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Fracture behaviour of polypropylene-fibre-reinforced concrete: Modelling and computer simulation
Composites
Science and Technology 56 (1996) 947-956 0 1996 Elsevier Scknce Limited Printed in Northern Ireland. All rights reserved
ELSEVIER
PII:
SO266.353?3(95)00058-9
0266-3538/96/$15.00
FRACTURE BEHAVIOUR OF POLYPROPYLENE-FIBREREINFORCED CONCRETE: MODELLING AND COMPUTER SIMULATION
M. Elser,” E. K. Tschegg,“* N. Finger” & S. E. Stanzl-Tscheggb “Institute of Applied and Technical Physics, Technical University of Vienna, A-1040 Vienna, Austria “Institute of Meterology and Physics, University of Agriculture, Vienna, Austria
(Received
5 February
1996; accepted
Abstract The proportion of fracture energy which is consumed by jibres during fracturing in uniaxial and biaxial loading of a composite (matrix plus fibre) can be calculated by means of a modified model. By the method of variation of parameters, the calculated load/displacement curve is adapted to the measured load/displacement curve. The jibre component of the measured load/displacement curves is obtained by subtracting the experimentally determined matrix component from the measured load/displacement curve of the composite material. By these parameters it is possible both to represent the load/displacement curve of every single fibre for each relative position, and to determine the change of the fracture energy with increasing compressive loading, (TV, of the specimens. 0 1996 Elsevier Science Limited
19 March
GIG, @‘Pt) @F‘520
f NGwet zibre
Gjmodel)
Keywords:
polypropylene-fibre-reinforced concrete, biaxial fracture behaviour, specific fracture energy
c;(F1(=)
NOTATION
H LIG
a aI, a2 A LIG a+ e b, B LIG CMO&m
d d(z) df e
Distance between force application and surface of the specimen Taylor expansion coefficients Ligament area Distance between point of attachment and notch root Coefficient when taking compressive load into account Ligament thickness of the splitting specimens Cutting parameters of the FH versus
k = s/L, 1 L, m = l/L, m max n AN
CMOD curve
P, P’
Crack Crack Fibre Notch
P,,,(d)
width for parallel fracture surfaces width for splitting specimens diameter depth of the splitting specimen
rf S Vf
* To whom correspondence
should
be addressed.
Vi
947
1996)
Fibre elastic modulus Matrix elastic modulus Snubbing factor Compression strength of cube Distribution function of the fibres Horizontal splitting force Snubbing coefficient Specific fracture energy of the uniaxial fracture test Specific fracture energy of the biaxial fracture test Normalised specific fracture energy Optimum specific fracture energy Fracture energy of composite Fracture energy of matrix Fracture energy of fibre component in the composite Fracture energy of fibre component in the model curve Fracture energy of fibre component in the case where there is no fibre rupture Ligament height of the splitting specimens Normalized fibre end slippage Embedded length of the fibre in the matrix Fibre length Normalised embedded length Integral limit for calculation of the mean friction stress or shear stress, respectively Number of fibres per unit volume Number of fibres in a zone width Lf around the fracture surface Forces on a single fibre Total bridging force for parallel fractured surfaces Fibre radius Fibre end slippage Fibre volume fraction Matrix volume fraction
M. Elser et al.
948 w(x) w (R)d2Q
x Z
Zrnid
Probability that centre of the fibre lies between x and dx Probability that the fibre points towards the solid angle element d2R around solid angle R Axis for position of the centre of the fibre Axis of symmetry of the splitting specimens Spherical co-ordinates Fibre tensile strength Normalised compressive stress Compressive stress Bridging normal stress stress shear Friction and stress, respectively Mean shear stress
1 INTRODUCTION The fracture energy that is absorbed during crack propagation in fibre-reinforced materials is provided mainly by the following mechanisms: matrix cracking, fibre rupture, fibre/matrix interface debonding and fibre pull-out. There have been numerous studies regarding theoretical models that deal with the fibre pull-out process and the fibrelmatrix interface debonding process in various fibre-reinforced materials. Some of these studies are mentioned below. The theoretical analysis of both the fibre/matrix interface debonding process and the fibre pull-out problem’32 are essentially based on two generally different models. l
l
Model I. The maximum shear stress criterion for fibre/matrix interface debonding (shear-lag model): if the shear stress exceeds the shear strength of the fibre/matrix interface, the result along the is unstable crack propagation fibre/matrix interface.‘-’ Model II. This model is based on the concept of fracture mechanics, where the debonded zone is regarded as a crack in the fibre/matrix interface, a so-called interfacial crack. The propagation of this crack depends on an energy criterion that has to be satisfied.2-‘2
Several studies2,13,14 have tried to combine these two models into one. Two models, namely those of Gao et al.,’ which is based on model I, and of Hsueh,‘5-1y which is based on model II, are investigated. Whereas the model of Gao predicts the maximum debond composite stress for long fibres, Lf, in epoxy-matrix systems very precisely, the model of Hsueh is more suitable for experiments with short fibres. For ceramic-matrix composites both models provide results that are identical with those of experiments for a large range of fibre lengths, Lf.2*‘3,14
All these models deal with fibre-reinforced materials, mostly cement-bound materials under uniaxial tensile load. Models regarding the biaxial loading of fibre-reinforced materials, such as thermoplastic material compounds, E-glass/epoxy laminates, fibrereinforced aluminium compounds, orthotropic fibrereinfoced laminates, multiaxial fibre-reinforced metal compounds, etc., are discussed below. reported on uniaxial and biaxial Hart-Smith2” frcture criteria of reinforced-plastic laminates. Makinde et a1.2’ deduced a biaxial strength criterion for orthotropic fibre-reinforced laminates under biaxial loading by analysing the elongations in the material Two material (strain-based strength criterion). provide an analysis of multiaxial fibremodels2* reinforced metal compounds by means of a finite element simulation. Mallik23 reports on different models, with regard to thin-walled pressure vessels consisting of fibre-reinforced material compounds, where the internal compressive load in the pressure vessel creates a biaxial stress field in the wall material. The simulations of load/displacement curves that have been determined by means of experiments on fibre-reinforced biaxially loaded concrete specimens, require a model that accounts for both the random distribution of fibres and the fibre fracture that occurs when the shear stress exceeds the fibre strength at the fibre/matrix interface. The model also has to account for the fibre pull-out process from the matrix owing to the fact that the biaxial state of stress increases or decreases the friction at the fibre/matrix interface depending on the effective shear component. The model used in this investigation for the simulation of load/displacement curves and for the determination of the specific fracture energy, G,, of fibre-reinforced concrete (FRC) is the model of Li and co-workers.24-27 This model deals with the fracture behaviour of brittle cement-bound materials that are reinforced by short fibres with random distribution under uniaxial tensile load. In the present paper the model is extended for the following cases: l
l
for the biaxial loading (compression + tension); for the splitting test on cubic specimens.
The drop in fracture energy with an increasing biaxial compressive loading of the materials, which has been determined experimentally in an investigation of the fracture behaviour of polypropylene FRC,28 can be exclusively simulated in this model by a decrease in fibre pull-out events or the increase in fibre fractures, respectively. Mechanisms (such as fibre/matrix debonding, matrix spalling, formation of fibre bundles, multiple cracking, etc.) that have been studied previously’,28 are also responsible for the increase or decrease in fracture energy under biaxial loading of the fibre-reinforced composites with an increasing compressive load. They cannot be taken into
Fracture behaviour of FRC: Model&g
and computer simulation
949
account in the extended model of Li and coworkers.24-27 This model does not take account of the elastic and plastic deformation behaviour of matrix and fibres. The fractured surface is considered to be a plane and the matrix is assumed to be ideally brittle. The fibres are not resistant to bending. Under these model assumptions the fracture behaviour of polypropylene FRC is simulated as described below. 2 MODEL
DESCRIPTION
2.1 Individual
fibre
A fibre of length Lf is considered, which is at an angle 8 to the fractured surface (Fig. 1) and is anchored in the matrix with an embedded length E (see Fig. 2). The fibre is loaded vertically to the fractured surface with a force P(k = s/Lf = 0). No fibre/matrix interface debonding has occurred and the fibre-end slippage, S, normalized to the fibre length, L,, is zero. The force P’(k = 0), which acts immediately on the fibre within the matrix, yields the following result:3”
Fig. 2. Force P(s/LJ, acting on a single fibre, the specimen being subjected to an additional compressive stress ((TV= compressive stress).
calculated as follows: P(k) = 2mf(l - s)z(k,q)ef” (3) It can be assumed that the transverse strain, r, depends on s/L, and (T, in the following manner (Taylor expansion at k = 0 and truncated after the square term): z(k,aI) = r(O)(l + bIcTIcosO)(l + a,k + a,k’)
P’(k = 0) = P(k = O)epfe
O-7 Sf 5 O-9
(1)
The factor S (snubbing factor) in the exponential function roughly corresponds to the effect of friction of a rope around a wheel and allows for the so-called snubbing effect.30 This coefficient was first introduced and determined experimentally by Tschegg et a1.,29the range of values mentioned above were determined for Nylon and polypropylene fibres.25 From a certain threshold value, P(k = 0) = PO, onward the fibre starts slipping from the matrix anchorage, s (see Fig. 3). This threshold value is proportional to the surface area of the fibre and to the maximum possible transverse strain, z(k = 0, u,) at the matrix/fibre interface, z(k = 0, (7,) depending on the additional compressive stress, v,, under biaxial loading. The following expression applies: PO= 27crflefez(0,a,) The force P(k)
required
Fig. 1. Fibre distribution
(2)
for further
in spherical
extraction
co-ordinates.
