Peridynamics simulation of crack propagation of ring-shaped specimen like rock under dynamic loading

International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104093

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Peridynamics simulation of crack propagation of ring-shaped specimen like rock under dynamic loading

T

Yanan Zhanga,∗, Hongwei Denga,∗∗, JunRen Denga,∗∗∗, Chuanju Liua, Bo Keb a b

School of Resources and Safety Engineering, Central South University, Changsha, 410083, China College of Resources and Environmental Engineering, Wuhan University of Technology, Wuhan, 430070, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Peridynamics theory Split Hopkinson pressure bar (SHPB) Ring-shaped specimen Crack propagation

The split Hopkinson pressure bar (SHPB) system with a spindle-shaped striker is widely used to test the characteristics of rock-like materials subjected to dynamic loading, and the crack propagation of ring-shaped specimens under dynamic loading is a research focus in rock mechanics. In this paper, a model of the SHPB system, including the striker, incident bar, specimen, and transmitted bar, is established based on peridynamics theory and is validated by comparing two kinds of stress waves obtained by a peridynamics simulation and experiment. Based on these results, the crack propagation of a ring-shaped specimen with an aperture is simulated, and the influences of the apertures of the specimen on the failure patterns are discussed. The results show that there are four cracks distributed on the specimen with an aperture of 24 mm: crack-1 first initiates at the end of the inner diameter near the incident bar, crack-2 then occurs at the end of the inner diameter near the transmitted bar, and then crack-3, also called a secondary crack, initiates on the upper and lower outer boundaries of the specimen; these secondary cracks are perpendicular to the other cracks. The sequence of starting velocities and average propagation velocities of cracks from high to low is crack-3, crack-2, and crack-1, and the failure of the ringshaped specimen occurs due to tension. Controlled by the structure effect of the ring-shaped specimen, the loading rate and the peak stress decrease gradually, and the failure patterns of the specimens transform from splatted by one initial crack to broken by two cracks that are perpendicular to each other and then to asymmetrical failure with an increase in the apertures of the specimen. The comparison results of peak stresses and failure patterns between the peridynamics simulation and the experimental results verify the accuracy of peridynamics theory.

1. Introduction There are many micro-holes in rock and concrete, and the Brazilian splitting test has a shortcoming in testing the tensile strength of soft rock; therefore, many researchers use specimens containing holes to study crack propagation1–4 or to analyse the strength properties of specimens5 Ring-shaped specimens, as a sample type with a hole, are widely used in obtaining the tensile strength and fracture toughness of rock and analysing the crack propagation mechanism. Hobbs6 found that the tensile strength is related to the ring dimensions and proposed a formula for calculating the tensile strength of ring-shaped specimens. Mellor and Hawkes7 found that the calculated strength tends to the modulus of rupture with increasing hole size. Kourkoulis and Markides8

explored the stresses and displacements in a ring under parabolic diametral compression and thought this loading method is closer to the externally imposed boundary condition. Tokovvy et al.9 and Chen et al.10 obta ined the distribution of the stresses and displacements in a ring-shaped specimen subjected to diametral compression by the Fourier series, single-domain BEM, and complex variable function method and found that the indirect tensile strength of anisotropic rock is controlled by various factors. Hanson et al.11 proposed using the ring test to determine the fracture toughness of soils. Chen et al.12 and Aliha et al.13 also used ring-shaped specimens containing cracks to analyse the fracture toughness and mixed fracture patterns of brittle rock material. Steen et al.14 analysed the fracture patterns of a disk with an eccentric hole under static loading by experiments and numerical simulations and found that macro fracture starts at the hole. Jiao et al.15

∗

Corresponding author. Corresponding author. ∗∗∗ Corresponding author. E-mail addresses: [email protected] (Y. Zhang), [email protected] (H. Deng), [email protected] (J. Deng). ∗∗

https://doi.org/10.1016/j.ijrmms.2019.104093 Received 10 September 2018; Received in revised form 14 August 2019; Accepted 1 September 2019 Available online 17 September 2019 1365-1609/ © 2019 Elsevier Ltd. All rights reserved.

