Damage zone length limit during the dislocation-crack interaction under shearing mode

Abstract

In this paper, limiting the damage zone length in mode II during the interaction between a main crack and a surrounding dislocation in a brittle material is considered. This study is mainly based on the strain and stress field generated by varying the distance between the dislocation and a main crack by taking into consideration several cracks’ length.The proposed model is a dish element having an edge crack surrounded by an arbitrarily dislocation anduniformly loaded under the shearing mode II. The problem is then analyzed using Finite Element Method (FEM) along with the software ABAQUS. For each distance between the main crack and the dislocation, stress and strains fields are determined and then, the limiting length of damage zone is drawn. The results are compared with those found for the opening mode I.

Keywords

Damage zone dislocation main crack stress strain cracking

1. Introduction

During the propagation of a crack, several researchers found that there are three zones in the vicinity of a crack tip [14,19,23,26]. The first zone is located at the end of the main crack, called “damage zone” being very difficult to study, because the strains and stress concentration is very high [15,18,20]. The following zone is known as a “singular zone” in which displacements and stress fields are continuous and infinite at the proximity of the crack tip. The third zone is very far from the crack tip, called “external zone”, in which the stress, strains and displacement fields can be determined, while respecting the boundary conditions. For the damage zone (DZ) with a very small size, researchers find that this area is difficult to determine by a classical elasticity approach [2,6,13]. Based on the theory of dislocations, Benzahar and Chabaat [12] limited the length of DZ as a function of the main crack’s length under the first mode of rupture (Mode I). In this present paper, a deep study is devoted to determining the length of DZ under a loading shear which corresponds to the second mode of rupture (Mode II). Numerically, the length of DZ can be obtained using the presence of a neighbouring dislocation at the crack tip. In this research work, we consider a dislocation that varies in the vicinity of a main crack. The model chosen is an element made of a brittle material with a rectangular shape (B = 700 mm, H = 500 mm and e = 2 mm). The element is initially cracked at one of its ends under a shearing load according to mode II. The model is discretized by finite elements in small squares and other triangular forms located at the end of the main crack. By applying a load that varies from 5 to 35 N/mm, for each crack length and distance between the dislocation and the main crack, plane strain and stress field (𝜀₁₁, 𝜀₁₂, S ₁₁, S ₁₂) are assessed all over the material. Based on these strain and stress fields, a length of DZ is determined function of the main crack’s length. Figure 1 represents the cracked model loaded in modes II.

Fig. 1.

Fracture model under Mode II loadings.

Here, H and B are the width and length of the element, respectively. d is the distance between the main crack and the dislocation and l is the length of the macro crack. b _x and b _y are the dimensions of the dislocation along the x and y axis, respectively. +𝜎 and −𝜎 correspond to the tensile and compression loading applied according to the mode II.

2. Stress distribution about the crack tip

The stress, strain and displacement fields can be characterized in the vicinity of a main crack [1,24]. In the case of a planar element with a thin thickness, the stress distribution is given by the following relation [3]: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}S_{11}\\ S_{22}\\ S_{21}\end{array}\right.=\frac{K_{I}}{\sqrt{2{\pi}\mathscr{r}}}\left\{\begin{array}{@{}l@{}}\cos ({\alpha}/2)[1-\sin ({\alpha}/2)\sin (3{\alpha}/2)]\\ \cos ({\alpha}/2)\sin ({\alpha}/2)\cos (3{\alpha}/2)\\ \cos ({\alpha}/2)[1+\sin ({\alpha}/2)\sin (3{\alpha}/2)]\end{array}\right.+\frac{K_{II}}{\sqrt{2{\pi}\mathscr{r}}}\left\{\begin{array}{@{}l@{}}-\sin ({\alpha}/2)[2+\cos ({\alpha}/2)\cos (3{\alpha}/2)]\\ \cos ({\alpha}/2)[1-\sin ({\alpha}/2)\sin (3{\alpha}/2)]\\ \sin ({\alpha}/2)\cos ({\alpha}/2)\cos (3{\alpha}/2)\end{array}\right. & & \displaystyle\end{eqnarray}$ (1) where S _ij is the plane stress, K _I and K _II are stress intensity factors of the Mode I and Mode II, respectively, r is the distance between DZ and the end of the main crack, 𝛼 is the orientation angle of the stress application point about the main crack [11].

