Fracture behavior of stitched warp-knit fabric composites - CiteSeerX

International Journal of Fracture 108: 73–94, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

Fracture behavior of stitched warp-knit fabric composites F. G. YUAN and S. YANG Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA Received 26 November 1998; accepted in revised form 30 August 2000 Abstract. Pilot studies are conducted to characterize the macroscopic fracture resistance behavior using linear elastic fracture mechanics and attempt to quantify the fracture parameters in which may govern the fracture and failure patterns of stitched warp-knit fabric composites. Methods based on the J -integral method and Betti’s reciprocal theorem in extracting the fracture parameters, critical stress intensity factors, T -stress, and the second term of σy (r, 0) near the crack tip prior to fracture initiation are formulated. Two fracture criteria, [σc , rc ] and [εc , rc ] are attempted to characterize the failure initiation for the fiber-dominated failure mode and self-similar crack extension in a given thickness of the laminate. Based on linear elastic fracture mechanics principle, these criteria are transformed into crack-driving forces [KQ , T ] and [KQ , g32 ]. The two-parameter fracture criteria, [KQ , T ] and [KQ , g32 ] provide a good correlation for the CCT and SENT specimens, but not for the high constraint CT specimens. With the limited experimental data, the results tend to show that the large tensile Tstress and large magnitude of negative g32 may inhibit the crack extension in the same crack plane and promote crack kinking. Key words: Betti’s reciprocal theorem, crack turning, J -integral, linear elastic fracture mechanics, stress intensity factors, T-stress.

1. Introduction Due to heterogeneity in fiber reinforced composites and inhomogenity in the composite laminates, fracture processes emanating from crack-like flaws are far more complicated than those occurred in metals. Wu [1] tested unidirectional composites with the crack placed parallel to the fibers and extended the crack tip field analysis for the linear isotropic case to linear elastic anisotropic solids. With this crack orientation, the crack extension followed in a selfsimilar manner. He found that critical stress intensity factors were virtually independent of crack length under pure tension and pure shear loading. Under these two combined loads a simple function of the stress intensity factors was developed to describe the fracture initiation. In order to determine the applicability of linear elastic fracture mechanics to crack extension perpendicular to the fibers, Beaumont and Philips [2] and Philips [3] tested center-cracked tension and tapered DCB specimens with side grooves along the crack line to promote the self-similar crack extension. Although the non self-similar crack extension has been observed experimentally even from the identical layup and geometry with different initial crack lengths (e.g., Konish et al. [4] and Prewo [5]), no explanation was made. Motivated by the effect of crack size on the fracture strength of the cross-ply laminates observed in the experiments, Waddoups et al. [6] introduced an inherent flaw concept in the intense energy region with a damage zone length a, much like Irwin’s plastic zone correction factor. Although the two-parameter model gave reasonable correlation within the experimental data range, the length a is not a constant for all crack sizes and the model reveals little phys-

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F. G. Yuan and S. Yang

ical insight of the fracture processes. The point stress and average stress models of Whitney and Nuismer [7] used two parameters, unnotched strength and a characteristic dimension, to characterize the fracture strength affected by the crack size. The point stress criterion is analogous to WEK model and the characteristic dimension depends on the stacking sequence of the laminate. In order to comply with the damage tolerance airworthiness requirements for the future commercial transport aircraft using new composite materials, the quantitative understanding of the parameters which control composite’s fracture behavior is essential to assess the residual strength and the direction of crack growth of aircraft structures. Under NASA’s ACT Composite Wing Program, Langley Research Center has recently performed fracture toughness tests for through-the- thickness cracks in evaluating fracture resistance of stitched warp-knit fabric composites. The tests were performed based on the standard fracture toughness testing procedures for metals. In this paper the composites with macro through-the-thickness cracks are modeled as linear, anisotropic, homogeneous materials using linear elastic fracture mechanics (LEFM). The effects of microscopic failure mechanisms such as fiber bridging, matrix cracking, and delamination, which accompany fracture are discussed as appropriate, but not taken into account in the modeling. The influence of stitching pattern and knit architecture on the failure initiation of the composites may be of minor importance thus is not considered. The macroscopic phenomena accompanying these microstructural features are reflected in the COD curves. The purpose of this pilot study is to examine if the fracture parameters based on LEFM principle can reveal a meaningful insight in correlating failure of this material. Test methods and experimental results performed by NASA [8] are briefly discussed. The stress state in the crack tip vicinity prior to fracture initiation, characterized by critical stress intensity factor, T stress, and the second-order term in σy (r, 0) at the onset of fracture, is evaluated numerically for three different specimen geometries: compact tension, center-cracked tension, and singleedge notched tension specimens. Using finite element analysis under the fracture initiation load, the critical stress intensity factors are calculated by the J -integral and a virtual crack closure method respectively. The T -stress and the second-order term in σy (r, 0) and other higher-order terms are derived and computed from two approaches. the J -integral method and Betti’s reciprocal theorem. The effects of geometrical attributes and loading configuration on these parameters and fracture behavior are investigated. For the failure being fiber-dominated and crack extension in a self-similar manner, two sets of two-parameter failure criteria are proposed to take into account the effects of geometrical attributes and loading configuration for the center-cracked tension and single-edge notched tension specimens. 2. Test methods and results The composites selected in this study are AS4 carbon warp-knit fabric, designed by the Boeing Company, Long Beach for the all-composite wing skin in a commercial transport aircraft. Each layer of fabric with fiber volume content of 59.4% contains AS4 fibers, 44% in the 0◦ direction, 44% in the ±45◦ directions, and 12% in the 90◦ direction. Layers of fabric are stacked and stitched together with spacing 0.5 in. along the 0◦ direction to form the laminate. The resulting materials properties are: E1 = 10.4 Msi,

