Crack propagation with different radius local random damage based on peridynamic theory

Abstract

Drawing from the advantages of Classical Mechanics, the peridynamic theory can clarify the crack propagation mechanism by an integral solution without initially setting the factitious crack and crack path. This study implements the peridynamic theory by subjecting bilateral notch cracked specimens to the conditions of no local damage, small radius local damage, and large radius local damage. Moreover, to study the effects of local stochastic damage with different radii on the crack propagation path and Y-direction displacement, a comparison and contact methodology was adopted, in which the crack propagation paths under uniaxial tension and displacement in the Y-direction were compared and analyzed. This method can be applied to steel structures under similar local random damage conditions.

Graphical abstract

Keywords

peridynamics / stochastic damage / bilateral notch crack

1 Introduction

Fatigue and fracture are among the main causes of structural failure [1]. Since the early 20th century, fatigue accidents have received increasing attention owing to the serious damage that they cause. Various methods and theories have been proposed to solve the problems associated with material fracture, including fracture mechanics theory, finite element method (FEM), and extended finite element method (XFEM).

In the area of fracture mechanics, Griffith [2] proposed the theory of fracture mechanical energy, Paris and Erdogan [3] proposed the crack extension model, and Rice [4] put forward the J integral theory. Through the continuous efforts of both domestic and foreign scholars, Clough established the FEM; Gonzalez-Herrera and Zapatero [5] and McClung and Sehitoglu [6] identified some elements of simulated crack closure in the FEM; Amiri et al. [7] applied a fourth-order phase-field model to crack propagation analysis; Amiri et al. [8] presented the linear elastic phase model of thin plates; Areias et al. [9] proposed a two-stage discrete crack model; and Thai et al. [10] built a model based on high-order stress damage. In the area of multi-scale fractures, Budarapu et al. [11,12] proposed an adaptive multiscale model for quasi-static crack propagation, and demonstrated the effectiveness of coarse-grained particles in multiscale fracture models. Silani et al. [13] applied a multiscale model to simulate the local damage of epoxy nanocomposites. Talebi et al. [14] presented a multiscale model of three-dimensional cracks and chaotic diffusion, while Silani et al. [15] proposed a model of semi-concurrent versus scale for nanocomposites. Talebi et al. [16] provided a multiscale model calculation library for material fracture, and introduced the molecular dynamics and extended finite element coupling in their study on dynamic brittle fracture by means of a three-dimensional extension bridge [17]. Several studies have also been conducted on the XFEM. For instance, Belytschko and Black [18] proposed the FEM independent of the grid part to solve a crack propagation problem. Daux et al. [19] used multiple extended functions to simulate crack branching. Fang et al. [20] built a cohesive crack model l based on the XFEM, while Zhuang and Cheng [21] conducted a study on the crack propagation path at the sub-interface of two kinds of materials.

The peridynamic theory, which combines the fracture mechanics approach, FEM, and XFEM, has improved the accuracy of integral solutions and facilitated progress in this field. Silling and Askari [22] solved the problem of the energy release rate with a three-dimensional peridynamic model. They suggested the use of the weight function within the local area of particles to identify the damage in the peridynamic field [23]. Ayatollahi and Aliha [24] demonstrated the applicability of the critical elongation value of the failure parameters of linear elastic materials through experimental studies. Foster et al. [25] proposed that the critical energy density should be considered as a material failure standard. Zeng [26] simulated the bilateral crack growth path of Q345 steel under tensile load. Gu and Zhou [27] analyzed the processes of crack propagation and connection. Ren et al. [28], Silani et al. [29], and Talebi et al. [30] conducted many studies highlighting multiscales. Feng et al. [31] simulated the dynamic response of material particles. Rabczuk et al. [32–34] proposed an alternative approach by expanding their views into refinement and variable horizons. In contrast to peridynamic’s (PD) approach, the version of cohesive crack model (CPM) approaches not only enabled the modeling of complex fractures without providing specific criteria for complex crack patterns, such as crack branching or coalescence, but also allowed traditional constitutive models to be applied in the bulk.

On the basis of the peridynamic theory, this study comprehensively analyzes the crack propagation mechanism of a Q345 steel plate with bilateral notch cracks under the conditions of uniaxial tension without local damage, with small radius local damage, and with large radius local damage, respectively. Then, the influence of the radius size of local stochastic damage on the propagation path of bilateral cracks is discussed.