(4)
~(O,(T,) = r(O)(l + b,a,cose) (5) The shear stress or friction stress, z, along the fibre/matrix interface increases in a linear manner with an increasing u, load and depends on the angle 8 and on the shape factor of the fibre cross-section. Because of the (TVload, compression of the cement matrix occurs next to the fibre/matrix interface. Pinchin and Tabor3’ investigated steel-fibre-reinforced concrete under compressive load, and they found that the matrix surrounding the fibre was compressed, thus increasing the friction stress along the fibre/matrix interface. Furthermore it was observed that the pull-out force increased in a linear manner with the inclusion of fibres in the matrix. Therefore a linear function was chosen for z(O,(~i) in order to be able to take the additional compressive load, (T,, into account. In eqns (4) and 1.5)the coefficient b, is introduced to account for the
is
Fig. 3. Geometric representation of a single fibre embedded in the matrix with the two parallel fractured surfaces.
M. Elser et al.
950 additionally applied compressive loading in wedge splitting tests: bi was estimated to be:”
biaxial
b, 9 O-3 MPaa’
(6)
This approximation results from a comparison of a constant compressive load in the radial direction on a cylinder, which is assumed to represent a fibre, with a constant plane-parallel load on a cylinder in a horizontal position. The first load results in a radial homogeneous stress distribution at the surface of the cylinder and the second load causes an elliptical stress distribution in the cross-sectional area of the cylinder. As a next step a fibre with lengths I, and 1, is investigated that bridges a crack of width d and which was originally embedded in the matrix (Fig. 3). The existence of this crack of width d implies that there is already a matrix crack. Under the assumption of a tensile load on the specimen both fractured surfaces are parallel. In order to obtain the P/d curve, the problem is first subdivided. Each fibre embedded length, 1, or 12, is analysed separately (fl(k,), i = 1,2) (Fig. 3). The resulting curves are combined, so that: P, = P2 The resulting
and
s1 fs,
force depends
2 7vfUfu
(9)
2.2 Probability density function for uniform random distribution If a zone with a width Lr/2 around the fractured surface is analysed, it contains AN = nLfALro fibres (Fig. 4). The probability that a fibre has its mid-point at a
w(Q)d2Q
= $
- LJ2 5 x 5 Lr/2
= & sinf3dedp
The condition force transmission is the following:
for this fibre to contribute between the two fractured 8 5 arccos(2x/L,l
to the surfaces (12)
This results in the following fibre probability function for uniform random distribution integration has already been carried out):
density f (cp
i_ I
(13)
0%8sarccos/2x/L,]
AN = nLfALIG =
LfAmi v, =pAm vt 7UfL, nr:
Fig. 4. Representation
of the effective zone in which the to the bridging, with the centre of the fibres surface.
(14)
2.3 Calculation of the bridging force The entire bridging force for parallel fracture surfaces (see Fig. 4), at a distance d from each other yields the following result:
” def(x,
The bridging
normal
O)P
stress, X, is: =+ LIG
x to the fractured
(10)
(11)
x(d)
being at a distance
fractured
(8)
The fibre is then considered to have been broken, and in the subsequent calculation of the bridging force, which integrates over all fibres, such fibres are not taken into account.
fibres contribute
the
The probability that the fibre points towards the solid angle element d*Q around the solid angle R = (8,(p) is:
-LJ21xSLr/2,
parameters:
Furthermore it has been assumed that the fibre tensile strength is not unlimited, but has a maximum permissible uniaxial value, a,,. If P(d,&,O) exceeds this, then =
w(x)dx = dxlL,,
dx from
(7)
p, = P(d,&, 0)
Pmax
x and
f(x, 0) = ANsin
= d
on several
distance of between surface is:
(16)
2.4 Calculation of the load/displacement curve for crack propagation caused by separation (splitting) Both in uniaxial and in biaxial wedge splitting is not separated with parallel tests2G2y the specimen fractured surfaces, as it is with an ideal tension test, but as shown in Fig. 5(a). Because of the model assumption that the specimen is rotated around the point 0 as the hinge (Fig. 5(b)), an additional force, F,, has to be assumed in the centre of motion in order to maintain force equilibrium.
Fracture behauiour of FRC: Model&g
and computer simulation
951
3 MATHEMATICAL EVALUATION OF THE MODEL PARAMETERS AND OF THE SPECIFIC FRACTURE ENERGY
Apart from an experimental determination of the Gt values for uniaxially and biaxially loaded specimens there is also the possibility of performing a simulation by means of the model introduced in the last section, which also provides Gf values. A third possibility would be a theoretical calculation of the Gi values. A shear-lag analysis carried out previously’ regarding the fibre pull-out process and the debonding supplies an optimum value for the specific fracture energy:
FRC split specimens, both halves of the specimen being rotated about point 0.
Fig. 5. Geometry and forces acting on
Therefore the crack width, d, is replaced by a linear function d(z): d(z)=CMODH
The load/displacement F,(CMOD)
= H
Llcl
:,+,
(17)
g=+(l+e@‘)
curve is calculated by: ‘;I
+ e
LIG
(3 2.5 Calculation
of specific fracture energy, Gf
The fracture energy resulting from integration of the load/displacement curve per unit of the fractured surface is given by: CMoD,,,
1 GJmode’)=
BL,GHL,G
where 7 = z(k,a,) is the friction stress and shear stress, respectively, and G& is the value for the specific fracture energy that is exclusively consumed by the fibre pull-out process, under the assumption that there is no fibre fracture (in case of a fibre fracture Gf would be considerably reduced). Equation (20) contains the snubbing coefficient, g, which has also been derived in elsewhere2 and which can be represented as a function of the snubbing factor, f:
Io
F,(CMOD)d(CMOD) (19)
For a parallel movement of both halves of the specimen away from each other, the horizontal force, FH, for CMOD = Lf (see Fig. 5(a) and (b)) would become zero, since all fibres with a fibre length L, have been pulled out. Under the assumption of rotation of both halves of the specimen around the 0 point (Fig. 5(a)) even in an extreme case of a rotation angle of 180” not all fibres would be pulled out of the matrix, namely those that are close to the centre of motion. This results in an asymptotic tapering off of the modelled load/displacement curve and requires a maximum integral limit (CMOD,,,) when calculating eqn (19).
(21)
For values of f between 0 and 1, g varies between 1 and 2.32. The snubbing factor, f, depends on the matrix and fibre materials and has been found experimentally7~3z to be approximately O-7 for polyprolyene fibres in a cement matrix.* Equation (20) only applies to aligned-fibre composites and to constant transverse strain, 7. However, no constant transverse strain, 7, is used in the model under consideration, but a linear or square dependence of 7is used instead, and eqns (4) and (5) cannot be directly inserted into eqn (19). Instead of the constant transverse strain, 7, a constant transverse strain with the average taken over the fibre distribution, 7mid(average taken of all solid angles and of all embedded lengths of the fibres, as well as of all transverse strain distributions) can be introduced into eqn (20), which can be determined in the following way:
i’”
dx
(“”f (x, 0)d0
(22)
rrnaX dk
with the following integral limits applying: 8 max
=
arccos)2x/Lf[;
mmax = i
=k- & f
f
(23)
M. Elser et al.
952 If the formulation
is used for the transverse (eqns (4) and (5)), eqn (22) results in a mean transverse strain: Tmid=T(0)
(
l+?+Z
1
The constant transverse average taken of the uniform in the matrix mainly depends and a2 of Taylor’s series (eqns the initial value, r(O), as well load, (TV,and it is independent (8,(p) of the fibres. Equation for the theoretical calculation
strain. r constant
(l+$,a,)
(24)
strain, rmid, with the random-oriented fibres on the coefficients a, (4) and (5)) and also on as on the compressive of the relative position (24) (rmid) can be used of qp’ in eqn (20).