E2 = 5.22 Msi,

E3 = 1.45 Msi,

ν13 = ν23 = 0.49,

ν12 = 0.403,

G12 = 2.54 Msi,

Fracture behavior of stitched warp-knit fabric composites

75

where subscript 1 refers to the direction parallel to the 0◦ fibers of the composite, subscript 2 refers to the transverse direction or perpendicular to the 0◦ fibers. The subscript 3 represents the out of plane direction of the composite. Note that the in-plane material properties are calculated using classical lamination theory and lamina constants from tests. The values of E3 , ν13 , and ν23 are assumed in order to perform plane-strain fracture analyses. The various types of specimens tested at NASA can be categorized into the following variables to assess their dependence on the fracture behavior: • in-plane geometry, • crack length, • specimen size, • specimen thickness, • loading configuration, • crack orientation to material orientation. Variation of in-plane geometry and crack length was evaluated from three specimen geometries: compact tension specimens (CT) with crack length-to-width ratios 0.46 6 a/W 6 0.69; center-cracked tension specimens (CCT) with 0.26 6 2a/W 6 0.42; and single-edge notched tension specimens (SENT) with 0.25 6 a/W 6 0.34. The specimen size was assessed by three geometrically similar specimens of each type. All dimensions of these three types of specimens, except laminate thickness, were scaled by a factor of 2 within the precision error. Note that the same pinhole size with 0.375 in. diameter was drilled for both small and large CT specimens. Due to the machining error, only a pair of specimens, CT-32 and CT-1L, can be considered as geometrically similar CT specimens. To address the role of the thickness effect, 12-inch-wide CCT specimens were made at NASA Langley Research Center with two layers of the warp-knit fabric (≈ 0.11 in.). The rest of specimens were cut from a wing skin five-stringer test panel fabricated by the Boeing Company with six layers of the warpknit fabric (≈ 0.34 in.). The pinhole loading was applied to the CT specimens; the uniform displacement applied to both CCT and SENT specimens. For most of the specimens, the crack is perpendicular to the 0◦ fiber (stiffer) direction. The effect of crack orientation to material orientation on the fracture behavior was investigated by two sets of specimens. The first set of CT specimens (CT-∗ L90) was made with the crack plane parallel to the 0◦ fiber direction; another set of inclined CCT specimens with the crack oriented 45◦ from the 0◦ fiber direction. Except the inclined CCT specimens under mixed-mode loading, the rest of the specimens were subjected to pure Mode-I loading. The specimen geometries of each type and specimen designation are shown in Figures 1–3. The standard ASTM Test Method for Plane Strain Fracture Toughness of Metallic Materials in brittle fracture (E-399-90) [9] follows the requirements of a finite-size window listed below: (a) The initial crack length should be 0.45 6 a/W < 0.55; (b) The specimen thickness > 2.5(KI C /σys )2 ; (c) Self-similar crack growth (8.2.4 of E-399); (d) Pmax /PQ 6 1.10 (9.1.2 of E399). Since there are no such test procedures for composite materials, the tests of the CT specimens were followed by the recommendations from E-399, except that a wider range of a/W was used and the crack tip was not sharpened by fatigue. The load versus crack mouth opening displacement (COD) was recorded for all the three types of specimens except the inclined

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Figure 1. Compact Tension (CT) Specimen Geometry; all dimensions given in inches. (a) Crack perpendicular to the 0◦ fiber direction, CT-∗∗ : W = 1.4, 0.46 6 a/W 6 0.56; CT-∗ L, W = 2.8, 0.57 6 a/w 6 0.69. (b) Crack parallel to the 0◦ fiber direction, CT-∗ L90, W = 2.8, 0.57 6 a/W 6 0.69.

center-cracked specimens. Nonlinear load-COD curves as a result of crack extension preceded by significant crack tip damage were observed. Based on the ASME E-399-90 test method, P5 is the load defined as the intersection point between a line drawn through the origin with a 5% offset from the initial slope of the load-COD curve and the load-COD record. In the case where the initial region exhibits increasing or decreasing in slope, the 5% offset from the linear region is used to determine P5 . If the load at every point of the curve which precedes P5 , is smaller than P5 , then P5 , is set to PQ ; if any load maximum precedes P5 , then PQ is equal to this maximum load. When the ratio of Pmax /PQ is less than 1.1 and the specimen satisfies the minimum specimen thickness and crack length, the critical stress intensity factor KQ computed from PQ is equal to KI C . By counting the localized plasticity correction near the crack tip, the value of PQ corresponds well to minimal level of crack extension in metals for brittle fracture. In the absence of suitable guidelines for fracture toughness testing in composites, the same procedure used in metals is adopted for these three specimen geometries in interpreting the fracture initiation for composites with damage zone. It is believed that this consistent procedure will probably provide conservative measure of fracture initiation load. In addition to the loads PQ and Pmax determined from the COD curves, failure patterns observed during the course of the tests are summarized below: 1. For all the small size of CT specimens (CT-∗∗ ), the crack kinked out of its own plane and extended parallel to the loading direction. 2. For the two closely geometrically similar CT specimens, CT-3∗ and CT-1L, the distinct fracture behavior, non self-similar for CT-3∗ and self-similar crack growth for CT-1L, occurred. 3. For the inclined CCT specimens, the crack extended approximately perpendicular to the loading direction. 4. The rest of specimens followed a self-similar crack growth pattern, collinear with the crack plane.


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Figure 2. Center-Cracked Tension (CCT) Specimen Geometry; all dimensions given in inches (a) CN∗ : L = 4 and W = 2; 2a/W = 0.26 and 0.34; CN∗ L: L = 8 and W = 4; 2a/W = 0.25 and 0.34. (b) 2A#∗ and 2B#∗ : L = 28 and W = 12; 0.25 6 2a/W 6 0.42. (c) ICN∗ : L = 5 and W = 2; 2a/W = 0.25 and 0.33. 2a is the projected crack length.

Figure 3. Single-Edge Notched Tension (SENT) Specimen Geometry; all dimensions given in inches (a) EN∗ G: L = 6 and W = 2; a/W = 0.25 and 0.34. (b) ENT∗ L: L = 12 and W = 4; a/W = 0.25, 0.33 and 0.50.

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Figure 4. A contour around the crack tip.

