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.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
Abbreviations
- 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
- B w :
-
Enhanced strain-nodal displacement matrix
- c :
-
Absolute value of the jump
- c 0 :
-
Cohesion
- d :
-
Scalar damage
- D :
-
Constitutive matrix
- D s0 :
-
Initial elastic shear stiffness
- D sκ :
-
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
- f t :
-
Tensile strength
- f w :
-
External vector force at the additional nodes
- G F :
-
Fracture energy
- h s :
-
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
- k n , k s :
-
Normal and shear penalty parameters respectively
- κ :
-
Scalar variable depending on the normal and shear jump components
- k 0 :
-
Parameter denoting the beginning of the softening
- \({{\bf K}_{aa}, {\bf K}_{\hat{a}\hat{a}}}\) :
-
Bulk stiffness matrices for the DSDA and GFEM
- K aw , K wa , K ww :
-
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
- K d :
-
Discontinuity stiffness matrix
- l :
-
Measure of distance around the tip
- l ch :
-
Hillerborg’s characteristic length
- l d :
-
Discontinuity length
- L :
-
Differential operator matrix
- L w :
-
Matrix used to compute the difference between top and bottom displacements for a discrete-interface
- m :
-
Jump direction vector
- M w :
-
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
- n w :
-
Number of additional nodes located at the discontinuity for jump interpolation
- N w :
-
Shape function matrix for the jumps
- P:
-
External load
- P h :
-
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
- T el :
-
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
- u v :
-
Vertical displacement
- w :
-
Nodal jump vector
- w i :
-
Weight for the integration point i
- x :
-
Global coordinates of a material point
- x 1, x 2 :
-
Global frame
- α :
-
Discontinuity angle
- β :
-
Shear contribution parameter
- Γ:
-
Boundary
- Γ d :
-
Discontinuity surface
- Γ t :
-
Boundary with natural forces
- Γ u :
-
Boundary with essential conditions
- ε :
-
Total strain tensor
- \({\hat{\bf \varepsilon}}\) :
-
Regular strain tensor
- ρ :
-
Dead –weight
- σ :
-
Stress tensors
- σ I :
-
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
References
Alfaiate J, Sluys L (2005) Discontinuous numerical modelling of fracture using embedded discontinuities. In: Owen DRJ, Nate EO, Suárez B (eds) Computational plasticity, fundamentals and applications. COMPLAS VIII, Barcelona, Spain
Alfaiate J, Sluys LJ (2004) On the use of embedded discontinuities in the framework of a discrete crack approach. In: Yao Z, Yuan WZM (eds) Sixth international congress on computational mechanics in conjunction with the second asian-pacific congress of computational mechanics. WCCMVI, Beijing, China
Alfaiate J, Pires EB, Martins JAC (1992) A finite element model for the study of crack propagation. In: Aliabadi MH, Cartwright DJ, Nisitani H (eds) 2nd international conference on localised damage, Computational Mechanics Publications and Elsevier Applied Science. Southampton, United Kingdom, pp 261–282
Alfaiate J, Pires EB, Martins JAC (1997) A finite element analysis of non-prescribed crack propagation in concrete. Comput Struct 63(1): 17–26
Alfaiate J, Wells GN, Sluys LJ (2002) On the use of embedded discontinuity elements with crack path continuity for mode-I and mixed-mode fracture. Eng Fract Mech 69(6): 661– 686
Alfaiate J, Simone A, Sluys LJ (2003) Non-homogeneous displacement jumps in strong embedded discontinuities. Int J Solids Struct 40(21): 5799–5817
Alfaiate J, Sluys LJ, Pires EB (2005) Mixed mode fracture in concrete and masonry. In: Ferro AC, Mai Y, Ritchie RG (eds) ICFXI, 11th international conference on fracture, Torino, Italy
Alfaiate J, Moonen P, Sluys LJ (2007a) On the use of DSDA and X-FEM for the modelling of discontinuities in porous media. In: Oñate E, Owen DRJ (eds) IX International Conference on Computational Plasticity.. COMPLAS IX, Barcelona, Spain
Alfaiate J, Moonen P, Sluys LJ, Carmeliet J (2007b) On the modelling of preferential moisture uptake in porous media. In: EPMESC XI, Kyoto, Japan
