INTRODUCTION
Fibers are most often added to cement composites to improve their fracture behavior. This concept of reinforcement was used in building materials in the old days. There is an evidence that asbestos fiber was used to reinforce clay posts about 5,000 years ago.
The behavior of composite materials formed of a given binder and a matrix presenting
high shrinkage properties is much related to the state of interlocking, mechanical
or chemical between the fiber and matrix. If the bond is perfect the shrinkage
will be inhibited. In this case the behavior of the material visàvis
the tensile strength and consequently the shrinkage induced cracking is not
known, whereas if plain fibers are used the shrinkage would not be inhibited
but the tensile strength of the resulting composite material would be improved
(Toledo Filho et al., 2005).
The modeling of fibers and the bond properties for fibers in fiber reinforced
concrete has received much attention in the literature in the past several decades.
Nammur and Naaman (1989) derived an analytical model
of the bond shear stresses at the fibermatrix interface in a pure tensile fiber
reinforced concrete specimen. The derived model predicted the shear stress distribution
along the fibermatrix, the slip distribution and the normal tensile stresses
in the fiber and the matrix. Perfect alignment of the fibers as well as square
packing was assumed. The model was finally used to predict numerically the bond
shear stress in a given tension composite using a specific bondslip curve.
In a previous work performed by Hanayneh (1994), a parametric
analysis of the fibermatrix interface in materials with high shrinkage properties
was studied numerically. The study showed the influence of some parameters on
the behavior of an elementary model. However, the applications of this type
of analysis were of limited use. Li and Li (2001) modeled
the behavior of fiber reinforced concrete based on the Continuum Damage Mechanics.
In the material, a cementsandcoarseaggregatewater mix was used as the matrix
and short steel fibers were used as the reinforcement. The quasibrittleness
of the matrix and the fibermatrix interfacial properties were taken into consideration.
Results show that the modelpredicted stressstrain curves agreed well with
those obtained experimentally.
Lately, many researchers applied experimental programs to study the use of
fiber reinforced to reduce plastic shrinkage in concrete (Boghossian
and Wegner, 2008; Sivakumar and Santhanam, 2007;
Passuello et al., 2009).
On the other hand, Boulekbache et al. (2010) investigated
the fiber distribution and orientation by using a translucent fluid model with
a yield stress. The observation confirmed the ability of the developed method
to provide data on the orientation and distribution of steel fibers within concrete.
It was showed that orientation and distribution are dependent on the yield stress
of the fluid material. The flexural strength depends on the fiber distribution
and orientation and is significantly improved when the fibers are oriented in
the direction of the tensile stresses fresh (concrete with good workability).
On the contrary, for concrete with poor workability, an inadequate orientation
of fibers occurred, leading to a poor contribution of the fibers to the flexural
behavior of the tested specimens, despite the relatively higher compression
strength of the tested concrete material.
Jay Kim et al. (2008) constructed reinforcing
fibers with three different geometries, i.e., embossed, straight and crimped,
from waste polyethylene terephthalate (PET) bottles and used them to control
plastic shrinkage cracking in cementbased composites. Pullout tests evaluated
how the fiber geometry and fraction by volume (0.11.00%) affected the rate
of moisture loss and controlled the plastic shrinkage cracking characteristics
Barluenga (2010) demonstrated that inclusion of small
amounts of short fibers has an effective solution to control cracking due to
drying shrinkage of concretes at early ages. The key point of fiber effectiveness
is their capacity to sew the crack sides, preventing crack opening, because
cracking of concrete matrix induces fiber actuation. The results showed that
as concrete mechanical capacity develops with age, while fibers have full properties
before being included in concrete matrix, the interphase between matrix and
fibers evolves during setting and hardening and affects cracking control effectiveness,
due to stresses induced by fibers into the matrix during concrete hydration.
Other researchers such as Leung et al. (2006)
used shotcrete and fiber reinforced shotcrete to produce layers or linings with
large surface area versus volume ratios to restrain shrinkage cracking. A new
testing configuration, consisting of a shotcrete specimen bonded to a steel
Isection and angles was proposed. From the results, the proposed setup is
shown to be a practical and viable approach for investigating the shrinkage
cracking behavior of shotcrete and fiber reinforced shotcrete.
Most of the existing models presented in the literature are relatively complicated. On the other hand, the extensive use of fiber reinforced composite materials made it necessary to come up with a material model that can be easily incorporated into existing design procedures for professional engineers. This paper proposes a simple analytical approach that was developed by the authors in Jordan on a research project that started on July 2008 and ends on March 2010, which would allow the analysis and the study of the behavior of fiberreinforced materials with high shrinkage properties by varying the parameters related to the fibers and the matrix. A model formed of a rigid fiber embedded in a cylindrical elastic matrix presenting high shrinkage properties is analyzed. Deformations of the matrix are assumed not to lead to deformations in the fibers. Thus, the analytical model presented in this study is most appropriately applicable to cases of stiff fibers.

