Shielding due to aligned microcracks in anisotropic media

E L S E V I E R Mechanics of Materials 22 (1996) 203-217

M E C H A N I C S OF

M A T E R I A L S

Shielding due to aligned microcracks in anisotropic media Sandeep Muju a, Peter M. Anderson b, Daniel A. Mendelsohn a a Department of Applied Mechanics, The Ohio State University, Columbus, OH 43210, USA

b Department of Materials Science & Engineering, The Ohio State University, Columbus, OH 43210, USA

Received 15 August 1995; accepted 13 October 1995

Abstract

This work presents a continuum damage analysis of the effect of aligned microcracks on a dominant crack in an otherwise anisotropic medium. This geometry is particularly relevant for composite media with an aligned reinforcement phase, where aligned cracking can change the magnitude and principal axes of anisotropy. Provided that the density of aligned microcracks reaches a stable saturation value near the crack tip, the near-tip stress intensity factor, K~l, may be related to the remote value, KPI, in terms of the anisotropic properties of the undamaged and saturated media.

Results show that of all types of damage, that which increases compliance in a direction normal to the crack plane typically reduces K~/Kp~ near the crack tip most significantly, while comparable increases in compliance parallel to the crack growth direction produce a relatively small change. Reduction in K~x/KPI due to increased shear compliance may be 80% of that due to increased compliance normal to the crack plane.

For damage consisting of aligned microcracks, the most effective reduction in K~//~ occurs when the microcracks are oriented parallel, rather than perpendicular to the main crack. This conclusion holds even for alumina/aluminum and graphite/epoxy systems that display large anisotropy. In those cases, optimal reduction in K~I/K~I occurs when in addition to parallel macro and microcracks, the stiffer direction is perpendicular to the crack surfaces. This case corresponds, for example, to macrocrack extension perpendicular to aligned fibers or layers in composite materials, with pinned microcracking through fiber or layer cross sections.

Keywords: Composites; Microcracking; Crack shielding; Aligned microcracks; Continuum damage; Anisotropic damage; Directional compliance; Fracture mechanics

1. Introduct ion

Experimental observations demonstrate that a near- tip region of microcracks may reduce the driving force on a dominant crack (Gu et al. 1992; Evans and Fu, 1985). This shielding effect is attributed to the increased compliance o f the material, due to microcrack opening (transformation) strains associated with creation of new internal surfaces (Ortiz, 1987; Hutchin- son, 1987; McMeeking and Evans, 1981). As a simple thought experiment, consider a damaged material

with elastic shear modulus/Zd and Poisson's ratio Vd in the near-tip region of a macrocrack. As/Zd is lowered relative to that of the far-field material, compatible deformation between regions requires that stresses in the near-tip region must reduce and hence, the near-tip stress intensity factor must decrease below the applied far-field value (Hutchinson, 1968; Rice and Rosen- gren, 1968). In particular, Hutchinson (1987) showed that if/-to, 1'0 are the remote material properties, the stress intensity f ac to r , / ( ap, at the crack tip is changed from the remote value,/~I according to

0167-6636/96/$15.00 (~) 1996 Elsevier Science B.V. All fights reserved SSDI 0 167-6636(95)00039-9

204

+ + "~1-'1 - 1 8(1 -- ivO) ~dd - 1

_3 [~'a/z0 1 + 4(1 ~'0) / ~ - ~ ' ° ] •

(1)

This paper acknowledges that microcracking in materials, particularly engineering composites, may have a preferred orientation due to inhomogeneities in fracture toughness and constitutive properties. For example, layered media may fracture consistently across the reinforcing phase, trapping microcracks between more ductile layers. Alternately, the interfaces may have a sufficiently low fracture toughness as to trap and contain microcracks along them. In both of these cases, the microcracks change the magnitude and per- haps the principal axes of anisotropy of the material. Thus, one purpose of this paper is to develop a counterpart to Eq. ( 1 ) for anisotropic materials.

The assumption embodied in Eq. ( 1 ) that the damaged region of reduced modulus extends to the crack tip presents a conceptual difficulty. In particular, it pro- poses that finite amounts of damage are possible even in the near crack-tip region where stress components are singular. It is difficult to envision continuum damage models of materials which sustain stable microcrack configurations and hence non-zero stiffness in the limit of unbounded stress. For example, Ortiz proposed that intergranular microcracks may be pinned from continued propagation by triple junctions between grains, and in the layered media examples cited here, microcracks which propagate through a brittle phase may be pinned by adjoining layers of higher fracture toughness or by branching along the interface. Further, for microcrack densities that are sufficiently high so that the local load-shedding zones of neigh- boring microcracks start to interact, the stress required to nucleate new microcracks increases rapidly, imply- ing microcrack saturation (Muj u et al., 1994). Despite pinning and local unloading, such arrangements can not remain stable right up to the crack tip unless/d ip is zero.

Theories of discrete crack interaction acknowledge that the continuum picture of damage must be aban- doned at some distance from the crack tip and they identify a damage-free region or ligament between the main crack and microcracks, as shown in Fig. 1. In the event that fracture progresses by linking up of the main crack and microcracks, the dimension of the ligament

S. Muju et aL/Mechanics of Materials 22 (1996) 203-217

r:?

Ktip I

Fig. 1. Zones of increased compliance and corresponding K-dominant regions in the vicinity of a fracture process zone.

is on the order of some microstructural length scale - for example, layer spacing, grain size, or particle size - that dictates the spacing between microcracks. Within this ligament, a singular/dip field dominates, and presumably, the macrocrack extends when K ap equals the fracture toughness of the phase in which the macrocrack tip resides. For some specific geometries, a microcrack tip may have a larger energy release rate than the macrocrack tip, so that crack growth occurs by extension of a microcrack back to the macrocrack. (Shum and Hutchinson, 1990)

Fig. 1 acknowledges that outside of the discrete microcrack region, a continuum description of damage may be applicable. Within the continuum damage region, it is assumed that there is a region in which local pinning, local load shedding, and flaw charac- teristics suppresses any existing or new microcrack growth without a substantial increase in local stress. The relatively uniform damage level in this saturated damage region will permit a local stress intensity, K s, to be identified. Beyond the saturated region, the density of microcracking reduces until at sufficient distance from the crack tip, the material is undamaged and a remote stress intensity, K °, is recovered. Both of these K-dominant regions represents a uniform damage state within that region.

