Concepts for bridged Mode ii delamination cracks

\ PERGAMON

Journal of the Mechanics and Physics of Solids 36 "0888# 0154Ð0299

Concepts for bridged Mode II delamination cracks Roberta Massaboa\\ Brian N[ Coxb a

Department of Structural and Geotechnical En`ineerin`\ University of Genova\ Via Montalle`ro 0\ 05034 Genova\ Italy b Rockwell Science Center\ 0938 Camino Dos Rios\ Thousand Oaks\ California\ 80259\ U[S[A[ Received 0 December 0886^ received in revised form 17 September 0887

Abstract Limiting cases and length scales are detailed for Mode II delamination cracks bridged by through!thickness reinforcement[ Analytical results are found for two limits] a steady!state con_guration indicative of noncatastrophic failure and a small!scale bridging con_guration indicative of catastrophic failure[ General large!scale bridging conditions are studied numeri! cally using bending theory for anisotropic plates[ The e}ects of the mechanical properties of the laminate and the reinforcement\ notch length\ and plate thickness on the transition between the two limiting con_gurations\ notch sensitivity and mechanical behavior are analyzed[ All of these e}ects can be expressed succinctly in terms of a few length scales which are material! structure parameters involving the plate thickness[ Þ 0888 Elsevier Science Ltd[ All rights reserved[ Keywords] Delamination^ Crack propagation and arrest^ Fracture toughness^ Layered material^ Strength! ening mechanisms^ Polymeric material^ Asymptotic analysis^ Stitched laminates

0[ Introduction Twenty years of experiments and theory have demonstrated that through!thickness reinforcement greatly improves the damage tolerance of composite laminates "see Drans_eld et al[ "0883# for a review and Dickinson "0884# for a bibliography#[ Means of reinforcing through the thickness include stitching or weaving continuous _ber tows and inserting discontinuous _brous or metal rods[ Adequate through!thickness reinforcement prevents the unstable growth of delaminations created by impact\

 Corresponding author[ Tel[] 28 909 242 1845^ fax] 28 909 242 1423^ e!mail] massaboÝscostr[unige[it 9911Ð4985:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reserved PII] S 9 9 1 1 Ð 4 9 8 5 " 8 7 # 9 9 0 9 6 Ð 9

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loading a part containing a free edge\ or during fabrication[ Through!thickness reinforcement bridges the delamination cracks\ compensating for the poor intrinsic interlaminar fracture toughness of the laminate[ While in!plane sti}ness\ fatigue life and strength might be degraded\ compressive strength after impact\ work of fracture\ ductility and notch insensitivity are much enhanced[ In composite laminates\ delaminations are con_ned to the matrix!rich zones between plies\ which are low!toughness fracture paths[ Thus\ propagation is controlled by the interlaminar fracture toughness and the crack path is known in advance[ The crack path is not such as to maintain pure Mode I conditions at the crack tip\ as it is in a homogeneous solid^ the mode ratio depends on the loading con_guration[ Bridged delamination cracks can be studied by a bridged crack model[ The action of the reinforcement is represented as distributed tractions acting on the crack faces and opposing their relative displacement[ The bridging tractions shield the crack tip and reduce the crack driving force[ An assigned bridging law links the bridging tractions to the crack displacement[ The bridging law can be inferred from test data or from micromechanical models[ It depends on the mechanism of load transfer from the through!thickness reinforcement to the surrounding material[ While a linear bridging law is probably a good approximation for stitches bridging Mode I cracks "Cox\ 0881^ Shu and Mai\ 0882^ Cox\ 0883^ He and Cox\ 0887#\ the form of the law for Mode II delamination cracks is still a subject of study[ It might generally be nonlinear[ In this paper\ plane strain crack growth is examined in a thin anisotropic laminate plate\ symmetric about the mid!plane[ The plate is loaded by uniform applied shear tractions\ t\ acting along the crack faces "Fig[ 0#[ Since the crack propagates along the mid!plane of the plate\ antisymmetry exists and the crack tip stress _eld is pure Mode II[ The bridging mechanisms are represented as nonuniform bridging shear tractions\ tb\ acting along the bridged portion of the delamination[ Figure 0 shows the most general edge delamination problem in a plate under shear loading\ which will be the subject of this study[ Solutions for particular specimens can be constructed from it by superposing a trivial solution for a plate containing no crack or notch and subject to external loads "Bueckner\ 0868#[ The applied shear

Fig[ 0[ The general problem of a delaminated plate loaded in shear[

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stress\ t\ would then be proportional to some other measure of the external load[ Solutions for the plate will also be reasonable approximations for wide beams\ e[g[ three or four point bending beams\ which satisfy d Ł 1h\ where d is the beam width "the dimension normal to the plane of Fig[ 0#[ Applicability to the End!Notched Flexural specimen\ which is frequently used to test Mode II delamination in stitched and non!stitched laminates\ is discussed in Appendix A[ In particular\ the question of whether delamination cracks propagate in Mode II in stitched laminates and the in~uence of friction between the crack surfaces will be addressed through supporting experimental and theoretical results[ Through!thickness reinforcement can radically change delamination fracture[ Nei! ther linear elastic fracture mechanics nor small scale yielding conditions generally apply and possible growth characteristics are therefore quite complex[ However\ very useful parallels with bridged Mode I cracks in large specimens allow fracture regimes to be classi_ed fairly simply[ This paper develops concepts and scaling properties for the Mode II case[ Progress is based on recognizing two limiting crack con_gurations\ namely the small!scale bridging limit "SSB# and the steady state ACK limit "after Aveston et al[\ 0860#[ A numerical model is formulated for general large!scale bridging conditions\ where neither of the two limits prevails and the length of the bridged zone and the structural response are strongly a}ected by geometry[0 The formulation is based on a bending theory developed by Whitney and Pagano "0869# for anisotropic plates\ whose sim! plicity allows scaling properties to be deduced in analytical form[ The same theory has been applied by Jain and Mai "0883# to analyze the ENF specimen in the case of intact stitches with a linear bridging law[ Here e.cient solutions are sought for general\ nonlinear bridging laws in _nite specimens[ The numerical model has been applied to stitched polymer matrix laminates by Massabo et al[ "0887#[

1[ Large!scale bridging conditions Consider the anisotropic laminate plate of depth 1h and length 1L shown in Fig[ 0[ A delamination crack of length\ a\ subject to uniform shear tractions\ t\ lies on the mid!plane[ A system of Cartesian coordinates xÐyÐz is introduced\ with the plane xÐ y the mid!plane of the upper delaminated region[ The laminate is assumed to be symmetric about its mid!plane "z  −h:1# with a Ł h and L Ł h[ Distributed shear tractions\ tb "w#\ applied along the crack surfaces\ represent bridging mechanisms in a continuous approximation[ Bridging will be supplied primarily by the through! thickness reinforcement\ but friction and cross!over _bers peeling o} the matrix may also contribute "see Appendix A#[ The parameter w represents half the crack sliding displacement[ The length of the unbridged portion of the delamination\ e[g[ an initial

0

Large scale bridging most generally refers to cracking with a bridging zone that is not much smaller than the crack length\ notch size\ or specimen dimensions[ Thus it signi_es the absence of scaling according to LEFM[ Large scale bridging conditions include the ACK limit as a special case[

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notch\ is a9[ Figure 1a shows an enlargement of the area around the delamination after deformation[ Chatterjee "0880# analyzed this laminate problem in the absence of bridging\ tb  9[ His model was extended to include linear bridging tractions\ tb "w#  tbi¦bw\ by Jain and Mai "0883\ 0884#[ More general solutions are sought here[ A state of plane strain is assumed in the plane xÐz[ To ensure that plane strain exists\ planes normal to the y!axis must be planes of elastic symmetry\ both in the laminate and in the sublaminates formed by the delamination crack[ Laminates satisfying this restriction exactly include multiple isotropic laminates and multiple specially orthotropic laminates "comprising arbitrarily ordered layers whose principal material directions align with the laminate axes\ e[g[ 9:89> laminates#[ Laminates satisfying them as a good approximation include various many!layered laminates\ such as regular\ symmetric angle!ply laminates "comprising plies of equal thickness arranged in pairs around a symmetry plane\ with the members of each pair having principal axes in the same orientation#[ However\ the approximation is good only if the number of layers is high "Jones\ 0864#[ The problem is analyzed by _rst order shear deformation theory for anisotropic laminated plates in the case of cylindrical bending "Whitney and Pagano\ 0869#[ The laminate plies are treated as orthotropic laminae\ with the axes of material symmetry being parallel and orthogonal to the _ber direction in each ply[ The material is assumed linear!elastic[ The through!thickness stress\ sz\ is neglected[ The in!plane displacements "x!component# are assumed to vary linearly and the through!thickness displacements "z!component# to be constant through the thickness[ Thus linear elements that are perpendicular to the mid!surface remain straight and inextensible during deformation\ but they do not necessarily remain perpendicular to the mid! surface[ Only small deformations are considered[ Two di}erent regions of the laminate must be examined[ In the delaminated region\ 9 ¾ x ¾ a\ the plate is represented as the assemblage of two disbonded sublaminates\

Fig[ 1[ Schematic of the laminate plate in general large!scale bridging conditions[

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placed above and below the mid!plane[ Displacement and stress _elds above and below the mid!plane are antisymmetric and solutions need only be derived explicitly for one sublaminate\ solutions for the other following by symmetry[ In the bonded region\ a ¾ x ¾ 1L\ the entire thickness of the laminate plate is examined[ The solution for the whole plate is then assembled by imposing continuity between the bonded region and the two sublaminates at x  a[ The displacement _eld of the upper sublaminate in 9 ¾ x ¾ a is characterized by the in!plane displacement\ ud9 \ the bending rotation\ 8d\ and the transverse "out!of! plane# displacement\ nd\ on the mid!plane of the sublaminate "z  9#[ The crack sliding displacement\ w\ is given by w  ud9 −8d h:1[ The stress _eld is described by the stress resultants over a unit width] normal force\ Nd\ bending moment\ Md\ and shear force\ Qd "Fig[ 1b#[ In the bonded region\ a ¾ x ¾ 1L\ the generalized displacements ui9 \ 8i and ni and the stress resultants Ni\ Mi and Qi refer to the laminate mid!plane "z  −h:1#[ Following plate theory\ the tractions t and tb acting along the crack are replaced by a combination of axial tractions and bending moments\ which depend on x alone and act along the mid!planes of the sub!laminates "Fig[ 1b#[ With partial di}erentiation denoted by a comma\ the constitutive equations are] 9 Nd "x#  Ad ud\x ¦Bd 8d\x