International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104093

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transmitted wave. Based on the SHPB theory31, the stress, strain and strain rate of the specimen can be derived as

developed a two-dimensional discontinuous deformation analysis (2DDDA) contact constitutive model and simulated the crack initiation and propagation of Brazilian disks with a central hole and an eccentric hole. Wang et al.16 simulated the failure modes and failure mechanism of disks with a central hole and an eccentric hole by a 3D numerical model (RPFA3D). The failure patterns and the mechanical properties of a ring under dynamic loading are also a research focus. Li et al.17 used the DIC technique to test the deformations and strengths of sandstone with different diameters of central holes under impact loading and found that the loading stress is continuously reduced and that the ring is broken into four parts as the diameter of the hole increases. They also compared the failure modes of marble with different diameters of central holes under static loading and dynamic loading by means of experimental and numerical methods18,19 Wang et al.20 analysed the dynamic fracture toughness and failure patterns of holed-cracked flattened Brazilian discs with different specimen sizes and different inner diameters and found that the failure mode has a size effect. Peridynamics theory, as a non-local theory, has unique advantages in the analysis of crack initiation and propagation because it can simulate the spontaneous formation process of cracks21,22 and has been widely applied in the areas of crack propagation, hydraulic fracturing, and material damage23–25 Many researchers have applied peridynamics theory to analyse dynamic crack propagation, branching, coalescence, and the associated mechanisms23,26,27 by applying the boundary conditions on the specimen directly. However, the dynamic properties of the rock material are obtained by the SHPB test. Therefore, it is novel and interesting to simulate the failure process of ring-shaped specimens under dynamic loading using peridynamics theory. In fact, peridynamics theory has been used to simulate the SHPB test. Jia and Liu28 first used peridynamics theory to simulate wave propagation in SHPB, but they did not consider wave dispersion. Gu et al.29 found that the improved PMB (prototype micro-elastic brittle) model can effectively reduce the numerical dispersion in one dimension and simulate the failure process of the concrete Brazilian disc in the SHPB test; however, they used peridynamics theory to simulate the one-dimensional SHPB test, which generates a rectangular wave, and wave dispersion is hard to avoid in the SHPB test with rectangular wave loading. Li et al.30 found that a spindle-shaped striker can generate a half-sine wave and avoid wave dispersion in the SHPB test. In this paper, SHPB tests with a rectangular striker and a spindle-shaped striker will be numerically simulated first based on peridynamics theory, and the simulated stress waves will be compared with the experimental stress wave. Then, the crack propagation of ring-shaped specimen like rock under dynamic loading will be analysed.

σ (t ) =

Ae Ee [εI (t ) + εR (t ) + εT (t )] 2As

(1)

ε (t ) =

Ce Le

(2)

ε˙ (t ) =

Ce [εI (t ) − εR (t ) − εT (t )] Le

∫0

t

[εI (t ) − εR (t ) − εT (t )] dt

(3)

where σ (t ) , ε (t ) and ε˙ (t ) are the axial compressive stress, strain and strain rate of the specimen, respectively, Ae , Ee and Ce are the crosssectional area, Young's modulus and 1D elastic wave speed of the bar, respectively, As and Ls are the cross-sectional area and length of the specimen, respectively, and εI , εR and εT are the incident strain, reflected strain and transmitted strain of the bar, respectively. 2.2. Introduction of peridynamics theory Peridynamics theory is a new theory in continuum mechanics that dispenses with the assumption that the body remains continuous during deformation. In peridynamics theory, the body consists of material points with finite volume and mass, and each material point interacts with other material points within some neighbourhood of the point, as shown in Fig. 3. These neighbour material points that do not exceed a distance δ from the material point form the subdomain of the material point, denoted by Hx . This interaction is also called the bond force, pairwise force or constitutive force, denoted by f (η , ξ ) . According to Newton's second law, the peridynamics equation of motion of a material point at time t is formulated as

ρ (x ) u¨ (x , t ) =

∫H f (η, ξ ) dVx′ + b (x, t ),

∀ x′ ∈ H

(4)

where η and ξ denote the relative displacement and relative position between material point x and material point x ′, respectively. ρ is the material density, u¨ is the acceleration of the material point at time t , dVx ′ is the infinitesimal volume of material point x ′ within the subdomain of material point x , and b is the body force density exerted on material point x . The constitutive force depends on the relative displacement η and relative position ξ of the two interacting material points; thus, the functional form of the force is f (η , ξ ) . Silling and Askari32 proposed the constitutive force of a Prototype Micro-elastic Brittle Material (PMB) material for isotropic materials, and its calculation equation is as follows.