5. The 12-in center-cracked tension specimens with thinner thickness exhibited dispersed progressive damage. 3. Theoretical background If the fracture behavior is a stress (or strain)-driven process and the scale of the damage zone is sufficiently smaller than the crack length and specimen dimensions, the state of stress near the crack tip dominates over the damage zone where microscopic failure mechanisms occur. A fundamental understanding of the fracture behavior requires an accurate analysis of the crack tip stress field. In this section, all the coefficients in the eigenfunction expansion terms are determined. In a fixed Cartesian coordinate system xi (i = 1, 2, 3), consider a two-dimensional deformation of an anisotropic elastic body in which the deformation field is independent of the x3 coordinate. Attention focuses on the material having three mutually perpendicular symmetry planes and one of the planes coinciding with the coordinate plane x3 = 0. In this case, the in-plane and out-of-plane deformations are uncoupled. For the in-plane deformation the strain and stress relations can be written as ε = s 0 σ,

(1)

where ε = [ε1 , ε2 , γ12 ]T , σ = [σ11 , σ22 , σ12 ]T . or εi = sij0 σj ,

i, j = 1, 2, 6,

where sij0 = sj0 i are reduced compliance coefficients defined by sij0 = sij − si3 sj 3 /s33 . Referring to a local coordinate system at the crack tip shown in Figure 4, general solution expressions of the displacement vector u, the stress function 8, and stresses σ for crack with traction-free surfaces, according to Stroh formalism (Ting [10]), can be represented by X X

u= Re[A zδn +1 B −1 g n ], 8 = B zδn +1 B −1 g n , (2) n=1

σi1 = −Re[8i,2 ],

n=1

σi2 = Re[8i,1 ],

where 8i are the components of the vector 8,

(3)

Fracture behavior of stitched warp-knit fabric composites δn = (n − 2)/2,

n = 1, 2, 3, . . . , (

g n = [gn1 , gn2 ]T =

79 (4)

real,

n = 1, 3, 5, . . . ,

pure imaginary,

n = 2, 4, 6, . . . ,

(5)

f (z) = diag f (z1 ), f (z2 ) , zα = x1 + µα x2 , µα > 0, g n are unknown constant vectors or coefficients to be determined. µα , a α , and bα are the Stroh eigenvalues and corresponding eigenvectors determined by elastic constants only. µα are given by the roots of the characteristics equation 2 0 0 0 0 0 0 s11 µ − 2s26 µ4 − 2s16 µ3 + 2s12 + s66 µ + s22 = 0, (6) with positive imaginary parts. From energy consideration, Lekhnitskii [11] showed that the roots are either complex or pure imaginary and cannot be real. The Stroh eigenvalue µ in the xi coordinates having angle ϕ with the principal material coordinates (1,2) may be obtained by Lekhnitskii [11] according to µ=

µ0 cos ϕ − sin ϕ , cos ϕ + µ0 sin ϕ

(7)

where µ0 is the eigenvalue in the principal material coordinates. A and B are Stroh matrices given by p 1 p2 A = [a 1 a 2 ] = , q1 q2

B = b1 ,

b2 =

−µ1 −µ2 1 1

0 0 0 pα = s11 µ2α − s16 µα + s12 ,

,

B

−1

1 = µ1 − µ2

−1 −µ2 1 µ1

(8) ,

0 0 0 qα = s12 µα − s26 + s22 /µα .

(9) (10)

It can be proved that B −T B −1 = −2iL−1 , L−1 = Im[AB −1 ], where L is a real, symmetric, and positive-definite tensor of rank two and 1 b d e −d −1 0 L = s11 , L= 0 , d e s11 (be − d 2 ) −d b µ1 + µ2 = a + i b, µ1 µ2 = c + i d, e = ad − bc = Im[µ1 µ2 (µ1 + µ2 )].

(11)

(12)

(13)

Performing algebraic caculation and defining, the stress and displacement components can be written as

80

F. G. Yuan and S. Yang h i X 1 δn δn δn 2 δn 2 δn σ11 = gn1 µ2 ζ2 − µ1 ζ1 + gn2 µ1 µ2 µ2 ζ2 − µ2 ζ1 , (δn + 1)r Re µ1 − µ2 n=1 h i X 1 δn δn δn δn δn σ22 = gn1 ζ2 − ζ1 + gn2 µ1 ζ2 − µ2 ζ1 , (δn + 1)r Re (14) µ1 − µ2 n=1 h i X 1 σ12 = gn1 µ1 ζ1δn − µ2 ζ2δn + gn2 µ1 µ2 ζ1δn − ζ2δn , (δn + 1)r δn Re µ − µ 1 2 n=1 h i X 1 δn +1 δn +1 δn +1 δn +1 δn +1 u1 = r gn1 p2 ζ2 + gn2 µ1 p2 ζ2 − µ2 p1 ζ1 , Re − p1 ζ1 µ1 − µ2 n=1 h i X 1 δn +1 δn +1 δn +1 δn +1 δn +1 u2 = r gn1 q2 ζ2 + gn2 µ1 q2 ζ2 − µ2 q1 ζ1 Re − q1 ζ1 µ1 − µ2 n=1 (15)

where ζα = cos θ + µα sin θ. √ For the leading-order term, δ1 = − 12 , letting (g11 , g12 ) = 2/π(k2 , k1 ), the singular solution and T -stress term for the anisotropic solids can be written in the form:     σ11 1 1 σ =  σ22  = √ (16) k1 σ I + k2 σ I I + T  0  + O(r 1/2 ), 2π r 0 σ12 " 0 # " # r 0 cos θ + s sin θ s − sin θ 2r 11 16 u= kl uI + k2 uI I T r + ωr + O(r 3/2 ), (17) 0 π s12 sin θ cos θ where

(σ11)I = Re

µ1 µ2 µ1 − µ2

µ2 µ1 √ −√ ζ2 ζ1

,

(σ11 )I I = Re

1 µ1 − µ2

µ2 µ2 √2 − √1 ζ2 ζ1

,

µ1 1 1 µ2 1 1 II (σ22) = Re , (σ22 ) = Re , √ −√ √ −√ µ1 − µ2 µ1 − µ2 ζ2 ζ1 ζ2 ζ1 1 µ1 µ1 , µ2 1 1 µ2 I II (σ12) = Re , (σ12 ) = Re , √ −√ √ −√ µ1 − µ2 µ1 − µ2 ζ1 ζ2 ζ1 ζ1 p p 1 I (u1 ) = Re µ1 p2 ζ2 − µ2 p1 ζ1 , µ1 − µ2 p p 1 II (u1 ) = Re p2 ζ2 − p1 ζ1 , µ1 − µ2 p p 1 (u2 )I = Re µ1 q2 ζ2 − µ2 q1 ζ1 , µ1 − µ2 p p 1 II (u2 ) = Re q2 ζ2 − q1 ζ1 , µ1 − µ2

I


81

k1 , and k2 are stress intensity factors for mode-I and mode-II respectively, ω, is a constant representing rigid body rotation, and T = −i(g21 b + g22 d),

0 ω = is11 (g21 d + g22 e)

(18)

