Crack onset and growth at the fibre–matrix interface under a remote biaxial transverse load. Application of a coupled stress and energy criterion

Crack onset and growth at the fibre-matrix interface under a remote biaxial transverse load. Application of a coupled stress and energy criterion V. Mantiˇc, I.G. Garc´ıa Group of Elasticity and Strength of Materials, School of Engineering, University of Seville Camino de los Descubrimienos s/n, 41092 Seville, Spain Abstract A theoretical model for prediction of the critical load generating a crack onset at the fibre-matrix interface under a remote biaxial transverse load is presented. In particular, this work is focused on the tension dominated failure. After an abrupt onset the crack grows unstably up to achieving an arrest length. A simple plane strain model of a single circular inclusion surrounded by an unbounded matrix allows to obtain conclusions approximately valid for a dilute fibre packing. Linear isotropic elastic behaviour is assumed for both inclusion and matrix. Two classical elastic solutions for both perfectly bonded and partially debonded circular inclusions are used together with a coupled stress and energy criterion, proposed recently in the framework of Finite Fracture Mechanics, and a phenomenological law for fracture toughness of interface cracks growing in fracture mixed mode. The obtained analytical and semi-analytical expressions make easy to study the influence of all the dimensionless parameters governing the fibrematrix system behaviour: Dundurs elastic bimaterial constants αand β, the interface brittleness number γ, the load biaxiality parameter η, and the fracture mode-sensitivity parameter λ. A size effect of the inclusion radius on the critical load is predicted, smaller inclusions being stronger and less dependent on the secondary load. Finally, an experimental procedure for measurement of the fibre-matrix interface fracture and strength properties is proposed. Keywords: composites, circular inhomogeneity, crack initiation, interface debond, size effect, finite fracture mechanics, fracture thoughness, damage mechanism, failure criteria, brittleness number 1. Introduction Composites reinforced by long fibres are commonly used as a structural material in lightweight structures at present. In aerospace applications, where lightweight is a key aspect of the design, their level of structural responsibility has significantly increased as they are massively used in primary structures. However, our understanding of the damage mechanisms occurring in these composites on different scales is still insufficient. Thus, it is necessary to generate more knowledge about these mechanisms in order to avoid the present high level of uncertainty in the failure loads predicted in the design. One of the most complex failure mechanisms on micro scale in these composites is associated to the matrix failure, also called inter-fibre failure. In particular, the tension dominated mechanism follows a well described sequence of stages, see Hull and Clyne (1996) and Par´ıs et al. (2007): i) failure is initiated at the fibre-matrix interface as small debonds, ii) the interface cracks grow along the interface until a certain arrest angle and then iii) kink out the interface towards the matrix, iv) coalescence of growing matrix cracks generates a macrocrack which may cause the failure of the unidirectional lamina. The present work is focused on the two first steps of this failure mechanism: crack initiation and growth at the fibre-matrix interface. Problem of a single fibre embedded in a matrix including a partial debond has been intensively studied by many authors for a long time, see Par´ıs et al. (2007) and Mantiˇc Email addresses: [email protected] (V. Mantiˇc), [email protected] (I.G. Garc´ıa) Preprint submitted to International Journal of Solids and Structures (2009) for comprehensive reviews. Nevertheless, the debond onset has not attracted sufficient attention up to the last decade. Results presented in bibliography have usually been obtained by computational methods as cohesive zone or weak interface models, see for instance Carpinteri et al. (2005), Xie and Levy (2007), and T´avara et al. (2011)). Mantiˇc (2009) proposed a theoretical model to predict the crack initiation along the fibre-matrix interface under an uniaxial remote tension. This theoretical model is based on the coupled criterion introduced by Leguillon (2002) in the framework of the Finite Fracture Mechanics, see also Cornetti et al. (2006) and for a review Taylor (2007). This model proposes to apply both the stress and energy criteria simultaneously as a sufficient condition for the interface crack onset. In the present work, an extension of the theoretical procedure developed by Mantiˇc (2009) to the case of tension dominated remote biaxial transverse loads is developed. Note that the effect of the secondary compression in addition to a dominating tension has been demonstrated to be important by Par´ıs et al. (2003) and Correa (2008). This analysis is carried out for dilute fibre packing where the influence of adjacent inclusions is almost negligible, whereas for dense fibre paking, a numerical method should be employed based on other solutions, e.g. Kushch et al. (2010). In addition, several models developed under the assumption of dilute fibre packing have demonstrated being useful to explain experimental results even for densely packed composites, see Correa (2008). After a revision of the solution of the elastic inclusion perfectly bonded to the matrix in Section 2, the elastic problem of an inclusion with a debond is analysed in Section 3, where a general analytical solution is simplified for this problem. Coupled criterion is developed and applied in Section 4 and the results for the critical crack length and remote load at the onset are obtained. Section 5 describes the influence of the remote secondary load and the inclusion size on the results obtained. Finally a new experimental procedure for an indirect measurement of the strength and fracture interface properties is presented in Section 6. 