Interface crack onset at a circular cylindrical inclusion under a remote transverse tension. Application of a coupled stress and energy criterion

Interface crack onset at a circular cylindrical inclusion under a remote transverse tension. Application of a coupled stress and energy criterion V. Mantiˇc1 Group of Elasticity and Strength of Materials, School of Engineering University of Seville, Camino de los Descubrimientos s/n, 41092 Seville, Spain November 4, 2008 Abstract The plane strain problem of a single circular cylindrical inclusion embedded in an unbounded matrix subjected to a remote uniform uniaxial transverse tension is studied. A theoretical model for the simultaneous prediction of the initial size of a crack originated at the inclusion/matrix interface (or equivalently the initial polar angle of this crack) and of the critical remote tension required to originate this crack is developed. Isotropic and linear elastic behaviour of both materials, with the inclusion being stiffer than the matrix, is assumed. The interface is considered to be strong (providing continuity of displacements and tractions across the interface surface) and brittle. The model developed is based on the classical analytic solutions for the above-mentioned inclusion problem without and with a crack situated at the inclusion/matrix interface and a recently introduced coupled stress and energy criterion of failure by Leguillon (Eur. J. Mech. A/Solids, 21, pp. 61–72, 2002). A new dimensionless structural parameter γ, depending on bimaterial and interface properties together with the inclusion radius a, which plays a key role in characterizing the interface crack onset, is introduced. Asymptotic behaviour of the predicted critical remote tension and the interface crack length/polar angle at the onset are characterized for small and large values of γand a. A size effect inherent to this problem is predicted and analysed. The following asymptotic characteristics of this size effect are noteworthy: i) for small inclusion radii a, the polar angle of the crack at onset is constant (independent of a), whereas the critical remote tension increases with decreasing a, being inversely proportional to the square root of a;ii) for large inclusion radii a, the length of the crack at onset and the critical remote tension are approximately constant. Keywords: inclusion debond, interface crack, crack nucleation, strength, fracture toughness, finite fracture mechanics, size effect, scaling, brittleness number, composites. 1 Introduction It is well known that stiff circular cylindrical inclusions embedded in a compliant matrix subjected to a remote transverse tension act as stress concentrators of radial and shear stresses, σrand σrθ (referring to a polar coordinate system centered at each inclusion), at the inclusion/matrix interfaces. The most representative industrial application of this inclusion/matrix system are Fibre Reinforced Composites (FRC). The failure mechanism in FRC under traverse tension loads, called ‘matrix cracking’ or ‘interfibre failure’, typically initiates as partial debonds at fibre/matrix interfaces (or as matrix voids near this interface) due to the above-mentioned stress concentrations therein. These debonds (voids) grow as cracks along (close to) the fibre/matrix interface, eventually, under certain conditions, leaving the interface and penetrating into the matrix, where the coalescence of these cracks originates macrocracks in the composite. Any improvement in the capability of predicting the development of these cracks in FRC would be of great importance for the design and evaluation of composite structures. The above-described failure mechanism, one of the common modes of failure of FRC, has been studied from the micromechanical point of view, considering a single fibre embedded in a matrix, in many previous theoretical, numerical and experimental works, see for instance England (1966); Toya (1974); Par´ıs et al. (1996); Chao (1997); Varna et al. (1997a,b); Zhang et al. (1997); Prasad (2003); Par´ıs et al. (2007). Nevertheless, the question of the partial debond initiation at the originally undamaged fibre/matrix interface has still not been addressed in a satisfactory manner, to the best knowledge of the author. In general, either a stress based criterion or an energy based criterion is used to analyse a failure initiation and its further growth. Nevertheless, the crack onset at the (originally undamaged) inclusion/matrix interface 1On leave at the Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A. 1 cannot be correctly predicted by an individual application of either of these two criteria. Each one has its own difficulties, when used in the present problem, to predict the critical remote load originating a partial debond at the originally undamaged inclusion/matrix interface. The normal stress criterion predicts rupture at those interface points where the local value of the normal traction is greater than the tensile strength of the interface. Hence, due to the non constant distribution of the normal tractions along the undamaged inclusion/matrix interface (known from works by Goodier (1933); Hardiman (1954); Honein and Herrmann (1990)), the predicted polar angle of the crack at onset depends on the predicted critical value of the remote load and viceversa. Thus, the dilemma of this criterion lies in the fact that it provides only one equation for two unknowns. The energy criterion, in the framework of classical Fracture Mechanics, compares the Energy Release Rate (ERR) evaluated for a hypothetical infinitesimal crack at the inclusion/matrix interface with a finite value of the interface fracture toughness. However, a consequence of the fact that no stress singularity is present at the originally undamaged interface is that this ERR vanishes for an infinitesimally small crack (see Toya (1974)). This paradoxically implies that an infinitesimal crack nucleation is not possible according to this criterion. To solve difficulties of this kind Leguillon (2002) proposed a coherent approach combining both criteria in the framework of Finite Fracture Mechanics (for a review see Taylor et al. (2005)), which, as opposed to the usual Fracture Mechanics models, does not need an initial crack length to work properly. Two necessary conditions, given by a pointwise stress criterion and an incremental global energy criterion, for an abrupt formation of a crack of a finite extension, are established in this approach. In this way both above-described difficulties of the individually applied criteria are solved, providing two equations for two unknowns, the critical load and the finite crack length at onset. Although similar approaches based on coupling stress and energy criteria to predict fracture or void nucleation have been