Clearly the T -stress term depends on the material properties in anisotropic solids. Some auxiliary fields with higher-order singularities are needed in order to determine the coefficients in the expansion of the crack tip field by the use of conservation laws of elasticity and Betti’s reciprocal theorem. These auxiliary fields may be obtained by choosing the values of k in Equations (2) as negative integers, that is,

ua = Re[A z1m +1 B −1 hm ], (19)

8a = B z1m +1 B −1 hm , σi1a = −Re[8ai,2 ], σi2a = Re[8ai,1 ], 1m = −m/2, m = 1, 3, 4, . . . , real, T hm = [hm1 , hm2 ] = pure imaginary,

(20) m = 1, 3, 5, . . . , m = 4, 6, 8, . . . ,

(21)

where hmi are arbitrary constants. In the following two sections, the T-stress term and coefficient of second-order term in σy are determined using the J -integral and the Betti’s reciprocal theorem including the use of above auxiliary fields. All the coefficients in the eigenfunction expansion terms can also be evaluated using the same concept and the detailed derivation is reported (Yuan [1 2]). 4. J -integral 4.1. T - STRESS

TERM

The path-independent J -integral or the rate of energy release rate per unit of crack extension along the x1 -axis is given by Z J = σ T εn1 /2 − t T u,1 ds, (22) 0

where 0 is an arbitrary path which starts on the straight lower face of the crack, encloses the crack tip and ends on the upper straight face with the positive direction in a counterclockwise direction shown in Figure 4. In elastic materials, J is equal to the energy release rate G and relates to the stress intensity factors through G = 12 k T L−1 k.

(23)

Consider a cracked body under the two-dimensional deformation. The components of stress, strain, and displacement fields are represented by σij , εij , and ui , respectively. As r → 0, the asymptotic fields including the constant T -stress terms are given before. Now the coefficients of the T -stress terms and third-order terms are derived using conservation laws.

82


In general, for the purpose of determining the coefficients g n of the term r δn (δn > − 12 ) in the actual crack tip stress field, we may employ the J -integral method and follow the following procedure: (i) find an auxiliary (pseudo) field that has singularity σija ∼ r −δn −1 as r → 0. It is convenient to select auxiliary stress field which gives zero traction on the crack surfaces and contains only the stress singular term r −δn −1 ; (ii) superimpose the actual field (the mixed-mode boundary value problem, in general) on the auxiliary field and represent the J -integral for the superimposed state as Js = J + Ja + JM , where Js =

(24)

Z Z0

Ja =

[(σ + σ a )T (ε + ε a )n1 /2 − (t + t a )T (u,1 + ua,1 )] ds, [(σ a )T εa n1 /2 − (t a )ua,1 ]ds

0

and JM = Js − J − Ja Z = {[σ T εa + (σ a )T ε]n1 /2 − t T ua,1 − (t a )T u,1 } ds 0 Z [σ T εa n1 − t T ua,1 − (t a )T u,1 ] ds 0 Z = (σij uai,j n1 − ti uai,1 − tia ui,1 ) ds,

(25)

0

where the superscript or subscript ‘a’ denote quantities referred to the auxiliary field; Js is the J -integral for the superimposed state; J for the actual state; and Ja for the auxiliary field and JM is the interaction integral. In the sequel, we assume that the J -integral is path-independent for both the actual field and the selected auxiliary fields, denoted by J and Ja . Then the integral Js for the superimposed state, thus JM , is also path-independent. If the auxiliary fields given by Equation (19) are used, it is readily proved that Ja 6 = 0,

for 1m = − 12 ,

Ja = 0,

for 1m < − 12 ;

(26)

(iii) evaluate JM as 0 → 0. For simplicity, 0 may be taken as a circle with radius r, as r → 0, the only terms in the integrand that contribute to JM are the cross terms between r δn in the actual stress field and the auxiliary stress term with order r −δn −1 ; (iv) carry out the routine manipulation, the exact expression for JM can be obtained as JM = JM (g n ) when r → 0; (v) evaluate JM for a finite contour 0 using the computed actual field and the exact auxiliary solution; and determine the coefficients g n from the value of JM and the expression of JM = JM (g n ) as r → 0. In extracting the T -stress in Equation (2) or g 2 , we make use of another auxiliary field, that is the solution to a point f (per unit thickness) applied at the crack tip. Under this loading, σija ∝ r −1 . Note that the point force f must be resisted by traction t applied to some boundary C in achieving equilibrium. In the Stroh formalism (Ting [10]), the real form solutions due to the point-force application can be written as


83

ln r 2u = − I + S(θ) h, π

(27)

28a = L(θ)h,

(28)

a

where h = L−1 f , f = [f1 , f2 ]T ,

2 Re[A ln(cos θ + µ sin θ B T ], π

2 L(θ) = − Re[B ln(cos θ + µ sin θ B T ], π

S(θ) =

x1 = r cos θ,

x2 = r sin θ.

It assumes the values

ln(cos θ + µa sin θ) =

θ = 0, θ = ±π.

0 ±iπ

In a cylindrical coordinate system (r, θ, x3 ), let t r and t θ be the traction vectors on a cylindrical surface r = constant and on a radial plane θ = constant, then 1 t ar = − 8a,θ , r σra = nT t r ,

t aθ = 8a,r

(29)

a σrθ = mt t r = nT t θ ,

σθa = mT t θ

where nT = [cos θ, sin θ] and mT = [− sin θ, cos θ]. It follows from Equation (27) and (28) that t aθ = 8a,r = 0,

a σθa = σrθ = 0,

σra = −nT 8a,θ /r, or

f1 [(µ1 µ2 − 1) sin θ + (µ1 + µ2 ) cos θ] cos2 θ − sin θ = − Im 2π r ζ1 ζ2 f2 [(µ1 µ2 − 1) cos θ − (µ1 + µ2 ) sin θ] sin2 θ + µ1 µ2 cos θ − Im 2π r ζ1 ζ2

(30)

f1 0 1 p2 ln ζ2 − p1 ln ζ1 b u = − − Im s ln r d 2π 11 µ1 − µ2 q2 ln ζ2 − q1 ln ζ1 f2 1 µ1 p2 ln ζ2 − µ2 p1 ln ζ1 d 0 − − Im s ln r e 2π 11 µ1 − µ2 µ1 q2 ln ζ2 − µ2 q1 ln ζ1

(31)

σra

and

a

By superimposing the actual field on the auxiliary field, and using the path-independent J -integral for the monoclinic elastic cracked body, it can be proved that 0 Js = J + T s11 f1 + ωf2,