2. Stresses in a single inclusion under a remote biaxial transverse load Stress based failure criteria usually consider the stress state prior to the damage appearance. Hence the aim of this section is to study the tractions along the fibre-matrix interface under remote biaxial load transverse to the fibre axis. Assuming certain hypothesis, a classical elastic solution is particularized for this problem providing closed form expressions of tractions. Finally, the traction dependence on the key problem parameters is discussed. Consider a circular cylindrical inclusion of radius aembedded in an infinite matrix and perfectly bonded along its lateral interface. Let (x, y, z) and (r, θ, z) be suitably defined cartesian and cylindrical coordinate systems, the z-axis being coincident with the inclusion (longitudinal) axis. Remote uniform biaxial transverse load (σ∞ x, σ∞ y) is applied parallel to the the two axes, xand y, transverse to the cylindrical inclusion axis, see Figure 1. Figure 1: The inclusion problem configuration. An analytic solution for stresses in this problem was deduced by Goodier (1933). As was shown by 2 Hardiman (1954) the stresses inside the inclusion are constant. Following Mantiˇc (2009), these inclusion stresses are rewritten in terms of the Dundurs bimaterial constants αand β(Dundurs (1967, 1969)) as follows: σ(1) xσ(1) xy σ(1) xy σ(1) y!=σ∞ xk0 0k−m+σ∞ yk−m0 0k,(1) where k(α, β) = 1 2 1 + α 1 + β 2 + α−β 1 + α−2βand m(α, β) = 1 + α 1 + β,(2) and the Dundurs bimaterial constants are defined as α=µ1(κ2+ 1) −µ2(κ1+ 1) µ1(κ2+ 1) + µ2(κ1+ 1) and β=µ1(κ2−1) −µ2(κ1−1) µ1(κ2+ 1) + µ2(κ1+ 1),(3) with µk=Ek/(2(1 + νk)) and κk= 3 −4νk,Ekand νkdenoting the Young’s modulus and Poisson’s ratio, respectively. It can be shown that 0 ≤k≤5/3 and 0 ≤m≤2. Let a dimensionless load-biaxiality parameter ηbe defined as the ratio of remote stresses η=σ∞ y σ∞ x .(4) Then, normal and tangential tractions, σand τ, acting along the interface (r=a) can be expressed as a function of the polar angle θ(see Figure 1) and the parameter ηas σ(θ) σ∞ x =σr(θ) σ∞ x =k+ (k−m)η−(1 −η)msin2θ, (5a) τ(θ) σ∞ x =−σrθ(θ) σ∞ x = (1 −η) sin θcos θ. (5b) In view of the problem symmetry only angles 0◦≤θ≤90◦will be considered for the sake of simplicity. In the following and without loss of generality it will be assumed that σ∞ x>0 and σ∞ x≥σ∞ y, i.e. η≤1. The derivative of σ(θ) evaluated from (5a), ∂σ(θ) σ∞ x ∂θ = (η−1)msin (2θ).(6) This shows that normal traction is a decreasing function of θfor θ∈[0◦,90◦] and any η≤1. According to this and the expression in (5a), σ(θ) achieves its maximum value at θ= 0◦: σmax =σ(θ= 0◦) = σ∞ x·k+σ∞ y·(k−m).(7) Note that kand k−mrepresent, respectively, the relative contribution of remote stresses σ∞ xand σ∞ y to this maximum value of σ(θ). Following (5a) and (7), the influence of σ∞ yon the normal stresses σ(θ) is determined by the ratio k/m. In particular, tension σ(θ= 0◦)>0 is generated by a remote compression σ∞ y<0 (tension σ∞ y>0) for k/m < 1 (k/m > 1), assuming ::::: smallσ∞ x&0. Recalling that σ∞ x>0, the semiangle θ0for which the interface normal tractions vanish is given from (5a) by θ0(η;α, β) = arcsin sk m+k m−1η 1−η.(8) According to this expression and the analysis in Appendix A, the angle θ0∈[0◦,90◦] does not exist for 3 Table 1: Examples of isotropic bimaterials constants (1, inclusion; 2, matrix) Bimaterial E1(GPa) ν1E2(GPa) ν2 Glass/epoxy 70.8 0.22 2.79 0.33 Carbon/epoxy 13.0 0.20 2.79 0.33 α β ε E∗(GPa) Glass/epoxy 0.919 0.229 -0.074 6.01 Carbon/epoxy 0.624 0.136 -0.044 5.09 k m k/m θη(◦)η0 Glass/epoxy 1.44 1.56 0.9205 16.3 0.086 Carbon/epoxy 1.32 1.43 0.9200 16.4 0.087 all the considered values of η∈(−∞,1] and all admissible values of k m∈[3/4,+∞), see Mantiˇc (2009). The condition of vanishing derivative of (5a) with respect to ηgives the angle θηfor which the interface normal traction is independent of η, θη(α, β) = arccosrk m= 90◦−θ0(η= 0).(9) This expression makes sense only if k≤m. Then, θηdivides the interface sector 0 ≤θ≤90◦into two regions. In the first region (θ < θη) where ∂σ ∂η <0, σdecreases with increasing the remote secondary load σ∞ y. On the contrary, in the second region (θ > θη) where ∂σ ∂η >0, σincreases with increasing the remote secondary load σ∞ y. Hence, θηis a key angle to evaluate the influence of load biaxiality. The values of the above defined and used constants characterizing the interface traction distribution for two typical fibre reinforced composites are presented in Table 1. The glass/epoxy bimaterial will be used as an example in the present work. Plots of normal tractions distributions along the interface obtained from (5a) for glass/epoxy are shown in Figure 2. Η=1 Η=0.5 Η=0 ΘΗ Η=-2 Η=-1.5 Η=-1 Η=-0.5 Θ0HΗ=-1L Θ0HΗ=0L 0 20 40 60 80 -3 -2 -1 0 1 2 ΘHºL Σ Σx ¥ Figure 2: Distribution of the normal tractions along the interface for several values of ηand glass/epoxy. 