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Modelling post-crack tension-softening behavior of fiber-reinforced materials
Probabilistic Engineering Mechanics 45 (2016) 157–163
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Probabilistic Engineering Mechanics journal homepage: www.elsevier.com/locate/probengmech
Modelling post-crack tension-softening behavior of fiber-reinforced materials S. Matthes a,n, F. Ballani b, D. Stoyan b a b
Institut für Keramik, Glas- und Baustofftechnik, TU Bergakademie Freiberg, Leipziger Str. 28, 09599 Freiberg, Germany Institut für Stochastik, TU Bergakademie Freiberg, Prüferstr. 9, 09599 Freiberg, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 26 February 2015 Received in revised form 12 April 2016 Accepted 13 April 2016 Available online 20 April 2016
We present a general method for the traction-separation law for the cohesive model of fiber reinforced materials with brittle matrix. The proposed approach is based on results from the theories of marked point and fiber processes. The application of stochastic notions in the field of traction-separation laws and tension-softening curves for fiber reinforced composites allows the thorough investigation of the random effect of the fiber reinforcement on cohesive behavior. The presented method accounts for correlations between length and orientation as may be the case in real fiber reinforced composites. We study the influence of randomness of fiber length and degree of anisotropy on the post-crack tension softening curves. It turns out that fiber length and orientation distributions have a tremendous effect on the crack-opening behavior. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Short-fiber composites Stress/strain curves Fiber/matrix bond Statistics Fiber bridging
1. Introduction Traction-separation laws are used for numerical and analytic studies of the crack propagation for different load cases and various length scales. They describe the failure behavior of materials and are often deployed in finite element analyses with predefined crack paths. These traction-separation laws include the cohesive stress at the crack plane sc and the crack width opening d. The resulting coherence σc = σc (d ) is called in the following tensionsoftening curve (TSC). The measurement of such a TSC is a formidable task and was carried out in the past for various materials [1]. Due to the stochastic character of composite materials with fiber reinforcement [2] traction-separation laws represent a very helpful tool to investigate failure behavior of such materials. The classical paper [3] presented a model which allows the computation of force resistance of reinforcing fibers bridging cracks in brittle matrix fiber reinforced composites (FRC) under the assumption that the fibers have constant length and are isotropically orientated. This work has been continued in many subsequent scientific investigations (e.g. [4–7]). However, in real FRC the fiber lengths are never constant. Fiber length is affected by various factors such as processing of fibers (cutting, chemical and mechanical treatments), embedding and n
Corresponding author. E-mail addresses: [email protected] (S. Matthes), [email protected] (F. Ballani), [email protected] (D. Stoyan). http://dx.doi.org/10.1016/j.probengmech.2016.04.001 0266-8920/& 2016 Elsevier Ltd. All rights reserved.
processing of reinforced material (mixing, casting, etc.). Refs. [8] and [9] proposed several statistical distributions for random fiber lengths. In real structures also deviations from the isotropic orientation distribution of fiber directions appear. While it seems to be natural to expect some kind of anisotropy of fiber orientation in the case of long-fiber reinforced materials, recent studies showed that in some composite materials even short fibers are not always isotropically oriented [10–12]. Finally, due to casting process, buoyancy effects and sedimentation, it has to be assumed that in various composites lengths and orientations of fibers are correlated. However, the joint influence of randomness of length and orientation of fibers in composite materials on mechanical properties has never been studied so far, although the spatial distribution of fibers in composite materials can be determined statistically (see e.g. [13]). In view of the general aim of improving or optimizing material properties, the influence of randomness of fiber length and orientation on mechanical properties is of great interest. It is natural to ask: How will the post-crack TSC vary if the fiber length is not constant but the mean fiber length is fixed? To which extent does the TSC change if the fibers have some special direction distribution and are not isotropically oriented? Obviously, there is a simple qualitative answer: Random fiber lengths imply that there are fibers which are longer than the mean fiber length. Therefore it might be expected that random fiber lengths lead to a TSC higher than for constant fibers. Furthermore,
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due to friction of fibers during pull-out, TSC will indeed change if the orientation of the fibers is not isotropic. Both effects are studied in the present paper. In order to demonstrate the application of our theory we use results from the literature about the frictional bond between fiber and matrix such as [14–16]. With the aid of such measurements the post-crack TSC can be studied quantitatively. The paper is organized as follows. The methods and assumptions made in order to derive the traction-separation law are presented in Section 2. There we define the mathematical notions, present an equation for TSC for general (joint) distributions of fiber length and orientation and also point out further extensions of the model. Finally, we investigate the impact of correlation of fiber length and orientation on the post-crack TSC in Section 4.
2. Model assumptions and methods 2.1. Model assumptions and extensions Throughout the paper we make the following model assumptions, which are mainly standard in the relevant literature, but some extensions are new. We consider a statistically homogeneous [17, p. 28] matrix material with randomly distributed fibers under the following conditions, see e.g. [3]: The matrix is discontinued by a planar crack of width d, see Fig. 1, its deformation during the fiber pullout is neglected. The spatial distribution of the positions of fibers in the composite is homogeneous and is independent of fiber length and orientation, the fibers are straight with cylindrical geometry. They behave linear elastically and rupture if their axial stress reaches the fiber strength σf ,max . The Poisson effect of the fibers during pull-out is neglected, the fiber–matrix bond is frictional and the elastic bond strength is neglected. As the crack opening d increases, the fiber ends are pulled out of the matrix. Eventually one fiber end is pulled out completely or the fiber ruptures due to high tension. Additionally we assume that fiber length and orientation are random. We describe this randomness by a two-dimensional probability density function (p.d.f.). This means that fiber length and fiber orientation are allowed to depend on each other, which allows a realistic approximation of a wide class of composite materials where one phase is built of fibers. Our approach can easily be combined with various models concerning the fiber pullout mechanism.
described by a reference point, length and angle w.r.t. the normal of some given crack plane. For convenience the crack plane is assumed to be the (x,y)-plane and the reference point of each fiber is its top point (in the sense of the z-axis). The fiber angle is the polar angle β of the fiber. Fig. 1 shows the underlying geometry of a single fiber that intersects a crack plane Ac with inclining angle β and embedded residual lengths r1 and r2. The random set of fibers is described by the following summary characteristics: 1. Nsp – mean number of fibers (i.e. fiber reference points) per unit volume. 2. fsp, L, B (l, β ) – joint p.d.f. of fiber length l and polar angle β in space. These characteristics are often known a priori and they are measurable e.g. by computed tomography, see [18,19], and [13,20]. If angles β and lengths l are statistically independent the joint p.d.f. is the product of the univariate p.d.f. fsp, L (l ) and fsp, B (β ) of length and angle. The parameter Nsp belongs to a group of summary mean-value characteristics, which include also fiber volume fraction Vf, mean fiber length l and fiber cross-sectional area Af. They satisfy the equation
Nsp =
Vf lAf
.
(1)
After the formation of a planar crack, i.e. when d ¼0, we are interested in the random embedded residual fibre lengths and the inclination angle w.r.t. the normal of the crack plane of the fibers which intersect the crack, see Fig. 1. We describe these quantities by the characteristics 1. Npl – mean number of fibers intersecting the crack plane per unit area and 2. fpl, R1, R2, B (r1, r2, β ) – joint p.d.f. of residual lengths above and below the crack plane and polar inclination angle. There are close mathematical relationships between these plane-related characteristics and the space-related characteristics of the fiber system. In particular, we have (cf. [21, Section 8.4])
Npl = Nsp (L sp cos Bsp ).
(2)
In this equation the expression π
∫0 2 ∫0
∞
2.2. Theoretical background, distribution of intersecting fibers
(L sp cos Bsp ) =
In order to fix notation we describe fiber systems and their characteristics in what follows. A system of fibers is a spatial set of line segments. Each fiber is
denotes the mean of l cos β . (We used here the notation L sp and Bsp for the random variables of length and polar angle.) For isotropically oriented fibers of constant length l0, i.e. the case studied
l cos β fsp, L, B (l, β ) dl dβ
Fig. 1. Left: A fiber intersecting the crack plane Ac having length l = r1 + r2 and inclining angle β. The reference point of the fiber is denoted by (x, y, z ) . Right: At crack width d the embedded fiber is being pulled out of the matrix.