4 COMPUTER SIMULATION OF THE LOAD/DISPLACEMENT CURVES and biaxial load/ In a previous studyz8 uniaxial displacement curves of polypropylene-fibre-reinforced concrete specimens were described. Four test series with different fibre contents (Vf = 0.001 and O+OOS)and
4500
“C 1 I
*H
[
CO...-.i+.,
,\
Polypropylene
Lenend:
Fibers:
fibre lengths (Lr = 10 and 20 mm) were investigated. load/displacement curves of unreinFurthermore, forced concrete have been determined experias a mentally,2y and these shall be used subsequently reference. In Fig. 6(a)-(d) the measured load/displacement curves are represented for the test series PF520 (polypropylene-fibre-reinforced concrete, V, = 0.005, Lf = 20 mm)-the composite2’-as well as for the test series NG(wet) (concrete containing natural gravel, which was stored in water for 28 days)‘“-the matrix. Each curve is the average of 3-4 experimentally obtained load/displacement curves for the dimensionless compressive stress ratio cTi/fc of 0, 15, 30 and 50% where cr, is the compressive stress and fc is the compressive strength. Since this model only considers the fibres in the fracture process, only the contribution of the fibres to simulated experimentally determined the and load/displacement curves has to considered, as well as their contribution to the total fracture. The fibre contribution to the total fracture energy, Gyhrc), of the
4500 ,
F”
.
m
i I+
I
I
N
Model Paramelers:
\
0.
0
i
i
a,.ZQQQ t= 475
model \ /------
3
crack mouth
4
z+
$
7
opening displacement
a,=-540 bl= Q,I
a
9
n -0
10
1
CMOD [mm]
2
3
4
5
7
6
6
crack mouth opening displacement
9
10
CMOD [mm]
(b)
(4 Polypropylene
Fibers:
L,=ZOmm
df=15pm
“f=o,5%
f,
1
I
FH
i :
1
F
matrix
L
[NJ
4500
=24,3MPa
I..
I
1
Model Parameters:
3000
biaxial
q&
= 50%
0
0 0
1
2
3
crack mouth
4
5
6
7
opening displacement (c)
6
9
CMOD [mm]
10
0
12
3
crack mouth
4
5
6
7
opening displacement
6
9
10
CMOD [mm]
(4
Fig. 6. Load/displacement curves of the composite and of the matrix, and the fibre contribution for the composite as a result of subtracting the latter from the former (shaded area). (a) Simulation of the uniaxial (a,/_& = 0%) part of fibres of the load/displacement curves of the composite; (b) simulation of the biaxial (a,/fc = 15%) part of fibres of the load/displacement curves of the composite; (c) simulation of the biaxial (a,/fc = 30%) part of fibres of the load/displacement curves of the composite; (d) simulation of the biaxial (CT,/& = 50%) part of fibres of the load/displacement curves of the composite.
Fracture behaviour of FRC: Modelling and computer simulation Table 1.
Normalized compressive stress, a,/fc (%)
953
Specific fracture energy values from experiments, the computer simulation and the calculated for various compressive loads, (rI, and coefficients, b,
Measured
Computer
c;(FpF5*) C;(NGwet) (k/m) (NW
Gmode’) (N/m)
(&a)
simulation
evaluation
b,opt)
Calculated
Fibre rupture W)
G$‘““)
Gf(oP’)
%lid (MW
(N/m)
5
x
cp’)
(N/m)
0
1069.6
113.8
955.8
905.6
0.00
54.3
1981.6
0.7771
388.7
1943.5
1.5
424.1
686
355.5
708.0 596.4 509.3
O-10 0.15 0.20
59.0 61.0 62.6
1726.8 1529.3 1361.8
0.9659 1.0603 1.1547
312.7 284.9 261.6
1563.6 1424.4 1307.9
30
411.2
78.0
333.2
509.3 398.3 320.5
0.10 0.15 0.20
62.6 64.9 67.5
1361.8 1134.8 986.2
1.1547 1.3436 1.5354
261.6 224.8 197.1
1307.9 1124.1 985.5
50
472.8
89.2
383-6
375.6 280.8 221.1
O-10 0.15 0.20
65.3 69.0 71.6
1082-4 905.8 778.5
1.4065 1.7212 2.0360
214.8 175.5 148.3
1073.8 877.5 741.8
-
composite is obtained by subtracting the matrix component, GpGcwet)(matrix), from the fracture energy of the composite, GfPF520(composite). Table 1 contains these three experimentally determined fracture energy values and a simplified correlation can be represented as follows: GfPF520(composite)
- GpG6”e’)(matrix)
= Gifibre)
(25)
Since the area under a measured load/displacement curve represents the fracture energy, the fibre component (shaded area in Fig. 6(a)-(d)) in the load/displacement curve of the composite has to be
determined by subtracting the energy for the matrix (NG(wet)) from the load/displacement curve of the composite PF520 (as in eqn (25)). By means of the model described above, the fibre contribution to the load/displacement curve of the composite can be simulated. Thus the dependence of the fracture energy, Gf, of the fibres (Fig. 7(a)) as well as of the normalized fracture energy Gr/GfO of the fibres (Fig. 7(b)) on the compressive load, crl, of the specimens can be calculated from the simulated load/displacement curves. The fibre component of the Gt values can be obtained by subtracting the matrix fracture energy values from the Gf values of the composite material compound. Suitable initial values of the model parameters (z(0),a,,a2,b, and f) were obtained in the following way. (a) Modelling of the uniaxially determined fibre share of the uniaxially measured load/displacement curves (gI = 0): . l
put G$--) =
@,oPf;
from eqn (19) for qpt, the value of Zmidcan be determined by means of an assumed snubbing factor, f (thus also for the snubbing coefficient 8);
0
io
zb
So
normalized compressive stress ollfc [%I (4
“I
so
0 normalized compressive stress q/f,
90 loo
[%I
@I
versus u,/fc curves for the composite and the matrix, and the fibre component for the composite obtained by subtracting the latter from the former (shaded area). (a) Simulation of the Gf versus (r,/f= curves of the fibre component in the composite as well as the different model curves for b,; (b) simulation of the G,/G, versus ul/fc curves of the fibre component in the composite as well as the different curves for bl, and the corresponding fraction in the fibre rupture events. Fig. 7. G,
M. Elser et al.
954
l
.
.
from Zmid (eqn (23)) a value for a, is calculated for a2 = 0, assuming r(0) in a first step; after a first modelling of the uniaxial fibre part, both a2 and a, as well as r(O) can be determined more precisely by means of a better fit of the model curve to the measured curve; the compressive coefficient, b, (eqns (4) and (5)) has to be considered for non-uniaxial loading only, since under uniaxial loading the compressive loading of the specimen is zero.
(b) Modelling of the biaxially of the biaxially components displacement curves (pi > 0): .
.
.
determined measured
fibre load/
the best approximation of the calculated uniaxial curve to the measured uniaxial load/displacement curves can be obtained by using the model parameters for the uniaxial fibre component (Fig. 6(a)); the biaxial fibre component can now be simulated by varying compressive the coefficient, b, (Fig. 6(b)-(d)); at the same time the fracture energy, G,, as well as the normalized fracture energy, GJGr,,, is simulated for every value of b1 for increasing compressive load, vl, on the samples and is represented for comparison purposes with the experimentally determined curves of fibre fracture energies (Fig. 7(a) and (b)).
