A comparative study on the modelling of discontinuous fracture by means of enriched nodal and element techniques and interface elements

  • D. Dias-da-Costa
  • J. Alfaiate
  • L. J. Sluys
  • E. Júlio
Original Paper


In this paper, three different approaches used to model strong discontinuities are studied: a new strong embedded discontinuity technique, designated as the discrete strong embedded discontinuity approach (DSDA), introduced in Dias-da-Costa et al. (Eng Fract Mech 76(9):1176–1201, 2009); the generalized finite element method, (GFEM), developed by Duarte and Oden (Tech Rep 95-05, 1995) and Belytschko and Black (Int J Numer Methods Eng 45(5):601–620, 1999); and the use of interface elements (Hillerborg et al. in Cem Concr Res 6(6): 773–781, 1976). First, it is shown that all three descriptions are based on the same variational formulation. However, the main differences between these models lie in the way the discontinuity is represented in the finite element mesh, which is explained in the paper. Main focus is on the differences between the element enrichment technique, used in the DSDA and the nodal enrichment technique adopted in the GFEM. In both cases, global enhanced degrees of freedom are adopted. Next, the numerical integration of the discretised equations in the three methods is addressed and some important differences are discussed. Two types of numerical tests are presented: first, simple academic examples are used to emphasize the differences found in the formulations and next, some benchmark tests are computed.


Strong embedded discontinuity Discrete crack approach Generalized finite element method 

List of Symbols


Total displacement vector at the nodes

\({\hat{\bf a}}\)

Regular displacement vector at the nodes

\({\hat{\bf a}_2}\)

Regular displacement vector for the enriched layer at the nodes

\({\tilde{\bf a}}\)

Enhanced displacement vector at the nodes

\({\tilde{\bf a}_{rb}}\)

Rigid body motion part of the enhanced displacement vector at the nodes

\({\bar{\bf b}}\)

Body forces vector


Strain-nodal displacement matrix


Enhanced strain-nodal displacement matrix


Absolute value of the jump




Scalar damage


Constitutive matrix


Initial elastic shear stiffness


Shear stiffness for an advanced state of damage


Young’s modulus


Loading function

\({\hat{\bf f}}\)

Regular external vector force at the regular nodes

\({\tilde{\bf f}}\)

Enhanced external vector force at the regular nodes


Tensile strength


External vector force at the additional nodes


Fracture energy


Parameter defined by: –ln (D sκ/D s0)


Heaviside function

\({{\bf H}_{\Gamma_{d}}}\)

Diagonal matrix containing the Heaviside function evaluated at each degree of freedom


Identity matrix

kn, ks

Normal and shear penalty parameters respectively


Scalar variable depending on the normal and shear jump components


Parameter denoting the beginning of the softening

\({{\bf K}_{aa}, {\bf K}_{\hat{a}\hat{a}}}\)

Bulk stiffness matrices for the DSDA and GFEM

Kaw, Kwa, Kww

Enhanced bulk stiffness matrices for the DSDA

\({{\bf K}_{\hat{a}\tilde{a}}, {\bf K}_{\tilde{a}\hat{a}}, {\bf K}_{\tilde{a}\tilde{a}}}\)

Enhanced bulk stiffness matrices for the GFEM


Discontinuity stiffness matrix


Measure of distance around the tip


Hillerborg’s characteristic length


Discontinuity length


Differential operator matrix


Matrix used to compute the difference between top and bottom displacements for a discrete-interface


Jump direction vector


Rigid body motion matrix

\({{\bf M}_{w}^{k}}\)

Matrix composed by evaluating the rigid body motion matrix at each finite element node


Number of the finite element nodes


Unit vector normal to the boundary


Unit vector normal to the discontinuity surface


Number of additional nodes located at the discontinuity for jump interpolation


Shape function matrix for the jumps


External load


Horizontal external load


Distance between the integration point and the discontinuity tip

s, n

Unit vectors tangent and orthogonal to the discontinuity, respectively


Traction vector

\({\bar{{\bf t}}}\)

Natural forces vector


Discontinuity constitutive matrix


Elastic discontinuity constitutive matrix


Total displacement vector

\({\bar{\bf u}}\)

Essential boundary conditions vector

\({\hat{\bf u}}\)

Regular displacement field vector

\({\hat{\bf u}_2}\)

Regular displacement field vector for the enriched layer

\({\tilde{{\bf u}}}\)

Enhanced displacement field vector


Jump vector


Vertical displacement


Nodal jump vector


Weight for the integration point i


Global coordinates of a material point

x1, x2

Global frame


Discontinuity angle


Shear contribution parameter




Discontinuity surface


Boundary with natural forces


Boundary with essential conditions


Total strain tensor

\({\hat{\bf \varepsilon}}\)

Regular strain tensor


Dead –weight


Stress tensors


First principle stress


Poisson ratio


Elastic domain


Incremental variation of (·)


Symmetric part of (·)


Admissible or virtual variation of (·)


Dirac’s delta–function along the surface Γ d


(·) belonging to the finite element e

(·)+, (·)

(·) at the positive and negative side of the discontinuity, respectively

(·)n, (·)s

Normal and shear component of (·)

Dyadic product

\({\left < \cdot \right > ^{+}}\)

McAuley brackets


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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • D. Dias-da-Costa
    • 1
  • J. Alfaiate
    • 2
  • L. J. Sluys
    • 3
  • E. Júlio
    • 1
  1. 1.ISISE, Civil Engineering DepartmentUniversity of CoimbraCoimbraPortugal
  2. 2.ICIST, Civil Engineering DepartmentInstituto Superior TécnicoLisboaPortugal
  3. 3.Civil Engineering Department, Engineering and GeosciencesDelft University of TechnologyDelftThe Netherlands

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