## Abstract

This chapter presents a model of a stress and displacement fields around a rough crack tip in brittle and ductile materials using a pre-fractal model. This approach allows for a more realistic fractal scaling, of a real fracture. A special stress vector for rugged surfaces was defined, and it was also shown that the pre-fractal consideration also results in different asymptotic limits for the singularity degree of the stress field at the crack tip for brittle materials. The asymptotic limit obtained here differs from the singularity degrees presented by other authors. Therefore, other consequences such as fracture instability are also included in the mathematical model presented here. A generalization of the stress fields for brittle and ductile materials is proposed. Changes in geometric shapes of the stress field are due to the roughness at the crack tip as shown by the mapping of simulated stress fields around a crack tip. The fractal stress field functions were mapped at different singularity power degrees indicating that the qualitative aspects of these fields alone, do not sufficient to determine which model presents the best fit to experimental results. Therefore, the model need a validation based on quantitative experimental measurements.

### Keywords

- stress field
- fractal fracture mechanics
- fractal dimension
- Hurst dimension
- stress intensity factor

## 1. Introduction

In classical fracture mechanics theory, a brittle material has an asymptotic stress field, whose singularity exponent depends on the distance from the crack tip, which Hutchinson [1] define as :

where

Mosolov [2–4] attempted to explain crack growth under compression based on the fractality of cracks. He was the first researcher to study and theorize about changes in the stress field singularity magnitude according to the fractality of rough crack surfaces in mesoscale dimensions. He showed that the stress at the crack tip of a fractal crack under uniform compressive stress is unique. Mosolov [2–6] stated that the singularity in the stress field in front of a rough fractal crack should be corrected by means of a fractional exponent. He also suggested that the elastic field ahead of the crack tip should have a fractional singularity exponent associated to the asymptotic dependence on the distance, represented by [2–4]:

This is because the stress field is primarily responsible by generate the noise of a crack or fracture surface in a solid body subjected to the action of a strain loading which is transferred for the formation of new surfaces through of an elastic or elastic-plastic energy releasing rate in the fracture phenomenon, as follows:

After Mosolov’s observation [2–6], Borodich [7–8] began to develop a fracture criterion involving the roughness of a crack based on fractal theory. These two authors established the mathematical relationships between the elastic stress field around a crack or fracture surface and the fractal roughness exponent of a fracture surface, using the dependence of the fractional exponents of singularity of the field at the crack tip and the fractional dependence of the fractal exponents of the scaling of fracture surfaces. Using the Griffith criterion, and considering the fact that the actual length of a fractal crack is larger than its apparent size, they came up with the asymptotic expression for a self-similar fractal crack in Mode I. They did this by comparing the stress field, as shown in the following equation:

Where

Cherepanov and Balankin [11–13] attempted to define a relationship between the exponent *α* suggested by Mosolov [2–4] and the fractal topology of a rough crack. Later, Balankin [12, 13] published a series of papers proposing a complete modification of fracture mechanics, from linear elasticity theory to nonlinear elastic fracture mechanics. Based on a dimensional analysis such as that shown in (4), Balankin [13] found in self-affine fractal cracks the same degree of singularity of the stress field as that proposed by Mosolov for a self-similar fractal crack. The relationship proposed by Balankin [12, 13] was as follows:

where

Realizing the limitations of the Mosolov model [2, 3], Balankin [12, 13] and Yavari [14–16] proposed a new approach to stress field singularity exponents and developed his model to its ultimate consequences. He even proposed two new fracture modes and a discrete fracture mechanics that combined the fractal theory with quantum aspects of the fracture described by Pugno and Ruoff [17]. Yavari [14–16] also discussed the earlier expressions proposed by Mosolov [7] in (2), as well as the model proposed by Balankin [12]. However, unlike Mosolov [2, 3], he found a relationship between the singularity exponent using (4) and substituted the box-dimension

Yavari [14, 15] then proposed a systematic approach for calculating the magnitude of stress field and singularity degree for fractal cracks, using the force lines method, which is applicable to all fracture modes. He considered the three classical fracture modes for fractal cracks and made a more in-depth investigation of the problem of stress singularity at the crack tip. Alternatively, however, this chapter proposes a new approach to the problem of stress field singularity, considering a geometrical instability coefficient,

where the new singularity exponent

Eqs. (7) and (8) are consistent with the experimental measurements and mathematical developments made to date [18–25]. These mathematical results are considered the most acceptable and realistic corrections and are therefore adopted and developed throughout this chapter. The proposal presented here is based on the relationship between the rough crack length and its Euclidean projection, which is given by Alves [18–25], i.e., the rough crack length was considered as

