PSI - Issue 62

Daniela Fusco et al. / Procedia Structural Integrity 62 (2024) 895–902 Fusco et al./ Structural Integrity Procedia 00 (2019) 000 – 000

897

3

integration of the response of the fibers, which is derived by the constitutive law of the material. For prestressed beams, the constitutive behaviour of the section is given by the response of the concrete and steel fibers corresponding to tendons and reinforcement. In this work, the 3D damage-plastic model for concrete-like materials proposed in Addessi et al. (2002) is considered for concrete fibers, properly modified to consider the unilateral effect due to the re-closure in compression of tensile cracks. The stress-strain relation is defined as:

p

(1 ) 2 ( C

)

D = −



−

(1)

where σ is the stress vector,

and p are the total and plastic strain vector, respectively, D is the damage 1 D = for a completely damaged state), C is the constitutive matrix of the

0 D = for undamaged material,

variable (

undamaged material, which depends on the Young’s modulus and Poisson ratio. To consider the unilateral effect due to the re-closure in compression of tensile cracks, the model proposed in Addessi et al. (2002), Gatta et al. (2018), Di Re et al. (2018) and Fusco et al. (2023) introduces a damage variable for tension, t D , and one for compression, c D whose combination provide the overall damage variable D :

t t c c D D D   = +

(2)

where t  and c  are the following weighting factors:

2  + t

2 ,

1   = −

(3)

t 

=

c

t

2  

t

c

e

e

Y

Y

,

(4)

t 

=

c 

=

t

c

( Y Y b D  + + ) e

( Y Y b D  + + ) e

0

0

t

t t

t

c

t c

c

The evolution of damage is governed by associated variables defined as equivalent strain measures as:

3

3

3



2   i e − −

2 +

,

Y

e

Y

k e e

=

=

(5)

t

i

c

i

j

−

−

1

1

1

i

i

j

i  =

=

=

The material parameter k determines the shape of the limit function in compression and i e is related to the principal total strains ˆ i by the following relation: 3



ˆ   +

ˆ

(1 2 )

e

(6)

= −

i

i

j

1

j

=

0 c Y and 0 t Y are the damage strain threshold; e t Y and e

c Y are based on the principal

Regarding to expression (4),

elastic strains, adopting the same definition as that in equations (5) and (6). The evolution of the two damage variables is controlled by the following damage limit functions for tension and compression,

0 c c c fYD YY aYbD YD YY aYbD f = − − + = − − + 0 ( , ) ( ) , ( , ) ( ) t t t t t t t t t c c c c c

(7)

c

and ruled by the classical Kuhn – Tucker and consistency conditions. In (7) the parameters c b and t b influence the rate of damage growth in tension and compression and govern the maximum strength of the material. The parameters c a and t a control the gradient of the degradation curves in the post-peak softening regime. As for the plastic mechanisms in concrete fibers, the Drucker-Prager plasticity model with isotropic and kinematic hardening is used in this work. This is capable of accurately representing the asymmetric plastic behaviour of concrete