Rock Mechanics and Rock Engineering

, Volume 52, Issue 1, pp 1–18 | Cite as

Size Effect Analysis for the Characterization of Marcellus Shale Quasi-brittle Fracture Properties

  • Weixin Li
  • Zhefei Jin
  • Gianluca CusatisEmail author
Original Paper


The fracture characterization of shale rocks requires understanding the scaling of the measured properties to enable the extrapolation from small-scale laboratory tests to field applications. In this study, the fracture properties of Marcellus shale were obtained through size effect tests. Fracture tests were conducted on three-point-bending specimens with increasing size. The test results show that the nominal strength decreases with increasing specimen size and it can be fitted well by Bažant’s size effect law. This demonstrates that shale fracture behavior deviates from classical linear elastic fracture mechanics (LEFM), and it has quasi-brittle characteristics. This implies, in turn, that the fracture toughness (or fracture energy) computed according to LEFM is size-dependent and, in general, cannot be considered a material property. Furthermore, the size effect analysis allows one to accurately identify the quasi-brittle fracture properties, namely the initial fracture energy and the effective fracture process zone length. A significant anisotropy was observed in the fracture properties determined with three principal notch orientations.


Marcellus shale Size effect Fracture energy Fracture toughness Fracture process zone Quasi-brittle 

List of Symbols


Energy release rate

\(G_{\text {Ic}}\)

Effective LEFM fracture energy

\(G_{\text {f}}\)

Initial fracture energy

\(G_{\text {F}}\)

Total fracture energy

\(K_{\text {IcA}}\)

Apparent fracture toughness calculated using LEFM

\(K_{\text {I}}\)

Mode I stress intensity factor

\(K_{\text {Ic}}\)

Fracture toughness

\(K_{\text {IcA}}\)

Apparent fracture toughness calculated using LEFM

\(\ell _{\text {FPZ}}\)

Length of FPZ

\(c_{\text {f}}\)

Effective FPZ length

\(\sigma (\delta )\)

Cohesive stress as a function of crack opening \(\delta\)

\(f'_{\text {t}}\)

Tensile strength of cohesive law


Brittleness number


Transitional size


Normalized size


Hillerborg’s characteristic length

LD, t

Specimen length, depth, and thickness


Support span


Crack length


Notch length (equal to initial crack length)


Dimensionless crack length

\(\alpha _0\)

Dimensionless notch length (equal to the initial value of \(\alpha\))


Applied load

\(P_{\text {u}}\)

Peak load

\(\sigma _{N}\)

Nominal stress

\(\sigma _{Nu}\)

Nominal strength (nominal stress at peak load)

\(E, E'\)

In-plane and out-of-plane modulus of material

\(\nu , \nu '\)

In-plane and out-of-plane Poisson’s ratio of material


Out-of-plane shear modulus of material

\(E_x, E_y\)

Elastic constants (Young’s modulus) in the specimen coordinate system

\(\nu _{xy}, \nu _{yx}\)

Elastic constants (Poisson’s ratio) in the specimen coordinate system


Elastic constants (shear modulus) in the specimen coordinate system

\(\lambda , \rho\)

Dimensionless elastic constants


Effective elastic modulus

\(g, g'\)

Dimensionless energy release rate and its derivative

\(g_0, g'_0\)

Dimensionless energy release rate at \(\alpha _0\) and its derivative value


Dimensionless stress intensity factor


Dimensionless function

\(\sigma _0\)

A parameter in size effect law


Notch-machining error


Coefficient of determination


Root-mean-squared error


Standard deviation


Standard error



The authors would like to thank Professor Brad Sageman (Department of Earth and Planetary Sciences, Northwestern University) for providing the Marcellus shale samples used in this study and Professor Giuseppe Buscarnera (Department of Civil and Environmental Engineering, Northwestern University) for his assistance with the Mini-Tester. This work also made use of the Materials Characterization and Imaging Facility and the Center for Sustainable Engineering of Geological and Infrastructure Materials (SEGIM) at Northwestern University.


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Theoretical and Applied MechanicsNorthwestern UniversityEvanstonUSA
  2. 2.Department of Civil and Environmental EngineeringNorthwestern UniversityEvanstonUSA

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