The purpose of this paper is to adopt a continuum description of an initially anisotropic material which damages by nucleation of aligned microcracks, and to relate K s and K ° as a function of initial anisotropy, and the density and orientation of microcracks in the saturated region. The result will be a counterpart to Eq. (1), except that in the context of Fig. 1, K ap in that equation is regarded as K s. In principle, such a relation is an important step to identifying a critical value

S. Muju et al./Mechanics of Materials 22 (1996) 203-217

of K ° for fracture. Presumably, the K s field provides complete boundary conditions on the discrete microcrack region, so that discrete microcrack analyses may be used to phrase a fracture criterion in terms of a critical value of K s . The work presented here should then permit the fracture criterion to be phrased in terms of a critical K °.

2. A constitutive relation for microcracked materials

The constitutive relation to relate macrostrains, Eij, and macrostresses, ~kt, in a homogenized composite material may be written as

Eij = Sijkl~,kl . (2)

In particular, the effective compliance tensor, Sijkl, for a composite with alternating layers of different elastic properties and layer thickness has an initial value, S//jk t, which depends on the elastic properties and volume fraction of each layer type, as reported in Eqs. (A.4), (A.5) of Appendix A. Upon loading, microcracks with surface normal n are assumed to nucleate and grow as shown in Fig. 2, so that Sijkt evolves with damage. Although the macrocrack and microcrack orientations may be arbitrary, particular geometries of interest involve macrocrack propagation either parallel or perpendicular to the layer direction, so that DI = 0 or 90 °, with microcracks oriented either parallel (f12 = 0) or perpendicular (/32 = 90 °) to the macrocrack. Moreover, these two special cases bracket all other possible configurations.

The constitutive relation employed extends previous efforts by considering microcrack nucleation on planes with some component of shear stress and in originally anisotropic media. In particular, Ortiz (1987) considered the case of microcracks nucleating normal to the maximum tensile direction in isotropic media, so that no shear loading on the crack plane results, and proposed that

Eij = S~ijkt~,kl q" A( o t ) n i n j n k n t ~ k l , (3)

where n is in the direction of maximum principal stress, and A is a scalar that monotonically increases with the damage level, c~. For a general problem, microcrack nucleation may also depend on local ma-

205

Fig. 2. Macrocrack-microcrack geometry in a layered composite material.

terial features such as heterogeneous fracture toughness, so that preferred planes of microcracking are not perpendicular to the direction of maximum principal stress. For such cases, a generalization of Eq. (3) for plane problems is proposed as

Ely = ~iijkl~kl + AE( ot)ninjnknl~kl

q- AG( Ot) (nisjnkSl -b sinjSknl) ~,kl

+ a~(a ) (sisjnknl q- ninjskst) ~kt, (4)

where n is the unit normal and s is the unit tangent vector to the microcrack plane. Thus, AE, Ac, and A~ are scalar damage parameters which represent the increase in the normal, shear and Poisson components of the compliance tensor as a function of microcrack density, a; thus, the Ortiz form (Eq. 3) is a special case of Eq. (4), where ~e = A, ac = 0 and A~ = 0. For the anisotropic case where the microcracks and macrocrack are parallel (f12 = 0),

= S 22 - S 22, = ½ ( 66 - - S 66) ,

(5a)

and in the case where the microcracks are perpendicular to the macrocrack (f12 = 90°),

- - - (sg66 - : 6 6 ) ,

Here, S//j and ~ represent Voigt components of the undamaged and damaged compliance tensors, i.e., $11 = S l l l l , S22 = $2222, S12 = S1122, and S66 = 4S1212. For convenience, the components are referred to the

206 S. Muju et al./Mechanics of Materials 22 (1996) 203-217

2~ Initial / l o a d i n g ~ , '

, , ," unloading

li- E

Fig. 3. Constitutive relation for damaging material that displays saturation.

macrocrack (1-2) axes in Fig. 2. For an arbitrary orientation angle/31, the components S~jit, measured in the material ( I ' - U ) axes, must be transformed to the components, Si#l, referred to the macrocrack axes. Equation (A.6) in Appendix A describes the transformation in more detail. Forms for ~ as a function of microcrack density will be discussed in a subsequent section, where/31 and/32 are assumed to equal 0 or 90 ° .

3. Anisotropic crack tip fields in the saturated region

Fig. 3 illustrates a generic macroscopic stress-strain relation for a damaging material which exhibits damage saturation and hence linear-elastic behavior with reduced stiffness. This permits a linear-elastic crack tip field, characterized by stress intensity factor(s) K s, to be used in the saturated region. In particular, the constitutive relation, Eq. (4), has a hyperelastic form, derivable from a complementary energy potential, ~k, according to

Eij_ c9~ O£~j " (6)

The potential O satisfies the integrability conditions with respect to £ij, so that

= 1 ~ll('Y., Ol) 7S~ijklXijXkl + f ( X , Or) , (7a)

where

f (E,ot ) = ½AE (ninjnknt) XijXkt

+ AG (nisjnkSl + sinjsknl) Xij•kl + hv (sisjnknt + ninjskst) Xij~,kl. (7b)

At saturation, AE, Ao, and a~ reach saturation values that are independent of macrostress, and ~b is a homogeneous function of degree two of the macrostress tensor. Further, since $ must be singular as 1/r at the crack tip (Hutchinson, 1968; Rice and Rosengren, 1968), the singular terms in the saturated region must maintain the same singularity present in an undamaged material, even though the amplitude and angular variations may be different. In particular, the stress distribution in the saturated region may be written in the form for anisotropic linear-elastic materials (Sih and Leibowitz, 1968; Cherepanov, 1979),

K s K s II Xijs = I2v,,~_~fo(O,.. Gs) + ~ g i j ( O , Gs) , (8/

where K~ and K~I are the Mode I and Mode II stress- intensity factors, respectively, in the saturated region, and (r, 0) are polar coordinates centered about the macrocrack tip, as shown in Fig. 2. The angular variations fij(O, Gs ) and gij(O, Gs) are reported in Eq. (B.1) of Appendix B, and they depend on the roots, Gs, to the characteristic equation,

~11 l~s -- 2S~16G~ -'l- (2~12 + ~66)G 2

-- 2~26Gs + ~22 = 0 , (9)

where the components, ~j, of compliance in the saturated region are expressed in the macrocrack coordi- nate system.