2. Introduction of SHPB and peridynamics theory

f (η , ξ ) =

2.1. Introduction of SHPB

∂w (η , ξ ) ξ+η = sc (ξ , δ ) ∂η ξ+η

(5)

where w (η , ξ ) is a scalar-valued micropotential, s is the bond stretch, and c is the micromodulus of the given material, which is related to the elastic modulus E as33

The SHPB technique is widely used in the testing of the dynamic properties of materials. The tester consists of a striker, incident bar, specimen, and transmitted bar and is shown in Fig. 1. A rectangular striker is generally used when testing the general materials; to eliminate wave dispersion, however, a spindle-shaped striker is used when testing the dynamic properties of rock materials. The size of the strikers is shown in Fig. 2. Strain gauges are placed on the incident bar and transmitted bar to derive the incident wave, reflected wave and

c=

9E πhδ 3

(6)

where h is the thickness of the plate. In this paper, the thickness h is the discretized size. The bond stretch is calculated by Fig. 1. The diagram of Split Hopkinson Pressure Bar.

2

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Fig. 2. The sizes of two strikers.

s=

ξ+η − ξ ξ

u˙ n + 1 = u˙ n +

(7)

The μ (x , ξ , t ) is a scalar-valued function used to determine whether there is an interaction between material points. When the stretch exceeds the critical stretch, the bond breaks, and then there is no constitutive force between two corresponding material points.

1 sc ≤ s ≤ st μ (x , ξ , t ) = ⎧ ⎨ else ⎩0

(9)

3. Peridynamics simulation of the SHPB test with strikers 3.1. Numerical scheme and algorithms 3.1.1. Numerical scheme To solve the peridynamics equation of motion, the body needs to be uniformly discretized into nodes with discretized size Δx . After discretization, the motion equation is written as m

ρ (x ) u¨ n =

∑ f (ηn , ξ ) Vx′ + bn (x ) H

(10)

where n is the number of time steps and m is the total number of material points x ′ in the subdomain of material point x . Correspondingly, the calculation equation of local damage at material point x can be expressed as

ϕ (x , t ) = 1 −

∑H μ (t ,

ηn ,

v‾(tk+) Δ t =

(Δt )2 [∑H f (ηn , ξ ) Vx ′ + bn (x )] 2ρ

u‾(tk+) Δ t − u (tk ) Δt

The reaction force

ξ ) dVx ′

∑H dVx ′

(12)

(13)

3.1.2. The contact algorithms The implementation of loading stress in the peridynamics simulation of SHPB follows two methods29 One method is the force boundary, in which the striker does not appear and the loading stress is directly exerted on the incident bar. Another method is the velocity boundary, in which the striker appears directly with an initial velocity vimpact . To compare the waveform of the stress wave under different striker shapes, the velocity boundary is used. There are two contact algorithms that are suitable for the contact between the rigid body and deformable body and for the contact between deformable bodies in the research of impact problems34,35 Because the stiffness of the bar is far higher than that of the rock material, the contact interaction between the bar and specimen is analysed according to the contact algorithm between the rigid body and the deformable body as shown in Fig. 4. At a particular time step t , the rigid impactor moves with velocity v , the specimen is governed by the PD equation of motion, and there are no interactions between the rigid impactor and the specimen. At the next time step t + Δt , the rigid impactor and the specimen partially overlap because the velocity of the rigid impactor exceeds the velocity of the specimen, as illustrated in Fig. 4(b). The material point exerts a reaction force F((kt +) Δ t ) on the impactor. To reflect the physical contact, the overlapping material points inside the impactor are relocated to their new positions outside the impactor and nearest to the impactor (see Fig. 4(c)). The material points have a modified displacement marked as u‾(tk+) Δ t . Hence, the modified velocity v‾(tk+) Δ t of the material point can be calculated as

(8)

∫H μ (t , η, ξ ) dVx ′ ∫H dVx ′

ρ

un + 1 = un + u˙ n Δt +

where sc and st are the compressive critical stretch and the tensile critical stretch, respectively. After the bond breaks, local damage occurs in the material points, and the local damage of any material point in the material can be calculated.

ϕ (x , t ) = 1 −

Δt [∑H f (ηn , ξ ) Vx ′ + bn (x )]

(11)

F((kt +) Δ t ) = −1 × ρ(k )

Based on the forward difference technique, the velocity and displacement at the next time step are given by

(14)

F((kt +) Δ t )

(v‾(tk+) Δ t

− Δt

can be computed from

v (tk+) Δ t )

V(k )

(15)

Then, the total reaction force, F (t + Δ t ) , on the impactor by the

Fig. 3. The schematic diagram of peridynamics theory. 3

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Fig. 4. Relocation of material points inside specimen to represent contact with the incident bar.

