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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

Abstract

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.

Keywords

Strong embedded discontinuity Discrete crack approach Generalized finite element method 

List of Symbols

a

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

B

Strain-nodal displacement matrix

Bw

Enhanced strain-nodal displacement matrix

c

Absolute value of the jump

c0

Cohesion

d

Scalar damage

D

Constitutive matrix

Ds0

Initial elastic shear stiffness

Dsκ

Shear stiffness for an advanced state of damage

E

Young’s modulus

f

Loading function

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

Regular external vector force at the regular nodes

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

Enhanced external vector force at the regular nodes

ft

Tensile strength

fw

External vector force at the additional nodes

GF

Fracture energy

hs

Parameter defined by: –ln (D sκ/D s0)

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

Heaviside function

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

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

I

Identity matrix

kn, ks

Normal and shear penalty parameters respectively

κ

Scalar variable depending on the normal and shear jump components

k0

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

Kd

Discontinuity stiffness matrix

l

Measure of distance around the tip

lch

Hillerborg’s characteristic length

ld

Discontinuity length

L

Differential operator matrix

Lw

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

m

Jump direction vector

Mw

Rigid body motion matrix

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

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

n

Number of the finite element nodes

n

Unit vector normal to the boundary

n+

Unit vector normal to the discontinuity surface

nw

Number of additional nodes located at the discontinuity for jump interpolation

Nw

Shape function matrix for the jumps

P

External load

Ph

Horizontal external load

r

Distance between the integration point and the discontinuity tip

s, n

Unit vectors tangent and orthogonal to the discontinuity, respectively

t

Traction vector

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

Natural forces vector

T

Discontinuity constitutive matrix

Tel

Elastic discontinuity constitutive matrix

u

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

〚u〛

Jump vector

uv

Vertical displacement

w

Nodal jump vector

wi

Weight for the integration point i

x

Global coordinates of a material point

x1, x2

Global frame

α

Discontinuity angle

β

Shear contribution parameter

Γ

Boundary

Γd

Discontinuity surface

Γt

Boundary with natural forces

Γu

Boundary with essential conditions

ε

Total strain tensor

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

Regular strain tensor

ρ

Dead –weight

σ

Stress tensors

σI

First principle stress

ν

Poisson ratio

Ω

Elastic domain

d(·)

Incremental variation of (·)

(·)s

Symmetric part of (·)

δ(·)

Admissible or virtual variation of (·)

\({\delta_{\Gamma_{d}}}\)

Dirac’s delta–function along the surface Γ d

(·)e

(·) 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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