It is clear from the classical arguments (Hutchin- son, 1968; Rice and Rosengren, 1968) that the remote stress field also has the same functional form as Eq. (8),

/G 0, 2~0 = 2X/~.~fij( GO) + ~ g i j ( O , GO), (10)

with fij and gij given by the same functional form, Eq. (B.1), except that the roots G0 to Eq. (9) are based on components, S//j, of undamaged material far away from the crack tip. The solution for the cracktip fields in between the K s and K ° dominated regions is unavail- able since the material is inhomogeneous anisotropic. However, further development will demonstrate that the principal goal to relate K~/K~I and K~l/K~u in terms of microcrack density and orientation in the saturated zone, as well as initial undamaged anisotropy, does not require the solution in the inhomogeneous region.

S. Muju et al./Mechanics of Materials 22 (1996) 203-217 207

4. Local stress intensity factor near the crack tip

This effort is based on a deformation approach to damage in which the damage-induced strains are assumed to depend only on the current state of macrostress. In reality, as Fig. 3 illustrates the unloading paths from damaged states tend to be linear with reduced modulus, so that in a closed cycle of loading and unloading, some work has been consumed in the creation of microcrack surfaces. The formalism of the J-integral for deformation theory leads to path independence of the contour to compute J (Rice, 1968), but the work associated with microcrack surface gen- eration is neglected. This will be discussed in more detail in the last section.

The path independence of J provides a relation between K s and K ° in terms of the roots ~:s and (0 to the characteristic equation. In particular, j0, which is furnished by a contour lying entirely in the K ° dominant zone, must be equal to js, furnished by a contour lying entirely in the damage saturated K s zone. The result of each contour takes the form

J* ( / f f ) z • . 2 S~2R e [.~:l + ( 2 ] - /' ~ - - - ~ - / L 1 2 J

r K*'~2 + ~ S T l I m ( ~ : ~ + sc~), (11)

where the superscript * denotes either "s" or "o", depending on the contour taken. Equality of J from each contour may be expressed as

a s ( l - p ) /" K~'~ 2

ao( -f =-p--f + bop k, K~i .}

= 1, (12 ) + ao(1 - p) +bop

where the values for a . and b. (* may be replaced by "s" or "0") are given by

1 . a . = ~S~2Re i

b. 1 • • = 7SIllm(s¢l + sc~). (13)

The fraction, p, which is a measure of the of the mode mixity of the applied loading is defined by

( ~ ) 2 p = (K~l 12 + (/~H) 2 ( O _ < p < 1) . (14/

In the case of pure Mode I (p = 0) and pure Mode II (p = 1) problems, Eq. (12) reduces to

K[ a~s ~-I = (pure Mode I: p = 0 ) ,

K~I _ b~0b0 - V b s (pure M o d e I I : p = l ) . (15)

Clearly, the change in stress intensity between the remote and saturated regions depends on features of the remote and damage-induced compliance, as expressed in the roots to the characteristic equation.

5. Cracking al igned to principal mater ia l axes

Fracture in aligned-fiber or layered composite materials may involve macrocrack growth parallel to or normal to the layers or fibers, so that fll = 0 or 90 °. In a corresponding manner, tunneling or delamination- type microcracks depicted in Fig. 2 may form in the vicinity of the macrocrack tip, so that/32 = 0 or 90 °. In such cases, the macrocrack and microcracks are parallel to one of the principal material axes, so that St6 and $26 are zero in both remote and saturated regions.

When Sl6 = $26 = 0, the corresponding roots to the quartic equation (9) are expressed in terms of the dimensionless material parameters

822 866 a t- 2S12 = = (16) A ~ , 6 2Sll

This ,l is not to be confused with the oned in Section 2. In the subsequent discussion, it will be useful also to express results in terms of the material parameters

x/~ 2 ~ S12 "r/ ~ ~ 866 + 2S12 Y = 866 (17)

For isotropic materials, a = 6 = r/ = 1. Further, the bounds, A > 0 and - 1 + S66/2V~llSzz < 1/71 < 1 + $ 6 6 / 2 ~ are required to help ensure that the compliance tensor is positive definite.

The roots of Eq. (9) are of two types. When ~7 2 > 1, the roots are pairs of complex conjugates, and those with a positive imaginary part are

7/2 > 1 : ~:1, ~:2 = -4-Al/4e4-idp/2,

cos,/, = - ( l / n ) . (18)


When v2 < 1, the roots are pure imaginary, and those with a positive imaginary part are

q* < 1: [I,& =iA’14 yy)“‘. (19)

The physical bounds on r] mentioned above exclude the range, -1 < 77 < 0, which corresponds to pure

real roots. The roots in Eqs. (IS), (19) may be substituted

into Eqs. ( 15) to yield the ratio of stress intensity factor in the saturated zone to that in the remote zone.

For the pure Mode I case, p = 0,

1 ($)li8 (i>,, ($“” [ Mo]‘i’

forqo2,q,2 > 1,

A$ = ( (g (# ($4

X

112

I forq~,vf < 1, (20)

and for the pure Mode II case, p = 1,

1 (g (g)“* (z?)“” [MO]“’

EJ = 1 (t%)3’8 (gy$

I r,/G7m+ ,JKi=a ‘I2

I forqij,q~ < 1,

(21)

Eqs. (20) and (21) show the explicit dependence of K”/@ on various components of compliance, and in- dicate that the pure Mode I and Mode II results have a

1 1.5 2 2.5 3 Normalized Compliance, Ss/Se

Fig. 4. Effect of near-tip compliance on KS/q.

parallel structure. In particular, Eqs. (2 1) may be obtained from Eqs. (20) simply by exchanging the $1 and S22 dependence, i.e., by a 90’ degree rotation of the material axes with respect to the macrocrack axes. Thus, the effect of S&/g2 or q,/$‘, on Kf/q is identical to the effect of q,/.C$, or sS,2/S$2, respectively, on KiI/qI. Further, increases in S& produce the same reduction in KS/@ as for K;,/g,, since the expressions for each have the same dependence on vs and qo.

