3. Fracture toughness calculation of the chevron-notched specimens

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 determine fracture toughness under plane strain conditions [17].

Standard tests of the chevron-notched specimen are conducted according to the scheme presented in Figure 2.

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.

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

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

The ductility derivative by the crack length is equal to

Figure 1. Straight-through-notched specimen.

218 Contact and Fracture Mechanics

dη dl <sup>¼</sup> <sup>24</sup><sup>l</sup>

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

the specimen in thickness of a with a crack length l is provided by the load:

<sup>G</sup> <sup>¼</sup> <sup>12</sup>P<sup>2</sup>

Eb<sup>3</sup>

According to Ref. [14], the displacement λ<sup>е</sup> of the cantilever end under elastic deformation for

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]:

> <sup>G</sup> <sup>¼</sup> <sup>3</sup>λ<sup>2</sup> e b3 E 4l

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.

<sup>P</sup> <sup>¼</sup> <sup>E</sup>λe<sup>a</sup> 4

2

l 2

b l <sup>3</sup>

Eab<sup>3</sup> : (4)

<sup>a</sup><sup>2</sup> : (5)

: (5a)

<sup>4</sup> : (6)

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

should be carried out. For this reason, the calculation formulas include adjustable coefficients considering the non-linear behavior of the material and a complex geometric shape of the notch.

This section provides a new method for calculating the fracture toughness characteristics of materials when testing the chevron-notched specimens. The novelty is in the fact that the calculation of SFE is based on calculation of the energy and power parameters of the specimen taking into account the complex geometry of a chevron notch according to the strict laws of solid mechanics. It is convenient to make calculations and experiments for the elongated rectangular specimen (Figure 3b). In this case, the chevron-notched specimen has a shape of a double cantilever beam configuration.

Let us determine the relation of the external force Р with an elastic deflection λ<sup>e</sup> of the single cantilever of the specimen. The cantilever can be represented as a set of elementary cantilevers (minicantilevers) of the infinitely small width dx. Figure 4 shows the projections of the chevron-notched specimens. The minicantilever length at a distance х from the symmetry axis of the specimen is l(x) = l<sup>0</sup> + x�ctg(α/2), where l<sup>0</sup> is the minimum distance from the load application point to the chevron notch, α is the chevron angle (Figure 4). For each minicantilever in the set, the well-known elasticity theory formula is valid [14]:

$$
\lambda\_{\rm e} = \frac{4dP(\mathbf{x})}{Ed\mathbf{x}} \left(\frac{l(\mathbf{x})}{b}\right)^3,\tag{7}
$$

where λ<sup>e</sup> is the elastic deflection of the minicantilever, dP is the elementary load, under the action of which the cantilever in thickness of dx is deflected to the value of λe, b is the thickness of the minicantilever. In view of this, from Eq. (6) we obtain the elementary load dependence dP, applied to the end of the minicantilever on its width dx:

$$dP(\mathbf{x}) = \frac{\lambda\_{\mathbf{e}} E}{4} \left(\frac{b}{l(\mathbf{x})}\right)^3 d\mathbf{x}.\tag{8}$$

Basically, this specimen with a crack is a set of two specimens: with a straight-through notch in width of h and with a chevron notch in width of а � h (Figure 5). Let us determine the forces Р<sup>0</sup> and Р<sup>00</sup> for these specimens, respectively, causing identical deflection λe. Using Eq. (5), we find an expression for Р<sup>0</sup> acting on a straight-through notched specimen in width of h = 2Δl�tg(α/2):

Figure 5. Presentation of the specimen with a crack in the form of straight-through and chevron-notched specimens.

<sup>4</sup> tg

According to Eq. (9), for the chevron-notched specimen in width of а � h, we obtain an

l0 4 þ a l0 ctg α 2 þ 2Δl l0 <sup>2</sup> <sup>þ</sup>

l b <sup>3</sup> k�<sup>1</sup>

α 2 b l <sup>3</sup>

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

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221

: (11)

a l0 ctg α 2

, (13)

<sup>2</sup>

: (12)

<sup>P</sup><sup>0</sup> <sup>¼</sup> <sup>λ</sup>eEΔ<sup>l</sup>

expression for Р00:

P00

<sup>¼</sup> <sup>λ</sup>eΔ<sup>l</sup> � tg <sup>α</sup>

4

Figure 4. Projections of chevron-notched specimen.