known for a relatively long time, e.g. Fisher and Gurland (1981) (see also Tszeng (1993) for a review), it seems that only the recent Leguillon’s (2002) proposal has been widely accepted, and several new successful applications of this approach to fracture onset in problems with singularities and stress concentrations have appeared in recent years, see e.g. Leguillon and Siruguet (2002); Cornetti et al. (2006); Leguillon et al. (2007); Carpinteri et al. (2008). Nevertheless, to the author’s best knowledge, the full potential of this coupled stress and energy criterion to characterize the fracture onset at the inclusion/matrix interface, and particularly in the present case of circular cylindrical inclusion, has still not been fully explored. Moreover, Leguillon et al. (2007) have shown that this coupled stress and energy criterion can explain a size effect for blunt notches and cavities. With reference to composites, the effect of inclusion size on their mechanical performance has been addressed by many authors (see e.g. Leidner and Woodhams (1974); Fisher and Gurland (1981); Cho et al. (2006)), demonstrating that, in general, the tensile strength of composites increases as the size of inclusions decreases, which is associated to the fact that a higher level of load is required to nucleate cracks at smaller inclusions than at larger ones. The interest in this inclusion size effect has recently been revived with the appearance of nanocomposites, where higher failure loads are expected in comparison with traditional composites. Thus, an analysis of the expected size effect predictions by the coupled stress and energy criterion can significantly contribute to understanding composite strength. The purpose of the present work is to contribute to modeling the abrupt formation of an inclusion/matrix debond in FRC. In particular, it aims to provide a new look (based on the coupled stress and energy criterion) at the debond onset at a circular cylindrical inclusion embedded in a matrix subjected to a remote uniaxial transverse tension. The crack onset is assumed to happen in a symmetric situation with respect to the load direction and at one side of the inclusion only. This appears to be a frequently observed configuration in real unidirectional plies subjected to a sufficiently large transverse tension load (see experimental results shown in Zhang et al. (1997); Par´ıs et al. (2007)). Recall that a further debond growth as an interface crack, not studied here, can be analysed in a similar way to that carried out by Par´ıs et al. (2007). In the present work, isotropic and linear elastic behaviour of both materials, with the inclusion being stiffer than the matrix, is assumed. The inclusion/matrix interface is considered to be strong (providing continuity of displacements and tractions across the interface surface) and brittle. The interface is characterized by its tensile strength and fracture toughness curve. The debond onset at this interface is assumed to have the form of a sharp crack. Hence, the present inclusion/matrix debonding is treated in the framework of Interface Fracture Mechanics. Two basic models have been developed in the past for the analysis of interface cracks: the open model, which assumes traction free crack faces, Williams (1959), and the contact model, which assumes a contact zone adjacent to each crack tip, Comninou (1977). For a comprehensive review of these models, in particular of their relations and limitations, see Rice (1988); Hutchinson and Suo (1992); Hills and Barber (1993); Hills et al. (1996); Gerberich and Yang (2003); Mantiˇc et al. (2006). With reference to the present problem of the interface crack onset, hence considering relatively small debond angles, for instance semidebond 2 angles below 60◦, the open model is considered appropriate for the required fracture assessment. This is due to the very small zone of non-compatible interpenetrations adjacent to the crack tip (inherent to the open model) for such semidebond angles, as follows from the analysis performed by several authors, e.g. Toya (1974); Par´ıs et al. (1996); Chao (1997); Varna et al. (1997a). Nevertheless, a study of the further debond growth (not intended in the present work), corresponding to greater semidebond angles, may require the application of the contact model as well, cf. Par´ıs et al. (2007). The procedure developed in the present work, applying the approach introduced by Leguillon (2002) to the present problem, uses several solutions, enumerated below, which are available in closed analytic form in the literature: (i) The analytic solution for stresses along the undamaged inclusion/matrix interface (Goodier (1933); Hardiman (1954); Honein and Herrmann (1990)), to be applied in the stress criterion. (ii) The analytic solution of the open model for stresses ahead of a crack at the inclusion/matrix interface (England (1966); Toya (1974)), to be applied for the evaluation of the fracture mode mixity of the interface crack in the energy criterion. (iii) The ERR in the open model for a crack at the inclusion/matrix interface (Toya (1974)), to be used in the energy criterion. (iv) A phenomenological law estimating the fracture toughness curve—the variation of the interface fracture toughness as a function of the fracture mode mixity of an interface crack (Hutchinson and Suo (1992)), to be used in the energy criterion. It has been useful for the purposes of the present work to rewrite the solutions in i)–iii) in terms of the Dundurs (1969) bimaterial parameters αand β, which in addition to an easy parametric study has allowed some interesting and useful relationships between these solutions to be elucidated. After a short review of the elastic bimaterial constants in Section 2, the stress solution along the undamaged interface (i) is presented and studied in Section 3. The stress solution ahead of an interface crack (ii) and the associated ERR (iii) are shown and analysed in Section 4. Although the present work is focused on the stiff inclusion embedded in a compliant matrix (α, β > 0), the results shown in Sections 3 and 4 cover in fact the whole range of isotropic bimaterials. The coupled stress and energy criterion is applied to the present problem in Section 5, providing the critical values of the remote load and the crack angle/length as functions of a dimensionless structural parameter and of the inclusion radius. The latter functions are in fact representations of the predicted size effect studied in Section 6. 