4. Theoretical determination of elastic crack opening δ<sup>е</sup>

The specimen presentation in the form of a double cantilever configuration allows us to determine the elastic crack opening initiated at the tip of the chevron. It is known that the elastic displacements of the cantilever points obey equation [14]:

$$\upsilon\_{\mathbf{e}}(\mathbf{x}) = \frac{P}{Ea} \left[ 2\left(\frac{\mathbf{x}}{b}\right)^3 - 6\frac{\mathbf{x}}{b}\left(\frac{l}{b}\right)^2 + 4\left(\frac{l}{b}\right)^3 \right],\tag{15}$$

where the x axis is directed along the cantilever. At x = 0, Eq. (15) determines the elastic deflection λ<sup>e</sup> of the load application point P.

Eq. (15) is also valid for the double cantilever beam specimen with a straight-through notch. Then elastic notch opening in the point x is δe(x)=2υe(x). The cantilever displacement in the point x = l<sup>0</sup> is equal to

$$\upsilon\_{\mathbf{e}}(l\_0) = \frac{2Pl\_0 \Delta l^2}{Eab^3} \left[ 3 + 2 \frac{\Delta l}{l\_0} \right]. \tag{16}$$

Substituting Eq. (5a) for P into Eq. (16), we obtain the cantilever displacement in the point x = l<sup>0</sup> for the straight-through notched specimen:

$$\upsilon\_{\mathbf{e}} = \lambda\_{\mathbf{e}} \frac{\Delta l^2 [3l\_0 + 2\Delta l]}{2(l\_0 + \Delta l)^3}. \tag{17}$$

Let us find the cantilever displacement in the point x = l<sup>0</sup> for the chevron-notched specimen. For this, we use Eq. (16), where we place the value of Р<sup>0</sup> instead of P according to Eq. (11), and the width of the specimen central part h = 2Δl�tg(α/2) shown in Figure 5 instead of a. Let us find the cantilever deflection in the load application point Р<sup>0</sup> :

$$
\lambda\_{\mathbf{e}} = \frac{2P'}{E\Delta l \cdot \text{tg}\left(\alpha/2\right)} \left(\frac{l\_0 + \Delta l}{b}\right)^3. \tag{18}
$$

Alternately, as follows from Eq. (17),

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

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

4. Theoretical determination of elastic crack opening δ<sup>е</sup>

P Ea <sup>2</sup> <sup>x</sup> b � �<sup>3</sup>

υeð Þ¼ l<sup>0</sup>

υ<sup>e</sup> ¼ λ<sup>e</sup>

displacements of the cantilever points obey equation [14]:

deflection λ<sup>e</sup> of the load application point P.

for the straight-through notched specimen:

υeð Þ¼ x

4 þ a l0 ctg α 2 þ 2Δl l0

> l 2

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

lar, if α = π, Eq. (14) goes over into Eq. (5) for the straight-through notch, since then k is 1.

This equation differs from the similar one for the straight-through notch only by k

Eb<sup>3</sup> a2 k�<sup>2</sup>

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 specimen presentation in the form of a double cantilever configuration allows us to determine the elastic crack opening initiated at the tip of the chevron. It is known that the elastic

> � 6 x b l b � �<sup>2</sup>

where the x axis is directed along the cantilever. At x = 0, Eq. (15) determines the elastic

Eq. (15) is also valid for the double cantilever beam specimen with a straight-through notch. Then elastic notch opening in the point x is δe(x)=2υe(x). The cantilever displacement in the

> 2Pl0Δl 2 Eab<sup>3</sup> <sup>3</sup> <sup>þ</sup> <sup>2</sup>

Substituting Eq. (5a) for P into Eq. (16), we obtain the cantilever displacement in the point x = l<sup>0</sup>

Let us find the cantilever displacement in the point x = l<sup>0</sup> for the chevron-notched specimen. For this, we use Eq. (16), where we place the value of Р<sup>0</sup> instead of P according to Eq. (11), and

½ � 3l<sup>0</sup> þ 2Δl

Δl 2

� �<sup>3</sup> " #

<sup>þ</sup> <sup>4</sup> <sup>l</sup> b

Δl l0 � �

� �

.