5 DISCUSSION When using identical model parameters in the range a,lfo = 1550%, there is qualitative and partly quantitative agreement between the mean load/displacement curves represented in Fig. 6(a)-(d) and the simulated curves and also between the experimentally determined and also the simulated Gf versus crl/fC curves (Fig. 7(a) and (b)). The deviation arises from the delamination process described previously,2x which is mainly responsible for the rapid decrease of the Gr value with increasing compressive load, cl/fC, as well as other mechanisms (fibre/matrix debonding, matrix spalling, formation of fibre bundles, multiple cracking, etc.) as described elsewhere,5%2y that are not taken into consideration. The increase in fibre fractures, together with the increase in average friction stress or transverse strain Z,i, (Fig. S), with increasing compressive load, (TV/&., results in a decrease in fracture energy when applying this model. Figure 8 represents the transverse strain, r(s/L,, gl), (eqns (4) and (5)) as a function of the normalized fibre-end slippage, s/L,, for a fibre that is embedded with I = LJ2 in the matrix and is normal to the fracture surface. The calculated rmid values mainly lie in the range of
tMPa1
0
0.1
0.2
0.3
0.5
0.4
normalized fiber-end slippage s/Lf
Fig. 8. Shear stress ~(s/L,,u~) along a fibre that is normal to the fractured surface embedded in the matrix with L,/2, for various compressive loads, (T,, and compressive coefficients b,.
the values determined experimentally by means of pull-out tests on individual fibres.“.30,33 As can be seen from eqn (20), the Gr value is indirectly proportional to the average transverse strain, Zmid (z = rmid). Equation (19) supplies considerably lower values for qp’ (calculated Gf values in Table 1) compared to the measured Gjfibrc) values and the simulated G$m”de’) values. A Gr’“OX/“)value can be calculated from the Gimode’) value, the former resulting from the simplifying assumption that all fibres, also those that have been broken, contribute to the fracture energy of the fibres:
,f$oo”)
=
,$mdel)
100 100 - fibre rupture
(%)
(26)
It is only this value which basically supplies a good correspondence with the’five-times higher Gip’ value (see Table l), the Ggmodel) value also being derived under the assumption that there is no fibre fracture. The simulated model curves represented in Fig. 6(a)-(d) (load/displacement curves) show a special characteristic with regard to the position of the load peaks. These peaks all lie at CMOD values of 2 mm. The increase in the experimentally determined fibre energies is determined by the deformation behaviour of the matrix, of the fibres and of the composite. Since the elastic deformation of the matrix and of the fibres has not been taken into account in the model, the invariable position of the load peaks mentioned above and the increase in the model area curves are exclusively caused by the force relationship taken into consideration at the fibre/matrix interface. In the evaluation (eqn (ll)), an overall homogeneous stress distribution at the cylinder surface and a fibre/matrix interface thickness of zero were assumed. A uniform homogeneous matrix was also assumed, so that no difference can be discerned within the matrix
Fracture behaviour of FRC: Mode&g material immediately surrounding the fibre. This is not the case in reality, however, since the matrix material surrounding the fibre is heterogeneously distributed, owing to the existence of aggregate grains and it therefore does not surround the fibre completely, so that the stress resulting from the compressive loading applies in a rather concentrated manner at the surface of the fibre. All these facts reduce the estimated value of b, (and therefore the < sign is introduced in eqn (11)). The simulation does in fact show that this parameter is smaller than the estimated value that is required in order to model the experimentally determined load/displacement curves and the Gr versus allfc dependence. The coefficient b, can be interpreted as a qualitative measure for the degree of compression of the matrix surrounding the fibre under compressive loading, i.e. it represents a measure for the approximation to a homogeneous stress distribution that applies overall at the fibre/matrix interface. The shape factor of the fibre cross-section and the deviation from the circular section of the fibre also influence the value of bI.
and computer simulation 4.
5.
955
The mean shear stress, rmid, increases with increasing u1 /fc values (increasing compressive load). This results in lower specific fracture energies of the simulated Gp’ values, which is in accordance with the experimental results. The GglOo%) value results agree with the simulated values, G@‘odel), if all fibres are subjected to a pull-out process, even those that would suffer a fibre fracture. Only the five times a good corresponhigher Gp’ values provides dence with the G$looo’) value. qp’ was computed under the assumption that there is no fibre fracture.
ACKNOWLEDGEMENTS This study was financed by the National Bank of Austria (osterreichische Nationalbank) together with the Austrian Science Foundation (Fonds zur Forderung der Wissenschaftlichen Forschung) under project number P8885-TEC.
REFERENCES 6 CONCLUSIONS The model proposed by Li and co-workersI is extended with regard to a biaxial state of stress (compression + tension) in splitting a specimen made of FRC. Simulations are performed under various assumptions and simplifications on the fracture behaviour of brittle materials that are reinformced by means of short, uniform, randomly oriented fibres, after the occurrence of a macro-matrix crack. 1.
2.
3.
With this model the fibre contribution to the load/displacement curve of the composite can be simutlated for both uniaxial and biaxial loading. The load peaks in the measured and simulated curves are shifted towards higher CMOD values for all loads. The experimentally determined and simulated peak values coincide for uniaxial (Fig. 6(b)) conditions with aI/fc (30 and 50%) quite well, while they differ for rl/fc (15%). When using identical model parameters a semi-qualitative agreement between the experimentally determined and the simulated Gf versus aI/fc curves (Fig. 7(a) and (b)) can also be achieved by means of a variation of parameter b 1. The reason that experimental and simulated behaviour (as mentioned in 1 and 2), coincide only partly, is that delamination occurs during compressive loading of the material and this fact is not taken into account in this model.
1. Gaa, Y. C., Mai, Y. W. & Coterell, B., Fracture of fiber reinforced materials. ZAMP, 39 (1988) 550-572.
2. Kim, J. K., Baillie, C. & Mai, Y. W., Interfacial debonding and fiber pull-out stresses: I. Critical comparison of existing theories with experiments. 1. Mater. Sci., 27 (1991) 3143-3154.
3. Greszczuk, L. B., Theoretical studies of the mechanics of fiber matrix interface in composites. ASTM STP 452, 1968, pp. 42-58. V. S. & Shah, S., Tensile fracture of 4. Gopalaratnam, steel fiber reinforced concrete. J. Engng Mech. Div., AXE,
113 (1987) 635-652. R. J., Analysis of the effect of embedded fiber length on fiber and pull-out from an elastic matrix: I. Review of theories . J. Mater. Sci., 19 (1984) 861-870. fiber 6. Gray, R. J., Analysis of the effect of embedded length on fiber debonding and pull-out from an elastic matrix: II. Application to a steel fiber-cementitious matrix composite system. J. Mater. Sci., 19 (1984) 1680-1691. 7. Laws, V., Langley, A. A. & West, J. M., The glass fiber/cement bond. J. Mater. Sci., 21 (1986) 289-296. 8. Li, S. H., Shah, S., Li, Z. & Maura, T., Micromechanical analysis of multiple fracture and evaluation of debonding behaviour for fiber reinforced composites. ht. J. Solids Struct., 30 (1993) 1429-1459. _. 9. Morrison. J. K., Shah, S. & Jeng, Y. S., Analysis of tlber debonding and pullout in composites. J. Eigng Mech., 114 (1988) 277-294. 10. Atkinson, C., The rod pull-out problem, theory and experiment. J. Mech. Phys. Solids, 30 (1982) 97-120. of fiber reinforced 11. Stang, H. & Shah, S., Failure composites by pull-out fracture. J. Mater. Sci., 21 (1985) 953-957. 12. Stang, H. & Shah, S., Fracture mechanics interpretation of fiber/matrix debonding process in cementitious composites. Proc. Int. Conf: on Fracture Mechanics of Concrete, ed. F. H. Wittman. Elsevier, Amsterdam, 1985, pp. 339-349.