0 JM = T s11 f1 + ωf2 .

(32)

84


Note that because σija ∝ r −1 , uai,j ∝ r −1 Z W a n1 − tia uai,1 ds = 0. Ja = 0

Kfouri [13] used the method to calculate the T-term for isotropic materials. Wang et al. [14] and Wu [15] applied the J -integral to determine the stress intensity factors for rectilinear anisotropic solids and general anisotropic materials respectively. In this report, the method is extended to determine all the coefficients in the crack tip field expansion for materials. From Equation (32), it follows that JM f =0 JM f =0 2 1 T = 0 and ω = . s11 f1 f2 Detailed proof of Equation (32) is given below: For general anisotropic linear elastic solids, we have the following relations σij εija = cij kl εkl εija = σija εij

and

σij εija = σij uai,j ,

where cij kl = cklij . From Equation (25), Z Z duai a a a a JM = (σij ui,j n1 − ti ui,1 − ti ui,1 )ds = σi2 − ti ui,1 ds, ds 0 0

(33)

where duai /ds is the tangential derivative of uai . As r → 0, we evaluate JM and note that the only terms that contribute to JM are the cross terms between T and f . After substituting these fields into the integral and performing the routine algebra, the integral JM may be evaluated as Z dua JM = lim σi2 i − tia ui,1 ds 0→0 0 ds Z a (2) dui a (2) = σi2 − ti ui,1 ds ds 0 Z a = − u(2) i,1 ti ds 0 (34) Z T a = − (u(2) ,1 ) t ds 0

Z T = −(u(2) ,1 )

t a ds 0

T = (u(2) ,1 ) f .

From Equations (2) and (17) via (11)2 , 0 T s11 (2) −1 u,1 = iL g 2 = . ω Insertion Equation (35) into (34) leads to

(35)


85

JM = ig T2 L−1 f = −if T L−1 g 2

(36)

0 JM = T s11 f1 + ωf2 .

(37)

and

Equations (36) and (37) can be used to calculate g 2 and T . As f is arbitrary, it is convenient to choose f to be the following values [1, 0]T ≡ e1 ,

[0, 1]T ≡ e2

respectively. Note that ek has a dimension force/length. Correspondingly, Equation (36) yields two linear equations and they are, in matrix notation, J˜ M = −iL−1 g 2 , where J˜ M = [JM(1) , JM(2) ]T and JM(k) is the value of JM when f = ek . Therefore, g 2 = iLJ˜ M .

(38)

Using Equation (37), The T-stress and ω can be obtained as 0 T = JM(1) /s11 ,

4.2. T HE SECOND

ω = JM(2) . TERM IN

(39)

σy

The second term in σy or the third-term coefficient in the eigenfunction expansion of the stress field can be also obtained from the J -integral method. An auxiliary field with singularity σija ∼ O(r −3/2 ) can be introduced by selecting m = 3 in Equation (19). By superimposing the actual field (the mixed mode boundary value problem) on the auxiliary field, the interaction integral JM may be evaluated as JM = − 32 π hT3 L−1 g 3 .

(40)

Following in a similar manner, g3 = −

2 ˜ LJ M , 3π

(41)

where J˜ M = [JM(1), JM(2) ]T and JM(k) is the value of JM when h3 = ek , (k = 1, 2) · ek has a dimension of force/(length)1/2 . In general, superimposing of an auxiliary field with σija ∝ r 1n on the actual field and applying the J -integral to this combined state for n 6 = 2, we can get the interaction integral JM denoted by JMn , that is JMn = −2π δn (δn + 1)hTn L−1 g n , Then

n = 1, 3, 4, 5, . . . .

(42)

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F. G. Yuan and S. Yang   LJ˜ Mn     − 2π δn (δn + 1) , gn =   iLJ˜ Mn    , 2π δn (δn + 1)

n = 1, 3, 5, . . . , (43) n = 4, 6, 8, . . . ,

where (1) (2) T J˜ Mn = [JMn , JMn ] (k) and JMn is the value of JMn when n = 1, 3, 5, . . . , ek , hn = iek , n = 4, 6, 8, . . . .

Here, ek possesses dimension force√/ (length)1−δn . For the first singular term, introducing √ stress intensity factors, k = [k2 , k1 ]T = π/2g 1 for the actual field and k a = [k2a , k1a ]T = π/2h1 for the auxiliary field, JMI and k can be rewritten from Equation (42), (43) as JM1 = (k a )T L−1 k, k = LJˆ M1 ,

(44)

(1) (2) T (k) where Jˆ M1 = [JM1 , JM1 ] and JM1 is the value of JM1 when k a = ek .

5. Betti’s reciprocal theorem 5.1. T - STRESS

TERM

For a linear elastic plane problem, Betti’s reciprocal theorem can be stated as Z t · ua − t a · u ds = 0,

(45)

C

where C is an any closed contour enclosing a simple connected region in the elastic body; u is the displacement vector and t the traction on C corresponding to the solution of any particular elastic boundary value problem for the elastic body; ua and t a are corresponding quantities of the solution of any other problem for the body. Considering a crack in an anisotropic linear elastic material, and suppose the crack surfaces are free of tractions for both elastic states. If the closed contour C encloses the crack tip and extends along the crack surfaces, then it can be shown that the integral Z I= t · ua − t a · u ds (46) 0

is path independent where 0 is an any path which starts from the lower crack face and ends on the upper. Let (t, u) be an actual state for the crack under consideration, then Equation (46) provides sufficient information for determining the amplitude for each term in the asymptotic crack-tip fields if proper auxiliary solutions (t a , ua ) are provided. In this section the Betti’s reciprocal work contour integral is used for computing stress intensity factors, T -stress and


87

other higher-order coefficients for monoclinic materials. The procedure can be evaluated from the analysis as follows. For determining the coefficients g n of the term r δn (δn > − 12 ) in the actual crack tip stress field, an auxiliary (pseudo) field with σija ∝ r −δn −2 or uai ∝ r −δn −1 can be chosen. As r → 0, take a 0 as a circle around the crack tip and evaluate integral I . When r → 0, the only product between g n and the auxiliary terms in the integrand given above can contribute to the integral I . Therefore, the expression for I = I (g n ) can be obtained as r → 0. The value of I for a finite contour 0 shown in Figure 2 is available from the numerical solutions for t and u of the boundary value problems and the exact auxiliary solution. The g n can be computed from the expression for I = I (g n ) and the value of I . To determine the T-stress or g2 for the crack-tip field from Equation (14), the auxiliary elastic field with stress singularity σija ∝ r −2 as r → 0 is used and can be obtained from Equation (19) by choosing m = 4, that is, in Stroh formalism,

ua = Re[A z−1 B −1 h4 ], (47)