2 þ a l0 ctg α 2

� ��<sup>2</sup>

: (14)

:

�2

, (15)

: (16)

<sup>2</sup>ð Þ <sup>l</sup><sup>0</sup> <sup>þ</sup> <sup>Δ</sup><sup>l</sup> <sup>3</sup> : (17)

. In particu-

<sup>k</sup> <sup>¼</sup> <sup>2</sup>Δ<sup>l</sup> <sup>a</sup> tg α 2 þ l l0

222 Contact and Fracture Mechanics

the Young's modulus.

point x = l<sup>0</sup> is equal to

$$
\lambda\_{\mathfrak{e}} = \frac{\upsilon\_{\mathfrak{e}} \mathfrak{L} (l\_0 + \Delta l)^3}{(\mathfrak{A}l\_0 + 2\Delta l)\Delta l^2}. \tag{19}
$$

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

$$P' = \frac{\upsilon\_{\text{e}} E b^3 \text{tg}\left(\alpha/2\right)}{(\Im l\_0 + 2\Delta l)\Delta l}.\tag{20}$$

As follows from Eq. (12), <sup>λ</sup><sup>e</sup> <sup>¼</sup> <sup>4</sup><sup>l</sup> <sup>2</sup>� ctg ð Þ <sup>α</sup>=<sup>2</sup> <sup>P</sup>�P<sup>0</sup> ð Þ Eb<sup>3</sup> ½ � a�2Δl tg ð Þ α=2 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> :

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

$$\upsilon\_{\rm e} = \frac{P(l\_0 + 2l)\Delta l}{Eb^3 \operatorname{tg}(\alpha/2)} - \frac{\lambda\_{\rm e} \Delta l (l\_0 + 2l)[a - 2\Delta l \operatorname{tg}\left(\alpha/2\right)][4l\_0 + a \operatorname{ctg}\left(\alpha/2\right) + 2\Delta l]}{4l^2 \operatorname{tg}\left(\alpha/2\right)[2l\_0 + a \operatorname{ctg}\left(\alpha/2\right)]^2}. \tag{21}$$

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

During the crack propagation, the increment of the single cantilever elastic deflection occurs in the load application point P. The increment

$$
\Delta\lambda\_{\mathbf{e}} = \lambda\_{\mathbf{e}}(l\_0 + \Delta l) - \lambda\_{\mathbf{e}}(l\_0) \tag{22}
$$

corresponds to the crack length Δl.

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 written as υ<sup>е</sup> = АΔl 2 , where А is the constant, which depends on the assignment of concrete 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 values of В is 5.969.

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 specimens.

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

The above equations are derived without any assumptions about the plastic properties of the material. Therefore, they can be used to calculate fracture toughness of any structural materials wherein the crack initiation at the tip of the chevron notch is observed. The product of <sup>Е</sup>G/(1 � <sup>ν</sup><sup>2</sup>

does not depend on the Young's modulus since SFE G is inversely proportional to the Young's modulus value E (see Eq. (14)). Therefore, irrespective of plastic properties of the material, the

> <sup>¼</sup> <sup>2</sup>Pl ab<sup>3</sup>=<sup>2</sup> k

ffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 � ν<sup>2</sup>

r

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

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG <sup>1</sup> � <sup>ν</sup><sup>2</sup> ð Þ <sup>s</sup>

determines the stress intensity factor (SIF) for the small-size chevron-notched specimens.

The standard test on ductility change of the chevron-notched specimens is made using the "loading-unloading" operations [7, 8, 18–20]. As a result, the fracture toughness of the material is determined under the low-cycle loading conditions, rather than under constant loading. It is known that the curve type "load-displacement" changes significantly depending on the previous loading history. This is due to the fact that the plastic deformation rate and strain hardening of the material essentially depend on the external load and time during which the load is acting. To define the mechanisms of failure of structural materials, first of all, the values of SFE and SIF under continuous loading are significant. Eqs. (13) and (15) allow us to calculate these characteristics with-out using the load-unload condition. It is enough to know the crack length

The examples of the fracture toughness analysis of a number of structural materials, which differ in their ability to crack formation and the Young's modulus, are presented below.