Fig. 5. The strain-time curve after the strikers impact the incident bar and transmitted bar together.

Fig. 6. The peridynamics simulation results of the crack propagation process of the specimen.

specimen at time step t + Δt is

F t+Δt =

∑

F((kt +) Δ t ) λ (tk+) Δ t

where λ (tk+) Δ t = 1 wise, λ (tk+) Δ t = 0 .

the contact algorithm between the deformable bodies because they consist of the same material. A repelling short-range force fsh (y(j) , y(k ) ) arises between two different deformable bodies when they come into contact with each other to avoid sharing the same locations36

(16)

when the material points inside the impactor; other-

The contact interactions between the striker and incident bar and between the incident bar and transmitted bar are analysed according to

fsh (y(j) , y(k ) ) = 4

y(j) − y(k ) y(j) − y(k )

min ⎧0, csh ⎛ ⎨ ⎝ ⎩

y(j) − y(k ) d

− 1⎞ ⎫ ⎠⎬ ⎭

(17)

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measured point is set in the middle of the incident bar to record the stress wave during impact. The value of m has an important influence on the calculation results of crack propagation and branch23. To study the m-convergence, the value of Δx1 is fixed at 2 mm, and the value of m varies from 2 to 4. The stress wave curves generated by the spindleshaped striker and rectangular striker under different m values are compared in Fig. 5. In Fig. 5, Sse and Sss are the stress-time curves generated by the spindle-shaped striker impact test and its peridynamics simulation, respectively, and Sre and Srs are the stress-time curves generated by the rectangle striker impact test and its peridynamics simulation, respectively. The peridynamics simulation results show that the rectangular striker generates a rectangular stress wave and that the spindle-shaped striker generates a half-sine stress wave. According to one-dimensional stress wave theory, the constant pressure magnitude in the SHPB test with a rectangular striker is σ = −ρe Ce vimpact /2 , where vimpact is the initial velocity of the striker. The theoretical value of the stress wave is 216 MPa when the initial velocity of the rectangular striker is 10 m/s. Comparing the peridynamics simulation results with the theoretical results and the experimental results in the references37, it can be seen that the waveforms and sizes of the stress wave obtained by the peridynamics simulation are approximately the same as those obtained by experiments and theories, which indicates that the established peridynamics model for the SHPB test is correct. Comparing the stress waves under different m values, it can be seen that the curves of the stress wave generated by both the spindle-shaped striker and rectangle striker are approximately superpositioned when m = 3 and m = 4 , which indicates that the calculated result has converged when m = 3. Therefore, the value of m is always 3 in later calculations.

Table 1 The length and growth rate of three cracks at different timestep. Timestep

3840 3860 3880 3900 3920 3940 3960 3980 4000 4020 4040 4060 4080 4100 4120 4140 4160 4180 4200 4220 4240 4260

Crack length(m)

Crack growth rate(m/s)

Crack-1

Crack-2

Crack-3

Crack-1

Crack-2

Crack-3

0 0.0005 0.002 0.004 0.0055 0.007 0.008 0.009 0.01 0.0105 0.011 0.0115 0.01356 0.01356 0.01356 0.01406 0.01406 0.01406 0.01406 0.01406 0.01406 0.01406

– – – – 0 0.001 0.0025 0.0045 0.0065 0.008 0.0085 0.0095 0.01 0.011 0.0115 0.012 0.01271 0.01271 0.01271 0.01271 0.01271 0.01271