2 Constants for isotropic bimaterials Following Dundurs (1969) the stress solution of a wide class of elastic plane strain problems for piecewise homogeneous isotropic bimaterials depends only on two dimensionless parameters: α=µ1(κ2+ 1) −µ2(κ1+ 1) µ1(κ2+ 1) + µ2(κ1+ 1) =E0 1−E0 2 E0 1+E0 2 ,(1) β=µ1(κ2−1) −µ2(κ1−1) µ1(κ2+ 1) + µ2(κ1+ 1),(2) where µk= 0.5Ek/(1 + νk) and κk= 3 −4νkare the shear modulus and the Kolosov’s constant of material k= 1,2, with Ekand νkdenoting the Young elasticity modulus and Poisson ratio, respectively. Effective elasticity modulus is defined as E0 k=Ek/(1 −ν2 k), and the harmonic mean of the effective elasticity moduli as 1 E∗=1 21 E0 1 +1 E0 2.(3) Physically admissible values of αand βare restricted to a parallelogram in (α, β) plane enclosed by lines defined as α=±1, and by 4β=α±1. Hence, |α| ≤ 1 and |β| ≤ 0.5. The so-called oscillation index ε=1 2πln 1−β 1 + β(4) appears in the elastic solution for an interface crack. Notice that cosh−2(πε) = 1 −β2. 3 Two bimaterial systems, representing typical fibre reinforced composites, will be used as examples in the present work: glass fibre/epoxy resin and carbon fibre1/epoxy resin, see Table 1 and Par´ıs et al. (2007). For illustration purposes, other bimaterial systems with extreme values of the Dundurs constants will be considered if useful. Bimaterial E1(GPa) ν1E2(GPa) ν2α β ε E∗(GPa) glass/epoxy 70.8 0.22 2.79 0.33 0.919 0.229 -0.074 6.01 carbon/epoxy 13.0 0.20 2.79 0.33 0.624 0.136 -0.044 5.09 Table 1: Examples of isotropic bimaterial constants (1 - inclusion, 2 - matrix). 3 Stresses in a single inclusion under a remote transverse tension Consider an infinitely long cylindrical inclusion, with a circular transversal section, embedded in an infinite matrix and perfectly bonded along its lateral surface. Both inclusion and matrix are assumed to be linearly elastic and isotropic materials identified by numbers 1 and 2 respectively. A uniform uniaxial remote tension σ∞>0 is applied perpendicularly to the inclusion direction. Thus, a plane strain state is generated in both inclusion and matrix. 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 and the x-axis being parallel to the load direction, see Figure 1. x y σ τ θ r=a ∞ σ ∞ σ 2 - matrix 1 - inclusion Figure 1: The inclusion problem configuration. An analytic solution for stresses in this problem was deduced by Goodier (1933). As shown by Hardiman (1954), the stresses inside the inclusion, denoted here as σ(1) ij , are constant. A compact expression of these stresses was presented by Honein and Herrmann (1990), using two bimaterial constants. Rewriting the aforementioned expression in terms of the Dundurs bimaterial constants αand βgives2 σ(1) xσ(1) xy σ(1) xy σ(1) y!=σ∞ 2 1 + α 1 + β 1 1 + α−2β2 + α−β0 0 3β−α.(5) 1The carbon fibre is in fact transversely isotropic. Nevertheless, for a plane strain state, equivalent isotropic Young elasticity modulus Eand Poisson ratio ν(appearing in Table 1) can be defined in the transversal isotropy plane as follows. Let the coordinate plane 12 be the plane of transversal isotropy, and the axis 3 the rotational symmetry axis. The elastic constants of the carbon fibre assumed are (in this footnote, subscripts denote axes): ˆ E1=ˆ E2= 13.5GPa, ˆ E3= 201GPa, ˆν31 = 0.22 and ˆν12 = 0.25. Then, considering a plane strain state, first the effective elasticity modulus and Poisson ratio are defined as E0=ˆ E1/(1 −ˆν13 ˆν31) and ν0= (ˆν12 + ˆν13 ˆν31)/(1 −ˆν13 ˆν31). Then, using definitions E0=E/(1 −ν2) and ν0=ν/(1 −ν), the equivalent isotropic Young elasticity modulus and Poisson ratio in the transversal isotropy plane are evaluated as E=ˆ E1(1 + 2ˆν12 + ˆν13 ˆν31)/(1 + ˆν12)2and ν= (ˆν12 + ˆν13 ˆν31)/(1 + ˆν12). These equivalent isotropic constants for the carbon fibre, Eand ν, will be used hereinafter in the present work. 2The bimaterial constants αHH and βHH used by Honein and Herrmann (1990) can be expressed as αHH =− α+β 1−βand βHH =− α−β 1+β. 4 According to the bounds for αand β, (2 + α−β)>0 and (1 + α−2β)≥0 (the equality holds only in the limit case α=−1 and β= 0), whereas the expression 3β−αcan be positive or negative. From (5), normal and tangential tractions, σand τ, acting along the inclusion/matrix interface (r=a), can be easily expressed as functions of the polar angle θ, σ(θ) = σr(θ) = σ∞k−msin2θ, τ(θ) = −σrθ(θ) = σ∞msin θcos θ, (6) where3 k(α, β) = 1 2 1 + α 1 + β 2 + α−β 1 + α−2β≥0, m(α, β) = 1 + α 1 + β≥0.(7) Looking at the left expression in (6), k(α, β) can be seen as the concentration factor of the normal tractions defined along the inclusion/matrix interface, their maximum being achieved for θ= 0. Thus, it is useful to check its behaviour, shown in Figure 2, as a function of αand β. In the particular case of equal inclusion and matrix materials, k(0,0) = 1. For α > 0 and β > 0 (the case the present work is focused on), k > 1, whereas for α < 0 and β < 0, k < 1. The maximum value is k(1,0.5) = 5 3= 1.¯ 6. The minimum value k(−1, β) = 0 is achieved if E0 1= 0, the inclusion in fact representing a void. A discontinuity of kcan be observed at the point (α, β) = (−1,0), where the limits of ktake values between 0 and 1 depending on the direction approaching this point. -1 -0.5 0 0.5 1 -0.5 -0.25 0 0.25 0.5 0 0.5 1 1.5 0 .25 0 0.25 Goodier3101.nb 1 Figure 2: kas a function of αand βin the Dundurs parallelogram. The behaviour of m(α, β), defined by the right expression in (7), is easier to characterize: m= 1 for α=β and m≶1 for α≶β. The maximum and minimum values are m(1,0) = 2 and m(−1, β) = 0, respectively. It is instructive to rewrite σ(1) ij in (5) in terms of kand mas σ(1) xσ(1) xy σ(1) xy σ(1) y!=σ∞mk m0 0k m−1.(8) This expressions shows that the character of the inclusion stress state is basically determinate by the ratio k m, in particular when referring to the position of its Mohr circumference with respect to the vertical τaxis. Let α > −1, and consequently k > 0 and m > 0. Then, the angle θ0for which the interface normal traction vanishes, i.e. σ(θ0) = 0, is given by θ0(α, β) = arcsin rk m.(9) The expression on the right hand side of (9) makes sense only if k≤m, which is equivalent to the condition σ(1) y≤0 or, in terms of the Dundurs constants, to 3β≤αand α > −1. It will be convenient to define θ0for k > m as θ0=∞. 3For the sake of simplicity, the dependence of different functions on bimaterial or adjustable parameters will be shown explicitly where the function is defined for the first time, but in subsequent usage of these functions these parameters will usually be omitted. 5 According to the left expression in (6), see also (8), in inclusion/matrix bimaterial systems for which k > m, only positive normal tractions take place along the whole interface, which at first sight can be a surprising result. Again, in view of the importance of the ratio k mfor the interface traction distribution it seems useful to check its behaviour, shown in Figure 3, as a function of αand β. It is easy to check analytically that the ratio k mis unbounded ( k m→ ∞) at the corner point (α, β) = (−1,0) of the Dundurs parallelogram, whereas its minimum value k m=3 4is achieved at the side 4β=α−1 of this parallelogram. The latter result implies that θ0defined in (9) has a minimum value θ0(α, 0.25(α−1)) = arcsin √3 2= 60◦. Hence, 60◦≤θ0≤90◦,for k≤m. (10) -1 -0.5 0 0.5 1 -0.5-0.25 00.25 0.5 0 0.5 1 1.5 2 2.5 0 0. 