Computational Mechanics

, Volume 62, Issue 4, pp 783–801 | Cite as

Numerical analysis of a main crack interactions with micro-defects/inhomogeneities using two-scale generalized/extended finite element method

  • Mohammad MalekanEmail author
  • Felício B. Barros
Original Paper


Generalized or extended finite element method (G/XFEM) models the crack by enriching functions of partition of unity type with discontinuous functions that represent well the physical behavior of the problem. However, this enrichment functions are not available for all problem types. Thus, one can use numerically-built (global-local) enrichment functions to have a better approximate procedure. This paper investigates the effects of micro-defects/inhomogeneities on a main crack behavior by modeling the micro-defects/inhomogeneities in the local problem using a two-scale G/XFEM. The global-local enrichment functions are influenced by the micro-defects/inhomogeneities from the local problem and thus change the approximate solution of the global problem with the main crack. This approach is presented in detail by solving three different linear elastic fracture mechanics problems for different cases: two plane stress and a Reissner–Mindlin plate problems. The numerical results obtained with the two-scale G/XFEM are compared with the reference solutions from the analytical, numerical solution using standard G/XFEM method and ABAQUS as well, and from the literature.


Generalized/extended FEM Fracture mechanics Two-scale analysis Micro-defects Stress intensity factor 


\( \varvec{{\bar{t}}} \)

External traction vector

\( \varvec{{\bar{u}}} \)

Prescribed displacement

\( \varvec{{\hat{n}}} \)

Unit outward normal

\( \varvec{b} \)

Body force vector

\( \varvec{m}_j, n_j \)

Outward unit normal vectors in contour integral

\( \varvec{u} \)

Displacement field vector

\( \varvec{v} \)

Test function

\( \delta \)

Kronecker delta

\( \epsilon _b, \epsilon _s \)

Bending and shear strains

\( \epsilon _{ij} \)

Strain tensor

\( \Gamma \)

Inner J-integral boundary

\( \kappa \)

Material constant

\( \mu \)

Shear modulus

\( \mu _d \)

Inhomogeneities shear modulus

\( \mu _m \)

Main problem shear modulus

\( \nu _d \)

Inhomogeneities Poisson’s ratio

\( \Omega \)

Problem domain

\( \omega _j\)

Cloud of node j

\( \partial \Omega \)

Problem boundary

\( \partial \Omega _c \)

Crack surface

\( \partial \Omega _t \)

Surface traction boundary

\( \partial \Omega _u \)

Displacement boundary

\( \phi _i \)

Angle of micro-defect i and main crack-tip connection line with respect to the x-axis

\( \psi \)

Section rotation of the plate middle plane

\( \sigma \)

Cauchy stress tensor

\( \sigma _{ij} \)

Stress tensor

\( \theta _i \)

Micro-defect i direction angle

\( \varepsilon \)

Linear strain tensor

\( a_{mc} \)

Main crack length

\( C_+, C_- \)

Upper and lower crack surfaces

\( C_0 \)

Outer J-integral boundary

\( d_i \)

Distance of the micro-defect i to the main crack-tip


Determinant of the Jacobian

\( h_{elem} \)

Square root of the crack-tip element area


Interaction integral



\( l_{md} \)

Characteristic length of the micro-defect


Bending moment

\( N_{gp} \)

Number of Gauss points


Shear load


Weighting function

\( q_j \)

A set of linearly independent functions defined at each nodal cloud

\( r_m \)

Interaction integral scalar multiplier


Strain energy density


Transverse displacement of the plate

\( w_{gp} \)

Weight of each Gauss point


Nodal parameters associated with G/XFEM


Hook’s tensor

\({\mathbb {R}}^{2}\)

Bi-dimensional domain

\({\mathcal {N}}_j\)

FE Shape function

\(\nu \)

Poisson ratio

\(\Omega _{G}\)

Global domain

\(\Omega _{L}\)

Local domain

\(\phi _{ji}\)

G/XFEM shape function

\(\theta \)

Angle between x coordinate and direction of the crack front


G/XFEM displacement approximation


Young’s modulus

\(K_I, K_{II}, K_{III} \)

Mode-I, II and III stress intensity factors


Local approximation function



The first author (CNPq Scholarship, Brazil—Grant No. 151003/2017-3) and the other author gratefully acknowledge the important support of the Brazilian research agencies CNPq (National Council for Scientific and Technological Developments—Grant No. 308932/2016-1), CAPES (Coordination for the Improvement of Higher Education Personnel), and FAPEMIG (Foundation for Research Support of the State of Minas Gerais—Grant No. APQ-02460-16).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Graduate Program in Structural Engineering (PROPEEs), School of EngineeringFederal University of Minas Gerais (UFMG)Belo HorizonteBrazil

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