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Research Papers

# Micromechanical Interpretation of the Dissipation Associated With Mode I Propagation of Microcracks in Brittle Materials

[+] Author and Article Information
Bernhard Pichler1

Laboratory for Materials and Structures (LMSGC), Ecole Nationale des Ponts et Chaussées (ENPC), 6 et 8, Avenue Blaise Pascal, F-77455 Marne-la-Vallée, Francebernhard.pichler@lmsgc.enpc.fr

Luc Dormieux

Laboratory for Materials and Structures (LMSGC), Ecole Nationale des Ponts et Chaussées (ENPC), 6 et 8, Avenue Blaise Pascal, F-77455 Marne-la-Vallée, Franceluc.dormieux@lmsgc.enpc.fr

Microcracking is used as a synonym for progressive microcrack propagation.

To show that the energy released is equal to the area under the softening curve, one can also use the $J$-integral concept, which goes back to the pioneering work of Rice (28). Employing this concept in the present context would involve integrals leading around the cohesive zone. Hence, the $J$-integral approach would not allow for direct identification of the source of dissipation. The aim of the submitted manuscript, however, is to explain how dissipation occurs. This provided the motivation to calculate the dissipation at the very domain where it occurs, namely, inside the cohesive zone.

1

On leave from Institute for Mechanics of Materials and Structures, Vienna University of Technology (TU Wien), Karlsplatz 13/202, A-1040 Vienna, Austria.

J. Appl. Mech 76(4), 041003 (Apr 22, 2009) (12 pages) doi:10.1115/1.3086594 History: Received February 22, 2007; Revised September 15, 2008; Published April 22, 2009

## Abstract

This paper deals with the dissipation associated with quasistatic microcracking of brittle materials exhibiting softening behavior. For this purpose an elastodamaging cohesive zone model is used, in which cohesive tractions decrease (during crack propagation) with increasing displacement discontinuities. Constant cohesive tractions are included in the model as a limiting special case. Considering a representative volume element containing a dilute distribution of many parallel microcracks, we quantify energy dissipation associated with mode I microcrack propagation. This is done in the framework of thermodynamics, without restricting assumptions on the size of the cohesive zones. Model predictions are compared with exact solutions, which are accessible for constant cohesive tractions. The proposed model reliably predicts both onset of crack propagation and the dissipation during microcracking. It is shown that the energy release rate is virtually equal to the area under the softening curve, if the microscopic tensile strength is at least twice as large as the macroscopic tensile strength. This result justifies approaches relying on the concept of constant energy release rate, such as those frequently used in the engineering practice.

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Copyright © 2009 by American Society of Mechanical Engineers
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## Figures

Figure 7

Comparison of model prediction (Eq. 51) with exact solution 66: dimensionless energy release rate during crack propagation, see Table 1; to render the curves distinguishable, the ordinate shows the interval [0.75,1.10] only

Figure 6

Comparison of model prediction (Eq. 50) with exact solution 56: relationship between remote stresses and crack opening at the inner edge of the cohesive zone (consider: m=⟦ξn⟧r=a/⟦ξncr⟧), see Table 1

Figure 5

Comparison of model prediction (Eq. 46) with exact solution 55; size of the cohesive zone during its evolution and during crack propagation, see Table 1

Figure 4

Microcrack comprising an annulus-shaped cohesive zone

Figure 3

Two-dimensional illustration of a three-dimensional representative volume element comprising an elastic matrix and parallel cohesive microcracks of identical size and orientation

Figure 2

Evaluation of elastodamaging cohesive zone law 4, for different values of the power law exponent β, and for maxt⟦ξn⟧=⟦ξncr⟧/3

Figure 1

Elastodamaging cohesive zone law (defined in Eqs. 1,2), relating cohesive normal tractions Tn to the normal displacement discontinuities ⟦ξn⟧ (also referred to as “softening curve”); plotted for different values of the power law exponent β

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