Crack Paths 2012

opening

(hence, for a linear curve

V

u w ) is equal to the ModeI fracture energy

f G

against w,

). This concept can be translated to the truss elements of the

t f u f G w 2

lattice model, if one assumes to smear the crack opening along the length of the truss.

t f l G 4 3 [6]. f

H

The ultimate cracking strain

uH turns out to be

u

M E C H A N IOCFSF I B R E - R E I N F O RMC EADT E R I A L S

In this Section, the modelling of the reinforcing effects (due to fibres) within the

framework of the two theoretical approaches being compared is described. Unless

otherwise specified, details of such a modelling can be found in Refs [5, 17] and Refs

[2, 6] for the continuum model and the lattice model, respectively.

Mesomechanical Model in the Continuum Approach

The heterogeneous (composite) material is supposed to consist of a matrix phase

(denoted by the subscript m) and a fibre phase (denoted by the subscript f) embedded in

the matrix (Fig. 1). The composite material (having characteristic size D) is assumed to

have macroscopically the same mechanical characteristics of a small Representative

VolumeElement (RVE, having characteristic size D ). d

The tangent elastic tensor of the macroscopically homogeneous composite can be

determined through an energy balance between the composite and the equivalent

macroscopically homogeneous material:

« « ¬ ª ˜ ˜ ˜ S S T 0 0 ) ( ' ' d d dsd s E f mfmf m f m f f f m F F C C C (5) M T H K ) ( ) ( ) (

³ ³ ˜ ˜ ˜ » » ¼ º

'

eq ' ,' ,' C Care the tangCent elastic tensors (of the matrix, of the fibres and of f m

where

the equivalent material, respectively);

f E is the Young modulus of the fibre phase;

k kis a secFond-order tensor, where k is the unit vector parallel to the fibre axis,

^ `T M T M T cos sinsin cossin ˜ ˜ (Fig. 1). In Eq.(5),

mfH is the strain (in the matrix)

k

* u

RVE

d

z

T

k

o

y

M

x

D

Figure 1. Scheme of the R V Eand

:

Z

identification of the fibre orientation

x

* t

in the 3D space.

O

* u * * U t

Y

X

306