– – – – – – – – – – – – – 0 0.0015 0.0035 0.006 0.0075 0.009 0.01 0.0115 0.012

250 750 1000 750 750 500 500 500 250 250 250 1030.776 0 0 250 0 0 0 0 0 0

– – – – 500 750 1000 1000 750 250 500 250 500 250 250 353.5534 0 0 0 0 0

– – – – – – – – – – – – – 750 1000 1250 750 750 500 750 250

4. Peridynamics simulation of crack propagation of ring-shaped specimen under dynamic loading To research crack propagation of ring-shaped specimen under dynamic loading, the 2D peridynamics model of the ring-shaped specimen SHPB test is established. The inner diameter and outer diameter of the ring are 50 mm and 24 mm, respectively. The material parameters of the specimen are as follows: the material point size Δx1 of the specimen is 0.5 mm, the density is 2400 kg/m3 and Young's modulus is 50 GPa, and the tensile critical stretch st and compressive critical stretch sc are 1.0 × 10−3 and − 1.0 × 10−2 , respectively. The parameters of the bars and striker are invariable. The initial velocity of the spindle-shaped striker is 6.5 m/s. Fig. 6 shows the crack propagation process of the ring-shaped specimen under dynamic loading. The crack marked as 1 first initiate at the inner diameter of the specimen near the incident bar and spreads outside in the direction of the parallel incident bar. Then, at the inner diameter of the specimen near the transmitted bar, the crack marked as 2 also initiates and propagates outside of the specimen in the direction of the parallel incident bar. We call these two cracks initial cracks. It can be seen that the initial cracks are basically essentially coincident with the centreline of the specimen, and there is local crushing at the end of the specimen near the incident bar. After the initial cracks penetrate through the whole ring-shaped specimen, the secondary cracks marked as 3 begin to develop on both sides of the specimen at the same time. The secondary cracks initiate in the outside of the specimen and extend from the outside to the inner side of the specimen, in which the propagation direction is exactly opposite to that of the initial cracks. The secondary cracks are perpendicular to the initial cracks. The simulated failure mode of the specimen is the same as that of the experimental results17, which indicates that the established peridynamics model of the specimen is correct. To analyse the propagation rules of the initial cracks and the secondary cracks, the lengths and growth rates of the cracks are recorded and shown in Table 1 and Fig. 7. It can be seen from the diagram that the length growth curves of the three cracks all appear in the shape of an “S”, in which the length

Fig. 7. The lengths and the growth rates of the cracks.

where csh is the short-range force constant, csh = 3.5c and 10c when using the spindle-shaped striker and rectangular striker, respectively, and d is the initial distance of adjacent material points in different objects. In addition, there is a repelling short-range force between the damaged material points in order to prevent them from penetrating each other; in this case, d = min(0.9 x (j) − x (k ) , 1.35 Δx ) . 3.2. Comparison of two stress waves To analyse the stress wave generated by the impact of two kinds of strikers, as evaluated by peridynamics theory, the peridynamics simulation of SHPB without specimens is carried out based on the above algorithm. In this simulation, the sizes of the test devices are based on the size of the SHPB equipment at Central South University. The lengths and diameters of the incident and transmitter bars are 2000 mm and 50 mm, respectively, and the sizes of the strikers are shown in Fig. 2. The material parameters of the bars are as follows: Young's modulus Ee = 240 GPa and mass density ρ = 7800 kg , the material point size Δx1 of the SHPB bar is 2 mm, the horizon size of δ1 is m times as large as the material point size Δx1 , and the time step is Δt = 1.0 × 10−7s . The 5

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Fig. 8. The horizontal and vertical displacement fields of the specimen during the crack propagation.

Fig. 9. The tensile strain energy density t fields of the specimen during the crack propagation. Table 2 The compared results of the peak stresses of peridynamics simulation and experimental. Diameter(mm) Test peak stress(MPa) Test loading rate(GPa/s) Simulated peak stress(MPa) Simulated loading rate(GPa/s) Peak stress error rate (%)

0 20.00 592.90 27.28 722.54 36.41

8 18.93 530.00 20.72 695.06 9.48

12 15.55 574.10 15.94 595.83 2.53

16 11.67 441.20 12.39 518.75 6.14

20 9.18 357.50 9.35 426.25 1.87

24 – – 6.28 279.27 –

28 – – 3.68 165.31 –

32 – – 1.89 77.66 –

36 – – 0.09 5.30 –

specimen during crack initiation and propagation. It can be seen that both the horizontal and vertical displacement fields are symmetrically distributed along the direction of stress wave propagation, but their variation laws are different during the dynamic loading. When the initial crack initiates, the whole specimen moves towards the transmitted bar, but the horizontal displacement of the specimen decreases gradually from the end of the specimen near the incident bar to the other end near the transmitted bar, and the horizontal displacement gradient at the left end of the specimen is far higher than that at the right end. With continued loading, the horizontal displacement of the specimen still decreases gradually from left to the right, but the distance between both ends of the specimen increases, and the horizontal displacement gradient at both ends of the specimen is high. After secondary crack

