Crack Paths 2012

Assuming a uniform distribution for the fibres, the peak stress due to the fibre crack

bridging is [2]:

¸¸¹·¨¨©§ 22 0 1 1 fef f f f f S W K where f = snubbing coefficient (usually ranging from 0.7 to 0.9), 2 DL

0 W = frictional bond

(7)

V

0

f L 2 = fibre length,

f K = volume fraction of fibres.

stress,

f D = fibre diameter,

depends on the angle

Dˆ

between the fibre

If unidirectional fibres are examined,

0 V

direction and the truss direction in the lattice:

ˆ 0

K

W

L

(8)

D

2

f

f f

f

V

D

e

0

The peak stress due to the fibre crack bridging in the truss of the lattice normal to a

0 ) 2 ( 3 V A (el.g.

V

putative ModeI crack plane is equal to

see, in Eq. 4, the stress

0

in the truss (1) when x is the loading axis).

The characteristic cracking strain values of the crack bridging curve due to fibres can

be determined by smearing the crack opening along the length of the truss, namely:

> @ f f f D E L w U W 1 0 2 0 (with

>@fffEEKKU 1

lw0

, lwfu f u , , H , where

)

0 H

is the crack opening at the peak stress of the crack bridging law due to fibres, and

f f u L w ,

is the ultimate crack opening of the bridging law due to fibres.

C O M P A R I SBOENT W E EN NU M E R I C ANLDE X P E R I M E N TR AE SLU L T S

The fracture behaviour of plain and fibre-reinforced concrete coupons under tensile

loading is examined herein by means of the two computational models being compared.

The first one, based on a continuum approach, has been implemented in an in-house

code, whilst the second one, based on a lattice approach, has been implemented in the

subroutine U M A oTf the commercial FEcode ABAQUS.

An experimental campaign related to prismatic Reactive Powder Concrete (RPC)

specimens subjected to tensile stress is examined [18]. The specimens have a total

length equal to 700 m mand cross section of 50x20 m m (Fig. 2). The mechanical

parameters of the R P C concrete are the following: Young modulus

E 50 GPa ,

[19]. The relevant

ultimate tensile strength

0.8 M P a

, fracture energy

f G

30 N/m

tf

%0.2

f K

parameters for unidirectional steel fibres are: fibre volume

, Young modulus

and diameter

f E

210 GPa

, fibre length

2

f L

m m 1 3

f D

mm16.0

, while the limit

matrix-fibre shear stress is equal to

0.2 M P a

. The analysis is performed under

0 W

displacement control by imposing a progressive upward vertical displacement at the top

of the specimens. A plane stress condition is assumed. Five orientations of fibres are

q q q q q 9 0 , 6 0 , 4 5 , 3 0 , 0 M

considered (

) together with the cases of random fibres and of

plain concrete (no fibres).

In Fig. 2, some crack paths of the continuum and the lattice models are shown.

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