2 E

b l <sup>3</sup>

<sup>1</sup> � <sup>2</sup>Δ<sup>l</sup> <sup>a</sup> tg α 2

On the basis of Eqs. (11) and (12), an expression for λ<sup>е</sup> is determined:

l

<sup>λ</sup>eðÞ¼ <sup>l</sup> <sup>4</sup><sup>P</sup> Ea

Integration of elementary forces (7), affecting each minicantilever across the specimen width а, will clearly determine the actual load Р, providing the minicantilever's deflection by λe:

$$P = \frac{\lambda\_e b^3 E}{4} \int\_{-\frac{a}{2}}^{\frac{c}{2}} d\mathbf{x} / \left(l\_0 + \mathbf{x} \cdot \text{ctg } \frac{a}{2}\right)^3,\\ \text{or}\\ P = \frac{E \lambda\_e a}{4} \left(\frac{b}{l\_0}\right)^3 \left[4 + \frac{a}{l\_0} \text{ ctg } \frac{a}{2}\right] \\ \Big/ \left[2 + \frac{a}{l\_0} \text{ ctg } \frac{a}{2}\right]^2 = \frac{E \lambda\_e a}{4} \left(\frac{b}{l\_0}\right)^3 k' \tag{9}$$

This equation differs from Eq. (5) from the straight-through notch only by the factor

$$k'=l\_0[4l\_0+a\coth\left(\alpha/2\right)]\left[2l\_0+a\coth\left(2\right)\right]^{-2}.\tag{10}$$

According to Eq. (9), a single cantilever elastic deflection of a double cantilever beam specimen with a chevron notch is λeð Þ¼ l<sup>0</sup> 4P Ea l0 b � �<sup>3</sup> k 0�<sup>1</sup> :

During the loading, the moment of crack initiation occurs at the tip of the chevron notch. Propagation of the initiated crack to the distance Δl increases the effective fracture length. Let us present the crack front as a straight line (Figure 5). The length of this line is h = 2Δl�tg(α/2). Determination of Fracture Toughness Characteristics of Small-Size Chevron-Notched Specimens http://dx.doi.org/10.5772/intechopen.72643 221

Figure 4. Projections of chevron-notched specimen.

should be carried out. For this reason, the calculation formulas include adjustable coefficients considering the non-linear behavior of the material and a complex geometric shape of the notch. This section provides a new method for calculating the fracture toughness characteristics of materials when testing the chevron-notched specimens. The novelty is in the fact that the calculation of SFE is based on calculation of the energy and power parameters of the specimen taking into account the complex geometry of a chevron notch according to the strict laws of solid mechanics. It is convenient to make calculations and experiments for the elongated rectangular specimen (Figure 3b). In this case, the chevron-notched specimen has a shape of a

Let us determine the relation of the external force Р with an elastic deflection λ<sup>e</sup> of the single cantilever of the specimen. The cantilever can be represented as a set of elementary cantilevers (minicantilevers) of the infinitely small width dx. Figure 4 shows the projections of the chevron-notched specimens. The minicantilever length at a distance х from the symmetry axis of the specimen is l(x) = l<sup>0</sup> + x�ctg(α/2), where l<sup>0</sup> is the minimum distance from the load application point to the chevron notch, α is the chevron angle (Figure 4). For each minicantilever in the

> <sup>λ</sup><sup>e</sup> <sup>¼</sup> <sup>4</sup>dP xð Þ Edx

dP xð Þ¼ <sup>λ</sup>e<sup>E</sup> 4

, or<sup>P</sup> <sup>¼</sup> <sup>E</sup>λe<sup>a</sup> 4

4P Ea l0 b � �<sup>3</sup> k 0�<sup>1</sup> :

where λ<sup>e</sup> is the elastic deflection of the minicantilever, dP is the elementary load, under the action of which the cantilever in thickness of dx is deflected to the value of λe, b is the thickness of the minicantilever. In view of this, from Eq. (6) we obtain the elementary load dependence

Integration of elementary forces (7), affecting each minicantilever across the specimen width а, will clearly determine the actual load Р, providing the minicantilever's deflection by λe:

> b l0 � �<sup>3</sup>

<sup>k</sup><sup>0</sup> <sup>¼</sup> <sup>l</sup>0½ � <sup>4</sup>l<sup>0</sup> <sup>þ</sup> <sup>a</sup> ctg ð Þ <sup>α</sup>=<sup>2</sup> ½ � <sup>2</sup>l<sup>0</sup> <sup>þ</sup> <sup>a</sup>ctgα=<sup>2</sup> �<sup>2</sup>

According to Eq. (9), a single cantilever elastic deflection of a double cantilever beam specimen

During the loading, the moment of crack initiation occurs at the tip of the chevron notch. Propagation of the initiated crack to the distance Δl increases the effective fracture length. Let us present the crack front as a straight line (Figure 5). The length of this line is h = 2Δl�tg(α/2).