Fig. 4 shows the dependence of KS/e on components of compliance in the saturated region, for three material cases that represent a range of ~0 and yo values reported in Table 1. Only the features for pure Mode I loading are presented. However, the parallel structure discussed permits the result for KS, to be inferred from this figure. It should be noted that, at the micromechanical level, for microcracks symmet- ric about the crack plane KfI = 0 if @I = 0, i.e., the damage does not induce mode mixity.

The results show that Kf/e monotonically decreases with increases in q,, g2 or A&. Typically, increases in Ss,, provide the most reduction in KS/@, followed by increases in A&, then 3,. Variations from this ordering do occur in extreme cases. In particular, Eqs. (20) reveal that for r]o -+ 0, KS/@ is equally dependent on S&/S& and S$,,/S$,, and independent of q,/q,. However, for 70 > 1, KS/@ is most dependent on sS,,/g2, less dependent on q, /$,, and independent of S&/S$.

The initial properties, ~0 and ‘yo, of the composite affect the value of Ki/g by as much as 15% for the cases shown. Typically, as 70 is increased, KS/@ becomes more sensitive to changes in q, /$‘, and


Table 1 Non-dimensional composite parameters

209

Fiber composite C ° ao 60 "q0 Yo

Graphite/epoxy (v f = 0.6) a Woven glass/epoxy (vy = 0.45) Carbon-epoxy (v f = 0.63) E-glass-epoxy (v f = 0.55) SiC/CAS (v f = 0.39)

1/4.5 14.16 18.59 0.202 0.008 1 / 5.3 1.0 2.63 0.379 0.06 1/7.2 13.79 9.59 0.387 0.0285 1/3.8 4.534 4.85 0.439 0.0545 I /44 1.08 1.175 0.884 0.145

Crystal ~ 8o 77o Yo

NaCI 1.0 1.53 0.653 - 0 . 0 6 AI 1.0 0.756 1.3226 -0 .162 Au 1.0 0.053 18.64 --0.447 Cu 1.0 0.0227 44.06 -0 .474 Ni 1.0 0.173 5.78 -0 .341

Laminates Sll ) <CVa) SlI' /SI~' SI~' /SlI' SI~' /SI~' V, ao 6o "1o ~'o

Alum-alumina 1/400 0.175 -0 .25 --0.3 0.5 2.26 1.7 0.57 --0.055 Near isotropic 1/400 0.975 - 0 . 3 - 0 . 3 0.5 1.0 1.0 0.999 --0. l 15 Cu-Ni 1/66.75 2.04 - 0 . 4 2 -0 .37 0.5 0.137 1.03 7.42 -0 .373

a vf denotes volume fraction of fibers.

Shl~2, but less sensitive to changes in ~ 6 / ~ 6 - The form of Eqs. (20) indicates that only 170 is required to determine the effect of H1 /~1 and ~22/~2 on K~/l~i. However, both ~70 and V0 are required to determine the effect of ~ 6 / ~ 6 on K~/KPI. Based on the materials selected in Fig. 4, K~/~II is more sensitive to ~ 6 / ~ 6 when the magnitudes of Y0 and *70 are smaller.

Table 1 provides a limited assessment of values of ~70 and T0 for composite and monolithic materials. For fiber-reinforced composite materials, 0 < ~70 < 1 is typical, based on reported measurements of composite elastic moduli. For layered composite materials comprised of isotropic elastic components, the effective elastic compliance may be computed, and the re- suits are summarized in Figs. A.1-A.4. Over the en- tire range of isotropic properties and volume fractions considered in those figures, 0 < *70 < 1. The qual- ification that the layers are isotropic restricts the application to layered composite materials consisting of polycrystalline layers, in which the grain size is much smaller than the layer thickness.

For materials with single crystal layers, such as the copper/nickel example reported in Table 1, ~72 > 1 is possible. Such cases typically involve Y0 values that approach -0 .5 , so that the denominator of r/ in Eq. (17) approaches zero. Magnitudes of Y0 range from less than 0.15 for the fiber-reinforced materials considered and for laminates with relatively isotropic components to as large as 0.47 for relatively anisotropic single crystals. All of the results presented are for material systems shown in Table 1.

Closed-form expressions for the limiting cases, r/02 --~ 0, Yo = 0 and 7/2 >> 1, Yo = 0 are obtained from Eqs. (20),

,i4 ,r#o 2 0, To=0: '

K~ (S~I l '~ 1/8 (~22~ 3/8 no 2 > > l , T 0 = o : \ 22) . (22 )

Eqs. (22) show that, in these limits, the effect of si- multaneous increases in more than one component of

210 s. Muju et al./Mechanics of Materials 22 (1996) 203-217

1 °9 i:-o0o8t

o _ 07 \ 0 . 6 ~

O. 4 -0.49

0 . 3 - 0 . 4 9 9 "

0 . 2 E i i J , i , ~ - - = 1 1.2 1 .4 1.6 ' 1 . 8 2

9s66/S°66

Fig. 5. Effect of near-tip compliance on K~/K~I for particular values of Y0 (r/0 = 0.2).

S~/is multiplicative. For intermediate values of r/0 and Y0, the multiplicative feature is approximate.

In principle, large reductions in K[ are possible even in cases where the saturated region still has non-zero components of stiffness. Fig. 5 presents K[/K~I as a function of increasing compliance, ~66/~6, for r/0 = 0.2. Decreases in K~/K~I occur since r/s in Eqs. (20) decreases with increasing ~66/S~66. The amount of decrease is dependent on Y0. For example, when To is between 0 and -0 .2 , as is representative of many systems reported in Table 1, increases of 200% in ~66 produce decreases in K~/K~I of less than 20%. How- ever, the decreases become more pronounced as To approaches - 0 . 5 ; in particular, up to 75% reduction in K[ is achieved for the limiting case y0 = -0 .499 shown. The cases shown in Fig. 5 are thermodynami- cally possible, in the sense that the eigenvalues of the compliance matrix remain positive, and hence strain energy remains a positive definite quantity.

