Debonding at the fibre–matrix interface under remote transverse tension. One debond or two symmetric debonds?

European Journal of Mechanics A/Solids 53 (2015) 75e88

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Debonding at the fibreematrix interface under remote transverse tension. One debond or two symmetric debonds? I.G. García a, b, *, V. Manti
c b, E. Graciani b a nica y Disen diz, Avenida de la Universidad de Ca diz 10, ~ o Industrial, Escuela Superior de Ingeniería, Universidad de Ca Departamento de Ingeniería Meca diz, Spain 11519 Puerto Real, Ca b Grupo de Elasticidad y Resistencia de Materiales, Escuela T ecnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimienos s/n, 41092 Sevilla, Spain

a r t i c l e i n f o

a b s t r a c t

Article history: Received 4 June 2014 Accepted 26 February 2015 Available online 7 March 2015

The controversy about the symmetry of the debond onset at the fibreematrix interface in single-fibre specimens under transverse tension is studied here applying the coupled stress and energy criterion of the finite fracture mechanics. This criterion enables to compare the two different post-failure configurations found in the literature studying this problem: an asymmetric configuration with a single debond and a symmetric one with two debonds. The coupled criterion applied here predicts that an asymmetric post-failure configuration is originated by a lower critical remote tension than the symmetric one, the difference being above 10% in some cases. Thus, the asymmetric debond onset is the preferred solution to the initially symmetric problem, which agrees with the experimental evidences found in the literature. This result is a consequence of the shielding effect between the two debonds in the symmetric solution. Numerical results are obtained for two common composites, glass/epoxy and carbon/epoxy, and three virtual bimaterials corresponding to the extreme values of the Dundurs elastic parameters. The stressestrain curves predicted for the two post-failure configurations are also compared. Eventually, simplified models using springs and breakable elements, which capture the essence of the problem studied and explain the energetic origin of the loss of symmetry, are introduced. © 2015 Elsevier Masson SAS. All rights reserved.

Keywords: Fibreematrix debond Finite fracture mechanics Symmetry breaking

1. Introduction Structural applications of composites have recently increased dramatically in industries where light-weight is a key aspect of design. Moreover, composites are now often applied in structurally relevant parts, e.g., in primary structures in aerospace applications. Thus, a substantial improvement of the existing failure criteria for composites is highly demanded by the industry. In particular, the failure criteria still do not predict satisfactorily failures of unidirectional laminae under transverse loads (with loads perpendicular to the fibre-axis), as was highlighted in a series of coordinated studies (known as the world-wide failure exercises) currently underway (Hinton et al., 2004; Hinton and Kaddour, 2013). A reason for this is that the current failure criteria are still not sufficiently physically based from the microscopic point of view. nica y Disen ~ o Indus* Corresponding author. Departamento de Ingeniería Meca diz, Avenida de la Unitrial, Escuela Superior de Ingeniería, Universidad de Ca versidad de C adiz 10, 11519 Puerto Real, C adiz, Spain. E-mail address: [email protected] (I.G. García). http://dx.doi.org/10.1016/j.euromechsol.2015.02.007 0997-7538/© 2015 Elsevier Masson SAS. All rights reserved.

The so-called matrix failure or inter-fibre failure is one of the most complex failure mechanisms in these laminae due to its strong dependence on the microstructure. At micromechanical level, the stages of the initiation of this failure are well known (Hull and Clyne, 1996; París et al., 2007): (i) failure is initiated as microdebonds at the fibreematrix interfaces, (ii) subsequently these microdebonds grow along the interface and (iii) kink out the interface towards the matrix, where (iv) coalescence of several microcracks generates a macrocrack which may cause the failure of the lamina. A simplified model given by a single long fibre surrounded by an infinite matrix is often used as a reasonable first approximation to study some failure mechanisms at the micromechanical level in fibre-reinforced composites, at least for dilute packing. Additionally, a few experimental studies of debond onset and growth in single fibre or inclusion specimens subjected to transverse loads € gren have been presented by Zhang et al. (1997), Gamstedt and Sjo (1999), Contreras (2000), and Martyniuk et al. (2013). In the case of matrix failure under remote transverse tension, the single fibre model leads to a plane strain problem of a circular

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inclusion embedded in an infinite matrix. The stress solution of such a problem for perfect and linear-elastic inclusion-matrixinterface, respectively, was deduced by Goodier (1933) and Gao (1995). Assuming the open model of interface cracks at perfect interfaces, analytic solutions for stresses, displacements and energy release rate (ERR) were obtained by England (1966) and Toya (1974) in the presence of a debond at the inclusionematrix interface. Although, assuming a more realistic contact model of interface cracks at perfect interfaces, no analytic solution is available for the present problem, accurate numerical solutions were presented in París et al. (1996) and Manti
c et al. (2006). Debond growing along the fibreematrix interface and kinking out, both originated by transverse loading, have been intensively analysed in many works, see e.g. París et al. (2007), Hasebe and Yamamoto (2014), and also He and Hutchinson (1989) for a general theory of interface crack kinking. However the initiation of the debond has not called so much attention. After some pioneering articles (Xu and Needleman, 1993; Levy, 1994, 1995), a few works studied the debond onset at the fibreematrix interface under transverse uniaxial loading by using different interface laws, e.g., Legarth (2004), Carpinteri et al. (2005), Han et al. (2006), Kushch vara et al. (2011). This idea has also been et al. (2011) and Ta applied to related problems, e.g., Tvergaard (2006) studied the effect of the triaxiality on the debond growth when the composite is mainly loaded along the fibre direction. Several of these works agree on some relevant results such as a size effect on the critical load, see García et al. (2014) for a recent comparison study. However, there is a disagreement on a key issue of the post-failure configuration (or post debond-onset configuration). For instance, some works (Carpinteri et al., 2005; Kushch et al., 2011) predict a symmetric debond onset whereas others (Levy vara et al., 2011) predict and Hardikar, 1999; Han et al., 2006; Ta breaking the symmetry of the original configuration due to the onset of a single debond. The loss of symmetry from perfectly symmetric initial conditions has also been predicted in other problems of fracture mechanics. For instance, the problem of a long strip containing a transverse crack symmetrically situated with respect to the strip edges (Ba
zant and Cedolin, 1991). The authors show that the asymmetric crack growth is preferential even if the initial state is perfectly symmetric. Some examples of loss of symmetry can also be found in Bigoni (2012). A model with (asymmetric) imperfections at the interface was studied in Legarth (2004) by assuming a non-uniform interface strength leading also to an asymmetric failure. The present work aims to analyse these two post-failure configurations observed in previous studies, an asymmetric debond and two symmetric debonds, by means of the coupled criterion of the finite fracture mechanics (FFM). The objective is to assess which of them is to be expected to appear in experiments and clarify why. It is assumed that the lower critical remote tension is predicted for the preferential post-failure configuration. The initial state, previous to the crack onset, is assumed to be perfectly symmetric in geometry, material properties and loading. Thus, the results obtained should be understood as a qualitative tendency in more realistic cases with initial asymmetries due to the natural randomness of the problem conditions. The natural presence of small defects or other sources of asymmetry would slightly modify the quantitative results, but no consequence is expected on the qualitative results about the failure symmetry. Actually, the objective of this work is not only to provide predictions about the failure in the present particular problem but also to give a conceptual answer, which can be applied to many other problems. The coupled stress and energy criterion was proposed by Leguillon (2002) in the context of crack initiation, see also Cornetti et al. (2006), and has been applied in a semianalytical manner by