2 Peridynamic theory

2.1 Theoretical description

In peridynamic theory, objects are discretized into many particles, and the distance between particles changes constantly with the external loads. The relative values of the peridynamic force and the variation of object properties are defined by the changes in the distance between particles. As shown in Fig. 1, {\mathit{u}}_{(k)} and {\mathit{u}}_{(j)} are the position changes of particles x_{(k)} and x_{(j)} under the external forces, while y_{(k)} and y_{(j)} are the new coordinate values of particles x_{(k)} and x_{(j)}, respectively.

The theory of peridynamics assumes that the interaction force density vector {\mathit{t}}_{(k)(j)}\left(j=1,2,\cdots ,\mathrm{\infty }\right), between particle x_{(k)} and any particle in the circle with the center at x_{(k)} and radius of \delta, is stored in the force vector \underset{\_}{\mathit{T}}:

(1)\underset{\_}{\mathit{T}}\left(x_{(k)},t\right)=\left\{\begin{array}{c}{\mathit{t}}_{(k)(1)}\\ ⋮\\ {\mathit{t}}_{(k)(\mathrm{\infty })}\end{array}\right\}.

The peridynamic force density vector of particle x_{(j)} on particle x_{(k)} is defined as

(2){\mathit{t}}_{(k)(j)}\left({\mathit{u}}_{(j)}-{\mathit{u}}_{(k)},x_{(j)}-x_{(k)},t\right)=\underset{\_}{\mathit{T}}\left({\mathit{x}}_{(k)},t\right)⟨x_{(j)}-x_{(k)}⟩.

The elongation relative to the distance between particle x_{(j)} and particle x_{(k)} under the action of the external force is defined as

(3)s_{(k)(j)}=\frac{y_{(j)}-y_{(k)}-x_{(j)}-x_{(k)}}{x_{(j)}-x_{(k)}}.

Then, the peridynamic force exerted by particle x_{(j)} on particle x_{(k)}, {\mathit{f}}_{(k)(j)}, which includes the material properties, can be expressed as

(4){\mathit{f}}_{(k)(j)}=\left[c_{1}s\left({\mathit{u}}_{(j)}-{\mathit{u}}_{(k)},x_{(j)}-x_{(k)}\right)-c_2\mathit{T}\right]\cdot \frac{y_{(j)}-y_{(k)}}{y_{(j)}-y_{(k)}},

where c_1 and c_2 are peridynamic parameters.

The dynamic equation of the peridynamic theory is

(5){\rho }_{(\mathrm{x})}\ddot{\mathit{u}}(x,t)={\int }_H\mathit{f}\left({\mathit{u}}_{(j)}-{\mathit{u}}_{(k)},x_{(j)}-x_{(k)},t\right)\mathrm{d}H+\mathit{b}(x,t),

where H is the particle action zone, and \mathit{b}(x,t) is the body force density vector.

2.2 Damage theory and crack propagation criterion

To apply the material damage model in the peridynamic context, Silling and Bobaru [23] proposed the use of s_{(k)(j)} to express the scalar-valued function based on the history \mu, which helps introduce the weight function of the interaction between particles \phi (x_{(k)},t), a function that can express the local damage value of a particle.

The scalar-valued function based on the history \mu can be expressed as

(6)

where s_{\mathrm{c}} is the critical elongation. It was assumed that if the elongation of two particles s, is less than s_{\mathrm{c}}, then there exists a peridynamic force, {\mathit{f}}_{(k)(j)} and \mu =1; otherwise, there is no peridynamic force and \mu =0.

The weight function of the interaction between particles is expressed as

(7)\phi (x_{(k)},t)=1-\frac{{\int }_H\mu \left(x_{(j)}-x_{(k)},t\right)\mathrm{d}{V}_{(j)}}{{\int }_H\mathrm{d}{V}_{(j)}},

where \phi (x_{(k)},t)\in \left[0,1\right].

When \phi (x_{(k)},t)=0, there is no damage between particle x_{(k)} and the other particle x_{(j)} in the action area, and {\mathit{f}}_{(k)(j)} is equal to the interaction force between the two particles. When \phi (x_{(k)},t)\in \left(0,1\right), {\mathit{f}}_{(k)(j)} gradually decreases with the increase of \phi (x_{(k)},t) from 0 to 1; the damage also continuously deepens. When \phi (x_{(k)},t)=1, {\mathit{f}}_{(k)(j)}=0, the damage is complete, and the crack forms and propagates.