6. Compliance of cracked bodies

The effect of second phase particles on the compliance of elastic media has received much attention, especially when the second phase consists of voids or cracks (Mura, 1987; Nemat-Nasser and Hori, 1993; Norris, 1985; Hashin 1988)A recent comprehensive compilation of research work is provided in (Nemat- Nasser and Hori, 1993). Of the methods presented, the differential scheme has proven effective in predict- ing the compliance of a homogenized, micro-cracked

composite material. In particular, the results of Hashin (1988) for stiffness reduction of a unit cell contain- ing a single crack in an orthotropic medium are used. Those results use the solution for a tunnel crack in a homogeneous medium, so only effective rather than discrete microcrack interaction is incorporated.

For the case of aligned microcracks with surface normal in the 2-direction, the damaged compliance components may be expressed as

l+ 2T_o_o_o_o_o_o_o 0/ ,

(23)

+ ~ , (24)

where ce = aZ/A is the microcrack density, defined in terms of an area, A, per microcrack in the 1-2 plane and the half-crack width, a, shown in Fig. 2. Increases in other components are considered to be negligible in comparison. For the case of aligned microcracks with surface normal in the 1-direction, increases in ~ l and ~6 are described by Eqs. (23) and (24), respectively, but with $22 and Sl I interchanged in those expressions. Regardless of microcrack orientation, the cracked material is assumed to be orthotropic initially, with the microcrack plane as the isotropic plane. This description is most applicable to through-cracking of aligned fiber reinforced materials and to interfacial cracking in layered materials.

Fig. 6 shows the increase in ~ 2 / ~ 2 and ~66/~6, as a function of a and elastic properties, r/0 and ~0 = ~z2/~l , of the undamaged material. Both components increase monotonically with microcrack density ce. However, the increase is most pronounced for the smallest values of Ao, corresponding to microcracks with surface normal in the direction of larger stiffness (c.f. curves 2 and 4). This observation is consistent with the functional dependence on A0 in Eqs. (23) and (24).

Increases in *1 appear to have different results, depending on the component. In particular, the effect of damage on ~66/~6 increases monotonically with r/, while the effect on ~22/~2 decreases monotonically with r/.


e ~ E o

; =

o z

8 ' I ' ' ' I ' ' I ' ' ' I ' ' '

7 L i n e ( ' q o , k o ) - C o m p o s i t e . , ~ 4

1 ( 0 . 2 0 , 1 4 . 2 ) - G r a p h i t e / E p o x y . .

6 2 ( 0 . 9 9 , 1 . 0 2 ) - N e a r l y I s o t r o p i c . , "

3 ( 0 . 5 7 , 5 . 2 ) - A l / A l u m i n a ([~1 = 0 ) , " ! 5 4 ( 0 . 5 7 , 1 / 5 . 2 ) - A l / A l u m i n a ( [ ~ I = 9 0 ) . " 2

4 s ea . " , 3 - - - 8 2 2 / o 2 2 . . . . .

3

2 _ . - . : = . - I " ' ' : ; ; : = - 2

0 , r , 1 , , , t , L , l , , L 1 , , ,

0 0 . 2 0 . 4 0 . 6 0 . 8

M i c r o c r a c k d e n s i t y p a r a m e t e r , ~ .

Fig. 6. Effects of microcracks on compliance. The microcrack normal is parallel to x2.

7. Effect o f a l igned microcracks on the local stress intensi ty

Thus far, the decrease in K~/K~I produced by increases in individual components of compliance, irre- spective of the cause, has been addressed, rather than the decrease in K~/I(~i produced by discrete aligned microcracks. The results of the previous section indi- cate that discrete aligned microcracks increase more than one component of compliance. For microcracks aligned parallel to the macrocrack plane (/32 = 0), the components ~22 and S~66 increase while for microcracks aligned perpendicular to the macrocrack plane (f12 = 90°), the components S]tt and SS~ increase. The corresponding reduction in /~/ /~I is provided by substituting into Eq. (20) the expressions in Eqs. (23), (24).

0 . 9 ~ ' ' ' ' ' ' ' ' (132=°)

0.8

m~l - 0 . 6 L i n e (

0.5 _--_ I (0.2,14.2)- ([31=0)~ 4 ~ 1 / 1 4 . 2 ) - ( 1 3 , = 9 0 ) ~ ^

0 . 4 ~ 34 ( ( : : :77 ' 51125!2 7 (_.~;~0_)90)

0 . 3 , I J , , I , , , I , , , I , r ,

0 0 . 2 0 . 4 0 . 6 0 . 8

O~ Fig. 7. Effects of microcracks parallel to the macrocrack on K~/I~I.

Although the general results are relatively detailed and not reproduced in closed form, simple expressions may be obtained for the limiting cases, ri02 ---+ 0 and ri2 >> 1, by substituting Eqs. (23), (24) into the respective limiting expressions reported in Eqs. (22). For the case of aligned microcracks oriented parallel to the macrocrack, with density ors in the saturated region near the crack tip,

(Parallel micro- and macrocracks (Y0 = 0 assumed) with f12 = 0 °. )

,7/02--+0:

K~ ~ 1 + C~s e, ~ 2Z4/4

( )-'/= - v o x 1 +

ri~ >> 1:

K~ ~ 1 + us c~ s ~-1 ~ 2V/2 al/---'-'----~ + . (25)

"L 0

In either limit of ri0, the decrease in K~/K~I is more pronounced when A0 decreases, or equivalently, when the initial stiffness normal to the cracks is made larger relative to the stiffness parallel to the cracks. Such cases would correspond, for example, to macrocraeks oriented perpendicular to the stiffer fiber or layer direction in composite materials, with microcracking through the cross section of individual fibers or layers. Further, the reduction in K~t/K~I appears to be independent of ri0 for ri0 >> 1, but the reduction grows monotonically as rl0 approaches zero.

Fig. 7 shows the results for parallel macro- and microcracks, and addresses cases where the cracks are oriented perpendicular to (A0 < 1) and parallel to (ao > 1 ) the stiffer direction of the undamaged material. The former case consistently produces lower values of K[/K~I, reflecting the same feature observed in the limiting cases described in Eqs. (25). For 90 ° ro- tations of the material relative to the cracks, only the value of A0 inverts, with no change in ri0.

