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

3.3. Fracture energy-based model for plain mortar/concrete interfacewhere ˙u = [ ˙u, ˙v] t is the rate of the relative joint displacement vector, decomposedinto the elastic and plastic components, ˙u el and ˙u cr , respectively. C defines a fullyuncoupled normal/tangential elastic stiffness matrixC =()k N 0(3.13)while ṫ i = [ ˙σ N , ˙σ T ] t is the incremental stress vector defined in the interface coordinates,being σ N and σ T the normal and shear components, respectively.σ Τc tan (φ )tan (β )χ− σ dil− σ Ν-150Mohr-Coulomb CriterionPlastic Potential-250Cracking SurfaceModified flow ruleFigure 3.6: Failure hyperbola by Carol et al. [1997], Mohr-Coulomb surface, plastic potentialand the modified flow rule according to Eq. (3.19) of the interface model.The vector of the plastic displacement rate, according to a non-associated flow rule, isdefined as˙u cr = ˙λm (3.14)where ˙λ is the non-negative plastic multiplier derived by means of the classical Kuhn-Tucker loading/unloading and consistency conditions which take the following form˙λ ≥ 0, f ≤ 0, ˙λ · f = 0 Kuhn − Tuckerf ˙ = 0Consistency(3.15)where f = f [σ N ,σ T ] defines the yield condition of the model on the bases of thefollowing three-parameter formulation (outlining the hyperbola represented in Figure51