3. Presence of dislocations

The dislocation is a micro-crack having an infinitesimal size compared to the main crack [21]. It is easily detected by an electronic microscope device with a high resolution [5,8]. It is known that dislocations or defects can influence the main crack by reducing or amplifying the stresses generated during the crack dislocation interaction. Consequently, this phenomenon can accelerate or delay the crack propagation [10]. Generally, each micro-crack is endowed with an action force created by the movement of atoms and their interactions [4,25]. The displacement of a dislocation is determined by a Burgers vector noted by b _e = b _x + b _y [16]. In case of existing dislocation nearby a crack tip, the following complex analytical functions can be used: $\begin{eqnarray}\displaystyle {\omega}_{d}\text{} & =\text{} & \displaystyle {\eta}\ln (z-z_{d})\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle x_{d}\text{} & =\text{} & \displaystyle \overline{{\eta}}\ln (z-z_{d})-{\eta}[z_{d}/(z-z_{d})],\end{eqnarray}$ (3) where $\begin{eqnarray}\displaystyle {\eta}={\mu}b_{e}/[\text{i}{\pi}(k+1)]. & & \displaystyle\end{eqnarray}$ (4) Here, 𝜔_d and 𝜒_d are the complex analytical functions of dislocation. z is the complex number, z and 𝜂 are the conjugate of z and 𝜂, respectively. 𝜈 is the Poisson’s ratio and k is the coefficient which can be k = 3–4𝜈 in the case of a plane stress.

4. Main crack-dislocation interaction

An existing dislocation in the vicinity of a main crack can have an effect on the crack propagation. The interaction problem is formulated during the propagation of a main crack in the DZ [7,22]. The components of strain, stress and displacements of a dislocation near the main crack can be obtained using the theory of the complex potentials of Muskhelishvili [17]. Using a superposition of complex analytical functions, the interaction between the main crack and the dislocation is expressed by the following functions: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}S_{11}+S_{22}=2[{\omega}_{T}^{\prime }(z)+\overline{{\omega}_{T}^{\prime }(z)}]\\[7.20007pt] S_{22}-S_{11}+\text{i}S_{12}=2[\overline{z}{\omega}_{T}^{\prime \prime }(z)+{\chi}_{T}^{\prime }(z)]\\[7.20007pt] U(z)=u_{1}+\text{i}u_{2}=\displaystyle \frac{1}{2{\mu}}[K{\omega}_{T}(z)-\overline{z{\omega}_{T}^{\prime }(z)}-\overline{{\chi}_{T}(z)}]\end{array}\right. & & \displaystyle\end{eqnarray}$ (5) with $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\omega}_{I}={\omega}_{MC}+{\omega}_{d}\\ {\chi}_{I}={\chi}_{MC}+{\chi}_{d},\end{array}\right. & & \displaystyle\end{eqnarray}$ (6) where 𝜔_MC and 𝜒_MC are analytical complex functions of a main crack. 𝜔_I and 𝜒_I are analytical complex functions of interaction between a dislocation and a macro crack.

Using Eqs (2), (3) and (5), plane stresses generated during the interaction between the dislocation and the main crack are determined as follows [9]: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}s_{11}=\displaystyle \frac{2{\mu}b\cos ({\beta})}{{\pi}(k+1)l}[3-\sin (2{\beta})\tan ({\beta})]\\[12.0pt] s_{22}=\displaystyle \frac{2{\mu}b\cos ({\beta})}{{\pi}(k+1)l}[1+\sin (2{\beta})\tan ({\beta})]\\[12.0pt] s_{11}=-\displaystyle \frac{4{\mu}b\cos ({\beta})}{{\pi}(k+1)l}[\sin ^{2}{\beta}\tan ({\beta})+1],\end{array}\right. & & \displaystyle\end{eqnarray}$ (7) where l is the distance between dislocation and the main crack. 𝛽 is the orientation angle of the dislocation around itself.