1 1 Goodier3101.nb 1 Figure 3: k mas a function of αand βin the Dundurs parallelogram. For the bimaterials defined in Table 1, the values of the above defined constants characterizing the inclusion/matrix interface traction distribution are presented in Table 2. In view of (9), very similar values of ratio k m, shown in Table 2, imply that the values of θ0for these bimaterials are also very similar. Bimaterial k m k/m θ0[◦] glass/epoxy 1.44 1.56 0.9205 73.63 carbon/epoxy 1.32 1.43 0.9200 73.57 Table 2: The values of k,m,k mand θ0for the examples of isotropic bimaterials. 4 The solution for a crack at the interface of a single inclusion under a remote transverse tension Consider now the single inclusion problem configuration from the previous section altered by the presence of a debond—interface crack, symmetrically situated with respect to the load direction, with a semidebond angle (θd≥0) and an infinite length in the z-axis direction, see Figure 4. This problem has been studied analytically and numerically by many authors, using both basic models of Interface Fracture Mechanics (open model and contact model), see Par´ıs et al. (2007) for a review. A fully analytic solution of this problem (in fact of a more general problem) was deduced by Toya (1974), using the open model. A concise presentation and a parametric analysis of the Toya’s solution can be found in Murakami (1988). Toya’s general expressions for the interface tractions ahead of the crack tip and for the Energy Release 6 x y σ τ ψ θ θ d θ l r=a ∞ σ ∞ σ 1 2 Figure 4: The interface crack problem configuration. Rate (ERR) are particularized here for the present problem and newly rewritten in terms of the bimaterial constants4defined in Section 2. These expressions will be later used in Section 5. The interface tractions at a point placed ahead of the crack tip at the polar angle θ=θd+θ`, where θ`>0, can be evaluated by the complex variable expression: σ(θ)−iτ(θ) = −σ∞ 2 1−α 1−βχ(θ)p(θ),(11) where i=√−1 is the imaginary unit, and χ(θ) = eiθ −eiθd−1 2−iε eiθ −e−iθd−1 2+iε ,(12) p(θ) = q(θd)eiθ −(cos θd−2εsin θd)−1 + α 1−αe−2ε(θd−π)(cos θd+ 2εsin θd)e−iθ −e−2iθ,(13) with q(θd) = 1−(cos θd−2εsin θd)e2ε(θd−π)+1 2(1 + α)(1 + 4ε2) sin2θd 3 + α−(1 −α)(cos θd−2εsin θd)e2ε(θd−π)−1 1−α.(14) The ratio of the interface shear and normal tractions ahead of the crack tip at a small (either geometryor material-based) reference length `gives a measure of fracture mode mixity of an interface crack, see Rice (1988); Hutchinson and Suo (1992); Banks-Sills and Ashkenazi (2000); Mantiˇc and Par´ıs (2004). An approach to experimentally determine a material-based characteristic reference length `mhas recently been proposed by Agrawal and Karlsson (2007) and discussed by Mantiˇc (2008). Nevertheless, in the present study it seems appropriate to adopt a small geometry-based reference length defined as `g=θ`a, where θ`is a small fixed reference angle (independent of the debond angle θd). Thus, it will be assumed that the angle ψgiven by tan ψ(θd;θ`) = τ(θd+θ`) σ(θd+θ`),(15) provides a suitable measure of the fracture mode mixity. Figure 5 shows the evolution of ψ(θd) for the bimaterials defined in Table 1, considering θ`= 0.1◦. Taking different values of θ`, e.g. θ`= 0.01◦or 1◦also considered in Section 5.3, would result in curves of the function ψ(θd) similar to those shown in Figure 5, with the straight part shifted an angle. It should also be noticed that interface tractions ahead of the crack tip, given by (11), and consequently also the angle ψ(θd), are independent of the inclusion radius a. The ERR of the interface crack propagating, for example, at its upper crack tip at an angle θdalong the interface can be expressed as G(θd) = (σ∞)2a E∗b G(θd;α, β),(16) 4The bimaterial constants νT,βT,kTand λTused by Toya (1974) can be expressed as νT=1−β 1+β,βT=1−α 1+β,kT=1−α 2and λT=−ε. 7 0 10 20 30 40 50 60 70 80 90 100 110 120 0 102030405060708090 θ d[º] ψ [º] glass/epoxy carbon/epoxy glass/epoxy Figure 5: Examples of the evolution of the fracture mode mixity angle ψtaking θ`= 0.1◦. where the dimensionless function b G(defining a normalized form of ERR) depends only on θd,αand β. It is defined as b G(θd;α, β) = π1+4ε2(1 + α)2sin θdd(θd)(d(θd)−2c(θd) cos θd) + c(θd)2/8c(θd),(17) and c(θd) = 2e−2ε(θd−π),(18) d(θd) = −4−(1 −α)(1 + 4ε2) sin2θd 3 + α−(1 −α)(cos θd−2εsin θd)e2ε(θd−π).(19) It should be noticed that, according to (16), the ERR G(θd) varies linearly with the inclusion radius a. Sometimes it is useful to approximate b G(θd) for small values of θdby its Taylor series expansion at θd= 0: b G(θd) = db G dθdθd=0 θd+O(θ2 d),(20) where the fact that ERR vanishes for an infinitesimal crack extension, b G(0) = 0, has been taken into account. From (17) it can be shown that b G0(0; α, β)def =db G dθdθd=0 =k2π(1 + 4ε2) cosh2(πε).(21) Then the derivative of Gwith respect to the crack semilength, evaluated for the infinitesimal crack length, is expressed as dG d(aθd)θd=0 =(kσ∞)2π(1 + 4ε2) E∗cosh2(πε).(22) This expression agrees with the analogous derivative of ERR for a crack at a straight interface between two half-spaces, see Rice (1988), except the factor k2. In fact, this result could be expected as kσ∞is the value of the normal tractions acting at θ= 0 before the infinitesimal crack appears therein. Figure 6 shows evolution of b G(θd) and its first order Taylor series expansion, obtained from (21), for the bimaterials defined in Table 1. 5 Interface crack onset at a single inclusion under a remote transverse tension. A novel approach to solve the problem of a crack onset at a stress concentration (e.g. a blunt notch) or at a weak stress singularity (e.g. a reentrant corner with traction free faces), where originally there is no crack, has 8 0 1 2 3 4 5 0 102030405060708090 θ d glass/epoxy carbon/epoxy asymptotes [º] Figure 6: Examples of the evolution of the normalized ERR and its linear approximation. been developed by Leguillon (2002); Leguillon and Siruguet (2002); Leguillon et al. (2007) in the framework of Finite Fracture Mechanics. The key idea of this approach is to use two coupled criteria, one strength based and one energy based, either of them representing a necessary but not sufficient condition for the crack onset. Applying each criterion individually leads to unresolvable questions, requiring the definition of a characteristic length, in the following sense: •The stress-strength criterion determines the minimum value of the applied load leading to rupture, but it is unable to determine unambiguously the size of the crack originated. •An application of the (infinitesimal) Griffith type energy criterion for crack growth requires an a priori existing crack. Therefore, no fundament for determining the initial crack length is given. However, when assuming an abrupt onset of a crack of a finite length, and applying both criteria simultaneously, they provide a system of two (typically nonlinear) equations for two unknowns: the length of the crack originated and the critical load required for its onset. In what follows, a generalization of the Leguillon’s coupled criterion for the interface cracks, where the fracture toughness is a function of fracture mode mixity, is applied to the present problem of the interface crack onset at a single circular cylindrical inclusion embedded in a matrix subjected to a remote transverse uniaxial tension. 