8

9 Md "x#  Bd ud\x ¦Dd 8d\x

8

9 Ni "x#  Ai ui\x

if 9 ¾ x ¾ a\ Mi "x#  Di 8i\x

Qd "x#  Kd "nd\x ¦8d #

if a ¾ x ¾ 1L\

Qi "x#  Ki "ni\x ¦8i # "0#

and the equilibrium equations are]

8 8

Nd\x "x#  t−tb "w#x

Md\x "x#  −9[4h"t−tb "w#x#¦Qd

if 9 ¾ x ¾ a\

Qd\x "x#  9 Ni\x "x#  9

Mi\x "x#  Qi

if a ¾ x ¾ 1L\

"1#

Qi\x "x#  9

where x is the unit step function "x  9 for x ³ a9 and x  0 for x − a9#^ Ad\ Dd and Kd are the axial\ bending and shear sti}ness of the upper sublaminate\ which depend on the stacking sequence of the laminae^ and Bd is the axial!bending coupling sti}ness\ which is zero only in symmetric sublaminates[ Ai\ Di and Ki are the sti}ness com! ponents of the laminate in the bonded region[ The sti}ness terms are elaborated in Appendix C[ For a specially orthotropic homogeneous laminate\ Ad  Eh\ Dd  Eh2:01\ Kd  4Gxzh:5 and Bd  9\ where E  Ex:"0−nxynyx#\ Ex is the axial Young|s modulus\ nxy and nyx are Poisson|s ratios in the plane xÐy and Gxz is the shear modulus in the plane xÐz[ The sti}ness terms for the bonded region are obtained from these by substituting 1h for h[

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Equations "0# and "1# de_ne a boundary!value problem in terms of the displacement variables[ The boundary conditions consist of] Nd "9#  Md "9#  9\ nd "9#  9^ Mi "1L#  9\ ui9 "1L#  ni "1L#  9^ ¦ ¦ ¦ Nd "a− Md "a− Qd "a− 9 #  Nd "a9 #\ 9 #  Md "a9 #\ 9 #  Qd "a9 #\ 9 ¦ ¦ ¦ 8d "a− nd "a− ud9 "a− 9 #  ud "a9 #\ 9 #  8d "a9 #\ 9 #  nd "a9 #^

nd "a#  ni "a#\ 8d "a#  8i "a#\ ui9 "a#  ud9 "a#−8d "a#h:1  9^ Ni "a#  9\ Qi "a#  1Qd "a#\ Mi "a#  1Md "a#¦Nd "a#h[ The last follow from antisymmetry "N dl  −N ud \ Q dl  Q ud and M dl  M ud \ where u and l denote the upper and lower sublaminate# and the equilibrium conditions at x  a\ i[e[ Ni  N ud ¦N dl \ Qi  Q ud ¦Q dl and Mi  M ud ¦ M dl ¦"N ud −N dl #h:1[ Since t and tb are self!balanced\ Qd and Qi are always zero in the problem of Fig[ 0\ which is therefore independent of the shear sti}nesses\ Kd and Ki[ The shear sti}ness enters the problem only when particular loading con_gurations are analyzed "see Appendix A#[ 1[0[ Linear brid`in` law and intact li`aments A closed form solution for linear bridging\ tb "w#  bw\ and intact ligaments is due to Jain and Mai "0883#[ They evaluated the critical shear stress for crack propagation\ t  tcr\ by applying Gri.th|s energy criterion\ GII  GIIC\ where GIIC is the intrinsic interlaminar fracture energy "in the absence of bridging mechanisms# and GII the strain energy release rate\ GII  −dW:da\ with W the total potential energy per unit width[ Jain and Mai|s expression for tcr is shown here as a standard for accuracy] zGIIC b cosh tcr  sinh

0X

0X

A Þb "a−a9 # h

1

\

"2#

1 X

Þb Þb A A "a−a9 # ¦a9 h h

ÞA Þd is given in eqn "C[4#[ If tb "w#  9\ where the sublaminate sti}ness function A then a  a9\ since an unbridged crack is equivalent to a sharp notch[ If the laminate Þ  3:E and tcr  zGIIC Eh:1a\ is also specially orthotropic and homogeneous\ then A in concurrence with classical beam theory "Carlsson and Gillespie\ 0878#[ It is worth comparing eqn "2#\ obtained via bending theory with only three dis! placement variables\ u"x#\ n"x# and 8"x#\ and neglecting the stress singularity at the crack tip\ with rigorous elasticity solutions[ For an isotropic homogeneous material\ the Mode II crack tip stress intensity factor in a short plate of depth 1hP\ with an edge crack of length a subject to uniform shear stresses\ t\ was found by Fett and Munz

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"0883# via the boundary collocation method\ KII 3 tzpað0[573"a:hP # 1 ¦0[01043 Ł 9[14 [ The crack propagation condition resulting from setting KII  zGIIC E is virtually identical to eqn "2# in the absence of bridging as long as hP ³ 9[4a[ For a homogeneous specially orthotropic laminate\ analogous conclusions hold[ In this case however\ the range of validity of bending theory is also a}ected by the anisotropy ratio Ex:Ez\ Poisson|s ratios nxz and nzx\ and the shear modulus Gxz "which do not enter the one!dimensional solution#[ Finite element calculations on a typical non!stitched\ unidirectional graphite!epoxy composite "Ex:Ez ½ 09\ zEx Ez :1Gxz −znxz nzx ½ 2[4# show that the strain energy release rate of the ENF specimen deviates from the predictions of bending theory by at most a few percent provided h ³ 9[0a "He and Evans\ 0881#[ Commensurately small variations ensue in predictions of tcr[ Thus applying bending theory to the fracture analysis of thin polymer!matrix laminates with a long pre!crack is validated[ Jain and Mai "0883# also examined the conditions for stitch failure[ They assumed that\ after the _rst stitch failure at x  a9\ the bridged zone simply translates forward with a constant size and the contribution from the stitches to the composite fracture energy becomes constant[ Complete analysis for large!scale bridging will contradict these assumptions "see also Suo et al[\ 0881#[ 1[1[ Nonlinear brid`in` law A numerical analysis is presented here for the case of nonlinear bridging stresses\ with tb9 the peak value and w9 the critical sliding displacement beyond which tb vanishes "Fig[ 1a#[ Equations "0# and "1# are then nonlinear\ but they can be solved conveniently as follows[ A grid of points is de_ned along the bridged crack\ "x#  "x0\ [ [ [ \ xn# T\ where x0  a9 and xn  a[ The corresponding vectors of the undetermined crack sliding displacements and bridging stresses are denoted "w#  "wb0\ [ [ [ \ wbn# T and "tb#  "tb0\ [ [ [ \ tbn# T\ respectively[ The continuous bridg! ing stress\ tb ðw"x#Ł\ is represented by a set of n−0 cubic splines\ "fk "x#\ k  0\ [ [ [ \ n−0#\ which are functions of position along the crack] 2

fk "x#  s alk "x−xk # l

xk ¾ x ¾ xk¦0 \

"3#

l9

with fk "xk#  tbk[ The 3"n−0# coe.cients\ alk\ are related to the bridging vector\ "tb#\ by "a#  ðQŁ −0 "tb#\ the coe.cients of ðQŁ being _xed once the grid is _xed "Cox and Marshall\ 0880a#[ The undetermined coe.cients alk are calculated by iteration[ Given an estimate of the crack pro_le\ "w# "i#\ for the ith iteration\ the corresponding vector of the bridging tractions\ "tb# "i#\ and the coe.cient\ "a# "i#\ become set[1 With "tb# "i# temporarily _xed

1

Rationales for choosing the initial guess\ "w# "0#\ for "w# and schemes for iterating to self!consistency can be found in Cox and Marshall "0880a#[ For the problems solved here\ convergence is always found very easily^ the initial guess and iterative scheme are unimportant[

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and the bridging tractions given by eqn "3#\ eqns "0# and "1# reduce to the following linear equations for the displacement variables]

0

1

n−0 2 0 Bd h¦1Dd F 9 u t− s s alk"i# "x−xk # l xk d\xx "x#  G 1 Ad Dd −B d1 k0 l9 G n−0 2 g 0 Ad h¦1Bd t− s s alk"i# "x−xk # l xk G8d\xx "x#  − 1 1 Ad Dd −B d k0 l9 G fn "x#  −8

0

d\x

1

9 ¾ x ¾ a\

d

8

9 ui\x "x#  9

8i\x "x#  9 a ¾ x ¾ 1L\

"4#

ni\x "x#  −8i

where xk  0 if xk ¾ x ³ xk¦0 and xk  9 elsewhere[ The boundary conditions are as stated above\ along with continuity conditions for stress resultants and displacements at x  xk "k  0\ [ [ [ \ n#[ The vectors "w# "i¦0# and "tb# "i¦0# for the next iteration follow immediately[ Iteration continues to self!consistency in the generalized displacements\ when the coe.cients a0k are determined[ The stress resultants\ generalized displacements\ and bridging stresses are recorded as functions of the applied shear stress\ t\ and crack and notch lengths\ a and a9[ The applied shear stress at the onset of crack propagation\ t  tcr\ is determined most accurately and conveniently by an application of the J!Integral "Rice\ 0857^ Jain and Mai\ 0884#] J

g0 G

U dz−T =

1

1u ds \ 1x

"5#

where G is a path circumscribing the crack tip\ U is the strain energy density\ T the surface traction vector\ and u the displacement vector[ The J!Integral evaluated along the closed path shown with dashed lines in Fig[ 1a must be equal to zero] J0 ¦J1 ¦Jb ¦Jtip  9[

"6#

The term J0 is calculated along the traction!free upper and lower boundaries of the plate and is therefore zero "T  dz  9#[ The term J1 is calculated along the vertical end boundaries of the plate\ where T  U  9[2 The term Jtip is evaluated along a 2 In plane strain\ the only potentially non!zero stress!strain product along these boundaries is that between the through!thickness stress and strain\ sz and oz[ In thin plate theory\ sz is assumed zero^ and therefore as far as plate theory is accurate\ J1 vanishes[ In more complete descriptions of the stress _elds\ through!thickness stresses and strains may be signi_cant near a free edge in a laminate comprising anisotropic plies laid up with di}erent orientations[ However\ they only extend a distance from the free edge similar to the laminate thickness[ From Saint Venant|s principle\ such edge stress _elds\ which possess zero resultant\ cannot signi_cantly a}ect the crack tip stress _eld when a Ł h[ Therefore\ their contribution to J must be negligible[