S. Matthes et al. / Probabilistic Engineering Mechanics 45 (2016) 157–163
in [3], Eq. (2) simplifies to
Npl =
2.3. Bridging forces
1 l N . 2 0 sp
(3)
By the way, if length L sp and angle Bsp are independent of each other then also residual lengths and inclining angle are independent. In this case the expression for Npl simplifies to
Npl = Nsp (L sp ) (cos Bsp ).
(4)
Furthermore, [22] introduced a relationship between the p.d.f. of residual lengths r1 and r2 and inclination angle β and the p.d.f. of spatial fiber length l and angle β. In our notation this reads as
cos β fsp, L, B (r1 + r2, β )
fpl, R1, R 2, B (r1, r2, β ) =
(L sp cos Bsp )
(5)
π . 2
for r1, r2 ≥ 0 and 0 ≤ β ≤ In the particular case of isotropically oriented fibers the marginal p.d.f. of Bpl then equals π fpl, B (β ) = sin 2β, 0 ≤ β ≤ , 2 see also [21, Eqn. (8.64)]. We remark that an isotropically distributed spatial orientation of a fiber implies the p.d.f.
fsp, B ( β ) = sin β,
0≤β≤
159
π , 2
(6)
of its polar angle Bsp , see also [3, Eqn. (4)] and [21, Eqn. (8.61)]. Obviously the angles Bpl and Bsp do not have the same distribution (as erroneously assumed in [9]). The difference between these distributions can be explained as follows: The p.d.f. fpl, B accounts only for those fibers that intersect the crack plane whereas fsp, B takes every fiber of the system into consideration. A fiber is more likely hitting the crack plane if its polar angle is small, see Fig. 2. In many approaches to model single fiber pull-out the minimum embedded fiber length plays a decisive role. Therefore we may also consider the distribution of the minimum and maximum embedded length of an inclining fiber, rmin = min {r1, r2 } and rmax = max {r1, r2 }, respectively. Eq. (5) implies
fpl, m, M (rmin, rmax, β ) ⎧ 2 cos β fsp, L, B (rmin + rmax, β )r,min ≤ rmax ⎪ = ⎨ (L sp cos Bsp ) . ⎪ ⎩ 0, otherwise
(7)
In order to model the TSC for a FRC the mechanics of a single fiber intersecting the crack plane is considered. Here two different approaches exist: a stress-based approach (e.g. [23,24]) and an energy-based bond-slip approach (e.g. [25,26]). These are characterized by specific stress-slip relations which depend on the crack width d, the inclination angle β and the residual fiber lengths r1 and r2, or the minimum and maximum embedded length rmin and rmax . The fiber–matrix debonding is described by a rate-independent contact law. Such laws have been proposed by [14,27,3,5,16] for fiber-reinforced cementitious composites. These papers established various expressions for the pull-out force P = P (r1, r2, β , d ) of a single fiber. The influence of the inclination angle on the pull-out behavior was discussed in [28,29], which both used the Euler–Eytelwein or Capstan formula
P (r1, r2, β, d) = P (r1, r2, 0, d) e βμs .
(8)
In this equation μs is the so-called static friction or snubbing coefficient. This well-known representation of the dependence of pull-out force on inclination angle will be henceforth used in this paper. However, more complex formulations and detailed physical concepts could be easily integrated into our general approach, see Section 3. For instance, [28] measured the pull-out force of steel fibers out of a cementitious matrix at various inclination angles and found significant deviations from the Euler–Eytelweyn formula which is caused by matrix spalling and crack plane deflection during fiber pull-out. In [30] these effects are theoretically investigated.
3. The proposed composite tension-softening model In this section we combine some of the ideas above in order to derive a model for the post-crack TSC, which describes the behavior of sc, the mean resistance force per unit area of the FRC material as a function of the crack width d. Each fiber which crosses the crack plane contributes to the crack-opening resistance force. Integrating the single-fiber pullout force contributions P (r1, r2, β , d ) over all (r1, r2, β ) weighted by their likelihood fpl, R1, R2, B (r1, r2, β ) and multiplying by the mean number of such fibers per unit area, Npl , yields
Fig. 2. Comparison of the spatial p.d.f. of the polar angle Bsp of isotropically oriented fibers (left) and the p.d.f. of the inclining angle Bpl of isotropically oriented fibers hitting the crack plane (right).
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∞
π
∞
∫0 ∫0 ∫0 2
σc (d) = Npl
P (r1, r2, β, d) fpl, R1, R 2, B (r1, r2, β ) dβ dr1
(9)
dr2,
as the mean resistance force per unit area at crack width d. At this point randomness of residual lengths and inclination angles is directly accounted for. Rewriting Eq. (9) in terms of spatial p.d.f. fsp, L, B and mean number of fibers per unit volume, Nsp , by inserting (2) and (5) into (9) yields the equation ∞
σc (d) = Nsp
∞
π
∫0 ∫0 ∫0 2
P (r1, r2, β, d) fsp, L, B (r1 + r2, β )
× cos β dβ dr1 dr2.
(10)
Using (7), Eq. (10) can be further rewritten in terms of minimum and maximum embedded length as ∞
σc (d) = 2Nsp
∫0 ∫0
rmax
π
∫0 2
P (rmin, rmax, β, d)
× fsp, L, B (rmin + rmax, β ) × cos β dβ drmin drmax.
(11)
Likewise we obtain ∞
σc (d) = 2Nsp
π
l
∫0 ∫0 2 ∫0 2
P (rmin, l − rmin, β, d) × fsp, L, B (l, β )
cos β drmin dβ dl
(12)
from (11) after applying the change of variables l = rmin + rmax and changing the order of integration. The derived relationships generalize earlier approaches such as the mathematical formulation of TSC in [3] for constant fiber lengths and isotropic orientation. Indeed it can be shown that the equation
Vf σc (d) = Af
l0 2
∫0 ∫0
⎛ 2z ⎞ arccos ⎜ c ⎟ ⎝ l0 ⎠
⎛ l zc l zc P⎜ 0 − , 0 + , β, ⎜2 cos β 2 cos β ⎝
× fsp, B (β ) fsp, Z (z c ) dβ dz c ,
Fig. 3. Schematic diagram of the distribution of the axial force in a fiber during two-sided pull-out using the Euler–Eytelweyn formula.
⎞ d⎟ ⎟ ⎠ (13)
which can be found in [3], is equivalent to (10) if fiber lengths are constant, the orientations of fibers are independent of their lengths and fsp, B (β ) = sin β , e.g. the fibers are isotropically oriented.
4. Numerical study of the effect of randomness on the TSC We now examine the behavior of TSC if fiber lengths are random by applying Eq. (11). For convenience we use established models for the distributions of fiber length and orientation in composite materials (see [8,31]). Due to the process of manufacturing of fibers, specific types of length distributions ensue. Empirical investigations were carried out for different combinations and products (e.g. [32–34]) and various models for these distributions have been conceived. The mechanical model for the single fiber pull-out force is chosen to be the same as in [3] in order to compare our results with the older ones. The pull-out force of a single fiber then reads as
⎧ τ (s ) πdf (rmin − d) e μs β , d ≤ rmin P (rmin, rmax, β, d) = ⎨ , else ⎩ 0, ⎪
(14)
where df is fiber diameter, μs is snubbing friction coefficient, τ (s ) is bond strength of fiber–matrix interface and s is the slippage of fiber out of matrix. We choose τ (s ) = τ (d ), i.e. expression (14) describes a fiber pull-out mechanism without elastic deformation and fiber slippage only at the shorter embedded end (Fig. 3). The function τ (s ) can be measured by single fiber pull-out
experiments and was investigated extensively in the literature (e.g. [14,28,35,36,16]). In order to study different forms of τ (s ) we use data of the matrix-fiber bond strength measured in single fiber pull-out experiments published in [3]: τ (s ) = 1.02 MPa for constant interface strength, τ (s ) = 1.02 + 0.2s + 0.2s2 MPa for a sliphardening model (matches certain synthetic fibers) and τ (s ) = 1.02 − 0.2s + 0.01s2 MPa for a slip-weakening model (matches certain steel fibers). These three interface strength models and the corresponding TSCs based on (11) are shown in Fig. 4 for a constant fiber length of 12.7 mm as in [3], where also fiber breakage at an ultimate fiber strength of σf ,max = 2.6 GPa is included. The corresponding TSCs are compared to those in the case of random lengths in the next section. 4.1. Effect of randomness of fiber lengths The following results show that in the case of random fiber lengths the initial post-peak strength and shape of the post-peak TSC changes and deviates significantly from the corresponding curve for constant fiber length. A commonly used model is the Weibull distribution which has the p.d.f.
