International Journal of Fracture manuscript No. (will be inserted by the editor)
Interface fracture toughness of a multi-directional woven composite Leslie Banks-Sills · Chaim Ishbir · Victor Fourman · Liran Rogel · Rami Eliasi
Received: date / Accepted: date
Abstract The aim of this investigation is to measure the interface fracture toughness of a woven composite. For this purpose, double cantilever beam (DCB) specimens are tested to measure the load as the delamination grows. The specimen is composed of 15 layers of a carbon-epoxy, balanced weave with alternate layers containing fibers in the 0◦ /90◦ directions and the +45◦ / − 45◦ directions. A thin piece of Teflon is placed between two layers of differing directions. The specimens are analyzed by means of the finite element method and an interaction energy or M -integral to determine the stress intensity factors, interface energy release rate and phase angles. The first term of the asymptotic solution for the stress and displacement fields obtained by means of the Stroh and Lekhnitskii formalisms are used as auxiliary solutions for the M -integral. The critical interface energy release rate is found and exhibits a slowly increasing resistance curve. Comparisons are made to a simple expression from the literature. Keywords Delamination toughness · Woven composites · Double cantilever beam specimens · Interaction energy integral L. Banks-Sills School of Mechanical Engineering, Tel Aviv University, Ramat Aviv, 69978, Israel; Division of Solid Mechanics, Lund Institute of Technology, Lund University, SE-221 00, Lund, Sweden Tel.: +972-3-640 8132 Fax: +972-3-640 7617 E-mail:
[email protected] C. Ishbir, V. Fourman, L. Rogel, R. Eliasi School of Mechanical Engineering, Tel Aviv University, Ramat Aviv, 69978, Israel Appears in: International Journal of Fracture, 182, (2013) 187-207, DOI: 10.1007/s10704-013-9868-6.
1 Introduction Composites are an alternative to traditional metals because of their high stiffness and strength, low weight and adjustable properties. Carbon fiber reinforced polymer composites are widely used in the automotive and aerospace industries reducing the structural weight for fuel saving and improved performance. The type of material considered in this investigation is a plain, balanced weave carbon/epoxy composite which has wide use in the aircraft industry. Although composite materials have appeared in various forms throughout the history of mankind, the history of modern composites goes back to 1930 with the first production of fiberglass. In 1961, a patent was issued for experimentally producing the first carbon fiber, and several years later, carbon fibers were commercially produced. Most practical composite structures are fabricated from multidirectional (MD) laminates and delaminations tend to occur at the interface of plies with different orientations. Delaminations may be viewed as an interface crack between two anisotropic materials in which the material properties of the layers are homogenized. Accurate measurement of the interlaminar fracture toughness and growth rate are necessary in order to assure the integrity of the composite structure. Since the introduction of high strength, stiff fibers, there has been much progress in the field of composites. The type of material considered in the current investigation is a plain weave, MD composite material. A woven composite is a fabric reinforced polymer material in which the fibers have been prefabricated to form a textile. The textile is built from yarn in which fibers are bundled together as rovings or twisted into threads. Woven fabrics are produced on a loom by the interlac-
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Leslie Banks-Sills et al.
g
a
x1 x3 Fig. 1 Plain weave composite.
ing of warp and weft yarns in a regular pattern. Textiles can be classified according to the manufacturing method (weaving, braiding, stitching or knitting) or the pattern in which the yarns are interlaced. Among the various textile forms, woven fabrics are the most widely used in composites. The most common two-dimensional woven composite is a plain weave as shown in Fig. 1. The variety of manufacturing methods have made textile composites cost-competitive with unidirectional laminated (UD) composites. For high in-plane specific stiffness and high in-plane specific strength applications, two-dimensional textile composites, especially two-dimensional woven fabric composites can be competitors to laminated composites made of UD layers. Woven fabric composites provide balanced properties in the fabric plane and higher impact resistance than unidirectional composites. The interlacing of yarns provides higher out-of-plane strength which can take up secondary loads resulting from load path eccentricities, local buckling, etc. It is easier to build thick laminates with woven fabrics than with UD tapes and handling of woven fabrics is easier which brings a reduction in the labor costs and manufacturing errors involved in making thick laminates. Woven fabric composites have higher fatigue resistance, fracture toughness and damage tolerance characteristics than UD layers (Niak , 1994, 2003). A unidirectional laminate is a reinforced plastic layer in which substantially all of the fibers are oriented in the same direction. The majority of research on fracture toughness of composite materials has been concerned with UD layers in which all fibers are in the 0◦ -direction. In addition, the only standardized test method for interlaminar fracture toughness testing of fiber reinforced composite materials is for mode I deformation of unidirectional composites ( ASTM Standard D 5528-01 , 2007; ISO 15024 , 2001). However, most composite materials in practical applications con-
sist of plies with fibers oriented at optimized directions forming MD laminates. It has been observed that intraply (within the ply) damage instead of pure interlaminar crack propagation can occur for MD laminates. In UD laminates where the crack propagation is in the fiber direction, the initial interlaminar crack plane is maintained. However, when the intended delamination growth direction does not coincide with the fiber direction, as happens in MD laminates, intraply damage may be observed. As a result of such damage morphology, propagation fracture toughness values may be considerably higher than those obtained for the UD laminate. Tests have been performed on DCB specimens fabricated from both UD and MD composites to determine the mode I energy release rate GIc (Nicholls and Gallagher , 1983; Chai , 1984, 1986; Robinson and Song , 1992; de Morais et al. , 2003; Andersons and K¨onig , 2004; Gong et al. , 2010). In many cases, graphite/epoxy material was tested. For 0◦ /90◦ , a low fracture toughness of GIc = 140 N/m was found by Nicholls and Gallagher (1983) where the delamination propagated along the interface. For angle ply laminates higher values were obtained because of the damage that occurred. Chai (1984, 1986) found for MD composites that the mode I fracture toughness was independent of ply orientation when the delamination propagated along the interface. Values as low as 86 N/m were found. In reviewing the literature, Andersons and K¨onig (2004) found that mode I initiation toughness was affected moderately by the interface lay-up and delamination growth direction relative to the fiber direction. For MD specimens it was found by Gong et al. (2010) that as specimen width increased the distribution of GI along the interface became more uniform. The directions of the plies adjacent to the interface were shown to affect the GIc values. Banks-Sills et al. (2005, 2006) developed a methodology for measuring the delamination toughness of MD composites. Fracture tests were conducted with a Brazilian disk specimen as it allows for a wide range of mode mixities. The 0◦ /90◦ and +45◦ / − 45◦ interfaces were examined. The minimum interface energy release rate was found to be 26.5 N/m for the first interface and 90.3 N/m for the latter. A three-dimensional failure curve was derived. On the other hand, there have also been measurements of fracture toughness for woven composities. As a result of the interlaced yarn, woven composites present complicated fracture mechanisms. The fracture toughness of a woven composite depends on the deformation type, the weave pattern, the fabric geometry and process, the volume fraction of the fibers and the crack propagation direction. Values of GIc may be obtained by
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conducting mode I fracture tests, although these may produce small shear components for MD lay-ups. Toughness tests on woven composites with all plies in the same directions were carried out by Alif et al. (1997, 1998); Niak et al. (2002); Gill et al. (2009); Nikbakht and Choupani (2009); Blake et al. (2011). Both carbon/epoxy and glass/epoxy woven materials were considered. In addition to mode II and mixed mode tests, DCB specimens were tested by Alif et al. (1997, 1998). For the latter R-curve behavior was found for 5H-satin specimens made with carbon fibers. The behavior was influenced by the direction of the delamination relative to the warp and weft directions. Two DCB specimen sets consisting of plain weave E-glass/epoxy were tested by Niak et al. (2002). In one set, the fibers were in the 0◦ /90◦ directions, and in the other set, the weave was rotated by 45◦ in the plane with respect to the loading direction. Values of GIc were less for the rotated laminate. In Gill et al. (2009), 5H-satin carbon/epoxy woven DCB specimens were tested. It was found that higher fiber volume fraction and more transverse tows along the delamination plane result in higher fracture toughness. DCB specimens were tested in Blake et al. (2011) to investigate the fracture behavior of E-glass woven fabric composites. The weave was unbalanced with 55% of the fibers in the warp direction and 45% in the weft direction. There was great variability between the results. Tests were carried out for multi-directional woven composites by Pereira et al. (2005). The material was a crowsfoot satin woven fabric glass reinforced epoxy. DCB specimens with different stacking sequences including the following delamination interfaces were tested: between 0◦ /90◦ and θ/(θ + 90◦ ) weaves, and between θ/(θ + 90◦ ) and −θ/(−θ + 90◦ ) weaves. It was observed that the GIc values calculated for delamination initiation for the 0◦ /90◦ and θ/(θ + 90◦ ) were independent of θ. In this investigation, fracture tests have been carried out on double cantilever beam, multi-directional, woven composite specimens containing a delamination. The specimen is shown in Fig. 2. The geometric parameters consist of the specimen width b, the height h, the overall length L, the length of the delamination from the specimen edge a and the delamination length ai (i = 0, 1, ..., 5) from the piano hinge. The load P is applied through the piano hinges. Each ply in the specimen is a balanced plain weave of carbon fiber yarn in an epoxy matrix; for woven composites, a balanced weave is composed of yarn which has the same fiber volume fraction in the weft and warp directions. The plies are laid up so that they alternate between a 0◦ /90◦ and a +45◦ / − 45◦ weave. There are 15 plies with the delamination located between the seventh and eighth plies.