3. The solution for a crack at the interface of a single inclusion under a remote biaxial transverse load Energy based fracture criteria consider a cracked configuration. Hence the aim of this section is to analyse the problem of a partial debond at the fibre-matrix interface. Under certain assumptions, a classical elastic solution particularized for the present problem provides closed form expressions for a fracture mode mixity and the energy release rate, their dependence on the key problem parameters being pointed out. 4 Consider now the problem configuration from the previous section altered by the presence of a crack at the interface. In view of the fact that the maximum of σ(θ) is achieved at θ= 0◦, see (6), it will be assumed that this crack is symmetrically situated with respect to the x-axis, with a semidebond angle θd≥0 and an infinite length in the z-axis direction, see Figure 3. Figure 3: The interface crack problem configuration. A more general problem has been analytically studied by Toya (1974) and several other authors using the open model of interface cracks. As will be seen later on, the validity of the analytic solutions based on the open model is limited and, thus, computational methods should sometimes be used employing the contact model of interface cracks, see Par´ıs et al. (2007) for a review. The interface tractions at a point placed ahead of the crack tip at the polar angle θ=θd+θl,θl>0, see Figure 3, can be expressed by particularizing Toya’s solution for stresses1and rewriting it in terms of the Dundurs parameters, σ(θ, θd, η, β)−iτ(θ, θd, η, β) = −σ∞ x 2 1−α 1−βχ(θ, θd, β)p(θ, θd, η, β),(10) where χ(θ, θd, β) and p(θ, θd, η, β) are defined in Appendix B). It should be noticed that the tractions along the interface are independent of the inclusion radius a. Ratio of the interface shear and normal tractions ahead the crack tip at a small reference length (either geometry or material based) gives a measure of fracture mode mixity of an interface crack. Thus, the angle ψat a reference angle θl, measured from the crack tip (see Mantiˇc (2009) for the discussion about this reference angle for a similar problem), is defined as: tan ψ(θd;θl, η, ε) = τ(θd+θl) σ(θd+θl).(11) This angle will be used as a suitable measure of the fracture mode mixity. Figure 4 shows the evolution of ψ(θd) for different values of the load-biaxiality parameter η. The ERR of the interface crack propagating at its upper crack tip at an angle θdcan be expressed, rewriting Toya’s expression as previously, by G(θd;σ∞ x, σ∞ y;a;E∗, α, β) = (σ∞ x)2a E∗ˆ G(θd;η;α, β),(12) 1The following values of the parameters used by Toya: φ= 0 and ε∞= 0 are taken. 5 Ψ = 90º Η=-2 Η=-1.5 Η=-1 Η=-0.5 Η=0 Η=0.5 Η=1 0 20 40 60 80 0 20 40 60 80 100 120 ΘdHºL ΨHºL Figure 4: Examples of the evolution of the fracture mode mixity angle ψ(obtained from Toya’s solution of the open model of interfacial cracks) taking θl= 0.1◦, for different values of ηand glass/epoxy. where E∗is the harmonic mean of the effective elasticity moduli 1 E∗=1 21−ν2 1 E1 +1−ν2 2 E2(13) and ˆ Gis a dimensionless normalized ERR whose expression is presented in Appendix C. According to expression (12), the ERR varies linearly with the ratio a/E∗and quadratically with the remote load σ∞ x. Figure 5 shows the evolution of the normalized ERR ˆ G(θd) , and also of its asymptotes for θd≈0◦ given by (D.1), for different values of the load-biaxiality parameter η, see Appendix D. Validity of these plots is limited by the validity of the open model of interface cracks. Notice, in relation to Figure 4, that compressions ahead of the crack tip correspond to |ψ|>90◦and can become relevant for η < 0. These compressions may have associated a relevant overlapping of crack faces close to the crack tip. This is not physically admissible, so it may invalidate Toya’s solution for some values of θdand η. Due to the above mentioned overlapping, an additional fictitious term, corresponding in some sense to Mode I, appears in the computation of the ERR, causing some overestimation of ˆ G. This overestimation can be studied by using the relation between the ERR based fracture mode mixity and the stress based fracture mixity according to Mantiˇc and Par´ıs (2004). This relation allows partitioning ˆ Ginto two components ˆ G(θd) = ˆ GI(θd, δθ) + ˆ GII (θd, δθ), for a given virtual-crack-step angle δθ, as shown in Appendix E, leading to, ˆ GI,II (θd, δθ) = 1 2ˆ G(θd)(1 ±F(ε) cos(2(ψ(θd, θl) + ψ0(δθ/θl, ε))),(14) where θlis the reference angle for ψ(11) and the oscillation index εis given in terms of βin (B.4). Figure 6 shows the individual components of the ERR corresponding to a small virtual-crack-step angle δθ = 0.5◦. These plots allow to clarify the range of validity of Toya’s expression of ERR for larger values of θdand different values of η. Decreasing values of ηdecreases the range of the values of θd, where Toya’s expression of ERR is valid. This figure also confirms that the cause of the strongly increase of ˆ G for large values of θdand η < 0 is associated to a fictitious contribution of ˆ GIdue to a large overlapping. In order to clarify the influence of the remote secondary load σ∞ yon the values of ˆ G, it is useful to study the variation of the derivative of ˆ G(D.1) with respect to the load-biaxiality parameter ηat θd∼ =0◦, d2ˆ G dηdθdθd=0 =2π(k−m) (k+ (k−m)η)1+4ε2 cosh (πε).(15) This expression shows again the importance of the parameter k/m. In fact, the sign of the variation of 6 Η=-2 Η=-1 Η=0 Η=1 asymptotes 0 20 40 60 80 0 2 4 6 8 10 ΘdHºL G ` Figure 5: Examples of the normalized ERR (obtained from Toya’s solution of the open model of interfacial cracks) and its asymptotes for different values of ηand glass/epoxy. the asymptotic slope of ˆ Gat θd∼ =0◦with ηis directly characterized by sign d2ˆ G dηdθdθd=0 = sign k m−1k m+k m−1η.(16) According to (16), a change of monotonicity of the asymptotic slope of ˆ Gat θd∼ =0◦occurs for (see Appendix A) η= k m 1−k m =1 η0 .