Manti
c (2009), Manti
c and García (2012) and Carraro and Quaresimin (2014) to the present problem assuming a single debond, the configuration usually observed in composites damaged under transverse loads (Hull and Clyne, 1996; Zhang et al., 1997; €gren, 1999). In addition, it has been applied sucGamstedt and Sjo cessfully to predict crack onset in related problems by Quesada et al. (2009) and Camanho et al. (2012). This criterion is based on the assumption that the crack or debond onset occurs if two criteria are fulfilled simultaneously: a stress condition on the stresses along the future crack path, and an energetic condition based of the first law of thermodynamics for the energetic balance between the states before and after the crack onset, corresponding to an asymmetric debond and two symmetric debonds, in order to compare the predicted critical remote tension for each of them. The plane strain problem under study is shown in Fig. 1. In the initial state, a single fibre is perfectly bonded to an infinite matrix. The matrix is loaded by a uniaxial remote tension s∞ in the x-direction. For a priori two different critical values of s∞, one or two debonds appear as showed in Fig. 1. As both initial state and postfailure configurations are symmetric with respect to the x-axis, only the upper-half (y
0) of the geometry is considered. Hence, the polar angle q and also other angles are defined as
0. Glass/ epoxy composite is taken as a reference bimaterial in the present work, see Table 1 for its linear elastic properties, although, a summary of key results is also shown for carbon/epoxy composite (Table 2) and for a few virtual materials with extreme values of the main parameters governing the problem. First, stress and energy criteria are studied and applied separately in Sections 2 and 3, respectively. Both criteria are applied in a parallel manner for the two post-failure configurations. The combination of both criteria is described in Section 4. Finally, the results about the loss of symmetry are discussed in Section 5 for several bimaterials, and a novel representation of crack onset by the stressestrain curves predicted for the two post-failure configurations is introduced and interpreted.

2. Stress criterion A stress criterion is usually invoked for brittle or quasi-brittle materials when no pre-existing damage exists. In the framework of the FFM, the stress criterion defines a condition on stresses along the assumed future crack surface in the elastic state prior to the crack onset. Several stress criteria have been used for this purpose. While the tensile criterion requires that normal tractions at all the points along the future crack surface exceed a critical value (Leguillon, 2002; Manti
c, 2009), the mixed stress criterion (García and Leguillon, 2012; García et al., 2014; Carraro and Quaresimin, 2014) takes into account the influence of shear tractions as well. On the other hand, a weaker condition was proposed by Cornetti et al. (2006) assuming that only the mean value of normal tractions along the future crack extension has to exceed the critical value, see also Camanho et al. (2012). The tensile criterion is used in the present study for the sake of simplicity. Assuming a critical value sc > 0 for normal tractions s(q) along the interface before the debond onset, the condition for the onset given by this criterion can be expressed as,

sðqÞ
sc ;

cq2½0; Dq;

(1)

where q is the polar angle, see Fig. 1, and Dq is the debond semiangle immediately after the onset. Analytical expression of the normal tractions s along the fibreematrix interface in the undamaged state can be extracted from the classical solution by Goodier (1933), cf. Manti
c (2009),

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77

Fig. 1. Schema of the problem under study.

Table 1 Elastic properties of the composites used. Ei: Young's modulus (i ¼ 1, fibre; i ¼ 2, matrix). ni: Poisson's ratio. a and b: the Dundurs elastic parameters, k and m: Elastic parameters characterizing the elastic solution around a fibre perfectly bonded to an infinite surrounding matrix, defined in Manti
c (2009). Bimaterial

E1 (GPa)

n1

E2 (GPa)

n2

a

b

k

m

Glass/epoxy Carbon/epoxy

70.8 13.0

0.22 0.20

2.79 2.79

0.33 0.33

0.919 0.624

0.229 0.136

1.44 1.32

1.56 1.43

sðqÞ ¼ k  msin2 ðqÞ; s∞

(2)

where k and m are dimensionless elastic bimaterial properties defined by Manti
c (2009) in terms of the Dundurs (1969) parameters a and b in plane strain,

kða; bÞ ¼

1 1þa 2þab ; 2 1 þ b 1 þ a  2b

mða; bÞ ¼

1þa : 1þb

(3)

Two identical maxima of s(q), see Fig. 2, are found at q ¼ 0 and 180 . These two points are the preferred points for the debond initiation which justifies the two post-failure configurations assumed here. In the following, due to the symmetry of the initial state with respect to the y-axis, only the half part for x
0 is studied in order to unify the analysis for the two post-failure configurations. In the case of the symmetric configuration, the two debond

Fig. 2. Normal tractions s along the fibreematrix interface at the initial state for glass/ epoxy.

onsets are obviously allowed when the stress condition is fulfilled on one of the sides. Taking into account that s(q) is a decreasing function for 0 q90 , see (2), and assuming that Dq90 , the condition in (1) is verified if s(Dq)
sc. Then, introducing (2) into (1), the expression of the stress condition for the debond onset is obtained

s∞ 1 :
sðDqÞ ¼ sc k  msin2 Dq

(4)

This gives a minimum remote tension s∞ necessary to originate the debond as a function of the debond semiangle Dq. Fig. 3 shows how the stress criterion splits the (Dq,s∞/sc) plane into unsafe and safe region where the debond onset is allowed or not, respectively. The stress criterion curve for glass/epoxy in Fig. 3 has a vertical asymptote at q0 ¼ 73.63 , thus no debond onset with Dq
q0 is possible for any remote tension. The reason is that s(q)0 for q
q0, see Fig. 2. A discussion of this issue for any bimaterial can be found in Manti
c and García (2012). As the stress criterion is based exclusively on the initial (undamaged) elastic state, it is quite obvious that the two post-failure configurations are equivalent with respect to this criterion.

Fig. 3. Graphical representation of the stress criterion for the two post-failure configurations and glass/epoxy.

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3. Energy criterion An incremental energy criterion, in contrast to the classical (infinitesimal) Griffith criterion, is considered in FFM by evaluating the energetic balance between the elastic states before and after the debond onset for the two post-failure configurations. The first law of thermodynamics, neglecting heat transfer, gives

DP þ DEk þ DG ¼ 0;

(5)

where DP and DEk, respectively, are variations of the elastic potential and kinetic energy and DG is the energy dissipated in the irreversible processes associated to the debond onset. Assuming an initial static state implies DEk
0 since a part of the potential energy might go to the kinetic energy. Thus (5) can be rewritten as

DP
DG;

(6)

which means that the released energy has to exceed the dissipated energy during the debond onset. In the next subsections DP and DG will be obtained as functions of the debond semiangle Dq, the bimaterial and interface properties and the post-failure configuration. 3.1. Released energy The increment of the elastic potential energy DP as a function of the debond semiangle Dq can be efficiently and accurately evaluated by using the energy release rate G (ERR). Assuming a continuous debond growth sequence between the absence of crack and the state with a debond with semiangle Dq, the two magnitudes can be related by the next expression,

ZDq DP ¼ 2n

Gðqd ; nÞadqd ;