5. Gray,
M. Elser et al. 13. Kim, J. K., Zhou, L. M. & Mai, Y. W., Interfacial debonding and fiber pull-out stresses: II. A new model based on fracture mechanics approach. J. Mater. Sci., 27 (1992) 3155-3166. 14. Kim, J. K., Zhou, L. M. & Mai, Y. W., Interfacial debonding and fiber pull-out stresses: 111. Interfacial properties of cement matrix composites. J. Muter. Sci., 28 (1993) 3923-3930. 15. Hsueh, C. H., Muter. Sci. Engng, Al23 (1990) 1. 16. Hsueh, C. H., Interfacional debonding and fiber pull-out stresses of fiber reinforced composites: II. Non-constant interfacial bond strength. Mater. Sci. Engng, A125 (1990) 67-73. 17. Hsueh, C. H., Interfacional debonding and fiber pull-out stresses of fiber reinforced composites: VII. Improved analyses for bonded interfaces. Mater. Sci. Engng, A154 (1992) 125-132. 18. Hsueh, C. H., Interfacional debonding and fiber pull-out stresses of fiber reinforced composites: VIII. The energy-based debonding criterion. Mater. Sci. Engng, Al59 (1992) 65-72. 19. Hsueh, C. H., An asymptotic approach for debonding at the fiber-matrix interface. Mater. Sci. Engng, A174 (1994) L17-L20. 20. Hart-Smith, L. J., Role of biaxial stresses in discrimiating between meaningful and illusory composite failure theories. Composite Structures, 25 (1993) 3-20. 21. Makinde, A., Neale, K. W. & Sacharuk, Z., A strain-based biaxial failure criterion for fiber-reinforced composites. Polym. Camp., 13 (1992) 263-272. 22. Caiazzo A. A., Material model for the analysis of multiaxially continuous fiber composites. Winter Annual Meeting of the American Society of Mechanical Engineers: Enhancing Analysis Techniques for Composite Materials, Vol. 10. ASME, NDE, New York, 1991, pp. 267-277. 23. Mallik K., Failure of Laminated Composites in Biaxial Stress Fields: A Review, Vol. 146. American Society of
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
Mechanical Engineers, Pressure Vessels and Piping Division, New York, 1988, pp. 205-212. Li, V. C., Wang, Y. & Backers, S., A micromechanical mode1 of tension-softening and bridging toughening of short random fiber reinforced brittle matrix composites. J. Mech. Phys. Solids, 39 (1991) 607-625. Li, V. C., Postcrack scaling relations for fiber reinforced cementitious composites. J. Mater. Civil Engng, ASCE, 4 (1992) 41-57. Li, V. C. & Leung, C. K. Y. Steady state multiple cracking of short random fiber composites. J. Engng Mech. ASCE, 118 (1992) 2246-2264. Li, V. C. & Leung, C. K. Y., Tensile failure modes of random discontinuous fiber reinforced brittle matrix composites. In Fracture Processes in Concrete, Rock and Ceramics, ed. J. G. M. van Mier, J. G. Rots & A. Bakker. RILEM, 1991, pp. 285-294. Elser, M., Tschegg, E. K. & Stanzl-Tschegg, S. E., Biaxial mode I fracture behaviour of polypropylenefibre-reinforced concrete: Experimental investigation. Comp. Sci. Technol., 56 (1996) 933-945. Tschegg, E. K., Elser, M. & Tschegg-Stanzl, S. E., Biaxial fracture tests on concrete, development and experience. J. Cement Concrete Camp., 17 (1995) 57-75. Li, V. C., Wang, Y. & Backers, S., Effect of inclining angle, bundling, and surface treatment on synthetic fiber pullout from a cement matrix. Composites, 21, pp. 132-140. Pinchin. D. J. & Tabor, D., Interfacional contact pressure and frictional stress transfer in steel fiber cement. Testing and Test Methods of Fiber Cement Composites, RILEM Symposium. The Construction Press, Lancaster, PA, 1978, pp. 337-344. Leung, C. K. Y. & Li, V. C., Effects of inclination crack bridging stresses in fiber reinforced brittle matrix composites. J. Mech. Phys. Solids, 40 (1992). Leung, C. K. Y. 8r Li, V. C., New strength based model for the debonding of discontinuous fiber in an elastic matrix. J. Mater. Sci., 26 (1991) 5996-6010.
Science and Technology 56 (1996) 947-956 0 1996 Elsevier Scknce Limited Printed in Northern Ireland. All rights reserved
ELSEVIER
PII:
SO266.353?3(95)00058-9
0266-3538/96/$15.00
FRACTURE BEHAVIOUR OF POLYPROPYLENE-FIBREREINFORCED CONCRETE: MODELLING AND COMPUTER SIMULATION
M. Elser,” E. K. Tschegg,“* N. Finger” & S. E. Stanzl-Tscheggb “Institute of Applied and Technical Physics, Technical University of Vienna, A-1040 Vienna, Austria “Institute of Meterology and Physics, University of Agriculture, Vienna, Austria
(Received
5 February
1996; accepted
Abstract The proportion of fracture energy which is consumed by jibres during fracturing in uniaxial and biaxial loading of a composite (matrix plus fibre) can be calculated by means of a modified model. By the method of variation of parameters, the calculated load/displacement curve is adapted to the measured load/displacement curve. The jibre component of the measured load/displacement curves is obtained by subtracting the experimentally determined matrix component from the measured load/displacement curve of the composite material. By these parameters it is possible both to represent the load/displacement curve of every single fibre for each relative position, and to determine the change of the fracture energy with increasing compressive loading, (TV, of the specimens. 0 1996 Elsevier Science Limited
19 March
GIG, @‘Pt) @F‘520
f NGwet zibre
Gjmodel)
Keywords:
polypropylene-fibre-reinforced concrete, biaxial fracture behaviour, specific fracture energy
c;(F1(=)
NOTATION
H LIG
a aI, a2 A LIG a+ e b, B LIG CMO&m
d d(z) df e
Distance between force application and surface of the specimen Taylor expansion coefficients Ligament area Distance between point of attachment and notch root Coefficient when taking compressive load into account Ligament thickness of the splitting specimens Cutting parameters of the FH versus
k = s/L, 1 L, m = l/L, m max n AN
CMOD curve
P, P’
Crack Crack Fibre Notch
P,,,(d)
width for parallel fracture surfaces width for splitting specimens diameter depth of the splitting specimen
rf S Vf
* To whom correspondence
should
be addressed.
Vi
947
1996)
Fibre elastic modulus Matrix elastic modulus Snubbing factor Compression strength of cube Distribution function of the fibres Horizontal splitting force Snubbing coefficient Specific fracture energy of the uniaxial fracture test Specific fracture energy of the biaxial fracture test Normalised specific fracture energy Optimum specific fracture energy Fracture energy of composite Fracture energy of matrix Fracture energy of fibre component in the composite Fracture energy of fibre component in the model curve Fracture energy of fibre component in the case where there is no fibre rupture Ligament height of the splitting specimens Normalized fibre end slippage Embedded length of the fibre in the matrix Fibre length Normalised embedded length Integral limit for calculation of the mean friction stress or shear stress, respectively Number of fibres per unit volume Number of fibres in a zone width Lf around the fracture surface Forces on a single fibre Total bridging force for parallel fractured surfaces Fibre radius Fibre end slippage Fibre volume fraction Matrix volume fraction
M. Elser et al.
948 w(x) w (R)d2Q
x Z
Zrnid
Probability that centre of the fibre lies between x and dx Probability that the fibre points towards the solid angle element d2R around solid angle R Axis for position of the centre of the fibre Axis of symmetry of the splitting specimens Spherical co-ordinates Fibre tensile strength Normalised compressive stress Compressive stress Bridging normal stress stress shear Friction and stress, respectively Mean shear stress
1 INTRODUCTION The fracture energy that is absorbed during crack propagation in fibre-reinforced materials is provided mainly by the following mechanisms: matrix cracking, fibre rupture, fibre/matrix interface debonding and fibre pull-out. There have been numerous studies regarding theoretical models that deal with the fibre pull-out process and the fibrelmatrix interface debonding process in various fibre-reinforced materials. Some of these studies are mentioned below. The theoretical analysis of both the fibre/matrix interface debonding process and the fibre pull-out problem’32 are essentially based on two generally different models. l
l
Model I. The maximum shear stress criterion for fibre/matrix interface debonding (shear-lag model): if the shear stress exceeds the shear strength of the fibre/matrix interface, the result along the is unstable crack propagation fibre/matrix interface.‘-’ Model II. This model is based on the concept of fracture mechanics, where the debonded zone is regarded as a crack in the fibre/matrix interface, a so-called interfacial crack. The propagation of this crack depends on an energy criterion that has to be satisfied.2-‘2
Several studies2,13,14 have tried to combine these two models into one. Two models, namely those of Gao et al.,’ which is based on model I, and of Hsueh,‘5-1y which is based on model II, are investigated. Whereas the model of Gao predicts the maximum debond composite stress for long fibres, Lf, in epoxy-matrix systems very precisely, the model of Hsueh is more suitable for experiments with short fibres. For ceramic-matrix composites both models provide results that are identical with those of experiments for a large range of fibre lengths, Lf.2*‘3,14
All these models deal with fibre-reinforced materials, mostly cement-bound materials under uniaxial tensile load. Models regarding the biaxial loading of fibre-reinforced materials, such as thermoplastic material compounds, E-glass/epoxy laminates, fibrereinforced aluminium compounds, orthotropic fibrereinfoced laminates, multiaxial fibre-reinforced metal compounds, etc., are discussed below. reported on uniaxial and biaxial Hart-Smith2” frcture criteria of reinforced-plastic laminates. Makinde et a1.2’ deduced a biaxial strength criterion for orthotropic fibre-reinforced laminates under biaxial loading by analysing the elongations in the material Two material (strain-based strength criterion). provide an analysis of multiaxial fibremodels2* reinforced metal compounds by means of a finite element simulation. Mallik23 reports on different models, with regard to thin-walled pressure vessels consisting of fibre-reinforced material compounds, where the internal compressive load in the pressure vessel creates a biaxial stress field in the wall material. The simulations of load/displacement curves that have been determined by means of experiments on fibre-reinforced biaxially loaded concrete specimens, require a model that accounts for both the random distribution of fibres and the fibre fracture that occurs when the shear stress exceeds the fibre strength at the fibre/matrix interface. The model also has to account for the fibre pull-out process from the matrix owing to the fact that the biaxial state of stress increases or decreases the friction at the fibre/matrix interface depending on the effective shear component. The model used in this investigation for the simulation of load/displacement curves and for the determination of the specific fracture energy, G,, of fibre-reinforced concrete (FRC) is the model of Li and co-workers.24-27 This model deals with the fracture behaviour of brittle cement-bound materials that are reinforced by short fibres with random distribution under uniaxial tensile load. In the present paper the model is extended for the following cases: l
l
for the biaxial loading (compression + tension); for the splitting test on cubic specimens.
