PSI - Issue 47

Francesco Ascione et al. / Procedia Structural Integrity 47 (2023) 460–468 Author name / Structural Integrity Procedia 00 (2019) 000–000

463

4

The bulk elements of mesh, representing the undamaged material, possess a linearly elastic mechanical behavior, of the type = σ ε C , where σ , C , and ε represent the volumetric stresses, the fourth-order elasticity tensor, and the linear strain operator, respectively. The cohesive elements placed between the concrete bulk elements and along the concrete/FRP interface are governed by the following cohesive expression: ( ) δ δ δ = = with , coh i i f t i n s , (1) where coh i t and i δ are the components of the cohesive traction vector and effective displacement jump vector, while δ is the maximum effective displacement jump recorded during the entire deformation history. ( ) f δ is a softening function whose form depends on the type of simulated materials. In this work, an exponential and a trapezoidal softening function are adopted for the cohesive elements placed between the concrete bulk elements and along the concrete/FRP interface, respectively (see Fig. 1). The expression of the adopted exponential law is the follows:

f



t

0

< ≤

i δ

0 δ δ i

  = 

0 δ

coh i

t

(2)

,

  

  

f

  

t

exp

f

−

0 δ δ δ < i i

t

G

f

where t f , 0 δ , and f G are the critical tensile strength, critical displacement, and fracture energy of the concrete. On the other hand, the trapezoidal expression for the cohesive elements along the concrete/FRP interface is written in terms of tangential strength and slip occurring along this interface, due to the fact that such a zone is mainly influenced by the mode-II fracture condition:

τ

      

max 1 max δ

0

s < ≤

s δ

1 δ δ

τ

1 δ δ δ < ≤ 2 s

 = 

τ

(3)

,

2 δ δ − f δ δ − s f

τ

2 δ δ δ < ≤ s f

max

0

f δ δ < s

where max τ is the maximum tangential strength while 1 δ and 2 δ are the relative slips identifying the length of the plateau branch (see Fig. 1). The complete detachment of the reinforcement system occurs at the final slip f δ where the tangential stress drops to zero. The trapezoidal function has been chosen to describe the bond-slip behavior of the FRP system enhanced with the incorporation of nanomaterials. As a matter of fact, as highlighted in several experimental works, the carbon nanotubes embedded in the epoxy resin make the adhesive more ductile (Wu and Huang, 2008), and a trapezoidal law, similar to that employed for metals (He et al., 2021), results to be suitable to describe the reinforcing effect offered by the nanomaterials incorporation. 2.2. Embedded truss model for steel reinforcement The mechanical behavior of the steel reinforcement has been described by using an embedded truss model proposed by some of the authors in (De Maio et al., 2022a, 2022b). In particular rebars and stirrups, of the RC elements, have been modeled by 1D elastoplastic truss elements including a linear hardening to describe the steel yielding stage. Such elements are connected to the concrete mesh through zero-thickness interface elements, depicted in Fig. 2a, equipped by a bond-slip relation valid for ribbed bars and good bond conditions taken from the CEB-FIP Model Code (CEB FIP, 2013) and illustrated in Fig. 2b. The truss elements are constrained in the perpendicular direction to the rebar