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Procedia Materials Science 3 (2014) 357 - 362

20th European Conference on Fracture (ECF20)

Fatigue crack propagation simulating fibre debonding in cyclically

loaded composites

Roberto Brighentia, Andrea Carpinteria, Daniela Scorzaa

a Dept. of Civil-Environmental Engng & Architecture, Univ. of Parma, 43100 Parma - ITALY

Abstract

A partially debonded fibre can be analyzed as a 3-D mixed Mode fracture problem for which the fibre-matrix detachment growth - leading to a progressive loss of the composite's bearing capacity - can be assessed through classical fatigue crack propagation laws. In the present study, the above mentioned problem is firstly examined from the theoretical point of view, and the effects of the stress field in the matrix material on the SIFs (associated to the crack representing fibre-matrix detachment) are taken into account. Suitable fatigue crack propagation laws for mixed mode SIFs are employed in order to quantify the crack growth rate corresponding to the fibre-matrix debonding growth rate, while the matrix material undergoes a mechanical damage quantified through a Wohler-based approach to fatigue. A damage scalar parameter aimed at measuring the debonding severity during fatigue process is also introduced. Finally, some numerical simulations are performed, and the obtained results are compared with results found in the literature.

© 2014 Elsevier Ltd. Open access under CC BY-NC-ND license.

Selection and peer-reviewunderresponsibility ofthe NorwegianUniversity ofScience andTechnology (NTNU),Department of Structural Engineering

Keywords: Fibre-reinforced composites; Fatigue; Debonding; Damage; Crack growth.

1. Introduction

Composites can be defined as structural materials consisting of two or more constituents combined at a macroscopic level. Their classification is usually based on the kind of matrix material (polymers, metals, ceramics) and of reinforcing phase (fibres, particles, flakes). Fibre-reinforced composites (FRCs) are commonly used in advanced engineering applications due to their enhanced mechanical properties, such as tensile strength, fracture resistance, durability, corrosion resistance, wear and fatigue strength, with respect to the traditional plain materials (Jones, 1999; Mallick, 2007; Cheng, 2012). Such multiphase materials are characterized by mechanical properties

2211-8128 © 2014 Elsevier Ltd. Open access under CC BY-NC-ND license.

Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineering doi:10.1016/j.mspro.2014.06.061

depending on those of their constituents (the bulk material, or matrix, and the fibres), as well as on their reciprocal bonding at the fibre-matrix interface. As a matter of fact, typical damage phenomena (leading to a significant decrease of the mechanical performance), occurring in such materials under service loading, can be typically related to the fibre-matrix delamination (or debonding), fibre breaking and matrix cracking. Such damaging phenomena are particularly severe for structural components under repeated loading (Brighenti, 2004; Carpinteri et al., 2006).

The present research work deals with the development of a micromechanical approach for the assessment of the fatigue behaviour of short fibre-reinforced composites under fatigue loading. In order to take into account the main degrading effects occurring during cyclic loading, the progressive fibre-matrix detachment (considered as a progressive fracture phenomenon) as well as the decrease of the mechanical properties of the matrix material are evaluated. The proposed micromechanical model is used to perform some comparisons with experimental results, and the main degrading effects are quantified and discussed.

Nomenclature

Dc (ct*, R*, N) Damage parameter after N loading cycles with stress amplitude G * and stress ratio R *

Em, Ef Elastic modulus of the matrix and of the fibre, respectively

Em,0, Em (N) Young modulus for the undamaged matrix, and the reduced value after N load cycles

K ), Kn « °) , Kn (ct™) Mode I and Mode II SIFs due to the remote stresses <7™ , a7 , respectively

Lad , Lf Adhesion length of a partially debonded fiber, and fibre semi-length

l = Lf - Lad Fiber debonded length

N' Number of loading cycles to failure under stress amplitude <J *

y, ^f Fibre-matrix interface Poisson's ratio, and diameter of the fibre

o*,o*0 Generic stress amplitude, and fatigue limit of the matrix material

Tf (x) Fiber-matrix interface shear stress

rf ,f, rf,» Interface friction stress, and ultimate adhesion fiber-matrix interface shear stress

2. Mechanics of fiber-matrix detachment

2.1. Shear lag model

The fiber-matrix debondig could be examined through the classical shear lag model, initially proposed by Cox (1952) and widely used in the literature (Nairn, 1997), by considering a cylindrical portion of composite made by a fiber surrounded by a sufficiently large volume of matrix material, under remote tension stress acting parallel to the fiber direction (Fig. 1a). The corresponding fiber-matrix interface shear stress T^ (x) and the normal stress um (x) in the matrix acting in the fibre direction can be expressed as follows (Brighenti, 2004; Fig. 1b):

si: cos.

inh(/?- x) h (p- Lf)

°m (X) =

_ P - f (x) _ _F_

\a-E ■ A.-

1 cosh (/?• x) cosh (p ■ Lf j

(1a) (1b)

where c is the fiber perimeter, a = (Em ■ Am) 1 + (Ef ■ Af) \ /3 = 4 c ■ k-a , F = P /(a■ Em ■ Am), P being the total force applied to the composite region under study; Am, Af are the cross sections of the matrix and of the fibre.