8a = B z−1 B −1 h4 . It is clear that only those parts of the integrand in Equation (46) which behaves like O(l/r) as r → 0 can contribute this portion of the integral. Substituting these fields of the two states into Equation (46), performing the integration for the circle surrounding the crack tip and evaluating the results in the limit of vanishing radius, the results may be derived, and Z I = lim t · ua − t a · u ds = −2π hT4 L−1 g 2 (48) r→0 0

g2 =

i ˜ LI , 2π

(49)

where I˜ = [I (1) , I (2)] and I (k) is the value of I when h4 = iek (k = 1, 2). (Dimension of ek is force.) From Equations (12), (18), and (49), T =

I (1) , 0 2π s11

ω=

5.2. T HE SECOND

I (2) . 2π

TERM IN

(50)

σy

The second term in σy or the third-term coefficient in the eigenfunction expansion of the stress field can also be obtained using Betti’s theorem. Selecting m = 5 in Equation (19), an auxiliary field with stress singularity σija ∼ r −5/2 desired for this purpose can be obtained. Applying the Betti theorem of reciprocity to the actual field and the auxiliary field and evaluating the integral I as 0 → 0 near the crack tip, we obtain I = −3π hT5 L−1 g 3 .

(51)

Equations (51) will be used to calculate g3 for mixed-mode problem when the two proper auxiliary field solutions are provided. g 3 can be expressed in the form g3 = −

1 ˜ LI . 3π

(52)

88


Applying Betti’s reciprocal theorem to the actual fields and auxiliary fields with σija ∝ r 1n+2 , the path independent I denoted by In+2 can be evaluated by In+2 = −2π(δn + 1)hTn+2 L−1 g n . It follows from Equation (53)  LI˜ n+2   − , n = 1, 3, 5, . . . ,   2π(δn + 1) gn =   iLI˜ n+2    , n = 2, 4, 6, . . . , 2π(δn + 1)

(53)

(54)

where I j = [Ij(1) , Ij(2)]T and Ij(k) is the value of Ij when ( n = 1, 3, 5, . . . , ek , hn = iek , n = 2, 4, 6, . . . . Here, ek possesses dimension force × (length)δn . For the first singular term from Equations (53) and (54), I3 and k can be written as √ I3 = −π hT3 L−1 g 1 = − 2π hT3 L−1 k, 1 k = − √ LI˜ 3 . 2π

(55) (56)

6. Discussion An attempt at reducing the laminate fracture initiation load, PQ , is to directly use linear elastic fracture mechanics in anisotropic solids. Note that the values of PQ for CCT and SENT specimens which are not listed in [8] are determined from the COD curves and PQ values from CT specimens are re-evaluated to reflect more precise values [16]. In modeling the CT specimens, the geometry without pinholes is simulated. For the CCT and SENT specimens, uniform displacements are applied at the ends of the specimens. The fracture parameters are computed by the equivalent end displacements corresponding to the resulting forces, PQ . It is expected for CCT specimens that due to the applied displacements far from the ends and symmetric geometry the parameters calculated from uniform displacements and uniform remote stress are almost identical. Comparison of KQ values obtained from the J -integral and the virtual crack closure method [17] via Equation (23) is made in Table 1 using CT specimens modeled by finite element analysis. The T-stress and g32 term are also compared from two approaches: Betti’s theorem and the J-integral. It is clearly seen that there are in excellent agreement. In the following tables except the inclined CCT specimens, the KQ are calculated from the J-integral method; T-stress and the coefficient of the second term in σy , g32 , are computed from the Betti’s theorem. Since


89

Table 1. Comparison of fracture parameters using different approaches Specimen a/W PQ J |P Q GI P KQ Q √ number (kips) (psi in) (psi in) (ksi in)

J -integral method

Betti’s theorem T (ksi)

β

g32 √ (ksi/ in)

T (ksi)

β

g32 √ (ksi/ in)

CT-11 CT-21 CT-22 CT-31 CT-32

0.46 0.50 0.51 0.56 0.56

2.29 1.52 1.34 1.18 1.22

277.3 153.4 124.6 131.5 140.5

277.3 153.4 124.5 131.5 140.5

49.3 36.6 33.0 34.0 35.0

21.1 15.4 13.8 13.4 13.8

0.610 0.625 0.628 0.619 0.620

−46.2 −34.7 −31.4 −33.0 −34.1

21.1 15.4 13.7 13.4 13.8

0.610 0.625 0.626 0.621 0.620

−46.3 −34.8 −31.5 −33.1 −34.3

CT-1L CT-2L CT-3L CT-1L90 CT-2L90 CT-3L90

0.57 0.63 0.69 0.57 0.63 0.69

2.60 2.13 1.30 1.53 1.19 0.77

392.0 402.8 260.0 174.8 173.4 127.6

391.9 402.7 259.9 174.8 173.3 127.5

58.6 59.4 47.7 33.2 33.1 28.4

15.7 14.9 10.6 9.16 9.24 8.31

0.595 0.594 0.550 0.619 0.659 0.722

−28.7 −30.9 −28.1 −13.2 −15.2 −15.7

15.6 14.9 10.6 9.17 9.23 8.31

0.594 0.594 0.549 0.619 0.658 0.722

−28.8 −31.0 −28.2 −13.2 −15.3 −15.7

Note: biaxiality parameter: β = T

√ πa/KQ .