(17) Thus, the sign of derivative in (15), cf. (16), for k m>1 is positive for 1/η0< η ≤1 and negative for η < 1/η0, whereas for k m<1 it is negative for all the values of η≤1. Consequently, for bimaterials with k m<1, a remote secondary tension σ∞ y>0 will hinder the crack onset from the energetic approach point of view, while a secondary compression σ∞ y<0 will facilitate it. 4. Interface crack onset at a single inclusion under a remote biaxial transverse load The approach developed by Leguillon (2002) in the framework of the Finite Fracture Mechanics will be used to predict the crack onset, in a similar way as done in Mantiˇc (2009) for remote uniaxial load. The key idea of this approach is to combine a stress and an energy criterion to predict the critical load originating crack onset and the crack length at the onset. The reason for applying the coupled criterion is that it is not possible to obtain a solution of the crack onset problem by applying each criterion individually without some extra assumptions due to the following reasons: 7 (a) η=−2 G ` G ` II fictitious G `I 0 20 40 60 80 0 5 10 15 20 ΘdHºL G ` (b) η=−1 G ` G `I G ` II fictitious 0 20 40 60 80 0 2 4 6 8 10 ΘdHºL G ` (c) η= 0 G ` G ` II fictitious G `I 0 20 40 60 80 0 1 2 3 4 5 ΘdHºL G ` (d) η= 1 G ` G ` II G `I 0 20 40 60 80 0 1 2 3 4 5 ΘdHºL G ` Figure 6: Plots of the individual components of the ERR associated to δθ = 0.5◦for different values of the load-biaxiality parameter ηand glass/epoxy. (a) η=−2. (b) η=−1. (c) η= 0. (d) η= 1. •A stress criterion can determine the minimal load but it cannot determine the size of the crack originated at the onset. •An application of the infinitesimal Griffith criterion requires an existing crack to obtain a value of Gdifferent from 0 in order to fulfill the condition G≥Gc>0. However, coupling both criteria and releasing the Griffith condition of infinitesimal crack growth by allowing finite increments of crack advancing, permits to obtain the length of the crack originated and the critical load required for the onset. In view of the remote tension σ∞ xdominating the crack onset and the problem symmetry with respect to the x-axis, onset of a crack situated as shown in Figure 3 will be assumed in agreement with experimental observations. Reasons why, in spite of the symmetry with respect to the y-axis, only one crack appears, resulting in an asymmetric configuration after the crack onset will be discussed in a forthcoming paper by Garc´ıa et al. (2012). First, in Subsection 4.1, the stress criterion is presented and applied to the stress state analysed in Section 2. Second, a condition imposed by the incremental energy criterion is obtained in Subsection 4.2 with the aid of the analysis introduced in Section 3. Then, both conditions are combined in Subsection 4.3 leading to the prediction of the critical load and semiangle. Finally, the post-crack-onset evolution and the applicability of the open model of interface cracks in the present problem are discussed in Subsections 4.4 and 4.5, respectively. 8 4.1. Stress criterion A stress criterion is usually invoked if no crack exists a priori. The present stress criterion is based on the idea of the existence of an interface tensile-strength σc, defined as the maximum tension that the interface can sustain. Thus, in the present problem, the inclusion-matrix interface can break at the points defined by an angle θwhere, σ(θ)≥σc,(18) defining a tensile criterion. According to Figure A.1 and Section 2, this criterion cannot be fulfilled for η≤1/η0and k/m > 1 because the whole interface is under compression and no crack onset can be predicted following the stress criterion. Hence, in the following analysis, it will be assumed that either η > 1/η0or k/m ≤1. Then, combining (18) and (5a), the stress criterion can be expressed as σ∞ x σc≥1 k+ (k−m)η−(1 −η)msin2θ=s(θ, η, α, β).(19) Assuming a sufficiently large remote loading, given by (19) for θ= 0◦, σ∞ x σc≥min θs(θ, η) = s(0◦, η) = 1 k+ (k−m)η>0,(20) an angle θσ c∈[0◦,90◦] can be defined by σ(θσ c) = σc. Then, due to the decreasing character of σ(θ) (see (6) and discussion in Section 2), condition (18) is verified for all θ∈[0◦, θσ c], θσ c= arcsin sk+ (k−m)η−σc σ∞ x (1 −η)m,(21) According to a discussion in Section 2, for a given value of ηan angle θ0(8) may exist where the traction is zero. Then, condition (19) leads to an infinite load for θ=θ0, which is an upper limit for the values of θσ c, θσ c< θ0(η;α, β) (22) Then, combining all the conditions related to the stress criterion, the maximum angle of a debond and the function sare defined in a rigorous manner suitable for computational proposes in Appendix F. Figure 7 shows a representation of the stress criterion for glass/epoxy as defined in Table 1 for different values of η. As predicted, all the curves of the stress criterion are increasing. Thus, for a load (values of σ∞ x/σcand η) two zones can be defined in this diagram if θσ cexists, a zone where a debond is possible [0◦, θσ c], and another where it is not possible (θσ c,180◦]. Note that an angle θη(see Figure 7) can be defined where the stress criterion is independent of the remote secondary load σ∞ yas demonstrated in Section 2. This semiangle separates the interface into two regions, a region (θ < θη) where a secondary compression σ∞ y<0 facilitates a debond onset and another region (θ > θη) where it hinders a debond. Finally it should be noticed that the stress criterion is not sufficient to uniquely characterize the debond onset, as it provides only one inequality for two unknowns, the critical remote load and debond angle after the onset. 4.2. Incremental Energy criterion An incremental Griffith criterion is used here with the aid of expressions developed in Section 3. First, an energy balance for the onset of an interface crack of a finite length is introduced and its different terms are particularized for this problem and analysed. Finally, a condition for the minimum load originating an energetically allowed fibre-matrix debond is deduced by means of a dimensionless function of the crack length representing the ratio of the dissipated to the released energy. 