⎧ ⎛ ⎞m − 1 ⎛ l ⎞m ⎪m⎜ l⎟ exp (−⎜ ⎟ ), l ≥ 0 fsp, L (l) = ⎨ k ⎝ k ⎠ ⎝ k⎠ ⎪ else ⎩ 0,
(15)
with positive parameters m and k, see [37]. It becomes narrower with increasing m (the shape parameter, or Weibull modulus) while the mean becomes larger with increasing k (the scale parameter).
S. Matthes et al. / Probabilistic Engineering Mechanics 45 (2016) 157–163
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Fig. 4. Three models for fiber-matrix slippage (left) and predicted TSCs for composites with two-sided fiber pull-out during crack opening (right). The following parameters are used: df = 38 μm , μs = 0.7 rad−1, the constant fiber length is 12.7 mm as in [3].
Fig. 5. Weibull probability density functions for fiber lengths with fixed mean l = 12.7 mm (left) and the corresponding TSCs based on (11) in comparison with the constant fiber length model (right). In all cases a constant interface strength of τ (s ) = 1.02 MPa was assumed.
We study the impact of randomness of fiber lengths on the mechanical response by comparing TSCs for various Weibull distributions with fixed mean l = 12.7 mm (as in [3]). The underlying probability density functions and the corresponding TSCs are shown in Fig. 5. Here we used isotropically oriented fibers with a constant interface strength τ (s ) = 1.02 MPa and the constituent parameters df = 38 μm and μs = 0.7 rad−1 and assumed that fiber length and orientation are independent. As the distribution of fiber length becomes broader the contribution of long fibers to the TSC becomes more significant and the composite strength increases. This effect is also visible for d ¼0, i.e. for the strength of the composite material at crack formation.
and fiber breakage is neglected. These assumptions are fulfilled in composites with a weak matrix reinforced with strong fibers. The TSC of this composite material can then be written as
4.2. Effect of anisotropy and preferential directions of the fiber orientation
which compares the TSC for a specific orientation distribution to the TSC for isotropically oriented fibers. If K ¼1 the post-peak strength of a fiber reinforcement with orientation distribution fsp, B (β ) is equal to that of isotropic fibers for same length dis-
In the following we discuss how the post-peak TSC of a FRC material is influenced by the kind of randomness of the 3D fiber orientations, where we are interested in the case when the fiber orientation becomes anisotropic. In order to illustrate this effect in a better way we assume the following: fiber angle and length are independent of each other, the single fiber pull-out force P is the product of a term PB which only depends on β and a term PR which only depends on the residual lengths and the crack opening width
σc (d) = Kσc,iso (d),
(16)
see e.g. [38]. Here K is an orientation factor π
K=
∫02 PB ( β ) fsp, B ( β ) cos β dβ π
∫02 PB ( β ) sin β cos β dβ
,
(17)
tribution. If K > 1 or K < 1 the fiber orientation leads to a postpeak strength higher or lower than isotropic reinforcement respectively. The orientation factor K depends only on PB and fsp, B and is independent of the fiber length distribution. It therefore describes the impact of anisotropy on the TSC. The TSC for a isotropically oriented fiber system is
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Fig. 6. The parametric fiber orientation density function fV , B (β ) as proposed by [19] for different choices of c. Note that for c ¼1 the fibers are isotropically oriented.
Fig. 7. Values of K in dependence of the parameter c, i.e. for different orientation distributions and different snubbing friction coefficients μs.
π
σc,iso (d) = 2Nsp
∫0 2
∞
∫0 ∫0
PB ( β ) sin ( β ) cos ( β ) dβ ×
rmax
PR (rmin, rmax, d) PB ( β )
× fsp, L (rmin + rmax ) drmin drmax.
(18)
In order to study the behavior of K we assume in the following that the angle-dependence of the single-fiber pull-out force is described by the Capstan formula, i.e. PB (β ) = e μs β , and use the following orientation distribution:
fsp, B (β ) =
c sin β
(1 +
(c2
− 1)
cos2
β)
3 2
,
0≤β≤
π , 2
c > 0, (19)
see [19]. This family of p.d.f. represents isotropically oriented fibers for c¼ 1, fibers which are nearly parallel to the crack plane for large c and nearly aligned fibers orthogonal to the crack plane for small c. In Fig. 6 the family of p.d.f. is depicted for different choices of the parameter c. Fig. 7 shows the behavior of the orientation factor K in dependence of the orientation parameter c. For μs = 0 rad−1 the snubbing friction of fibers at the crack plane vanishes. The coefficient K therefore increases for decreasing c because fibers orthogonal to the crack plane intersect it more likely, see e.g. (4). For μs > 0 rad−1 the effect of snubbing friction of the fiber at the crack plane becomes more dominant, i.e. fibers with a large intersection angle have a stronger effect on the TSC although they intersect the crack plane less likely. 4.3. Effect of the dependence between length and orientation Finally, we apply Eq. (11) to the case of dependent fiber length and orientation. The following example is motivated by fiber reinforced autoclaved aerated concrete (AAC). During the swelling process of this material the swelling direction appears to be the preferential direction of longer fibers for certain volume fractions, see [18]. The orientation distribution of shorter fibers remains isotropic. In the following we use the process parameters for glass fiber reinforced AAC: we use (19) in order to model the anisotropy of long fibers for different parameters c. We choose the preferential direction of long fibers to be orthogonal to the crack plane.
Fig. 8. Result of a numerical simulation of TSC of a FRC with dependence between fiber length and orientation. Model parameters from Table 1 were used. The parameter c describes the degree of anisotropy of long fibers, see (19), while short fibers are isotropically oriented. Table 1 Model parameters for the TSC for dependent length and orientation. Short fibers Long fibers Ratio of the number of short fibers–long fibers Fiber diameter Fiber strength Fiber volume fraction μs τ Short fibers Long fibers
0 mm < l ≤ 3 mm 3 mm < l ≤ 15 mm 1:2 20 μm 600 MPa 1% 1 rad 1 0.7 MPa Isotropically oriented Anisotropically oriented
Fig. 8 shows the resulting TSC for the model parameters mentioned in Table 1 and different parameters c. For small c short fibers are isotropic while long fibers have a preferential direction orthogonal to the crack plane. The curve c¼ 1 represents a
S. Matthes et al. / Probabilistic Engineering Mechanics 45 (2016) 157–163
reference curve where all fibers, long and short, are isotropically oriented. If the angle of a fiber which crosses the crack plane is large it contributes to a high peak strength and eventually ruptures at crack formation. It does not contribute to crack opening resistance after peak strength. Therefore as the parameter c decreases TSC increases because more long fibers tend to intersect the crack plane at a small angle and do not rupture at crack formation. Naturally, the contribution of the short fibers remains constant. Therefore the number of long fibers contributing to the post-peak TSC increases with decreasing c. For c¼ 1 the slope of the TSC decreases as soon as most of the short fibers are pulled out. As c increases this effect vanishes because the contribution of long fibers becomes dominant. The interplay of correlation of fiber length and orientation and its effects on the TSC cannot be studied without the stochastic approach presented in this paper.
