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REVIEW ARTICLES

Scaling Laws and Multiscale Approach in the Mechanics of Heterogeneous and Disordered Materials

[+] Author and Article Information
Alberto Carpinteri1

Department of Structural and Geotechnical Engineering, Politecnico di Torino, 10129 Torino, Italyalberto.carpinteri@polito.it

Pietro Cornetti, Simone Puzzi

Department of Structural and Geotechnical Engineering, Politecnico di Torino, 10129 Torino, Italy

1

Corresponding author.

Appl. Mech. Rev 59(5), 283-305 (Sep 01, 2006) (23 pages) doi:10.1115/1.2204076 History:

The present paper is a review of research carried out on scaling laws and multiscaling approach in the mechanics of heterogeneous and disordered materials in the last two decades, especially at the Politecnio di Torino. The subject encompasses theoretical, numerical and experimental aspects. The research followed two main directions. The first one concerns the implementation and the development of the cohesive crack model, which has been shown to be able to simulate experiments on concrete like materials and structures. It is referred to as the dimensional analysis approach, since it succeeds in capturing the ductile-to-brittle transition by increasing the structural size owing to the different physical dimensions of two material parameters: the tensile strength and the fracture energy. The second research direction aims at capturing the size-scale effects of quasibrittle materials, which show fractal patterns in the failure process. This approach is referred to as the renormalization group (or fractal) approach and leads to a scale-invariant fractal cohesive crack model. This model is able to predict the size effects even in tests where the classical approach fails, e.g., the direct tension test. Within this framework and introducing the fractional calculus, it is shown how the Principle of Virtual Work can be rewritten in its fractional form, thus obtaining a scaling law not only for the tensile strength and the fracture energy, but also for the critical strain.

Copyright © 2006 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Dimensionless load of crack instability versus relative crack depth

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Figure 2

Dimensionless load of crack instability as a function of the dimensionless deflection

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Figure 3

Dimensionless load of crack instability versus dimensionless crack mouth opening displacement

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Figure 4

Constitutive laws of the cohesive crack model: (a) undamaged material, (b) process zone

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Figure 5

Limit-situation of complete fracture with cohesive forces

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Figure 6

Load-deflection diagrams: (a) ductile and (b) brittle condition

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Figure 7

Stress distribution across the cohesive zone (a) and equivalent nodal forces in the finite element mesh (b)

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Figure 8

Finite element nodes along the potential fracture line

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Figure 9

Cohesive crack configurations at the first (a) and (l−k+1)th (b) crack growth increment

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Figure 10

Dimensionless load versus deflection diagrams by varying the brittleness number sE,a0∕h=0.0

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Figure 11

Decrease in apparent strength by increasing the specimen size

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Figure 12

Constant distribution of cohesive stresses

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Figure 13

Fictitious crack depth at the maximum load as a function of the specimen size

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Figure 14

Dimensionless load versus deflection diagrams by varying the brittleness number sE,a0∕h=0.5

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Figure 15

Increase of fictitious fracture toughness by increasing the specimen size

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Figure 16

A concrete specimen subjected to tension (a) Fractal localization of the stress upon the resistant cross section (b) and of the energy dissipation upon crack surface (c)

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Figure 17

Fractal localization of the strain (a) and of the energy dissipation inside the damaged band (b)

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Figure 18

Fractal cohesive model

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Figure 19

Tensile tests on dog-bone shaped concrete specimens (a) by Ferro (151): stress-strain diagrams (b), cohesive law diagrams (c), fractal cohesive law diagrams (d)

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Figure 20

Tensile strength (a) and fracture energy (b) versus specimen size: experimental data by van Vliet (152) and linear interpolation in the bilogarithmic plot

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Figure 21

Tensile tests on dog-bone shaped concrete specimens (a) by van Vliet (152): stress-strain diagrams (b), cohesive law diagrams (c), fractal cohesive law diagrams (d). The dashed lines refer to extrapolated values.

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Figure 22

Details of the peak (a) and of the tail (c) of the cohesive law diagrams. Details of the peak (b) and of the tail (d) of the fractal cohesive law diagrams. The dashed lines refer to extrapolated values.

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Figure 23

Direct tensile test on a concrete specimen (a) and fractal stress distribution over the resistant ligament (b)

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Figure 24

Three point bending test concrete specimen (a) and fractal stress distribution over the resistant ligament (b)

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Figure 25

The fractal bar subjected to an axial load: the displacement field

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Figure 26

Multifractal scaling laws for tensile strength and fracture energy

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Figure 27

Homogeneously diffused strain (a) and extremely localized deformation (b) along the bar, valid, respectively, for small and large structures

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Figure 28

Multifractal scaling law for the critical strain

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