Skip to main content
SpringerLink
Log in

Fracture propagation in a cracked semicircular bend specimen under mixed mode loading using extended finite element method

  • Original Paper
  • Published:
Arabian Journal of Geosciences Aims and scope Submit manuscript

Abstract

The semicircular specimen under three-point bending (SCB) has been widely used to investigate mode I, mode II, and mixed mode I/II fracture behavior in brittle rocks. Compared to the other numerical methods, the extended finite element method (X-FEM) models a crack independently of the finite element mesh without any remeshing step in fracture propagation. In this regard, a numerical code called MEX-FEM, based on X-FEM, has been developed for modeling crack propagation in rock material. Since dimensionless stress intensity factors (i.e., Y I and Y II) in crack modeling of SCB specimen depend on crack length, crack angle, and span ratio, these factors and crack propagation trajectory are determined for different combinations of crack angle, crack length ratio, and span ratio. A very good agreement exists between the calculated factors Y I and Y II in this research work and those reported in the literature. The results show that the values of Y I and Y II increase by increasing the span ratio. The angle in which the pure mode II occurs decreases with increasing of the crack length ratio and it increases when the span ratio increases. The crack growth is along the initial crack when the specimen is subjected to pure mode I (i.e., the crack angle is 0), whereas in mixed mode, the crack is deviated toward the upper loading point. The results of this study demonstrate the utility and robustness of the X-FEM to simulate the crack growth in rock materials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Abbreviations

A :

Crack length

A ω :

Nodal support area

A aboveω :

Nodal support area above the crack

A belowω :

Nodal support area below the crack

a j :

Additional degrees of nodal freedom associated with the Heaviside function

b l k :

Additional degrees of nodal associated with the elastic asymptotic crack tip functions

CB:

Chevron edge-notched round bar in bending

CCBD:

Central cracked Brazilian disk under diametric compression

D :

Diameter of specimen

H :

Modified Heaviside step function

I :

Set of all nodes in the mesh

I (1,2) :

Interaction integral

J :

Set of nodes enriched with discontinuous enrichment

K :

Set of nodes enriched with asymptotic enrichment

K I :

Stress intensity factor mode I

K II :

Stress intensity factor mode II

N(x):

Shape function

P :

Compressive applied load

R :

Radius of specimen

r, θ :

Crack tip coordinates

S :

Half span length

SCB:

Semicircular bend

SIF:

Stress intensity factor

SR:

Chevron-notched short rod

t :

Thickness of specimen

u i :

Nodal displacement

U x :

Horizontal displacement

W (1,2) :

Interaction strain energy

X-FEM:

Extended finite element method

Y I :

Dimensionless stress intensity factor mode I

Y II :

Dimensionless stress intensity factor mode II

α :

Crack angle

θ c :

Crack initiation angle

Δa :

Crack length increment

References

  • Aliha M, Ayatollahi M (2011) Mixed mode I/II brittle fracture evaluation of marble using SCB specimen. Proc Eng 10:311–318

    Article  Google Scholar 

  • Aliha M, Ayatollahi M (2013) Two-parameter fracture analysis of SCB rock specimen under mixed mode loading. Eng Fract Mech 103:115–123

    Article  Google Scholar 

  • Atkinson C, Smelser R, Sanchez J (1982) Combined mode fracture via the cracked Brazilian disk test. Int J Fract 18:279–291

    Google Scholar 

  • Awaji H, Sato S (1978) Combined mode fracture toughness measurement by the disk test. J Eng Mater Technol 100:175

    Article  Google Scholar 

  • Ayatollahi M, Aliha M (2006) On determination of mode II fracture toughness using semi-circular bend specimen. Int J Solids Struct 43:5217–5227

    Article  Google Scholar 

  • Ayatollahi M, Aliha M (2007) Wide range data for crack tip parameters in two disc-type specimens under mixed mode loading. Comput Mater Sci 38:660–670

    Article  Google Scholar 

  • Ayatollahi M, Aliha M, Hassani M (2006) Mixed mode brittle fracture in PMMA—an experimental study using SCB specimens. Mater Sci Eng A 417:348–356

    Article  Google Scholar 

  • Ayatollahi M, Aliha M, Saghafi H (2011) An improved semi-circular bend specimen for investigating mixed mode brittle fracture. Eng Fract Mech 78:110–123

    Article  Google Scholar 

  • Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45:601–620

    Article  Google Scholar 

  • Belytschko T, Gracie R, Ventura G (2009) A review of extended/generalized finite element methods for material modeling. Model Simul Mater Sci Eng 17:043001

    Article  Google Scholar 

  • Chahine E, Laborde P, Renard Y (2008) Crack tip enrichment in the XFEM using a cutoff function. Int J Numer Methods Eng 75:629–646

    Article  Google Scholar 

  • Chang S-H, Lee C-I, Jeon S (2002) Measurement of rock fracture toughness under modes I and II and mixed-mode conditions by using disc-type specimens. Eng Geol 66:79–97