Fig. 1. (a) Geometrical parameter of a cylindrical fiber. (b) Stress distributions along the fiber in a partial debonding stage. (c) Debonded extremity (3D cylindrical crack) of a fiber under remote radial ((J^ ) and axial ( (7^° ) stresses.

According to the shear lag model, the debonding phenomenon takes place when the limit value of the interface shear stress is attained at the extremities of the adhesion region, i.e. x = Lad (x = Lf for a complete bonded

fiber, Fig. 1c). In such a situation, Eqs (1) are still valid as long as the fibre semi-length Lf is replaced with half of

the bonded length Lad. The critical condition for debonding extension can be written as (Brighenti et al., 2012):

f ,max

= *f (x = Lad ) = — ■ tanhCtf- Lad)

= ст.

c ■ (a-Em)

• tanh(£- Lad ) = r

Tfи ■ С -a- Em p- tanh(fl- Lad)

(2a) (2b)

where Eq. (2a) represents the critical condition rewritten in terms of the remotely applied stress. Along the debonded length Lad < x < Lf (l = Lf -Lad), the shear stress arising at the interface can be assumed to be equal to

the friction stress zf f . However, such a residual shear strength can be reasonably neglected, as is done in the following, because its contribution to the composite bearing capacity is usually limited.

2.2. Fracture mechanics approach

The problem of an elastic bi-material plane with an interface crack has been widely examined in the literature (Rice, 1988; Hutchinson et al., 1987). The extension of such a problem to a 3D case can be used to describe the above-mentioned cylindrical crack arising in a partially debonded cylindrical fibre (Fig. 1c).

From the above remark, the detachment phenomenon can be deduced to be studied as a fracture mechanics problem: the debonded zone can be assumed as a 3D cylindrical crack lying between two different materials (Zbib et al., 1995; Chaudhuri, 2006). By considering the generic case of an elastic fiber embedded in an elastic matrix under

remote axial ( a™ ) and radial (CT^ ) stresses (Fig 1c), a mixed mode of fracture arises, and the energetically equivalent SIF can be defined as follows (Brighenti et al., 2013):

=Jv к v;

Ku К™) + Ki (p\ Ku «)

> 0 < 0

Note that, in first approximation, the remote axial stress produces only a Mode II SIF, while the remote radial stress is mainly responsible for Mode I and Mode II SIFs. The above SIFs, KI (0"™), Kn (0"™) , Kn (ct™), can be conveniently rewritten in a dimensionless form as K*Mw = KM ) , where KMw = KM(0-™) indicates

the generic Mode M SIF (M = I,II ) due to the remote stress cr™ ( w = r, z ), and K *Mw is the corresponding dimensionless value. The equivalent SIF Kt at the fiber-matrix interface crack front (see Eq. (3)) can be also employed to define the condition of unstable crack propagation, that is to say: K{ = Kcc =t]E{ ■ Gic plane stress; E i ■ G ic /(1 plane strain, where Gic is the interface fracture energy and

Kic is the corresponding fracture toughness, whereas Et and ia are the Young modulus and the Poisson ratio of the interface, respectively (Brighenti et al., 2013). Such a fracture mechanics approach allows us to use the classical crack growth rate equations for the assessment of the stable crack propagation due to repeated loading.

3. Mechanical model of fiber-reinforced composite under cyclic loading and numerical implementation

The fatigue life assessment of fiber-reinforced materials is a complex task due to the different damaging mechanisms occurring in the matrix, in the fibers and at the fiber-matrix interface. The cyclic loading reduces the mechanical properties of the composite by decreasing the matrix mechanical properties and reducing the effectiveness of the fiber-matrix bond. As is mentioned above, the progressive fiber-matrix debonding can be evaluated through a fatigue propagation law applied to the growth of the detached length l of the fibers (Fig. 1c):

dl / dN = C -AK"

where C," are the Paris constants of the interface and AKi is the equivalent stress intensity factor range produced by the remote cyclic stresses. For the sake of simplicity, such stresses are assumed to be in phase in order to easily define an equivalent SIF range. On the other hand, the fatigue effect in a homogeneous material under uniaxial constant amplitude cyclic loads can be assessed by using the experimental Wohler diagrams (S-N curves), which determine the number of load cycles to failure for a given value of the load ratio R = CTmin /ct (Fig. 2a).

Such curves can be approximated through the following relationship:

N J {<y / A) 1/b = const • c

where A, B > 0 are Wohler fatigue constants of the material, and ct*0 is the fatigue limit under cyclic stress with

stress ratio equal to R * . The number N * of loading cycles to failure can be written as follows: N* = A1bct *~1b . Further, the damage parameter Dc quantifying the damage severity can be written as the ratio between the number

N of loading cycles, for a given stress amplitude and stress ratio, and the corresponding number N * of cycles to failure. Taking into account the fatigue history subdivided in several blocks with Ncycc^les for each block, the damage increment at the end of the i-th block can be written as follows, by using the corresponding stress amplitude

R = 1.0

Ncycles Ncycles ^ Ncycles ^ Nccycles Ncycles

Fig. 2. (a) Wohler's curves for different stress ratios R . (b) Constant amplitude stress cycles subdivided in Nbiocks with N l for each block.