(7)

0

where G gives the ERR of n debonds growing from q ¼ 0 symmetrically with respect to the x-axis, with qd being the semiangle of an intermediate debond employed only to compute DP. Note that the choice of the sequence of growing of the intermediate debond to calculate DP in (7) is arbitrary since DP for a Dq and n given is independent of the sequence chosen because DP is derived from a potential. To evaluate DP by (7) we need to know G(qd,n) for any 0qdDq. Toya (1974) deduced an analytical solution for the case n ¼ 1 using the open model of interface cracks. However, to the best of our knowledge no analytic solution is available for the case n ¼ 2, though accurate numerical results, assuming the open model of interface cracks, were presented by Chen and Nakamichi (1997) and Murakami (2001). Nevertheless, following París et al. (1996) and Manti
c et al. (2006), the values of G computed assuming the open model of interface cracks are valid only for sufficiently small debond lengths. Since relatively large closed zones are expected for large debonds, the possibility of frictionless contact at the interface crack tips is considered here, which requires a computational code able to solve the pertinent nonlinear receding contact problems and accurately compute G, for any debond length. For given bimaterial elastic properties, G depends, in addition to n and qd, on the remote tension s∞ and the geometry, completely defined by the fibre radius a. In view of the large amount of physical variables and in order to reduce the number of numerically solved cases required to obtain G, the VaschyeBuckingham P theorem (Vaschy, 1892; Buckingham, 1914) is applied. Thus, as demonstrated b can be defined as, in Appendix A, a dimensionless ERR G

b Gðq d ; n; a; bÞ ¼

E aðs∞ Þ2

Gðqd ; n; s∞ ; a; E ; a; bÞ;

(8)

where E* is the harmonic mean of the effective elastic moduli,

! 1 1 1  n21 1  n22 ¼ þ : E 2 E1 E2

(9)

As the first part of the present study is carried out for glass/ epoxy, the dependence on a and b will be omitted for the sake of brevity. The elastic problem for each value of qd and n is solved by the Boundary Element Method (BEM) code developed by Graciani et al. (2005). This code uses continuous linear boundary elements for the solution of frictionless contact problems of interface cracks. The BEM model is defined by a square cell of matrix significantly larger than the circle at its centre representing the fibre. Nodes outside the debonded zone are considered “perfectly bonded” whereas “possible frictionless contact” is assumed at nodes inside the debonded zones, which means that these nodes can either be open or in frictionless contact. Due to the unilateral character of contact, the sizes of the open and closed zones are not a priori known and are calculated by the code using an iterative algorithm. A mesh suitably refined at the crack tips is used, with element lengths at the interface varying from 2 , far from the crack tips, to 0.00004 , at the crack tips. Normal traction s∞ > 0 is imposed at two parallel external edges of the square cell, whereas the other two edges are free. A detailed description of the geometry, mesh and boundary conditions can be found in García et al. (2012). The values of interface tractions and relative displacements computed by BEM in the vicinity of the crack tips are used to esb by the Virtual Crack Closure Technique (VCCT) with the aid timate G of a Chebyshev-Gauss quadrature, see Graciani et al. (2010) and García et al. (2012) for details. b as a function of qd and n, Fig. 4 shows the computed values of G and includes the analytical solution by Toya (1974) for checking b computed for q (65+ and n ¼ 1 are very purposes. The values of G d similar to the analytical values. However, for larger values of qd, a significant contact zone appears at the debond tips, that is not captured by the analytic solution assuming the open model of b by Toya's interface cracks, leading to an overestimation of G solution.

b as a function of the debond semiangle qd Fig. 4. Dimensionless energy release rate G for glass/epoxy, obtained by the VCCT applied, with a virtual crack increment angle of dq ¼ 0.5 , to the BEM results for each value of qd and n, compared with the solution by Toya (1974) for n ¼ 1.

I.G. García et al. / European Journal of Mechanics A/Solids 53 (2015) 75e88

b values for n ¼ 1 and 2 shows that they are A comparison of G larger for n ¼ 1 at any value of qd > 0. The difference is quite relevant for large values of qd due to the shielding effect between the two debonds for n ¼ 2. This effect is a consequence of the interaction between the two debonds, which may be quite strong for sufficiently large values of qd. The presence of a second debond produces a weaker stress field around the debond in comparison with the single-debond case. Consequently, the near tip stresses and the energy release rate are reduced with respect to the single-debond b for n ¼ 2 tends to zero for qd ¼ 90 , since it corcase. Moreover, G responds to a symmetric situation of two shear cracks approaching each other along the same line. The previous qualitative arguments are independent of the bimaterial properties, so the differences between the cases n ¼ 1 and 2 observed in Fig. 4 can be extrapolated to other bimaterials. 3.2. Dissipated energy The energy dissipated during the debond onset due to the associated irreversible processes can be estimated by integrating the interface fracture toughness corresponding to an intermediate debond semiangle qd as proposed in Manti
c (2009),

ZDq DG ¼ 2n

Gc ðjðqd ; nÞÞdqd ;

(10)

0

where the angle j is a stress-based measure of fracture mode mixity at the debond tip for qd. This expression is analogous to that for the released energy (7). However, in contrast with the previous one, in this case the value of the integral does depend on the growth sequence chosen for the intermediate debond. In the present case, the debond growth sequence described above (a debond symmetrically situated with respect to x with the intermediate debond semiangle qd growing from 0 to Dq) for the computation of DP is assumed. The estimation of the dissipated energy in presence of fracture mode mixity has been discussed by García and Leguillon (2012) and Carraro and Quaresimin (2014). The dependence of Gc on j can be approximated by the phenomenological law due to Hutchinson and Suo (1992), which provides a good agreement with available experiments for interface cracks,





b c ¼ G 1 þ tan2 ½ð1  lÞj ; Gc ¼ G1c G 1c

(11)

where G1c is the interface fracture toughness in pure mode I, l is a fracture mode-sensitivity parameter, with a typical (but not b c is a dimensionless interface exclusive1) range 0.2  l  0.35, and G fracture toughness. The fracture-mode-mixity angle j can be calculated by using the interface tractions obtained from the computational model

tðqd þ ql Þ ; tanjðqd ; ql Þ ¼ sðqd þ ql Þ

(12)

where ql is the distance from the debond tip where the fracture mode mixity is measured. Following a previous work (Manti
c, 2009), ql ¼ 0.1 is taken. Fig. 5(a) shows the value of j as a function of qd for the two postfailure configurations. In general the angle j is very similar for the two configurations, though somewhat larger for n ¼ 2, except for the extreme where qd /90 . At this extreme, a closed crack with ja90 is expected for n ¼ 1, whereas j/180 for n2 > 0 and n ¼ 2 due to the symmetry of the limit configuration and the Poisson effect of the matrix. The ratio of the interface fracture toughness to its value in pure b c is plotted in Fig. 5(b) as a function of qd, for the opening mode G above values of j and l ¼ 0.3. As could be expected, in view of b c is somewhat greater for n ¼ 2 than for n ¼ 1 Fig. 5(a) and (11), G and this difference increases with qd. Fig. 5(b) also shows the perb c between n ¼ 1 and n ¼ 2, confirming that centage difference in G for typical values of l, the above observations hold. Anyway, the b c appears to be in general smaller than in G, b relative difference in G the latter providing the main contribution to the difference in the energetic balance between the two configurations.