5.1 Tensile stress criterion A stress criterion is usually invoked if no pre-existing debond exists at the inclusion surface. The tensile stress criterion adopted here assumes the existence of a critical value σc>0 of interface normal tensions— interfacial tensile strength, defined as the maximum tension that the interface can sustain without fracture. Thus, according to this stress criterion, the inclusion/matrix interface will break at those points where tension exceeds σc: σ(θ)≥σc.(23) In the present problem of remote uniaxial tension, the normal stress σ(θ) is changing along the interface with the position angle θas shown in (6). Combining the left expression in (6) with (23) leads to the necessary condition for rupture at an interface point defined by the angle θ: σ∞ σc≥1 k−msin2θ.(24) Inasmuch as the function on the right hand side of (24) increases with increasing |θ|, the interface debond is expected to happen, for a sufficiently large particular value of ratio σ∞/σc, along the entire portion of the inclusion surface, symmetrically with respect to the load direction, limited by the maximum angle θσ cfor which (24), and equivalently the stress criterion (23), are fulfilled, |θ| ≤ θσ c.(25) 9 0 1 2 3 4 5 6 7 8 9 10 0 1020304050607080 Δθ energy and stress criteria stress criterion curve energy criterion curves [º] (B) (A) (a) 0 1 2 3 4 5 6 7 8 9 10 0 1020304050607080 Δθ energy and stress criteria energy criterion curves stress criterion curve [º] (A) (B) (b) Figure 11: Plots of coupled stress and energy criteria, scenario Adefined by (51) with γ= 1, and scenario B defined by (53) with γ= 5, taking λ= 0.3 and θ`= 0.1◦. (a) glass/epoxy, (b) carbon/epoxy. Let a threshold value of γbe defined as γth(α, β;λ, θ`) = 1 pg(θE min) 1 k−msin2θE min .(56) Then, scenario Acorresponds to 0 < γ ≤γth. In simple terms, small values of γare associated to relatively small values of the interface fracture toughness G1cor of the harmonic mean of the effective Young moduli E∗, or to relatively large values of the interface strength σcor of the inclusion radius a. It can be deduced from Figure 11 that lim γ→0+ θc= 0.(57) Then, θccan be approximated by a quadratic function of γ, θc∼ =2k2 b G0(0)γ2=2 cosh2(πε) π(1 + 4ε2)γ2,for sufficiently small values of γ > 0,(58) obtained by neglecting msin2θcwith respect to kon the right hand side of (51), and by approximating g(θc) by ˜g(θc) defined in (46). In view of these results, it is obtained from (51) that σ∞ c σc &k−1,for sufficiently small values of γ > 0,(59) σ∞ cthen being essentially strength governed. Notice that the upper bounds of ranges where (58) and (59) are valid are not sharply defined. 16 Find min ∆θg(∆θ)def =g(θE min) If θE min < θ0and γpg(θE min)<1 k−msin2θE min or θE min ≥θ0then Solve the following equation for ∆θ < min θE min, θ0: γpg(∆θ) = 1 k−msin2∆θ θc=the solution ∆θof this equation Else θc=θE min Endif Compute the critical load σ∞ cby σ∞ c σc=γpg(θc)or equivalently by σ∞ c=qG1cE∗ ag(θc) End Figure 12: Computational procedure for the evaluation of θcand σ∞ c. Scenario Bcorresponds to γ > γth. In simple terms, large values of γare associated to relatively large values of the interface fracture toughness G1cor of the harmonic mean of the effective Young moduli E∗, or to relatively small values of the interface strength σcor of the inclusion radius a. In this case, the interface crack onset is essentially governed by the energy criterion. Then, θc=θE min,for γ > γth,(60) and, in view of (43), σ∞ c σc =σ∞,E c σc =γqg(θE min),for γ > γth.(61) Thus, while θcis constant for γ > γth,σ∞ cis a linear function of γ. The values of the characteristic parameters θE min,g(θE min) and γth for the examples of isotropic bimaterials defined in Table 1 are presented in Table 3. To check how the choice of the values of the fracture-mode sensitivity parameter λ, used in (30), and of the reference angle θ`, used in (15), may affect these characteristic parameters, different values of λand θ`are considered in Table 3. Bimaterial λ θE min g(θE min)γth glass/epoxy 0.2 37.7◦/42.5◦/46.8◦0.92/0.73/0.63 1.2/1.6/2.1 0.3 43.8◦/48.2◦/52.3◦0.77/0.64/0.56 1.7/2.2/2.9 carbon/epoxy 0.2 43.0◦/45.6◦/47.8◦0.85/0.77/0.71 1.7/1.9/2.2 0.3 48.3◦/50.7◦/52.7◦0.76/0.70/0.65 2.2/2.6/3.0 Table 3: The values of θE min,g(θE min) and γth for the examples of isotropic bimaterials, and two values of λ (0.2/0.3) and three values of θ`(0.01◦/0.1◦/1◦). The applicability of the Toya (1974) solution for the bimaterials considered can be easily checked by means of a formula, deduced by Hills and Barber (1993) and generalized by Graciani et al. (2007), for the estimation of the extension of the interpenetration zone adjacent to an interface crack tip, always existent in the open model solution. Rewriting this formula to the present case, the angle θI(θd) defining the length of this interpenetration zone can be obtained as the largest value of θI(θd) = θ`exp 2n−1 2π−ψ(θd;θ`)sgnε+ arctan(2|ε|)/|ε|,(62) which is lower than the debond angle 2θd, with nbeing an integer. By substituting into this formula the values of the fracture mode mixity angle ψcomputed by (15) (see also Figure 5) and corresponding to the values of θ`and θd=θE min from Table 3, it is obtained that for glass/epoxy θI<0.052◦, whereas for carbon/epoxy θI<0.00046◦. Thus, these interpenetration zones are sufficiently small to validate the open model solution from Toya (1974) in the procedure presented. The fact that θI(θd) is an increasing function of θdfor the range of θdconsidered has been taken into account in the interpenetration zone estimations. 