9 of γ(recall that k/m < 1 and η0>0 for glass/epoxy). However, a non-monotonic boundary curve of the safe region is observed for greater values of γin Figure 12. In fact, it is observed that the curve which joins the points with the maximum critical remote load σ∞ cx shows that the maximum is situated at η= 1 just for γ→0+. For moderate values of γ, the maximum critical remote load σ∞ cx corresponds to η < 0. The reason for this behaviour is clarified in Figure 13 where two situations are explained. Γ=0.2 Γ=0.4 Γ=0.6 Γ=0.8 Γ=1 Γ=1.2 Γ=1.4 Γ=1.6 Γ=1.8 Γ = 2 SAFE Scenario A Scenario B FAILURE Scenario A Threshold curve Maximum critical load Asymptote for Γ ® 0+: Hk-mLΣcy +kΣcx = Σc -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Σcx ¥ Σc Σcy ¥ Σc Figure 12: Critical biaxial loads originating a crack for different values of γ, taking λ= 0.3, θl= 0.1◦and glass/epoxy. In the first case, the value of γis relatively small, Figure 13(a), which is associated to small values of θc, as previously demonstrated. For small values of θc, the effect of the secondary load σ∞ yis the same for both the stress criterion and energy criterion curves, both curves descending when ηreduces. In the second case, the value of γis larger, Figure 13(b), and the values of θcare greater than θη, see (9) and discussion below. Then, for values of γoriginating θc> θη, an increase of the remote secondary load σ∞ y makes less restrictive the stress criterion and more restrictive the energy criterion. Thus, the monotony of the function σ∞ cx(σ∞ cy ) can be broken down as observed in Figure 12 for moderate values of γ. The straight line defined by (44) represents, according to Figure 12, a limit of failure envelope curves for γ→0+. Note that the failure envelope curves for γ→0+in Figure 12 show the most relevant influence of the secondary load σ∞ yon the value of the critical load σ∞ cx. Figure 12 also shows the ”threshold curve” which separates scenarios A and B. It is interesting to remark that greater values of γcorrespond to a larger range of failure behaviour governed by scenario B. On the contrary, the presence of a remote secondary compression σ∞ y<0 leads to scenario A, for small 16 (a) Η = 0.9 Stress criterion Energy criterion Η = -1 Η = 0.9 Η = 0 Η = -1 Η = 0 Η = 0.9 Γ=0.4 0 10 20 30 40 50 60 70 0.0 0.5 1.0 1.5 2.0 DΘHºL Σcx ¥ Σc (b) Η = -1 Η = 0 Η = 0.9 Stress criterion Stress criterion Stress criterion Η = 0.9 Η = -1 Η = 0 Energy criterion Η = 0 Η = -1 Η = 0.9 Γ=1 0 10 20 30 40 50 60 70 0.0 0.5 1.0 1.5 2.0 DΘHºL Σcx ¥ Σc Figure 13: Stress and energy criteria curves, taking λ= 0.3, θl= 0.1◦and glass/epoxy for two different values of γ. (a) γ= 0.4, (b) γ= 1 and moderate values of γ. Figure 14 studies the influence of αand βvalues on the biaxial safe region for γ→0+, for selected theoretical (but possible) bimaterials and also for usual composites. From Dundurs’ α−βparallelogram it is seen that the most common bimaterials have very similar properties in the debond onset problem. A more extensive list of α−βvalues for real bimaterials can be found in Suga et al. (1988) and Schmauder and Meyer (1992). The safe region in the limit case γ→0+is defined by the intersection of the semiplanes including the origin of coordinates and limited by the straight line defined by (44) and the symmetric one with respect to the bisector of the coordinate axes. From (44), the position of the corner point of the safe region (η= 1) is given by σ∞ cx σc =σ∞ cy σc =1 2k−m.(45) 17 (a) 1 3 -1 0.6 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 2 ¥ 0.7 AA BB DD EE FF 0.5 Carbon/epoxy Boro/epoxy Aramid/epoxy Glass/polyester Glass/epoxy -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 C -1.0 -0.5 0.0 0.5 1.0 -0.4 -0.2 0.0 0.2 0.4 Α Β (b) A B C SAFE FAILURE -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Σcx ¥Σc Σcy ¥ Σc SAFE FAILURE D E F -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Σcx ¥Σc Σcy ¥ Σc FAILURE SAFE Glass/epoxy Boro/epoxy Aramid/epoxy Glass/polyester Carbon/epoxy -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Σcx ¥Σc Σcy ¥ Σc Figure 14: (a) α−βdiagram for bimaterials in plane strain with isovalue curves for γ→0+corresponding to: solid lines with values of σ∞ cx/σc=σ∞ cy /σc, and dashed lines with values of ∂σ∞ cx/∂σ∞ cy . (b) Some biaxial failure envelopes for γ→0+for selected points in the α−βdiagram and usual composites. A: k/m = 0.75, B: k/m = 1, C: k/m →+∞, D: k/m = 1.25, E: k/m = 1.25, F: k/m = 1.25, Carbon/epoxy: k/m = 0.9200, Glass/polyester: k/m = 0.9201, Glass/epoxy: k/m = 0.9205, Aramid/epoxy: k/m = 0.9217, Boro/epoxy: k/m = 0.9204 The slope of the linear relation in (44) characterizes the influence of the secondary load σ∞ yon the critical load σ∞ cx. This slope can be expressed as: lim γ→0+∂σ∞ cx ∂σ∞ cy =η0(46) where η0is defined in (A.1) and its range in (A.2), see also Figure A.1. Hence, this slope is only dependent on the elastic bimaterial properties and does not depend on the interface properties. In view of the range of possible values for the slope (A.2), the safe region is always convex, cf. Figure 14(b). For k/m < 1 the slope (46) is positive and an increase in the secondary load σ∞ yincreases the critical load σ∞ cx necessary to originate a debond, see Figure 14. This is the case of the two bimaterials defined in Table 1, i.e. glass/epoxy and carbon/epoxy. However, an opposite effect is predicted for k/m > 1, see Figure 14. In fact, this dependence matches up with the effect of the secondary load σ∞ ypredicted by 18 Goodier’s solution for the interface point θ= 0◦, see (1), which shows that a compression or tension is expected at θ= 0◦when a secondary load σ∞ y>0 is applied for k/m < 1 or k/m > 1, respectively. The present results show that the influence of the secondary load σ∞ yon the critical load σ∞ cx is at most moderate in usual composites. Taking into account that in the case γ→0+, analysed in Figure 14, the values of σ∞ cx are the most sensitive to the values of σ∞ y, in general for k/m < 1 the influence of σ∞ yon the value of σ∞ cx exists but it is small or at most moderate. For bimaterials as glass/epoxy and carbon/epoxy with k m<1, these results agree with the hypothesis introduced by Par´ıs et al. (2003). According to this hypothesis a secondary compression σ∞ ymakes easier the debond onset for these bimaterials and reduces the critical load σ∞ cx as shown in Figure 14(b). However, for bimaterials with k m>1 the effect is opposite, a secondary compression σ∞ y<0 increases the critical load σ∞ cx. 4.4. Post-onset evolution After the onset of a new crack, an unstable growth of the crack is possible depending on the relation between G(θd) and Gc(θd) for θd≥θcand according to the criterion of the classical (infinitesimal) interface fracture mechanics (see, e.g. Par´ıs et al. (2007); Mantiˇc et al. (2006)). Thus, the condition for the further crack growth will be G(θd, η)≥Gc(ψ(θd, η)), θd≥θc.