INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS
Int. J. Numer. Anal. Meth. Geomech., 2006; 30:1173–1199
Published online 27 April 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.518
Smeared crack approach: back to the original track
and M. Chiumenti
International Center for Numerical Methods in Engineering (CIMNE), Technical University of Catalonia (UPC),
Ediﬁcio C1, Campus Norte, Jordi Girona 1-3, 08034 Barcelona, Spain
This paper briefly reviews the formulations used over the last 40 years for the solution of problems
involving tensile cracking, with both the discrete and the smeared crack approaches. The paper focuses on
the smeared approach, identifying as its main drawbacks the observed mesh-size and mesh-bias spurious
dependence when the method is applied ‘straightly’. A simple isotropic local damage constitutive model is
considered, and the (exponential) softening modulus is regularized according to the material fracture
energy and the element size. The continuum and discrete mechanical problems corresponding to both the
weak discontinuity (smeared cracks) and the strong discontinuity (discrete cracks) approaches are analysed
and the question of propagation of the strain localization band (crack) is identiﬁed as the main difﬁculty to
be overcome in the numerical procedure. A tracking technique is used to ensure stability of the solution,
attaining the necessary convergence properties of the corresponding discrete ﬁnite element formulation.
Numerical examples show that the formulation derived is stable and remarkably robust. As a consequence,
the results obtained do not suffer from spurious mesh-size or mesh-bias dependence, comparing
very favourably with those obtained with other fracture and continuum mechanics approaches. Copyright
# 2006 John Wiley & Sons, Ltd.
: tensile cracking; strain softening; strain localization; damage; tracking algorithms; mesh
Cracking is an essential feature of the behaviour of concrete structures and, therefore, tensile
cracking must be taken into account in predicting their ultimate load capacity as well as service
The tensile fracture of concrete is regarded as (quasi)brittle. Concrete has no yield behaviour
as exhibited by metals. Its tensile stress–strain diagram is nearly linear up to the peak stress,
whereupon it immediately starts to descend. In spite of this, concrete shows considerable
toughness. This toughness is related to the existence of a descending branch in the nominal
stress–strain curve. This is known as strain softening.
Received 20 December 2005
Revised 6 February 2006
Accepted 21 February 2006Copyright # 2006 John Wiley & Sons, Ltd.
Correspondence to: M. Cervera, International Center for Numerical Methods in Engineering (CIMNE), Technical
University of Catalonia (UPC), Ediﬁcio C1, Campus Norte, Jordi Girona 1-3, 08034 Barcelona, Spain.