    Article  Google Scholar 

  • Chong K, Kuruppu M (1984) New specimen for fracture toughness determination for rock and other materials. Int J Fract 26:R59–R62

    Article  Google Scholar 

  • Chong KP, Kuruppu MD (1988) New specimens for mixed mode fracture investigations of geomaterials. Eng Fract Mech 30:701–712

    Article  Google Scholar 

  • Chong KP, Kuruppu MD, Kuszmaul JS (1987) Fracture toughness determination of layered materials. Eng Fract Mech 28:43–54

    Article  Google Scholar 

  • Dai F, Chen R, Xia K (2010) A semi-circular bend technique for determining dynamic fracture toughness. Exp Mech 50:783–791

    Article  Google Scholar 

  • Dolbow JE (1999) An extended finite element method with discontinuous enrichment for applied mechanics. PhD thesis, Northwestern University

  • Eftekhari M, Baghbanan A, Hashemolhosseini H (2014) Determining stress intensity factor for cracked Brazilian disc using extended finite element method. Int J Sci Eng Technol 3:890–893

    Google Scholar 

  • Erdogan F, Sih G (1963) On the crack extension in plates under plane loading and transverse shear. J Basic Eng 85:519

    Article  Google Scholar 

  • Fleming M, Chu Y, Moran B, Belytschko T, Lu Y, Gu L (1997) Enriched element-free Galerkin methods for crack tip fields. Int J Numer Methods Eng 40:1483–1504

    Article  Google Scholar 

  • Kuruppu M, Obara Y, Ayatollahi M, Chong K, Funatsu T (2014) ISRM-suggested method for determining the mode I static fracture toughness using semi-circular bend specimen. Rock Mech Rock Eng 47:267–274

    Article  Google Scholar 

  • Kuruppu MD, Chong KP (2012) Fracture toughness testing of brittle materials using semi-circular bend (SCB) specimen. Eng Fract Mech 91:133–150

    Article  Google Scholar 

  • Lancaster I, Khalid H, Kougioumtzoglou I (2013) Extended FEM modelling of crack propagation using the semi-circular bending test. Constr Build Mater 48:270–277

    Article  Google Scholar 

  • Lim I, Johnston I, Choi S (1993) Stress intensity factors for semi-circular specimens under three-point bending. Eng Fract Mech 44:363–382

    Article  Google Scholar 

  • Mahmoud E, Saadeh S, Hakimelahi H, Harvey J (2014) Extended finite-element modelling of asphalt mixtures fracture properties using the semi-circular bending test. Road Mater Pavement Des 15:153–166

    Article  Google Scholar 

  • Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150

    Article  Google Scholar 

  • Mohammadi S (2008) Extended finite element method: for fracture analysis of structures. Wiley, Chichester

    Book  Google Scholar 

  • Nuismer R (1975) An energy release rate criterion for mixed mode fracture. Int J Fract 11:245–250

    Article  Google Scholar 

  • Richardson CL, Hegemann J, Sifakis E, Hellrung J, Teran JM (2011) An XFEM method for modeling geometrically elaborate crack propagation in brittle materials. Int J Numer Methods Eng 88:1042–1065

    Article  Google Scholar 

  • Shetty DK, Rosenfield AR, Duckworth WH (1987) Mixed-mode fracture in biaxial stress state: application of the diametral-compression (Brazilian disk) test. Eng Fract Mech 26:825–840

    Article  Google Scholar 

  • Sih GC (1974) Strain-energy-density factor applied to mixed mode crack problems. Int J Fract 10:305–321

    Article  Google Scholar 

  • Stazi F, Budyn E, Chessa J, Belytschko T (2003) An extended finite element method with higher-order elements for curved cracks. Comput Mech 31:38–48

    Article  Google Scholar 

  • Sukumar N, Prévost J-H (2003) Modeling quasi-static crack growth with the extended finite element method. Part I: computer implementation. Int J Solids Struct 40:7513–7537

    Article  Google Scholar 

  • Whittaker BN, Singh RN, Sun G (1992) Rock fracture mechanics. Elsevier, Amsterdam

    Google Scholar 

  • Zhou Y et al (2012) Suggested methods for determining the dynamic strength parameters and mode-I fracture toughness of rock materials. Int J Rock Mech Min Sci 49:105–112

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mining Engineering, Isfahan University of Technology (IUT), Isfahan, 8415683111, Iran

    M. Eftekhari, A. Baghbanan & H. Hashemolhosseini

Authors
  1. M. Eftekhari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Baghbanan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Hashemolhosseini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Eftekhari.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eftekhari, M., Baghbanan, A. & Hashemolhosseini, H. Fracture propagation in a cracked semicircular bend specimen under mixed mode loading using extended finite element method. Arab J Geosci 8, 9635–9646 (2015). https://doi.org/10.1007/s12517-015-1906-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12517-015-1906-4

Keywords

Search

Navigation