h P Piano hinge L ai
b
a
P
Teflon insert P
P Fig. 2 Double cantilever beam woven composite specimen.
Thus, the interface is between a 0◦ /90◦ weave and a +45◦ / − 45◦ weave. Each of these plies is taken to be effectively anisotropic. In Section 2, the first term of the asymptotic stress and displacement fields are developed for this interface by means of the Stroh (1958) and Leknitskii (1963) formalisms. Mechanical and thermal properties of the woven layers are obtained by means the micromechanical model High-Fidelity Generalized Method of Cells (HFGMC) presented in Aboudi (2004) and described in Section 3. Use is made of the asymptotic displacement field as auxiliary solutions in the interaction energy or M -integral which is presented in Section 4. In Section 5, the test procedure and analysis of the specimens are described. Finite element analyses are carried out on the DCB specimens with stress intensity factors calculated by means of the M -integral. The critical interface energy release rate Gic and phase angles are presented in Section 6, together with Gic -R curves. Some comparisons are made of the interface energy release rate values with those obtained using a simple formula found in the ASTM Standard D 5528-01 (2007). In Section 7, SEM studies of delamination surfaces are presented. Finally, a summary and conclusions are given in Section 8.
2 Asymptotic stress and displacement fields Each layer of material is a woven composite with fibers in different directions. The delamination is along an
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Leslie Banks-Sills et al.
The 3 × 3 matrix
X2
˘ = D −1 W , S
Material (1)
−1 D = L−1 1 + L2 ,
weave
W =
Material (2) +45 -45 weave
interface between a 0◦ /90◦ and a +45◦ / − 45◦ weave as illustrated in Fig. 3. The material properties of each layer, as well as the total laminate, are homogenized by means of HFGMC. These results are presented in Section 3. With the mechanical properties for each layer, the Stroh (1958) and Leknitskii (1963) formalisms are used to obtain the first term of the asymptotic stress and displacement fields. Use is made of expressions given in Ting (1996). The in-plane stress components are given by 1 [ ( iϵ ) (k) (1) σαβ = √ ℜ Kr k Σαβ (θ) 2πr ] ( ) (2) + ℑ Kriϵ k Σαβ (θ) ; (1) whereas, the out-of-plane stress components are found as KIII (III) =√ k Σα3 (θ) . 2πr
(2)
−
(7)
S 2 L−1 2
(8)
−1 −1 −Ak B −1 k = S k Lk + iLk ,
(3)
where K1 and K2 are real and, respectively, the modes 1 and 2 stress intensity factors; KIII is the mode III stress intensity factor. The oscillatory parameter ε is given by ( ) 1 1+β 1 ln . (4) ε = tanh−1 β = π 2π 1−β The expression for β may be found from the BarnettLotte tensors as { }1 1 ˘2 2 0 ≤ β = − tr(S ) 0. From eqs. (5) and (12), it is found that W12 β=√ . D11 D22
(13)
so that ε in eq. (4) is positive. The first term of the asymptotic expression for the in-plane displacement components may be written as √ ] ( iε ) r [ ( iε ) (2) (1) (k) uα = ℜ Kr k Uα (θ) k Uα (θ) + ℑ Kr 2π (14) with α = 1, 2; whereas, the out-of-plane displacement is given by √ r (k) (III) u3 = KIII k U3 (θ) . (15) 2π (1)
(2)
In eqs. (14) and (15), the functions k Uα (θ), k Uα (θ) (III) and k U3 (θ) are presented in Appendix B. It may be noted that they have units of L2 /F through the elements of the matrix Ak .
5 Table 1 Mechanical (for T300 see Miyagawa et al. , 2005) (for epoxy 913 see Yee and Pellegrino , 2005) and thermal (for T300 and epoxy 913 see Torayka Data Sheet , 2012; Bowles and Tompkins , 1989) properties of the graphite T300 fibers and epoxy 913. material graphite T300 epoxy 913
EA (GPa)
ET (GPa)
GA (GPa)
νA
νT
αA (×10−6 /◦ C)
αT (×10−6 /◦ C)
230 3.4
8
27.3
0.26 0.41
0.30
-0.41 43.92
10.08
3 Mechanical and thermal properties of the woven layers The specimens are composed of woven composite material. The fibers are carbon T300 which are transversely isotropic; hence, they are described by five independent elastic constants and two thermal constants. These include EA and ET which are the axial and transverse Young’s moduli, respectively, νA and νT , the axial and transverse Poisson’s ratios, respectively, GA , the axial shear modulus and the transverse shear modulus, GT = ET /2(1+νT ). The thermal properties are αA and αT being, respectively, the axial and transverse thermal coefficients of expansion (CTE). The epoxy matrix is 913 and is isotropic and described by two independent elastic constants: Young’s modulus E, Poisson’s ratio ν and the coefficient of thermal expansion α. Their mechanical and thermal properties are presented in Table 1. There is a range of properties in the literature for carbon T300. The mechanical properties of the T300 carbon fibers may be found in Miyagawa et al. (2005); those of epoxy were taken from Yee and Pellegrino (2005). The axial CTE of the T300 carbon is given in Torayka Data Sheet (2012); its transverse component was taken from Bowles and Tompkins (1989). For the epoxy, the value given for three other epoxies in Bowles and Tompkins (1989) was taken here for the epoxy 913. The fiber volume fraction for this material was not obtained experimentally. The material was produced by Hexcel with an apparent fiber volume fraction within the weave of 51% (HexCel , 2012). Each layer of the specimen is a balanced, plain weave; a sketch of the 0◦ /90◦ layer is shown in Fig. 1. Mechanical properties measured by Israel Aerospace Industries (IAI) (Freed , 2009) are presented in Table 2. The subscripts on the mechanical properties are with reference to the coordinate system in Fig. 1. Note that the x2 coordinate is perpendicular to the plane of the weave. The mechanical and thermal properties were calculated by means of HFGMC (for details of the method see Aboudi , 2004; Decad et al. , 2007). To achieve this value, it was assumed that the fiber volume fraction within the yarn is 63% and the yarn within the weave is 81%. Multiplying these together gives the desired fiber
volume fraction of 51% within the weave. In addition, measurements of the geometric properties of the weave were made. The parameters are shown in Fig. 1 where a is the yarn width, g is the distance between yarns and h is the thickness of the layer. They were taken as a = 1.7 mm, g = 0.4 mm and h = 0.24 mm. Using the mechanical properties in Table 1 and the geometrical parameters as input into the HFGMC Graphical User Interface (GUI) (Decad et al. , 2007) produces the mechanical properties shown in Table 3. In Table 3, E11 = E33 are the Young’s moduli in the x1 and x3 directions, respectively, that is, in-plane elastic moduli (see Fig. 1); E22 is the out-of-plane Young’s modulus; G13 is the in-plane shear modulus and G23 = G21 are the out-of-plane shear moduli; ν13 is the in-plane Poisson’s ratio and ν23 = ν21 are the out-of-plane Poisson’s ratios. Note that νij νji = Eii Ejj
(16)
where summation is not implied. The layer type which is denoted as 0◦ /90◦ is the weave as shown in Fig. 1. It may be observed from Table 3 that the material in this direction is tetragonal, so that there are six independent elastic constants. Comparison may be made to values which were measured and presented in Table 2. The in-plane Young’s moduli differ by less than 4%; the in-plane shear moduli agree; and there are great differences between the Poisson’s ratios. It may be noted that this is a very small quantity and difficult to measure. In the specimens, there are also layers in the +45◦ / − 45◦ direction. By rotating the properties about the x2 -axis in Fig. 1 by 45◦ , they are obtained as shown in Table 3; this layer is also represented by six independent mechanical properties. In the finite element model, two layers on each side of the delamination are modeled with these properties. In addition, to save on computa-
Table 2 Mechanical properties of the woven composite measured by Israel Aerospace Industries (Freed , 2009). E11 = E33 (GPa)
G13 (GPa)
ν13
59.6
3.9
0.07
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Leslie Banks-Sills et al.