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path segment circumscribing the crack tip and is equal to the strain energy release rate\ GII "Rice\ 0857#[ The term Jb is the contribution from the path along the crack surfaces] Jb  1

g

wb0

tb "w# dw−1twcmsd \

"7#

9

in which wcmsd  =w"x  9#=\ wbl  =w"x  a9#= and the factor of 1 accounts for the two surfaces of the crack[ The integration in eqn "7# is performed numerically[ At the onset of crack propagation t  tcr and Jtip  GII must equal the delamination fracture energy\ GIIC[ The bridging reinforcement _rst fails when wb0 0 =w"x  a9#= equals w9 "Fig[ 1a#[ With further crack growth\ the unbridged crack propagates and the length of the bridging zone is found by maintaining the condition =w"x  ar#=  w9\ with ar the new length of the unbridged crack "a9 : ar in the governing equations#[ The inverse formulation of the numerical model can be used to deduce the bridging law\ tb "w#\ from experiments[ Massabo et al[ "0887# applied this method to charac! terize stitched ENF specimens[

2[ Limiting con_gurations 2[0[ The mode II ACK limit Consider the problem of Fig[ 0 in the absence of a notch\ a9  9\ so that both t and tb act along the entire crack[ Assume that tb "w# is an increasing function\ at least over some interval of w[ The power law tb "w#  bwa will be used as a prime example\ although the concepts to be illustrated are applicable to much more general functions[ When it is long enough\ such a delamination crack can reach a limiting con_guration analogous to the ACK limit in Mode I cracks "Aveston et al[\ 0860^ see Appendix B for a summary of concepts for Mode I cracks#[ The bridging tractions\ tb\ and crack sliding\ w\ increase monotonically with distance behind the crack tip up to the crack mouth and approximate the asymptotic values t and wt  tb−0 "t# over most of the crack^ i[e[ the bridging tractions and the applied load approach equilibrium in the far crack wake "Fig[ 2#[ Then the critical applied shear stress for crack propagation approaches a constant value\ tcr  tACK\ independent of crack length[ This is the Mode II ACK limit[ The steady state stress\ tACK\ is easily evaluated via the J!Integral "Rice\ 0857^ Marshall and Cox\ 0877#[ The J!Integral evaluated along the closed path shown in Fig[ 2 is given by eqn "6#\ with J0  J1  9 and Jtip  GII[ The term Jb is now Jb  −1

g

wt

ðt−tb "w#Ł dw[

9

Substituting eqn "8# into eqn "6# yields

"8#

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Fig[ 2[ Schematic of the laminate plate in steady!state conditions "ACK limit#[

$ g

wt

GII  1 twt −

% g

t

tb "w# dw  1

w"t# dt\

"09#

9

9

i[e[ the strain energy release rate\ GII\ equals the complementary energy of the ligaments "Fig[ 2#[ For power law bridging GII 

1 a 0¦a t a [ b 0¦a

"00#

The crack growth criterion\ GII  GIIC\ leads to the steady!state crackin` stress\ tACK\ and the corresponding sliding displacement\ wACK  tb−0 "tACK #]

$

0¦a tACK  GIIC b 1a

$

0 a

wACK  GIIC b

−0

a 0¦a

%

0 0¦a

%

0¦a 1a

\

[

"01#

The ACK limit is also attainable when a notch exists "a9  9# provided the bridging ligaments are strong enough to survive the stress concentration at the notch tip[ When a Ł a9 and tb ¼ t over most of the wake\ the net tractions acting on the fracture plane are signi_cantly di}erent from zero only near the crack tip and over the notch itself[ Consideration of weight functions "e[g[ Tada et al[\ 0874# shows that the e}ect of the notch on crack tip stress intensity becomes insigni_cant as the crack becomes very large^ while the e}ect of the net tractions near the crack tip becomes constant[

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Therefore\ the steady state shear stress\ tACK\ will take the same value as in the absence of a notch[ In other words\ tACK is a material property[ 2[0[0[ Noncatastrophic brid`in` len`th scale 2[0[0[0[ Len`th scale for linear brid`in`[ The noncatastrophic length scale in Mode II will be introduced for linear bridging\ tb "w#  bw[ In this case\ tACK given in eqn "01# is the limit for large bridged crack lengths\ a Ł a9\ of tcr given in eqn "2#[ Normalized with respect to tACK\ eqn "2# becomes cosh tcr tACK

 sinh

0

0

1

a−a9

zaIIm h

1

a−a9

zaIIm h

¦

a9

[

"02#

zaIIm h

Thus tcr:tACK : 0 as "a−a9# :  whatever a9\ con_rming that the steady state stress is una}ected by a notch[ Crack growth when the bridging ligaments remain intact is then controlled by the product of two parameters] the noncatastrophic len`th scale\ aIIm\ given by aIIm 

0 bA Þ

"03a#

and the depth of the plate\ h[ The length scale aIIm is a material constant "see eqns "C0#Ð"C4##[ For a homogeneous orthotropic material\ A Þ  3:E and aIIm 

E [ 3b

"03b#

The product lACK  zaIIm h

"04#

is a material!structure property\ which is proportional to the crack advance required to approach the steady state con_guration[ 2[0[0[1[ Len`th scale for power law brid`in`[ An analytical expression for the non! catastrophic length scale cannot be de_ned for a generic bridging law\ as a closed form solution of tcr is not achievable[ Nevertheless\ comparing tACK for Mode I and Mode II cracks\ eqns "B0# and "01#\ and the corresponding length scales\ eqns "B1# and "03a#\ for linear bridging laws suggests the following de_nition for power law bridging]

0

0 0¦a aIIm  GIIC A Þ 1a

0−a 0¦a

1

−1

b0¦a \

which reduces to eqn "03a# for a  0[

"05#

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The utility of lACK can be illustrated by calculations using the numerical formulation of Section 1 of the critical shear stress for various cases[ Figure 3 shows crack growth curves in the normalized form\ tcr:tACK vs a:lACK\ with tACK given by eqn "01# and lACK by eqns "04# and "05#\ for a very long specimen in which the bridging ligaments remain intact[ Figure 3a shows results for a linear bridging law and various notch sizes\ a9[ The dotted curve describes the unstable crack propagation of an unbridged laminate[ All of the solid curves\ which are obtained from eqn "02#\ tend to the limit value tcr  tACK\ the Mode II ACK limit[ The ACK limit is reached only after the crack has propagated

Fig[ 3[ Normalized shear cracking stress as a function of the normalized crack length for a laminate with intact through!thickness reinforcement] "a# linear bridging law^ "b# power law bridging[

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over a distance that is typically several times lACK  "aIImh# 0:1[ Thus the ACK limit can be approached in a specimen of _nite size only if its intact length\ L−a9\ is at least several times lACK[ In a specimen of limited length\ L ³ lACK\ bridging e}ects remain negligible and the crack growth curve lies close to that for an unstable\ unbridged crack[ Even in large specimens\ initial crack growth is unstable for smaller notch sizes "those satisfying a9 N 4[9"aIImh# 0:1#[ However\ for all but the smallest notches "those satisfying a9 N 0[9"aIImh# 0:1# the unstable phase of growth is brief and gives way to stable growth through the rest of the specimen[ The stable phase dominates exper! imental observations of noncatastrophic crack growth "Massabo et al[\ 0887#[ Figure 3b shows similar curves for power law bridging\ tb "w#  bwa\ with a  0\ 1 and 9[4[ The length scale\ "aIImh# 0:1\ clearly serves the same role as for linear bridging\ supporting the assignment of eqn "05#[ In fact\ the curves tcr:tACK vs a:"aIImh# 0:1 remain invariant for each a if A Þ\ GIIC\ b\ and h are changed to represent di}erent composite laminates[ Moreover\ when the notch is short "e[g[ a9  9#\ the curves for di}erent values of a collapse approximately onto one curve[ This echoes the behavior of Mode I cracks with power law bridging in in_nite specimens "Cox\ 0882#[ When a notch exists\ the curves for di}erent values of a are no longer quite as close\ but the length scale is still a useful guide to the stability of the crack growth[ It is easy to see that eqn "05# is also relevant to semi!in_nite specimens[ If h:a becomes very large while individual plies remain very thin\ the crack con_guration becomes an edge crack in a semi!in_nite homogeneous medium[ Now the bridged crack problem could always be solved using weight functions\ rather than via cal! culations of energy release rates as in this paper\ and the weight functions for Mode I and Mode II edge cracks in semi!in_nite media have identical forms "Tada et al[\ 0874#[ Therefore\ the noncatastrophic crack evolution for the two modes must scale in the same way in semi!in_nite specimens] for the Mode II crack of Fig[ 0 in the limit h:a : \ aIIm of eqn "05# will characterize crack growth just as am of eqn "B1# controls Mode I bridged cracks "see Appendix B#[ In other words\ for a semi!in_nite "or in_nite# medium\ one would have lACK  aIIm\ and lACK would be a material constant\ rather than a material!structure parameter[ There is no contradiction in having a length scale that depends on a structural measure\ the plate thickness\ for thin plates and one that is a material constant for in_nite specimens[ Plate theory breaks down when h is much greater than the crack length\ a[

2[1[ The Mode II small!scale brid`in` limit Consider again the general shear loading problem of Fig[ 0[ If the crack dis! placement at the notch root\ =w"x  a9#=\ exceeds the critical value\ w9\ the unbridged segment of the crack will extend\ with ar denoting its evolving length[ In a long enough specimen\ the bridged zone size\ a−ar\ will eventually become constant and much smaller than the crack length\ a[ This is the small!scale brid`in` con_`uration[ In this limit\ Linear Elastic Fracture Mechanics applies and the delamination fracture energy of the unbridged composite\ GIIC\ is enhanced by a constant amount\ Gb\ given by

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twice the area under the bridging curve\ as shown in the following crack propagation criterion "Rose\ 0876^ Budiansky et al[\ 0875#]

g

w9

GII  GIIC ¦Gb  GIIC ¦1

tb "w# dw\

"06#

9

where GII is now the strain energy release rate in the absence of bridging] GII 

t1 A Þ a1 \ h

"07#

which follows from eqn "2# with tb "w#  9 and a  a9[ The critical stress in small! scale bridging\ tcr  tSSB\ is tSSB 

