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

Chapter 8. Elasto-plastic microplane formulation for FRCC8.2.2 Homogenization between macro- and microplane stress spaceIn Eqs. (8.3) and (8.4), σ N and σ T,k are the microplane components of stress. Theequilibrium between micro- and macroscopic stress tensor can be achieved by meansof the application of the virtual work principle applied in the spherical microplaneregion∫2Ω(σN δε N + σ Tr δε Tr)dΩ =4π3 σ i j δε i j (8.3)nε Nmicnσ Nmicε T ,2tσ T ,2tε T ,1 Tσ T ,1 TFigure 8.2: Strain and stress components at the microplane level.where σ i j denotes the components of the macroscopic stress tensor while Ω the boundarysurface of one hemisphere. By combining Eqs. (8.2) and (8.3), the following relationfor the macroscopic stress tensor can be derivedσ i j = 3 ∫2πΩ(σ N n i n j + σ T,k [ ] )ni δ k j + n j δ ki dΩ. (8.4)2Figure 8.2 shows the microplane strain ε mi c and stress vectors σ mi c , with the correspondingprojections on the microplane direction, n, and its orthogonal.8.3 Composite constitutive formulation for FRCCA smeared crack microplane model for FRCC, based on the well-known “Mixture Theory”[Trusdell and Toupin, 1960], is formulated by means of the composite combinationof three internal constitutive laws whose main features are detailed in sections 8.4 - 8.5and briefly outlined below:156