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

Chapter 4. Bond behavior of fibers in cementitious materials: a unifiedformulation∂τ y∂κ= − τ y,0G f(4.35)∂κ∂s p = τ (4.36)being τ y,0 the shear strength while G f the fracture energy under mode I I of crack.4.4.2 Algorithmic tangential operatorThe non-linear behavior, within a finite increment step, is solved by adopting theclassical Newton-Raphson solution to solve the non-linear FE equations. The model isformulated by means of the construction of the algorithmic tangent operator to ensurea higher convergence rate than the continuous consistent one [Simo and Hughes, 1998].Considering the differentiated form of the incremental shear-slip law, a linearizedtangential format of Eq. (4.4) can be obtained(∆τ = k E · ∆s − ∆λ ∂f )⇒∂τ(d∆τ = k E · d∆s − d∆λ ∂f )∂f− ∆λd∂τ ∂τ(4.37)in whichd ∂f∂τ = ∂2 f · d∆τ = M · d∆τ (4.38)∂τ2 Substituting the Eq. (4.38) into (4.37) and solving for d∆τ(d∆τ = k E ,2 · d∆s − d∆λ ∂f )∂τ(4.39)being d∆λ the linearized plastic multiplier.Based on the first-order differential form of the consistency condition, the linearizedtangential format of the plastic multiplier d∆λ can be derived as82d f = ∂f ∂f ∂κ· d∆τ +∂τ ∂κ ∂s p · d∆sp = 0 (4.40)