2. Fracture toughness calculation of straight-through notched specimens

To determine the global failure conditions, irrespective of the material state and geometrical dimensions of the specimen, the energy approach is appropriate [12]. The key point of the energy fracture criterion in fracture mechanics is formulated as follows: the crack growth can occur if the system can release energy required to initiate crack propagation to the elementary distance dl. The energy needed for crack growth, appears only due to the elastic deformation energy that occurs in-side the material under the applied external force.

The reliable fracture toughness characteristic is the critical value of the elastic energy release rate (ERR) during the crack propagation Gc. In the two-dimensional version, this characteristic is equivalent to the value of the J-integral [13, 14]. For brevity, the value of G will be called specific fracture energy. The specific fracture energy (SFE) is defined as energy that is spent on the formation of the crack surface with area 1 m2 . The unit of measurement for SFE is J/m<sup>2</sup> .

In practice, there is a decrease in specimen stiffness or structure at initiation and propagation of the crack. The specimen stiffness M is defined as the ratio of load P, applied to the specimen, to the displacement of load application point λ<sup>е</sup> at elastic deformation of the specimen: М = P/λе.

The reciprocal of the stiffness is defined as a specimen compliance η: η = λе/P.

(SIF) of the 1st kind КI<sup>с</sup> (for cleavage crack) is taken for the main fracture toughness characteristics of the material. The plane strain state condition of the loaded specimen is required in the experiments. In this regard, standard tests are conducted on specimens at least 10 mm thick.

1. There is no possibility to assess the fracture toughness of the material when testing the

5. Use of significant amount of the complex-shaped specimens (cut out by layers, holes,

8. Need for periodic unloading of the specimen to determine the change in specimen com-

9. Availability of phenomenological constants in constitutive equations, taking into account

As a rule, during a fracture toughness test of small-size specimens, the chevron-notched specimens are used [7–11]. The specimens with this configuration do not require the preliminary fatigue crack. When testing the small-size chevron-notched specimens, many of the

This chapter proposes a new method for fracture toughness determination of structural materials using the small-size chevron-notched specimens. The method allows us to determine fracture toughness characteristics without severe restrictions on the specimen ability to plastic deformation. There are no phenomenological dependencies and empirical constants in the

The fracture toughness characteristics comply with the conditions of continuous loading of

The important calculation works were carried out associated with the use of chevron-notched

2. Fracture toughness calculation of straight-through notched specimens

To determine the global failure conditions, irrespective of the material state and geometrical dimensions of the specimen, the energy approach is appropriate [12]. The key point of the energy fracture criterion in fracture mechanics is formulated as follows: the crack growth can

2. Complexity of the mechanical processing and manufacturing of specimens;

Disadvantages of used methods are as follows:

3. Requirement for the fatigue pre-cracking on the notch;

the geometric shape and boundary loading conditions.

specimens, without using the "loading-unloading" operation.

small thickness specimens;

notches);

216 Contact and Fracture Mechanics

calculations.

specimens during testing.

4. Availability of special test equipment;

6. Need for the high power load device; 7. High steel intensity of test specimens.

pliance under loading.

above-mentioned problems are absent.

The necessary condition for through-crack propagation in the flat specimen of unit thickness obeys the equation [12–14]

$$G = \frac{P^2 d\eta}{2dl},\tag{1}$$

where dη/dl is the change in specimen ductility during crack propagation, dl is the short distance, to which a straight-line crack front propagates. At the stage of stable crack propagation, this value characterizes the fracture toughness G<sup>c</sup> of the material. As follows from Eq. (1), elastic energy per unit of new crack surface at its propagation to dl in the specimen in thickness а is equal to

$$G = \frac{P^2 d\eta}{2a dl} = \frac{P^2 d\eta}{dS},\tag{2}$$

where dS = 2adl is the elementary increment of the crack surface area.

There is a classic example for calculating the stress intensity factor KI to test a double cantilever beam specimen with a straight-through notch [15]. The relation between G and KI for the plane stress state obeys the equation

$$G = \mathbf{K}\_{\rm I}^2 \left(1 - \mathbf{v}^2\right) / E. \tag{3}$$

Let us consider the case of double-cantilever beam specimens with a straight-through notch in detail, since the result will be used in the calculation of G for the chevron-notched specimens.