(47) The crack will stop growing at an arrest angle θa≥θcverifying G(θa, η) = Gc(ψ((θa, η)) if for angles θd&θacriterion (47) is not fulfilled. The stability of the post-onset growth of the crack is different for two scenarios A and B separated by γth two post-onset scenarios being possible: •For γ < γth,G(θc, η)> Gc(ψ(θc, η)) and the crack is expected to continue growing in an unstable manner up to an arrest angle θa> θE min, which can be shown similarly as in Mantiˇc (2009). •For γ≥γth,θc=θE min,G(θc, η) = Gc(ψ(θc, η)) and dG/dθd|θd=θE min ≤dGc/dθd|θd=θE min , see (31) and related discussion in Subsection 4.2. Therefore, assuming strict inequality in (31) (which, in fact, has been verified in all present calculations), no unstable crack growth is usually expected after the crack onset and θa=θE min. Figures 11(a) and 15 were computed by implementing the above ideas. The values of Gand Gcand their derivatives are compared for θd≥θcin order to find the arrest semiangle θa. According to these figures, a long unstable crack growth after the crack onset is predicted for small values of γ(brittle configurations), whereas short (or zero) unstable crack growth is predicted for large vales of γ(tough configurations). 4.5. Applicability of the open model of interface cracks The applicability of the theoretical model developed is limited by its assumptions, perhaps the most restrictive being the usage of the open model of interface cracks. Toya’s (1974) solution assumes negligible overlapping of traction-free crack faces. The angle of the overlapping zone at the crack tip can be estimated by the formula deduced by Hills and Barber (1993) and generalized by Graciani et al. (2007), which rewritten for the present case is defined as the largest value of θI(θd, η) = θl·exp [((2n−1/2) π−ψ(θd, η, θl)signε+ arctan(2 |ε|)) /|ε|],(48) lower than the semidebond angle θd, with nbeing an integer. Figure 16 shows the evolutions of ˆ Gfor glass/epoxy and different values of the load biaxiality parameter η. Additionally, for γ= 1.5, corresponding to relatively tough configurations, the values of θc,θE min and θacomputed by the present model are indicated. Finally, angles θI,1% for which the overlapping zone represent 1% of the crack length, i.e. θI/2θd= 0.01, providing a reasonable limit of validity of the open model, are also presented in Figure 16. This figure shows that all the values of θcare lower than the reference limit θI,1%, therefore the open model is acceptable for the evaluation of θcand the critical load σ∞ cx. However, it might not be fully acceptable when computing θafor large negative values of η. Note that, the above discussed limit on the semiangle θdis mainly due to somewhat inaccurate evaluation of 19 Γ=0.2 Γ=0.6 Γ=1 Γ=1.4 Γ = 2 Γ = 2 Γ=1.4 Γ=1 Γ=0.6 Γ=0.2 Arrest semiangle Critical semiangle Θmin EHindependent of ΓL -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 0 20 40 60 80 Η ΘcHºL Θmin EHºL ΘaHºL Figure 15: Semiangles θc,θE min and θaas functions of the biaxiality parameter ηfor different values of γ, taking θl= 0.1◦, λ= 0.3 and glass/epoxy. ˆ Gbecause of a large overlapping zone at the crack tip, see also Figure 6 and the related discussion. A correct procedure for the evaluation of ˆ Gis such cases would require employing the contact model of interface cracks as in Par´ıs et al. (2007) and Correa (2008). 5. Size effect of the inclusion radius aon the crack onset and its variations with the biaxiality A size effect in the present debond onset problem can be understood as a dependence of the critical remote load (σ∞ cx,σ∞ cy ) and critical semiangle θcon the only geometrical parameter in the present problem, the inclusion radius a. As will be seen, this size effect is directly related to the brittle-to-tough transition governed by the brittleness number γ, see Figure 11. Let a bimaterial characteristic length a0be defined in terms of the interface properties σcand G1c and the harmonic mean of effective Young moduli E∗, a0=G1cE∗ σ2 c .(49) Then, the normalized radius a a0and γare related by γ=ra0 a.(50) Analogously to the threshold value γth (41), a threshold value of ais defined as ath =a0 γ2 th .(51) In fact, all the analysis presented above taking as reference γcan be expressed as a dependence on a in an analogous manner. For sufficiently large values of a, which correspond to small values of γ, the 20 Θc Θmin E Θ0 Θa Η = -2 Η = 1 Θdwith ΘÈ 2Θd =0.01 Η = 0 Η = -1 increasing Ηwith DΗ = 0.2 0 20 40 60 80 0 2 4 6 8 10 12 ΘdHºL G ` Figure 16: Plots of evolution of values of dimensionless ERR limited by the semiangles which estimate the validity of the model and representation of important values of semiangles of the results, taking λ= 0.3, θl= 0.1◦,γ= 1.5 for glass/epoxy. critical semiangle θcand the critical remote load σ∞ cx can be approximated by the following expressions, see Section 4, θc∼ =2cosh2(πε) π(1 + 4ε2) a0 aand σ∞ cx σc∼ =1 k+ (k−m)η,(52) whereas for a≤ath, θc=θE min and σ∞ cx σc =qg(θE min, η)ra0 a.