Table 3 Mechanical properties of the plain woven composites (Vf = 0.51). layer type ◦
E11 = E33 (GPa)
E22 (GPa)
G13 (GPa)
G23 = G21 (GPa)
ν13
ν23 = ν21
57.3 13.6 41.2
7.6 7.6 7.6
3.9 27.6 15.7
2.5 2.5 2.5
0.04 0.77 0.31
0.07 0.07 0.07
◦
0 /90 +45◦ / − 45◦ effective material
tional resources, effective properties of the woven laminate are found and presented in Table 3. This material is quasi-isotropic in the x1 -x3 plane. The coefficients of thermal expansion were also calculated by HFGMC from those of the constituents in Table 1. These results are presented in Table 4. As expected, the CTEs in the x1 and x3 -directions are identical. In addition, it may be observed that rotation of the values for the 0◦ /90◦ layer by 45◦ about the x2 -axis leads to identical results. Hence, thermal stresses are not caused by curing.
4 Interaction energy integral In the DCB specimen, there is a through interface crack (Fig. 2). Recall that the upper layer adjacent to the crack has yarn in the 0◦ /90◦ directions; the lower layer has yarn in the +45◦ / − 45◦ directions. In order to determine the stress intensity factors, energy release rate and the phase angles associated with this crack, a conservative integral for mechanical loading is used in conjunction with a finite element solution of the body. This is the M -integral or interaction energy integral which was first derived in Chen and Shield (1977) and implemented in Yau et al. (1980) for a homogeneous isotropic material. By means of this integral, it is possible to separate mixed mode stress intensity factors. The integral has been extended for many applications, in particular for an interface crack between two anisotropic materials in three-dimensions (see for example, Freed and Banks-Sills , 2005; Nagai et al. , 2007). For the M -integral, two equilibrium solutions are required. Solution (1) is the sought after solution in which the fields are obtained by means of the finite element method. For solution (2), the auxiliary solution, the first term of the asymptotic solution for the disTable 4 Coefficients of thermal expansion of the plain woven composites (Vf = 0.51).
placement field is used. These expressions are given in eqs. (14) and (15). A particular choice of stress intensity factors are made for solution (2). The M -integral in three dimensions is given by (Freed and Banks-Sills , 2005) [ 2 ∫ (1) (2α) 1 ∑ (1,2α) (2α) ∂kui (1) ∂kui MN = σ + σ k ij k ij A1 ∂x1 ∂x1 k=1 Vk ] ∂q 1 − k W (1,2α) δ1j dV . (17) ∂xj In eq. (17), the superscript (1) represents the desired solution, whereas the superscript (2α), α = a, b, c, represents three auxiliary solutions. For the auxiliary solutions, specific values are used for the stress intensity factors which are given in Table 5. Integration takes place in a volume Vk of finite elements which is one element thick through the specimen thickness (see Fig. 4). The subscript N denotes the element along the crack front in which the calculation takes place as illustrated in Fig. 5. The area A1 is given by ∫ A1 ≡
LN
(N )
ℓ1 (x3 ) dx3
(18)
0
where LN is the length of the element along the crack front, the crack front is taken along the x3 -axis and (N ) ℓ1 (x3 ) is the virtual crack extension which is parabolic. The subscript k = 1, 2 represents the upper and lower materials, respectively. The displacement components (1) are obtained from the finite element solution. The k ui (1) (1) stress and strain components k σij and k ϵij , respectively, are also determined by means of a finite element formulation through the derivatives of the shape functions. As mentioned previously, the displacement com(2α) ponents k ui are obtained from the first term of the Table 5 Stress intensity factors for the three-dimensional auxiliary solutions.
layer type
α11 = α33 (×10−6 /◦ C)
αT (×10−6 /◦ C)
0◦ /90◦ +45◦ / − 45◦ effective material
2.9 2.9 2.9
52.1 52.1 52.1
solution
K1
K2
KIII
2a 2b 2c
1 0 0
0 1 0
0 0 1
7 (1,2α)
Another expression for MN is sought. To this end, the interface energy release rate may be found as
S x2
V1
Gi =
1 D22 = , H1 4 cosh2 πε
Fig. 4 In-plane cross-section of volumes Vk and outer surface S. Note that the integral begins at the crack front.
asymptotic solution of a crack between the two materials considered here. The stress and strain compo(2α) (2α) nents k σij and k ϵij , respectively, are determined by means of a finite element formulation through the derivatives of the shape functions. The mutual strain energy density k W (1,2α) of the two solutions in the upper and lower half planes is given by (1,2α)
(1)
= k σij
(2α) k ϵij
(2α)
= k σij
(1) k ϵij
(19)
and δij is the Kronecker delta. The parameter q1 is defined as
q1 =
20 ∑
Ni (ξ, η, ζ) q1i
(20)
i=1
where Ni (ξ, η, ζ) are the shape functions of a twenty noded, isoparametric element and q1i is a vector which determines the virtual displacement of the element nodal points. Further details may be found in Appendix A of Freed and Banks-Sills (2005).
crack plane LN
l1(N)(x3)
x1
n1
Fig. 5 Virtual crack extension along the crack front denoted on the finite element mesh.
(22)
For the materials used here, values of the parameters ϵ, D11 , D22 , D33 , H1 and H2 are presented in Table 6. Using the auxiliary solution for case (2a) in eqs. (17) and (23) with the aid of Table 5, it may be shown that [ 2 ∫ (2a) (1) H1 ∑ (1) (1) ∂kui (2a) ∂kui K1 = + k σij k σij 2A1 ∂x1 ∂x1 k=1 Vk ] ∂q 1 − kW (1,2a) δ1j dV ; (24) ∂xj using solutions (2b) and (2c) in eqs. (17) and (23) with the aid of Table 5, leads to, respectively, [ 2 ∫ (2b) (1) H1 ∑ (1) (1) ∂kui (2b) ∂kui K2 = + k σij k σij 2A1 ∂x1 ∂x1 k=1 Vk ] ∂q 1 (25) − kW (1,2b) δ1j dV ∂xj and
[ 2 ∫ (2c) (1) H2 ∑ (1) ∂kui (2c) ∂kui = + k σij k σij 2A1 ∂x1 ∂x1 k=1 V ] ∂q 1 dV . − kW (1,2c) δ1j (26) ∂xj (1)
x3 crack front
1 D33 = . H2 4
In eq. (22), D22 and D33 are diagonal elements of the matrix D appearing in eq. (10) and ε is given in eq. (4). From eq. (21), it is possible to derive another expression for the M -integral given by ] 2 [ (1) (2α) (1,2α) (1) (2α) = MN K1 K1 + K2 K2 H1 2 (1) (2α) + K K . (23) H2 III III
(1) KIII
N
(21)
where
x1 V2
kW
1 1 2 (K12 + K22 ) + K H1 H2 III
(1)
(1)
In eqs. (24) through (26), K1 , K2 and KIII are average local stress intensity factors along the crack front within element N (see Fig. 5). Next, the integration domains are described. In calculating the integral in eq. (17), elements surrounding the crack front are used. One layer of elements in the thickness direction is employed. As a result of the function q1 in eq. (17) which is a normalized virtual crack extension, all integration domains must include the elements adjacent to the crack front which generally produce inaccurate results. Cross-sections of the four paths used here are shown in Fig. 6. Quarter-point, twenty
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Leslie Banks-Sills et al.