Figure 1 presents the double-cantilever beam specimens with a straight-through notch. Distance from the load application point P to the crack front is the initial crack length l. As follows from the cantilever bending theory, displacement of the load application point λ<sup>e</sup> (Figure 1) is equal to <sup>λ</sup><sup>e</sup> <sup>¼</sup> <sup>4</sup><sup>P</sup> Ea l b <sup>3</sup> , where E is the Young's modulus, b is the cantilever thickness. For the double cantilever beam specimen, displacement of load application points ζ is 2λe. Therefore, the specimen ductility <sup>η</sup> is <sup>η</sup> <sup>¼</sup> <sup>ζ</sup> <sup>P</sup> <sup>¼</sup> <sup>8</sup> Ea l b <sup>3</sup> :

Figure 1. Straight-through-notched specimen.

The ductility derivative by the crack length is equal to

$$\frac{d\eta}{dl} = \frac{24l^2}{Eab^3}.\tag{4}$$

3. Fracture toughness calculation of the chevron-notched specimens

determine fracture toughness under plane strain conditions [17].

Figure 2. Loading configuration of the chevron-notched specimen.

presented in Figure 2.

When testing the small-size specimens, generally, the chevron-notched specimens are used [7, 17]. For the first time, a chevron-notched specimen was proposed by L.M. Barker in 1977 to

Determination of Fracture Toughness Characteristics of Small-Size Chevron-Notched Specimens

http://dx.doi.org/10.5772/intechopen.72643

219

Standard tests of the chevron-notched specimen are conducted according to the scheme

Figure 3 shows examples of the chevron-notched specimens. The specimens of this configuration do not require the preliminary guidance of a fatigue crack to the tip of the notch. From the moment of loading, there is a high stress concentration at the tip of the chevron notch that is sufficient to crack initiation. It is assumed that the development of plastic deformation in the chevron zone satisfies the plane strain state condition. The crack initiated at the tip of the chevron, can propagate only along the notch plane. At the same time, there is a high probability that the crack front during propagation, at the average, maintains a straight shape. The chevron notch geometry allows us to fix and extend the stable crack propagation stage and,

thus, to calculate the beginning of the specimen catastrophic failure more accurately.

Figure 3. The chevron-notched specimens: short circular specimen (a) and elongated rectangular specimen (b).

As a rule, short circular specimens are used in the experiments (Figure 3a). The disadvantage of the standard method for measuring fracture toughness of the chevron-notched specimens is that in order to determine the change in specimen ductility, the "loading-unloading" condition

Substitution of Eq. (4) into Eq. (1) leads to the following expression for SFE [13, 15]

$$G = \frac{12P^2l^2}{Eb^3a^2}.\tag{5}$$

According to Ref. [14], the displacement λ<sup>е</sup> of the cantilever end under elastic deformation for the specimen in thickness of a with a crack length l is provided by the load:

$$P = \frac{E\lambda\_{\text{e}}a}{4} \left(\frac{b}{l}\right)^3. \tag{5a}$$

Substituting this expression into Eq. (4), we obtain the equation for G that allows us to calculate fracture energy by the crack length l and by the elastic opening value of the notch tips λ<sup>е</sup> [16]:

$$G = \frac{3\lambda\_{\text{e}}^2 b^3 E}{4l^4}.\tag{6}$$

In the given representation, the value of G does not depend on the specimen thickness а.

Eq. (5) determines SFE by the crack length l, and by the external load value Р, at which spontaneous crack propagation begins. Basically, SFE can be calculated according to Eq. (6) when testing the small-size specimens. It should be noted that there are no any empirical constants in Eq. (6). All necessary values can be taken from the experiment. To maintain the experimental integrity, one can grow a fatigue crack at the tip of the notch. However, this method has several disadvantages.

Eq. (5) gives only a rough approximation of the SFE value. There is a certain divergence due to the fact that the cantilevers' ends are fixed not absolutely rigidly, as in the cantilever embedded in the rigid base. But this is not the most important thing. The main disadvantage is that in practice crack propagation along the notch plane is not guaranteed. Consequently, deviations in crack propagation direction cause shear deformations. Besides, crack front straightness is not preserved. To some extent, this problem is solved by using the chevron-notched specimens.