3 Calculation example 1

The specimen is Q345B steel, a type of low-alloy high-strength structural steel with an elastic modulus of E=203\mathrm{G}\mathrm{P}\mathrm{a}, Poisson’s ratio of \nu =0.3, elongation ratio of \delta =27.96\mathrm{\%}, and density of \rho =7850\mathrm{k}\mathrm{g}/{\mathrm{m}}^3. The size is 40 mm × 70 mm, with longitudinal spacings of 0, 10, and 20 mm. The uniaxial elongation rate of the load along the long side of the steel is 2.217\times {10}^{-5}\mathrm{m}/\mathrm{s}. The cubic particles selected have a peridynamic unit side length equal to the longitudinal spacing \mathrm{\Delta }x=0.5\mathrm{m}\mathrm{m}. The load was applied to the nodes with a boundary width of d=3\mathrm{\Delta }x and load deflection of 2.217\times {10}^{-5}\mathrm{m}/\mathrm{s}. The time step was less than 0.1 s to satisfy the quasi-static test condition.

In order to obtain more realistic fracture results, the fracture results of three specimens under load obtained from the experiment in Guangxi University in Fig. 2 are quoted. It can be seen from the test that the fracture structure of the specimen with a longitudinal crack spacing of 0 mm is a horizontal crack through, and that of the specimen with a longitudinal crack spacing of 10 mm is a 45° oblique crack.The fracture results of the specimen with the longitudinal crack spacing of 20 mm are two penetrating horizontal cracks without influence on each other.

Figure 3 shows the crack propagation in undamaged specimen 10-00 under the load, and the cracks eventually merged into a horizontal crack. Figure 4 shows the displacement distribution of undamaged specimen 10-00 along the Y-direction under the load. It can be seen that for the specimen with the longitudinal spacing of 0 mm, the displacement decreased in the direction from the two ends to the middle of the specimen. After the specimen was totally torn, the displacements of the two ends are equal in value but opposite in directions.

Figure 5 shows the crack propagation in undamaged specimen 10-10. The cracks eventually merged into a 45° oblique crack. Figure 6 shows the change of the Y-direction displacement during the crack propagation process. It can be seen that the distribution of the Y-direction displacement is irregular near the crack, but regular and even along the long side of the steel away from the crack, which is consistent with Saint-Venant’s principle. Finally, the Y-direction displacements of the upper and lower ends reached the maximum values after the specimen was completely torn.

Figure 7 shows the crack propagation in undamaged specimen 10-20, with two cracks that are unaffected by each other and extend toward two parallel cracks. Figure 8 shows the Y-direction displacement under the load. The displacement distribution is similar to that of specimen 10-10, which is regular near the crack, and even but increasing stepwise away from the crack.

4 Calculation example 2

This example uses the same material and specimen as in example 1; however, there are two circular locally damaged areas with a radius of 2 mm. The center coordinates are x_1=0.005,y_1=-0.0085 and x_2=-0.005, y_2=0.0085. The distribution of fractures among particles in the damaged area is stochastic.

Figure 9(a) illustrates two locally damaged specimens with longitudinal crack spacing of 0 mm. Figures 9(b) and 9(c) show the crack propagation process under the load. As seen in Figs. 3 and 9, the local damage has no effect on the crack propagation path of the specimen with a longitudinal spacing of 0 mm. A comparison of Figs. 4 and 10 shows that the Y-direction displacement of the specimen with a local damage radius of 2 mm is basically the same as that of the undamaged specimen. When the crack completely penetrates the specimen, the maximum value of the Y-direction displacement in the damaged specimen is slightly lower than that of the undamaged specimen.

Figure 11(a) shows the model of two locally damaged specimens with longitudinal crack spacing of 10 mm and damage radius of 2 mm. Figures 11(b) and 11(c) show the crack propagation process under the load, in which no 45° oblique crack is formed due to the effect of local damage. However, at the location where the 45° oblique crack is expected, the crack propagates along a straight line, eventually forming two parallel cracks with a longitudinal spacing of less than 10 mm. A comparison of Figs. 5 and 11 shows that the crack propagation is significantly affected by the local damage with radius of 2 mm.