Table 6 Some material parameters for the 0◦ /90◦ – +45◦ / − 45◦ interface. ε
D11 (GPa−1 )
D22 (GPa−1 )
D33 (GPa−1 )
H1 (GPa)
H2 (GPa)
0.00862
0.212
0.501
0.443
7.99
9.03
noded, brick elements are used along the delamination front. These elements produce the dominant squareroot singularity. Finally, it may be noted that three test cases were solved using the M -integral and finite elements with excellent results obtained except, as expected, in domain 1 in Fig. 6 where calculations take place in only the singular elements.
5 Delamination toughness testing and analysis Eight delamination toughness tests have been carried out on DCB specimens illustrated in Fig. 2. The specimens were fabricated by IAI. The tests are being performed in the spirit of the ASTM Standard D 5528-01 (2007). Recall that the geometric parameters consist of the specimen width b, the height h, the overall length L, the length of the delamination from the specimen edge a and the delamination length ai (i = 0, 1, ..., 5) from the piano hinge. The load P is applied through the piano hinges. Each ply in the specimen is a balanced plain weave of carbon fiber yarn in an epoxy matrix. The plies are laid up so that they alternate between a 0◦ /90◦ and a +45◦ / − 45◦ weave. There are 15 plies with the delamination located between the seventh and eighth plies. Thus the interface is between a
0◦ /90◦ weave and a +45◦ / − 45◦ weave. Each of these plies is taken to be effectively anisotropic. The test methodology is presented in Section 5.1. In Section 5.2, the test results and specimen analyses are described. 5.1 Test methodology An Instron loading machine (model number 8872; High Wycombe, England) with a load cell of maximum load 250 N and a resolution of ±0.5 % of the reading is employed. This small load cell is required since the maximum load at initial fracture is less than 60 N. A video microscope including a PixeLink PL-A686M 6.6 MP FireWire microscope camera and software (Ontario, Canada) with a Navitar modular optical microscope lens (Zoom 6000 High Mag Zoom Lens NAVITAR with 3.5X adapter and lens attachment 0.5 X, Rochester, NY) is employed to take pictures of the crack as it propagates. In addition, use is made of a video camera (Sony DCR-HC32) to monitor the Instron controller in order to synchronize the load, the crack opening displacement and the crack length (see Fig. 7). The COD was measured by means of an additional camera connected to a computer which was behind the Instron. An image processing technique was developed to detect the COD. A correlation was found between the crosshead displacement d and the COD, ∆, as ∆ = 0.95d .
2
The relation in eq. (27) is used to convert the crosshead displacement to the COD. Since d is part of the
1
(a )
(27)
( b) video camera
4 3
video microscope Instron
(c)
(d )
Fig. 6 Two-dimensional cross-sections of the integration volumes or domains. The domains have a thickness of one element orthogonal to the figure.
Fig. 7 Experimental set-up.
9
specimen
support
Fig. 8 Crack in the DCB specimen.
output from the Instron, the applied force and the crosshead displacement are synchronized, so that, the COD is synchronized with the applied force.
Fig. 9 Close-up of DCB specimen.
In carrying out an experiment, the cross-head displacement of the Instron is increased quasi-statically. The displacement rate for loading was 1.02 mm/min and unloading was 2.04 mm/min. This is in accordance with the ASTM Standard D 5528-01 (2007) in which the displacement rate for loading should be between 1 and 5 mm/min; the unloading rate may be increased to 25 mm/min. The applied load and the cross-head displacement are obtained from a computer which monitors the Instron. In addition, the crack jumps ahead at certain cross-head displacements. The crack length is measured and synchronized with the applied load.
In Fig. 9, a close-up of the DCB specimen in the Instron is shown. The white paper and paint are used to monitor crack propagation. The grips were designed according to the ASTM Standard D 5528-01 (2007). The support at the right end of the specimen allows the specimen to lean on it but the latter is free to move up during the test. The specimen height h (see Fig. 2) is measured at the center-line of the specimen at five locations along its length. These were done at the end of the Teflon, at the end of the specimen and at three equally spaced positions in between. The ASTM Standard D 5528-01 (2007) recommends measurement in both the crack and uncracked regions at a total of three locations. However, it is recommended to measure h only in the region of the uncracked ligament so as not to take the Teflon thickness into consideration. For specimens FT-3 through FT-5, measurements were made with an electronic digital caliper of resolution 0.01 mm and accuracy 0.02 mm. For the remainder of the specimens, a micrometer was used which has a resolution of 0.01 mm, the same as the caliper. The standard prescribes a finer accuracy of 2.5 µm for thickness measurement. The width of the specimen b is measured in the same locations with the electronic digital caliper. Accuracy of the width measurement is 0.2 mm; the standard requires 0.25 mm. The initial crack length a0 (i = 0 in Fig. 2) is measured on the front and back sides of the specimen with a caliper. The length L of the specimen is measured with a ruler on the two outer surfaces. The length a between the end of the specimen and the crack front is also measured by means of a caliper on both sides of the specimen. The room temperature and humidity are measured at the beginning of, during and at the end of the test.
To this end, the applied force data as a function of time are obtained from the Instron and crack length as a function of time is obtained by means of image processing. The crack length is determined using a LabVIEW (2008) tool for the pictures obtained from the video microscope placed in front of the Instron. An example of such a photograph is exhibited in Fig. 8. Data which includes applied force, cross-head displacement and time are recorded by a computer. The time recorded by the computer is incorrect. The time data is corrected via that obtained with the video camera from the Instron controller. In this way, the applied load and the crosshead displacement are synchronized with the correct time. In addition, the video microscope is controlled to take pictures of the crack every 3 seconds. The camera does not work exactly according to this command. A computer program was developed which determines exactly the rate at which the images are taken. This time, together with the time from the Instron controller are used to synchronize the load, crack length and crosshead displacement. In addition, all sounds emanating from the specimen were recorded indicating internal cracking within the specimen.
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Leslie Banks-Sills et al.