The case of aligned microcracks perpendicular to the macrocrack (f12 = 90 °) appears to have a less pronounced effect on K~/K~I. Again, simple expres-

212 S. Muju et aL /Mechanics o f Materials 22 (I 996) 203-217

1

0 .95

~, - 0.9

0 . 8 5

0.8

0.75 0

Line (rl o, X o) ~ 1 (0.2, l 4.2) - (!31_--0) ~ ' ~ ' ~ 2 (0~, l/1&2) - (~=90) " ~ . ~ 3 (0.57, 5.2) - ( [ ~ 1 = 0 )

4 (0.57, 1/5.2)- ([3=90)

0.2 0 .4 0 .6 0 .8 or.

Fig. 8. Effects of microcracks perpendicular to the macrocrack on X~ / X~'.

sions may be obtained for the limiting cases, r/0 --~ 0 and r/0 >> 1, by substituting Eqs. (23), (24) into the second of Eqs. (22). The corresponding limits are given by

(Perpendicular micro- and macrocracks (Y0 = 0 assumed) with/32 = 90°.)

rj~ ~ O:

i1 / ~-I ~ ~ j 1 + 2 V'2r/o 1/Z ]

"qo2 >> 1: Q ,.,~1/4 ~ ) -1/4

K~ ~"o ~-i ~ I + - ~ + (26)

These expressions differ from those for parallel micro and macrocracks in that the dependence on A0 is inverted, and for r/0 >> 1, the overall exponent is - 1/4 rather than - 3 / 4 . The first difference infers that decreases in K~/K~I with a are more pronounced for larger a0. Such cases correspond to macrocracks oriented along the stiffer fiber or layer direction in composite materials, with microcracking through the cross section of individual fibers or layers. As with the parallel crack orientation (/32 = 0), reductions in K~/K~I are more pronounced as r/0 approaches zero; for r/0 >> 1, the reductions are considerably smaller and are also independent of r/0.

Fig. 8 shows the results for perpendicular micro and macrocracks. The largest reductions occur when the

macrocrack is oriented parallel to the stiffer direction in the composite material (A0 > 1 ), and when ~70 ---+ 0. The same features discussed for the limiting cases, Eqs. (26), are reflected in this figure.

8 . C o n c l u s i o n s

The anisotropy in material compliance induced by near-crack-tip damage, such as aligned microcracking, can produce varied reductions in local stress intensity near the crack tip, depending on the elastic anisotropy of the undamaged compliance components far away from the crack tip and the change in anisotropy state in the damaged region near the crack tip. However, such conclusions are based on an assumption that the level of damage saturates to a spatially uniform value near the crack tip. Such saturated regions may be possible due to limited numbers of defect nucleation sites and pinning of defects so that stable arrangements of defects are maintained near the crack tip. Although these conclusions apply to situations of arbitrary mixed mode loading, cases of pure Mode I remote and local loading are addressed in greater detail, since macroscopic crack extension is frequently along paths of pure Mode I. Special attention is given to crack propagation along principal directions of anisotropy, corresponding to propagation perpendicular or parallel to fibers or layers in composite materials, or on planes of mirror symmetry in single crystals.

Over the range of elastic material parameters for many fiber and laminate composite materials, the largest reductions in K~ in the saturated region near the crack tip occur from increases in the compliance, ~22, normal to the macrocrack surfaces. In more typical situations where 0.2 < r/o < 1, increases in the in-plane shear compliance, ~66, and compliance, ~ l , in the direction of crack extension provide, respectively, smaller reductions in K~I. However, in other cases for which r/o >> 1, such as single crystal layered structures or some single crystals, the order of importance of ~66 and all I may reverse.

When solutions for the effect of aligned microcracking on elastic compliance are incorporated, the largest reduction in KI near the crack tip is typically produced by microcracks aligned parallel to the macrocrack, with the direction of largest stiffness normal to the crack surfaces, similar to the observation in Mauge


and Kachanov (1992). The remaining order of de- creasing effects is microcracks parallel to the macrocrack, with the direction of largest stiffness parallel to crack extension; microcracks perpendicular to the macrocrack, with the direction of largest stiffness parallel to macrocrack extension; and finally perpendicular microcracks, with the direction of largest stiffness perpendicular to macrocrack extension.

There are some qualifications for use of the continuum damage approach presented. First, near-tip regions must exist which contain a uniform density of damage. Such regions may exist; for example, constitutive relations for layered materials which damage by tunnel cracking in the lower toughness layers suggest that microcrack saturation may occur when microcrack spacing is on the order of the thickness of the layer in which the cracks reside. Second, use of a continuum damage approach near the tip can only identify K~, which is not the same as K~ p. Such pre- dictions are provided only from solution of the stress field in the crack tip region, based on a discrete defect analysis in regions near the crack tip, for which K~ may supply the far-field boundary conditions. Finally, the continuum analysis based on path independence of J assumes that the damaged material translates with the macrocrack tip. In fact, steady state crack extension does involve self-similar translation of an active damage zone with the crack tip, but path independence of J neglects the energy consumed by the wake of damage produced behind the crack front. Thus, esti- mates provided here of K~ in the saturated region near the crack tip are overestimates, i.e., a lower bound on shielding. The fraction of energy consumed by damage in the continuum damage region, per unit area ad- vance of the macrocrack, may be written as Gd/Go. If this contribution were included, K~/K~I would then be given by

-

~ - I - l - --'as ( 2 7 )

rather than Eq. (15). The results presented interface with the discrete

crack-microcrack interaction zone bounded by K ~ dominant zone. As depicted in Fig. 1, the local stress intensity factor, K s , presented here serves as the boundary condition for a discrete damage analysis inside the saturated region. When coupled with a

micromechanical analysis of microcrack-macrocrack interaction (Hori and Nemat-Nasser, 1987; Kachanov et al., 1990) in the discrete zone this local stress intensity factor, K s, may be used to infer Kc ~p, the inherent toughness of the material in the tip region, Fig. 1.

Acknowledgements

The first author wishes to thank Dr. Carl H. Popelar for helpful discussions. Support for this work was provided in part by the Center for Materials Research, ,The Ohio State University, Grant 220179, and by the Office of Naval Research, Contract No. N00014-91-J- 1988.

R e f e r e n c e s

Cherepanov, G.E (1979), Mechanics of Brittle Fracture, McGraw Hill.