3.3. Final expression for the energy criterion Both terms in the energetic balance (6) have been analysed and computed above. This subsection aims to collect the main results in order to achieve an expression analogous to that obtained for the stress criterion in (4). Introducing the expression of the released energy (7) and the dissipated energy (10) in the energetic balance (6) and taking into b in (8) and G b c in (11), this balance leads account the definitions of G to

ðs∞ Þ2 a 2n E

ZDq

b Gðq d ; nÞdqd
2nG1c

0

Values out of this range can be found in the literature. For instance, experiments with interface cracks in glass/epoxy specimens by Banks-Sills and Ashkenazi (2000) showed a good agreement with the HutchinsoneSuo law taking l ¼ 0. In the context of FFM, Carraro and Quaresimin (2014) assumed the fracture toughness to be independent of the fracture-mode mixity, which corresponds to l ¼ 1, leading also to a good agreement with experiments in composites. Finally, the analysis by García et al. (2015) of experimental results carried out by Cho et al. (2006) showed that for the glass/polyester interface studied l z 0.11.

ZDq

b c ðjðq ; nÞÞdq : G d d

(13)

0

This inequality can be rewritten to the form of a condition for the normalized remote tension s∞/sc necessary to originate a debond of semiangle Dq, giving a minimum remote tension semiangle

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s∞
g gðDq; nÞ; sc

(14)

where

g¼

1 sc

rffiffiffiffiffiffiffiffiffiffiffiffi G1c E a

(15)

is a brittleness number, which modulates the influence of the stress and energy criteria as will be seen later on, cf. Manti
c (2009), and

Z gðDq; nÞ ¼

Dq

0

Z 0

1

79

b c ðjðq ; nÞÞdq G d d Dq

(16) b Gðq d ; nÞdqd

is a dimensionless function which measures the ratio of the dimensionless dissipated to released energies of the debond and depends on the parameters that determines the post-failure situation, i.e. Dq and n, and also on the dimensionless material parameters a, b and l. According to (14), the function g can be interpreted as the resistance against the debond onset when the energy criterion is considered only. The function g, computed by numerical integrations, is plotted in Fig. 6, showing that g(D,1) < g(D,2) with both

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b c ¼ Gc =G as a function of the debond semiangle qd Fig. 5. (a) Fracture-mode-mixity angle j and (b) interface fracture toughness normalized using its value in pure opening mode G 1c ð2 debondsÞ

for the two post-failure configurations and l ¼ 0.3, ql ¼ 0.1 and glass/epoxy. Percentage difference DGc ¼ 100$ðGc toughness for the symmetric

ð2 debondsÞ Gc

and asymmetric

ð1 debondÞ Gc

ð1 debondsÞ

 Gc

ð1 debondÞ

Þ=Gc

between the interface fracture

configurations.

the absolute and relative difference of these two values increasing with Dq (at least for the analysed Dq
1 ). It is interesting to observe that g has a minimum for Dq ¼ DqEmin < 90 . In the limit Dq/0 , g/ þ ∞ in accordance with the classical Griffith criterion for an infinitesimal advance of a crack.

4. Coupled criterion Both stress and energy criteria have been analysed previously in an independent manner leading to two conditions on the remote tension s∞ required for a debond onset. The two criteria are combined in this section assuming Leguillon's hypothesis (Leguillon, 2002): the critical remote tension s∞ c originating a debond onset is given by the minimum remote tension s∞ for which both criteria are fulfilled simultaneously for some value of Dq. An analysis of the monotonicity of the functions on the righthand-sides of (4) and (14), corresponding to the stress and energy criteria, respectively, is required to implement Leguillon's hypothesis in a semianalytical procedure. For the two post-failure configurations, the function s(Dq) in (4) is increasing with Dq, whereas g(Dq,n) in (16) is decreasing with Dq up to a minimum DqEmin . At the plane (Dq,s∞/sc) for Dq  DqEmin, the two curves given by these criteria have either one or none at all intersection point, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sðDqÞ ¼ g gðDq; nÞ, depending on the value of g, which leads to the following two scenarios:  Scenario A: If the two curves have one intersection point for Dq  DqEmin , the minimum remote tension fulfilling both criteria

for the debond onset s∞ c is given by this intersection point pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðDqc ; g gðDqc ; nÞÞ, Dqc being the critical semiangle of debond pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and s∞ c =sc ¼ g g ðDqc ; nÞ the critical remote tension for the debond onset.  Scenario B: If the two curves have none intersection point for Dq  DqEmin , the remote tension fulfilling both criteria for the debond onset s∞ c is given by the minimum value of g, the critical semiangle of debond being given by DqEmin, and the critical qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E remote tension by s∞ c =sc ¼ g gðDqmin ; nÞ. The scenario is determinated by the value of g since the energy criterion curve is proportional to this value. A threshold value of g can be defined as

  s DqEmin

; gth ¼  g DqEmin ðnÞ; n

(17)

which separates the scenario A (g  gth) from B (g > gth). Thus, for low values of g, the debond onset (qc and s∞ c =sc ) is governed by the combination of the two criteria, the critical remote tension being essentially given by the stress criterion, see (Manti
c and García, 2012), whereas the debond semiangle by the energy criterion. On the contrary, for large values of g, the debond onset is governed by the energy criterion only. An unstable growth of the debond is possible after its onset. This is studied here by means of the classical LEFM. Thus, the condition for a further growth of the debond is given by the Griffith criterion as

  G s∞ c ; qd ; n
Gc ðjðqd ; nÞÞ;

for

Dqc  qd  qa ;

(18)

where this unstable growth will be arrested for an arrest semiangle qa verifying still

Gðqa ; nÞ ¼ Gc ðjðqa ; nÞÞ; but

  dGðqd ; nÞ dGc ðjðqd ; nÞÞ < :   dqd dqd qd ¼qa qd ¼qa (19)

The critical values characterizing the debond as Dqc and qa are computed by algorithms developed in Manti
c (2009) and s∞ c =sc ,

Fig. 6. Dimensionless resistance g against the debond onset due to the energy criterion as a function of the debond semiangle Dq, for l ¼ 0.3, ql ¼ 0.1 and glass/epoxy.