5. Summary and conclusions We presented a general stochastic approach to the micromechanical model for the post-peak TSC of FRC with brittle matrix. It is motivated by the need to consider the random nature of fiber reinforcement in such materials and allows even the incorporation of correlations between fiber length and orientation. Our main result is Eq. (11), which yields the mean value of the random TSC. Due to the modular nature of our approach, various results of micromechanical analyses (single fiber pull-out models (e.g. [36]), snubbing coefficients (e.g. [28])) can be employed in order to obtain composite post-peak TSCs for different materials and physical effects such as matrix spalling and snubbing as well as tension softening or hardening during fiber pull-out. In view of numerical material simulation the proposed model extends the study of failure behavior of FRC using representative anisotropic volume elements with prescribed fiber distributions. This allows simulations of materials with fiber-reinforcement with spatially dependent characteristics, which refines the use of stochastic finite element method or Monte Carlo methods [39]. The impact of the randomness of length and orientation on TSC was studied numerically. We compared TSCs of various configurations of fiber reinforcement and found that consideration of randomness of fiber length and orientation leads to post-peak TSCs which deviate significantly from curves obtained for models with constant fiber lengths and isotropic orientation. Randomness of fiber length tends to increase the TSCs while the influence of orientation is more complex. We note that it is not trivial to measure the parameters of the presented model. Our theoretical investigations and results should therefore be supplemented by experimental data. Our results show which effects may occur and may lead to effective methods of measurement for the TSC.
Acknowledgements The financial support of the ESF #100088076 programme is gratefully acknowledged. We also thank Dr. Uwe Mühlich for his valuable comments and hints.
References [1] Y. Wang, V. Li, S. Backer, Tensile properties of synthetic fiber reinforced mortar, Cem. Concr. Compos. 12 (1990) 29–40. [2] A. Ionita, A. Weitsman, Randomly reinforced composites properties, failure and aspects of material design, Probab. Eng. Mech. 21 (2006) 64–72. [3] V. Li, Y. Wang, S. Backer, A micromechanical model of tension-softening and bridging toughening of short random fiber reinforced brittle matrix
163
composites, J. Mech. Phys. Solids 39 (1991) 607–625. [4] M.F. Bäuml, Steigerung der Dauerhaftigkeit selbstverdichtender Betone durch den Einsatz von Polymerfaserkurzschnitt (Ph.D. thesis), Eidgenössische Technische Hochschule Zürich, 2002. [5] T. Pfyl, Tragverhalten von Stahlfaserbeton (Ph.D. thesis), Eidgenössische Technische Hochschule Zürich, 2003. [6] A. Paglia, A fiber pull-out based model for synthetic fiber reinforced concrete beams under a flexural load, Open J. Civ. Eng. 3 (2013) 202–217. [7] V. Li, S. Wang, Microstructure variability and macroscopic composite properties of high performance fiber reinforced cementitious composites, Probab. Eng. Mech. 21 (2006) 201–206. [8] W.K. Chin, Effects of fiber length and orientation distribution on the elastic modulus of short fiber reinforced thermoplastics, Polym. Compos. 9 (1988) 27–35. [9] S. Fu, B. Lauke, Effects of fiber length and fiber orientation distributions on the tensile strength of short-fiber-reinforced polymers, Compos. Sci. Technol. 56 (1996) 1179–1190. [10] N. Neves, G. Isdell, S. Pouzada, P. Powell, On the effect of the fiber orientation on the flexural stiffness of injection molded short fiber reinforced polycarbonate plates, Polym. Compos. 19 (1998) 640–651. [11] G. Zak, C. Park, B. Benhabib, Estimation of three-dimensional fibre-orientation distribution in short-fibre composites by a two-section method, J. Compos. Mater. 35 (2001) 316–339. [12] R. Stadie, Festigkeits- und Verformungsverhalten von kurzfaserverstärktem Porenbeton (Ph.D. thesis), Technische Universität Berlin, 2008. [13] C. Redenbach, I. Vecchio, Statistical analysis and stochastic modelling of fibre composites, Compos. Sci. Technol. 71 (2011) 107–112. [14] Y. Wang, V. Li, S. Backer, Analysis of synthetic pull-out from a cement matrix, Mater. Res. Soc. Symp. Proc. 114 (1988) 159–164. [15] M. Butler, V. Mechtcherine, S. Hempel, Experimental investigations on the durability of fibre-matrix interfaces in textile-reinforced concrete, Cem. Concr. Compos. 31 (2009) 221–231. [16] A. Caggiano, E. Martinelli, A unified formulation for simulating the bond behaviour of fibres in cementitious materials, Mater. Des. 42 (2012) 204–213. [17] S. Torquato, Random Heterogeneous Materials. Microstructure and Macroscopic Properties, Springer, New York, 2002. [18] G. Weidemann, R. Stadie, J. Goebbels, B. Hillemeier, Computed tomography study of fibre reinforced autoclaved aerated concrete, Mater. Test. 50 (2008) 278–285. [19] J. Ohser, K. Schladitz, 3D Images of Materials Structures, Wiley-VCH, Weinheim, 2009. [20] M. Teßmann, S. Mohr, S. Gayetskyy, U. Haßler, R. Hanke, G. Greiner, Automatic determination of fiber-length distribution in composite material using 3d ct data, J. Adv. Signal Process. 545030 (2010) 1–9. [21] S. Chiu, D. Stoyan, W. Kendall, J. Mecke, Stochastic Geometry and its Applications, 3rd ed., Wiley Chichester, UK, 2013. [22] S. Matthes, D. Stoyan, Planar sections through three-dimensional line-segment processes, Image Anal. Stereol. 33 (2014) 55–64. [23] A. Katz, V. Li, Inclination angle effect of carbon fibers in cementitious composites, ASCE J. Eng. Mech. 12 (1995) 1340–1348. [24] A. Ghavami, A. Abedian, M. Mondali, Finite difference solution of steady state creep deformations in a short fiber composite in presence of fiber/matrix debonding, Mater. Des. 31 (2010) 2616–2624. [25] P. Shah, C. Ouyang, Mechanical behavior of fiber-reinforced cement-based composites, ASCE J. Eng. Mech. 74 (1991) 2727–2738. [26] A. Fantilli, P. Vallini, A cohesive interface model for the pullout of inclined steel fibers in cementitious matrixes, J. Adv. Concr. Technol. 5 (2007) 247–258. [27] A. Naaman, G. Namur, H. Najm, J. Alwan, Bond Mechanisms in Fiber Reinforced Cement-based Composites, Technical Report, Department of Civil Engineering, University of Michigan, 1989. [28] V.C. Li, Y. Wang, S. B, Effect of inclining angle, bundling and surface treatment on synthetic fibre pull-out from a cement matrix, Composites 21 (1990) 132–140. [29] J. Zhang, V. Li, Effect of inclination angle on fibre rupture load in fiber reinforced cementitious composite, Compos. Sci. Technol. 62 (2002) 775–781. [30] A.M. Brandt, On the optimal direction of short metal fibres in brittle matrix composites, J. Mater. Sci. 20 (1985) 3831–3841. [31] M. Xia, H. Hamada, Z. Maekawa, Flexural stiffness of injection molded glass fiber reinforced thermoplastics, Int. Polym. Proc. 10 (1995) 74–81. [32] V. Gupta, R. Mittal, P. Sharma, Some studies on glass-fibre-reinforced polypropylene: I. Reduction in fibre length during processing, Polym. Compos. 10 (1989) 8–15. [33] K. Ramani, D. Bank, N. Kraemer, Effect of screw design on fiber damage in extrusion compounding and composite properties, Polym. Compos. 16 (1995) 258–266. [34] Å.L. Døssland, Fibre reinforcement in load carrying concrete structures (Ph.D. thesis), Norwegian University of Science and Technology, Trondheim, 2008. [35] J. Alwan, A. Naaman, W. Hansen, Pull-out work of steel fibers from cementitious composites analytical investigation, Cem. Concr. Compos. 13 (1991) 247–255. [36] G. Bao, Y. Song, Crack bridging models for fiber composites with slip-dependent interfaces, J. Mech. Phys. Solids 41 (1993) 1425–1444. [37] S.-Y. Fu, C.-Y. Yue, X. Hu, Y.-W. Mai, Characterization of fiber length distribution of short-fiber reinforced thermoplastics, J. Mater. Sci. Lett. 20 (2001) 31–33. [38] L. Jain, R. Wetherhold, Effect of fiber orientation on the fracture toughness of brittle matrix composites, Acta Metall. Mater. 40 (1992) 1135–1143. [39] P. Spanos, A. Kontsos, A multiscale Monte Carlo finite element method for determining mechanical properties of polymer nanocomposites, Probab. Eng. Mech. 23 (2008) 456–470.