Areias P, Belytschko T (2005) Analysis of three-dimensional crack initiation and propagation using the extended finite element method. Int J Numer Methods Eng 63: 760–788
Armero F, Garikipati K (1996) An analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids. Int J Solids Struct 33(20–22): 2863–2885
Barenblatt G (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7: 55–129
Barpi F, Valente S (2000) Numerical simulation of prenotched gravity dam models. J Eng Mech 126(6): 611–619
Bazant ZP, Oh BH (1983) Crack band theory of concrete. Mater Struct 16: 155–177
Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45(5): 601–620
Bocca P, Carpinteri A, Valente S (1986) Mixed mode fracture of concrete. Int J Solids Struct 27: 1139–1153
Bolzon G (2001) Formulation of a triangular finite element with an embedded interface via isoparametric mapping. Comput Mech 27(6): 463–473
Borja RI (2008) Assumed enhanced strain and the extended finite element methods: a unification of concepts. Comput Methods Appl Mech Eng 197(33–40): 2789–2803
Carey GF, Ma M (1999) Joint elements, stress post-processing and superconvergent extraction with application to Mohr–Coulomb failure. Commun Numer Methods Eng 15(5): 335–347
Carol I, Prat P (1995) A multicrack model based on the theory of multisurface plasticity and two fracture energies. In: Owen DRJ, Oñate E, Hinton E (eds) Computational Plasticity, fundamentals and applications. Pineridge Press, Barcelona, Spain, pp 1583–1594
Cervera M, Chiumenti M (2006) Smeared crack approach: back to the original track. Int J Numer Anal Methods Geomech 30(12): 1173–1199
Dias-da-Costa D, Alfaiate J, Sluys LJ, Júlio E (2009) A discrete strong discontinuity approach. Eng Fract Mech 76(9): 1176–1201
Daux C, Moës N, Dolbow J, Sukumar N, Belytschko T (2000) Arbitrary branched and intersecting cracks with the extended finite element method. Int J Numer Methods Eng 48(12): 1741–1760
Duarte CAM, Oden JT (1995) H-p clouds-an h-p meshless method. Tech Rep 95–05
Duarte CAM, Babuška I, Oden JT (2000) Generalized finite element methods for three-dimensional structural mechanics problems. Comput Struct 77(2): 215–232
Dugdale DS (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8(2): 100–104
Dvorkin EN, Cuitiño AM, Gioia G (1990) Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions. Int J Numer Methods Eng 30(3): 541–564
Gasser TC, Holzapfel GA (2006) 3d crack propagation in unreinforced concrete: a two-step algorithm for tracking 3d crack paths. Comput Methods Appl Mech Eng 195(37–40): 5198–5219
Goodman RE, Taylor RL, Brekke TL (1968) A model for the mechanics of jointed rock. J Soil Mech Found Div 99: 637–659
Herrmann LR (1978) Finite element analysis of contact problems. ASCE J Eng Mech Div 104(5): 1043–1057
Hillerborg A, Modeer M, Petersson PE (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 6(6): 773–781
Ingraffea A (1989) Shear cracks. In: Elfgren L (eds) Fracture mechanics of concrete structures—from theory to applications, report of the Technical Committee 90-FMA Fracture Mechanics of Concrete-Applications. Chapman and Hall, London, United Kingdom, pp 231–233
Ingraffea A, Saouma V (1985) Numerical modelling of discrete crack propagation in reinforced and plain concrete. In: Sih GC, DiTommaso A(eds) Engineering Application of Fracture Mechanics. Martinus Nijhoff Publishers
Jirásek M, Belytschko T (2002) Computational resolution of strong discontinuities. In: Mang HA, Rammerstorfer JEFG (ed) WCCM V, Fifth world congress on computational mechanics, Vienna, Austria
Kikuchi N, Oden JT (1988) Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM Studies in Applied Mathematics 8, Philadelphia
Klisinski M, Runesson K, Sture S (1991) Finite element with inner softening band. J Eng Mech 117(3): 575–587
Larsson R, Runesson K (1996) Element-embedded localization band based on regularized displacement discontinuity. J Eng Mech 122(5): 402–411
Linder C, Armero F (2007) Finite elements with embedded strong discontinuities for the modeling of failure in solids. Int J Numer Methods Eng 72(12): 1391–1433