5. Numerical analysis

For the numerical study, a model of a rectangular element with a thin thickness and an edge crack is shown in Fig. 2. A dislocation of a circular shape having an infinitesimal size with respect to the macro crack size is considered. To proceed in this research work, several cracks’ length (l ₁ = 50 mm, l ₂ = 100 mm, l ₃ = 150 mm, l ₄ = 200 mm, l ₅ = 250 mm, l ₆ = 300 mm, l ₇ = 350 mm) are taken in consideration. In order to find the stress and strains fields, for each crack length, one changes the position of the dislocation with a distance d. The numerical problem is then analyzed by FEM using ABAQUS where the cracked model is discretized by square and triangular elements. The chosen material has a Young’s modulus E = 70000 N/mm² and a Poisson’s ratio 𝜐 = 0.20. The shearing fracture Mode II requires a loading parallel to the faces of the openings of the main crack.

Fig. 2.

Loaded model according to Mode II.

The chosen model is subjected to a shearing load that grows from 0 to 50 N/mm. By varying the dislocation’s position with respect to the main crack, the plane stress and strain are determined for each length of the main crack. On the other side, varying the length of the main crack, Figs 3 and 4a, b, c, d represent the plane stress and strain in case of a crack’s length of 50 mm and 350 mm subjected to a loading of 10 N/mm.

Fig. 3.

Stress and strain values for a short main crack (l = 50 mm). (a) S ₁₁, (b) S ₁₂, (c) 𝜀₁₁, (d) 𝜀₁₂.

Fig. 4.

Stress and strain values for a long main crack (l = 350 mm) (a) S ₁₁, (b) S ₁₂, (c) 𝜀₁₁, (d) 𝜀₁₂.

The values of the stress and strain can be positive and sometimes negative delimiting the zones between compression and tension. Applying a shearing load from 0 to 50 N/mm and then changing the dislocation’s position with respect to the main crack, values of stress and strain are obtained for each main crack’s length as illustrated in Figs 5, 6, 7 and 8. These figures represent the variation of strain and stress as a function of the distance of main crack dislocation for each main crack’s length.

Fig. 5.

Variation of stress (S ₁₁) versus the distance (d) between the main crack and the neighboring dislocation. (a) l = 50 mm, (b) l = 100 mm, (c) l = 150 mm, (d) l = 200 mm, (e) l = 250 mm, (f) l = 300 mm, (g) l = 350 mm.

Fig. 6.

Variation of stress (S ₁₂) versus the distance (d) between the main crack and the neighboring dislocation. (a) l = 50 mm, (b) l = 100 mm, (c) l = 150 mm, (d) l = 200 mm, (e) l = 250 mm, (f) l = 300 mm, (g) l = 350 mm.

Fig. 7.

Variation of strains (𝜀₁₁) versus the distance (d) between the main crack and the neighboring dislocation. (a) l = 50 mm, (b) l = 100 mm, (c) l = 150 mm, (d) l = 200 mm, (e) l = 250 mm, (f) l = 300 mm, (g) l = 350 mm.

Fig. 8.

Variation of strains (𝜀₁₂) versus the distance (d) between the main crack and the neighboring dislocation. (a) l = 50 mm, (b) l = 100 mm, (c) l = 150 mm, (d) l = 200 mm, (e) l = 250 mm, (f) l = 300 mm, (g) l = 350 mm.

6. Results and discussion

In this research paper, the length of the DZ (a) is limited by the area whose stresses and strain increase sharply in the vicinity of the main crack. Strain and stress components generated during the main crack-dislocation interaction are represented in Figs 3 and 4.