17 Figures 13(a) and (b) show θcand σ∞ c σcas functions of the dimensionless structural parameter γfor the bimaterials defined in Table 1, taking λ= 0.3 and θ`= 0.1◦. Inasmuch as some features of these functions, in particular the asymptotic behaviour, are more easily identified in log-log scale, the corresponding log-log plots are presented as well, Figure 14. One can easily check from these plots how the variation of one of the problem parameters, e.g. σcor G1c, may affect the initial debond angle and the critical remote load value. According to Table 3, choosing different values of λand θ`from physically reasonable ranges results in, at most, moderate variations of the characteristic parameters θE min,g(θE min) and γth. Thus, also in view of the fact that the asymptotes shown in Figures 13 and 14 are independent of λand θ`, the curves plotted in these figures will vary only a little when different values of λand θ`are chosen. These curves can be easily approximated taking the pertinent values of θE min,g(θE min) and γth from Table 3. 0 10 20 30 40 50 60 0 0,5 1 1,5 2 2,5 3 γ θc[º] asymptotes γ th carbon/epoxy glass/epoxy (a) 0 0,5 1 1,5 2 2,5 3 0 0,5 1 1,5 2 2,5 3 γ γ th asymptotes linear function carbon/epoxy glass/epoxy k 1 (b) Figure 13: (a) Critical semiangle θcand (b) Critical remote tension as functions of the dimensionless structural parameter γ, taking λ= 0.3 and θ`= 0.1◦. After the abrupt crack onset of a semiangle θcpredicted by the coupled stress and energy criterion, a further growth of the sharp interface crack can be assessed by the criterion of the classical ‘infinitesimal’ Interfacial Fracture Mechanics, in a similar way to that carried out, for instance, in Par´ıs et al. (2007). This means that the crack of a semidebond angle θdis assumed to grow along the interface if G(θd)≥Gc(ψ(θd)).(63) Let θa(θa≥θc) denote the arrest angle, defined as the maximum angle θdfor which (63) holds. According to the analysis of relationships between Gand Gcin Section 5.2, and in particular according to the examples shown in Figure 10, the following two post crack-onset scenarios can be expected: a) If θc< θE min (scenario Awithout the upper limit case), then G(θc)> Gc(ψ(θc)) and also G(θE min)> 18 0,1 1 10 100 0,1 1 10 γ θ c[º] asymptotes γ th carbon/epoxy glass/epoxy 1 2 (a) 0,1 1 10 0,1 1 10 γ γ th asymptotes glass/epoxy carbon/epoxy 1 1 k 1 (b) Figure 14: (a) Critical semiangle θcand (b) Critical remote tension as functions of the dimensionless structural parameter γin log-log scale, taking λ= 0.3 and θ`= 0.1◦. Gc(ψ(θE min)). Thus, the interface crack is expected to continue growing unstably from θcto θa> θE min, cf. Figure 10(a). b) If θc=θE min (scenario Band the upper limit of scenario A), then G(θc) = Gc(ψ(θc)) and dG(θd)/dθd|θd=θc <dGc(ψ(θd))/dθd|θd=θc. Thus, no further unstable crack growth along the interface is expected and θa=θE min, cf. Figure 10(b). 6 Size effect A size effect in the problem under study can be understood as a variation of the critical value of the remote tension σ∞ cwith variations of the inclusion radius a, keeping all other problem parameters, namely the bimaterial properties (α, β, E∗) and the interface properties (σc, G1c), constant. Thus, according to the previous analysis, and particularly in view of the dependence of the key dimensionless parameter γdefined in (52) on the inclusion radius a, some size effect governing the crack onset at the cylindrical inclusion/matrix interface can be expected. This predicted size effect appears basically due to the fact that from the four basic magnitudes appearing in the coupled stress and energy criterion, interface traction distribution (6), interface strength σc, ERR (16), and the interface fracture toughness (30), the only magnitude dependent on the inclusion radius a (assuming a physically reasonable range for a) is the ERR G(θd). Let a bimaterial characteristic length a0be defined in terms of the interface properties σcand G1cand of 19 the elastic bimaterial property E∗as follows6: a0=G1cE∗ σ2 c ,thus γ=ra0 a.(64) Hence, a=a0is equivalent to γ= 1. Additionally, let a threshold value of abe defined as ath =a0 γ2 th .(65) For a < ath, the interface crack onset is essentially governed by the energy criterion, and relationships (60) and (61) directly imply, in view of (43), that θc=θE min,and σ∞ c σc =σ∞,E c σc =qg(θE min)ra0 a.(66) Thus θcis constant, whereas σ∞ c∼1 √a. According to the left equation in (66), the initial semilength of the interface crack, aθc, is varying linearly with afor a < ath. For sufficiently large a, relationships (58) and (59) yield θc∼ =2 cosh2(πε) π(1 + 4ε2) a0 a,and σ∞ c σc &k−1,(67) thus θc∼1 a, whereas σ∞ cis essentially governed by the stress criterion and is approaching a constant. From the left equation in (67) it follows that the initial semilength of the interface crack is approaching a constant for sufficiently large a, aθc∼ =2 cosh2(πε) π(1 + 4ε2)a0.(68) This result corresponds to the fact that, for a very small θc, the crack onset problem at the concentration point of normal tensions at the inclusion/matrix interface is locally similar to the problem of a crack situated at an infinite straight interface subjected to a remote tension kσ∞. For the bimaterials defined in Table 1, Figure 15 shows θc,aθcand σ∞ c σcas functions of the inclusion radius a, taking θ`= 0.1◦and λ= 0.3. Notice that ath a0= 0.21 and 0.15 for glass/epoxy and carbon/epoxy, respectively. The corresponding plots in log-log scale are presented in Figure 16. From the above analytic results and from Figures 15 and 16, one can conclude that, for the same bimaterial and the same quality of the interface, - the critical crack semiangle θcis constant for small a, and decreases, proportionally to 1 a, for increasing and large a, - the critical crack semilength aθcvaries linearly for small a, and approaches a constant for large a, - the critical remote tension σ∞ cis increasing as 1 √afor decreasing and small a, and is approaching a constant for large a. Finally, for the bimaterials defined in Table 1, values of the parameters defined in the procedure developed are shown in Table 4. These parameters have been computed assuming the inclusion radius a= 7.5µm (taken from Par´ıs et al. (2007)), with λ= 0.3 and θ`= 0.1◦. As the interface strength σcand fracture toughness G1cparameters are difficult to know precisely, roughly estimated minimum and maximum values of these parameters from the data available in Varna et al. (1997b); Zhang et al. (1997); Soden et al. (1998) are used in Table 4. In fact, σcvalues are only estimated from the bulk epoxy tensile strength values given in these references. Taking either the minimum σcand maximum G1cor viceversa, the minimum and maximum values of each parameter presented are obtained. It can be seen that for both bimaterials γ < γth (cf. Table 3), or equivalently a > ath, thus in all the cases shown the critical values θcand σ∞ care determined by combining both stress and energy criteria (the situation corresponding to scenario Adescribed in Section 5.3). 