0 a

X

h"GIIC ¦Gb # [ A Þ

"08#

Thus stable crack growth can be obtained in this limit only if the applied load is reduced while the crack displacement is controlled directly[ 2[1[0[ Catastrophic brid`in` len`th scale The length of the bridged zone\ a−ar\ is evaluated by imposing the condition\ w  w9\ at the boundary of the unbridged crack segment\ x  ar[ In the small!scale bridging limit\ this length becomes a material!structure property\ lSSB\ which varies as h0:1\ just as does lACK[ The analytical result\ lSSB  "9[4aIIsh# 0:1\ obtains for a rectilinear bridging law\ tb "w#  tb9 "w ¾ w9# and tb "w#  9 "w × w9# "see Appendix D#\ where aIIs is a material constant de_ned as the Mode II catastrophic len`th scale] aIIs  3

w9 tb9 A Þ

0X

0¦

GIIC 1 GIIC − [ Gb Gb

X 1

"19#

In the limit GIIC : 9\ aIIs reduces to w9 \ lim aIIs  3 tb9 A Þ GIIC :9

"10a#

which is an upper bound estimate for aIIs[ Inspection of the scaling properties of the theory of Section 1 shows that lSSB also scales with h0:1 for general bridging laws[ A fundamental di}erence is then evident between Mode I bridged cracks in in_nite specimens\ for which lSSB is a material constant "Appendix B#\ and thin plates\ for which lSSB is a material!structure constant[ A Mode II characteristic length\ lIIch\ is de_ned from eqn "10a# in analogy to the case of Mode I cracks\ eqn "B5# of Appendix B "Hillerborg et al[\ 0865#]

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lIIch  3

Gb 1 tb9 A Þ

0168

"10b#

[

For a homogeneous and orthotropic material lIIch 

Gb E 1 tb9

"10c#

[

It will be shown in Section 4 that for cohesive zone models "GIIC  9#\ the quantity "lIIchh# 0:1 controls the notch sensitivity of ultimate strength of the laminate[ When GIIC  9\ notch sensitivity and other fracture characteristics will depend on both "lIIchh# 0:1 and GIIC:Gb[ Numerical calculations con_rm that\ in analogy to the Mode I case "Smith\ 0878#\ as long as tb "9#  9\ lSSB in the cracked plate under Mode II loading will be given to order of magnitude by lSSB ½ "aIIsh# 0:1\ with aIIs given by eqn "19#[ When tb "9#  9\ two di}erent cases arise[ In the brittle limit\ when the bridging toughness becomes insigni_cant and GIIC:Gb : \ lSSB ½ "aIIsh# 0:1 is still a fair estimate of the bridging zone size\ because in this limit aIIs : 9\ being proportional to Gb:GIIC[ The ratio lSSB:"aIIsh# 0:1 will depend on the shape of the bridging law\ but will always remain of order unity[ For a power law\ tb "w#  bwa with a − 9\ numerical cal! culations show that lSSB  lim GIIC :Gb :

X

0¦a zaIIs h[ 1

"11#

This result is similar to that found analytically for a Mode I crack in an in_nite body "Cox and Marshall\ 0883#[ On the other hand\ if tb "9#  9 and GIIC:Gb : 9\ lSSB will become much larger than "aIIsh# 0:1[ Pursuing the analogy with the Mode I case\ it is appealing to guess that lSSB will be given to a factor of order unity by "lIIh# 0:1\ with lII 

w9 Gb [ tb9 A Þ GIIC

"12#

The dimensionless diagram of Fig[ 4 shows the computed ratio lSSB:lACK "thick curve# as a function of GIIC:Gb for a linear bridging law\ tb "w#  bw with tb "9#  9[ As both lSSB and lACK scale with h0:1\ the same curve is obtained for any plate depth\ h[ In the same diagram\ the two thin curves represent "lIIh# 0:1:lACK and "aIIsh# 0:1:lACK and con_rm the observations of the previous paragraph] when GIIC:Gb :  "GIIC:Gb × 0[4 in this case#\ lSSB  "aIIsh# 0:1\ as predicted by eqn "11#^ when GIIC:Gb : 9\ lSSB is bounded by "lIIh# 0:1[ The thin plate solution is correct only within the range of validity of the bending theory\ i[e[ for small h:a and large lSSB:h "or equivalently large aIIs:h\ which is always the case in laminates reinforced through the thickness\ see Table 0#[ Again\ for h:a : \ the solution of an edge crack in a semi!in_nite sheet must hold and lSSB becomes a material constant^ i[e[ lSSB ¼ aIIs or aIIs ³ lSSB ³ lII as appropriate[ The fact that the equilibrium zone length\ lSSB\ is a material constant for in_nite or semi!in_nite bodies

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Fig[ 4[ Normalized length of the bridged zone in SSB\ lSSB:lACK\ as a function of the normalized interlaminar intrinsic fracture toughness\ GIIC:Gb "linear bridging#[

Table 0 Quasi!isotropic carbon!epoxy laminates Reduced axial Young|s modulus\ E Intrinsic interlaminar fracture energy\ GIIC Individual stitch area\ 1Af

E  49 GPa GIIC  1[9 kJ m−1 1Af  9[53 mm1

Stitching pattern

Laminate L0

Laminate L1

Stitch pitch "stitches per inch in each row0# Stitch row density "rows per inch#

7 7

3 3

ACK shear stress\ tACK "eqn 01# lACK  "aIImh# 0:1 "eqn 04# lSSB "Fig[ 4#

L0 tACK ¼ 05 MPa L0 l ACK ¼ 08 mm L0 l SSB ¼ 25 mm

L1 tACK ¼ 7 MPa L1 l ACK ¼ 26 mm L1 l SSB ¼ 34 mm

0

0 in−0 ¼ 9[93 mm−0[

while it scales with the plate depth\ h\ for thin plates has already been noted by Suo et al[ "0881#[

3[ Transition from noncatastrophic to catastrophic failure Which of lSSB or lACK is greater or smaller in order of magnitude will generally determine whether the crack will ultimately grow towards the ACK limit "non!

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catastrophic failure# or the SSB limit "catastrophic failure# in an unnotched specimen[ If a notch exists\ catastrophic failure will be favored more heavily[ If lSSB ¼ lACK or the specimen is not long enough\ large!scale bridging conditions will prevail\ with neither analytical limit reached[ The numerical solutions of Section 1 are then required[ Some illustrative examples are presented here[ The calculations treat the general problem of Fig[ 0 and focus on the transition from noncatastrophic to catastrophic growth when bridging ligaments fail\ possibly after some crack growth[ Once again\ the specimen length is taken to be very large[ Whether qualitative di}erences would exist in a _nite specimen would depend on how its length\ L\ compares with lACK or lSSB[ The bridging law is assumed linear\ tb "w#  bw[ 3[0[ Numerical examples A quasi!isotropic carbon!epoxy laminate is considered\ with a total of 37 plies in the lay!up ð34:9:Ð34:89Ł ns\ depth 1h  6 mm\ and two representative patterns of glass stitches\ L0 and L1\ inserted in parallel rows according to the details shown in Table 0[ The laminate is approximately transversely isotropic and homogeneous\ with xÐy the plane of isotropy\ so that A ÞA Þd  3:E[ The bridging sti}ness parameter b is written b  P9ns:w9\ with ns the area density of stitches and P9 and w9 the ultimate shear load and corresponding displacement of a single stitch[ Representative values of P9 and w9 are P9  499 N and w9  9[3 mm "Turrettini\ 0885#[ The assumed law could represent stitches uniformly stretching along their lengths[ Thus bL0 ¼ 014 N mm−2 and bL1 ¼ 20 N mm−2 for L0 and L1\ respectively[ The steady state cracking stresses and noncatastrophic length scales\ from eqns "01# and "04#\ are shown in Table 0[ The catastrophic length scales\ deduced from Fig[ 4 for GbL1 :GIIC ¼ 09 and GbL1 :GIIC ¼ 1[4\ are also shown[ Both lSSB and lACK are much greater than the stitch spacing in the two laminates\ validating the representation of discrete stitching by continuous tractions in the two limits[ In general large scale bridging conditions\ the continuous approximation de_nes a response which on average re~ects the actual discontinuous one\ provided the length of the crack is much higher than the stitch spacing "see Carpinteri and Massabo\ 0886\ for Mode I#[ Moreover\ both lSSB and lACK are much greater than the beam depth\ so that when the crack is long enough for bridging to be e}ective\ it will necessarily satisfy a × h as required in plate theory[ Further\ for both laminates\ the specimen length required to reach the ACK limit is large[ Figure 3a shows that\ for an initial notch of the same size as the length scale "aIImh# 0:1 "or 08 mm for laminate L0#\ the ACK limit would be reached only in a specimen of length L × 099 mm[ From this observation and since lSSB and lACK are comparable in the two laminates\ L0 L0 L1 L1 l SSB :l ACK ¼ 0[8 and l SSB :l ACK ¼ 0[1 "see Fig[ 4#\ large scale bridging will usually domi! nate in these materials\ necessitating full numerical solution of the problem[ Experi! ments on typical ENF specimens con_rm the prevalence of large scale bridging conditions in stitched polymer!matrix laminates "Cox et al[\ 0886^ Massabo et al[\ 0887#[

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3[1[ Crack `rowth with li`ament failure Figure 5 shows the e}ects of rupture of the through!thickness reinforcement on tcr and the mode of failure[ The abscissa is the normalized length of the bridged crack\ "a−ar#:"aIImh# 0:1\ with ar  a9 at initiation[ The bridging tractions vanish beyond the critical crack displacement\ w9\ where the bridging traction is tb9  bw9[ The dotted curves mark ligament failure for various values of the ratio\ tb9:tACK[ For each notch size\ crack extension follows the solid curve up to the dotted curve corresponding to the prevailing value of tb9:tACK\ which thus controls the transition from noncatastrophic to catastrophic failure[ Then the behavior is controlled by the dotted curve\ which de_nes the length of the bridged zone for each value of the critical load[ For power law bridging\ tb "w#  bwa\ eqn "01# shows that specifying tb9:tACK is equivalent to specifying GIIC:Gb] a 0¦a

0 1

Gb tb9  a tACK GIIC

[

"13#

Moreover\ when GIIC:Gb is large\ aIIs of eqn "19# reduces to lII of eqn "12#\ so that\ from eqns "05#\ "12#\ and "13#\ 1:a

0 1

tACK aIIm a aIIs tb9

[ "14#

Fig[ 5[ Solid curves] normalized shear cracking stress as a function of the normalized bridged crack length for a laminate with intact through!thickness reinforcement and a linear bridging law[ Dotted curves] loci of ligament failure for various values of the ligament strength[