The drop in fracture energy with an increasing biaxial compressive loading of the materials, which has been determined experimentally in an investigation of the fracture behaviour of polypropylene FRC,28 can be exclusively simulated in this model by a decrease in fibre pull-out events or the increase in fibre fractures, respectively. Mechanisms (such as fibre/matrix debonding, matrix spalling, formation of fibre bundles, multiple cracking, etc.) that have been studied previously’,28 are also responsible for the increase or decrease in fracture energy under biaxial loading of the fibre-reinforced composites with an increasing compressive load. They cannot be taken into
Fracture behaviour of FRC: Model&g
and computer simulation
949
account in the extended model of Li and coworkers.24-27 This model does not take account of the elastic and plastic deformation behaviour of matrix and fibres. The fractured surface is considered to be a plane and the matrix is assumed to be ideally brittle. The fibres are not resistant to bending. Under these model assumptions the fracture behaviour of polypropylene FRC is simulated as described below. 2 MODEL
DESCRIPTION
2.1 Individual
fibre
A fibre of length Lf is considered, which is at an angle 8 to the fractured surface (Fig. 1) and is anchored in the matrix with an embedded length E (see Fig. 2). The fibre is loaded vertically to the fractured surface with a force P(k = s/Lf = 0). No fibre/matrix interface debonding has occurred and the fibre-end slippage, S, normalized to the fibre length, L,, is zero. The force P’(k = 0), which acts immediately on the fibre within the matrix, yields the following result:3”
Fig. 2. Force P(s/LJ, acting on a single fibre, the specimen being subjected to an additional compressive stress ((TV= compressive stress).
calculated as follows: P(k) = 2mf(l - s)z(k,q)ef” (3) It can be assumed that the transverse strain, r, depends on s/L, and (T, in the following manner (Taylor expansion at k = 0 and truncated after the square term): z(k,aI) = r(O)(l + bIcTIcosO)(l + a,k + a,k’)
P’(k = 0) = P(k = O)epfe
O-7 Sf 5 O-9
(1)
The factor S (snubbing factor) in the exponential function roughly corresponds to the effect of friction of a rope around a wheel and allows for the so-called snubbing effect.30 This coefficient was first introduced and determined experimentally by Tschegg et a1.,29the range of values mentioned above were determined for Nylon and polypropylene fibres.25 From a certain threshold value, P(k = 0) = PO, onward the fibre starts slipping from the matrix anchorage, s (see Fig. 3). This threshold value is proportional to the surface area of the fibre and to the maximum possible transverse strain, z(k = 0, u,) at the matrix/fibre interface, z(k = 0, (7,) depending on the additional compressive stress, v,, under biaxial loading. The following expression applies: PO= 27crflefez(0,a,) The force P(k)
required
Fig. 1. Fibre distribution
(2)
for further
in spherical
extraction
co-ordinates.
(4)
~(O,(T,) = r(O)(l + b,a,cose) (5) The shear stress or friction stress, z, along the fibre/matrix interface increases in a linear manner with an increasing u, load and depends on the angle 8 and on the shape factor of the fibre cross-section. Because of the (TVload, compression of the cement matrix occurs next to the fibre/matrix interface. Pinchin and Tabor3’ investigated steel-fibre-reinforced concrete under compressive load, and they found that the matrix surrounding the fibre was compressed, thus increasing the friction stress along the fibre/matrix interface. Furthermore it was observed that the pull-out force increased in a linear manner with the inclusion of fibres in the matrix. Therefore a linear function was chosen for z(O,(~i) in order to be able to take the additional compressive load, (T,, into account. In eqns (4) and 1.5)the coefficient b, is introduced to account for the
is
Fig. 3. Geometric representation of a single fibre embedded in the matrix with the two parallel fractured surfaces.
M. Elser et al.
950 additionally applied compressive loading in wedge splitting tests: bi was estimated to be:”
biaxial
b, 9 O-3 MPaa’
(6)
This approximation results from a comparison of a constant compressive load in the radial direction on a cylinder, which is assumed to represent a fibre, with a constant plane-parallel load on a cylinder in a horizontal position. The first load results in a radial homogeneous stress distribution at the surface of the cylinder and the second load causes an elliptical stress distribution in the cross-sectional area of the cylinder. As a next step a fibre with lengths I, and 1, is investigated that bridges a crack of width d and which was originally embedded in the matrix (Fig. 3). The existence of this crack of width d implies that there is already a matrix crack. Under the assumption of a tensile load on the specimen both fractured surfaces are parallel. In order to obtain the P/d curve, the problem is first subdivided. Each fibre embedded length, 1, or 12, is analysed separately (fl(k,), i = 1,2) (Fig. 3). The resulting curves are combined, so that: P, = P2 The resulting
and
s1 fs,
force depends
2 7vfUfu
(9)
2.2 Probability density function for uniform random distribution If a zone with a width Lr/2 around the fractured surface is analysed, it contains AN = nLfALro fibres (Fig. 4). The probability that a fibre has its mid-point at a
w(Q)d2Q
= $
- LJ2 5 x 5 Lr/2
= & sinf3dedp
The condition force transmission is the following:
for this fibre to contribute between the two fractured 8 5 arccos(2x/L,l
to the surfaces (12)
This results in the following fibre probability function for uniform random distribution integration has already been carried out):
density f (cp
i_ I
(13)
0%8sarccos/2x/L,]
AN = nLfALIG =
LfAmi v, =pAm vt 7UfL, nr:
Fig. 4. Representation
of the effective zone in which the to the bridging, with the centre of the fibres surface.
(14)
2.3 Calculation of the bridging force The entire bridging force for parallel fracture surfaces (see Fig. 4), at a distance d from each other yields the following result:
” def(x,
The bridging
normal
O)P
stress, X, is: =+ LIG
x to the fractured
(10)
(11)
x(d)
being at a distance
fractured
(8)
The fibre is then considered to have been broken, and in the subsequent calculation of the bridging force, which integrates over all fibres, such fibres are not taken into account.
fibres contribute
the
The probability that the fibre points towards the solid angle element d*Q around the solid angle R = (8,(p) is:
-LJ21xSLr/2,
parameters:
Furthermore it has been assumed that the fibre tensile strength is not unlimited, but has a maximum permissible uniaxial value, a,,. If P(d,&,O) exceeds this, then =
w(x)dx = dxlL,,
dx from
(7)
p, = P(d,&, 0)
Pmax
x and
f(x, 0) = ANsin
= d
on several
distance of between surface is:
(16)
2.4 Calculation of the load/displacement curve for crack propagation caused by separation (splitting) Both in uniaxial and in biaxial wedge splitting is not separated with parallel tests2G2y the specimen fractured surfaces, as it is with an ideal tension test, but as shown in Fig. 5(a). Because of the model assumption that the specimen is rotated around the point 0 as the hinge (Fig. 5(b)), an additional force, F,, has to be assumed in the centre of motion in order to maintain force equilibrium.