Manti
c and García (2012). The critical and arrest semiangles are plotted in Fig. 7 as functions of g and the post-failure configuration. Note that Dqc is very similar for the two post-failure configurations in scenario A whereas for scenario B, the two corresponds to the value of DqEmin , which is quite different for the two post-failure configurations. On

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5. Analysis and discussion of results This section focuses on the main question which motivated the present work: the loss or conservation of symmetry in the postfailure configuration in a single-fibre specimen under transverse tension. First, the two configurations are compared by computing the difference of their critical tensions predicted by the coupled criterion of the FFM for glass/epoxy and also for other bimaterials. Then, the stressestrain curves predicted by this criterion for the two post-failure configurations are plotted and interpreted. Finally, some experimental results found in the literature are briefly revisited. 5.1. Difference in critical loads for symmetric and asymmetric postfailure configurations Fig. 7. Critical and arrest semiangles, Dqc and qa, for l ¼ 0.3, ql ¼ 0.1 and glass/epoxy, as functions of the brittleness number g modulating the influence of the stress and energy criteria.

the contrary, the arrest angle is very different in the two cases due b for large qd, see Fig. 4. to the very different values of G Fig. 8 shows s∞ =s as a function of g for the two post-failure c c configurations. s∞ c is quite constant for small g, but it increases strongly for ga1. In particular, it is a linear function of g in scenario B, as shown above. Qualitatively the behaviour of s∞ c is similar for the two post-failure configuration in both scenarios, although a larger absolute difference is found in scenario B, as will be pointed out in Section 5. For given material properties, the dependence of the results on g can be interpreted as a size effect by defining a reference radius

a0 ¼

G1c E ; s2c

(20)

and relating a dimensionless radius a/a0 and g by

a 1 ¼ : a0 g2

(21)

A strong size effect can be observed in Fig. 9, showing s∞ c =sc as a function of a/a0, where s∞ c =sc increases strongly for small values of a/a0, i.e. predicting a strong size effect, cf. Manti
c (2009); García et al. (2014).

Fig. 8. Critical remote tension s∞ c for the two post-failure configurations, l ¼ 0.3, ql ¼ 0.1 and glass/epoxy, as a function of the brittleness number g modulating the influence of the stress and energy criteria.

We assume that the preferential post-failure configuration, which is expected to appear in experiments, corresponds to the configuration requiring a lower critical remote tension for the debond onset. Thus, in view of Figs. 8 and 9, the asymmetric configuration is preferential for glass/epoxy. The percentage difference between the critical tensions predicted for the two configurations,

Ds∞ c ð%Þ ¼ 100$

∞ s∞ c ðn ¼ 2Þ  sc ðn ¼ 1Þ ; ∞ sc ðn ¼ 1Þ

(22)

is shown in Fig. 10, where it is clearly seen that ∞ s∞ c ðn ¼ 2Þ > sc ðn ¼ 1Þ for g > 0. Therefore, the asymmetric configuration is preferential independently of g. However, Ds∞ c varies between two limit values. In the limit g/0þ , which corresponds to scenario A, Ds∞ c ¼ 0. The reason is that in this limit case the critical remote tension is governed by the stress criterion. As discussed in Section 2, the stress criterion is equivalent for the two post-failure configurations. At the other extreme, for large values of g, corresponding to scenario B, Ds∞ c tends to its maximum value

 ∞ g
max maxDs∞ ð%Þ ¼ Ds g ðnÞ th c c n¼1;2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0v 1 ffi u  E ug Dq ðn ¼ 2Þ; n ¼ 2 min Bu   1C ¼ 100$@t  A; E g Dqmin ðn ¼ 1Þ; n ¼ 1

(23)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E obtained from the expression s∞ c =sc ¼ g gðDqmin ; nÞ for scenario B. The maximum difference is found in scenario B because this is

Fig. 9. Critical remote tension s∞ c as a function of the fibre radius a for the two postfailure configurations, l ¼ 0.3, ql ¼ 0.1 and glass/epoxy.

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governed by the energy criterion, which is the source of difference between the two configurations. The main reason for this is the shielding effect between the two debonds in the symmetric configuration, as discussed in Section 3. In view of the previous discussion and Fig. 10, the difference Dsc is bounded by the two limit values described above. Thus, this difference for a particular bimaterial is essentially characterised by the upper limit (23), as the lower limit is null independently of the bimaterial. To generalise the previous study to other materials, this upper limit is computed for several bimaterials and values of the fracture-mode-mixity sensitivity parameter l. Table 2 shows the maximum percentage difference, maxDs∞ c , predicted for two usual composites, glass/epoxy and carbon/epoxy, and three virtual bimaterials corresponding to some extreme values of the Dundurs parameters a and b. The bimaterial A corresponds to the extreme case of identical elastic materials, whereas the bimaterials B and C correspond to cases with the fibre much stiffer than the matrix. The difference between them is given by Poisson's ratio of the matrix, n2 ¼ 0 for B and 0.5 for C. This table also shows the variation of the maximum difference with l in order to evaluate the influence of the b c on the fracture-modesensitivity of the fracture toughness G mixity angle j. Note that, the bimaterial A corresponds to a limit value considered here to study the tendency of the difference when the two materials become elastic similar. Additionally, the bimaterial A can represent the case in which two pieces of the same material are bonded together, thus creating a weak surface that may promote failure modes completely different to those of a homogeneous material. According to Table 2, the parameter l has a strong influence on the values of maxDs∞ c . The reason for this is that a variation of l b c -curve. For modifies the position of the steeply increasing part of G b c to large values of l, which correspond to a lower sensitivity of G b c -curve moves tovariations of j, the steeply increasing part of G b c is a constant, ward large values of qd. In fact, at the extreme l¼1, G see Carraro and Quaresimin (2014) for an application using this limit. Therefore, in view of the definition of g in (16), if the b c is moved towards large values of qd, the poincreasing part of G sition DqEmin of the minimum of g, see Fig. 6, moves also towards large values of qd, predicting a larger debond in scenario B. The reason is that the increasing part of g is mainly due to the steeply b c . Thus, since the shielding effect between two symincreasing G metric debonds increases for large debonds, maxDs∞ c increases

with increasing l. This argument will repeatedly be used in the following. The influence of the elastic bimaterial properties on the value of maxDs∞ c is elucidated in Table 2 by analysing the extreme cases A, B and C of a and b, cf. Manti
c and García (2012). This influence decreases with increasing l. In fact, the value of Ds∞ c for l ¼ 1 is very similar for all the bimaterials studied. It means that the main source of difference between bimaterials is the different evolution of the fracture mode mixity j with the debond semiangle qd for each bimaterial. This is particularly remarkable by comparing the results for the virtual bimaterials with a rigid fibre (a ¼ 1): B (b ¼ 0) and C (b ¼ 0.5). It can be shown, following Dundurs (1967), that for a bimaterial with a ¼ 1, stress solution only depends on Poisson's ratio of the matrix, which is n2 ¼ 0.5 and n2 ¼ 0 for B and C, respectively. In view of this and the values in Table 2, n2 has a strong influence on maxDs∞ c , at least for the extreme a ¼ 1. This is due to the influence of n2 on the evolution of j with qd for small debonds, see Fig. 11. The constraining effect of Poisson's ratio in the zones of the matrix near the crack tip reduces the fracture-mode-mixity. This effect countervails the main source of increasing of j with qd: the variation with j of the angle between the normal to the interface at the crack tip and the direction of the remote tension. Consequently, j grows more steeply with qd when the constraining effect is vanishing (material C: n ¼ 0) than for incompressible mab c moves trix (material B: n ¼ 0.5). Thus, the increasing part of G toward small values of qd when reducing n2, thus maxDs∞ c decreases in a similar way as described previously in the analysis of the influence of l. For identical elastic materials of the fibre and matrix (a ¼ b¼ 0), j increases less steeply than for the cases with a ¼ 1. The reason is that a more compliant fibre allows the crack tip to move towards the symmetry plane, reducing the fracture-mode mixity. This explains that, according to the previous discussion about the relation ∞ between j and maxDs∞ c , the largest maxDsc is predicted for the extreme case when the fibre and matrix materials are identical. The behaviour of Ds∞ c for carbon/epoxy and glass/epoxy can be understood as an intermediate case of the extreme cases discussed previously. Thus, for these two common bimaterials, the two effects described previously act combined, maxDs∞ c being slightly higher for carbon/epoxy than for glass/epoxy, which could be foreseen from the above analysis. 5.2. Stressestrain curves predicted by the coupled stress and energy criterion of the FFM

∞ð2 debondsÞ

∞ð1 debondÞ

∞ð1 debondÞ

Fig. 10. Percentage difference Ds∞  sc Þ=sc c ¼ 100$ðsc between the critical remote tensions for the symmetric and asymmetric post-failure  configurations, taking l¼0.3, ql ¼ 0.1 and glass/epoxy, as a function of the brittleness number g modulating the influence of the stress and energy criteria.

