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

Chapter 7. Cracked hinge numerical model for fiber-reinforced concretel f /2 l f /2l f /23 232 12hl f /23 23bFigure 7.3: The three orientation zones for the square cross section beam: b × h × l (base ×height × length) having l ≥ b and l ≥ h.The stress-crack opening relationship, σ[u cr ], of plain concrete matrix is based on theinterface law proposed by Carol et al. [1997] and considering the case of tension only.Particularly, the interface loading criterion, the flow rule and the softening (evolution)law are defined asf (σ,κ) = σ 2 − σ 2 y ≤ 0˙u cr =˙λ∂f∂σ = 2 · ˙λ ·)σσ y = f t(1 − w crG I floading criterionplastic flowevolution law(7.6)where σ y is the current tensile strength and κ the internal state variable. The incrementalcracking opening, ˙u cr , is defined by means of the classical flow rule, being ˙λ the rateof the non-negative plastic multiplier. The variation of σ y is assumed to be linear, fromits maximum value f t (tensile strength) to zero, based on the ratio between the workspent and the available fracture energy, w cr. The rate of fracture work, ẇG I cr , is definedfas followsẇ cr = σ · ˙u cr . (7.7)Based on these hypothesis, the employed model exhibits the following closed-formsolution under the well-known Kuhn-Tucker and consistency conditions [Carol et al.,144