Figure 12 presents the crack propagation and displacement of the specimen along the Y-direction. It can be seen from Figs. 6 and 12 that the two local damages with a radius of 2 mm have a small influence on the displacement distribution along the Y-direction, but a large influence on the maximum displacement value. The maximum displacements are 1.4 and 1.7 mm, respectively, in the damaged and undamaged specimens.

Figure 13 shows the crack propagation in specimen 10-20 with a damage radius of 2 mm. The bilateral crack with a longitudinal spacing of 20 mm transformed into two parallel cracks under the load. It can be seen from the comparison of Fig. 7 and 13 that the local damage barely influences the bilateral crack propagation path with the longitudinal spacing of 20 mm.

Figure 14 shows the Y-direction displacement in specimen 10-20 with a damage radius of 2 mm. A comparison of Figs. 8 and 14 shows that the damage does not influence the displacement distribution; however, it does influence the maximum displacement value along the Y-direction in the specimen with a longitudinal crack spacing of 20 mm. The maximum displacements are 1.4 and 1.8 mm, respectively, in the damaged and undamaged specimens.

5 Calculation example 3

This example is similar to example 2, except that the damage radius is 5 mm. Figure 15(a) illustrates two locally damaged specimens with longitudinal crack spacing of 0 mm. Figures 15(b) and 15(c) display the crack propagation process under the load. It can be seen from Figs. 3, 9, and 15 that the local damage radius has little effect on the crack propagation path of the specimen with a longitudinal spacing of 0 mm. A comparison of Figs. 4, 10, and 16 shows that the Y-direction displacement distribution is related to the local damage radius. The distribution range of the minimum Y-direction displacement increased with the damage radius after the specimen is broken.

Figure 17(a) shows the model of two locally damaged specimens with a longitudinal crack spacing of 10 mm and damage radius of 5 mm. Figures 11(b) and 11(c) show the crack propagation process under the load. A comparison of Figs. 5, 11, and 17 shows that the influence of local damage on the crack propagation path increased with the damage radius. Similar to the crack propagation in the undamaged specimen with a longitudinal spacing of 20 mm, the two bilateral notch cracks propagated separately into two parallel horizontal cracks penetrating the specimen.

Figure 18 presents the crack propagation and the displacement of the specimen along the Y-direction. It can be seen from Figs. 6, 12, and 18 that the local damage radius significantly influences the Y-direction displacement. The larger damage radius generally corresponds to the wider distribution range of the minimum Y-direction displacement.

Figure 19 shows the model of two locally damaged specimens with a longitudinal crack spacing of 20 mm and damage radius of 5 mm. A comparison of Figs. 7, 14, and 19 shows that the local damage radius barely influences the crack propagation path of the specimens with a longitudinal crack spacing of 20 mm, and the two cracks propagated separately into two parallel horizontal cracks along a straight line.

Figure 20 shows the crack propagation and the displacement of the specimen along the Y-direction. It can be seen from Figs. 14 and 20 that the local damage radius significantly influences the Y-direction displacement. The larger damage radius generally corresponds to the wider distribution range of the minimum Y-direction displacement when the specimen is broken.

6 Conclusions

This study implemented the peridynamic theory to examine the propagation of bilateral notch cracks with various longitudinal spacings by following the different propagation paths. The experimental results showed that adopting the peridynamic theory accurately reflected the simulation process of the crack propagation path. The local damage radius influenced the crack propagation path in the following ways. When the longitudinal spacing was 0 mm, the crack propagation path and the Y-direction displacement were not affected by the local damage radius. When the longitudinal spacing was 10 mm, the crack propagation path and the distribution range of the minimum Y-direction displacement were significantly affected by the local damage radius. When the longitudinal spacing was 20 mm, the minimum Y-direction displacement increased with the local damage radius. The crack propagation path was hardly influenced by the longitudinal spacing, and two bilateral notch cracks propagated along a horizontal straight line until they penetrated the whole specimen. In conclusion, the simulation experiment conducted on the crack propagation path indicates a strong correlation between the crack propagation path and the various radii and longitudinal distances. The results of this study can be applied in the testing of steel structures for future engineering quality assurance.

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