Table 7 Geometric properties of tested fracture specimens. specimen no.
h1 (mm)
h2 (mm)
h3 (mm)
h4 (mm)
h5 (mm)
¯ (mm) h
SE (mm)
FT-3 FT-4 FT-5 FT-8 FT-9 FT-10 FT-11 FT-13
3.86 3.60 3.80 3.73 3.69 3.69 3.70 3.78
3.74 3.62 3.85 — — 3.64 3.83 3.85
3.71 3.51 3.63 3.65 3.63 3.64 3.62 3.75
3.75 3.72 3.66 3.67 3.64 3.64 3.66 3.74
3.65 3.58 3.71 3.65 3.60 3.68 3.66 3.77
3.74 3.61 3.73 3.67 3.64 3.66 3.69 3.78
0.034 0.034 0.042 0.017 0.017 0.011 0.036 0.019
width
b1 (mm)
b2 (mm)
b3 (mm)
b4 (mm)
b5 (mm)
¯b (mm)
SE (mm)
FT-3 FT-4 FT-5 FT-8 FT-9 FT-10 FT-11 FT-13
24.58 25.12 25.32 25.31 25.00 25.50 25.12 24.77
25.01 25.09 25.87 25.79 25.28 25.65 24.83 25.60
25.03 24.90 25.56 25.35 25.34 25.71 25.19 25.12
25.29 24.77 25.40 25.78 25.25 25.42 25.24 24.86
25.63 24.98 25.48 25.37 25.49 25.80 25.00 25.38
25.11 24.97 25.53 25.52 25.27 25.62 25.08 25.15
0.173 0.064 0.095 0.109 0.080 0.069 0.074 0.156
crack length
a0f (mm)
a0b (mm)
a ¯0
af (mm)
ab (mm)
a ¯
FT-3 FT-4 FT-5 FT-8 FT-9 FT-10 FT-11 FT-13
56.92 55.30 55.00 54.70 54.65 55.64 55.20 55.54
54.87 55.88 56.81 54.87 54.59 55.64 55.20 57.30
55.90 55.59 55.91 54.79 54.62 55.64 55.20 56.42
81.97 81.30 82.63 80.37 80.16 80.79 80.15 81.02
81.36 81.60 82.64 80.41 79.88 80.79 80.26 81.85
81.67 81.45 82.64 80.39 80.02 80.79 80.21 81.44
5.2 Test results and specimen analyses Eight DCB specimens containing a delamination were tested based on the methodology presented in Section 5.1. The geometric properties h, height, and b, width, (see Fig. 2) of the tested specimens are presented in Table 7. The average quantities are given; they appear with an over-bar. The standard error presented in the table is obtained as s SE = √ (28) n where s is the standard deviation and n are the number of samples; in this case 4 or 5. For specimens FT-8 and FT-9, the measurement h2 was neglected. This measurement is at the far end of the specimen. In these two specimens, these values were rather small, about 3.35 mm. This result for h in this location should not be included in the calculation of the average height since one is interested in the height in the region in which the crack propagates. Hence, in these two cases, the standard error is based on four measurements. The width of the specimen b is presented in Table 7. According to the standard, the width should be between 20 and 25 mm and the height should be between 3 and 5 mm. The variation of the height should not be more than
0.1 mm. These values are exceeded here, but the standard is for UD laminates and not woven composites. The length L of the specimens was seen to be approximately 244 mm for each specimen which conforms to the ASTM Standard D 5528-01 (2007) that requires the length to be at least 125 mm. The initial delamination length a0 on the front (f ) and back (b) of the specimens, as well as the average is given in Table 7. In addition, the length of the initial delamination to the edge of the specimen is presented as a for the front and back of the specimens together with the averages as shown in Fig. 2. According to the standard, a0 should be 50 mm. It was somewhat longer here. The thickness of the Teflon was measured to be between 30 µm and 40 µm by a micrometer with a resolution of 10 µm. According to the standard, the thickness should be 13 µm. The room temperature and relative humidity (RH) measured for each specimen are presented in Table 8. The tests lasted between 1 and 1.5 hr. It should be pointed out that according to the ASTM Standard D 5528-01 (2007), the specimens should be conditioned before the tests. This means that they should be placed in a container at a specific temperature and relative humidity for a certain period of time. Since the container was only obtained several months after receiving
11 Table 8 Temperature and relative humidity during the tests. specimen no.
FT-3
FT-4
FT-5
FT-8
FT-9
FT-10
FT-11
FT-13
initial T ( C) mid-test T (◦ C) final T (◦ C)
22.9 23.9 24.6
26.4 27.6 27.7
25.5 – 26.8
24.5 24.8 24.9
25.9 26.1 26.1
25.7 26.0 26.4
24.9 25.0 25.2
23.6 24.0 24.5
initial RH (%) mid-test RH (%) final RH (%)
29.8 27.3 28.1
67.3 68.4 68.1
44.5 – 43.0
33.0 30.5 26.0
36.6 36.6 36.7
52.1 55.7 48.7
50.5 47.8 49.4
55.8 58.5 57.8
◦
the specimens, specimens FT-3 through FT-5 were not conditioned. After that, all specimens were put in the container at 23◦ C and 50% relative humidity. The uniformity of the conditions in the container are ±1◦ C and ±3% relative humidity. The container is maintained continuously. This is in keeping with the ASTM Standard D 5229 (2010). The specimens should be tested at standard laboratory atmosphere of 23 ± 3◦ C and 50 ± 10 % relative humidity. As may be seen from Table 8, there is a problem with maintaining the proper humidity during a test. Use of a humidifier during a test does not appear to alleviate the problem. The relative humidity during the tests for specimens FT-10, FT-11 and FT-13 were within the required range. The temperatures were somewhat higher than specified. The effect of these quantities on the results will be examined. The load/crack opening displacement curve from the three first tests reported here is shown in Fig. 10. In all cases, initially the curve rises linearly until it reaches a maximum value which is between 50 and 60 N. Typically, clicking sounds are heard which probably indicate some delamination propagation within the specimen. These sounds were recorded by the video camera. This may be observed for specimen FT-3 in which there is a slight drop in the load two times before the first maximum is reached. Although delamination is taking place internally, there was no observation of delamination on the specimen surface. For example, for specimen FT3, when the load reached P = 51.8 N, the delamination propagated 0.6 mm (see Table. 9) and the load decreased (see Fig. 10a). As the delamination propagated, its length was determined using the LabVIEW tool for image processing. The load was increased to its maximum value of 54.9 N and the delamination propagated by 6.8 mm. At this point, the specimen was unloaded as seen in the figure. According to the standard, the specimen should be unloaded when the delamination grows between 3 and 5 mm. In this test, the delamination propagated 7.4 mm. After that, reloading takes place. A maximum load is reached once again as the delamination propagates and the load drops (a2 in Fig. 10a). Since displacement control is used, the load begins to rise again as the cross-head displacement is increased.
There are a series of load steps as the delamination propagates ahead. It appears that the delamination is probably running within the specimen before it runs on the outer surfaces. Similar tables were established for all specimens. In Table 9, additional data is presented, namely the force P , the crack opening displacement (COD), and the time t which is obtained from the controller which gives force vs. time. The delamination images are found on another computer in which synchronization has been performed as explained previously. The increment in delamination length ∆a is measured from the correspond-
60 P (N) 40
a1 a0 a2 a 3
FT-3 a4
a5
20 0 60 P (N) 40
20
40 (a)
a0 a1 aa32 a4 a5
60
80
COD(mm)
FT-4
20 0 60 P (N) 40
20 a a a0 1a2 3
40 (b) a4
60
80
COD(mm)
FT-5
a5
20 0
20
40 (c)
60
80
COD(mm)
Fig. 10 Load/displacement curve for specimens (a) FT-3, (b) FT-4 and (c) FT-5.
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Leslie Banks-Sills et al.
Table 9 Synchronization of force, crack opening displacement and delamination length vs time for specimen FT-3. delamination no.
