Acta Geotechnica

, Volume 12, Issue 6, pp 1207–1227 | Cite as

Rate-independent fracture toughness of gray and black kerogen-rich shales

  • Pooyan Kabir
  • Franz-Josef Ulm
  • Ange-Therese AkonoEmail author
Research Paper


The objective of this investigation is to characterize the influence of the loading rate, scratch speed, mineralogy, morphology, anisotropy, and total organic content on the scratch toughness of organic-rich shale. We focus our study on a gray shale, Toarcian shale (Paris basin, France) and a black shale, Niobrara shale (northeastern Colorado, USA). Microscopic scratch tests are performed for varying scratch speeds and loading rates. We consider several orientations for scratch testing. For all gas shale specimens, the scratch toughness is found to increase with increasing scratch speed. In the asymptotic regime of high speeds, there is a convergence toward a single constant value irrespective of the loading rate. To understand this evolution of the scratch toughness, a nonlinear fracture mechanics model is built that integrates fracture dissipation with the various forms of viscous processes. In particular, a coupling is shown between the fracture energy and the viscoelastic characteristics. An inverse approach which combines scratch and indentation testing makes it possible to represent all tests in a single curve and retrieve the rate-independent fracture toughness of kerogen-rich shale materials. The presence of organic matter drastically alters the creep and fracture properties at the microscopic length-scale. The fracture behavior is anisotropic with the divider orientation yielding the highest fracture toughness value and the short transverse orientation yielding the lowest fracture toughness. Elucidating the fracture-composition-morphology relationships in organic-rich shale will promote advances in science and engineering for energy-related applications such as hydraulic fracturing in unconventional reservoirs or \(\hbox {CO}_2\) sequestration in depleted reservoirs.


Fracture Kerogen-rich shale Scratch test Viscoelasticity 

List of symbols


Scratch projected load-bearing contact area


Indentation contact radius


Contact creep modulus


Closed contour including the crack tip


Material constant

\(\mathbb {C}_0\)

Initial stiffness tensor


Scratch penetration depth

\(\mathcal {D}\)

Energy dissipated

\(\varepsilon _{xx}\), \(\varepsilon _{zz}\)

Strain tensor components

\(\underline{e}_x\), \(\underline{e}_y\), \(\underline{e}_z\)

Unit base vectors in Cartesian reference frame


Scratch horizontal force


Scratch vertical force


Scratch loading rate

\(\mathcal {F}_2\)

Dimensionless function

\(\mathcal {G}\)

Energy release rate

\(G_{\rm f}\)

Fracture energy


Indentation depth

\(\mathcal {H}(t)\)

Viscoelastic correction factor


Blade-material scratch interface

\(K_{\rm s}\)

Scratch toughness

\(K_{\rm c0}\)

Rate-independent fracture toughness

\(\mathcal {L}^{-1}\left( \cdot \right)\)

Inverse Laplace operator


Relaxation plane strain modulus


Initial plane strain modulus


Outward unit vector

\(n_x\), \(n_z\)

Components of outward unit vector on x-axis and z-axis


Scratch perimeter


Prony series constant


Tip radius


Stress vector


Volume density of frozen energy


Critical speed

\(\bar{\gamma }\)

Energy ratio


Fracture surface

\(\underline{\underline{\varepsilon }}\)

Strain tensor

\(\underline{\underline{\varepsilon }}^{el}\)

Elastic strain tensor

\(\underline{\underline{\varepsilon }}^v\)

Viscous strain tensor


Deviatoric portion of strain tensor


Volumetric strain


Probe half-apex angle

\(\widehat{\kappa }(t)\)

Relaxation bulk modulus

\(\kappa _0\)

Initial bulk modulus

\(\bar{\lambda }\), \(\bar{\lambda }_i\)

Plane strain viscoelastic factors


Crack length

\(\widehat{\mu }(t)\)

Relaxation shear modulus

\(\mu _0\)

Initial shear modulus

\(\underline{\xi }\)

Displacement field vector

\(\underline{\underline{\sigma }}\)

Stress tensor


Deviatoric portion of stress tensor

\(\sigma _{xx}\), \(\sigma _{yy}\)

Stress tensor components

\(\sigma _m\)

Volumetric stress

\(\tau\), \(\tau _i\)

Relaxation characteristic time

\(\tau _{\rm cr}\)

Creep characteristic time


Volume density of Helmholtz free energy

\(\psi ^{el}\)

Volume density of elastic strain energy


Material volume

\(P_{\max }\)

Maximum indentation load


Indentation unloading stiffness


Time increment


Scratch speed



The authors would like to thank Total Corp., Paris, France, for providing the gas shale specimens tested and analyzed in this investigation. The research was funded by Prof. Akono’s Start-Up fund account which was provided by the Department of Civil and Environmental Engineering as well as the College of Engineering at the University of Illinois at Urbana-Champaign. In addition, we acknowledge the Distinguished Structural Engineering Fellowship that supported Pooyan Kabir during his Ph.D. studies. We are thankful to Total Corp. (France) and the MIT X-Shale Project for providing the Toarcian shale and Niobrara shale specimens. The work was carried out in part in the Frederick Seitz Materials Research Laboratory Central Research Facilities, University of Illinois at Urbana-Champaign.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Civil and Environmental EngineeringMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Department of Mechanical Science and EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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