In addition to different critical tensions predicted for the two post-failure configurations, the coupled criterion predicts different paths in the stressestrain plane for these configurations. Figs. 12 and 13 shows the dimensionless remote tension s∞/sc versus a homogenized strain between two pairs of matrix points, for two arbitrarily chosen values of g corresponding to the two scenarios A and B described in Section 4. This homogenized strain is evaluated uB’ x as an adequately normalized difference between the displacements ux at these reference pairs of points. The first pair of points A’A is given by the two poles at the fibreematrix interface and located on the matrix side. The points B’B are situated at the intersection of the symmetry axis and the external edges of matrixcell. The pair A’A is chosen because the distance between the points A’ and A leads to a stressestrain curve with the debond onset affecting significantly the represented stiffness, therefore all the steps described below are easily visible, see Fig. 12. However, the areas enclosed by the curves do not provide a useful interpretation in terms of the work or energy. The reason is that the external forces are applied at the remote matrix-edges. That is why it is also useful to plot s∞/sc versus the homogenized strain for the pair B’B,

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83

Table 2 Maximum percentage difference in critical tensions max Ds∞ c (23) between the symmetric and asymmetric configurations for several bimaterials and values of l. Virtual bimaterials corresponds to: A: Identical elastic properties for fibre and matrix. B: Rigid fibre (E1 [E2 ) and incompressible matrix (n2 ¼ 0.5). C: Rigid fibre (E1 [E2 ) and matrix with n2 ¼ 0.

l¼ Bimaterial

a

b

Glass/epoxy Carbon/epoxy A B C

0.919 0.624 0 1 1

0.229 0.136 0 0 0.5

0

0.1

0.2

0.3

0.4

0.5

1

2.53% 4.26% 8.29% 4.56% 0.85%

3.16% 5.03% 9.06% 5.26% 1.65%

4.02% 5.97% 9.91% 6.11% 2.63%

5.13% 7.08% 10.82% 7.18% 4.31%

6.63% 8.39% 11.81% 8.48% 7.16%

8.78% 9.93% 12.84% 10.03% 9.84%

15.09% 15.77% 16.56% 16.15% 14.29%

shown in Fig. 13. Actually, since a uniform tension is applied at the B external edge, strictly speaking uB’ x and ux represent the mean displacements along the two external edges in order to be able to interpret the areas enclosed by the curves in terms of work or energy. The problem of this pair of points is that the influence of a debond is insignificant on the stiffness evaluated at these points due to the large dimensions of the matrix square-cell with respect to the fibre radius, specifically 400/3 larger than the fibre radius in the numerical model used. This is solved here, following Ba
zant and Cedolin (1991), by subtracting from the term uBx  uB’ x the homogenized strain due to a purely linear elastic deformation DuB’B;elast x between the same points with the whole interface perfectly bonded. This makes visible the stages of the debond onset predicted by the model used, while keeping the meaning of the represented areas in terms of work or energy. Dashed curves shown in Figs. 12 and 13 represent the equilibrium paths given by the classical LEFM, and dashed straight lines their asymptotes corresponding to the elastic solutions with the whole fibreematrix interface either bonded or debonded. We can consider that the straight equilibrium path of the fully bonded configuration ends up at infinity by a bifurcation point, which is followed by the two equilibrium paths of the classical LEFM. These equilibrium paths predicted by the LEFM are defined by the points which verify Griffith's energy criterion,

Gðqd ; n; a; s∞ ; E ; a; bÞ ¼ Gc ðG1c ; qd ; n; lÞ;

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u b c ðq ; n; lÞ u G s∞ d : ¼ gt b sc Gðq d ; n; a; bÞ

(25)

Then, the relative displacements uPx ðqd Þ  uP’ x ðqd Þ at certain reference pair of points P'P can be obtained by evaluating the ratio of the remote tensions to the associated displacements extracted from the computational model for each value of qd,

(24)

and using the dimensionless expressions defined in (8)e(11) for G and Gc respectively, the expression of the remote tension s∞ for this equilibrium path takes the form

Fig. 11. Fracture-mode-mixity angle j, obtained with ql ¼ 0.1, as a function of the debond semiangle qd for the two post-failure configurations and the virtual bimaterials corresponding to rigid fibre (E1 [ E2) with either (B) incompressible matrix (n2 ¼ 0.5) or (C) matrix with n2 ¼ 0.

Fig. 12. Curves of the normalized remote stress (s∞/sc) versus normalized homoge nized strain ððuAx  uA’ x Þ=ðasc =E ÞÞ for the reference pairs of points A’A predicted by the coupled stress and energy criterion, for l ¼ 0.3, ql ¼ 0.1 and glass/epoxy.

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Fig. 13. Curves of the normalized remote stress (s∞/sc) versus normalized homogenized B’B;elast inelastic strain ððuBx  uB’ Þ=ðasc =E ÞÞ for the reference pairs of points B’B x  Dux predicted by the coupled stress and energy criterion, for l ¼ 0.3, ql ¼ 0.1 and glass/epoxy.

uPx ðqd Þ  uP’ x ðqd Þ ¼

~ P’ ~ Px ðqd Þ  u u x ðqd Þ ∞ s ; ∞ ~ s

(26)

~ Px , e.g., represents the displacement value extracted from where u ~∞ applied. the numerical model for a tension of reference s By an adequate normalization of the previous expression, the homogenized strain for a reference pair of points P'P is given by

uPx ðqd Þ

 uP’ x ðqd Þ asc =E

¼

~ Px ðqd Þ u

~ P’ u x ðqd Þ ∞ a~ s =E

s∞ sc

;

(27)

and using (25) to substitute the ratio s∞/sc, the next expression is obtained for the homogenized strain as a function of qd,

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u b ~ P’ ~ Px ðqd Þ  u uPx ðqd Þ  uP’ u x ðqd Þ x ðqd Þ t G c ðqd ; n; lÞ : ¼ g ∞   b asc =E a~ s =E Gðq d ; n; a; bÞ

(28)

The parametric equation of the equilibrium path predicted by the LEFM and shown in Fig. 12(a) is given by (25) and (28), where qd is the parameter. As expressions on the right-hand-side of (25) and