Table 2. Fracture parameters for compact tension (CT) specimens Specimen a/W Thickness PQ Pmax /PQ J |PQ KQ T β √ number (in) (kips) (psi in) (ksi in) (ksi)

g32 √ (ksi in)

CT-11 CT-21 CT-22 CT-31 CT-32

0.46 0.50 0.51 0.56 0.56

0.354 0.351 0.353 0.352 0.352

2.29 1.52 1.34 1.18 1.22

1.17 1.52 1.73 1.59 1.56

277.3 153.4 124.6 131.5 140.5

49.3 36.6 33.0 34.0 35.0

21.1 15.4 13.8 13.4 13.8

0.610 0.625 0.628 0.619 0.620

−46.2 −34.7 −31.4 −33.0 −34.1

CT-1La CT-2La CT-3La CT-1L90b CT-2L90b CT-3L90b

0.57 0.63 0.69 0.57 0.63 0.69

0.330 0.331 0.331 0.326 0.325 0.326

2.60 2.13 1.30 1.53 1.19 0.77

1.07 1.22 1.45 1.10 1.04 1.18

392.0 402.8 260.0 174.8 173.4 127.6

58.6 59.4 47.7 33.2 33.1 28.4

15.7 14.9 10.6 9.16 9.24 8.31

0.595 0.594 0.550 0.619 0.659 0.722

−28.7 −30.9 −28.1 −13.2 −15.2 −15.7

a Drawing error, intended to be 0.45, 0.5, and 0.55. b Drawing error, intended to be the crack perpendicular to the 0◦ fiber (stiffer) direction.

the fracture parameters are evaluated from the line integrals away from the crack tip, refining the mesh size in the vicinity of the crack tip is not necessary. In the Tables 2–5, PQ , Pmax /PQ ratios, and fracture parameters for all the specimens tested are listed. The data indicate that the majority Pmax /PQ ratios exceed 1.1, even higher values occurred in the small CT specimens with non self-similar crack extension. The magnitude of the T -stress, a stress parallel to the crack flanks, varies with applied load and its geometrical dependence is indexed by a non-dimensional geometry factor, introduced by Leevers and Radon [18] in isotropic solids, known as the biaxiality parameter β: √ T πa β= . KQ

90

F. G. Yuan and S. Yang Table 3. Fracture parameters for center-cracked tension (CCT) specimens Specimen 2a/W Thickness PQ Equivalent End Pmax /PO J |PQ KQ T √ number (in) (kips) Dips (×10−3 in) (psi in) (ksi in) (ksi) CN2 CN3

0.26 0.34

0.343 0.346

40.3 37.2

23.4 22.4

1.10 1.00

349.3 408.2

CN1L CN2L 2A#1a 2B#2a 2A#2a 2B#3a 2A#3a

0.25 0.34 0.42 0.38 0.33 0.29 0.25

0.326 0.325 0.113 0.113 0.113 0.112 0.112

56.4 51.9 – 47.2 40.4 54.5 59.8

34.2 33.4 – 89.7 74.4 98.5 106.8

1.36 1.14 – 1.11 1.18 1.06 1.08

361.7 450.8

β

g32 √ (ksi in)

−44.8 −0.732 42.7 −42.8 −0.740 35.4

55.3 59.8

−32.8 −31.8 – −28.5 −23.6 −31.4 −34.1

56.3 62.8 – – 1196 102.4 724.4 79.6 1138 99.8 1158 100.7

−0.732 −0.740 – −0.745 −0.739 −0.735 −0.731

22.6 18.6 – 9.07 8.09 11.5 13.6

a The 12-inch-wide CCT specimens were fabricated at NASA Langley Center with two layers of the warp-

knit fabric. The 2A#1 specimen exhibited instability behavior. The rest of the specimens were constrained in the out-of-plane direction to prevent buckling. The fracture of these thin specimens was accompanied by progressive damage (matrix cracking and delamination) in the outer layers in which also reflected macroscopically in the load-COD curve. Table 4. Fracture parameters for inclined center-cracked tension (CCT) specimens Specimen 2a/W Thickness Pmax Equivalent End number (in) (kips) Disp. (×10−3 in.) ICN1 ICN2

0.25 0.33

0.323 0.324

39.8 37.7

60.2 59.2

For anisotropic cracked solids, the β values are also function of anisotropic material properties. p 0 In0 the CCT specimens, β is strongly dependent on the material anisotropy ratio, − s22 /s11 . Since the stress intensity factor defined from σy (r, 0) as r → 0 does not reflect the anisotropic properties, the T -stress or β parameter may provide a usefull measure in anisotropic solids. The values of KQ , T-stress, biaxiality parameter β, and g32 values for the specimens are shown under Mode-I loading. In the case of inclined CCT specimens, mixed mode stress intensity factors, (k1 )max and (k2 )max , and g32 and g31 under Pmax are tabulated. In general, geometries with a tensile T -stress corresponding to a positive β and negative g32 exhibit high constraints in the crack region. . Specimen Jmax (GI )max (GI I )max (k1 )max (k2 )max √ √ number (psi in) (psi in) (psi in) (ksi in) (ksi in) ICN1 ICN2

268.9 353.3

142.6 193.6

126.3 159.7

35.2 40.9

33.4 37.7

T (ksi) −11.2 −12.9

1. Only Pmax was measured. 2. 2a/W is the ratio of projected crack length to specimen width.

g32 g31 √ √ (ksi/ in) (ksi/ in) 19.4 15.7

19.2 16.5


91

Figure 5. Variation of KQ versus crack length/width ratio. Table 5. Fracture parameters for single-edge notched tension (SENT) specimens Specimen a/W Thickness PQ Equivalent end Pmax /PQ J |PQ KQ √ number (in) (kips) Disp. (×10−3 in.) (psi.in) (ksi. in)

T (ksi)

β

g32 √ (ksi/ in)

EN1G EN2G

0.25 0.34

0.346 0.338

39.6 34.3

67.7 63.8

1.30 1.17

519.4 686.6

67.4 77.5

−12.0 −0.316 −11.1 −0.296

5.55 2.51

ENT1L ENT2L ENT3L

0.25 0.33 0.50

0.322 0.321 0.319

28.0 24.6 13.6

51.4 47.8 32.4

1.13 1.03 1.24

594.1 735.0 532.9

72.1 80.2 68.3

−18.1 −0.316 −16.5 −0.298 −10.4 −0.272

11.8 5.91 −4.37

As previously described in the CT specimen testing, the failure patterns exhibited distinct characteristics affected by the specimen size, perhaps by the a/W ratios. In the small CT specimens (CT-∗∗ ) where the crack kinked and extended parallel to the applied loading direction, the average critical stress intensity factor under non self-similar crack extension (≈ 41.7 ksi √ in) is markedly smaller than √ that of the crack extension in a self-similar manner for large CT specimens (≈ 55.2 ksi in) shown in Figure 5. In addition, the magnitudes of tensile T stress (on the average), β, and negative g32 are greater than those of large CT specimens. The high constraint due to relatively large tensile T-stress and large magnitude of negative g32 term may be expected to inhibit the crack extension in the same plane and promote crack kinking. Further, among the small CT specimens, the greater T and less g32 exhibit higher KQ . From the testing of CCT specimens, the failure mechanisms are influenced by the laminate thickness. The progressive damage such as matrix cracking and delamination developed on the outer layers of the thin laminate relieves the stress in the crack tip region and renders calculation of KQ based on LEFM invalid. The failure mechanisms caused by less constraint in the thickness direction alleviate the stress intensification near the crack tip and result in elevated apparent KQ .

