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
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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
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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
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