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

Chapter 8. Elasto-plastic microplane formulation for FRCCfunction of three material parameters: i.e., the tensile strength χ mi c , the cohesion c mi cand the internal friction angle φ mi c . tan ( mic )tan ( mic ) mic dilmic N,CAP -200Cracking Surface -250Modified flow rulePlastic PotentialCAP surfaceFigure 8.3: Maximum strength criteria and plastic flow rules.On the contrary, if σ N < σ N ,C AP an elliptical function is taken into account as followsf(˙σ mi c) = ‖σ T ‖ 2 − 1 [ (σN ) 2 ( ) ] 2R 2 − σ N ,0 − σN ,C AP − σ N ,0C AP(8.14)where R C AP , σ N ,C AP and σ N ,0 are model parameters. Particularly σ N ,C AP representsthat value of σ N situated on the boundary between the hyperbola and the ellipse.The plastic flow rule, defining the direction of inelastic strains, is represented by ageneral non-associated law for the formulation based on the hyperbolic model and anassociated one for the elliptical CAP formulation.Particularly, the rate of plastic strains can be generally written as˙ε p,mi c = ˙λm mi c (8.15)where ˙λ is the non-negative plastic multiplier derived by means of the classical Kuhn-Tucker and consistency conditions as follows˙λ ≥ 0,f(˙σ mi c) ≤ 0, ˙λ f(σ mi c) = 0,(f ˙ σ mi c) = 0, (8.16)160