P (N)
COD (mm)
a (mm)
∆a (mm)
image no.
t (s)
0 1 2 3 4 5
51.8 54.9 47.8 47.4 41.0 37.4
13.4 15.1 17.6 22.3 28.9 30.9
56.9 57.5 64.3 70.1 78.6 83.1
0.6 6.8 5.8 8.5 4.5 11.3
288 327 922 1025 1167 1212
870 979 2607 2915 3339 3475
ing picture by image processing using the LabVIEW tool. With the information from Tables 7 and 9, it is possible to carry out three-dimensional finite element analyses to determine the critical value of the interface energy release rate Gic , the normalized stress intensity ˆ 1, K ˆ 2 and K ˆ III , as well as the phase angles factors K ψ and ϕ, along the delamination front. The values of Gic along the delamination front are obtained from the J-integral. The values of the stress intensity factors are obtained from the mechanical M -integral presented in Section 4 and are normalized as ˆ iε ˆ = K√L K σ πa
(29)
and ˆ III = K√III K σ πa
(30)
ˆ =K ˆ 1 + iK ˆ 2 and L ˆ is taken to be 1 mm. The where K oscillatory parameter ε is calculated by means of eq. (4) with the mechanical properties given in Table 3. This value, together with values for other relevant parameters, is given in Table 6. The delamination length a is denoted as ai (i = 0, 1, ..., 5) in Fig. 2 and P σ = ¯¯ . bh
(31)
¯ are, respecIn eq. (31), P is the applied load, ¯b and h tively, the average values of the specimen width and height given in Table 7. Finally, the phase angles are defined as ] [√ ] [ ˆ iϵ ) ℑ(K L D11 σ12 = arctan ψ = arctan ˆ iϵ ) D22 σ22 θ=0,r=Lˆ ℜ(K L
The specimens were analyzed by means of the finite element method using ADINA, Bathe (2007). An example of a mesh used in the analyses is shown in Fig. 11. It contains 83,240 20-noded isoparametric elements and 351,103 nodal points. About the delamination front, quarter-point, square-root singular elements are used. The dominant singularity of the analytical solution is square-root. There are 60 elements along the delamination front. The model consists of six layers. The 0◦ /90◦ layers are green (see Fig. 11b); the +45◦ / − 45◦ layers are red. The delamination is between a 0◦ /90◦ and a +45◦ / − 45◦ layer. The mechanical properties are given in Table 3. The rest of the specimen consists of alternate 0◦ /90◦ and +45◦ /−45◦ layers. In order to decrease the number of elements in the mesh, effective properties of the overall composite were determined by means of HFGMC and are also presented in Table 3. These elements are the shown in pink in Fig. 11. To examine convergence, four meshes were used containing between 246,613 and 568,695 nodal points. The crack length ai = 83.11 was chosen for study since it is relatively long leading to difficulty in obtaining accurate values of Gi . A 3% difference was found between values obtained with the finest and coarsest meshes. For the other two meshes of 351,103 and 446,241 nodal points, the differences with values obtained with the finest mesh was less than −0.1%. The two meshes used in the analyses of the specimens were that containing 351,103 nodal points (specimens FT-3 through FT-5) and 446,241 nodal points (specimens FT-8 through FT13). In both cases, the element along the delamination
delamination
00 / 900
(32) and ϕ = arctan
[√
H1 K √ III H2 K12 + K22
] .
(33)
The values of D11 , D22 , H1 and H2 may be found in Table 6. The first two parameters are found in eq. (10) and H1 and H2 are given in eq. (22).
+450 / -450
(a)
(b)
Fig. 11 (a) Mesh of the DCB specimen. (b) Detail near the delamination front.
13 ∧
1200
14
Gic
K1
12
1000
10 800
8 a0 a1 a2 a3 a4 a5
6 4 2
600 a a01 a2 a3 a4 a5
400 200
0
0.2
0.6
0.4
0.8
(a) ∧
x3 / b
1 0
0.6
K2
0.2
0.4
0.6
0.8
x3 / b
1
Fig. 13 Critical interface energy release rate Gic as a function of the normalized delamination front coordinate x3 /b for different delamination lengths for specimen FT-3.
0.5 0.4 0.3 0.2 0.1 0
0.2
0.6
0.4
0.8
(b) ∧
K III
x3 / b
1
2 1 0 0.2
0.6
0.4
-1
0.8
x3 / b
1
-2 (c) ˆ 1 , (b) K ˆ2 Fig. 12 Normalized stress intensity factors (a) K ˆ III as a function of normalized delamination front and (c) K coordinate x3 /b for different delamination lengths for specimen FT-3.
front had a ratio of in-plane dimensions of 1.02. The ideal ratio is unity (Banks-Sills and Bortman , 1984). In addition, path independence of the mechanical conservative M -integral was examined. In Fig. 6, four in-plane integration domains are shown. The thickness of the domains is one element along the crack front. Several cases were considered for the interface energy release rate Gi . Differences between domains 1 and 2 were less than 2%. For domains 2 and 3, the differences were less than 0.1%. For specimen FT-3, values of the normalized stress intensity factors are presented in Fig. 12 as a function of the normalized delamination front coordinate x3 /b. The expressions for these quantities are given in eqs. (29) and (30). Results were obtained from integration do-
main 3 shown in Fig. 6. It may be observed that the curves increase monotonically as delamination length ˆ 1 and K ˆ 2 are symmetric increases. The results for K with respect to the specimen center-line, whereas those ˆ III are antisymmetric. This behavior is expected. for K The increase of mode III deformation at the specimen edges is frequently observed with interface delaminations. Note the difference in the scales of the plots. Next, values of the critical interface energy release rate Gic are plotted as a function of the normalized delamination front coordinate x3 /b for different delamination lengths and presented in Fig. 13. It may be observed that the values have an approximately parabolic behavior with a maximum occurring at the center of the specimen. This would seem to imply that delamination should take place first within the specimen. Of course, the phase angles must be considered, as well. But it is probable that the crack front is curved rather than straight through. It may be noted that the ordering of the curves with respect to delamination length differs from that of the normalized stress intensity factors. It was observed that the applied load has an important effect on this ordering. In addition, the phase angles ψ and ϕ in eqs. (32) and (33) are presented in Fig. 14. It may be observed that there are negligible differences for each angle for different delamination lengths except near the specimen edges. In other words, the phase angles have the same value along most of the delamination front for different delamination lengths. The phase angle ψ is rather small ˆ 1 dominates K ˆ 2 . Moreover, from results implying that K ˆ 2 is in Figs. 12a and 12b it may be observed that K ˆ 1 . On the other hand, near the much smaller than K specimen edges, ϕ increases in value implying that in this region there is a relatively large contribution of mode III deformation.
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Leslie Banks-Sills et al.
0.05
ψ
0.4
0.04
φ a0 a1 a2 a3 a4 a5
0.02 0.01 0
0.4
0.2
0.2
0.4
0.6
0.8
-0.2
0.6
0.8
x3 / b
1
x3 / b
1
(b)
max
800
G ic (N/m)
x
x
x
x
x
x
x
xx FT3 FT4 FT5 x FT8 x FT9 FT10 FT11 FT13
600 400 200 0 55
60
x
x x
xx FT3 FT4 FT5 x FT8 x FT9 FT10 FT11 FT13
200 0 55
60
65
70
75
80 85 a(mm)
90
95
Fig. 16 Average critical interface energy release rate G¯ic as a function of delamination length.
With the average value of Gic , a GR curve was determined which is shown in Fig. 16. The overall behavior in this graph is similar to that for the maximum values of Gic in Fig. 15 but, as expected, are somewhat lower. By using a multiple regression, it is possible to show that as humidity decreases, the values of either the average or maximum values of Gic increase. For temperature it is the opposite. In fact, for every percentage point decrease of relative humidity, the average Gic values increase by 2.7 N/m; whereas, for each increase of 1◦ C, these values increase by 24 N/m.
x x
x
400
In this section, two GR (resistance) curves are presented for all eight specimens tested. In Fig. 15, the maximum critical interface energy release rate at the center of the specimen is plotted as a function of delamination length. The points are for each delamination length analyzed (see, for example, Table 9 for specimen FT-3) with a linear least square fit through all the data (solid line). A slowly rising line is observed. Another version of the curve is obtained by determining the average value of Gic over the specimen width b. That is, ∫ 1 b ¯ Gic = Gic (x3 )dx3 . (34) b 0
x
x
x
600
-0.4
6 Results
1000
800 x x
Fig. 14 Values of the phase angles (a) ψ and (b) ϕ as a function of the normalized delamination front coordinate x3 /b for different delamination lengths for specimen FT-3.
1200
x
x
0
(a)
G ic (N/m)
1000
0.2
0.03
65
70
75
80 85 a(mm)
90
95
max Fig. 15 Maximum critical interface energy release rate Gic at the center of the specimen as a function of delamination length.