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Thus specifying tb9:tACK is also equivalent to specifying the relative magnitudes of the length scales\ aIIs and aIIm[ The intercepts of the dotted curves on the horizontal axis de_ne the normalized lengths of the bridging zone in small!scale bridging conditions for the given tb9:tACK[ For small tb9:tACK "small Gb:GIIC#\ these intercepts indicate lSSB ¼ "aIImh# 0:1 = tb9:tACK[ This result in conjunction with eqn "14# leads to lSSB ¼ "aIIsh# 0:1\ again con_rming eqn "11#[ For low values of tb9:tACK "low values of Gb:GIIC or aIIs:aIIm# and large notch sizes\ the dotted curves are almost vertical[ In this case the length of the bridged crack at _rst ligament failure\ a−a9\ approaches the equilibrium value\ lSSB\ and the crack will attain the small!scale bridging limit almost immediately upon ligament failure[ These are the only cases for which LEFM is applicable for analyses of crack growth[ For high values of the ratio tb9:tACK "high values of Gb:GIIC or aIIs:aIIm#\ the dotted curves turn over for increasing crack extension and\ for small notch sizes\ never intersect the solid curves[ Then ligament failure never occurs\ the bridged zone length\ a−ar  a−a9\ increases during crack growth\ and crack growth is noncatastrophic and stable[ For higher notch sizes the dotted curves may intersect the solid curves at some large crack extension\ indicating catastrophic failure[ But the crack still propa! gates in large!scale bridging conditions\ a−ar progressively decreasing as a increases[ Figures 6 and 7 describe crack growth after ligament failure for di}erent values of tb9:tACK[ Figure 6 refers to notch size a9:"aIImh# 0:1  0[9[ It shows that the ACK limit "no ligament failure# is reached for tb9:tACK − 1[9\ indicating signi_cant ductility if the specimen is long enough[ This case applies to the laminate L0 of Table 0\ for which tb9:tACK 3 2[9[ For slightly lower values of tb9:tACK\ the ligaments do fail and the crack grows unstably in large!scale bridging conditions "e[g[ tb9:tACK  0[4#[

Fig[ 6[ Transition from stable to unstable crack growth for decreasing values of the ligament strength in a laminate plate with a notch of length a9:"aIImh# 0:1  0[9 "linear bridging law#[

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Fig[ 7[ Transition from stable to unstable crack growth for decreasing values of the ligament strength in a laminate plate with a notch of length a9:"aIImh# 0:1  1[9 "linear bridging law#[

Laminate L1 "Table 0#\ for which tb9:tACK ¼ 0[5\ would show this response[ For smaller tb9:tACK\ the curves tend to the small!scale bridging limit[ For very small tb9:tACK the response is that for an unbridged composite "dotted curve^ e[g[ the case tb9:tACK  9[14#[ Then noncatastrophic failure is favored by small h and large Gb:GIIC[ Figure 7 refers to notch size a9:"aIImh# 0:1  1[9 and highlights the e}ects of the length of the unbridged delamination on the failure mode[ In the case shown\ unstable response " failed ligaments# is predicted for all of the values of tb9:tACK considered[ Very similar trends are predicted for a square root bridging law\ tb "w#  bw9[4[ In materials in which lACK and lSSB are smaller than in the examples of Table 0\ it may be relatively easy to machine a notch satisfying a9 Ł lSSB and a9 Ł lACK[ Then the noncatastrophic:catastrophic transition could be located approximately by com! paring the limit stresses tACK and tSSB of eqns "01# and "08#] noncatastrophic cracking will occur if tSSB × tACK^ catastrophic if tSSB ³ tACK[

4[ Notch sensitivity and failure modes Fracture mode and strength in brittle materials depend strongly on the size of ~aws "notch sensitivity#[ In the absence of through!thickness reinforcement\ the ultimate shear strength of a laminate is de_ned by the small!scale bridging limit of eqn "08#\ with Gb  9 and a  a9 the size of the ~aw[ The laminate has the notch sensitivity of a brittle solid and its strength falls monotonically with ~aw size[ Mode II crack propagation is unstable and causes complete fracture of the plate[ Through!thickness reinforcement generally improves strength\ reduces notch sensitivity\ and changes the mode of failure[

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Table 1 Two measures of ultimate strength Ultimate strength parameter

Assumed ~aw

"max# tult \ Maximum notched strength "min# tult \ Conservative notched strength

Notch of length a9 Notch a9¦matrix ~aw of unknown length

Two forms of ~aw should be distinguished in discussing notch sensitivity\ viz unbridged and bridged delamination cracks] "0# An unbrid`ed delamination crack is equivalent to a tractionless\ sharp notch\ de_ned formally by a domain of the crack within which tb  9[ An unbridged delamination crack could be a machined notch in a specimen or a region in a structure where the through!thickness reinforcement is either absent because of a manufacturing error or has been destroyed in service\ especially by impact[ For all these situations\ the term {notch| will be used for brevity[ "This is the sense in which notch was used earlier in the paper[# The notch size is usually known[ "1# A brid`ed delamination crack\ which will be referred to here as a matrix delamination ~aw "or simply a matrix ~aw#\ is a delamination crack that remains bridged by intact through!thickness reinforcement[ If a matrix ~aw is the only damage in the laminate\ it can be di.cult to detect and measure[ Conservative design rules for ultimate strength will therefore be based on the assumption that the most damaging possible matrix ~aw is present\ which will usually be one of in_nite length[ The following discussion considers two measures of the notch sensitivity of ultimate strength\ which is the maximum load sustained before a specimen separates into two unconnected halves "see Table 1#[ The _rst is the maximum load sustained if there is no initial matrix ~aw^ i[e[ the notch is the only ~aw[ This will be called the maximum "max# notched strength\ tult [ The second is the maximum load if a matrix ~aw of unknown length\ perhaps the most deleterious length\ is present[ This will be called the con! "min# servative notched strength\ tult [ The applied load may attain the maximum notched strength either prior to or at the onset of ligament rupture\ depending on the notch length\ the shape of the bridging law\ and the ratio Gb:GIIC[ The conservative notched strength always corresponds to the onset of ligament failure[ The shape of the bridging law\ tb "w#\ controls the fracture behavior[ In particular\ if the ACK limit can be approached\ complex behavior ensues[ The consequences for notched strength will be analyzed for di}erent bridging laws[ Numerical solutions will be obtained for e}ectively in_nite specimens\ but are valid for any specimen whose unnotched length is at least several times the maximum of lSSB and lACK[ 4[0[ Notched stren`th for a linear brid`in` law and Gb:GIIC ¾ 0 Figure 8 refers to a composite with a linear bridging law and Gb:GIIC ¾ 0 "e[g[ the toughness due to bridging is relatively small#[ Under these conditions\ tACK of eqn

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Fig[ 8[ Maximum and conservative notched strength as functions of the normalized unbridged delamination "notch# length for laminates with Gb:GIIC  0 "linear bridging law#[

"01# is higher than the ultimate strength of the ligaments\ tb9 "see eqn "13##^ the "min# ligaments break and the ACK limit cannot be reached[ The diagram depicts tult "max# "dashed curves# and tult "solid curves# vs notch length\ a9\ for Gb:GIIC  0[9\ 9[4 and 9[14[ The shear stresses are normalized with respect to tb9\ while a9 is normalized by "lIIchh# 0:1 = "0¦GIIC:Gb# 0:1\ with lIIch de_ned by eqn "10b#[ With this normalization\ the single dotted curve describes the small!scale bridging solution of eqn "08# for all cases[ "min# "max# is bounded by tb9 while tult tends to in_nity\ being controlled by For small a9\ tult the intrinsic interlaminar toughness of the unbridged laminate[ The small circle on each maximum notched strength curve separates distinct regimes of failure for large and small notches[ For small notches\ the maximum stress will be attained prior to ligament failure[ It coincides with the initiation of a delamination crack[ The crack will propagate unstably\ with catastrophic bridging ligament failure requiring no further increase in load[ For large notches\ unstable delamination may again occur\ but ligament failure does require further load increase[ "max# For all notch sizes\ ligament failure on the curves for tult "no pre!existing matrix ~aw# will occur while the delamination crack is still _nite "typically a few times "max# "min# "aIImh# 0:1*see Fig[ 5#[ Thus the curves for tult lie above those for tult ^ the latter correspond to ligament failure when an in_nite matrix ~aw "delamination crack# exists in advance[ The di}erence will be much the same if the pre!existing matrix ~aw is more than ½09 times "aIImh# 0:1\ rather than in_nite\ because the applied load required to fail ligaments at the notch root becomes very nearly independent of matrix crack length for cracks satisfying this bound[

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"max# As Gb:GIIC : 9 "negligible bridging toughness#\ the curves for tult tend to the small!scale bridging solution[ This limit de_nes the highest notch sensitivity "i[e[ the most rapid change of strength with notch size#\ which for applications requiring damage tolerance is the least desirable behavior[ The small scale bridging limit is "max# approached by all of the curves for tult for large notch sizes[

4[1[ Notched stren`th for a linear brid`in` law and Gb:GIIC × 0 If Gb:GIIC × 0\ the response is more complex because the ACK limit can now be reached[ Figure 09 describes the behavior of three plates of in_nite length\ with "min# Gb:GIIC  3\ 7 and 05[ All of the curves for the conservative notched strength\ tult \ fall within the dotted band limited by the cases Gb:GIIC  3 and 05[ For small notches\ "max# the maximum notched strength\ tult \ rises above this band\ as indicated by the solid lines[ Here the applied stress required to initiate a crack exceeds that required sub! sequently to fail bridging ligaments[ For larger notch sizes "i[e[ any notch size to the right of the intersection of the solid line for some value of Gb:GIIC and the conservative "min# "max# and tult coincide[ In this regime\ the stress required notched strength band#\ tult to propagate a matrix crack to in_nite length\ tcr"# \ is less than the minimum stress required to rupture the bridging ligaments at the notch root[ Therefore\ the ultimate notched strength is una}ected by whether or not a matrix ~aw exists in advance] in either case\ it will exist prior to the ultimate load[ The dashed lines in Fig[ 09 show how tcr"# varies\ illustrating the various routes to ultimate failure in the absence of an initial matrix ~aw[ For the smallest notches\ e[g[ "max# the segment AB of the curve for Gb:GIIC  05\ tcr"# is identical to tult [ Peak load

Fig[ 09[ Maximum and conservative notched strength as functions of the normalized unbridged delami! nation "notch# length for laminates with Gb:GIIC × 0 "linear bridging law#[