<7 * acting on the matrix (note that, even for constant amplitude cyclic stress <y * , the damage increment is not constant due to the variable bearing effect of the fibres that progressively debond from the surrounding material):

\N , / N * = N , / (a*. / A)_1/5 < 1 if <r* >ct0

AD^ok(a*,,R*,N) = \ cycles cycles ^ ■ > *' ° (6)

' [o VN if <r *,. < a0

In the case of cyclic loads with er* < <r0, the damage is assumed to be equal to zero. The failure condition (fully damaged material) is reached when the damage parameter is equal to 1, i.e. Dc(o*,R*,N) = 1. The

mechanical properties of the matrix, such as the Young modulus, are worsened by the effect of fatigue loading (Brighenti, 2004; Avanzini et al., 2011):

Em(N) = Em,o • [1 - DEm (a*,R*, N)] (7)

where Em0 is the undamaged Young modulus of the material, and DE = Dc is the Young modulus damage written

as a piecewise linear function (see Eq. (6)) of the number of loading cycles. For multiaxial stress states, the previous equations can be applied by replacing cr * with the combined stress <req related to the yielding criteria of the matrix material. Alternatively, the principal stress amplitude can be also applied for the damage evaluation.

The proposed model is implemented in a FE code in order to verify its capability to estimate the fatigue behaviour of fiber-reinforced composites under cyclic loads. For the sake of its applicability in a computational approach, the remotely applied cyclic stresses crZ? and cr" are evaluated at the Gauss points location in each finite element by considering the fibres pattern in the composite. In other words, in the case of randomly distributed fibres and in the case of nearly unidirectional fibres, we have = cry° = <7, /3 and

(7™ = (k ® k): G, c™ = <Jt , / 2 , with <Jt,, the normal stress tensor components in a plane containing the fibre axis identified by the unit vector k .

In the case of fatigue loading, Eq. (4) is applied by considering the interface SIF range evaluated through the above stresses. In order to numerically evaluate the fibre detachment increment, the fatigue growth equation is applied at the end of each block of the whole stress history (see dots in Fig. 2b):

M,-thblock = Ncycles ■ C, -AKm- (8)

Once the current debonded fibre length l(N) is known, the sliding function parameter ) = Sf / sj (given by the ratio between fibre strain and matrix strain measured in the fibre direction, Brighenti and Scorza, 2012) can be evaluated, and the tangent elastic tensor C'eq of the homogenized material can be finally obtained:

C'eq = ^"C'm +V E f

ds(e mf)

•J pv(fp) • pe(0) ■ F ® FdO (9)

where jU, T] are the fibre and matrix volume fractions, C'm ,-E'f are the tangent tensor and the elastic modulus of

the matrix material (evaluated with Ef (N) ) and the fibres, respectively. Further, pv(<p),Pg(&) are the

probability distribution functions describing the fibres arrangement in the space, and F = k 0 k is a second-order tensor (Brighenti and Scorza, 2012).

4. Numerical simulation of experimental tests

The fatigue behaviour of a 20% glass (randomly distributed) fibre-reinforced polycarbonate specimen under constant amplitude uniaxial cyclic stress is herein examined (Zago et al., 2001). The material parameters are as

Fig. 3. (a) Wohler's curves a glass fibre-reinforced polycarbonate specimen (dimensions in mm): experimental and present results. (b) Damage evolution in the matrix and dimensionless debonding in the fibres (at point P) vs the number of stress cycles.

follows: Ct = 1.01 -10~4,mt = 3.1 ( dl/dN in mm/cycle, AKi in MPaVm), a0 = 5MPa,N0 = 2-106,B = 0.293,

2Lf = 4 -10"4m,0f = 10 ^m. The attainment of the ultimate matrix strain value (su = 10%) is assumed to be the

fatigue failure condition (Fig. 3a). The dimensionless cyclic stress amplitude a* / au (with au = 75MPa = tensile

strength of the composite) against the number of stress cycles is shown in Fig.3a, where a good agreement with some experimental results can be noted. In Fig. 3b, the damage value for a given value of cr * in the matrix material and the dimensionless detached length are plotted against N at point P (see Fig. 3a).

5. Conclusions

In the present study, a micromechanical-based approach to assess the fatigue behaviour of FRCs is proposed. The matrix and the fibre-matrix interface damages due to fatigue loading are accounted for. In particular, a fracture mechanics-based approach is adopted to describe the fiber debonding, whereas a Wohler approach is used to quantify the damage in the matrix. A damage scalar parameter quantifying the debonding severity during fatigue is introduced. Finally, some numerical simulation results are compared with literature results. The present model seems to catch the main degrading mechanical effects of repeated loading on FRCs.

References

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