Evans, A.G. and Y. Fu (1985), Some effects of microcracks on the mechanical properties of brittle solids - 1, Acta Metall. Mater. 33, 1515.

Gu, W.H., K. Faber and R.W. Steinbrech (1992), Microcracking and R-curve behavior in SiC-TiB composites, Acta Metull. Mater. 40, 3121.

Hashin Z. (1988), The differential scheme and it application to cracked materials, J. Mech. Phys. Solids 36, 719.

Hori, M. and S. Nemat-Nasser (1987), Interacting microcracks near the tip in the process zone of a macrocrack, Z Mech. Phys. Solids 35, 601.

Hutchinson, J.W. (1968), Singular behaviour at the end of a tensile crack in a hardening material, J. Mech. Phys. Solids 16, 13.

Hutchinson, J.W. (1987), Crack tip shielding by micro-cracking in brittle solids, Acta Metall. 35, 1605.

Jones, R.M. (1975), Mechanics of Composite Materials, Scripta Book Company.

Kachanov, M., E.L.E. Montagut and J.E Laures (1990), Mechanics of crack-microcrack interactions, Mech. Mater. 10, 59.

Mange, C. and M. Kachanov (1992), Interacting arbitrarily oriented cracks in anisotropic matrix. Stress intensity factors and effective moduli, Int. J. Fracture 58, R69.

McMeeking, R.M. and A.G. Evans (1981), Mechanics of transformation-toughening in brittle materials, J. Amer. Cer. Soc. 65, 242.

Orfiz, M. (1987), A continuum theory of crack shielding in ceramics, J. Appl. Mech. 54, 54.

Rice, J.R. (1968), Mathematical analysis in the mech. of fracture, in: H. Leibowitz, ed., Fracture, Vol. II.

Muju, S., RM. Anderson and C.H. Popelar (1994), Phenomenological damage modeling based on micromechanics, Proc. Ninth Amer. Soc. for Composites Conf., University of Delawa~.


Mura, T. (1987), Micromechanics Of Defects In Solids, Martinus Nijhoff Pub., Dordrecht.

Nemat-Nasser, S. and M. Hori (1993), Micromechanics: Overall Properties Of Heterogeneous Materials, Elsevier Science Publishers B.V..

Norris A.N. (1985), A differential scheme for effective moduli of composites, Mech. Mater. 4, 1.

Rice, J.R. and G.E Rosengren (1968), Plane strain deformation near a crack tip in a power-law hardening material, J. Mech. Phys. Solids 16, 1.

Shum, D.K.M. and J.W. Hutchinson (1990), On toughening by microcracks, Mech. Mater. 9, 83.

Sih, G.C. and H. Leibowitz (1968), Mathematical theories of brittle fracture, in: H. Leibowitz, ed., Fracture, Vol. il.

Appendix A. Effect ive elastic compl iance o f laminated compos i te mater ia ls

The linear-elastic constitutive relation for each phase may be written for components ( 1 ) and (2) as

~l l 0-11

~22 °'22

~tl ; {0-j}(*) = o"11 . (A . I ) {ei}(*) = [Si j]*{o' i}*; {~i}(*) =-- 2e23 0"23

2~t3 o'13 2et2 o'12

where (*) corresponds to (1) or (2) , depending on the component (1) or (2) considered. For well-bonded interfaces across which no relative sliding or opening occurs, certain components of internal stress and strain in each layer must be equal to components, E~j and E[j, of the macroscopic stress and strain, respectively,

,(1) _,(2) E~ t ,(1) _ o,(2) t , ( l) ,(2) = E~3, 11 = ~ II = , ~33 - - ° 3 3 =E33, ~13 = ~ 1 3

"1(I) ,(2) ---- V2.~2, ,(I) ,(2) ---- ~ 2 , , ( l ) 1(2) = ~'t3 ( A . 2 ) 22 = 0" 22 0" 12 = Or 12 0- 13 -- 0" 13 ,

where ( ' ) denotes that the components are referred to the composite material axes 1 I - 2 ' - 3 ' shown in Fig. 2. Eqs. (A. 1 ) and (A.2) may be used to obtain a corresponding macroscopic constitutive relation of the form,

E; = S~j~ ~, (A.3)

where E~ is defined in terms of components E~j and ~ is defined in terms of components of ~ j analogous to the definitions of ei and 0-i in Eqs. (A.1). For general cases, the components S~j are given by

IOll -- S ) "F v2S ) S ) 2 S )

s'~ ' 1- -333 ~

S~2 =

S13 =

2VlV2 ~ 1 1 ~11 °12 °12 ]

2 (,) {[¢(2,~2~_[¢(2,'~2~.t..2e(2)[o(,)--o(I)~t-- ((¢(I)'~2~_~(I)¢(1) 0{¢(l)~2a_a¢(l,¢(2)a_(~(2)~2~(2)~(2) 9{¢(2)'~2~ VlSI, ~12 ] " ~ l , ] 7--2 'l ~ 12 ,l ) ~ " l ,] - 'l "'2 --'~'12 ,] " "12 ~12 "~" l ] "'11 "12 --'~'1' ] /]

(2 (2 v (1) (1) v (s,, _s,~)+ ~(~,, +,s,~ )

vial011) ( ( S l ~ ) ) 2 - ( a l l ) ) 2 ) "4- V2o12c(2)((all)) 2 -- (511) ) 2 )

{~(t)~(2) ~(t)~(2)~ _ + _ + ,.,, .,,_


S~6 = S~6 S~6 = 0 S~6 -- {" ¢,(1) ) ) = , ~,V1066 + V2 S(2 ,

. c,(') {c,(2) (2)) . c , ( 2 ) ( ( ' ) .~,(1)) Stl2 = S~ 3 = Vl°12 ~° l l + S12 + v2012 SI2 + ~ll ( A . 4 )

VI ( S ~ ) -~ ~'12 } -~- v2

For the special case of isotropic component materials with Young's moduli, E. , Poisson's ratios 1"., and volume fractions v., values of S~j (= 5~. i) are

s ' , , = s ;~ = E2v2(1 -- 1"~)+ElVl(1 -- 1"5)

( E l V l + E 2 v 2 ) 2 - ( g l v l v 2 + E 2 v 2 1 " l ) 2 '