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Probabilistic Engineering Mechanics journal homepage: www.elsevier.com/locate/probengmech
Modelling post-crack tension-softening behavior of fiber-reinforced materials S. Matthes a,n, F. Ballani b, D. Stoyan b a b
Institut für Keramik, Glas- und Baustofftechnik, TU Bergakademie Freiberg, Leipziger Str. 28, 09599 Freiberg, Germany Institut für Stochastik, TU Bergakademie Freiberg, Prüferstr. 9, 09599 Freiberg, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 26 February 2015 Received in revised form 12 April 2016 Accepted 13 April 2016 Available online 20 April 2016
We present a general method for the traction-separation law for the cohesive model of fiber reinforced materials with brittle matrix. The proposed approach is based on results from the theories of marked point and fiber processes. The application of stochastic notions in the field of traction-separation laws and tension-softening curves for fiber reinforced composites allows the thorough investigation of the random effect of the fiber reinforcement on cohesive behavior. The presented method accounts for correlations between length and orientation as may be the case in real fiber reinforced composites. We study the influence of randomness of fiber length and degree of anisotropy on the post-crack tension softening curves. It turns out that fiber length and orientation distributions have a tremendous effect on the crack-opening behavior. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Short-fiber composites Stress/strain curves Fiber/matrix bond Statistics Fiber bridging
1. Introduction Traction-separation laws are used for numerical and analytic studies of the crack propagation for different load cases and various length scales. They describe the failure behavior of materials and are often deployed in finite element analyses with predefined crack paths. These traction-separation laws include the cohesive stress at the crack plane sc and the crack width opening d. The resulting coherence σc = σc (d ) is called in the following tensionsoftening curve (TSC). The measurement of such a TSC is a formidable task and was carried out in the past for various materials [1]. Due to the stochastic character of composite materials with fiber reinforcement [2] traction-separation laws represent a very helpful tool to investigate failure behavior of such materials. The classical paper [3] presented a model which allows the computation of force resistance of reinforcing fibers bridging cracks in brittle matrix fiber reinforced composites (FRC) under the assumption that the fibers have constant length and are isotropically orientated. This work has been continued in many subsequent scientific investigations (e.g. [4–7]). However, in real FRC the fiber lengths are never constant. Fiber length is affected by various factors such as processing of fibers (cutting, chemical and mechanical treatments), embedding and n
Corresponding author. E-mail addresses: [email protected] (S. Matthes), [email protected] (F. Ballani), [email protected] (D. Stoyan). http://dx.doi.org/10.1016/j.probengmech.2016.04.001 0266-8920/& 2016 Elsevier Ltd. All rights reserved.
processing of reinforced material (mixing, casting, etc.). Refs. [8] and [9] proposed several statistical distributions for random fiber lengths. In real structures also deviations from the isotropic orientation distribution of fiber directions appear. While it seems to be natural to expect some kind of anisotropy of fiber orientation in the case of long-fiber reinforced materials, recent studies showed that in some composite materials even short fibers are not always isotropically oriented [10–12]. Finally, due to casting process, buoyancy effects and sedimentation, it has to be assumed that in various composites lengths and orientations of fibers are correlated. However, the joint influence of randomness of length and orientation of fibers in composite materials on mechanical properties has never been studied so far, although the spatial distribution of fibers in composite materials can be determined statistically (see e.g. [13]). In view of the general aim of improving or optimizing material properties, the influence of randomness of fiber length and orientation on mechanical properties is of great interest. It is natural to ask: How will the post-crack TSC vary if the fiber length is not constant but the mean fiber length is fixed? To which extent does the TSC change if the fibers have some special direction distribution and are not isotropically oriented? Obviously, there is a simple qualitative answer: Random fiber lengths imply that there are fibers which are longer than the mean fiber length. Therefore it might be expected that random fiber lengths lead to a TSC higher than for constant fibers. Furthermore,
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due to friction of fibers during pull-out, TSC will indeed change if the orientation of the fibers is not isotropic. Both effects are studied in the present paper. In order to demonstrate the application of our theory we use results from the literature about the frictional bond between fiber and matrix such as [14–16]. With the aid of such measurements the post-crack TSC can be studied quantitatively. The paper is organized as follows. The methods and assumptions made in order to derive the traction-separation law are presented in Section 2. There we define the mathematical notions, present an equation for TSC for general (joint) distributions of fiber length and orientation and also point out further extensions of the model. Finally, we investigate the impact of correlation of fiber length and orientation on the post-crack TSC in Section 4.
2. Model assumptions and methods 2.1. Model assumptions and extensions Throughout the paper we make the following model assumptions, which are mainly standard in the relevant literature, but some extensions are new. We consider a statistically homogeneous [17, p. 28] matrix material with randomly distributed fibers under the following conditions, see e.g. [3]: The matrix is discontinued by a planar crack of width d, see Fig. 1, its deformation during the fiber pullout is neglected. The spatial distribution of the positions of fibers in the composite is homogeneous and is independent of fiber length and orientation, the fibers are straight with cylindrical geometry. They behave linear elastically and rupture if their axial stress reaches the fiber strength σf ,max . The Poisson effect of the fibers during pull-out is neglected, the fiber–matrix bond is frictional and the elastic bond strength is neglected. As the crack opening d increases, the fiber ends are pulled out of the matrix. Eventually one fiber end is pulled out completely or the fiber ruptures due to high tension. Additionally we assume that fiber length and orientation are random. We describe this randomness by a two-dimensional probability density function (p.d.f.). This means that fiber length and fiber orientation are allowed to depend on each other, which allows a realistic approximation of a wide class of composite materials where one phase is built of fibers. Our approach can easily be combined with various models concerning the fiber pullout mechanism.
described by a reference point, length and angle w.r.t. the normal of some given crack plane. For convenience the crack plane is assumed to be the (x,y)-plane and the reference point of each fiber is its top point (in the sense of the z-axis). The fiber angle is the polar angle β of the fiber. Fig. 1 shows the underlying geometry of a single fiber that intersects a crack plane Ac with inclining angle β and embedded residual lengths r1 and r2. The random set of fibers is described by the following summary characteristics: 1. Nsp – mean number of fibers (i.e. fiber reference points) per unit volume. 2. fsp, L, B (l, β ) – joint p.d.f. of fiber length l and polar angle β in space. These characteristics are often known a priori and they are measurable e.g. by computed tomography, see [18,19], and [13,20]. If angles β and lengths l are statistically independent the joint p.d.f. is the product of the univariate p.d.f. fsp, L (l ) and fsp, B (β ) of length and angle. The parameter Nsp belongs to a group of summary mean-value characteristics, which include also fiber volume fraction Vf, mean fiber length l and fiber cross-sectional area Af. They satisfy the equation
Nsp =
Vf lAf
.
(1)
After the formation of a planar crack, i.e. when d ¼0, we are interested in the random embedded residual fibre lengths and the inclination angle w.r.t. the normal of the crack plane of the fibers which intersect the crack, see Fig. 1. We describe these quantities by the characteristics 1. Npl – mean number of fibers intersecting the crack plane per unit area and 2. fpl, R1, R2, B (r1, r2, β ) – joint p.d.f. of residual lengths above and below the crack plane and polar inclination angle. There are close mathematical relationships between these plane-related characteristics and the space-related characteristics of the fiber system. In particular, we have (cf. [21, Section 8.4])
Npl = Nsp (L sp cos Bsp ).
(2)
In this equation the expression π
∫0 2 ∫0
∞
2.2. Theoretical background, distribution of intersecting fibers
(L sp cos Bsp ) =
In order to fix notation we describe fiber systems and their characteristics in what follows. A system of fibers is a spatial set of line segments. Each fiber is
denotes the mean of l cos β . (We used here the notation L sp and Bsp for the random variables of length and polar angle.) For isotropically oriented fibers of constant length l0, i.e. the case studied
l cos β fsp, L, B (l, β ) dl dβ
Fig. 1. Left: A fiber intersecting the crack plane Ac having length l = r1 + r2 and inclining angle β. The reference point of the fiber is denoted by (x, y, z ) . Right: At crack width d the embedded fiber is being pulled out of the matrix.