In Figs 5 to 8, the main crack length varies from 50 mm to 350 mm while applied shearing load increases from 0 to 50 daN/cm. Taking in consideration the distance between the crack and a nearby dislocation (d), strain and stress components undergo change at the same rate. One can notice that value of s ₁₁, s ₁₂, 𝜀₁₁ and 𝜀₁₂ increases sharply in the vicinity of the main crack, which reflects that the dislocation has reached the DZ.

Figures 5 and 7 represent the variation of the plane stress and strain s ₁₁ and 𝜀₁₁, function of the distance (d), parallel to the applied load. It is seen that the stress field changes their speed from the distance d𝜖] 0,10] mm, d𝜖] 0,15] mm, d𝜖] 0,20] mm whose the length of the DZ a = 10 mm, a = 15 mm and a = 20 mm, respectively.

Figures 6 and 8 show the variation of S ₁₂ and 𝜀₁₂ function of the distance (d). Stress and strain components increase when the distance d𝜖] 0,10] mm and d𝜖] 0,15] mm whose the length of the DZ a = 10 mm and a = 15 mm, respectively.

For applying several shearing loads and varying the main crack’s length, the obtained lengths of the DZ are summarized in Table 1.

Table 1
DZ length size for different main crack lengths

Crack length l (mm) 50 100 150 200 250 300 350

Length of the DZ, a (mm) for each stress and strain. Figure 5 (s ₁₁) 10 10 15 20 15 15 10

Figure 6 (s ₁₂) 10 10 15 15 15 15 15

Figure 7 (𝜀₁₁) 10 10 15 20 10 15 10

Figure 8 (𝜀₁₂) 10 10 15 15 15 15 10

Crack length l (mm)	50	100	150	200	250	300	350
Length of the DZ, a (mm) for each stress and strain.	Figure 5 (s ₁₁)	10	10	15	20	15	15	10
	Figure 6 (s ₁₂)	10	10	15	15	15	15	15
	Figure 7 (𝜀₁₁)	10	10	15	20	10	15	10
	Figure 8 (𝜀₁₂)	10	10	15	15	15	15	10

The percentage of length of the DZ relative to the main crack can be calculated by the following formula: $\begin{eqnarray}\displaystyle P∼(\%)=[a/l]\times 100, & & \displaystyle\end{eqnarray}$ (8) where P is the ratio of the DZ length with respect to the main crack length.

Whatever the loading applied in mode II for a given crack’s length, the ratio of the length of the DZ with respect to the main crack, calculated by Eq. (8), is shown in Fig. 9.

Fig. 9.

Percentage variation of the length of the DZ with respect to the main crack as a function of the generated stress and strain (a) stress, (b) strain.

One can notice in Fig. 9 that whatever the loading applied, the percentage P (%) is inversely reduced with the length of the main crack. In each stress and strain, the mean value of P (%) is given as follows: P _m(S ₁₁) = 9.11%, P _m(S ₁₂) = 8.96%, P _m(𝜀₁₁) = 8.82% and P _m(𝜀₁₂) = 8.75%. This percentage is very comparable with that found in mode I where P (%) = 8% (refer to [12]). Considering all stresses and strains (s ₁₁, s ₁₂, 𝜀₁₁, 𝜀₁₁), it is proven that whatever the length of the crack and the applied force, the percentage does not exceed twenty percent (20%).

7. Conclusion

The objective of this research is to determine the percentage of the length of the DZ compared to that of the main crack in mode II. Using different main crack lengths subjected to loadings according to mode II and taking in consideration the orientation of an existing dislocation in the vicinity of the main crack tip, stresses and deformations are found to be very high compared to the other places in the material. This small area is considered as a damage area. The stresses and deformation generated by the interaction between the main crack and the dislocation vary at the same rates depending on the main crack’s length, the distance between the crack and the dislocation and also on the applied loading. In mode II, the length of the DZ gets closer to nine percent (9%) of the length of the main crack which is almost comparable to that found in mode I [12] where the percentage is equal to eight percent (8%). It is obvious that the DZ maintains its size regardless of the type of loadings.

Footnotes

Conflict of interest

None to report.

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