In addition to the resistance curves shown above, comparisons of the average energy release rate G¯ic for specimen FT-3 were made to an expression for the energy release rate given in the ASTM Standard D 552801 (2007) for unidirectional laminates. Three methods were presented with a modified beam theory (MBT) recommended. Hence, comparison will be made to this theory with Gic =
3Pc δc 2b(a + |∆|)
(35)
where Pc is the load at failure, δc is the delamination load point opening displacement at failure, b is specimen width and a is delamination length. The parameter ∆ is a correction to the delamination length to account for rotation at the delamination front. The parameter ∆ is determined experimentally by plotting the cube root of the compliance C 1/3 as a function of delamination length. The compliance is given by C=
δ . P
(36)
In this study, the value was determined to be ∆ = 2.44 when all of the data from all of the tests were plotted on one graph. Note that the expression in eq. (35) also gives average through the thickness values of Gic . This equation was used to calculate Gic in two ways. In the first case, the load point displacement was obtained from the experimental data. This calculation is denoted as Gδ−exp . In the second case, the load point displacement was obtained from the finite element analysis. This calculation is denoted as Gδ−F E . For specimen FT-3, values obtained from each of these calculations and that obtained from eq. (34) are presented in Table 10 for each value of crack length a. It may be observed when comparing the finite element result to that obtained with eq. (35) for δ found from the test data, that the percent difference is between 11% and 17%. When δ is
15 Table 10 Values of G¯ic obtained by various methods for specimen FT-3. G¯ic (N/m) a (mm)
eq. (34) (N/m)
Gδ−exp (N/m)
% diff
Gδ−F E (N/m)
% diff
56.9 57.5 64.3 70.1 78.6 83.1
653.2 747.5 702.5 816.8 764.8 706.2
732.2 865.0 788.5 913.9 915.7 847.4
-10.8 -13.5 -11.0 -10.6 -16.5 -16.7
647.3 739.0 697.3 812.3 761.0 703.1
0.9 1.1 0.7 0.6 0.5 0.4
obtained from the FE analysis, the differences decrease to less that 1.2%. Similar behavior is observed for other specimens. It appears that the finite element results underestimate the load point, delamination opening displacement δ, as well as the values of Gic . To this end, the effective mechanical properties of the woven layers and the effective properties of the entire composite were recalculated by means of HFGMC assuming a fiber volume fraction of 48% within the composite. Recall, that the fiber volume fraction was assumed to be 51% in the original calculations. In Table 11, results from an additional finite element calculation are presented. Using the data from specimen FT-8 with crack length a = 54.7 mm, a finite element analysis was carried out in conjunction with the conservative J-integral to obtain the average through the thickness value of G¯ic . This value is presented in Table 11 as 663.0 N/m. The original value obtained when Vf = 0.51 is also presented. It may be observed that the G¯ic value obtained with the lower volume fraction is higher. Using the value of the load point displacement δ = 12.46 obtained from the experiment in eq. (35), a value of G¯ic = 664.4 N/m is found, which now agrees well with that determined from the finite element calculation. Note also that the value of δ has also increased when a 48% fiber volume fraction is used. Although it may be that the fiber volume fraction is less than that used throughout this study, the results that have been presented here may be conservative.
7 Scanning electron microscope studies In looking at delamination surfaces after a test, it was not clear if the delamination propagated along the interface between the 0◦ /90◦ and +45◦ / − 45◦ interface. Observation of a +45◦ / − 45◦ surface, for example, appeared to show fibers from the 0◦ /90◦ surface. In order to more clearly interpret the behavior of these surfaces, scanning electron microscopy (SEM) was carried out on the delamination surfaces of specimen FT-8. The microscope is a Quanta 200FEG ESEM manufactured by FEI (Eindhoven, Netherlands). Figure 17 shows four views of a +45◦ / − 45◦ surface. In Fig. 17a, it appears as if the majority of the fibers are in the 0◦ /90◦ direction. Fibers which have been cut and are in the +45◦ -direction may be observed. Increasing the scale four times in Fig. 17b shows a clear view of fibers in the +45◦ -direction. It may now be observed that what appear to be fibers in the 90◦ -direction is simply an epoxy imprint of fibers in that direction. Enlarging that region further in Figs. 17c and 17d, the difference between fibers and the epoxy imprint is clearly seen.
epoxy imprint fibers
(a)
(b) epoxy imprint
Table 11 Values of the load line displacement δ and G¯ic obtained by various methods for specimen FT-8. Vf (%)
δ (mm)
G¯ic (N/m)
51 48
11.48 (FE) 12.25 (FE)
621.3 (FE) 663.0 (FE)
test
12.46
664.4 (eq. (35))
fibers
(c)
(d) ◦
◦
Fig. 17 SEM view of a +45 / − 45 surface; scale bar is (a) 2 mm, (b) 500 µm, (c) 200 µm and (d) 200 µm.
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Leslie Banks-Sills et al.
epoxy imprint
fibers
(a)
(b) ◦
◦
Fig. 18 SEM view of a 0 /90 surface; scale bar is (a) 2 mm and (b) 500 µm.
Next, a 0◦ /90◦ surface is shown in Fig. 18. In Figs. 18a and 18b, the fibers in the 0◦ -direction and 90◦ direction may be observed. In addition, an epoxy imprint in the +45◦ -direction is seen. This region is enlarged in Fig. 19. The fibers in the 0◦ and 90◦ -directions may be clearly seen in Fig. 19a. The fibers in the 90◦ direction are further enlarged in Fig. 19b. Hence, it may be concluded that the delamination for specimen FT-8 is within the 0◦ /90◦ –+45◦ / − 45◦ interface. Observation without micrographs may lead to erroneous conclusions.
be pointed out that the thermal properties found indicated that residual curing stresses did not occur in these specimens. In addition, the first term of the asymptotic stress and displacement fields were derived for use in the M -integral. The critical interface energy release rate was obtained from the J-integral. The phase angles were determined from the stress intensity factors. It may be noted that Gic found from a combination of the stress intensity factors was in close agreement with values calculated by means of the J-integral. Although the delamination was not symmetrically located within the specimen, nearly mode I deformation was produced. Slowly rising Gic − R curves were found. Scanning electron microscopy showed the delamination to propagate along the interface. Partial delamination of tows at the specimen edges were observed during the test. It is not clear if this phenomenon occurred within the specimens. The observed behavior accounts, at least in part, for the rising Gic − R curves. An unsuccessful attempt was made to obtain a threedimensional initiation failure curve such as that which may be found in Banks-Sills et al. (2006). To that end, mixed mode tests are required and suggested for a future study. Appendix A – Explicit expressions of the matrices A and B
8 Summary and conclusions In this study, DCB, multi-directional, woven composite specimens were tested to obtain the interface critical energy release rate between a 0◦ /90◦ ply and a +45◦ / − 45◦ ply of this material. Displacement control was used in testing the specimens. The load at failure and the corresponding crack length measured by image processing were used in a finite element analysis to obtain the stress intensity factors by means of an interaction energy integral. To this end, the effective mechanical and thermal properties were obtained through a homogenization procedure. The mechanical properties were used in the finite element calculations. It may
The complex 3 × 3 matrices A and B may be obtained in terms of the reduced elastic compliance components s′αβ which are given by s′αβ = sαβ −
sα3 s3β s33
with sαβ the components of the effective compliance matrix of each material and α, β = 1, ..., 6. For the materials studied here (see Ting , 1996, pp 170–172) k1 ξ1 (p1 ) k2 ξ1 (p2 ) 0 −1 A = k1 p−1 0 (38) 1 ξ2 (p1 ) k2 p2 ξ2 (p2 ) −1 0 0 k3 p3 η4 (p3 ) and
−k1 p1 −k2 p2 0 B = k1 k2 0 0 0 −k3
(a)
(b) ◦
Fig. 19 SEM view of a 0 /90 (a) 300 µm and (b) 100 µm.
◦
surface; the scale bar is
(37)
(39)
where pi are the eigenvalues of the elastic constants with positive imaginary parts. Furthermore, −1 −k1 −p2 k1−1 0 1 k2−1 p1 k2−1 (40) B −1 = 0 ∆ −1 0 0 −(p1 − p2 )k3 where ∆ = p1 − p2 .
