PSI - Issue 39

Umberto De Maio et al. / Procedia Structural Integrity 39 (2022) 677–687 Author name / Structural Integrity Procedia 00 (2019) 000–000

680

4

The theoretical formulation of the diffuse interface model, explained in detail in De Maio et al. (2020c), relies on a variational approach written for a discretized domain considering linearly elastic planar volumetric elements and nonlinear cohesive interface elements with zero thickness, under the assumption of small displacements, plane stress state, and negligible inertial forces. The nonlinear behavior of the cohesive forces t , acting along all mesh boundaries, is described by an intrinsic mixed-mode traction-separation law and written in the following matrix form considering the displacement jump between the two crack faces § ¨ δ and the second-order constitutive tensor 0 K :

n     s t t

n δ δ     s  

  

  

0

0

K

(

)

1 = −  

d

n

(1)

,

0

0

K

s

where the subscripts n and s refer to the normal and tangential components, respectively. In the uncracked configuration the interface possesses the initial stiffness parameters 0 n K and 0 s K , playing the role of penalty parameters in order to enforce the inter-element kinematic constraint, without having a physical meaning. Such values are computed, according to the micromechanics-based calibration technique proposed by some of the authors in De Maio et al. (2020c), by the following expressions:

0 K E L K K κ ξ = = 0 0 n

(2)

,

mesh

S

n

where E is the Young’s modulus of the considered material whereas κ and ξ are dimensionless stiffness parameters obtained by the calibration method, explained in detail in De Maio et al. (2020c), as a function of the desired Young’s modulus reduction and the Poisson’s ratio of the bulk material. The scalar damage variable d , reported in Eq. (1), with exponential or linear evolution, involves the following effective displacement jump:

2

2

=

δ δ +

δ

(3)

,

m

n

s

with the symbol g denoting the positive part of the enclosed quantity. The mixed-mode initiation and propagation are governed by the following stress- and energy-based criteria, respectively:

2

2

I G G G G     + =         + = II 1 n s nc sc t t t t

1

(4)

,

I

II

c

c

where the nc t , sc t and

I c G , II c G denote the normal/tangential critical interface stresses and the critical mode I/II

fracture energies, respectively.

2.2. Steel and FRP reinforcement modeling The integrated fracture model takes advantage of two sub-models: (i) an embedded truss model and (ii) a single interface model, useful to simulate the mechanical behavior of steel rebars, together with their interaction with the neighboring concrete, and the debonding mechanisms of the FRP system, respectively. According to the embedded truss model, the rebars are modeled by means of truss elements with an elastoplastic constitutive behavior 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 Figure 2a, equipped by a bond-slip relation