Fig. 1: 
Representation of the cylindrical model used in the study 
DESCRIPTION OF THE MODEL
The model consists of two coaxial cylinders of radii r_{f} and R, respectively, as shown in Fig. 1. The inner cylinder represents the rigid fibers with a radius of r_{f} and the outer cylinder represents the matrix at a radius equal to R. The length of the model is equal to L and the length of the bond between the fibers and the matrix is assumed to be equal to b. Since the cylinder is symmetrical, it will be sufficient for further calculations to consider one quarter of the cylinder as shown in Fig. 1.
Assumptions: To simplify the study of the model, the following assumptions
are adopted:
• 
Isotropy and homogeneity of the two constituents: the fiber
and matrix 
• 
Rigidity of the fiber, i.e., deformations of the matrix does
not necessarily lead to deformations in the fiber. This assumption limits
the applicability of the approach to cases of stiff fibers 
• 
Isotropy of the shrinkage of the matrix 
PROBLEM FORMULATION
Equilibrium equations: The deformations of the cylinder are symmetrical
around the zaxis. The stresses are independent of θ and consequently all
their derivatives with respect to θ are equal to zero. The shear stresses
σ_{rθ} and σ_{θz} cancel one another by
virtue of symmetry. The equilibrium equations in cylindrical coordinate system,
therefore, are limited to:
Constitutive lawsshrinkage: If a material is anisotropic, its shrinkage,
if any, will necessarily be anisotropic too. Depending upon the dimensions of
the sample, shrinkage can equally be heterogeneous and hence a shrinkage gradient
develops: the outer parts of the sample shrink more than the inner parts at
any given moment of time.
As far as the above model is concerned, it will be assumed that shrinkage is isotropic. On the other hand, since its dimensions are assumed to be small, shrinkage will be assumed to be homogeneous. Consequently, shrinkage does not depend on coordinates r, z, θ but only on time. Proceeding in a manner similar to classical thermo elasticity, the term for thermal expansion will be replaced by a term called shrinkage.
In the absence of stresses, anisotropic shrinkage results
in a contraction:
with considered
here to be dependent only on time. On the other hand at time t_{0},
for reference shrinkage ,
the material is assumed to behave as an elastic isotropic material with Poisson’s
ratio v and modulus of elasticity E. Therefore, in a simplified manner, the
shrinkage of the matrix becomes:
and inversely the stresses are:
where, 3k = 3λ + 2μ with μ= E/2(1+v), λ = v/(1  2v) (1+v),
3k = E /(12v) and in a system of cylindrical coordinates, the nonzero stresses
at time t are:
It is observed that the expressions for normal stresses cancel one another
for ε_{rr} = ε_{zz} = ε_{θθ}
=.
This is the case of free shrinkage. On the other hand, for a symmetrical deformation,
the deformationdisplacement relationships are:
The equilibrium Eq. 1 and 2, expressed in terms of displacements, are:
Variation of the state of link at the fiber matrix interface Bond length
and slip length: The study of the variations of the state of link at the
fibermatrix interface has its importance due to its effect on the global behavior
of fiberreinforced material. A case of perfect bond is assumed to take place
if the mean bond strength τ_{b}, determined experimentally, is
higher than the shear stress σ_{rz}; otherwise, slip along the
fiber prevails. The modulus of elasticity E and Poisson’s ratio v will
be assumed to be known and having constant values.
Determination of the shear stress σ_{rz}: The shear stress
σ_{rz} is developed in the matrix due to its inherent shrinkage
and to due to the presence of rigid fiber itself. The shear stress σ_{rz}
can be determined using the displacement Eq. 310.
Determination of the displacements U_{r}, U_{z }and the
shear stress σ_{rz}: An approximate solution of the equilibrium
Eq. 1112 which satisfies the boundary
conditions in displacements which impose that in the case of perfect bond at
the interface, U_{r} (r_{f}, z) = 0 and U_{z }(r_{f},
z) = 0 was found in the following forms:
where, the coefficients k_{1} to k_{4} can be determined using the results obtained from the numerical method. These values are: k_{1} = 1.1ε, k_{2 }= ε, k_{3} = 0.25 and k_{4} = 0.21. The shear stress σ_{rz} can be found using Eq. 6, which gives:
which yields at the interface (r = r_{f})
Determination of the bond length b: Following, perfect bond takes place in the case where:
which yields:
if the length of the cylindrical model L is introduced, Eq. 17 becomes:
Hence, the following three cases of the state of link can be identified:
• 