(41)
17
The remaining parameters are given by ξα (pβ ) = p2β s′α1 + s′α2 ,
(42)
η4 (p3 ) = −s′44
(43)
for α, β = 1, 2. The parameters ki are normalization factors which are not necessary for the case studied.
Appendix B – Functions defining the stress and displacement fields (1)
(2)
(III)
The functions k Σαβ (θ), k Σαβ (θ) and k Σαβ (θ) appearing in eqs. (1) and (2) are presented in this section. They give the full definition of the first term of the asymptotic expansion for the in-plane and out-of-plane (1) stress components. For σ11 (of the upper material) {√ } eπε D22 ′(1) (1) ′(1) ′(1) ′(1) T sin(γ11 ) − T12 cos(γ12 ) 1 Σ11 (θ) = C1 D11 11 } {√ e−πε D22 e′(1) ′(1) ′(1) ′(1) γ12 ) sin(e γ11 ) + Te12 cos(e T − C1 D11 11
equations take on the value 2 (i.e. k = 2), and eqs. (44) through (47) are modified by substitution of −π instead of π. (k) ′(k) (k) ′(k) (k) ′(k) The components Tst , Tst , Test , Test , γst , γst , (k) ′(k) γ est and γ est are given by √[ ( )] [ ( )] (k)
(k)
ℜ Mst
Tst = ′(k) Tst (k) Test
2
(k)
+ ℑ Mst
2
(49)
√[ ( )]2 [ ( )]2 ′(k) ′(k) = ℜ Mst + ℑ Mst √[ ( )]2 [ ( )]2 (k) (k) fst fst ℜ M + ℑ M =
(50) (51)
√[ ( )]2 [ ( )]2 ′(k) ′(k) fst fst ℜ M + ℑ M ( (k) ) ℑ Mst (k) ) γst = tan−1 ( (k) ℜ Mst ( ′(k) ) ℑ Mst ′(k) ) γst = tan−1 ( ′(k) ℜ Mst (44) ( (k) ) fst ℑ M and (k) −1 ( ) est = tan {√ } γ (k) fst πε ℜ M e D 22 ′(1) (2) ′(1) ′(1) ′(1) T cos(γ11 ) + T12 sin(γ12 ) 1 Σ11 (θ) = ( ′(k) ) C1 D11 11 fst ℑ M } {√ ′(k) −1 ( ) γ e = tan e−πε D22 e′(1) ′(1) ′(1) ′(1) st ′(k) γ12 ) . + γ11 ) − Te12 sin(e T11 cos(e fst ℜ M C1 D11 (45) (1)
(1)
′(k) Test =
f where s, t = 1, 2. The matrices M (k) , M ′(k) , M ′(k) f M are defined as
For σ12 and σ22 (of the upper material) {√ } 1 (k) − +iε eπε D22 (1) (1) (1) (1) (1) M (k) = B k ⟨ ζ∗ 2 (θ)⟩B k−1 , Σ (θ) = − T sin(γ ) − T cos(γ ) 1 α2 α1 α2 α2 C1 D11 α1 1 −1 } M ′(k) = B k ⟨(k) ζ∗− 2 +iε (θ)p(k) {√ ∗ ⟩B k , −πε e D22 e(1) (1) (1) (1) 1 (k) γα1 ) + Teα2 cos(e γα2 ) T sin(e + (k) − −iε f C1 D11 α1 M = B k ⟨ ζ∗ 2 (θ)⟩B k−1 ,
1 (46) f ′(k) (k) (k) − −iε M = B k ⟨ ζ∗ 2 (θ)p∗ ⟩B k−1 ,
and
(52)
(53)
(54)
(55)
(56) (k)
and
(57) (58) (59) (60)
(k)
{√
} where p∗ are the roots with positive imaginary parts D22 (1) e (2) (1) (1) (1) obtained from solution of T cos(γα1 ) + Tα2 sin(γα2 ) 1 Σα2 (θ) = − C1 D11 α1 (61) {√ } l2 (p)l4 (p) − l3 (p)l3 (p) = 0 D22 e(1) e−πε (1) (1) (1) T cos(e γα1 ) − Teα2 sin(e γα2 ) − or C1 D11 α1 (62) (47) Q + p(R + RT ) + p2 T = 0 . πε
where α = 1, 2 and C1 = 2 cosh πε .
In eq. (61) (48)
For the lower material, the subscript behind Σ and the superscript in parentheses on the right hand side of the
l2 (p) = s′55 p2 + s′44
(63)
l3 (p) = 0 l4 (p) =
s′11 p4
(64) +
(2s′12
+
s′66 )p2
+
s′22
(65)
18
Leslie Banks-Sills et al.
where s′αβ are the reduced compliance coefficients given in eq. (37) (for further details, see Ting (1996, p 121)). In addition, the 3 × 3 matrices Q, R and T may be found in Ting (1996, p 135). In eqs. (57) through (60), the angular brackets represent a diagonal matrix and (k)
ζ∗ = cos θ +
(k) p∗
sin θ .
=
1 (k) O 2 33
(67)
1 b(k) (68) =− O 2 33 where k = 1, 2 represents the upper and lower materib (k) are given als, respectively. The matrices O (k) and O by } { 1 (k) − O (k) = ℜ B k ⟨ ζ∗ 2 (θ)⟩B k−1 b O
(k)
χ est
(69)
} { 1 (k) (k) − = ℜ B k ⟨ ζ∗ 2 (θ)p∗ ⟩B k−1 .
(76)
(77)
e (k) are dewhere s, t = 1, 2. The matrices G(k) and G fined as
(III) k Σ31
(k)
(k)
χst
(66)
For the out-of-plane stress components in eq. (2) (III) k Σ32
( ) (k) ℑ Gst ) − tan−1 (2ε) = tan−1 ( (k) ℜ Gst ( (k) ) e st ℑ G ) + tan−1 (2ε) = tan−1 ( (k) e ℜ Gst
(70)
G(k) = Ak ⟨ e G
(k)
= Ak ⟨
(k) (k)
1
ζ∗2
+iε
1 2 −iε
ζ∗
(θ)⟩B k−1
(78)
(θ)⟩B k−1 .
(79)
(III)
The function k U3 (θ) appearing in eq. (15) defining the displacement component in the x3 -direction is given by (III) (θ) k U3
(k)
= S33
(80)
and the matrix S is given by } { 1 (k) S (k) = ℜ Ak ⟨ ζ∗2 (θ)⟩B k−1 . (k)
(81)
(1)
For the upper material (k = 1), the functions k Uα (θ) (2) and k Uα (θ) appearing in eq. (14) defining the displacement components are given by } {√ eπε D22 (1) (1) (1) (1) (1) N sin(χα1 ) − Nα2 cos(χα2 ) 1 Uα = − C2 D11 α1 } {√ e−πε D22 e (1) (1) (1) (1) e cos(e + N sin(e χα1 ) + N χα2 ) α2 C2 D11 α1 (71) and (2) 1 Uα
eπε =− C2 e−πε − C2
{√ {√
D22 (1) (1) (1) (1) N cos(χα1 ) + Nα2 sin(χα2 ) D11 α1 D22 e (1) (1) (1) e (1) sin(e N cos(e χα1 ) − N χα2 ) α2 D11 α1
} }
(72) where α = 1, 2 and √ C2 = 1 + 4ε2 cosh πε .
(73)
For the lower material, the subscript behind U and the superscript in parentheses on the right hand side of the equations take on the value 2 (i.e. k = 2), eqs. (71) and (72) are modified by substitution of −π instead of π. (k) e (k) (k) (k) The components Nst , N est are given by st , χst and χ √[ ( )] [ ( )] (k)
Nst = (k) est N =
(k)
ℜ Gst
2
(k)
+ ℑ Gst
2
√[ ( )]2 [ ( )]2 (k) e (k) e ℜ G + ℑ G st st
(74) (75)
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