## 2. Theoretical development: analysis of the influence of the crack roughness on the stress field at the crack tip

The use of fractal measure theory to model fracture phenomena is of great interest in the mathematical development of Fractal Fracture Mechanics. The roughness of a real crack extends within a finite limit ranging from a lower to an upper fractal cut-off scale. However, the fractal models for fracture proposed by several authors presuppose a fractal crack with an infinite range of self-affine scales that cannot be verified experimentally. It is known that a real crack has limits imposed by a finite number of fractal scales ranging from a lower fractal cut-off scale to an upper fractal cut-off scale. In a real fracture, the finite limits of fractal scales are always compatible with the minimum microcrack and with the length of the macroscopic crack in the sample. Thus, the lower fractal cut-off scale is given by the characteristic size of a microcrack, while the length of the macroscopic crack on the specimen indicates the upper fractal cut-off scale. Therefore, fractal models of fracture phenomena require a more realistic approach that takes into account the limits of scales and singularity exponents, and hence, the fractal scales that are actually found in a real fracture.

To model the real movement of the crack tip, one must consider a infinitesimal element of length of a rough crack with its corresponding flat Euclidean projection, as shown in Figure. Thus, all the quantities previously considered and associated with the projected length of a crack must be rewritten in terms of the actual rough crack length. Hence, the stress field will be modified by some function of the type:

Consequently, all the other quantities of the fracture mechanics will also be modified analogously. As can be seen in Figure 1, a crack grows with variations in the coordinates of its tip that follow a rough path, having both a microscopic and a macroscopic velocity, which produce the surface roughness and its macroscopic path, respectively.

Figure 2 presents experimental results obtained by Ohr in 1985 [26]. These results demonstrated experimentally that the aspect of the stress field in front of a crack appears deformed because of its roughness. Also, according to Xin [27], inclined crack models show that the appearance of the stress field ahead of the crack tip is distorted relative to a non-inclined crack. This supports the proposal of this chapter that roughness affects both the intensity of the field as its geometric aspect.

The geometric instability coefficient of a crack with a fractal roughness, which is given in (7), can be generalized based on the following arguments: (i) experimental observations; (ii) previous arguments about the influence of roughness on intensity; (iii) geometric aspect of the stress field at the crack tip; and (iv) the coherent mathematical development; and a correction can be proposed for the mathematical solution of the rough stress field, using a quantity *geometric instability coefficient of the crack*, as follows:

This coefficient reflects the interaction between the local and global aspects of the crack length.

Considering a self-affine fractal crack, according to Alves et al. [18, 22, 23], the coefficient of geometric instability, which depends on the roughness at the crack tip, is given as follows:

Since there is a relationship between the distance

In regions near the crack tip (

The geometric instability coefficient of a crack in its complex form can be defined as:

where the Cartesian coordinates

Therefore, one can propose that the complex function of Westergaard,

Its polar form

The graphic of

for

## 3. Modeling the fractal stress field around a rough crack

There are three independent movements that correspond to the three fundamental fracture modes, as pointed out by Irwin [29]. These basic fracture modes are usually called Mode I, Mode II, and Mode III, respectively, and any fracture mode in a cracked body may be described by one of the three basic modes, or by combinations thereof.

The Airy-Westergaard function for the stress field with fractal roughness can be determined based on the foregoing arguments about the influence of roughness on the intensity and the geometrical aspect of the stress field at the crack tip. Based on the mathematical development performed here, we propose that the Westergaard complex function,

where the three parameters

and grouping similar terms, one has:

Considering the above conditions, we propose the following stress function due to the dominant term of the stress function around the tip of a fractal crack. Therefore, the potential function

where

### 3.1. Calculation of the stress field with fractal roughness

#### 3.1.1. Solution of stress fields and displacement around the tip of a fractal crack in Mode I ( G I C → K I C )

In *Mode I,* one has a two-dimensional tensile stress at infinity

Note that we have applied a constant load stress in the

Analogous to what Westergaard discussed regarding various Mode I crack problems, these problems involving fractal roughness can be solved as follows:

where

If the elastic problem can be arranged so that the crack of interest extends over a line segment of the

Note that this particular Westergaard formulation is restricted to solutions that have the properties of

In addition,

where

The term

This term was used to relate

These quantities, in turn, can be substituted in Eq. (24), resulting in the asymptotic solution associated to the stresses, which is valid for both plane stress and plane strain in Mode I loading. By applying equations in (24), one has:

where

and

Similarly, following the same procedure, in plane stress and plane strain loading conditions, the asymptotic displacements or deflections

## 4. Results and analysis of a mapping of the stress field of a rough fractal fracture

To better illustrate the results obtained by Mosolov [2–4], Yavari [14, 15], and Alves, the classical fractal and stress fields at the tip of a crack in Mode I loading were mapped in this chapter in order to compare them qualitatively with the approach of other authors.