This section presents the calculation results of the fracture toughness characteristics of VT6

The specimens 21 � <sup>10</sup> � 6 mm<sup>3</sup> in size were cut from the work piece by the electroerosion method. Then a notch 0.3 mm thick was made with a chevron angle α = 60� (see Figure 4). The crack length at the pre-fracture stage was determined by the specimen images. Alloys with different ability to plastic deformation and with different values of the Young's modulus E were tested. The loading of specimens made of VT6 and Fe + 34.6%Ni alloys was performed by the intrusion of a narrow wedge into the notch at a motion rate of 5 μm/s (Figure 8). The 12GBA tube steel loadings were performed by application of opposite forces to the tips of the

Figure 8 shows the scheme of the specimen wedging. The constant motion rate of the wedge provides the prolonged stage of stable crack propagation initiated at the chevron. The equation for the calculation of P bending the cantilever is obtained from the condition of equilibrium of

KI ¼

5. Fracture toughness of structural materials

(Ti + 6%Al + 4%V) alloy, Fe-35.4% Ni and 12GBA tube steel.

equation

Δl initiated at the chevron.

notch (Figure 7).

forces:

)

225

(23)

Figure 6. Dependence υ<sup>e</sup> on Δl and Δλe.

cantilever deflection λ, except for λе, contain a part of the equation λ<sup>p</sup> = λ λ<sup>е</sup> that is not related to the change in specimen ductility. The crack opening values of υ measured in the same way contain the plastic deformation contribution υ<sup>p</sup> = υ υe. The values of λ<sup>p</sup> and υ<sup>p</sup> are very important when simulating the fracture process in the chevron-notched zone.

Using Eq. (13), according to the experimentally measured value of cantilever deflection λ (Figure 7), one can determine the relative value of (λλе)/λ<sup>е</sup> = λp/λ<sup>е</sup> as an additional fracture toughness characteristic. It is obvious that the more ductile a material is, the higher is its fracture toughness. The value of λ<sup>p</sup> is not associated with change in specimen ductility since it is determined only by the elastic deflection of the specimen. The stress distribution in the plastic deformation zone is significantly different from the stress field in an elastic medium with a crack. On the way of crack propagation, the material is always subjected to a certain degree of plastic deformation. This means that crack is always surrounded by a layer of the plastically deformed material. The calculations made in Ref. [21] by the method of relaxation elements showed that stress field in the plastic deformation zone differs significantly from the crack stress field in the elastically deformable medium. Plastic deformation leads to stress relaxation. For this reason, there is no singularity in the crack mouth. The maximum stress concentration is observed in the plastic deformation zone.

Figure 7. Scheme of the cantilever deflection.

The above equations are derived without any assumptions about the plastic properties of the material. Therefore, they can be used to calculate fracture toughness of any structural materials wherein the crack initiation at the tip of the chevron notch is observed. The product of <sup>Е</sup>G/(1 � <sup>ν</sup><sup>2</sup> ) does not depend on the Young's modulus since SFE G is inversely proportional to the Young's modulus value E (see Eq. (14)). Therefore, irrespective of plastic properties of the material, the equation

$$\mathbf{K}\_{\rm I} = \sqrt{\frac{EG}{(1-\mathbf{v}^2)}} = \frac{2Pl}{ab^{3/2}k} \sqrt{\frac{3}{1-\mathbf{v}^2}}\tag{23}$$

determines the stress intensity factor (SIF) for the small-size chevron-notched specimens.

The standard test on ductility change of the chevron-notched specimens is made using the "loading-unloading" operations [7, 8, 18–20]. As a result, the fracture toughness of the material is determined under the low-cycle loading conditions, rather than under constant loading. It is known that the curve type "load-displacement" changes significantly depending on the previous loading history. This is due to the fact that the plastic deformation rate and strain hardening of the material essentially depend on the external load and time during which the load is acting. To define the mechanisms of failure of structural materials, first of all, the values of SFE and SIF under continuous loading are significant. Eqs. (13) and (15) allow us to calculate these characteristics with-out using the load-unload condition. It is enough to know the crack length Δl initiated at the chevron.

The examples of the fracture toughness analysis of a number of structural materials, which differ in their ability to crack formation and the Young's modulus, are presented below.