6Analogous material characteristic lengths have previously been introduced by several authors in different contexts, e.g., the critical length for quasibrittle materials `c=EGc f2 t ,ftbeing the tensile strength, by Hillerborg et al. (1976). The length a0is also related to the plastic zone correction factor in a ductile material, rY S =1 2π K2 σ2 Y S , by Irwin (1960). Nevertheless, it seems that the present definition of a0is the first proposal of a characteristic length of this kind for interface cracks. In fact, a0could be considered as a generalization of the Hillerborg’s critical length `cto interface cracks. 20 Bimaterial σc[MPa] G1c[Jm−2]a0[µm] ath [µm] a a0γ θc[◦]σ∞ c σc glass/epoxy 60 10 16.7 3.5 0.4 1.5 39 1.2 90 2 1.5 0.3 5.1 0.4 7.4 0.7 carbon/epoxy 60 10 14.1 2.1 0.5 1.4 35.7 1.2 90 2 1.3 0.2 6.0 0.4 6.1 0.8 Table 4: Estimations of the maximum and minimum values of a0,ath,a/a0,γ,θc,σ∞ c/σcfor the examples of isotropic bimaterials. 7 Concluding remarks 1) A theoretical model has been developed for the prediction of the crack onset at the interface between a stiff circular cylindrical inclusion and a compliant unbounded matrix subjected to a remote uniaxial transverse tension. This model is based on a coupled pointwise stress criterion and an incremental energy criterion, an approach recently introduced by Leguillon (2002). The inclusion and matrix materials are assumed to be homogeneous isotropic linearly elastic, bonded along a strong and brittle interface. The interface is characterized by two failure parameters: the tensile strength σcand the fracture toughness curve Gc(ψ), ψbeing the fracture mode mixity angle. At onset, an abrupt crack formation of a finite extension is assumed to occur around the tensile traction concentration point at the interface. The present coupled stress and energy criterion predicts the critical value of the remote load σ∞ cand the initial crack angle θcat onset. 2) The predicted values of σ∞ c(normalized by σc) and θcare determined as functions of the Dundurs bimaterial parameters αand βand of a new dimensionless microstructural parameter γ(52). The parameter γhas shown to be suitable for characterization of the present crack onset problem. It is closely related to a characteristic length parameter a0(64), defined in terms of the interface tensile strength σc, the interface fracture toughness G1cassociated to the fracture Mode I, and the harmonic mean E∗of the effective elastic moduli of the inclusion and matrix. The parameter γcan be defined as the square root of the ratio of a0to the inclusion radius a. Therefore, a size effect, i.e. variations of predicted values of σ∞ cand θcwith akeeping all other problem parameters constant, is inherent to the predictions obtained. Basically, this size effect is associated to the fact that the crack Energy Release Rate (ERR) decreases as adecreases for the same remote load applied, whereas the effect of avariation (in a physically reasonable length range) on the interface fracture toughness is negligible. Notice also that the stress criterion alone is not able to predict any size effect, as the interfacial stress distribution is independent of a, considering that the effect of avariation on the interface strength is negligible. 3) The asymptotic behaviour of the predicted values of σ∞ cand θcfor small and large values of acan be characterized in simple terms as follows. For small values of a, the crack onset of a constant angle θc(independent of a) is expected to occur, while the critical remote tension σ∞ cis increasing as ∼1 √awith decreasing a. For large values of a, the semilength of the crack aθcat onset and the critical remote tension σ∞ care approximately constant. The former size effect feature seems to be in accordance with the experimental evidence that, in general, the tensile strength of composites increases as the inclusion size decreases. Nevertheless, it might be possible that although for very small values of athe interface debond will not occur, rupture of the matrix near the inclusion could instead become the preferred mode of failure. 4) With reference to the size effect studied, considering different inclusion radii a, one could argue that the application of a fixed material-based reference length `mto define the fracture mode mixity angle ψwould be a more consistent choice than the geometry-based reference length `g=θ`a(θ`being a small fixed reference angle) used in the present work. The reference length `gis adopted in the present work for the sake of simplicity and the universality of the analysis performed, being perfectly consistent for the study carried out in Section 5. Nevertheless, to check the influence of this choice on the size effect studied in Section 6, additional calculations have been performed for a small and physically reasonable `m= 0.1◦×7.5µm=0.013µm. The four limit cases of the characteristic length parameter a0presented in Table 4 have been analysed. Only slight deviations have been observed from the plots shown in Figures 15 and 16 for glass/epoxy, these deviations being even smaller for carbon/epoxy. In fact, in large parts of some plots the differences are hardly visible. Considering a physically reasonable range 0.1≤a/a0≤10, the maximum relative differences between the values of the three quantities presented in these plots (θc,θca/a0and σ∞ c/σc) obtained using `mand `ghave been less than 9% for glass/epoxy and less than 3% for carbon/epoxy. In particular, with reference to the predicted behaviour σ∞ c∼1 √afor small values of a, the least squares fitting of power law in the range 0.1≤a/a0≤ath/a0gives the exponents whose relative differences from −0.5 are less than 6% and close to 1%, respectively, for the 21 couples of large and small values of a0in Table 4. Thus, the choice of a small and physically reasonable `m affects only weakly the size effect predicted by using `g. It can be expected that the size effect characteristics extracted from the behaviour of θc,θcaand σ∞ cin Figures 15 and 16, obtained using `g, represent a good universal approximation of the behaviour of these quantities when obtained using other physically reasonable choices of `m. 5) It should be mentioned that, although the present work has mainly focused on the interface crack onset, a simple analysis of the post crack-onset behaviour has also been introduced. The key parameter of this analysis is the angle θE min (38), the upper bound for θc. If θc< θE min then a further unstable crack growth along the inclusion/matrix interface can be expected up to an arrest angle greater than θE min. However, if θc=θE min, no further crack growth along the interface is expected for the remote load value considered. A more detailed and realistic analysis of this post crack-onset behaviour can be carried out by the suitable analytical and numerical tools presented in Par´ıs et al. (2007). In fact, according to the results of the