PSI - Issue 66

Umberto De Maio et al. / Procedia Structural Integrity 66 (2024) 459–470 Author name / Structural Integrity Procedia 00 (2025) 000–000

462 4

where pl ε  is the rate of the plastic strain, p  p  is the rate of the scalar hardening variable, whose evolution is proportional to the norm of the plastic strain tensor, scaled by a hardening ductility measure and it follows the evolution equation:  is the rate of the plastic multiplier, and

pl h un x  



 2 2cos un 

(4)







  

p

The plasticity model is also described by the Kuhn-Tucker loading-unloading conditions:





(5)

0,

0,

0

F

p p F



p 







p

To describe the damage part of the model it is necessary to present the damage loading functions for tension and compression, which are linked to the equivalent strain of the material, and the Kuhn-Tucker loading-unloading conditions:     , tension and compression i i d eq un d F i t c      σ (6)

i

i  d

i 

i d d F

0,

0,

0

(7)

F











d

The equivalent strain is the same for both tension and compression, and the expression is derived from the plastic yield surface 0 p F  :



(8)

eq  

un

E

The damage model is also based on evolution laws of the damage variables for tension and compression, defined in the framework of isotropic damage mechanics:   1 2 , , i i i i i d d d d d g     (9)

where i

1 i d  , and

2 i d  are additional damage history variables.

d  ,

2.2. Dynamic damage indicators In the classes of damage detection methods, vibration-based techniques are based on the concept that the damage will alter the dynamic properties of the structures, and it is possible to detect the occurrence of damage phenomena by measuring the dynamic structural response. With the aim of detecting the presence and the location of the damage, some dynamic damage indicators are introduced. These indicators can be used as effective tools to investigate the effects of damage phenomena on the mechanical behavior of reinforced concrete elements, such as beams. Among several criteria, based on the changes in the modal properties of the system with respect to the undamaged configuration, it is possible to use as a damage indicator the so called inverse eigenvalue sensitivity method, which analyze the variation of natural frequencies between the undamaged UD i f and damaged configuration D i f , normalized by the undamaged frequency, for each mode shape: