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Study on the Coupling Effect of Sample Size and Strain Rate on Rock Compressive Strength

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Rock Mechanics and Rock Engineering Aims and scope Submit manuscript

Abstract

Deformation and failure of rock have size and strain rate effects. However, the coupling effect of sample size and strain rate on strength of rock is not well studied. At present, the experimental research in this area has accumulated definite experimental data, but there is a lack of theoretical model with solid theoretical basis. Therefore, in the present paper the coupling effect of sample size and strain rate on compressive strength of rock is studied theoretically. In the quasi-static loading regime, the formula for the coupling effect of sample size and strain rate on rock strength is obtained by combining the thermally activated mechanism and the static size effect law of rock strength. In the dynamic loading regime, the formula for the coupling effect of sample size and strain rate on the rock compressive strength is determined by using the Weibull’s activation law of rock defects and the shortest time condition of propagation and coalescence of cracks at different scale levels. Based on the conclusion that the strain rate sensitivity of the strength of rock is the result of competition between the coexisting thermally activated and macro-viscous mechanisms, which dominate at different ranges of strain rates, the formulae for the coupling effect of sample size and strain rate in static and dynamic regimes are superposed to obtain an unified formula for the coupling effect of strain rate and sample size effect on rock compressive strength. The critical strain rate for given sample size and the critical sample size for given strain rate are determined. The comparison with the existing theoretical models and experimental results shows that the present theoretical model is consistent with the existing theories and experimental results, indicating that the proposed model is reasonable. The present model is applicable to rocks of laboratory scale levels.

Highlights

  • The formulae for the coupling effect of sample size and strain rate on rock strength in the quasi-static and dynamic loading regimes are obtained.

  • The unified formula for the coupling effect of strain rate and sample size effect on rock strength is obtained.

  • The critical strain rate for given sample size and the critical sample size for given strain rate are determined.

  • The present model is applicable for rocks.

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Data availability

All data, models, or codes generated or used during the study are available from the corresponding author upon request.

Abbreviations

\(\sigma_{1}\) :

Model material parameter

Γ(*):

Gamma function

V :

Sample volume (m3)

V 0 :

Reference volume (m3)

U 0 :

Activation energy (kJ/mol)

γ :

Activation volume (m3)

K :

Boltzmann’s constant

T :

Absolute temperature (°C)

\(\dot{\varepsilon }\) :

Arbitrary strain rate (s−1)

\(\varepsilon_{0}\) :

The deformation limit at failure (s−1)

\(\dot{\varepsilon }_{I}\) :

Intensity of the strain-rate (s−1)

\(\dot{\varepsilon }_{cr}\) :

Critical strain rate (s−1)

\(\sigma_{t}\) :

Uni-axial tensile stress (MPa)

\(\sigma_{Y}\) :

Failure strength (MPa)

\(\sigma_{r}\) :

The strength corresponding strain rate \(\dot{\varepsilon }_{r}\) (MPa)

\(\sigma_{vis}\) :

Strength due to macro-viscosity (MPa)

\(\sigma_{c}\) :

Static compressive strength (MPa)

\(\sigma_{0}\) :

The reference stress (MPa)

F w :

Opening forces of crack (N)

r :

The size of cracks (mm)

\(v_{r}\) :

Maximum propagation speed (mm)

\(t\) :

Loading time (s)

\(t_{r}\) :

The required crack propagation time (s)

\(\Delta \sigma_{I}\) :

Intensity of residual stress deviator

θ :

Activated crack size distribution parameter

α 0 :

Material constant

n :

Activated crack density

K 1 :

Material constant

\(d\) :

Scale level of the inhomogeneity (mm)

\(t_{total}\) :

Total time required for the adjacent cracks to coalesce (s)

\(\tau_{0}\) :

A temporal parameter in the order of Debye’s vibration period of atoms (s)

\(\phi\) :

An angle between the initial crack surface and the load direction (º)

\(\sigma\) :

Internal stress (MPa)

\(\sigma_{r0}\) :

Material parameter of size effect (MPa)

\(D_{r0}\) :

Material parameter of size effect (mm)

\(l\) :

Distance between centers of adjacent cracks (mm)

\(\dot{e}_{ij}\) :

Deviator strain-rate components (s−1)

\(\rho\) :

The density of the medium (Kg/m3)

\(v\) :

Relaxation velocity (mm/s)

\(\Delta \sigma_{I}\) :

Intensity of residual stress deviator

\(\mu\) :

Poisson’s ratio

E :

Young’s elastic modulus (GPa)

\(G\) :

Shear modulus (GPa)

\(\alpha\) :

Strength model coefficient

λ 0 :

Maximum number of cracks in unit volume

k 0 :

A material constant

\(m\) :

Weibull’s modulus

D :

Specimen size (mm)

S :

The shape parameter

N :

Crack number

k 1 :

A constant

\(r_{c}\) :

First coalesced crack size (mm)

\(t_{i}\) :

The time to crack initiation (s)

\(t_{relax}\) :

Relaxation time (s)

\(t_{M}\) :

Failure time (s)

\(\eta\) :

Macro-viscosity (Pa·s)

c 1 ~ c 3 :

Model coefficients

d 1 ~ d 4 :

Material constants

a :

Initial radius of disk-shaped crack(mm)

\(n^{\prime}\) :

Spatial dimension of model

\(c_{s}\) :

The elastic shear wave speed (m/s)

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Acknowledgements

The study was supported by the National Natural Science Foundation of China (NSFC grants No. 12172036, 51774018), Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT, IRT_17R06).

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Authors and Affiliations

  1. Beijing Advanced Innovation Center for Future Urban Design, Beijing International Cooperation Base for Transportation Infrastructure Construction, Beijing University of Civil Engineering and Architecture, Beijing, 100044, China

    Chengzhi Qi, Zefan Wang & Xiaozhao Li

  2. State Key Laboratory of Disaster Prevention & Mitigation of Explosion & Impact, Army Engineering University of PLA, Nanjing, 210007, China

    Mingyang Wang

Authors
  1. Chengzhi Qi
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  2. Mingyang Wang
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  3. Zefan Wang
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  4. Xiaozhao Li
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Contributions

CQ: Conceptualization, Methodology, Formal analysis, Investigation, Writing—original draft, Writing—review & editing, Funding acquisition, Resources, Supervision. MW: Investigation, Writing—review & editing. Zefan Wang: Formal analysis, Investigation, Writing – review & editing. XL: Investigation, Writing—review & editing.

Corresponding author

Correspondence to Chengzhi Qi.

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Qi, C., Wang, M., Wang, Z. et al. Study on the Coupling Effect of Sample Size and Strain Rate on Rock Compressive Strength. Rock Mech Rock Eng 56, 5103–5114 (2023). https://doi.org/10.1007/s00603-023-03309-z

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  • DOI: https://doi.org/10.1007/s00603-023-03309-z

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