growth curves of crack-1 and crack-2 are parallel basically and the length growth curve of crack-3 is steeper. From the starting velocity of crack propagation, the starting velocity of crack-1 is less than that of crack-2, which is less than that of crack-3, and the starting velocities of three cracks are 250 m/s, 500 m/s, and 750 m/s. The various laws of the growth rate of the three cracks are basically similar and all of them change with the rule of first increasing and then decreasing. The growth rate of crack-1 increases sharply when local crushing occurs at the end of the specimen near the incident bar. The relationship among the average growth rate of the three cracks which is 550 m/s, 661 m/s, and 821 m/s, is the same as that of their starting velocities without considering the local crushing at the end of the ring-shaped specimen. Fig. 8 shows the horizontal and vertical displacement fields of the 6

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loading, as shown in Fig. 9. It can be seen that the area where the tensile strain energy density is relatively high is mainly distributed on the inner diameter near the incident bar when the left initial crack initiates. As the loading continues, the tensile strain energy density at the inner diameter near the transmitted bar is higher than that at the left side of the specimen. This is also the reason why the propagation velocity of crack 1 is lower than that of crack 2 when these two cracks propagate simultaneously. During the propagation of the initial crack, an obvious concentration area of tensile strain energy density appears at the upper and lower sides of the outer diameter, and this region continues to expand and concentrate in the middle of the specimen gradually. Over the entire crack propagation process, the location of the maximum tensile strain energy density is always at the tip of the crack. This result further indicates that the failure of the ring-shaped specimen occurs in tension. 5. Discussion on the effect of the hole diameter on rock failure The aperture size has an important influence on the failure mode of the ring-shaped specimen, so we conduct peridynamics simulations of the SHPB test of the specimen with a different central hole. The parameters of the specimen are invariable, and the striker is spindle-shaped with an initial velocity of 6.5 m/s. The loading stress curve, loading rate and peak stresses of specimens with different apertures are obtained. To compare the simulated results with the experimental results, the peak stress and loading rate of the loading stress of the simulation and experiment are recorded in Table 2. Figs. 10 and 11 show the compared results of the peak stress between the peridynamics simulation and experiment and the loading stress curves of the ring-shaped specimen with different central holes. It can be seen that the peak stress of the peridynamics simulation is very close to that obtained by experimental tests, especially for ring-shaped specimens with central holes with diameters of 12 mm, 16 mm and 20 mm. The differences between the peridynamics simulation results and the test results are less than 7%. For the specimen with an aperture of 8 mm, the difference between simulation and experiment is no more than 10%. The difference for the disk is relatively large: a 36.41% difference between the peridynamics simulation results and the experiment results. This result occurs because the stress concentration acts at the end of the specimen and the simulated results do not the dynamic tensile strength of the specimen. In spite of this factor, the comparison of results further indicates that the simulation based on peridynamics is correct. It can be seen that the loading capacity of the ring-shaped specimen and the loading rate of the stresses decrease, and the curves of loading stress tend to be gradual with increasing aperture of the ring-shaped specimen under the condition of the same initial velocity of the striker. Fig. 12 shows the failure patterns of the ring-shaped specimens under dynamic loading by experiment and peridynamics simulation. It can be seen that the peridynamics-simulated failure patterns of the ringshaped specimens are very similar to those of the experimental results, and they all show the same rules. When the apertures of the ring-shaped specimen are small, the failure patterns of the specimen are the same as those of the complete disc specimen, and there is only one initial crack that parallels the propagation direction of the stress wave distributed on the ring-shaped specimen that was divided into two parts. When the apertures of the ring-shaped specimen increase gradually, the specimens are divided into four parts by the initial cracks and secondary cracks that are perpendicular to each other. When the apertures of the ring-shaped specimen increase to a certain extent, the failure mode of the specimen changes to asymmetrical failure. Although the position of the initial crack is unchanged, the secondary cracks initiate near the incident bar, and the numbers of secondary cracks and broken parts may increase. This phenomenon is discovered in Ref. 19 and related to the structure effect of the ring-shaped specimen. According to previous analysis, an obvious concentration area of tensile strain energy density

Fig. 10. The compared results of the peak strength between the peridynamics simulation and experiment.