S~ 6 = 2E2v2(1 + Vl) + 2ELY2(1 + 1"2)

E1E2

S~2 = E~VlV2al+E~VlV2al+EiE2v~(1 - 1"2)+EiE2v~(1 - 1"l)+4E2v2V2ElVl1"l

E12E2vI ( 1 - 1"2 ) + El E22v2 ( 1 - 1"1 )

a . = ( 1 - 1 " . - 2 1 " 2 , )

5~3 --~ _ ElVlVl(1 - 1"~)+E2v21"2(1 - 1"~)

(E lVl + g 2 v 2 ) 2 - (E1v11"2 + E2v21"I) 2 '

Sll6 = S~6 = S;6 = O,

S~ 2 = S~ 3 = v i i ' l ( 1 -- 1','2) + v21"2(1 -- 1"1) E l V l ( 1 -- 1'2) + E2v2(1 - 1"1)

(A.5)

= ' ' ' 2S~2 ) for laminated composites with Figs. A.I-A.4 show the material parameter, 7/' 2 ~ / ( S ~ 6 + isotropic components as a function of the elastic properties, E. , v., and volume fractions v.. Over the large range of parameters considered, 0 < ~/~ < 1.

The components of compliance, Sij, referred to the macrocrack axes (1-2-3) rotated from the composite material axes (1 ' -2 ' -3 ' ) by angle ill, as shown in Fig. 2, are obtained from the components S~j according to (Jones, 1975),

al l = Stll c0s4 /~ l -{- (2S~2 n t- S66) sin 2/31 c os 2/31) + S~2 sin 4 i l l ,

Sl2 = S]2 ( sin4 fll + c0s4 i l l ) -[- (S~I + St~2 - $66) si n2 tim c0s2 i l l ,

$22 = S~1 sin 4 f l l + (2S~2 + S~6) sin 2 fll co s 2 fit + S~2 cos 4 i l l ,

2S16 = (2S~ 1 - S]2 - S~6) sin fl l co s 3 f l l - (2S~2 - S~2 - S~6) sin 3 fll co s i l l ,

2S26 = (2S~1 - S~ 2 - S~6) sin 3 fit co s fll - (2S~2 - 2S~2 - S~6) sin fl l co s a i l l ,

2S66 = 2 (2S~ 1 + 2S~2 - 4S~ 2 - $66) sin2 fll co s 2 f l l + S~6 ( s in4 fll + cos 4 fll ) - (A.6)

216 S. Muju et al./Mechanics of Materials 22 (l 996) 203-217

'/ 0.8 .-.

0.6

qo - - ~ 2 / ~ 1 = 2.0

0.4 - - - - - - " a 2 / ' a l = 1 . 0

. . . . " a2 /~1 = 0 . 5

0.2 V 2 / V I = 1 . 0

0 1 2 3 4 5 6

E2/E 1

Fig. 1. Laminate anisotropy of parameter r/0 as a function of moduli ratio.

1.1

0 .9

L

7

0 . 8 - -

1"1 o _-

0 . 7 --" ~ ~

0.6;_- ~ - - E 2 / E ] = 1 . 0

0 . 5 ~ . . . . . EJEI=3&II3 ~2/~1 =2"0"~ ~" "~" " "

- - - - Ez/E I = 10 & 1/10 v / v = 1.o

-0 .5 -0 .4 -0 .3 -0 .2 -0.1 0 0.1 0.2 1)]

F i g . 3 . L a m i n a t e a n i s o t r o p y o f p a r a m e t e r ~/0 a s a f u n c t i o n o f

P o i s s o n ' s r a t i o .

1.2

1

0 . 8

0 . 6

0 . 4

0 . 2

qo - . • • . •

Ez/E I = 1.0 " - .

E2/E 1 = 3 & 1/3 " • .

. . . . . . . E 2 / E l = 1 0 & 1 / 1 0

v z / v I = *O2 / * t ) I = 1 , 0

~ , I J , ~ I , , , I , , , I , I I J I I I [ , , , I

- 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 .4

1) 1

Fig. 2. Laminate anisotropy o f parameter r/o as a function o f P o i s s o n ' s r a t i o .

I . . . . I ' ' ' I ' ' ' L ' ' ' I ' ' '

. . . . "a2/~L = 0.5 / 0 . 9 5 ~ ' ~ _ "a2/'a =].0 /

'.x\ / r '3 ;{'

E , ) 5 0 . 8 ~ - 'o t = 0 . 2 5 " . " . ~ - - ~ . J ' "

L E2/EI = 3.0

0 . 7 5 j , , i , , , i , , , i , , i , ,

0 0.2 0.4 0.6 0.8 V

I

F i g . 4 . L a m i n a t e a n i s o t r o p y o f p a r a m e t e r ~/o a s a f u n c t i o n o f

v o l u m e f r a c t i o n .

Appendix B. Angular distribution functions for anisotropic stress fields

The crack-tip stress field in a n anisotropic material with components of elastic compliance referred to crack tip axes ( 1 - 2 - 3 ) shown in Fig. 2 is given by Eq. (8 ) , where (Sih and Leibowitz, 1968; Cherepanov, 1979)

[ ,2 ,, ]) f l l (8 , ~:) = Re [ st I _ ~2 x/cos 0 + st2 sin 8 - x/cos O + ~:1 sin #

{ l [ " ]/ f22(0, ~) = Re g:l - ~% x/cos 0 + ~:2 sin 0 - x/cos 0 + g:l sin 0

f , 2 ( 0 , ~ : ) = R e { ~:,~:z [ 1 1 ] } ~] - ~2 x/cos 0 + ~:l sin 0 - x/cos 0 + ~:2 sin 0'


gll (0, s ~) = Re ~ - ~:2 [ x/cos 0 + ~:2 sin 0 - x/cos 0 + Ct sin 0

, , ]} e,22(0,¢) =Re -~:2 ¢ c o s O + ¢ 2 s i n O - x / cosO+¢ls inO

g l 2 ( 0 ' " = R e { ~ l 1 [ x / c o s 0 ~ , l s i n 0 - - s~2

217

(B.1)

where ~:l and ~2 are the roots (with distinct and positive imaginary parts) of the characteristic equation (9) .

Shielding due to aligned microcracks in anisotropic media

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