(53) As follows from (52) and (53), the critical crack-semilength aθcis constant and independent of aand ηfor large a, whereas for small ait is linearly proportional a. The above described asymptotic behaviour of θcand σ∞ cx can be easily identified in Figure 17 where the variations of θc,θaand σ∞ cx as functions of aare plotted. In particular, in Figure 17(a) and (b) it is seen that θc=θa=θE min and σ∞ cx ≈1/√a, respectively, for a≤ath. Thus, σ∞ cx increases drastically for small inclusions while for large inclusions it tends to a constant value given by the stress criterion applied at θ= 0◦. As can be observed in Figure 17(b), the size effect on σ∞ cx is similar for different values of η, being quite independent of the combination of remote transverse loads. 6. Experimental procedure for the measurement of the brittleness number γ, interfacial tensile strength σcand fracture toughness G1c Fracture properties of the fibre-matrix interfaces are very important for the macroscopic behaviour of fibre reinforced composites. However the experimental measurement of these is very difficult to be carried out. An indirect experimental procedure is proposed here for obtaining first the value of γ, and subsequently the values of σcand G1c, for a bimaterial. Elastic properties of the bimaterial (E∗, α, β) are 21 (a) Critical semiangle Arrest semiangle Η=1 Η=0 Η=-1 Θmin E 0 1 2 3 4 5 6 0 20 40 60 80 aa0 ΘcHºL Θmin EHºL ΘaHºL (b) 1 k+Hk-mLΗ Η=1 Η=0 Η=-1 0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 aa0 Σcx ¥ Σc Figure 17: (a) Critical semiangle θcand (b) Critical remote tension in σ∞ cx as a function of the inclusion radius a, taking λ= 0.3, θl= 0.1◦and glass/epoxy. assumed to be known, as they can be measured by carrying out standard material tests for each material separately. To determine the interface properties, the sole measure of the critical stress for the case of remote uniaxial tension (η= 0) is not sufficient. The reason is that the critical stress depends not only on γbut also on σc, see the normalization used in (38). Nevertheless, if the critical stress is also measured for a biaxial load (η6= 0), then, the ratio of these critical stresses depends only on the value of γdue to the influence of γvalue on the solution θcof (37). This is the key idea behind the experimental procedure proposed. This procedure employs plots of the ratio of critical stresses σ∞ cx(η6=0) σ∞ cx(η=0) as a function of γor η. As an example, Figure 18 shows values of σ∞ cx(η6=0) σ∞ cx(η=0) for glass/epoxy, taking θl= 0.1◦and λ= 0.3. The steps of the experimental procedure are briefly explained in the following: 1. Determine the critical stress σ∞ cx in the uniaxial tension test (η= 0). This is a relatively easy test, thus a good accuracy is expected. 2. Determine another critical stress σ∞ cx in a biaxial test for η6= 0. Combining the plots in Figure 18(a) and a rough a priori estimation of the γvalue, choose the most suitable value of ηto test by looking 22 (a) Η = -2 Η = -1.75 Η = -1.5 Η = -1.25 Η = -1 ΓthHΗ= 0.95 L ΓthHΗ= -0.75 L ΓthHΗ= -0.5L ΓthHΗ= -0.25 L ΓthHΗ= 0L ΓthHΗ= 0.25 L ΓthHΗ= 0.5L ΓthHΗ= 0.75 L Η = -0.75 Η = -0.5 Η = -0.25 Η = 0 Η = 0.25 Η = 0.5 Η = 0.75 Η = 0.95 0 1 2 3 4 0.80 0.85 0.90 0.95 1.00 1.05 1.10 Γ Σcx ¥ Σcx ¥HΗ = 0L (b) Γ=0 Γ=0.2 Γ=0.4 Γ=0.6 Γ=0.8 Γ=1 Γ=1.4 Γ=1.8 Γ = 2 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 0.80 0.85 0.90 0.95 1.00 1.05 1.10 Η Σcx ¥ Σcx ¥HΗ = 0L Figure 18: Graphs for determination of γby applying the experimental procedure proposed. Predictions for σ∞ cx(η) σ∞ cx(η=0) as a function of (a) γand (b) ηfor glass/epoxy, θl= 0.1◦and λ= 0.3. for an invertible segment of the pertinent function plotted in Figure 18(a) and for its maximum slope. Capabilities of the testing machine may represent an additional constraint. 3. Evaluate the ratio of the measured critical stress σ∞ cx(η6=0) σ∞ cx(η=0) . Then, estimate a value of ˜γfrom the measured ratio of the critical stresses for the chosen value of ηfrom Figure 18(a). 4. Estimate a value of the critical semiangle θcby solving the following nonlinear equation, employing ˜γ, see (37): ˜γqg(˜ θc, η) = s(˜ θc, η) (54) 5. Estimate the interfacial tensile strength σcfrom (38), ˜σc=˜σ∞ cx(η) ˜γqg(˜ θc, η) ,(55) where ˜σ∞ cx is one of the two values measured, either for η= 0 or η6= 0. 23 6. Estimate the interfacial fracture toughness G1cfrom (36) ˜ G1c=(˜σc˜γ)2a E∗.(56) A few comments follow with reference to a possible stumbling block tackled in the 2nd step of the above procedure. As can be observed from Figure 18(a), the ratio σ∞ cx(η6=0) σ∞ cx(η=0) for a given ηis not an injective (one-to-one) function of γfor the whole range of γconsidered. Nevertheless, this ratio may become an injective function of γwhen restricted to a suitable interval of γ, e.g. to small values of γroughly in the range 0< γ .1. In general, an a priori estimate of γwill be very useful in choosing a suitable η6= 0 for the biaxial test and a pertinent interval of γwhere the above ratio is an injective function. Recall that for γ > γth the critical semiangle is constant (θc=θE min) and the energy criterion determines the critical remote tension, which is directly proportional to γ, see (40). Then, for a given value of η, the value of pg(θE min(η), η) is fixed. Thus, the ratio shown in Figure 18(a) for γ≥γth(η) and γ≥γth(η= 0) is σ∞ cx(η) σ∞ cx(η= 0) =pg(θmin(η), η) pg(θmin(η= 0), η = 0).