Crack Paths 2012

f V

is the actual fibre stress.

measured in the fibre direction,

Furthermore,

V V

V V , / K f

represent the matrix volume fraction and fibre volume

P

/

m m

f

fraction in the composite RVE, respectively.

Finally, )(mfs is the sliding scalar

>

@mf

sH

˜

function that quantifies the matrix-fibre strain jump, >@>@ m f H

, occurring in

H

1

mf

the case of matrix-fibre debonding. Preferential orientation of the fibres in one

particular space direction can be taken into account through suitable probability

TM,

distribution density functions

)(),(TMT M p p of the orientation angles

:

2 1 ) ( ) ( ) ( ) ( ) ( ¸¸¹·¨¨©§ ˜ ¸¸¹·¨¨©§ ˜ ¸¸¹·¨¨©§ ˜ G P S D D G P D 2 2 21 21 D

D

˜ D eD De D B with C B A ˜ ˜ ˜ G S ˜ ˜ D D D

D

D

D

D

2

G S

G S

D

D

2

D

(6)

2

D

G P S D

2 12

) (

) (

2 1

,

1

,

˜ ˜

˜

2

with T M D , .

T M D , , defined in the intervals

S D 0 d d, are

The above functions

)(D Dp

with

theoretically formulated by assuming a Gaussian-like expression, which attains the

maximumvalue at T M D , (mean values of the Euler angles), and the minimumvalue

2/ ,2/ S T S M D r r . Further, the cumulated probability over the function domain

at

P represents the mean value of the probability

is equal to one. The parameter

M P or

MG

T P ) , whereas

distribution density functions (i.e.

and

are the

TG

corresponding variances. The case of randomly oriented fibres can be obtained by

setting

and 2 / 1 ) ( S M

) s i n ( ) ( T T .

p

p

M

T

In order to take into account the fibre-matrix debonding, we introduce the sliding

function )(mfs that allows us to write the fibre strain fH

as a function of the matrix

sH

>@:)(kk˜mf

H

mfH, that is,

strain

[17]. the fibre-matrix debonding can be

f

taken into account properly multiplying the integral in Eq. (5) by a function depending

on such a relative fibre-matrix sliding.

Note that the maximumtensile stress along a fibre is always reached at its centre

ft f, )0( t ,

[17]. Whensuch a maximumstress reaches the fibre tensile strength, i.e.

f

the fibre is assumed to break in two parts having the same length.

Fibre Crack Bridging in the Lattice Approach

The tensile behaviour of a fibre-reinforced composite material is described according to

the cohesive crack approach. Hence, the stress-strain curve for the above cracked

matrix is combined with the crack bridging law due to fibres. The resulting stress-strain

curve is characterized by a perfectly-elastic

behaviour in compression; the tensile

behaviour is elastic up to the first cracking stress, and then a linear piecewise

postcracking curve with softening branches follows.

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