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coincides with crack initiation and ligament failure follows as the crack length a :  by unstable growth[ Over the segment BC\ initiation is again followed by unstable growth to in_nite a\ but a higher load is then required to fail bridging ligaments] "min# [ Over the segment CD\ crack growth is stable after initiation^ the applied tcr"# ³ tult stress increases to the steady state value\ tACK  tcr"# \ independent of notch size\ as a : [ Further stress increase must then follow before ultimate "ligament# failure[ For the largest notch sizes\ segment DE\ a stable delamination crack is again initiated\ but ligament failure occurs at the notch root while it is still _nite in length[ The small!scale bridging solution "dotted curve in Fig[ 09# is once again an upper bound to the ultimate strength\ but it is signi_cantly nonconservative for all but the largest notches[ For laminate L0 of Table 0\ Gb:GIIC ¼ 09 and "lIIchh# 0:1 = "0¦GIIC:Gb# 0:1 ¼ 39 mm[ Its strength follows a curve similar to that shown in Fig[ 09 for Gb:GIIC  7[ A long enough plate "L × 5"aIImh# 0:1 ¼ 019 mm# with 19 ¾ a9 ¾ 39 mm would sustain the ACK shear stress during most of the crack propagation and then fail at some higher load[ 4[2[ Notched stren`th for a rectilinear brid`in` law When the bridging law does not permit attainment of the ACK limit\ slightly simpler behavior occurs[ Figure 00 shows results for a rectilinear law\ tb "w#  tb9 "w ¾ w9# and tb "w#  9 "w × w9#[ The curves for the maximum notched strength\ "max# tult \ vs notch size show progression between two limits as Gb:GIIC is varied[ The _rst

Fig[ 00[ Maximum and conservative notched strength as functions of the normalized unbridged delami! nation "notch# length "rectilinear bridging law#[

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0178

limit is the small!scale bridging solution "dotted curve# and is approached as Gb:GIIC : "min# \ represented in Fig[ 00 9[ The second limit is the conservative notched strength\ tult by the shaded band of solutions\ which covers the range 0 ³ Gb:GIIC ³ [ For rectilinear bridging\ ultimate failure of a specimen with no initial matrix ~aw always occurs while the crack is still _nite[ The only transition in failure sequence is marked "max# by the circles on the curves for tult ^ for small notches\ the stress for the initiation of a delamination crack will be higher than that for subsequent ligament rupture and vice versa for large notches[ Similar results have been obtained by Bao and Zok "0882# for a bridged crack in an in_nite medium under Mode I loading[ 4[3[ In~uence of the shape of the brid`in` law on notched stren`th and ductility The in~uence of the shape of the bridging law on fracture behavior is further illustrated in Figs 01Ð03[ Figure 01 shows the notched strength\ tult\ for the special "max# "min#  tult #[ The notch case of vanishing intrinsic toughness\ GIIC  9 "in which case tult 0:1 −0 0:1 length is normalized by "lIIch h#  tb9 "3Gb h:A # \ which becomes Þ −0 "Gb Eh# 0:1 for a specially orthotropic homogeneous material "after "lIIch h# 0:1  tb9 Cottrell\ 0852 and Bao and Suo\ 0881\ whose results for notched strength under Mode I conditions were entirely restricted to GIIC  9#[ The three solid curves relate to three di}erent bridging laws\ viz linear tb "w#  bw\ square root tb "w#  bw9[4\ and rectilinear tb "w#  tb9 "w  w9# and tb "w#  9 "w × w9#[ The results are a}ected modestly by the shape of the law\ with variations in strength for the chosen examples always less than 19)[ The parameter "lIIchh# 0:1 controls notch sensitivity[ Notch sensitivity exists when the notch size\ a9\ satis_es a9 n "lIIchh# 0:1[ If a9 ³ 9[4"lIIchh# 0:1\ net section strength will

Fig[ 01[ Maximum and conservative notched strength as functions of the normalized unbridged delami! nation "notch# length for di}erent bridging laws in a laminate with a vanishing intrinsic interlaminar fracture toughness[

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Fig[ 02[ E}ect of the bridging law shape on the notched strength of a laminate with Gb:GIIC  3[

Fig[ 03[ E}ect of the bridging law shape on the shear cracking stress and the ductility of a laminate plate with Gb:GIIC  3\ as a function of the normalized crack length[

be approached "see Appendix B for Mode I#[ The three solid curves of Fig[ 01 also "min# describe the conservative notched strength\ tult \ of laminates with nonvanishing GIIC[ "max# Figure 02 shows the maximum notched strength\ tult \ for the same three bridging

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0180

laws when Gb:GIIC  3[ A rectilinear law leads to a higher normalized strength\ "max# :tb9 \ than those generated by linear or square root bridging[ On the other hand\ tult from the rectilinear to the square root to the linear laws\ ~aw tolerance improves "slower fall o} with increasing notch size#[ Finally\ Fig[ 03 demonstrates the in~uence of the shape of the bridging law on the failure mode and ductility[ The evolution of the critical stress for crack growth\ tcr "a#\ is compared for three laminates characterized by Gb:GIIC  3\ with normalized notch length a9:"lIIchh# 0:1 = "0¦GIIC:Gb# 0:1  9[34[ The diagram highlights the transition from stable fracture approaching the ACK limit "intact ligaments#\ here for the linear bridging law\ to unstable response approaching the small!scale bridging limit in the other two cases[ For the square root and rectilinear laws\ propagation is stable only up to the peak shown in Fig[ 03\ which corresponds to ligament failure and represents "max# tult [ Furthermore\ it is only when the curves of Fig[ 03 approach the dotted curve "SSB limit#\ that the bridged zone length\ lSSB\ approaches its asymptotic value\ which it does from above[

5[ Conclusions Mode II delamination crack growth in a composite plate with large scale shielding provided by through!thickness reinforcement has been analyzed by plane!strain plate theory[ In the limit of plate theory\ the scaling properties of the problem can be identi_ed analytically[ Two limiting con_gurations exist\ the steady!state ACK limit "noncatastrophic cracking# and the small!scale bridging limit "catastrophic cracking#\ with two associ! ated length scales\ the noncatastrophic length scale\ aIIm\ and the catastrophic length scale\ aIIs[ The length scales are analogous to those introduced previously for Mode I cracks[ In particular\ the catastrophic length scale\ aIIs\ is similar when GIIC : 9 to the characteristic length\ lch\ of Cottrell "0852#\ Hillerborg et al[ "0865# and Rice "0857\ 0879#[ For power law bridging\ which may often be realistic\ both aIIm and aIIs can be de_ned analytically[ Their ratio\ aIIm:aIIs\ depends on the ratio of the intrinsic fracture energy of the laminate and that due to the bridging mechanism\ GIIC:Gb[ In in_nite or semi!in_nite specimens\ aIIs approximates the length of the bridging zone in small scale bridging\ lSSB^ and aIIm the crack growth required to approach the ACK limit in noncatastrophic conditions\ lACK[ The magnitudes of the two length scales control the stability of crack growth\ the balance between catastrophic and noncatastrophic growth\ and the notch sensitivity of ultimate strength[ In thin plates\ which are the relevant limit for common stitched laminates\ the qualitative nature of fracture depends not only on aIIm and aIIs\ but also on the depth of the laminate\ h[ The crack growth required to approach the ACK limit\ lACK\ scales as "aIImh# 0:1[ If tb "9#  9 or GIIC:Gb is moderately large\ the equilibrium zone length\ lSSB\ is given approximately by "aIIsh# 0:1^ if tb "9#  9 and GIIC:Gb ¾ 0\ lSSB becomes much larger than "aIIsh# 0:1\ but still scales with h0:1[ In the latter case\ lSSB is bounded by known curves that depend on aIIm and aIIs[ The values of lACK and lSSB can be found for all cases through numerical analysis of the large!scale bridging problem[

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Knowledge of the orders of magnitude of lACK and lSSB prescribes the qualitative behavior expected in fracture[ For an unnotched specimen\ failure will be non! catastrophic and the ACK limit approached if lACK ð lSSB and the plate is long enough^ the small!scale bridging limit "catastrophic failure# will prevail if lSSB ð lACK[ A notch shifts the transition towards catastrophic failure[ Propagation under large!scale bridging characterizes laminates of _xed depth with lACK 3 lSSB\ or specimens whose length is similar to either lACK or lSSB\ where neither limiting solution may be attained[ Estimates for typical stitched laminates show that both lSSB and lACK are tens of mm\ so that experiments with specimens of common size will be conducted under large!scale bridging conditions[ By applying the numerical model\ the e}ects of the length of a pre!existing bridged or unbridged delamination\ of the shape of the bridging law\ and of the ratio Gb:GIIC on the stability of crack growth\ the mode of failure\ and the notch sensitivity have been investigated[ The possible failure sequences leading to ultimate failure have been mapped out[ Maximum and conservative limits for the notched strength have been de_ned[ The single material!structure parameter\ "lIIchh# 0:1\ controls the ~aw depen! dence of the conservative notched strength\ as well as that of maximum notched strength when GIIC : 9 "a commonly held condition in stitched polymer laminates#[ The maximum notched strength depends for general values of Gb:GIIC on "lIIchh# 0:1\ Gb:GIIC\ and the shape of the bridging law[

Acknowledgements R[M[ acknowledges the support of the Fulbright Program\ the Italian Department for the University and for Scienti_c and Technological Research\ and Rockwell Independent Research and Development funding[ B[N[C[ was supported by AFOSR Contract No[ F38519!83!C!9929[

Appendix A[ Delamination cracks in ENF specimens The End Notched Flexure "ENF# specimen is a three!point bending beam with an embedded through!width delamination on its mid!plane "Fig[ A0a#[ Typical ENF specimens are symmetrical about the mid!plane[ Free sliding of the fracture surfaces past one another\ where the loading pin exerts compression\ must be ensured[ Here\ the validity is examined of depicting the ENF test for stitched laminates as a Mode II delamination problem and analyzing it through the superposition of the two plane!strain problems of Fig[ A0b "the trivial problem of a three point bending intact plate# and A0c "the problem of Fig[ 0 examined in Section 1#[ The applied shear stress\ t\ in the problem of Fig[ A0c must be equal but opposite in sign to the stresses generated by the external load P at the mid!plane of the intact plate of Fig[ A0b[ For a specially orthotropic homogeneous laminate\ for instance\ t  2P:7h\ with P the load per unit width[

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Fig[ A0[ The end notched ~exure "ENF# test of "a# is created by superposing the two problems of "b# and "c#[

First\ both stitched and non!stitched _ber!reinforced laminates do indeed delami! nate along the mid!plane in ENF tests "Sharma and Sankar\ 0884^ Carlsson and Gillespie\ 0878^ Cox et al[\ 0886^ Massabo et al[\ 0887#[ Second\ if the specimen is wide enough and the laminate is of a special class\ plane!strain conditions exist[ But the validity of the solution also relies on pure shear existing on the mid!plane ahead of the crack[ Both theory and experiments con_rm this if L Ł h\ a Ł h and the two arms in the delaminated region bend with the same curvature[ Finite element calculations on ENF specimens of specially orthotropic homo! geneous laminates "He and Evans\ 0881# show non!zero compression stresses on the crack plane only very close to the loading point\ so that a negative Mode I stress intensity factor is discounted\ provided 1h ³ a ³ L−1h[ He and Evans further show that the opening displacement is zero along the crack[ They conclude that the delami! nation is a Mode II crack "He\ private communication\ 0886#[ Carlsson and Gillespie "0878# also show that the normal stress\ sz\ is identically zero along the centerline ahead of the crack tip[