### 4.1. Mapping based on analytical results of fractal fractures, varying Yavari’s exponent of singularity for Mode I loading

This section describes the calculation of the stress field for Mode I loading according to the model proposed by Yavari, where

where

Substituting the values of

where

Figures 4, 5, and 6 illustrate the stress fields around a crack tip with fractal singularity degree:

The idea of determining the degree of singularity of the stress field at the fractal crack tip proposed by Mosolov [2–4] and later by Yavari [14, 15] starts from the stress field expression. This, in turn, depends on the radius vector

The calculations performed by Yavari [14, 15] are mathematically correct, but do not correspond to physical reality, in the interval for Hurst exponent of

### 4.2. Mapping based on analytical results of fractal fractures, varying Alves’s exponent of singularity for Mode I loading

This section describes the calculation of the stress field for Mode I loading according to the model proposed in this chapter for

where

The stress field for

## 5. Discussions

A comparison of the fractal model proposed by Alves et al. [23] and the model proposed by Mosolov-Borodich [2–4, 7, 8] and Yavari [14, 15], published in the literature, leads to the following conclusions about these three theories.

In the Mosolov-Borodich model [2–4, 7, 8], cracks are treated as true mathematical fractals that extend within an infinite range of scales:

In the model of Yavari a non-root square was took at the Eqs. (15), (16) and (18) until (21), whereas in the model proposed by Alves et al. [23] an exact root square was took to maintain the symmetric dependence of the radius vector

The model of Mosolov-Borodich [2–4, 7–9], Yavari [14, 15], and Carpinteri [30, 31] at the limit of small scales, uses the renormalization theory to calculate infinitesimals in order to satisfy the Griffith criterion. Conversely, the fractal model proposed by Alves et al. [23], at the limit of small scales, uses the calculation on the *J‐R* curve is only locally independent of the path, because the integration of its radius depends on the form r^{−D}, Conversely, in the fractal model proposed by Alves et al. [23], the *J-R* curve is totally independent of the path because its integration does not depend on either the radius

### 5.1. Influence of the local roughness of a fracture surface on the stress field at the crack tip

The graph in Figure 10 was created by comparing the values of the Hurst exponent for the models proposed here by Alves with those of the Yavari model [14, 15]. This graph clearly shows a discontinuity in the value of the singularity of the field

When a ductile material undergoes stresses before fracturing, it produces defects that interact with pre-existing defects, causing it to become brittle before it fractures. If the material is subjected to a high loading rate it can harden even further, creating more defects and producing a higher level of stress than it displayed its initial stage. This strain hardening, which is the result of the pile-up and interaction of these defects, mainly dislocations, undoubtedly changes the degree of singularity,

On the other hand, a brittle material does not undergo strain hardening before it breaks. The material breaks upon reaching ultimate failure. In this case, roughness is simply an effect of the interaction between the crack and the microstructure of the material, where the crack may be intergranular, transgranular or mixed, depending on the material's internal mechanical strength relative to the energy imposed by the load as the crack propagates. However, if the loading rate on a brittle material is increased, roughness may increase due to the nonlinear effect of the interaction of the stress field on the microstructure of the material. In this case, cracks may appear rougher

In a more ductile material, as the stress level increases in response to the emergence of defects in the crack tip, i.e., as the material hardens, this stress level increases due to the increase in the degree of singularity

## 6. Conclusions

The proposed model describes the stress field around a crack tip for brittle materials.

The model proposed by Mosolov-Balankin-Yavari is insufficient to portray field intensity variations around the crack tip due to roughness in the interval for Hurst exponent of

The analysis of stress field fractal models based on the coefficient of geometric instability,

In the model presented here, the exponent

The fractal behavior of the stress field can be characterized as a fractal within a finite range of scales (

The asymptotic limit for the singularity of the stress field can lead to other results if the range of scales is not considered.

The model can provide a narrower reliability limit for the fracture stress of the brittle materials.