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Numerical Investigation of Dynamic Rock Fracture Toughness Determination Using a Semi-Circular Bend Specimen in Split Hopkinson Pressure Bar Testing

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Abstract

The International Society for Rock Mechanics (ISRM) has suggested a notched semi-circular bend technique in split Hopkinson pressure bar (SHPB) testing to determine the dynamic mode I fracture toughness of rock. Due to the transient nature of dynamic loading and limited experimental techniques, the dynamic fracture process associated with energy partitions remains far from being fully understood. In this study, the dynamic fracturing of the notched semi-circular bend rock specimen in SHPB testing is numerically simulated for the first time by the discrete element method (DEM) and evaluated in both microlevel and energy points of view. The results confirm the validity of this DEM model to reproduce the dynamic fracturing and the feasibility to simultaneously measure key dynamic rock fracture parameters, including initiation fracture toughness, fracture energy, and propagation fracture toughness. In particular, the force equilibrium of the specimen can be effectively achieved by virtue of a ramped incident pulse, and the fracture onset in the vicinity of the crack tip is found to synchronize with the peak force, both of which guarantee the quasistatic data reduction method employed to determine the dynamic fracture toughness. Moreover, the energy partition analysis indicates that simplifications, including friction energy neglect, can cause an overestimation of the propagation fracture toughness, especially under a higher loading rate.

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Abbreviations

DEM:

Discrete element method

ISRM:

International Society for Rock Mechanics

NSCB:

Notched semi-circular bend

SHPB:

Split Hopkinson pressure bar

SIF:

Stress intensity factor

a :

Crack length of the NSCB sample (m)

α a :

Dimensionless crack length of the NSCB sample

A b :

Cross-section area of the pressure bars (m2)

A s :

Area of the fracture surface (m2)

B :

Thickness of the NSCB sample (m)

dF s :

Increment of the shear force (N)

ds :

Increment of the relative displacement (m)

E b :

Young’s modulus of the elastic bars (MPa)

E bond :

Potential energy stored in bonds (J)

E contact :

Potential energy stored in contacts (J)

E friction :

Friction energy (J)

E kinetic :

Kinetic energy (J)

f Im :

Values of the contact force m on the bar–specimen incident interface (N)

f Tm :

Values of the contact force n on the bar–specimen transmitted interface (N)

F Ic :

Force on the specimen’s incident end (N)

F Tc :

Force on the specimen’s transmitted end (N)

F nbi :

Normal force applied on the bond i (N)

F sbi :

Shear force applied on the bond i (N)

F ni :

Normal force applied on the contact i (N)

F si :

Shear force applied on the contact i (N)

G :

Fracture energy dissipated per unit area (J/m2)

I bi :

Moment of inertia of the bond i (kg m2)

I i :

Moment of inertia of the particle i (kg m2)

k nbi :

Normal stiffness of the bond i (N/m)

k sbi :

Shear stiffness of the bond i (N/m)

k ni :

Normal stiffness of the contact i (N/m)

k si :

Shear stiffness of the contact i (N/m)

K I :

Quasistatic stress intensity factor (MPa m0.5)

K IC :

Mode I fracture toughness (MPa m0.5)

K dIC :

Mode I dynamic initiation fracture toughness (MPa m0.5)

K dpIC :

Mode I dynamic propagation fracture toughness (MPa m0.5)

\(\mathop {K_{\text{I}} }\limits^{ \cdot }\) :

Loading rate (GPa m0.5/s)

M bi :

Moment applied on the bond i (Nm)

m i :

Mass of the particle i (kg)

N I :

Number of contacts on the bar–specimen incident interface

N T :

Number of contacts on the bar–specimen transmitted interface

N b :

Number of bonds

N t :

Number of total steps

N broken :

Number of broken bonds

N p :

Number of particles

N c :

Number of contacts

R :

Radius of the NSCB sample (m)

P 1 :

Axial force applied on the incident end of the sample (N)

P 2 :

Axial force applied on the transmitted end of the sample (N)

S :

Span of the supporting pins (m)

t i :

Instant when the stress wave first arrives at the incident end of the specimen (s)

t t :

Instant when the stress wave first arrives at the transmitted end of the specimen (s)

t b :

Instant when the force equilibrium on both ends of the specimen is first achieved (s)

t p :

Instant when the forces on both ends of the specimen reach the peak value (s)

t d :

Instant when the specimen is destroyed (s)

t e :

Instant when the loading process ends (s)

U :

Strain energy (J)

v i :

Translational velocity of the particle i (m/s)

W :

Surface energy (J)

ω i :

Rotational velocity of the particle i (rad/s)

Y :

Dimensionless stress intensity factor

ε i :

Incident strain signal on the incident bar

ε r :

Reflected strain signal on the incident bar

ε t :

Transmitted strain signal on the incident bar

σ :

Tensile strength of the bond (MPa)

τ :

Shear strength of the bond (MPa)

θ m :

Angle between the bar axis and the direction vector of the contact force m

θ n :

Angle between the bar axis and the direction vector of the contact force n

μ :

Force equilibrium coefficient

E :

Young’s modulus of the specimen (MPa)

\(\nu\) :

Poisson’s ratio of the specimen

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Acknowledgments

The authors are grateful for the financial support from the National Program on Key Basic Research Project (no. 2015CB057903), National Natural Science Foundation of China (no. 51374149), Program for New Century Excellent Talents in University (NCET-13-0382), the Youth Science and Technology Fund of Sichuan Province (2014JQ0004) and the Doctoral Fund of Ministry of Education of China (no. 20130181110044).

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Authors and Affiliations

  1. State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resources and Hydropower, Sichuan University, Chengdu, 610065, Sichuan, China

    Y. Xu, F. Dai, N. W. Xu & T. Zhao

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  1. Y. Xu
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  2. F. Dai
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Correspondence to F. Dai.

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Xu, Y., Dai, F., Xu, N.W. et al. Numerical Investigation of Dynamic Rock Fracture Toughness Determination Using a Semi-Circular Bend Specimen in Split Hopkinson Pressure Bar Testing. Rock Mech Rock Eng 49, 731–745 (2016). https://doi.org/10.1007/s00603-015-0787-x

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