(57) Hence, this ratio is a constant independent of γ, as can be observed in Figure 18(a). Obviously for these rather large values of γ, the proposed experimental procedure (including the biaxial test for only one value of η) is not directly applicable. Nevertheless, repeating the biaxial tests for several adequately chosen values of ηand applying least square fitting to functions plotted in Figure 18(b) could provide a good estimation of γ, and subsequently of σcby (55) and G1cby (56) as well. 7. Concluding Remarks 1. The problem of the onset of a debond of a finite length at the initially undamaged interface of a circular cylindrical inclusion embedded in an infinite matrix subjected to a remote biaxial transverse load has been studied. A theoretical model has been developed assuming linear elastic plane strain states before and after the debond onset and sufficiently dilute packing. The critical biaxial load leading to the onset of a debond symmetrically situated with respect to the dominating remote tension is predicted together with the debond size. The present model is based on the Finite Fracture Mechanics approach introduced by Leguillon (2002) combining a pointwise normal tension criterion with an incremental energy criterion. It is expected that the present work will contribute to the knowledge of the governing parameters and to overall understanding of the failure mechanism in the fiber composites under tension dominated transverse loads. A special attention has been given to the influence of a secondary compression/tension on the value of the critical (primary) tension. Although the present work is focused on a stiff inclusion embedded in a compliant matrix (glass/epoxy composite has been used as a representative example), most results are generally valid for any combination of elastic bimaterial parameters. For the sake of simplicity a remote biaxial stress state (σ∞ x, σ∞ x) with σ∞ xy = 0 is assumed. Nevertheless, the present model and results may be easily adapted to a general remote in-plane stress state with σ∞ xy 6= 0 by working in its principal coordinate system and assuming that at least one principal stress is tension. 2. The predictions of the present model are governed by the dimensionless brittleness number γintroduced for interface cracks in Mantiˇc (2009). A consequence of this fact is that a size effect on the critical remote load and on the size of the debond at onset is predicted by the present model. It may be useful to realize that an alternative brittleness number given in terms of the critical Stress Intensity Factor (SIF) in fracture Mode I K1c, instead of the critical ERR G1c, can be proposed. Taking into account the relation between the complex Stress Intensity Factor Kand 24 Energy Release Rate Gin interfacial fracture mechanics (see Malyshev and Salganik (1965)) G= |K|2/(E∗·cosh2(πε)), this alternative brittleness number is expressed as γK=γ·cosh (πε) = K1c σc 1 √a.(58) Recall that the expression of γKin terms of K1creminds the classical definition of the brittleness number sin homogeneous materials by Carpinteri (1981). Note that, ε= 0 for a crack in a homogeneous material, thus γK=γin this case. 3. The way how the remote secondary load σ∞ yinfluence the critical value of remote tension σ∞ cx depends on the value of the ratio k/m =1 2 2+α−β 1+α−2β, defined in terms of the Dundurs elastic bimaterial parameters αand β. In particular, for γ1, a remote secondary compression σ∞ ydecreases or increases σ∞ cx if k/m < 1 or k/m > 1, respectively. This result, for k/m < 1, is coherent with the hypothesis proposed and experimentally verified by Par´ıs et al. (2003) for a particular carbon/epoxy composite. For moderate or larger values of γ,γ&1, this model predicts an almost negligible influence of the secondary compression σ∞ yon the critical tension σ∞ cx for the glass/epoxy bimaterial studied, having a somewhat flat maximum for η.0 (see Figure 12). This observation is related to the fact that the critical semidebond angles θcpredicted for these values of γare sufficiently large to make the influence of the secondary compression σ∞ ymore complex. However, the latter conclusions should be accepted with a caution in view of the range of model applicability, which appears to be very suitable for brittle configurations but to a lesser extent for tough ones. 4. In addition to the inclusion-matrix debond onset mechanism studied in the present work, other failure mechanisms can occur in the inclusion-matrix system under remote transverse loads. This is, for example, the case of the dominating compressive load studied by the coupled stress and energy criterion in Quesada et al. (2009), where parallel cracks in the inclusion and the matrix are predicted. Another example of inclusion-matrix debond configurations not allowed by the present assumption of the debond symmetrically situated with respect to the principal directions of the remote load were studied in Correa et al. (2008). Experimental tests of specimens subjected to remote transverse compressions show debonds originating at interface positions with large shear stresses. Thus, in order to complete the picture of failure envelopes shown in Figure 12, such configurations should be studied in a similar way as done in the present work. In order to take into account the influence of interface shear stresses, the stress criterion should be revised. The Mohr-Coulomb criterion (used in brittle materials and soil mechanics, see Carpinteri (1986) for a review) appears to be a suitable candidate as it shows a good agreement with the experiments carried out by Toda et al. (2001) and Ogihara and Koyanagi (2010) about failures at interfaces. Acknowledgements The authors thank to Prof. F. Par´ıs for his motivation and continuous support of the present work. The authors also thank to Dr. E. Correa for her Mathematica code of Toya’s solution used for checking proposes. This work was supported by the Junta de Andaluc´ıa and the Spanish Ministry of Science and Innovation, through the Projects TEP4051 and MAT2009-14022, respectively. I.G. Garc´ıa also acknowledges the support by the Spanish Ministry of Education through the FPU Grant 2009/3968. Appendix A. Domain of existence on the angle θ0where normal traction vanishes A threshold parameter η0can be defined as η0(α, β) = k m−1 −1.(A.1) 25