Dimensioning of Fractal Fracture on a Concrete Slab

*Francisco Casanova-del-Angel*

## **Abstract**

What is presented here is a methodology that allows to study fractures in any material, especially in concrete elements. The importance lies in the moment in which the fracture that occurs in a structural element (slab in our case study) causes a negative effect on its structural behavior. Traditionally, its study is developed from the perspective of fracture mechanics, with which the energy at the tip of the crack or fracture is calculated, and it cannot go further. But the combination of the theories of fracture mechanics and fractal geometry allows us to obtain the patterns of its behavior in the future. Thus, the research was based on tests made on structural elements of concrete, from a slab led to structural failure and choosing a fracture. This was vectorized in order to obtain a fractal axis, which was called state 0, taking three parts or sections of that state and repeating them on a smaller scale. Subsequently, the research was based on five of the methods used for the study and calculation of the fractal dimension: box dimension, perimeter-area dimension, information dimension, mass dimension, and ruler dimension. Most of the fractal dimensions obtained under these methods were similar to each other.

**Keywords:** curved fractal, mechanical fracture, dimension of fracture, concrete, fractal geometry

## **1. Introduction**

I think it is necessary to start this article with some comments that have to do with the icon that represents a fracture, fraction, and the concept of fractal. For those seasoned in fractures and fractals, they can be childish, but not for architects and civil engineers. Why this brief comment? Well, the research covers two gigantic areas of knowledge: fractals and fracture mechanics. The molecular composition in humans, that is, small structures in large quantities, is the same in a single being. Geometric shapes with the same structure, which if split into equal parts, result in the same smaller part that remains the same or many of them to create a larger one with the same features.

A fractal is a geometric object with a basic structure, whether fragmented or irregular, repeated at various scales. Mathematically, a fractal may be expressed as follows: Let *<sup>I</sup>* = [0, 1] and let *<sup>A</sup>*<sup>1</sup> <sup>¼</sup> *<sup>I</sup>* � <sup>1</sup> <sup>3</sup> , <sup>2</sup> 3 be that subset of *I* obtained by removing those points which lie in the open middle third of *<sup>I</sup>*; that is, *<sup>A</sup>*<sup>1</sup> <sup>¼</sup> 0, <sup>1</sup> 3 ∪ <sup>2</sup> <sup>3</sup> , 1 . Let *<sup>A</sup>*<sup>2</sup>

be that subset of *A*<sup>1</sup> obtained by removing the open middle third of 0, <sup>1</sup> 3 and of <sup>2</sup> <sup>3</sup> , 1 . Continue this process and define *<sup>A</sup>*3, *<sup>A</sup>*4, … *:* The set *<sup>C</sup>* <sup>¼</sup> <sup>∩</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> *An* is called the Cantor set. Some characteristics are (i) *C* is a compact set having Jordan content zero, (ii) *C* is uncountable, and (iii) the characteristic function of *C* is Riemann integrable on [0, 1], see [1].

We should bear in mind that the fractal dimension may be calculated in various ways. One is through the Hurst exponent: many structures in nature have the feature to be originated in two dimensions and end in a fractional dimension between two and three [2, 3]. A fractal object should have at least one of the following features: (i) it should show similarity among details on small and large scales; (ii) it should not be able to be represented through classical geometry; (iii) it should have a fractional dimension, that is, a non-whole dimension; and (iv) it should be able to be recursively defined; see [4].

The classical theory of fracture mechanics allows to predict the quick propagation of a macro crack through a homogeneous, isotropic, and elastic material. In this theory, the stress intensity factor, *KI*, which is a function of the geometry of the crack and stress, is applied. Failure occurs when *KI* reaches a critical value *KIC*, known as the critical stress intensity factor in plane strain conditions. Therefore, *KIC* is a measure of the fracture hardness of the material. To properly measure *KIC* in a material, the sample to be examined must be of a size that allows the ensuring of maximum restriction (plane strain) on the tip of the crack. To apply linear elastic fracture mechanics (LEFM), the value of *KIC* must be a constant of the material, regardless of the geometry of the specimen (as are other constants of materials, such as yield strength).

Regarding the material used in the laboratory, it must be said that it has the adhesion and cohesion properties required to bond inert aggregates and make a solid mass with proper resistance and durability, as well as that for the fabrication of structural concrete the so-called hydraulic cements are exclusively used. In the current case, Portland cement has been used since it is a grayish material finely powdered, mainly formed by calcium aluminum silicates, see [5, 6]. When the cement is mixed with water to create a soft paste, it gradually becomes rigid until becoming a solid mass. This process is known as casting and hardening.

## **2. Laboratory tests**

The test included applying a load concentrated on the gravity center of a 70 cm wide and 500 cm long portion of a prefabricated slab, 20 cm thick with two open web joists on the ends and a small vault covering the span, **Figure 1**. The test required the construction of a slab portion, 70 cm wide and 500 cm long with two joists on the ends and small vaults covering the spaces between them, with a 6\*6–10/10 electro-welded mesh and a 3 cm thick concrete layer in the compression zone. The slab was supported by two 70 cm long and 170 cm high concrete blocks, low walls with reinforcing bar grids, and closure slabs armored with *armex* 15\*20–4, while to confine such low wall headframes were placed on the ends with *armex* 15\*20–4. The resistance to the compression of the concrete was 200 kg/cm<sup>2</sup> , see [7].

The steel required for the area was calculated for the construction of the 5 m long joist, with a 17 cm slope. Once the steel areas were obtained, preassembling, armoring, and casting of the joints were carried out together with the curing process, waiting for

#### **Figure 1.** *View of the slab before the first load.*

the concrete to reach its maximum strength. The joints and the small vaults were taken to the laboratory where the load frames were placed to place the prefabricated slab. First, the low walls to support the slab were constructed with a concrete block, founded on a reinforcing bar grid with a 15 cm base and 20 cm slope, and armored with *armex* 15\*20–4. To confine the low wall, headframes were placed on the ends, 15 cm by 20 cm, armored with *armex* 15\*20–4, and the same was used for the closure slab. After that, the required formwork was placed, the joists were fitted into the closure, and the small vaults were put in place. In the center of the span, a temperature nerve was placed and for the compression layer a 6\*6–10/10 electro-welded mesh was used and a 3 cm thick layer was placed at the closure, allowing the concrete to harden, *ibid*.

After 7 days, the formwork was removed from the slab, the concrete having been cured for days previously. After this and after the concrete reached its maximum strength, which occurred 28 days after casting the slab, the test started. To be precise in obtaining the results, the load cell was first calibrated in the computer. To do this, a wooden pole was placed on which the load was applied. In the computer, it was observed if the load and strain were registered, which were correct. Then, the load was taken away and all data involved in the test were marked as zero. A smooth steel rod was placed on the central axis of the slab and with such a steel a square profile was placed to reach the height of the frame load piston, which in turn supported the 200 ton load cell. At last, we have the piston applying the load on the frame. Under the slab and on the sides of the low walls supporting the slab, micrometers were placed to measure warping and lateral strain. To easily observe the cracks, the slab's lower part was painted white.

To start, a 10-ton load was applied. When the first cracks appeared, they were colored with a green marker. Then, a second 10-ton load was applied. New cracks appeared and the existing ones were enlarged. Those were colored with a red marker. A 5-ton load was then applied. More cracks appeared and the existing ones were

**Figure 2.** *Evolution of cracking as the load was applied.*

enlarged, which were then colored blue. Finally, a final 5-ton load was applied. No new cracks appeared, only the existing ones were enlarged. Those were colored brown. Days after that, the test was continued applying load until the stab collapsed, **Figure 2**.
