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An Integrated Approach for Evaluation of Linear Cohesive Zone Model’s Performance in Fracturing of Rocks

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Abstract

Fracturing in rocks results in the formation of an inelastic region surrounding the crack tip called the fracture process zone (FPZ), which is often characterized using the Linear Cohesive Zone Model (LCZM). Various numerical studies have shown that the prediction of the FPZ characteristics is significantly influenced by variability in the input parameters of LCZM, such as crack tip opening displacement and tensile strength. In this study, an integrated approach was used for evaluating the LCZM performance in predicting fracture processes of Barre granite specimens, as a representative rock, under Mode I loading. The approach involved experimental testing, numerical simulation, uncertainty quantification of overall fracture behavior, and global sensitivity analysis. First, parameters of the LCZM were estimated from three-point bending tests on center notch Barre granite specimen using the two-dimensional digital image correlation (2D-DIC) technique. This was followed by the implementation of the LCZM in XFEM-based numerical model to simulate the evolution of the FPZ in tested geometry. The results from the deterministic numerical simulation showed that while LCZM can predict all stages of FPZ evolution, the variability in the experimental results, such as the FPZ size, cannot be accounted. The variability of the material response was quantified using a random variable analysis, which involved treating the LCZM’s parameters as random variables. This was followed by the global sensitivity analysis that revealed the most sensitive input parameters is the tensile strength for accurate prediction of the global response of rock specimens under Mode I loading.

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Abbreviations

DIC:

Digital image correlation

CZM:

Cohesive Zone model

LCZM:

Linear Cohesive Zone model

FPZ:

Fracture process zone

FPZ-I:

Fracture process zone initiation

TFZ:

Traction free zone

RBF:

Radial basis function

MCS:

Monte Carlo simulation

GSA:

Global sensitivity analysis

NSE:

Nash–Sutcliffe efficiency

COV:

Coefficient of variation

PDF:

Probability distribution function

CMOD:

Crack mouth opening displacement

H :

Specimen height

B :

Specimen thickness

\(a_{0}\) :

Notch length

\({\text{CTOD}}\) :

Crack tip opening displacement

\(x,y\) :

Coordinates

\(U\) :

Displacement in x direction

\(\delta\) :

Load-point displacement

\(\delta _{{{\text{peak}}}}\) :

Load-point displacement at peak load

\(\delta _{{{\text{norm}}}}\) :

Normalized load-point displacement

\(P_{{{\text{peak}}}}\) :

Peak load

\(P_{{{\text{norm}}}}\) :

Normalized applied load

\(w\) :

Crack opening displacement

\(w_{{\text{e}}}\) :

Elastic opening displacement

\(w_{{{\text{ne}}}}\) :

Inelastic crack opening displacement

\(w_{{\text{c}}}\) :

Critical crack opening displacement

\(w_{{{\text{ne}}}}^{{\text{c}}}\) :

Critical inelastic crack opening displacement

\(l,~L_{{{\text{FPZ}}}}\) :

Developing and fully developed fracture process zone

\(\sigma _{n}\) :

Cohesive stress

\(\sigma _{{\text{e}}}\) :

External tensile stress

\(\sigma _{{\text{t}}}\) :

Tensile strength of rock

\(\sigma _{{n,\max }}\) :

Cohesive strength

\(\sigma _{{\max }}\) :

Maximum principal stress

\(\partial {\text{CTOD}}/\partial \left( {\delta _{{{\text{norm}}}} } \right)\) :

CTOD derivative

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

Mode I fracture energy based on CZM

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

Mode I fracture energy based on LEFM

\(S_{i}\) :

First order Sobol’ index for ith input parameter

\(S_{{T_{i} }}\) :

Effects Sobol’ index for ith input parameter

\(u\left( x \right)\) :

Total displacement

\(u_{{{\text{cont}}}} \left( x \right),u_{{{\text{disc}}}} \left( x \right)~\) :

Continuous and discontinuous part of displacement

\(H\left( x \right)\) :

Heaviside enrichment function

\(F_{\alpha } \left( x \right)\) :

Crack tip enrichment functions

\(a_{J} ,b_{{I\alpha }}\) :

Degrees of freedoms of the enriched nodes

\(N_{I} \left( x \right)\) :

Standard nodal shape functions

\(u_{I}\) :

Nodal displacement for standard nodes

\(K_{\Gamma } ,~K_{\Lambda }\) :

Nodal subsets containing crack

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Acknowledgements

Funding for this research was provided by the National Science Foundation under Award number 1644326. The authors are grateful for this support.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Colorado School of Mines, Golden, CO, 80401, USA

    Prasoon Garg, Ahmadreza Hedayat & D. V. Griffiths

  2. Department of Civil Engineering, Indian Institute of Science, Bangalore, India

    Bhardwaj Pandit & Gandluri Lakshminarasimha Sivakumar Babu

Authors
  1. Prasoon Garg
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  2. Bhardwaj Pandit
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  3. Ahmadreza Hedayat
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  4. D. V. Griffiths
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  5. Gandluri Lakshminarasimha Sivakumar Babu
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Correspondence to Prasoon Garg.

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Garg, P., Pandit, B., Hedayat, A. et al. An Integrated Approach for Evaluation of Linear Cohesive Zone Model’s Performance in Fracturing of Rocks. Rock Mech Rock Eng 55, 2917–2936 (2022). https://doi.org/10.1007/s00603-021-02561-5

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  • DOI: https://doi.org/10.1007/s00603-021-02561-5

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