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The most detailed observations of stitched ENF specimens appear in two con! temporary papers "Cox et al[\ 0886^ Massabo et al[\ 0887#[ Experimental tests on stitched carbon:epoxy laminates using the ENF geometry and detailed measurement of the sliding and opening displacement pro_les by stereoscopic analysis of optical micrographs showed that the crack tip conditions are pure Mode II[ The opening displacement is undetectable in a region behind the crack tip on the order of ½1Ð2 mm[ In the further crack wake it rises to an approximately constant value due to the propping e}ect of the stitches when they bend plastically within the crack[ Friction e}ects in stitched laminates are con_ned to the zone of low sliding displacement near the crack tip and are approximately invariant during crack growth[ They can be interchangeably included in the bridging tractions\ tb "w#\ or in the interlaminar frac! ture energy\ GIIC[ Processes of microcracking and polymer crazing associated with arrays of S!shaped microcracks whose shape and orientation are such as to create pure Mode I conditions at each of their tips "Fleck\ 0880^ Xia and Hutchinson\ 0883^ Cox et al[\ 0883# may indeed occur in ENF tests[ But they occur at a scale "½09 mm# much smaller than the length of the zone where Mode I opening displacements are undetectable "½0 mm#\ let alone the length of the bridging zone "½09Ð099 mm#[ Thus they need not be explicitly represented in the bridged!crack models developed for macroscale analy! sis of the laminate and the shielding e}ects of stitching[ They may be associated with a stress _eld that propagates invariantly with the delamination crack tip and their e}ect lumped into a material constant characterizing the interlaminar toughness\ i[e[ GIIC[ Massabo et al[ "0887# do indeed _nd GIIC to be independent of crack length over several tens of mm[

Appendix B[ Review of limiting con_gurations for bridged Mode I cracks Consider a Mode I crack in a linear elastic isotropic body with bridging tractions\ p\ related to the crack opening displacement\ 1u\ according to p"u#[ Two limiting con_gurations exist with associated material length scales\ which vary by many orders of magnitude in di}erent material systems "Bao and Suo\ 0881^ Cox and Marshall\ 0883#[ B0[ The ACK limit The _rst limit is the steady!state limit implied "without explicit derivation as a limit# by Aveston et al[ "0860# for a crack in an in_nite medium that is wholly bridged by intact ligaments[ In this ACK limit\ the opening displacement u takes that uniform value in the far crack wake required for the bridging tractions to balance the applied load[ The ACK limit can exist only if the applied load is uniform and p"u# is an increasing function for at least some domain of u[ Such a matrix crack is non! catastrophic*the composite survives its growth across the entire specimen[ The critical applied load for crack propagation in this limit\ the steady!state crackin` stress\ s0\ is independent of a[ The amount of crack growth required to approach the

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limit scales as the noncatastrophic len`th scale\ am[ Both s0 and am are material properties[ For bridging tractions obeying the power law p"u#  bua\

$

0¦a s 0  GC b 1a 0 a

0

a 0¦a

%

pE? 0¦a G am  3 1a C

\

0−a 0¦a

1

"B0# −1

b0¦a \

"B1#

with E?  E in plane stress and E?  E:"0−n1# in plane strain\ E and n being the composite|s Young|s modulus and Poisson|s ratio and GC its intrinsic fracture tough! ness "the fracture toughness of a hypothetical unbridged crack# "Cox\ 0882#[ Once the bridging law is known to be an increasing function\ am can be computed and used as a guide to noncatastrophic fracture[ For the ACK limit to be approached\ the specimen length\ b\ must be somewhat larger than am[ If b ð am\ bridging e}ects will be small[ Further\ crack growth from a notch will be stable if a9 n am and unstable if a9 ³ am[ B1[ The small!scale brid`in` limit The second limit is the small!scale brid`in` limit[ In this limit most bridging liga! ments are broken and the length of the remaining bridged zone becomes constant and small compared to the dimensions of the crack and the body that contains it[ Bridging supplies a constant increment\ Gb\ to the intrinsic composite fracture toughness\ given by twice the area under the bridging law "Rose\ 0876^ Budiansky et al[\ 0875#[ The small!scale brid`in` crackin` stress\ sSSB\ follows from LEFM[ For a crack of length 1a embedded in a uniformly loaded in_nite medium] sSSB 

X

"GC ¦Gb #E? [ pa

"B2#

Since sSSB falls with a\ crack growth is unstable[ The size of the bridged zone\ lSSB\ is given analytically for a rectilinear bridging law\ p"u#  p9 "u ¾ u9# and p"u#  9 "u × u9# "Bao and Suo\ 0881^ Cox and Marshall\ 0883# by lSSB  asp:3\ where as is a catastrophic len`th scale] as 

u9 E? p9

0X

0¦

GC GC 1 − [ Gb Gb

X 1

"B3#

For GC : 9\ as reduces to u9 E? lim as  \ p9 GC :9

"B4#

which is also an upper bound to as for GC  9[ The quantity 1as in this limit may also be regarded as a particular instance " for rectilinear p"u## of the length scale for

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cohesive zone models "i[e[ GC  9# noted on many occasions "Cottrell\ 0852^ Rice\ 0879^ Hillerborg et al[\ 0865#] lch 

Gb E? p91

\

"B5#

where p9 is the maximum of p for all u and the notation lch is Hillerborg|s[ To within a factor of order unity\ lSSB ¼ lch for bridging laws with p"9#  9 when GC  9 "Smith\ 0878#[ When GC  9\ lSSB will decrease[ While extensive results are unavailable for this case\ eqn "B3# is probably representative of the decline of lSSB with GC:Gb in general\ especially for large GC:Gb\ where eqn "B3# gives as  "GC:Gb# −0^ brittle behavior is approached when Gb becomes insigni_cant[ For increasing laws with p"9#  9\ e[g[ for continuous bridging _bers\ lSSB is no longer always similar to as[ When p"9#  9 and GC is small\ a very large bridged zone will emanate from the notch at small applied stresses\ because neither the matrix nor the bridging ligaments o}ers strong resistance[ The equilibrium zone length\ lSSB\ will also be large] it diverges as GC : 9\ being approximately bounded by l 

0 u9 E? Gb [ 3 p9 G C

"B6#

Equations "B3# and "B6# provide rough lower and upper bounds to lSSB for many bridging laws\ with lSSB closer to the lower bound\ eqn "B3#\ as GC:Gb rises "Cox and Marshall\ 0883#[ Small scale bridging conditions can only be achieved if the specimen length\ b\ satis_es b Ł lSSB[ For nonincreasing bridging laws or in materials in which GC is not much smaller than Gb\ the length scale\ as\ is a good indicator of lSSB[ For materials in which p"9#  9 and GC is small\ lSSB is di.cult to estimate accurately] only the bounds of eqns "B3# and "B6# are available[ Full computational solutions may be needed to establish that small scale bridging has been achieved[ B2[ The transition from noncatastrophic to catastrophic crackin` Whether crack propagation across an unnotched and long specimen "b Ł lSSB\ am# will be noncatastrophic "no ligament failure*ACK limit attainable# or catastrophic "extensive ligament failure*small scale bridging limit approached# can be estimated by comparing the orders of magnitude of am and lSSB "Cox and Marshall\ 0883#[ If am Ł lSSB\ failure will be catastrophic^ if lSSB Ł am\ noncatastrophic[ The presence of a large notch will favor catastrophic cracking[ If a9 is large enough "a9 Ł lSSB#\ the transition can be estimated by comparing the ACK limit stress\ s0\ and the small scale bridging stress\ sSSB\ for the given notch size "eqns "B0# and "B2##[ Failure will be catastrophic if s0 Ł sSSB^ noncatastrophic if s0 ð sSSB[ If neither limit can be approached "e[g[ lSSB ¼ am and b ¼ am or b ¼ lSSB# then large! scale bridging conditions prevail and detailed calculations are required[

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B3[ Notch sensitivity of ultimate stren`th When GC  9 and p"9#  9\ e[g[ for rectilinear or decreasing bridging laws in a cohesive zone model\ lch "or equivalently for these cases\ lSSB# controls the notch sensitivity of ultimate strength "Cottrell\ 0852^ Bao and Suo\ 0881#[Notch sensitivity exists when a9 n lch[ If a9 ³ lch\ the net section strength will be approached and the ultimate strength will equal p9[ When GC  9 and p"9#  9\ the controlling scale for notch sensitivity depends on both lch and GC:Gb "Bao and Zok\ 0882#[ Similarly\ when GC  9 and p"9#  9\ notch sensitivity depends on lch and GC:Gb[ For large GC\ notch sensitivity also correlates with lSSB\ since lSSB  lchGb:GC "eqn "B6##[ However\ when p"9#  9 and GC is small\ lSSB becomes very large and can have no role in notch sensitivity[ Nevertheless\ in such cases of diverging bridging zone length\ lch remains a relevant length scale] lch measures a zone near the notch within which bridging ligaments su}er large strains\ while the opening displacements in the long bridged crack far ahead of the notch remain small "Suo et al[\ 0881#[

Appendix C[ Stiffness components for a laminated plate Sti}ness terms over a unit width and related quantities "Whitney and Pagano\ 0869#] m j tj \ A  s Q 00

ðFL−0 Ł\

"C0#

ðFŁ\

"C1#

j0 m j tj zj \ B  s Q 00 j0

0

m

j D  s Q 00 tj zj1 ¦ j0

tj2 \ 01

1

ðFLŁ\

"C2#

m j tj \ K  k s c44

ðFL−0 Ł\

"C3#

j0

A Þ

h0 AD−B

1

0

1

0 Bh0 ¦D¦ Ah01 \ 3

ðF−0 L1 Ł\

"C4#

where m  total number of plies^ tj  thickness of the jth ply^ h0  laminate depth\ m

h0  s tj ^ j0

zj  distance of the centroid of the jth ply from the laminate mid!plane^ j Q 00  plane!stress reduced sti}ness of the jth orthotropic lamina\ referred to the