(28) are proportional to g, the equilibrium path given by the LEFM is expanded (or shrinked) proportionally in the stressestrain plane when increasing (or decreasing) g, i.e. the effect of varying g is a uniform scaling of the curve (homothetic transformation whose centre is the coordinate origin). The evolution predicted by the coupled criterion of the FFM is represented by solid lines in Figs. 12 and 13. After following the linear elastic path without debond, when the critical tension is reached a jump in the value of displacements is predicted. This jump ends either at the corresponding equilibrium path defined by the LEFM, in scenario B (Figs. 12(b) and 13(b)), or at a lower value of the homogenized normal strain, in scenario A (Figs. 12(a) and 13(a)). In the latter case, an unstable debond growth is predicted to occur immediately up to reaching the corresponding equilibrium path. Afterwards, in both scenarios, a stable growth is predicted following the equilibrium path. The areas between the equilibrium paths defined by the LEFM and the jump-line predicted by the FFM are highlighted in Fig. 13. According to the energy criterion the highlighted areas are equal as indicated in these figures. Thus, the jump defined by the FFM can be understood as a kind of Maxwell line (strictly speaking, according to the classical definition of the Maxwell line, it is true only in scenario B), and the equilibrium paths above this line correspond to metastable states. According to this idea, since the jump-line predicted by FFM for the asymmetric post-failure configuration is below that for the symmetric one, the former is a preferential solution for the problem of fibreematrix debonding. Thus, once the critical remote stress for an asymmetric debond is reached, the equilibrium path ends up at this critical point and is followed by two options: i) a finite jump towards the other branch of the equilibrium path (in scenario B, it is actually a transition from one equilibrium branch to the other one) corresponding to an asymmetric debond onset and ii) an unstable (trivial equilibrium) path without any damage at the interface ending at other similar critical point corresponding to the critical remote stress for two symmetric debonds. At the first critical point, a symmetric crack onset is not possible because the system is not able to release enough energy to open two cracks. In a realistic case, the presence of perturbations avoids the possibility of following the unstable path above the first critical point. Therefore, assuming the presence of perturbations, the asymmetric crack onset is preferential. Following the ideas by Bigoni (2012) and Mielke and Roubí
cek (2015), the above loss of symmetry can be clarified by studying a simplified model with springs, see Appendix B. This simplified model is solved analytically and similar results as shown above are found again. Thus a numerical origin of the difference between the critical remote stresses for the symmetric and asymmetric failures is refuted.

5.3. Discussion of previous experimental results The above predicted fact that an asymmetric debond is generally the preferential post-failure configuration, although for some composites (with a small brittleness number g) this difference could be negligible, is in accordance with the experimental evidences presented in the literature for single-fibre tests under transverse loading. Single-fibre tests under transverse cyclic € gren loading at low frequency carried out by Gamstedt and Sjo (1999) showed in microscope observations a clear asymmetric initiation of the debond along the fibreematrix interface. In macroscopic tests of a circular inclusion embedded in a more compliant matrix carried out by Contreras (2000), the specimen was subjected to a monotonous loading and for a critical load showed an abrupt debond onset along one of two sides of the interface, the debond being arrested after the onset.

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Recently, related experiments were carried out by Martyniuk et al. (2013) for a single fibre specimen under transverse tension. However debonds appeared, in a genuine 3D situation at a circular fibreematrix interface edge, where a weak singularity of stresses takes place. Thus, the results of the present 2D study cannot directly be extrapolated to those experiments, although a very similar 3D study could be carried out.

85

effect, discussed here as the main source of the preference for asymmetry, could be observed in other (initially) symmetric problems. In view of the comparison with cohesive zone models, this is a key aspect to take into account when a massive application of the cohesive elements is used in computational analysis.

Acknowledgements 6. Concluding remarks The application of the coupled stress and energy criterion of the FFM to the classical problem of debonding at the fibreematrix interface shows that an asymmetric debond is preferential. The cause for this is found in the energy criterion, which provides a more restrictive condition for a symmetric post-failure configuration than for an asymmetric failure configuration. The reason for this is that the mean released energy per unit debond surface is higher for the asymmetric configuration. This is a consequence of the shielding effect between the two debonds in the symmetric configuration. This effect implies the existence of a critical point on the equilibrium path leading to a preferential transition to the other branch of the equilibrium path by a finite jump along a Maxwell line corresponding to an asymmetric failure. An alternative unstable path leading to a symmetric failure is not possible due to the presence of unavoidable perturbations in the system. A BEM code developed by some of the authors has been used to compute the required elastic states, assuming the possibility of contact between debond faces, and the energy release rates associated to different debond configurations. Despite the high accuracy of the numerical model used, the quantitative results presented here are affected by typical small inaccuracies due to the use of numerical methods. Nevertheless, the main conclusions are quite independent of the numerical results and can be deduced analytically employing suitable physical interpretations. The whole analysis was carried out assuming a perfectly symmetric initial state. We consider this initial situation as the most favourable for a symmetric failure. If, even for this initial situation, an asymmetric failure is preferential, in a more realistic situation, this prediction is expected to remain valid. The reason is that the presence of defects or other sources of asymmetry in the initial situation, would just favour an asymmetric failure. Moreover, even if the presence of defects was considered a tendency to the symmetry, the differences found between the two post-failure configuration could countervail the influence of defects. This difference is particularly high for quasi-similar materials and low sensitivity of the fracture toughness to the fracture-mode mixity. This agrees with some experimental evidence found in the literature as discussed previously. In addition, the prediction of the loss of symmetry agrees with some numerical works using cohesive vara et al., 2011). However, it interface laws (Han et al., 2006; Ta disagrees with other numerical works typically based on smoother cohesive interface laws (Carpinteri et al., 2005; Kushch et al., 2011). The introduction of cohesive elements makes the problem essentially nonlinear, therefore, the loss of symmetry is possible even in the absence of non-homogeneities as demonstrated in a recent work by Nguyen and Levy (2011). In these last cases, small sources of asymmetry, as a non totally symmetric mesh or the different position in the stiffness matrix of the degrees of freedom geometrically symmetric, triggers the asymmetric failure favoured by the tendency shown here. This effect is analogous to that promoted by the natural random variation of the properties. The conclusions of this work are confined to the problem of debonding at the fibreematrix interface. However, the physical discussion of the loss of symmetry presented here can be applied to many other problems. In particular, the presence of the shielding

V. Manti
c thanks to Prof. F. París (University of Seville) and Prof. J. Varna (Luleå University of Technology) who presented him the problem studied in this work in 1994, while E. Graciani acknowledges their support in the development of the BEM code used in vara for his support in the this work. I.G. García thanks to Dr. L. Ta use of the BEM code. V. Manti
c acknowledges discussions with Prof. T. Roubí
cek (Charles University of Prague) particularly useful for the simplified model proposed here. This work was supported by the Spanish Ministry of Education (FPU grant 2009/3968), the Spanish Ministry of Science and Innovation (Project MAT2009-14022), the Spanish Ministry of Economy and Competitiveness and European Regional Development Fund (Project MAT2012-37387), the Junta de Andalucía and the European Social Fund (Project P08-TEP-4051).