Fracture behauiour of FRC: Model&g
and computer simulation
951
3 MATHEMATICAL EVALUATION OF THE MODEL PARAMETERS AND OF THE SPECIFIC FRACTURE ENERGY
Apart from an experimental determination of the Gt values for uniaxially and biaxially loaded specimens there is also the possibility of performing a simulation by means of the model introduced in the last section, which also provides Gf values. A third possibility would be a theoretical calculation of the Gi values. A shear-lag analysis carried out previously’ regarding the fibre pull-out process and the debonding supplies an optimum value for the specific fracture energy:
FRC split specimens, both halves of the specimen being rotated about point 0.
Fig. 5. Geometry and forces acting on
Therefore the crack width, d, is replaced by a linear function d(z): d(z)=CMODH
The load/displacement F,(CMOD)
= H
Llcl
:,+,
(17)
g=+(l+e@‘)
curve is calculated by: ‘;I
+ e
LIG
(3 2.5 Calculation
of specific fracture energy, Gf
The fracture energy resulting from integration of the load/displacement curve per unit of the fractured surface is given by: CMoD,,,
1 GJmode’)=
BL,GHL,G
where 7 = z(k,a,) is the friction stress and shear stress, respectively, and G& is the value for the specific fracture energy that is exclusively consumed by the fibre pull-out process, under the assumption that there is no fibre fracture (in case of a fibre fracture Gf would be considerably reduced). Equation (20) contains the snubbing coefficient, g, which has also been derived in elsewhere2 and which can be represented as a function of the snubbing factor, f:
Io
F,(CMOD)d(CMOD) (19)
For a parallel movement of both halves of the specimen away from each other, the horizontal force, FH, for CMOD = Lf (see Fig. 5(a) and (b)) would become zero, since all fibres with a fibre length L, have been pulled out. Under the assumption of rotation of both halves of the specimen around the 0 point (Fig. 5(a)) even in an extreme case of a rotation angle of 180” not all fibres would be pulled out of the matrix, namely those that are close to the centre of motion. This results in an asymptotic tapering off of the modelled load/displacement curve and requires a maximum integral limit (CMOD,,,) when calculating eqn (19).
(21)
For values of f between 0 and 1, g varies between 1 and 2.32. The snubbing factor, f, depends on the matrix and fibre materials and has been found experimentally7~3z to be approximately O-7 for polyprolyene fibres in a cement matrix.* Equation (20) only applies to aligned-fibre composites and to constant transverse strain, 7. However, no constant transverse strain, 7, is used in the model under consideration, but a linear or square dependence of 7is used instead, and eqns (4) and (5) cannot be directly inserted into eqn (19). Instead of the constant transverse strain, 7, a constant transverse strain with the average taken over the fibre distribution, 7mid(average taken of all solid angles and of all embedded lengths of the fibres, as well as of all transverse strain distributions) can be introduced into eqn (20), which can be determined in the following way:
i’”
dx
(“”f (x, 0)d0
(22)
rrnaX dk
with the following integral limits applying: 8 max
=
arccos)2x/Lf[;
mmax = i
=k- & f
f
(23)
M. Elser et al.
952 If the formulation
is used for the transverse (eqns (4) and (5)), eqn (22) results in a mean transverse strain: Tmid=T(0)
(
l+?+Z
1
The constant transverse average taken of the uniform in the matrix mainly depends and a2 of Taylor’s series (eqns the initial value, r(O), as well load, (TV,and it is independent (8,(p) of the fibres. Equation for the theoretical calculation
strain. r constant
(l+$,a,)
(24)
strain, rmid, with the random-oriented fibres on the coefficients a, (4) and (5)) and also on as on the compressive of the relative position (24) (rmid) can be used of qp’ in eqn (20).
4 COMPUTER SIMULATION OF THE LOAD/DISPLACEMENT CURVES and biaxial load/ In a previous studyz8 uniaxial displacement curves of polypropylene-fibre-reinforced concrete specimens were described. Four test series with different fibre contents (Vf = 0.001 and O+OOS)and
4500
“C 1 I
*H
[
CO...-.i+.,
,\
Polypropylene
Lenend:
Fibers:
fibre lengths (Lr = 10 and 20 mm) were investigated. load/displacement curves of unreinFurthermore, forced concrete have been determined experias a mentally,2y and these shall be used subsequently reference. In Fig. 6(a)-(d) the measured load/displacement curves are represented for the test series PF520 (polypropylene-fibre-reinforced concrete, V, = 0.005, Lf = 20 mm)-the composite2’-as well as for the test series NG(wet) (concrete containing natural gravel, which was stored in water for 28 days)‘“-the matrix. Each curve is the average of 3-4 experimentally obtained load/displacement curves for the dimensionless compressive stress ratio cTi/fc of 0, 15, 30 and 50% where cr, is the compressive stress and fc is the compressive strength. Since this model only considers the fibres in the fracture process, only the contribution of the fibres to simulated experimentally determined the and load/displacement curves has to considered, as well as their contribution to the total fracture. The fibre contribution to the total fracture energy, Gyhrc), of the
4500 ,
F”
.
m
i I+
I
I
N
Model Paramelers:
\
0.
0
i
i
a,.ZQQQ t= 475
model \ /------
3
crack mouth
4
z+
$
7
opening displacement
a,=-540 bl= Q,I
a
9
n -0
10
1
CMOD [mm]
2
3
4
5
7
6
6
crack mouth opening displacement
9
10
CMOD [mm]
(b)
(4 Polypropylene
Fibers:
L,=ZOmm
df=15pm
“f=o,5%
f,
1
I
FH
i :
1
F
matrix
L
[NJ
4500
=24,3MPa
I..
I
1
Model Parameters:
3000
biaxial
q&
= 50%
0
0 0
1
2
3
crack mouth
4
5
6
7
opening displacement (c)
6
9
CMOD [mm]
10
0
12
3
crack mouth
4
5
6
7
opening displacement
6
9
10
CMOD [mm]
(4
Fig. 6. Load/displacement curves of the composite and of the matrix, and the fibre contribution for the composite as a result of subtracting the latter from the former (shaded area). (a) Simulation of the uniaxial (a,/_& = 0%) part of fibres of the load/displacement curves of the composite; (b) simulation of the biaxial (a,/fc = 15%) part of fibres of the load/displacement curves of the composite; (c) simulation of the biaxial (a,/fc = 30%) part of fibres of the load/displacement curves of the composite; (d) simulation of the biaxial (CT,/& = 50%) part of fibres of the load/displacement curves of the composite.
Fracture behaviour of FRC: Modelling and computer simulation Table 1.
