tesi A. Caggiano.pdf - EleA@UniSA - Università degli Studi di Salerno

8.4. Fracture energy-based cracking microplane model for plain concrete/mortarwhere E N and E T are the microscopic elastic stiffnesses, while the superscripts “el”and “p” represent the elastic and plastic parts for the microstrains. In analogy tomacroscopic plasticity-based models, the constitutive formulation of the microplaneproposal is introduced in an incremental form. Actually, the additive decompositioninto the elastic and plastic contributions takes place in both the normal (Eq. 8.8) andtangential strains (Eq. 8.9).The incremental rate stress vector, ˙σ mi c , and the rate of the microplane elastic strains,˙ε el ,mi c , are connected by means of the following elastic stiffness operator⎛C mi c ⎜= ⎝⎞E N 0 0⎟0 E T 0 ⎠. (8.10)0 0 E Tthen(˙σ mi c = C mi c · ˙ε el ,mi c ; ˙σ mi c = C mi c · ˙ε mi c − ˙ε p,mi c) , (8.11)As above mentioned, the model assumes a microplane strain decomposition into elasticand plastic components: i.e., ˙ε el ,mi c and ˙ε p,mi c , respectively.The microscopic elastic moduli are related to the macroscopic ones as demonstratedin Bazant and Prat [1988a,b]E N = 3K + 2G E T = 2G (8.12)being K and G the Bulk and shear macroscopic moduli, respectively.8.4.2 Post-cracking behaviorThe inelastic behavior is described at the microplane level in terms of normal/shearstress vs. strain relationship with the aim to characterize the non-linear fracture behaviorof concrete.The considered strength criteria f ( ˙σ mi c) are given on the microplane stress-space asshown in Figure 8.3. Its expressions are represented by the three-parameter hyperbolaby Carol et al. [1997] when σ N ≥ σ N ,C AP ,f(˙σ mi c) = ‖σ T ‖ 2 − [c mi c − σ N tan(φ mi c )] 2 + [c mi c − χ mi c tan(φ mi c )] 2 , (8.13)159