S. Matthes et al. / Probabilistic Engineering Mechanics 45 (2016) 157–163
in [3], Eq. (2) simplifies to
Npl =
2.3. Bridging forces
1 l N . 2 0 sp
(3)
By the way, if length L sp and angle Bsp are independent of each other then also residual lengths and inclining angle are independent. In this case the expression for Npl simplifies to
Npl = Nsp (L sp ) (cos Bsp ).
(4)
Furthermore, [22] introduced a relationship between the p.d.f. of residual lengths r1 and r2 and inclination angle β and the p.d.f. of spatial fiber length l and angle β. In our notation this reads as
cos β fsp, L, B (r1 + r2, β )
fpl, R1, R 2, B (r1, r2, β ) =
(L sp cos Bsp )
(5)
π . 2
for r1, r2 ≥ 0 and 0 ≤ β ≤ In the particular case of isotropically oriented fibers the marginal p.d.f. of Bpl then equals π fpl, B (β ) = sin 2β, 0 ≤ β ≤ , 2 see also [21, Eqn. (8.64)]. We remark that an isotropically distributed spatial orientation of a fiber implies the p.d.f.
fsp, B ( β ) = sin β,
0≤β≤
159
π , 2
(6)
of its polar angle Bsp , see also [3, Eqn. (4)] and [21, Eqn. (8.61)]. Obviously the angles Bpl and Bsp do not have the same distribution (as erroneously assumed in [9]). The difference between these distributions can be explained as follows: The p.d.f. fpl, B accounts only for those fibers that intersect the crack plane whereas fsp, B takes every fiber of the system into consideration. A fiber is more likely hitting the crack plane if its polar angle is small, see Fig. 2. In many approaches to model single fiber pull-out the minimum embedded fiber length plays a decisive role. Therefore we may also consider the distribution of the minimum and maximum embedded length of an inclining fiber, rmin = min {r1, r2 } and rmax = max {r1, r2 }, respectively. Eq. (5) implies
fpl, m, M (rmin, rmax, β ) ⎧ 2 cos β fsp, L, B (rmin + rmax, β )r,min ≤ rmax ⎪ = ⎨ (L sp cos Bsp ) . ⎪ ⎩ 0, otherwise
(7)
In order to model the TSC for a FRC the mechanics of a single fiber intersecting the crack plane is considered. Here two different approaches exist: a stress-based approach (e.g. [23,24]) and an energy-based bond-slip approach (e.g. [25,26]). These are characterized by specific stress-slip relations which depend on the crack width d, the inclination angle β and the residual fiber lengths r1 and r2, or the minimum and maximum embedded length rmin and rmax . The fiber–matrix debonding is described by a rate-independent contact law. Such laws have been proposed by [14,27,3,5,16] for fiber-reinforced cementitious composites. These papers established various expressions for the pull-out force P = P (r1, r2, β , d ) of a single fiber. The influence of the inclination angle on the pull-out behavior was discussed in [28,29], which both used the Euler–Eytelwein or Capstan formula
P (r1, r2, β, d) = P (r1, r2, 0, d) e βμs .
(8)
In this equation μs is the so-called static friction or snubbing coefficient. This well-known representation of the dependence of pull-out force on inclination angle will be henceforth used in this paper. However, more complex formulations and detailed physical concepts could be easily integrated into our general approach, see Section 3. For instance, [28] measured the pull-out force of steel fibers out of a cementitious matrix at various inclination angles and found significant deviations from the Euler–Eytelweyn formula which is caused by matrix spalling and crack plane deflection during fiber pull-out. In [30] these effects are theoretically investigated.
3. The proposed composite tension-softening model In this section we combine some of the ideas above in order to derive a model for the post-crack TSC, which describes the behavior of sc, the mean resistance force per unit area of the FRC material as a function of the crack width d. Each fiber which crosses the crack plane contributes to the crack-opening resistance force. Integrating the single-fiber pullout force contributions P (r1, r2, β , d ) over all (r1, r2, β ) weighted by their likelihood fpl, R1, R2, B (r1, r2, β ) and multiplying by the mean number of such fibers per unit area, Npl , yields
Fig. 2. Comparison of the spatial p.d.f. of the polar angle Bsp of isotropically oriented fibers (left) and the p.d.f. of the inclining angle Bpl of isotropically oriented fibers hitting the crack plane (right).
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∞
π
∞
∫0 ∫0 ∫0 2
σc (d) = Npl
P (r1, r2, β, d) fpl, R1, R 2, B (r1, r2, β ) dβ dr1
(9)
dr2,
as the mean resistance force per unit area at crack width d. At this point randomness of residual lengths and inclination angles is directly accounted for. Rewriting Eq. (9) in terms of spatial p.d.f. fsp, L, B and mean number of fibers per unit volume, Nsp , by inserting (2) and (5) into (9) yields the equation ∞
σc (d) = Nsp
∞
π
∫0 ∫0 ∫0 2
P (r1, r2, β, d) fsp, L, B (r1 + r2, β )
× cos β dβ dr1 dr2.
(10)
Using (7), Eq. (10) can be further rewritten in terms of minimum and maximum embedded length as ∞
σc (d) = 2Nsp
∫0 ∫0
rmax
π
∫0 2
P (rmin, rmax, β, d)
× fsp, L, B (rmin + rmax, β ) × cos β dβ drmin drmax.
(11)
Likewise we obtain ∞
σc (d) = 2Nsp
π
l
∫0 ∫0 2 ∫0 2
P (rmin, l − rmin, β, d) × fsp, L, B (l, β )
cos β drmin dβ dl
(12)
from (11) after applying the change of variables l = rmin + rmax and changing the order of integration. The derived relationships generalize earlier approaches such as the mathematical formulation of TSC in [3] for constant fiber lengths and isotropic orientation. Indeed it can be shown that the equation
Vf σc (d) = Af
l0 2
∫0 ∫0
⎛ 2z ⎞ arccos ⎜ c ⎟ ⎝ l0 ⎠
⎛ l zc l zc P⎜ 0 − , 0 + , β, ⎜2 cos β 2 cos β ⎝
× fsp, B (β ) fsp, Z (z c ) dβ dz c ,
Fig. 3. Schematic diagram of the distribution of the axial force in a fiber during two-sided pull-out using the Euler–Eytelweyn formula.
⎞ d⎟ ⎟ ⎠ (13)
which can be found in [3], is equivalent to (10) if fiber lengths are constant, the orientations of fibers are independent of their lengths and fsp, B (β ) = sin β , e.g. the fibers are isotropically oriented.
4. Numerical study of the effect of randomness on the TSC We now examine the behavior of TSC if fiber lengths are random by applying Eq. (11). For convenience we use established models for the distributions of fiber length and orientation in composite materials (see [8,31]). Due to the process of manufacturing of fibers, specific types of length distributions ensue. Empirical investigations were carried out for different combinations and products (e.g. [32–34]) and various models for these distributions have been conceived. The mechanical model for the single fiber pull-out force is chosen to be the same as in [3] in order to compare our results with the older ones. The pull-out force of a single fiber then reads as
⎧ τ (s ) πdf (rmin − d) e μs β , d ≤ rmin P (rmin, rmax, β, d) = ⎨ , else ⎩ 0, ⎪
(14)
where df is fiber diameter, μs is snubbing friction coefficient, τ (s ) is bond strength of fiber–matrix interface and s is the slippage of fiber out of matrix. We choose τ (s ) = τ (d ), i.e. expression (14) describes a fiber pull-out mechanism without elastic deformation and fiber slippage only at the shorter embedded end (Fig. 3). The function τ (s ) can be measured by single fiber pull-out
experiments and was investigated extensively in the literature (e.g. [14,28,35,36,16]). In order to study different forms of τ (s ) we use data of the matrix-fiber bond strength measured in single fiber pull-out experiments published in [3]: τ (s ) = 1.02 MPa for constant interface strength, τ (s ) = 1.02 + 0.2s + 0.2s2 MPa for a sliphardening model (matches certain synthetic fibers) and τ (s ) = 1.02 − 0.2s + 0.01s2 MPa for a slip-weakening model (matches certain steel fibers). These three interface strength models and the corresponding TSCs based on (11) are shown in Fig. 4 for a constant fiber length of 12.7 mm as in [3], where also fiber breakage at an ultimate fiber strength of σf ,max = 2.6 GPa is included. The corresponding TSCs are compared to those in the case of random lengths in the next section. 4.1. Effect of randomness of fiber lengths The following results show that in the case of random fiber lengths the initial post-peak strength and shape of the post-peak TSC changes and deviates significantly from the corresponding curve for constant fiber length. A commonly used model is the Weibull distribution which has the p.d.f.