Lofti HR, Shing PB (1995) Embedded representation of fracture in concrete with mixed finite elements. Int J Numer Methods Eng 38(8): 1307–1325
Lourenço P, Rots JG (1997) A multi-surface interface model for the analysis of masonry structures. ASCE J Eng Mech 123(7): 660–668
Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice-Hall International, Englewood Cliffs, New Jersey
Melenk JM, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139(1–4): 289–314
Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1): 131–150
Moës N, Sukumar N, Moran B, Belytschko T (2000) An extended finite element method (x-fem) for two- and three-dimensional crack modeling. In: ECCOMAS 2000, Barcelona
Nooru-Mohamed MB (1992) Mixed-mode fracture of concrete: an experimental approach. PhD thesis, Delft University of Technology
Ohlsson U, Olofsson T (1997) Mixed-mode fracture and anchor bolts in concrete analysis with inner softening bands. J Eng Mech 123(10): 1027–1033
Oliver J (1996) Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. part 1: fundamentals. Int J Numer Methods Eng 39(21): 3575–3600
Oliver J, Huespe AE, Pulido MDG, Chaves E (2002) From continuum mechanics to fracture mechanics: the strong discontinuity approach. Eng Fract Mech 69: 113–136
Oliver J, Huespe AE, Sanchez PJ (2006) A comparative study on finite elements for capturing strong discontinuities: E-FEM vs X-FEM. Comput Methods Appl Mech Eng 195(37–40): 4732–4752
Pivonka P, Ozbolt J, Lackner R, Mang HA (2004) Comparative studies of 3d-constitutive models for concrete: application to mixed-mode fracture. Int J Numer Methods Eng 60(2): 549–570
Remmers JJC (2006) Discontinuities in materials and structures: a unifying computational approach. PhD thesis, Delft University of Technology
Rots JG (1988) Computational modeling of concrete fracture. PhD thesis, Delft University of Technology
Sancho J, Planas J, Gálvez J, Cendón D, Fathy A (2005) Three-dimensional simulation of concrete fracture using embedded crack elements without enforcing crack path continuity. In: Owen DRJ, nate EO, Suárez B (eds) Computational plasticity, fundamentals and applications. COMPLAS VIII, Barcelona, Spain
Schlangen E (1993) Experimental and numerical analysis of fracture processes in concrete. PhD thesis, Delft University of Technology
Simo JC, Rifai MS (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29(8): 1595–1638
Simo JC, Oliver J, Armero F (1993) An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Comput Mech 12: 277–296
Simone A (2004) Partition of unity-based discontinuous elements for interface phenomena: computational issues. Commun Numer Methods Eng 20(6): 465–478
Simone A, Wells GN, Sluys LJ (2003) From continuous to discontinuous failure in a gradient-enhanced continuum damage model. Comput Methods Appl Mech Eng 192(41–42): 4581–4607
Simone A, Duarte CAM, Van der Giessen E (2006) A generalized finite element method for polycrystals with discontinuous grain boundaries. Int J Numer Methods Eng 67(8): 1122–1145
Tijssens MGA, Sluys LJ, Van der Giessen E (2000) Numerical simulation of quasi-brittle fracture using damaging cohesive surfaces. Eur J Mech, A/Solids 19(5): 761–779
Ventura G (2006) On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method. Int J Numer Methods Eng 66(5): 761–795
Wells GN, Sluys LJ (2000) Application of embedded discontinuities for softening solids. Eng Fract Mech 65(2–3): 263–281
Wells GN, Sluys LJ (2001a) A new method for modelling cohesive cracks using finite elements. Int J Numer Methods Eng 50(12): 2667–2682
Wells GN, Sluys LJ (2001b) Three-dimensional embedded discontinuity model for brittle fracture. Int J Solids Struct 38(5): 897–913
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dias-da-Costa, D., Alfaiate, J., Sluys, L.J. et al. A comparative study on the modelling of discontinuous fracture by means of enriched nodal and element techniques and interface elements. Int J Fract 161, 97–119 (2010). https://doi.org/10.1007/s10704-009-9432-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10704-009-9432-6