinates are obtained, wherein the th contour in the intersection area is subjected to discretization processing to obtain discrete points, the discrete points are determined on the th contour based on the equal polar angle difference between the adjacent points, and the intersection area represents a rectangle formed by the intersection of the th circumscribed rectangle and the second circumscribed rectangle.
In step S60, the determining whether the th contour and the second contour intersect each other in the intersection region formed by the th circumscribed rectangle and the second circumscribed rectangle includes (making the second particle be the master particle and the th particle be the slave particle) discretizing the contour of the second contour in the intersection region (i.e., EFGH) and equalizing the difference Δ θ between the polar angles θ between the adjacent points (e.g., making Δ θ equal to π 180), and determining the polar angle θ and the radius r in the th contour local coordinate system based on the coordinate values (x, y) of the points (e.g., 9 points, points a1 to a9) in the rectangular coordinate system1 z(theta), i.e.
Figure BDA0002218629600000054
If x-x01> 0 and y-y01> 0 (formula 5a)
Figure BDA0002218629600000055
If x-x01< 0 and y-y01> 0 (formula 5b)
Figure BDA0002218629600000061
If x-x01< 0 and y-y01< 0 (formula 5c)
Figure BDA0002218629600000062
If x-x01> 0 and y-y01< 0 (formula 5d)
Wherein x is01，y01Representing the coordinate value of the mass center in the rectangular coordinate system in the second profile, and simultaneously substituting the polar angle theta into the Fourier expansion equation (namely into the formula 1-2) of the th profile, namely
Figure BDA0002218629600000064
The radius of a point (e.g., B6 in FIG. 4) on the th profile with the same polar angle θ (e.g., A6 and B6 in FIG. 4) is obtained
Figure BDA0002218629600000065
Judgment ofWhether or not less thanIf it is
Figure BDA0002218629600000068
It means that the th contour and the second contour intersect and the embedding depth of the point is obtained
Figure BDA0002218629600000069
Whether discrete points (points A1-A9 of FIG. 4) of the second contour, which are located in the intersection area (i.e., EFGH), enter the th contour is sequentially judged, and the embedding depth is sequentially calculated for the entering points (points A4-A7 of FIG. 4).
After the embedding depth corresponding to the discrete point of the second contour in the intersecting region (i.e. EFGH) entering the th contour is calculated, the normal n of each point on the th contour in the intersecting region is calculated1I.e. by
Figure BDA00022186296000000610
Calculating the normal n of each point on the second contour in the intersection region2I.e. by
Figure BDA00022186296000000611
Wherein x is1And y1Is the coordinate of each point of the th contour on a rectangular coordinate, x2And y2The intersection area represents the rectangle formed by the intersection of the th bounding rectangle and the second bounding rectangle, such as EFGH in FIG. 4.
Step S70, calculating the direction of each contact pair contact force, i.e. the
Figure BDA00022186296000000612
The direction of the contact force of each contact pair is the average direction of the normal directions of the contact pairs, the contact pair comprises a th point on the th contour and a second point on the second contour, wherein the straight line connecting the th point and the second point passes through the mass center P1 of the th contour (therefore, the included angles formed by the straight line connecting the mass center P1 of different contact pairs and the mass center P1 of the th contour and the X-axis direction are different, and the included angle is the polar angle theta), for example, A6 on the th contour and B6 of the second contour, A6 and B6 are contact pairs.
Step S80, calculating the contact force F on each contact pairiI.e. Fi＝kSFinFiThe magnitude of the contact force of each contact pair is proportional to the area of each contact pair, with the area of the middle contact pair being -half the sum of the areas of the left and right regions thereofFiDenotes the area of each contact pair, k being the particle contact stiffness; in step S80, when Fi is a middle contact pair,
Figure BDA0002218629600000071
i.e. the area of each middle contact pair is half the sum of the areas of its left and right regions, where S isiObtained by integration, i.e.
Figure BDA0002218629600000072
Wherein the intermediate contact pair comprises pairs of discrete points (or may be understood as non-edge contact pairs) in the intersection region between the th discrete point and the last discrete points, such as illustrated by the 4 discrete points a4-a7 in fig. 5, the intermediate contact pair comprises A5B5 and A6B6, and for the intermediate contact-to-area calculation, for example, as shown in fig. 5, the contact pair A6B6 has an area SA6B6Hatched parts, i.e.
For the area of the edge contact pair, the area of the edge contact pair (e.g., A4B4, A7B7 in fig. 5) is the sum of the area of half of the middle region and the area of the edge region, and the effective area of the contact pair A4B4 in fig. 5 is:
Figure BDA0002218629600000075
in step S90, a contact force F between two particles is calculated, i.e. F ═ Fi. According to the method for calculating the contact force between the two-dimensional irregular particles, the contact force F between the two particles calculated finally is higher in precision, and the method has the following beneficial effects: the invention directly calculates the real contact area between irregular particles, calculates the contact force of each division area by dividing the embedded area, and then obtains the total contact force by superposition, and the obtained result has higher precision compared with the prior art.
The above description is only a preferred embodiment of the present invention, and not intended to limit the scope of the present invention, and all modifications of equivalent structures and equivalent processes, which are made by using the contents of the present specification and the accompanying drawings, or directly or indirectly applied to other related technical fields, are included in the scope of the present invention.

Claims (2)

1, A method for calculating contact force between two-dimensional irregular particles, which is characterized by comprising the following steps:
step S10, acquiring a th contour corresponding to the th particle, and acquiring a second contour corresponding to the second particle;
step S20, determining the th center of mass of the th contour, and determining the second center of mass of the second contour;
step S30, determining a distance from the point on the th contour to the th center of mass, determining a second distance from the point on the second contour to the second center of mass;
step S40, establishing a rectangular coordinate system, determining a th circumscribed rectangle corresponding to the th outline, and determining a second circumscribed rectangle corresponding to the second outline;
step S50, judging whether the circumscribed rectangle and the second circumscribed rectangle are intersected, if the circumscribed rectangle and the second circumscribed rectangle are intersected, executing step S60;
step S60, judging whether the th contour and the second contour in the intersection region formed by the th circumscribed rectangle and the second circumscribed rectangle are intersected, if the th contour and the second contour are intersected, performing discrete processing on the th contour in the intersection region to obtain each discrete point, and calculating the normal n of each discrete point1I.e. by
Figure FDA0002218629590000011
Calculating the normal n of points on the second contour associated with discrete points on the th contour in the intersection region2I.e. by
Figure FDA0002218629590000012
Wherein x is1And y1Is the coordinate of each point of the th contour on a rectangular coordinate, x2And y2Coordinates of each point of the second contour on the rectangular coordinate;
step S70, calculating the direction of each contact pair contact force, i.e. the
Step S80, calculating the contact force F on each contact pairiI.e. by
Figure FDA0002218629590000014
Wherein S isFiDenotes the area of each contact pair, k being the particle contact stiffness;
in step S90, a contact force F between two particles is calculated, i.e. F ═ Fi。
2. The two-dimensional irregular interparticle contact force calculation method of claim 1, wherein the contact pair comprises a th point on a th contour and a second point on a second contour, a straight line connecting the th point and the second point passes through a center of mass of the th contour, and the th point and the second point are both located in the intersection region.
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