Appendix A. Dimensional analysis of the energy release rate of a debond along the fibreematrix interface The value of the ERR of a debond at a loaded specimen is a functional depending on the stress tensor sij and the displacement vector ui in the whole specimen,

  G ¼ f sij ðx; yÞ; ui ðx; yÞ :

(A.1)

The solution for stresses and displacements depends on the problem geometry, which includes the debond angle, the elastic properties of the material and boundary conditions. Assuming that the length of the contact zone is independent of the remote tension s∞ > 0, since this is a frictionless receding contact problem, sij can be rewritten as the product of s∞ and a function b s ij depending on the point (x,y) and the bimaterial elastic properties (E1,n1,E2,n2)

s ij ðx; y; E1 ; n1 ; E2 ; n2 ; aÞ; sij ¼ s∞ b

(A.2)

and analogously

b i ðx; y; E1 ; n1 ; E2 ; n2 ; aÞ: ui ¼ s∞ u

(A.3)

It is known from the LEFM theory that the ERR value is a homogeneous and linear function of a product of displacements and stresses,

  b i ¼ ðs∞ Þ2 hðqd ; a; n; E1 ; n1 ; E2 ; n2 Þ; s ij ; u G ¼ ðs∞ Þ2 h b

(A.4)

where the terms x, y, i and j have been omitted since the ERR value cannot depend on them. Then, in order to obtain the dimensionless parameters which govern the value of G, we should select two independent dimensional parameters to normalize this expression since this is a static mechanical problem, see Barenblatt (1996) for details. Taking the fibre radius a and E* defined in (9) and taking into account that the elastic parameters can be reduced to two under the conditions discussed by Dundurs (1967), the value of G can be expressed as

G¼

ðs∞ Þ2 a b Gðqd ; n; a; bÞ: E

(A.5)

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Appendix B. Simplified model using springs Consider the simplified (discrete) model proposed in Fig. 14. In this model the matrix cell and fibre are modelled by springs. In particular, the stiffness of the matrix is represented by three sets of springs: i) two springs with the stiffness km1 connecting the external boundaries of the matrix cell with the fibre at its poles through a part of the interface which is breakable, ii) four springs with the stiffness km2 connecting the external boundaries with the fibre through a part of the interface which is assumed not to break, and iii) two springs with the stiffness km3 connecting the opposite external boundaries of the matrix without any connection with the fibre. Finally, a spring with the stiffness kf represents the contribution of the fibre to the global stiffness. This spring is connected through the interface to the springs with the stiffnesses km1 and km2. In what follows, this model is solved in order to clarify the loss of symmetry from an initial perfectly symmetric linear elastic system. The stress and energy criteria are applied in a similar manner as carried out previously. Displacement control is assumed, the displacement u at the external boundary of the matrix being the control parameter.

SC u
uSC c1 ¼ uc2 ¼

2kf þ km1 þ 2km2 fc ; km1 kf

(B.2)

SC where uSC c1 and uc2 , respectively, are the critical values for the displacement u verifying the stress criterion in the case of failure of one or two breakable elements. This value is obviously the same for the two cases analogously to the result obtained previously in the continuum problem.

Appendix B.2. Energy criterion The energy criterion can be evaluated by calculating the released energy and comparing it with dissipated energy when one or two of the breakable elements fail. The released energy is calculated as the difference of the stored elastic energies before and after the failure of the elements. The elastic energy before any failure is

U¼

    1 2km3 u2 þ 2km2 u2A þ ðu  uB Þ2 þ km1 u2A þ ðu  uB Þ2 2  þ kf ðuB  uA Þ2 (B.3)

and the elastic energies for one (specifically element B) or two elements broken are

U1 ¼

U2 ¼

  1 2km3 u2 þ 2km2 u2A þ ðu  uB Þ2 þ km1 u2A 2  þ kf ðuB  uA Þ2

(B.4a)

   1 2km3 u2 þ 2km2 u2A þ ðu  uB Þ2 þ kf ðuB  uA Þ2 2 (B.4b)

Fig. 14. Schematic of the simplified model using springs proposed to clarify the loss of symmetry in the problem of fibreematrix debonding.

Appendix B.1. Stress criterion

The change in the potential energy DP under displacement control is given by the change in the stored elastic energy DU. Thus, the change in potential energy due to the failure of either one or two elements, respectively, DP1 and DP2, can be written in terms of u by applying the pertinent equilibrium equations,

k2f km1 ðkm1 þ 2km2 Þ  u2 DP1 ¼ U1  U ¼   2 2kf þ km1 þ 2km2 2km2 ðkm1 þ 2km2 Þ þ kf ðkm1 þ 4km2 Þ

(B.5a)

The stress criterion in this case imposes that the force at the breakable rigid elements should exceed a certain critical value fc. From the equilibrium conditions and previously to any failure, the forces at the breakable rigid elements A and B, respectively, are

(B.5b)

km1 kf u; 2kf þ km1 þ 2km2 km1 kf u; fB ¼ km1 ðu  uB Þ ¼ 2kf þ km1 þ 2km2

fA ¼ km1 uA ¼

(B.1)

showing that fA ¼ fB. Thus, the condition of the stress criterion: fA
fc and/or fB
fc writes:

k2 k  f m1 u2 DP2 ¼ U2  U ¼   2 kf þ km2 2kf þ km1 þ 2km2

Assuming a constant value Gc of the energy necessary to break one element, the dissipated energy due to the failure of either one element (A or B) or two elements (A and B) is DG1¼Gc and DG2 ¼ 2Gc, respectively. Using the energy condition as discussed in Section 2, DP
DG, EC and defining uEC c1 and uc2 as the critical values for the displacement u verifying the energy criterion for either one or two elements broken, respectively, the corresponding expressions of the energy criterion take the form

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u
uEC c1

u


uEC c2

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u  u2 2k þ k þ 2k 2km2 ðkm1 þ 2km2 Þ þ kf ðkm1 þ 4km2 Þ u m1 m2 f Gc ; ¼t k2f km1 ðkm1 þ 2km2 Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ðkf þ km2 Þð2kf þ km1 þ 2km2 Þ ¼ Gc : k2f km1

(B.6b)

It is interesting to compute the ratio of these two critical values,

uEC c2 uEC c1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kf km1 ¼ 1þ 2km2 ðkm1 þ 2km2 Þ þ kf ðkm1 þ 4km2 Þ

(B.7)

It can be shown that

1

uEC c2 uEC c1



pffiffiffi 2;

(B.8)

where the lower bound is achieved for either km1 ≪km2 or kf ≪km2, whereas the upper bound corresponds to the case where both km1 [km2 and kf [km2 hold. Thus, the failure of only one element, leading to an asymmetric failure, requires a lower value of the displacement u from the point of view of the energy criterion. The only exception is, considering a non-vanishing stiffness of the fibre kf, the case of km1 ≪km2 . As the difference between km1 and km2 is given by the length of broken interface, it means by the debond semiangle Dq, the relationship km1 ≪km2 represents the case Dq/0, which agrees with the results obtained in the continuum model, cf. Figs. 7 and 10. It is easy to check pffiffiffi that the above upper bound coincides with the EC ¼ ratio uEC =u 2 in the simplest model of this kind consisting of c2 c1 one spring with two breakable elements at its extremes (in series) stretched under displacement control, as shown in Fig. B. 15. This simplest model captures the essence of the problem studied in this work. It is not difficult to realize, that actually this is the theoretical upper bound for any problem of this kind, as it represents the limit situation where all the stored energy is released at breakage and for breakage of the two elements the double stored-energy is required than for breakage of just one of these elements.

Fig. B. 15. Schematic of the simplest model of one spring showing the energetic origin of the loss of symmetry.

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