⎧ ⎛ ⎞m − 1 ⎛ l ⎞m ⎪m⎜ l⎟ exp (−⎜ ⎟ ), l ≥ 0 fsp, L (l) = ⎨ k ⎝ k ⎠ ⎝ k⎠ ⎪ else ⎩ 0,
(15)
with positive parameters m and k, see [37]. It becomes narrower with increasing m (the shape parameter, or Weibull modulus) while the mean becomes larger with increasing k (the scale parameter).
S. Matthes et al. / Probabilistic Engineering Mechanics 45 (2016) 157–163
161
Fig. 4. Three models for fiber-matrix slippage (left) and predicted TSCs for composites with two-sided fiber pull-out during crack opening (right). The following parameters are used: df = 38 μm , μs = 0.7 rad−1, the constant fiber length is 12.7 mm as in [3].
Fig. 5. Weibull probability density functions for fiber lengths with fixed mean l = 12.7 mm (left) and the corresponding TSCs based on (11) in comparison with the constant fiber length model (right). In all cases a constant interface strength of τ (s ) = 1.02 MPa was assumed.
We study the impact of randomness of fiber lengths on the mechanical response by comparing TSCs for various Weibull distributions with fixed mean l = 12.7 mm (as in [3]). The underlying probability density functions and the corresponding TSCs are shown in Fig. 5. Here we used isotropically oriented fibers with a constant interface strength τ (s ) = 1.02 MPa and the constituent parameters df = 38 μm and μs = 0.7 rad−1 and assumed that fiber length and orientation are independent. As the distribution of fiber length becomes broader the contribution of long fibers to the TSC becomes more significant and the composite strength increases. This effect is also visible for d ¼0, i.e. for the strength of the composite material at crack formation.
and fiber breakage is neglected. These assumptions are fulfilled in composites with a weak matrix reinforced with strong fibers. The TSC of this composite material can then be written as
4.2. Effect of anisotropy and preferential directions of the fiber orientation
which compares the TSC for a specific orientation distribution to the TSC for isotropically oriented fibers. If K ¼1 the post-peak strength of a fiber reinforcement with orientation distribution fsp, B (β ) is equal to that of isotropic fibers for same length dis-
In the following we discuss how the post-peak TSC of a FRC material is influenced by the kind of randomness of the 3D fiber orientations, where we are interested in the case when the fiber orientation becomes anisotropic. In order to illustrate this effect in a better way we assume the following: fiber angle and length are independent of each other, the single fiber pull-out force P is the product of a term PB which only depends on β and a term PR which only depends on the residual lengths and the crack opening width
σc (d) = Kσc,iso (d),
(16)
see e.g. [38]. Here K is an orientation factor π
K=
∫02 PB ( β ) fsp, B ( β ) cos β dβ π
∫02 PB ( β ) sin β cos β dβ
,
(17)
tribution. If K > 1 or K < 1 the fiber orientation leads to a postpeak strength higher or lower than isotropic reinforcement respectively. The orientation factor K depends only on PB and fsp, B and is independent of the fiber length distribution. It therefore describes the impact of anisotropy on the TSC. The TSC for a isotropically oriented fiber system is
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S. Matthes et al. / Probabilistic Engineering Mechanics 45 (2016) 157–163
Fig. 6. The parametric fiber orientation density function fV , B (β ) as proposed by [19] for different choices of c. Note that for c ¼1 the fibers are isotropically oriented.
Fig. 7. Values of K in dependence of the parameter c, i.e. for different orientation distributions and different snubbing friction coefficients μs.
π
σc,iso (d) = 2Nsp
∫0 2
∞
∫0 ∫0
PB ( β ) sin ( β ) cos ( β ) dβ ×
rmax
PR (rmin, rmax, d) PB ( β )
× fsp, L (rmin + rmax ) drmin drmax.
(18)
In order to study the behavior of K we assume in the following that the angle-dependence of the single-fiber pull-out force is described by the Capstan formula, i.e. PB (β ) = e μs β , and use the following orientation distribution:
fsp, B (β ) =
c sin β
(1 +
(c2
− 1)
cos2
β)
3 2
,
0≤β≤
π , 2
c > 0, (19)
see [19]. This family of p.d.f. represents isotropically oriented fibers for c¼ 1, fibers which are nearly parallel to the crack plane for large c and nearly aligned fibers orthogonal to the crack plane for small c. In Fig. 6 the family of p.d.f. is depicted for different choices of the parameter c. Fig. 7 shows the behavior of the orientation factor K in dependence of the orientation parameter c. For μs = 0 rad−1 the snubbing friction of fibers at the crack plane vanishes. The coefficient K therefore increases for decreasing c because fibers orthogonal to the crack plane intersect it more likely, see e.g. (4). For μs > 0 rad−1 the effect of snubbing friction of the fiber at the crack plane becomes more dominant, i.e. fibers with a large intersection angle have a stronger effect on the TSC although they intersect the crack plane less likely. 4.3. Effect of the dependence between length and orientation Finally, we apply Eq. (11) to the case of dependent fiber length and orientation. The following example is motivated by fiber reinforced autoclaved aerated concrete (AAC). During the swelling process of this material the swelling direction appears to be the preferential direction of longer fibers for certain volume fractions, see [18]. The orientation distribution of shorter fibers remains isotropic. In the following we use the process parameters for glass fiber reinforced AAC: we use (19) in order to model the anisotropy of long fibers for different parameters c. We choose the preferential direction of long fibers to be orthogonal to the crack plane.
Fig. 8. Result of a numerical simulation of TSC of a FRC with dependence between fiber length and orientation. Model parameters from Table 1 were used. The parameter c describes the degree of anisotropy of long fibers, see (19), while short fibers are isotropically oriented. Table 1 Model parameters for the TSC for dependent length and orientation. Short fibers Long fibers Ratio of the number of short fibers–long fibers Fiber diameter Fiber strength Fiber volume fraction μs τ Short fibers Long fibers
0 mm < l ≤ 3 mm 3 mm < l ≤ 15 mm 1:2 20 μm 600 MPa 1% 1 rad 1 0.7 MPa Isotropically oriented Anisotropically oriented
Fig. 8 shows the resulting TSC for the model parameters mentioned in Table 1 and different parameters c. For small c short fibers are isotropic while long fibers have a preferential direction orthogonal to the crack plane. The curve c¼ 1 represents a
S. Matthes et al. / Probabilistic Engineering Mechanics 45 (2016) 157–163
reference curve where all fibers, long and short, are isotropically oriented. If the angle of a fiber which crosses the crack plane is large it contributes to a high peak strength and eventually ruptures at crack formation. It does not contribute to crack opening resistance after peak strength. Therefore as the parameter c decreases TSC increases because more long fibers tend to intersect the crack plane at a small angle and do not rupture at crack formation. Naturally, the contribution of the short fibers remains constant. Therefore the number of long fibers contributing to the post-peak TSC increases with decreasing c. For c¼ 1 the slope of the TSC decreases as soon as most of the short fibers are pulled out. As c increases this effect vanishes because the contribution of long fibers becomes dominant. The interplay of correlation of fiber length and orientation and its effects on the TSC cannot be studied without the stochastic approach presented in this paper.
