Numerical Modeling of Concrete Cracking

The book presents the underlying theories of the different approaches for modeling cracking of concrete and provides a critical survey of the state-of-the-art in computational concrete mechanics. It covers a broad spectrum of topics related to modeling of

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INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 532

NUMERICAL MODELING OF CONCRETE CRACKING

EDITED BY GÜNTER HOFSTETTER UNIVERSITY OF INNSBRUCK, AUSTRIA GÜNTHER MESCHKE RUHR UNIVERSITY BOCHUM, GERMANY

This volume contains 185 illustrations

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned VSHFLÀFDOO\WKRVHRIWUDQVODWLRQUHSULQWLQJUHXVHRILOOXVWUDWLRQV broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2011 by CISM, Udine SPIN 80073533

All contributions have been typeset by the authors.

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called strong ellipticity condition, precludes any type of discontinuous bifurcation. Remark: IRU D UDWHLQGHSHQGHQW PDWHULDO +VWDELOLW\ ε˙ : DT : ε˙ > 0 ∀ ε˙ implies strong ellipticity.

106

A. Huespe and J. Oliver

1.3

Example of a Material Model Subjected to Stability Loss and Bifurcation: Isotropic continuum damage model for concrete

7DEOH  GHVFULEHV D VSHFLÀF LVRWURSLF FRQWLQXXP GDPDJH PRGHl, with a scalar internal variable D, the damage varaible, describing the elastic stiffness degradation due to micro cracking: D = 0, for the undamaged material, and D = 1 for the fully damaged material (for more generic continuum damage models, see for H[DPSOH /HPDLWUH DQG 'HVPRUDW   ,Q 7DEOH  DQG IROORZLQJ 2OLYHU   WKH GDPDJH YDULDEOH D depends on an internal strain-like variable r and its stress-like conjugate internal variable, q which depends on r. The free energy, denoted W , depends on the strain tensor ε and the damage variable D. The elastic strain energy for the undamaged material is denoted W0 , and E LV WKH +RRNH·V HODVWLF WHQVRU μ and λ DUH WKH /DP`H·V SDUDPHWHUV and I and 1 are the fourth and second order identity tensors respectively. ¯ LV WKH HIIHFWLYH VWUHVV ,WV SRVLWLYH FRXQWHUSDUW LV WKHQ GH,Q HTXDWLRQ   σ ÀQHG DV ¯ + = ¯ σi  pi ⊗ p i  σ where ¯ σi  VWDQGV IRU WKH SRVLWLYH SDUW 0F$XOH\ EUDFNHWV RI WKH LWK Srincipal σi  = σ ¯i for σ ¯i > 0 and ¯ σi  = 0 for σ ¯i ≤ 0 DQG pi stands effective stress σ ¯i ( ¯ for the i-th principal stress direction. (TXDWLRQ  GHÀQHV WKH GDPDJH IXQFWLRQ f and the initial elastic domain ¯ + = 0  as: f < 0. This domain is unbounded for compressive stress states (σ Therefore, damage evolution is only possible with tensile stress states, as it is usually observed in concrete