This equation differs from Eq. (5) from the straight-through notch only by the factor

l xð Þ b � �<sup>3</sup>

b l xð Þ � �<sup>3</sup>

<sup>4</sup> <sup>þ</sup> <sup>a</sup> l0 ctg <sup>α</sup> 2 � ��

, (7)

dx: (8)

<sup>¼</sup> <sup>E</sup>λe<sup>a</sup> 4

: (10)

b l0 � �<sup>3</sup>

k<sup>0</sup> (9)

<sup>2</sup> <sup>þ</sup> <sup>a</sup> l0 ctg <sup>α</sup> 2

� �<sup>2</sup>

double cantilever beam configuration.

220 Contact and Fracture Mechanics

<sup>P</sup> <sup>¼</sup> <sup>λ</sup>eb<sup>3</sup> E 4

ða 2 �a 2

with a chevron notch is λeð Þ¼ l<sup>0</sup>

dx<sup>=</sup> <sup>l</sup><sup>0</sup> <sup>þ</sup> <sup>x</sup> � ctg <sup>α</sup>

� �<sup>3</sup>

set, the well-known elasticity theory formula is valid [14]:

dP, applied to the end of the minicantilever on its width dx:

2

Figure 5. Presentation of the specimen with a crack in the form of straight-through and chevron-notched specimens.

Basically, this specimen with a crack is a set of two specimens: with a straight-through notch in width of h and with a chevron notch in width of а � h (Figure 5). Let us determine the forces Р<sup>0</sup> and Р<sup>00</sup> for these specimens, respectively, causing identical deflection λe. Using Eq. (5), we find an expression for Р<sup>0</sup> acting on a straight-through notched specimen in width of h = 2Δl�tg(α/2):

$$P' = \frac{\lambda\_{\text{e}} E \Delta l}{4} \text{tg } \frac{\alpha}{2} \left(\frac{b}{l}\right)^3. \tag{11}$$

According to Eq. (9), for the chevron-notched specimen in width of а � h, we obtain an expression for Р00:

$$P' = \frac{\lambda\_{\text{c}} \Delta l \cdot \text{tg } \frac{\alpha}{2} E}{4} \left(\frac{b}{l}\right)^3 \left(1 - \frac{2 \Delta l}{a} \text{tg } \frac{\alpha}{2}\right) \frac{l}{l\_0} \left[4 + \frac{a}{l\_0} \text{ctg } \frac{\alpha}{2} + \frac{2 \Delta l}{l\_0}\right] \Big/ \left[2 + \frac{a}{l\_0} \text{ctg } \frac{\alpha}{2}\right]^2. \tag{12}$$

On the basis of Eqs. (11) and (12), an expression for λ<sup>е</sup> is determined:

$$
\lambda\_{\rm e}(l) = \frac{4P}{Ea} \left(\frac{l}{b}\right)^3 k^{-1} \,\tag{13}
$$

where Р = Р<sup>0</sup> + Р00, l = l<sup>0</sup> + Δl and k is

$$k = \frac{2\Delta l}{a}\text{ tg }\frac{\alpha}{2} + \frac{l}{l\_0}\left(1 - \frac{2\Delta l}{a}\text{ tg }\frac{\alpha}{2}\right)\left(4 + \frac{a}{l\_0}\text{ ctg }\frac{\alpha}{2} + \frac{2\Delta l}{l\_0}\right)\left[2 + \frac{a}{l\_0}\text{ ctg }\frac{\alpha}{2}\right]^{-2}.$$

It is easy to verify that at Δl ! 0, the value of k ! k<sup>0</sup> .

Substituting Eq. (13) for λ<sup>е</sup> into Eq. (6), we obtain the expression for SFE:

$$\mathbf{G} = \frac{12P^2l^2}{Eb^3a^2}k^{-2}.\tag{14}$$

the width of the specimen central part h = 2Δl�tg(α/2) shown in Figure 5 instead of a. Let us find

EΔl � tg ð Þ α=2

<sup>λ</sup><sup>e</sup> <sup>¼</sup> <sup>υ</sup>e2ð Þ <sup>l</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup><sup>l</sup> <sup>3</sup> ð Þ 3l<sup>0</sup> þ 2Δl Δl