5. Summary and conclusions We presented a general stochastic approach to the micromechanical model for the post-peak TSC of FRC with brittle matrix. It is motivated by the need to consider the random nature of fiber reinforcement in such materials and allows even the incorporation of correlations between fiber length and orientation. Our main result is Eq. (11), which yields the mean value of the random TSC. Due to the modular nature of our approach, various results of micromechanical analyses (single fiber pull-out models (e.g. [36]), snubbing coefficients (e.g. [28])) can be employed in order to obtain composite post-peak TSCs for different materials and physical effects such as matrix spalling and snubbing as well as tension softening or hardening during fiber pull-out. In view of numerical material simulation the proposed model extends the study of failure behavior of FRC using representative anisotropic volume elements with prescribed fiber distributions. This allows simulations of materials with fiber-reinforcement with spatially dependent characteristics, which refines the use of stochastic finite element method or Monte Carlo methods [39]. The impact of the randomness of length and orientation on TSC was studied numerically. We compared TSCs of various configurations of fiber reinforcement and found that consideration of randomness of fiber length and orientation leads to post-peak TSCs which deviate significantly from curves obtained for models with constant fiber lengths and isotropic orientation. Randomness of fiber length tends to increase the TSCs while the influence of orientation is more complex. We note that it is not trivial to measure the parameters of the presented model. Our theoretical investigations and results should therefore be supplemented by experimental data. Our results show which effects may occur and may lead to effective methods of measurement for the TSC.
Acknowledgements The financial support of the ESF #100088076 programme is gratefully acknowledged. We also thank Dr. Uwe Mühlich for his valuable comments and hints.
References [1] Y. Wang, V. Li, S. Backer, Tensile properties of synthetic fiber reinforced mortar, Cem. Concr. Compos. 12 (1990) 29–40. [2] A. Ionita, A. Weitsman, Randomly reinforced composites properties, failure and aspects of material design, Probab. Eng. Mech. 21 (2006) 64–72. [3] V. Li, Y. Wang, S. Backer, A micromechanical model of tension-softening and bridging toughening of short random fiber reinforced brittle matrix
163
composites, J. Mech. Phys. Solids 39 (1991) 607–625. [4] M.F. Bäuml, Steigerung der Dauerhaftigkeit selbstverdichtender Betone durch den Einsatz von Polymerfaserkurzschnitt (Ph.D. thesis), Eidgenössische Technische Hochschule Zürich, 2002. [5] T. Pfyl, Tragverhalten von Stahlfaserbeton (Ph.D. thesis), Eidgenössische Technische Hochschule Zürich, 2003. [6] A. Paglia, A fiber pull-out based model for synthetic fiber reinforced concrete beams under a flexural load, Open J. Civ. Eng. 3 (2013) 202–217. [7] V. Li, S. Wang, Microstructure variability and macroscopic composite properties of high performance fiber reinforced cementitious composites, Probab. Eng. Mech. 21 (2006) 201–206. [8] W.K. Chin, Effects of fiber length and orientation distribution on the elastic modulus of short fiber reinforced thermoplastics, Polym. Compos. 9 (1988) 27–35. [9] S. Fu, B. Lauke, Effects of fiber length and fiber orientation distributions on the tensile strength of short-fiber-reinforced polymers, Compos. Sci. Technol. 56 (1996) 1179–1190. [10] N. Neves, G. Isdell, S. Pouzada, P. Powell, On the effect of the fiber orientation on the flexural stiffness of injection molded short fiber reinforced polycarbonate plates, Polym. Compos. 19 (1998) 640–651. [11] G. Zak, C. Park, B. Benhabib, Estimation of three-dimensional fibre-orientation distribution in short-fibre composites by a two-section method, J. Compos. Mater. 35 (2001) 316–339. [12] R. Stadie, Festigkeits- und Verformungsverhalten von kurzfaserverstärktem Porenbeton (Ph.D. thesis), Technische Universität Berlin, 2008. [13] C. Redenbach, I. Vecchio, Statistical analysis and stochastic modelling of fibre composites, Compos. Sci. Technol. 71 (2011) 107–112. [14] Y. Wang, V. Li, S. Backer, Analysis of synthetic pull-out from a cement matrix, Mater. Res. Soc. Symp. Proc. 114 (1988) 159–164. [15] M. Butler, V. Mechtcherine, S. Hempel, Experimental investigations on the durability of fibre-matrix interfaces in textile-reinforced concrete, Cem. Concr. Compos. 31 (2009) 221–231. [16] A. Caggiano, E. Martinelli, A unified formulation for simulating the bond behaviour of fibres in cementitious materials, Mater. Des. 42 (2012) 204–213. [17] S. Torquato, Random Heterogeneous Materials. Microstructure and Macroscopic Properties, Springer, New York, 2002. [18] G. Weidemann, R. Stadie, J. Goebbels, B. Hillemeier, Computed tomography study of fibre reinforced autoclaved aerated concrete, Mater. Test. 50 (2008) 278–285. [19] J. Ohser, K. Schladitz, 3D Images of Materials Structures, Wiley-VCH, Weinheim, 2009. [20] M. Teßmann, S. Mohr, S. Gayetskyy, U. Haßler, R. Hanke, G. Greiner, Automatic determination of fiber-length distribution in composite material using 3d ct data, J. Adv. Signal Process. 545030 (2010) 1–9. [21] S. Chiu, D. Stoyan, W. Kendall, J. Mecke, Stochastic Geometry and its Applications, 3rd ed., Wiley Chichester, UK, 2013. [22] S. Matthes, D. Stoyan, Planar sections through three-dimensional line-segment processes, Image Anal. Stereol. 33 (2014) 55–64. [23] A. Katz, V. Li, Inclination angle effect of carbon fibers in cementitious composites, ASCE J. Eng. Mech. 12 (1995) 1340–1348. [24] A. Ghavami, A. Abedian, M. Mondali, Finite difference solution of steady state creep deformations in a short fiber composite in presence of fiber/matrix debonding, Mater. Des. 31 (2010) 2616–2624. [25] P. Shah, C. Ouyang, Mechanical behavior of fiber-reinforced cement-based composites, ASCE J. Eng. Mech. 74 (1991) 2727–2738. [26] A. Fantilli, P. Vallini, A cohesive interface model for the pullout of inclined steel fibers in cementitious matrixes, J. Adv. Concr. Technol. 5 (2007) 247–258. [27] A. Naaman, G. Namur, H. Najm, J. Alwan, Bond Mechanisms in Fiber Reinforced Cement-based Composites, Technical Report, Department of Civil Engineering, University of Michigan, 1989. [28] V.C. Li, Y. Wang, S. B, Effect of inclining angle, bundling and surface treatment on synthetic fibre pull-out from a cement matrix, Composites 21 (1990) 132–140. [29] J. Zhang, V. Li, Effect of inclination angle on fibre rupture load in fiber reinforced cementitious composite, Compos. Sci. Technol. 62 (2002) 775–781. [30] A.M. Brandt, On the optimal direction of short metal fibres in brittle matrix composites, J. Mater. Sci. 20 (1985) 3831–3841. [31] M. Xia, H. Hamada, Z. Maekawa, Flexural stiffness of injection molded glass fiber reinforced thermoplastics, Int. Polym. Proc. 10 (1995) 74–81. [32] V. Gupta, R. Mittal, P. Sharma, Some studies on glass-fibre-reinforced polypropylene: I. Reduction in fibre length during processing, Polym. Compos. 10 (1989) 8–15. [33] K. Ramani, D. Bank, N. Kraemer, Effect of screw design on fiber damage in extrusion compounding and composite properties, Polym. Compos. 16 (1995) 258–266. [34] Å.L. Døssland, Fibre reinforcement in load carrying concrete structures (Ph.D. thesis), Norwegian University of Science and Technology, Trondheim, 2008. [35] J. Alwan, A. Naaman, W. Hansen, Pull-out work of steel fibers from cementitious composites analytical investigation, Cem. Concr. Compos. 13 (1991) 247–255. [36] G. Bao, Y. Song, Crack bridging models for fiber composites with slip-dependent interfaces, J. Mech. Phys. Solids 41 (1993) 1425–1444. [37] S.-Y. Fu, C.-Y. Yue, X. Hu, Y.-W. Mai, Characterization of fiber length distribution of short-fiber reinforced thermoplastics, J. Mater. Sci. Lett. 20 (2001) 31–33. [38] L. Jain, R. Wetherhold, Effect of fiber orientation on the fracture toughness of brittle matrix composites, Acta Metall. Mater. 40 (1992) 1135–1143. [39] P. Spanos, A. Kontsos, A multiscale Monte Carlo finite element method for determining mechanical properties of polymer nanocomposites, Probab. Eng. Mech. 23 (2008) 456–470.