Normalized compressive stress, a,/fc (%)
953
Specific fracture energy values from experiments, the computer simulation and the calculated for various compressive loads, (rI, and coefficients, b,
Measured
Computer
c;(FpF5*) C;(NGwet) (k/m) (NW
Gmode’) (N/m)
(&a)
simulation
evaluation
b,opt)
Calculated
Fibre rupture W)
G$‘““)
Gf(oP’)
%lid (MW
(N/m)
5
x
cp’)
(N/m)
0
1069.6
113.8
955.8
905.6
0.00
54.3
1981.6
0.7771
388.7
1943.5
1.5
424.1
686
355.5
708.0 596.4 509.3
O-10 0.15 0.20
59.0 61.0 62.6
1726.8 1529.3 1361.8
0.9659 1.0603 1.1547
312.7 284.9 261.6
1563.6 1424.4 1307.9
30
411.2
78.0
333.2
509.3 398.3 320.5
0.10 0.15 0.20
62.6 64.9 67.5
1361.8 1134.8 986.2
1.1547 1.3436 1.5354
261.6 224.8 197.1
1307.9 1124.1 985.5
50
472.8
89.2
383-6
375.6 280.8 221.1
O-10 0.15 0.20
65.3 69.0 71.6
1082-4 905.8 778.5
1.4065 1.7212 2.0360
214.8 175.5 148.3
1073.8 877.5 741.8
-
composite is obtained by subtracting the matrix component, GpGcwet)(matrix), from the fracture energy of the composite, GfPF520(composite). Table 1 contains these three experimentally determined fracture energy values and a simplified correlation can be represented as follows: GfPF520(composite)
- GpG6”e’)(matrix)
= Gifibre)
(25)
Since the area under a measured load/displacement curve represents the fracture energy, the fibre component (shaded area in Fig. 6(a)-(d)) in the load/displacement curve of the composite has to be
determined by subtracting the energy for the matrix (NG(wet)) from the load/displacement curve of the composite PF520 (as in eqn (25)). By means of the model described above, the fibre contribution to the load/displacement curve of the composite can be simulated. Thus the dependence of the fracture energy, Gf, of the fibres (Fig. 7(a)) as well as of the normalized fracture energy Gr/GfO of the fibres (Fig. 7(b)) on the compressive load, crl, of the specimens can be calculated from the simulated load/displacement curves. The fibre component of the Gt values can be obtained by subtracting the matrix fracture energy values from the Gf values of the composite material compound. Suitable initial values of the model parameters (z(0),a,,a2,b, and f) were obtained in the following way. (a) Modelling of the uniaxially determined fibre share of the uniaxially measured load/displacement curves (gI = 0): . l
put G$--) =
@,oPf;
from eqn (19) for qpt, the value of Zmidcan be determined by means of an assumed snubbing factor, f (thus also for the snubbing coefficient 8);
0
io
zb
So
normalized compressive stress ollfc [%I (4
“I
so
0 normalized compressive stress q/f,
90 loo
[%I
@I
versus u,/fc curves for the composite and the matrix, and the fibre component for the composite obtained by subtracting the latter from the former (shaded area). (a) Simulation of the Gf versus (r,/f= curves of the fibre component in the composite as well as the different model curves for b,; (b) simulation of the G,/G, versus ul/fc curves of the fibre component in the composite as well as the different curves for bl, and the corresponding fraction in the fibre rupture events. Fig. 7. G,
M. Elser et al.
954
l
.
.
from Zmid (eqn (23)) a value for a, is calculated for a2 = 0, assuming r(0) in a first step; after a first modelling of the uniaxial fibre part, both a2 and a, as well as r(O) can be determined more precisely by means of a better fit of the model curve to the measured curve; the compressive coefficient, b, (eqns (4) and (5)) has to be considered for non-uniaxial loading only, since under uniaxial loading the compressive loading of the specimen is zero.
(b) Modelling of the biaxially of the biaxially components displacement curves (pi > 0): .
.
.
determined measured
fibre load/
the best approximation of the calculated uniaxial curve to the measured uniaxial load/displacement curves can be obtained by using the model parameters for the uniaxial fibre component (Fig. 6(a)); the biaxial fibre component can now be simulated by varying compressive the coefficient, b, (Fig. 6(b)-(d)); at the same time the fracture energy, G,, as well as the normalized fracture energy, GJGr,,, is simulated for every value of b1 for increasing compressive load, vl, on the samples and is represented for comparison purposes with the experimentally determined curves of fibre fracture energies (Fig. 7(a) and (b)).
5 DISCUSSION When using identical model parameters in the range a,lfo = 1550%, there is qualitative and partly quantitative agreement between the mean load/displacement curves represented in Fig. 6(a)-(d) and the simulated curves and also between the experimentally determined and also the simulated Gf versus crl/fC curves (Fig. 7(a) and (b)). The deviation arises from the delamination process described previously,2x which is mainly responsible for the rapid decrease of the Gr value with increasing compressive load, cl/fC, as well as other mechanisms (fibre/matrix debonding, matrix spalling, formation of fibre bundles, multiple cracking, etc.) as described elsewhere,5%2y that are not taken into consideration. The increase in fibre fractures, together with the increase in average friction stress or transverse strain Z,i, (Fig. S), with increasing compressive load, (TV/&., results in a decrease in fracture energy when applying this model. Figure 8 represents the transverse strain, r(s/L,, gl), (eqns (4) and (5)) as a function of the normalized fibre-end slippage, s/L,, for a fibre that is embedded with I = LJ2 in the matrix and is normal to the fracture surface. The calculated rmid values mainly lie in the range of
tMPa1
0
0.1
0.2
0.3
0.5
0.4
normalized fiber-end slippage s/Lf
Fig. 8. Shear stress ~(s/L,,u~) along a fibre that is normal to the fractured surface embedded in the matrix with L,/2, for various compressive loads, (T,, and compressive coefficients b,.
the values determined experimentally by means of pull-out tests on individual fibres.“.30,33 As can be seen from eqn (20), the Gr value is indirectly proportional to the average transverse strain, Zmid (z = rmid). Equation (19) supplies considerably lower values for qp’ (calculated Gf values in Table 1) compared to the measured Gjfibrc) values and the simulated G$m”de’) values. A Gr’“OX/“)value can be calculated from the Gimode’) value, the former resulting from the simplifying assumption that all fibres, also those that have been broken, contribute to the fracture energy of the fibres:
,f$oo”)
=
,$mdel)
100 100 - fibre rupture
(%)
(26)
It is only this value which basically supplies a good correspondence with the’five-times higher Gip’ value (see Table l), the Ggmodel) value also being derived under the assumption that there is no fibre fracture. The simulated model curves represented in Fig. 6(a)-(d) (load/displacement curves) show a special characteristic with regard to the position of the load peaks. These peaks all lie at CMOD values of 2 mm. The increase in the experimentally determined fibre energies is determined by the deformation behaviour of the matrix, of the fibres and of the composite. Since the elastic deformation of the matrix and of the fibres has not been taken into account in the model, the invariable position of the load peaks mentioned above and the increase in the model area curves are exclusively caused by the force relationship taken into consideration at the fibre/matrix interface. In the evaluation (eqn (ll)), an overall homogeneous stress distribution at the cylinder surface and a fibre/matrix interface thickness of zero were assumed. A uniform homogeneous matrix was also assumed, so that no difference can be discerned within the matrix
Fracture behaviour of FRC: Mode&g material immediately surrounding the fibre. This is not the case in reality, however, since the matrix material surrounding the fibre is heterogeneously distributed, owing to the existence of aggregate grains and it therefore does not surround the fibre completely, so that the stress resulting from the compressive loading applies in a rather concentrated manner at the surface of the fibre. All these facts reduce the estimated value of b, (and therefore the < sign is introduced in eqn (11)). The simulation does in fact show that this parameter is smaller than the estimated value that is required in order to model the experimentally determined load/displacement curves and the Gr versus allfc dependence. The coefficient b, can be interpreted as a qualitative measure for the degree of compression of the matrix surrounding the fibre under compressive loading, i.e. it represents a measure for the approximation to a homogeneous stress distribution that applies overall at the fibre/matrix interface. The shape factor of the fibre cross-section and the deviation from the circular section of the fibre also influence the value of bI.
and computer simulation 4.
5.
955
The mean shear stress, rmid, increases with increasing u1 /fc values (increasing compressive load). This results in lower specific fracture energies of the simulated Gp’ values, which is in accordance with the experimental results. The GglOo%) value results agree with the simulated values, G@‘odel), if all fibres are subjected to a pull-out process, even those that would suffer a fibre fracture. Only the five times a good corresponhigher Gp’ values provides dence with the G$looo’) value. qp’ was computed under the assumption that there is no fibre fracture.
ACKNOWLEDGEMENTS This study was financed by the National Bank of Austria (osterreichische Nationalbank) together with the Austrian Science Foundation (Fonds zur Forderung der Wissenschaftlichen Forschung) under project number P8885-TEC.
REFERENCES 6 CONCLUSIONS The model proposed by Li and co-workersI is extended with regard to a biaxial state of stress (compression + tension) in splitting a specimen made of FRC. Simulations are performed under various assumptions and simplifications on the fracture behaviour of brittle materials that are reinformced by means of short, uniform, randomly oriented fibres, after the occurrence of a macro-matrix crack. 1.
2.
3.
With this model the fibre contribution to the load/displacement curve of the composite can be simutlated for both uniaxial and biaxial loading. The load peaks in the measured and simulated curves are shifted towards higher CMOD values for all loads. The experimentally determined and simulated peak values coincide for uniaxial (Fig. 6(b)) conditions with aI/fc (30 and 50%) quite well, while they differ for rl/fc (15%). When using identical model parameters a semi-qualitative agreement between the experimentally determined and the simulated Gf versus aI/fc curves (Fig. 7(a) and (b)) can also be achieved by means of a variation of parameter b 1. The reason that experimental and simulated behaviour (as mentioned in 1 and 2), coincide only partly, is that delamination occurs during compressive loading of the material and this fact is not taken into account in this model.
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