• 

• 

Effect of the bond strength τ_{b} on the bond length b:
The values considered in this paragraph are taken from a previous work (Hanayneh,
1994): E = 600 MPa, v = 0.175, r_{f} = 0.4 mm, L = 42 mm and ε
= 5x10^{4}. Thus, Eq. 18 becomes:
Figure 2 as well as Eq. 19 show that the bond length increases as the bond strength at the interface increases.
Effect of shrinkage on the bond length: Let the value of the bond strength τ_{b}= 0.2 MPa and keep the values of the other parameters E, v, r_{f}, L and ε the same as in the previous section, then Eq. 18 becomes:
This equation as well as Fig. 3 shows that the bond length is inversely proportional to shrinkage, given that the bond strength at the interface is constant.
Effect of age on the bond length: Although the modulus of elasticity and Poisson’s ratio undergo slight variation with time, they are assumed to be constant in this study. The following relationships relating the effect of age on the bond strength τ_{b} and free shrinkage ε are used:

Fig. 2: 
Effect of the bond strength on the bond length 

Fig. 3: 
Effect of shrinkage on the bond length 

Fig. 4: 
Effect of age on the bond length 
where, t is the time in days.
Reporting the previous relationships into Eq. 19 gives:
Equation 23 is represented in Fig. 4.
It shows that the bond length decreases as age increases, however, this behavior
does not mean that fiber reinforced materials become weaker. In fact, this depends
upon the tensile strength of the matrix and the bond strength at the interface.
Estimation of the equivalent mean shrinkage: The idea of introducing
the concept of equivalent mean shrinkage to give an adequate approximation of
the shrinkage of fiberreinforced material can be made through studying the
behavior of an elementary cell. In fact, the global material can be considered
as an assembly of elementary cells in a given direction. Figure
5 shows a representation of such an assembly. In this study, the shrinkage
is assumed longitudinal and the reinforcing fibers aligned.
In the model which consists of a rigid fiber embedded in a matrix, the longitudinal shrinkage is not uniform along the section of the cylinder. In fact, it has been shown that the shrinkage of the matrix increases as the distance from the fiber increases. In addition, the shrinkage of the rigid fiber is considered negligible.
The equivalent mean shrinkage will be calculated for the following two cases:
• 
Perfect bond, (ε_{eq})_{b} 
• 
Partial slip,(ε_{eq})_{s} 
Case of perfect bond along the whole length:This case is characterized
by the following relations:

Fig. 5: 
Assembly of the elementary cells containing the fibers 
and
Therefore, the equivalent mean shrinkage (ε_{eq})_{b}
can be written as:
and replacing (U_{z})_{b} by its expression and after integration, the equation becomes:
Introducing the volumetric fraction V_{f }= (r^{2}_{f}/R) Eq. 24 becomes:
Figure 6 represents the variation of the equivalent mean shrinkage in terms of Vf.
Case of partial slip Determination of the displacements (U_{z})_{
s} and (U_{z})_{b}: In this case, an expression of
σ_{rz }satisfying the matrixfiber interface conditions is derived.
Assuming that the displacement U_{r} keeps the same form as in the zone
z<b, the equations pertaining to the problem permit to get the expression
of (U_{z})_{s}. All the constants are determined by considering
the continuity conditions between the zones of perfect bond and slip.
The expression of (σ_{rz})_{s} can be written as in the following form:
and as U_{r} vary slightly whether the zone of perfect bond or slip is considered, the expression of (U_{r})_{ s }can be written as:
But
and comparing with Eq. 27, one gets:
Hence:
where:
Therefore:
where the constants can be determined by using the continuity of the displacements between the two zones (z = b) on one hand and the boundary condition on the other hand: (σ_{rz})_{s} = τ_{b} for r = r_{f}.
These constants have the following expressions:
In a similar manner, the expression (U_{z})_{b} can be given as shown previously by:
Where:
At this point, one should note that if the continuity of the displacement U_{z} is satisfied between the zones of perfect bond and partial slip. Then, the continuity of σ_{rz} will be automatically satisfied.
Finally the expression of (U_{z})_{b} and (U_{z})_{s} can be written as:
where,
as given by Eq. 18
Equivalent mean shrinkage: In the case of partial slip along one part
of the fiber and perfect bond along the other, the equivalent mean shrinkage
can be written:
with φ = π(R^{2}r^{2}_{f})
Replacing (U_{z})_{s} by its expression given in Eq. 33 and integrating, one gets:
and introducing V_{f}, it comes:

Fig. 6: 
Effect of Vf on the equivalent mean shrinkage (case of perfect
bond) 

Fig. 7: 
Equivalent mean shrinkage function of V_{f} (%) (Sliding
along L/2) 
If τ_{b} is replaced by its expression given by Eq.
18, the previous equation becomes:
By inserting the values of the parameters of the model used in this study, one gets:
The effect of the variation of (ε_{eq})_{s}/ε in
terms of V_{f} and the equivalent mean shrinkage for case of perfect
bond is shown in Fig. .6, while the effect of the variation
V_{f} and the equivalent mean shrinkage considering that slip takes
place along onehalf of the fiber length (b = L/2) is shown in Fig.
7. It can be seen from Fig. 6 that the equivalent mean
shrinkage of a composite material is inversely proportional to the volumetric
percentage of the fibers. On the other hand, Fig. 7 shows
that in case of slip that takes place along onehalf of the fiber length; the
value of equivalent mean shrinkage was 0.34 and 0.2 for V_{f} 0.2% and
2%, respectively, i.e., inversely proportional to the volumetric percentage
of the fibers This seems to be logical if the fibers are assumed to be rigid
and nondeformable as opposed to the matrix.
CONCLUSIONS
Based on the results of this analytical investigation, the following conclusions are drawn:
• 
The bond strength is proportional to the bond length 
• 
The bond length is inversely proportional to free shrinkage 
• 
The bond length decreases with age of the composite material 
• 
The equivalent mean length is inversely proportional to the
fiber volumetric fraction 