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j laminate coordinates xÐyÐz "sh  Q hk ok \ for h\ k  0\ 1\ 5\ where s0  sxx\ s1  syy\ and s5  txy#^ j c44  out of plane sti}ness of the jth orthotropic lamina referred to the laminate j ok \ for h\ k  3\ 4\ where s3  syz\ s4  sxz#^ coordinates xÐyÐz "sh  chk k  Mindlin|s shear correction factor[

Appendix D[ Solutions for a rectilinear bridging law For a rectilinear bridging law\ tb "w#  tb9 "w ¾ w9# and tb "w#  9 "w × w9#\ eqns "0# and "1# can be solved in closed form[ The crack sliding displacement\ w  ud9 −8d h:1\ is w−

A Þ ðt "a−ar # 1 ¦t"x1 −a1 #Ł\ if x ¾ ar \ 1h b9

w−

A Þ ðt "a1 −x1 ¦1ar x−1ar a#¦t"x1 −a1 #Ł\ if ar ¾ x ¾ a\ 1h b9

"D0#

where t is the applied shear stress\ x represents the position along the crack and ar is the length of the unbridged crack[ If the crack is growing in small!scale bridging conditions\ then t  tSSB 

0 a

X

h"GIIC ¦Gb # \ A Þ

"D1#

w"x  ar #  w9 \

"D2#

lSSB  a−ar ð a\ L[

"D3#

By assuming x  ar and substituting eqns "D1# and "D2# in eqn "D0#\ under the condition "D3#\ one _nds lSSB 

X

1

w9 tb9 A Þ

0X

0¦

GIIC 1 GIIC − h[ Gb Gb

X 1

"D4#

References Aveston\ J[\ Cooper\ G[A[\ Kelly\ A[\ 0860[ Single and multiple fracture[ The properties of _ber composites[ Conference Proceedings\ National Physical Laboratory\ IPC Science and Technology Press Ltd[\ pp[ 04Ð13[ Bao\ G[\ Suo\ Z[\ 0881[ Remarks on crack!bridging concepts[ Appl[ Mech[ Rev[ 13\ 244Ð255[ Bao\ G[\ Zok\ F[\ 0882[ On the strength of ductile particle reinforced brittle matrix composites[ Acta Metall[ Mater[ 30\ 2404Ð2413[ Budiansky\ B[\ Hutchinson\ J[W[\ Evans\ A[G[\ 0875[ Matrix fracture in _ber reinforced ceramics[ J[ Mech[ Phys[ Solids 23\ 056Ð078[

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Bueckner\ H[\ 0868[ Field singularity and related integral representations[ In] Sih\ G[C[ "Ed[#\ Fracture Mechanics[ I[ Methods of Analysis and Solutions of Crack Problems[ Noordho}\ Leyden\ pp[ 128Ð 203[ Carlsson\ L[A[\ Gillespie\ J[W[\ 0878[ Mode!II interlaminar fracture of composites[ In] Friedrich\ K[ "Ed[#\ Application of Fracture Mechanics to Composite Materials[ Elsevier Science Publishers\ Amsterdam\ pp[ 002Ð046[ Carpinteri\ A[\ Massabo\ R[\ 0886[ Continuous versus discontinuous bridged crack model for _ber! reinforced materials in ~exure[ Int[ J[ Solids Structures 23\ 1210Ð1227[ Chatterjee\ S[N[\ 0880[ Analysis of test specimens for interlaminar mode II fracture toughness\ part 0[ Elastic Laminates[ J[ Comp[ Mater[ 14\ 369Ð382[ Cottrell\ A[H[\ 0852[ Mechanics of fracture[ Proceedings of the Tewksbury Symposium on Fracture[ University of Melbourne\ Australia\ pp[ 0Ð16[ Cox\ B[N[\ 0881[ Fundamental concepts in the suppression of delamination buckling by stitching[ In] Soderquist\ J[R[\ Neri\ L[M[\ Bohon\ H[L[ "Eds[#\ Proceedings of the Ninth DoD:NASA:FAA Con! ference On Fibrous Composites in Structural Design[ Lake Tahoe\ Nevada\ November 0880\ "U[S[ Dept[ Transportation#\ pp[ 0094Ð0009[ Cox\ B[N[\ 0882[ Scaling for bridged cracks[ Mechanics of Materials 04\ 76Ð87[ Cox\ B[N[\ 0883[ Delamination and buckling in 2D composites[ J[ Comp[ Mater[ 17\ 0003Ð0015[ Cox\ B[N[\ Marshall\ D[B[\ 0880a[ Stable and unstable solutions for bridged cracks in various specimens[ Acta Metall[ Mater[ 28\ 468Ð478[ Cox\ B[N[\ Marshall\ D[B[\ 0880b[ The determination of crack bridging forces[ Int[ J[ Fracture 38\ 048Ð 065[ Cox\ B[N[\ Marshall\ D[B[\ 0883[ Concepts for bridged cracks in fracture and fatigue[ Acta Metall[ Mater[ 31\ 230Ð252[ Cox\ B[N[\ Massabo\ R[\ Kedward\ K[B[\ 0885[ Suppression of delaminations in curved structures by stitching[ Composites] Part A 16A\ 0022Ð0027[ Cox\ B[N[\ Dadkhah\ M[S[\ Morris\ W[L[\ Flinto}\ J[G[\ 0883[ Failure mechanisms of 2D woven composites in tension\ compression\ and bending[ Acta Metall[ Mater[ 31\ 2856Ð2873[ Cox\ B[N[\ Massabo\ R[\ Mumm\ D[R[\ Turrettini\ A[\ Kedward\ K[B[\ 0886[ Delamination fracture in the presence of through!thickness reinforcement[ In] Scott\ M[ "Ed[#\ Proceedings of the 00th International Conference on Composite Materials[ Technomic Publishing\ Lancaster\ PA\ pp[ 048Ð066[ Dickinson\ L[C[\ 0884[ Trans!laminar!reinforced composites] a review of the literature as of April 0884[ College of William and Mary\ Applied Science Department Report\ Williamsburg\ VA[ Drans_eld\ K[\ Baillie\ C[\ Mai\ Y[!W[\ 0883[ Improving the delamination resistance of CFRP by stitching* a review[ Composites Science and Technology 49\ 294Ð206[ Fett\ T[\ Munz\ D[\ 0883[ Stress Intensity Factors and Weight Functions for One!Dimensional Cracks[ KfK 4189\ Kernforschungszentrum Karlsruhe[ Fleck\ N[A[\ 0880[ Brittle fracture due to an array of microcracks[ Proc[ Roy[ Soc[ London A321\ 44Ð65[ He\ M[Y[\ Cox\ B[N[\ 0887[ Crack bridging by through!thickness reinforcement in delaminating curved structures[ Composites] Part A 18\ 266Ð282[ He\ M[Y[\ Evans\ A[G[\ 0881[ Finite element analysis of beam specimens used to measure the delamination resistance of composites[ J[ Composites Tech[ and Research 03\ 124Ð139[ Hillerborg\ A[\ Modeer\ M[\ Petersson\ P[E[\ 0865[ Analysis of crack formation and crack growth in concrete by means of fracture mechanics and _nite elements[ Cement and Concrete Research 5\ 662Ð 671[ Jain\ L[K[\ Mai\ Y[!W[\ 0883[ Analysis of stitched laminated ENF specimens for interlaminar mode!II fracture toughness[ Int[ J[ Fracture 57\ 108Ð133[ Jain\ L[K[\ Mai\ Y[!W[\ 0884[ Determination of mode II delamination toughness of stitched laminated composites[ Composite Science and Technology 44\ 130Ð142[ Jones\ R[M[\ 0864[ Mechanics of Composite Materials[ Hemisphere Publ[ Corp[\ U[S[A[ Lu\ T[!J[\ Hutchinson\ J[W[\ 0884[ Role of _ber stitching in eliminating transverse fracture in cross!ply ceramic composites[ J[ Amer[ Ceram[ Soc[ 67\ 140Ð142[ Marshall\ D[B[\ Cox\ B[N[\ 0877[ A J!Integral method for calculating steady!state matrix cracking stresses in composites[ Mechanics of Materials 6\ 016Ð022[

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R[ Massabo\ B[N[ Cox:Journal of the Mechanics and Physics of Solids 36 "0888# 0154Ð0299

Massabo\ R[\ Mumm\ D[\ Cox\ B[N[\ 0887[ Characterizing mode II delamination cracks in stitched composites[ Int[ J[ Fracture "in press#[ Rice\ J[R[\ 0857[ A path independent integral and the approximate analysis of strain concentration by notches and cracks[ J[ Appl[ Mechanics 24\ 268Ð275[ Rice\ J[R[\ 0879[ The mechanics of earthquake rupture[ In] Dziewonski\ A[M[\ Boschi\ E[ "Eds[#\ Physics of the Earth|s Interior\ Proceedings of the International School of Physics {E[ Fermi|\ Course 67\ 0868[ Italian Physical Society:North Holland Publ[ Co[ Rose\ L[R[F[\ 0876[ Crack reinforcement by distributed springs[ J[ Mech[ Phys[ Solids 24\ 272Ð394[ Sharma\ S[K[\ Sankar\ B[V[\ 0884[ E}ects of through!the!thickness stitching on impact and interlaminar fracture properties of textile graphite:epoxy laminates[ NASA Contractor Report 084931\ NASA Langley\ VA[ Shu\ D[\ Mai\ Y[!W[\ 0882[ E}ect of stitching on interlaminar delamination extension in composite laminates[ Comp[ Sci[ Tech[ 38\ 054Ð060[ Smith\ E[\ 0878[ The size of the fully developed softening zone associated with a crack in a strain!softening material*I[ A semi!in_nite crack in a remotely loaded in_nite solid[ Int[ J[ Engng[ Sci[ 16\ 290Ð296[ Suo\ Z[\ Bao\ G[\ Fan\ B[\ 0881[ Delamination R!curve phenomena due to damage[ J[ Mech[ Phys[ Solids 39\ 0Ð05[ Tada\ H[\ Paris\ P[C[\ Erwin\ G[R[\ 0874[ The Stress Analysis of Cracks Handbook[ Paris Productions Inc[\ St[ Louis\ MO[ Turrettini\ A[\ 0885[ An investigation of the Mode I and Mode II stitch bridging laws in stitched polymer composites[ Masters thesis\ Department of Mechanical and Environmental Engineering\ University of California\ Santa Barbara[ Whitney\ J[M[\ Pagano\ N[J[\ 0869[ Shear deformation in heterogeneous anisotropic plates[ J[ Applied Mechanics 26\ 0920Ð0925[ Xia\ Z[C[\ Hutchinson\ J[W[\ 0883[ Mode II fracture toughness of a brittle adhesive layer[ Int[ J[ Solids Structures 20\ 0022Ð0037[