THE PROGRAM FOR AUTOMATIC DETECTION OF THE UNIFORM DISTRIBUTION OF FIBERS IN A CONCRETE MATRIX

ABSTRACT

This article investigates the mathematical modeling of the uniform distribution of fibers within a concrete matrix. The study emphasizes the critical importance of fiber spatial arrangement in accurately and reliably assessing the mechanical properties of fiber-reinforced concrete. Mathematical formulations have been established for the central coordinates, orientation vector, and end points of fibers, while boundary conditions are defined to ensure that the fibers remain within the concrete volume. Based on this model, a specialized program was developed in Python to calculate the total number of fibers according to the given concrete dimensions and fiber parameters, as well as to visualize their random distribution in 3D space. The program not only enables comparison with experimental results but also provides opportunities for virtual experiments, thereby facilitating laboratory preparation, improving the evaluation of structural strength, and supporting the efficient design of new composite materials.

АННОТАЦИЯ

В данной статье исследуется математическое моделирование равномерного распределения волокон в бетонной матрице. Подчёркивается критическая важность пространственного расположения волокон для точной и надёжной оценки механических свойств фибробетона. Разработаны математические формулировки для центральных координат, вектора ориентации и конечных точек волокон, при этом определены граничные условия, гарантирующие нахождение волокон внутри объёма бетона. На основе данной модели была создана специализированная программа на языке Python, предназначенная для расчёта общего числа волокон в соответствии с заданными размерами бетона и параметрами волокон, а также для визуализации их случайного распределения в трёхмерном пространстве. Программа не только обеспечивает возможность сопоставления с экспериментальными результатами, но и открывает перспективы для проведения виртуальных экспериментов, что способствует подготовке лабораторных испытаний, улучшает оценку прочности конструкций и поддерживает эффективное проектирование новых композиционных материалов.

Keywords: fiber-reinforced concrete, mathematical modeling, uniform fiber distribution, concrete matrix, mechanical properties, Python program, 3D visualization, virtual experiments, composite materials.

Ключевые слова: фибробетон, математическое моделирование, равномерное распределение волокон, бетонная матрица, механические свойства, программа на Python, 3D-визуализация, виртуальные эксперименты, композиционные материалы.

Introduction

Mathematical modeling of the issue of uniform distribution of fibers in the concrete matrix is ​​of great importance in modern building materials research. This process is of great importance not only theoretically, but also practically, and plays a key role in correctly assessing and optimizing the mechanical properties of fiber-reinforced concrete[1]. If the fibers are located in concrete in a certain proportion and in a certain shape, their effect on the material will be different[2]. If the fibers are not evenly distributed in the matrix and are collected on one side, this will not ensure the strength of the concrete. Through mathematical modeling, it is possible to calculate the nature of the arrangement of fibers in concrete, their interaction, and direction[3]. Based on this modeling, the general mechanical properties of the material, including tensile, bending, and compression strength, are more accurately estimated. Uniform and arbitrary arrangement of fibers in concrete forms the isotropic or anisotropic properties of concrete. If it is possible to control this process through mathematical modeling, then accurate and reliable results can be achieved in calculations for structural elements[4].

In addition, mathematical modeling allows us to analyze in advance what changes will occur in the material by changing the geometry (length, diameter), elastic modulus and concentration of various fibers (e.g. steel, basalt, polypropylene). Studies have shown that fibers that are evenly distributed and whose length is adjusted to the dimensions of the concrete matrix increase the strength and durability of concrete at high tensile strength[5]. Otherwise, fibers that accumulate in one place can negatively affect the strength of the structure and local defects in some parts of the concrete[6]. Mathematical modeling also makes it possible to determine the optimal parameters for the method of introducing fibers into concrete in the real production process (e.g. manual mixing, machine mixing or various types of vibration). This reduces the number of experiments, saves time and costs, and allows the production of high-quality fiber-reinforced concrete[7, 8].

Method

Direction and distribution of fibers. The model randomly places each fiber in three coordinates in space and gives it a random orientation.

1. Center coordinates. The center of the fiber can be anywhere within the concrete:

,          ,                                    (1)

where: , , - coordinates, m; u(a,b) - this is a value distributed with the same probability in the interval [a, b], without unity.

2. Direction vector. The direction of the fibers in space should be arbitrary. Therefore, the vector obtained from the 3-dimensional normal distribution is reduced to unit length:

,                                                       (2)

where:

 – a random vector drawn from a three-dimensional normal (Gaussian) distribution, since it is a coordinate vector, its units are in units of length, m;

- vector length, m.

 –  direction vector of unit length (normalized), i.e., it indicates the direction in which the fiber lies. Since it is a direction vector, it has no units (dimensionless quantity). That is, it only indicates the direction.

3. Coordinates of fiber ends (start and end points). It is built based on the center and direction of the fiber:

,                                                 (3)

where:  - the point shifted from the center of the fiber by half a length back in the direction, m;

 - the point shifted forward from the center of the fiber by half a length in the direction, m.

  – coordinates of the central points of the fiber, m.

In the model, the fibers must remain within the concrete. Therefore, each fiber end must be checked to be within the following limits:

,               ,                                     (4)

If the fiber start or direction point extends outside the concrete volume, a new coordinate and direction are selected.

This mathematical model is based on physical-geometric vector formulas. This model involves drawing each fiber around a center, at a constant length and in a known direction, determining the fiber ends (beginning and ending), and placing them in a modeled 3D space with a mathematical basis.

Based on the experimental results and the developed mathematical model, a program was developed in the Python programming language to simulate the uniform distribution of fibers in concrete. This program calculates the total number of fibers and determines their coordinates based on the volume of concrete, fiber properties (length, diameter, density) and fiber percentage determined from the mathematical model. Each fiber is placed in three-dimensional space with random central coordinates and a random direction. The start and end points of the fiber are determined from this center by half the fiber length. The program controls that the two ends of the fibers do not extend beyond the boundaries of the concrete sample, thereby reflecting the real physical conditions in the model. The working window of the program that determines the uniform distribution of fibers in the concrete matrix is ​​shown in Figure 1.

Figure 1. View of the working window of the program that determines the even distribution of fibers in the concrete matrix

Using this program, it is possible to observe the spatial arrangement of fibers in a concrete sample not only theoretically, but also graphically. The number, density and orientation of fibers are visually depicted using a 3D graph. The program allows the user to create a model that is suitable for real conditions by entering any concrete dimensions, fiber parameters and percentages. This is very useful in preparing for laboratory tests, assessing the strength of structures and designing new composite materials. The program also allows for comparison with experimental results to validate the model. The fact that the distribution of fibers in the volume of concrete corresponds to experimental observations makes it possible to use it as a reliable tool. The program interface is simple and understandable, and in the future it can be improved and turned into an application with a more user-friendly graphical interface. The developed program can serve as a convenient platform for scientific research, and can be effectively used in virtual experiments and design work.

References:

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