92


In Tables 2–5, it is clearly demonstrated for a given stress intensity factor that the CT specimens give the highest constraint, the SENT next, and the CCT the lowest. For the crack perpendicular to the 0◦ fiber direction and extended in a self-similar manner, the average KQ ≈ √ 55.2 ksi in is obtained from large CT specimens (0.57 6 a/W 6 0.69), while the average √ KQ ≈ 58.6 ksi in CCT specimens (0.25 6 2a/W 6 0.34). From Figure 5, it is clearly √ indicated that with same range of crack length to width ratio the average KQ ≈ 73 ksi in from SENT specimens gives a higher value (0.25 6 a/W 6 0.50). The stress gradients induced by the bending moments across the uncracked ligament may be responsible for this trend. Critical Mode-I stress intensity factor is strongly dependent on the material orientation ◦ in composites. In the case of the initial √ crack parallel to the 0 fiber direction, the KQ values are considerably lower (≈ 31.6 ksi in). Among the test specimens, the specimens (CT-∗ L90) exhibited the self-similar crack extension and showed the largest β. It is evident the crack kinking criterion using the magnitude of β is questionable, at least for composites. As discussed before, if the fracture behavior is a stress (or strain)-driven process and the scale of the damage zone is sufficiently smaller than the crack length and specimen dimensions, the state of stress near the crack tip dominates over the finite damage zone where microscopic failure mechanisms occur. Since the stress intensity factor only represents the stress state in the eigen-expansion series solutions as r → 0, a fundamental understanding of the fracture behavior requires an accurate analysis of the crack tip field over the finite distance from the crack tip. For the fiber-dominated failure mode and a self-similar crack extension, it is postulated that crack extension will take place when the normal stress or normal strain over a critical distance rc directly ahead of the crack reaches a critical value σc or εc . rc is a material characteristic parameter. If the stress or strain field at the finite distance rc can be correctly represented by the first two terms of the eigen-expansion series solutions, the two-parameter failure criterion [σc , rc ] can be transformed into crack-driving forces [KQ , g32 ] KQ = (2π rc )1/2 (σc − 1.5g32 rc1/2 );

(56)

the two-parameter failure criterion [εc , rc ] results in crack-driving forces [KQ , T ] 0 0 0 0 1/2 KQ = (2π rc )1/2 (εc − T )/[s22 + s12 (s22 /s11 ) ].

(57)

For the CCT and SENT specimens which exhibit loose constraints, using a linear curve fitting technique, the two-parameter failure criteria are determined as [σc , rc ] = [85.8 Ksi, 0.117 in.] and [εc , rc ] = [0.00684, 0.0782 in.] shown in Figure 6. However, the failure criteria fail in the high constraint regions for CT specimens. This implies with limited experimental data that the two-parameter failure criteria seem to characterize the fracture strength with different geometries under loose constraints. 7. Conclusions This paper has summarized that the use of foregoing results to quantify the fracture criterion and crack turning is inconceivable by the given the currently available limited experimental data. However, the results, although limited, indicate that fracture parameters do provide some qualitative trends in predicting the fracture behavior of stitched warp-knit fabric composites. Several summaries and recommendations drawn from the pilot study are: (1) The applicability of using continuum-based fracture parameters in LEFM requires the heterogeneous damage zone at fracture to be sufficiently smaller than the macroscopic crack


93

Figure 6. Variation of KQ with T -stress and the coefficient of the second term σy , g32 .

length and specimen dimensions. For the composite with a crack perpendicular to the 0◦ fiber direction with the given laminate thickness and the failure is fiber-dominated and crack extends in a self-similar manner, the average KQ value from large CT specimens is slightly smaller than that in CCT specimens. The average KQ from SENT specimens gives a higher value. The stress gradients induced by the bending moments across the uncracked ligament may be a cause of this trend. (2) For the non self-similar crack extension observed in small CT specimens (CT-∗∗ ) where the crack kinked and extended parallel to the applied loading direction, the average critical stress intensity factor is smaller than the crack extension in a self-similar manner for large CT specimens. From the Tables, the magnitudes of average tensile T -stress, β, and negative g32 are greater than those of large CT specimens. The high constraint due to large tensile T-stress and large magnitude of negative g32 , term may be expected to inhibit the crack extension in the same plane and promote crack kinking. Further, among the small CT specimens, the greater T and less g32 exhibit higher KQ . The prediction of crack kinking direction determined by a mutual competition between the direction of maximum driving mechanical force and the weakest (anisotropic) material resistance path under Mode-I loading has yet to be quantified. The stress field near the crack tip with small kinks with different angles may provide a valuable insight in describing the fracture behavior (Yang and Yuan [19]). (3) From the testing of CCT specimens, the failure mechanisms are influenced by the laminate thickness. The progressive damage relieves the stress in the crack tip region and renders calculation of KQ based on LEFM invalid. The failure mechanisms caused by less constraint in the thickness direction alleviate the stress intensification near the crack tip and result in elevated apparent KQ . A three-dimensional study in quantifying another constant stress term in the out-of-plane direction may explain the thickness effect. (4) As expected, critical Mode-I stress intensity factor is strongly dependent on the material orientation in composites. In the case of the crack parallel to the 0◦ fiber direction, the KQ values are considerably lower. (5) The mixed mode fracture criterion and the criterion for direction of crack extension under mixed mode loading in composites remains to be established.

94


(6) With limited experimental test data, for the failure being fiber-dominated and crack extension in a self-similar manner, two sets of two-parameter failure criteria, [KQ , T ] and [KQ , g32 ] provide a good correlation for the CCT and SENT specimens, but not for the high constraint CT specimens. Acknowledgements The authors wish to acknowledge the financial supported provided by NASA Grant 98-0548 from NASA Langley Research Center under Advanced Composite Technology Program. The authors thank Mr. F. Liang for calculating and drawing some of the figures. The helpful discussions with the contract monitor, C. C. Poe, Jr., are also acknowledged. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

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