Fig. 11. The loading stress of ring specimen with a different diameter central hole under dynamic loading.

initiates, the specimen still moves towards the transmitted bar at the stage of crack initiation. When the crack has propagated to a certain length, the left end of the specimen still moves towards the transmitted bar, but the right end moves in the opposite direction. At the top and bottom of the specimen, the right-side example of a secondary crack also continues to move forward, but the left-side example moves back. The crack opening displacement of all cracks increases constantly. For the vertical displacement, the displacement in the middle of the ringshaped specimen is higher than that at the end of the specimen. At the initiation stage of the initial crack, most of the area with larger vertical displacement tends to distribute on the left end of the specimen. With continued loading, this area moves into the middle of the specimen gradually, and the vertical displacement on both sides is also approximately symmetrically distributed. Combined with the variation laws of the displacement fields and the propagation laws of the cracks, it can be judged that the failure modes of these cracks are tensile failures. To better understand the mechanism of initiation and propagation of the cracks, we obtained the tensile strain energy density fields of the ring-shaped specimen during dynamic 7

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Fig. 12. The comparison of failure patterns between the experiment and peridynamics simulation.

middle of the specimen. Therefore, the secondary cracks form in the specimen near the incident bar. In addition, it can also be found that a crack-branching phenomenon occurs in some specimens with a smaller inner diameter. This result is also caused by the structure effect of the ring-shaped specimen. When the inner diameter of the specimen is smaller, the ring width is larger, which means that the distance over which the horizontal crack can propagate is longer. At the initiation

appears at the upper and lower sides of the outer ring near the incident bar, and this area continues to expand and concentrate in the middle of the specimen gradually after the initial cracks penetrate. With the increase of the inner diameter of the specimen, the strength and the bearing capacity decrease rapidly. After the initial cracks penetrate, the tensile strain energy density at the outer diameter near the incident bar exceeds the capacity of the specimen before the area moves into the

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References

stage of the horizontal crack, the dynamic stress intensity factor is smaller, so the crack propagates only. With the increase of the crack length, the dynamic stress intensity factor also increases rapidly. When this factor exceeds a certain value, the crack branches. If crack branching occurs, fragments will form when the horizontal cracks propagate to the end of the specimen. At this moment, these fragments are still affected by the dynamic loading effect; they will continue to move with a higher velocity and lower velocity and will interact within the specimen, resulting in crack branching occurring at another end of the horizontal cracks. When the inner diameter is relatively large, the ring width and the dynamic stress intensity factor are relatively small, and the horizontal cracks penetrate before they branch. Therefore, we can find that the crack branch occurs only in specimens with a smaller inner diameter rather than specimens with a larger inner diameter. This phenomenon is similar to that observed in Ref. 18. The variation law of the failure patterns of the ring-shaped specimen also indicates that the end of the ring-shaped engineering structure should be strengthened for protection and regularly inspected.

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6. Conclusion In this paper, peridynamics theory is applied to simulate the SHPB experiment and the crack propagation of ring-shaped specimen like rock under dynamic loading. The numerical results show that the stress wave obtained by the peridynamics simulation is consistent with the experimental results. Under dynamic loading, the failure of the ringshaped specimen with an aperture of 24 mm occurs due to tension, and this tensile failure is mainly distributed on the upper and lower sides of the outer diameter and the left and right ends of the inner diameter. The crack marked as 1 initiates first near the incident bar inside the ringshaped specimen, and then the crack marked as 2 initiates near the transmitted bar inside the specimen, while the secondary cracks marked as 3 initiates on the outer edge of the two sides of the specimen. The sequence of the starting velocities and average propagation velocities of the cracks from high to low is crack-3, crack-2, and crack-1. Under the condition of the same initial velocity of the striker, the loading rates and the peak stresses decrease gradually, and the failure patterns of the specimens transform from splatted by one initial crack to broken by two cracks that are perpendicular to each other and then to asymmetrical failure with an increase in the apertures of the specimen. These variation laws are affected by the structure effect. The peridynamics simulation results are the same as the experimental results basically, which indicates that peridynamics theory is an accurate and effective method to analyse the failure processes of rock-like materials under dynamic loading.

Conflicts of interest The authors declare that they have no conflicts of interest.

Acknowledgments This work is financially supported by the National Natural Science Foundation of China (51874352, 51774323), the Fundamental Research Funds for the Central Universities of Central South University (2018zzts213), the Natural Science Foundation of Hunan Province (2018JJ3676) and the Key Research & Development Program of Hunan Province (2017GK2190).

Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ijrmms.2019.104093. 9

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