<sup>P</sup><sup>0</sup> <sup>¼</sup> <sup>υ</sup>eEb<sup>3</sup> tg ð Þ <sup>α</sup>=<sup>2</sup> ð Þ 3l<sup>0</sup> þ 2Δl Δl

Taking into account Eq. (20), we find a cantilever point displacement at a distance l<sup>0</sup> from the

During the crack propagation, the increment of the single cantilever elastic deflection occurs in

Figure 6 presents the curves of the υ<sup>е</sup> dependence on the crack length Δl and on the increment of the single cantilever elastic deflection Δλ<sup>e</sup> obtained using Eq. (21), at following values of the parameters: Е = 110 GPa, l<sup>0</sup> = 18.12 mm, Δl = 3.77 mm, P = 822 Н, α = π/9 (20�), а = b = 4.35 mm. The calculations show that there is a parabolic dependence between Δl and υе, which can be

parametric values in Eq. (21). In this case, А is equal to 2.58. As seen from the plot, there is a linear dependence Δλ<sup>e</sup> = Вυ<sup>е</sup> between Δλ<sup>e</sup> and υе. The proportionality factor for the assigned

The equations given above are derived from the constitutive equations of engineering fracture mechanics for the first time and can be used for the calculation of SFE for the chevron-notched

The processes of plastic deformation affect the cantilever deflection value and opening of crack sides in the point of its initiation. For this reason, the experimentally measured values of the

4l

Eb<sup>3</sup> tg ð Þ <sup>α</sup>=<sup>2</sup> � <sup>λ</sup>eΔl lð Þ <sup>0</sup> <sup>þ</sup> <sup>2</sup><sup>l</sup> ½ � <sup>a</sup> � <sup>2</sup>Δ<sup>l</sup> tg ð Þ <sup>α</sup>=<sup>2</sup> ½ � <sup>4</sup>l<sup>0</sup> <sup>þ</sup> <sup>a</sup> ctg ð Þþ <sup>α</sup>=<sup>2</sup> <sup>2</sup>Δ<sup>l</sup>

½ � <sup>2</sup>l0þ<sup>a</sup> ctg ð Þ <sup>α</sup>=<sup>2</sup> <sup>2</sup> ½ � <sup>4</sup>l0þactgα=2þ2Δ<sup>l</sup> :

<sup>λ</sup><sup>e</sup> <sup>¼</sup> <sup>2</sup>P<sup>0</sup>

<sup>2</sup>� ctg ð Þ <sup>α</sup>=<sup>2</sup> <sup>P</sup>�P<sup>0</sup> ð Þ

½ � a�2Δl tg ð Þ α=2

The value of δe(l0)=2υ<sup>e</sup> determines the crack opening initiated at the chevron.

Eb<sup>3</sup>

:

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

l<sup>0</sup> þ Δl b <sup>3</sup>

:

: (18)

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223

<sup>2</sup> : (19)

: (20)

<sup>2</sup> tg ð Þ <sup>α</sup>=<sup>2</sup> ½ � <sup>2</sup>l<sup>0</sup> <sup>þ</sup> <sup>a</sup> ctg ð Þ <sup>α</sup>=<sup>2</sup> <sup>2</sup> : (21)

Δλ<sup>e</sup> ¼ λеð Þ� l<sup>0</sup> þ Δl λеð Þ l<sup>0</sup> , (22)

, where А is the constant, which depends on the assignment of concrete

the cantilever deflection in the load application point Р<sup>0</sup>

From Eqs. (18) and (19), we obtain an expression for Р<sup>0</sup>

Alternately, as follows from Eq. (17),

As follows from Eq. (12), <sup>λ</sup><sup>e</sup> <sup>¼</sup> <sup>4</sup><sup>l</sup>

<sup>υ</sup><sup>e</sup> <sup>¼</sup> P lð Þ <sup>0</sup> <sup>þ</sup> <sup>2</sup><sup>l</sup> <sup>Δ</sup><sup>l</sup>

the load application point P. The increment

corresponds to the crack length Δl.

2

written as υ<sup>е</sup> = АΔl

values of В is 5.969.

specimens.

load application point P:

This equation differs from the similar one for the straight-through notch only by k �2 . In particular, if α = π, Eq. (14) goes over into Eq. (5) for the straight-through notch, since then k is 1.

As follows from Eq. (14), the characteristic of G depends on the Young's modulus E. The higher E is, the lower is the SFE value under all other conditions being equal. In contrast, according to Eq. (2), characteristic of KI does not depend on E, i.e. SIF is invariant in relation to the Young's modulus.