numerical study presented therein, it can be expected that the interface crack will grow unstably up to semidebonding angles of values 60◦−70◦, where it will either stop or continue growing along the interface or kink towards the matrix, then continuing its unstable growth as a matrix crack in the direction perpendicular to the load. If the latter scenario represents reasonably the real progression of damage in a unidirectional ply under transverse tension, assuming the coalescence of the matrix cracks initiated at the fibre/matrix interfaces, then the critical load for the interface crack onset predicted in the present work could be quite directly related to the critical transverse tension for the whole ply. 6) It is expected that the present work can contribute to clarifying which relations of the bimaterial and fibre/matrix interface properties play an important role in the resulting transverse tensile strength of a unidirectional ply. These relations can be very useful in the fibre/matrix interface characterization. In particular, knowledge of the value of the parameter γ(or equivalently of the characteristic length parameter a0), governing the interface crack onset, seems to be fundamental in this sense. Thus, it could be very useful to develop some specific experiments, for example, using single fibre specimens, to determine the value of γ for a particular fibre/matrix system. 7) The present formulation of the coupled stress and energy criterion can be easily modified by incorporating the average instead of the pointwise tensile stress criterion employed here, as suggested by Cornetti et al. (2006) and Carpinteri et al. (2008). It has been checked that visible differences between the predictions obtained by the coupled criteria, using one of these two tensile stress criteria, appear only in the transition regime between the two asymptotic regimes, corresponding to small and large inclusion radii a. In fact, there is no difference between the application of these stress criteria once sufficiently small inclusion radii aare considered, as for these athe debond onset is governed by the energy criterion only. Also, the differences between the predictions obtained using these stress criteria are hardly visible for large a, as these predictions are governed by the same asymptotes. The main difference between the application of these stress criteria is the threshold value ath, which is several times greater for the average than for the pointwise tensile stress criterion (e.g. about 3.3 times for glass/epoxy and 4.1 times for carbon/epoxy, taking λ= 0.3 and a small reference angle θ`= 0.1◦). An application of the Mohr-Coulomb pointwise stress criterion in the coupled stress and energy criterion could also be of interest, considering large shear tractions acting along the inclusion/matrix interface for the values of the polar angle θclose to 45◦. As the Mohr-Coulomb criterion can predict the position of the debond initiation at an angle θdifferent from 0◦, such an application would be more challenging, requiring to analyse asymmetric configurations of the load and an asymmetrically growing debond. Experimental evidence would be necessary to determine which of these criteria is the best suited to the present problem. 8) The present approach can also be extended to the cylindrical inclusion/matrix configuration subjected to other kinds of remote transverse loads, like compression, Correa et al. (2008a,b), or biaxial loads, Par´ıs et al. (2003), to obtain pertinent predictions of the critical load initiating an interface debond. Such studies could further contribute to a better understanding of the FRC strength under transverse loads. Acknowledgement The author thanks Prof. Federico Par´ıs for his motivation and continuous support of this work. Comments by Prof. Federico Par´ıs and Dr. Enrique Graciani have substantially improved the final version of the manuscript. Stimulating discussions with Dr. Elena Correa and use of her Mathematica code of the Toya’s solution are also gratefully acknowledged. This work was supported by the Spanish Ministry of Education, Culture and Sport through Project TRA2005-06764, and by the Junta de Andaluc´ıa, through the Project of Excellence TEP1207. Part of the present work was performed during a research stay at the Oak Ridge National Laboratory (ORNL) 22 in 2008. The support of this stay by Dr. Len Gray (ORNL) and by the Junta de Andaluc´ıa (Estancia de excelencia) is also gratefully acknowledged. A Proof of inequality dG(θd) dθd <dGc(ψ(θd)) dθdfor θd=θE c=θE min Let θE c=θE min. By the definition of θE c, as the minimum angle ∆θ > 0 for which equality holds in (35), Z∆θ 0 G(θd)dθd<Z∆θ 0 Gc(ψ(θd))dθd,for 0 <∆θ < θE min,(69) ZθE min 0 G(θd)dθd=ZθE min 0 Gc(ψ(θd))dθd.(70) Hence, by subtracting (70) from (69), ZθE min ∆θ G(θd)dθd>ZθE min ∆θ Gc(ψ(θd))dθd,for 0 <∆θ < θE min.(71) According to definitions (16) and (30), see also Figures 6 and 8, the functions −G(θd) and Gc(ψ(θd)) are strictly convex in the range of angles of interest for the present study, say θd≤80◦. Then, the left part of the Hermite-Hadamard inequality for strictly convex functions applied to the members of the inequality (71) yields the following inequality chain for 0 <∆θ < θE min: G∆θ+θE min 2>1 θE min −∆θZθE min ∆θ G(θd)dθd>1 θE min −∆θZθE min ∆θ Gc(ψ(θd))dθd> Gcψ∆θ+θE min 2. (72) Considering the first and the last terms in (72) leads directly to the following general inequality in the case θE c=θE min: G(θd)> Gc(ψ((θd)),for θE min 2< θd< θE min.(73) Applying a basic property of differentiable strictly convex functions to the members of (73) gives GθE min+dG dθdθd=θE min θd−θE min> G(θd)> Gc(ψ(θd)) > GcψθE min+dGc dθdθd=θE min θd−θE min,(74) for θE min 2< θd< θE min. Then, in view of the equality in (40), dG dθdθd=θE min <dGc dθdθd=θE min .(75) 23 0 10 20 30 40 50 60 0123456 a/a0 θ c carbon/epoxy asymptotes θEmin, carbon/epoxy θEmin, glass/epoxy glass/epoxy ath/a0 (a) 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0123456 a/a0 crack length/a0 glass/epoxy carbon/epoxy asymptotes ath/a0 linear function (b) 0 0,5 1 1,5 2 2,5 3 0123456 a/a0 ath/a0 asymptotes glass/epoxy carbon/epoxy k 1 (c) Figure 15: (a) Critical semiangle θc, (b) Critical semilength of the crack aθcand (c) Critical remote tension σ∞ cas functions of the inclusion radius a, taking λ= 0.3 and θ`= 0.1◦. 24 1 10 100 0,1 1 10 a/a0 carbon/epoxy asymptotes glass/epoxy ath/a0 1 1 θ c[º] (a) 0,1 1 0,1 1 10 a/a0 crack length/a0 glass/epoxy carbon/epoxy asymptotes ath/a0 1 1 (b) 0,1 1 10 0,1 1 10 a/a0 ath/a0 asymptotes glass/epoxy carbon/epoxy 10.5 k 1 (c) Figure 16: (a) Critical semiangle θc, (b) Critical semilength of the crack aθcand (c) Critical remote tension σ∞ cas functions of the inclusion radius ain log-log scale, taking λ= 0.3 and θ`= 0.1◦. 25