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

4.3. Elasto-plastic joint/interface model with isotropic linear softeningTable 4.1 describes the key aspects of both models. In particular, f (τ,κ) and g (τ,κ)represent the two yielding criteria based on the interface shear stress τ and the internal(strain-like) variable κ of each considered model; τ y represents the failure shearstrength.Table 4.1: Interface bond-slip models.Fracture energy-based modelStrain-softening elasto-plasticityLoading criterion f (τ,κ) = τ 2 − τ 2 y ≤ 0 g (τ,κ) = |τ| − (τ y,0 +Q) ≤ 0)Stress-like internal variables τ y = τ y,0(1 − w slG˙κ = ẇ f sl = τ · ṡ p˙κ = ˙Q = ˙λ · k HPlastic flowṡ p ∂f= ˙λ∂τ = 2 · ˙λ · τConstitutive equation ˙τ = k E (ṡ − ṡ p )ṡ p ∂g= ˙λ∂τ = ˙λ · si g n[τ]Loading-unloading condition ˙λ ≥ 0, f ≤ 0, ˙λ · f = 0 ˙λ ≥ 0, g ≤ 0, ˙λ · g = 0Constitutive tangent operator⎛k ept an = k E ,2 · ⎝1 −( ) ∂f 2+∆λ ∂f∂τ( ∂f∂τ˙τ = k ept an · ṡ( ∂f ∂κ) ⎞∂τ ∂κ ∂sp · ∂2 f) ∂τ2⎠2+H/kE,2k ept an = k Eel asti c/unloadi ngk ep (t an = k E 1 − k )Ek E +k HBoth bond-slip models can be directly implemented in plasticity-type constitutive lawsfor interface elements. The rate of elastic relative slip, ṡ e , is introduced and related tothe shear stress through the elastic stiffness, k E . In the framework of the incrementalplasticity theory, the following basic equation can be usedṡ = ṡ e + ṡ pṡ e = ˙τk E(4.4)where the inelastic, ṡ p , and the total interface slip ṡ are introduced in rate form.Integrating each constitutive model, the constitutive laws can be defined in terms ofthe tangent elasto-plastic constitutive operator, k ept an, which is specified for loading orunloading/elastic processes in Table 4.1.4.3 Elasto-plastic joint/interface model with isotropic linearsofteningThis section presents a classical one-dimensional plasticity model aimed at simulatingthe bond-slip behavior of fiber-to-concrete interface.71