Fracture in multilayers

Fracture in multilayers

Scripta METALLURGICA et MATERIALIA Vol. 27, p p . 687-692, 1992 Printed in the U.S.A. Pergamon Press Ltd. All rights reserved VIEWPOINT SET No. 19 ...

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Scripta METALLURGICA et MATERIALIA

Vol. 27, p p . 687-692, 1992 Printed in the U.S.A.

Pergamon Press Ltd. All rights reserved

VIEWPOINT SET No. 19

FRACTURE

IN MULTILAYERS

Peter A n d e r s o n Materials Science and Engineering D e p a r t m e n t , Ohio State University, Columbus, OH 43210 I . - H . Lin and Robb T h o m s o n Materials Science and Engineering Laboratory National Institute for Standards and Technology, Gaithersburg, M D 20899

(Received July 6, 1992)

1. Introduction Fracture in multilayers is a subject about which very little is known. One of the only papers on the mechanisms of interracial fracture of multilayers at the microstructural scale is that by Hirth and Evans[1 ]. General references to the experimental literature are given in the paper by Foecke and Lashmore in this series. The main thrust of this paper, however, will be the discussion and interpretation of a single observation of an interracial crack in a Cu/Ni multilayer by Lashmore[2], and since the theory is so little developed, we will often be projecting what we know about interracial cracking into the multilayer world. For presentation of the results by Lashmore, again, see the paper in this series by Lashmore and Foecke. Some preliminary interpretation of this micrograph is reported in Reference [2], based on extrapolation from dislocation emission theory in homogenous solids. It was suggested there that the fracture toughness of multilayers may be higher than in the corresponding homogeneous materials because emitted dislocations will shield the crack more forcefully due to the limited mobility of shielding glide dislocations in the multilayer. It was also proposed that a brittle transition might be caused by the very dense region of misfit dislocations emitted onto the cleavage plane in Mode II configuration. In this paper, the following two sections will deal with the continuum properties of dislocations and cracks in multilayers, and in §4, we will present some recent results of atomic modeling of cracks and dislocations--all in an attempt to gain some preliminary insight into failure mechanisms in multilayers. 2. C o n t i n u u m Crack/Dislocation Calculations in a Thin F i l m . There are in the literature calculations for a crack in a thin film[31, a crack in a thin film with an interface[4], and for dislocations in a thin film with an interface[5]. The problem of dislocation emission and shielding at cracks has also been treated by Shive, et a/.[6] for a surface crack at an interface between two blocks, but the problem of emission and shielding at a semi-infinite crack in a thin bilayer has not been treated to our knowledge. We will list here some results obtained by one of us (I.-I~.L) for anti-plane strain in this case. Conformal mapping was used to obtain the anti-plane stress potentials for a bilayer film with an interface through its middle. It is found that the critical value of the stress intensity, kIlle, for emission on the interface plane is governed by the effective elastic modulus, Pelf -- 2 p l p 2 / ( p l + p2). kZll~ = 22~/2g~o ~'

Ie is a factor depending on the overall film thickness, and x0 is the core size of the dislocation. Interaction between multiple dislocations on the interface is also governed by the same effective elastic constant. Free surface effects act to decrease kllle for films relative to an infinite interfacial system. Dislocation-dislocation 687 0956-716X/92 $5.00 + .00 Copyright (c) 1992 Pergamon Press Ltd.

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The energy function is bilinear (see the solid line in Fig. lb), with extrema at the interfaces given by 0, U~, where ~0 hI M1 al - a2 (7) Ua = baldy = bhl Mla2h' al 1 + M2ath2 The corresponding variation in energy due to image effects is addressed using the analysis of Kamat, et al.[7] for a screw rather than misfit dislocation, with b = b_% and _~= e_r, in a three layer system for which the interfaces at y = - h 2 , hi + h2 in Fig. l a are extended to 4-oo by setting h2 ---* oo. This approximation to a periodic multilayer is shown by Kamat, et a/.[7] to be reasonable for ratios of shear moduli between 1/3 and 3, with hi = h2. The shear stress at position y = c, due to image dislocations produced by the two interfaces present, is = - 4-;-

2.-1 rt=l

nh(- c

(s)

( n - 1)hi + c

where t¢ = (~t2 - btl )/(/~2 + #1). Again, integration of the energetic force acting per unit length of dislocation from a core cutoff distance, y = r0 to y = e, c < hi yields

Ui...e -

"1b2 [tcln (1-c')c' + ~-~ tc2"-lln ( n - c ' ) ( n - l +c')] 4~r

r-----~o

t1=2

- ~ (n "----i )

'

(9)

where prime (i) denotes normalization with respect to hi, and rh << 1 is assumed. The same procedure may be used to determine Uimage for a dislocation residing in material 2. For modest differences in the elastic moduli, the first term in eqn. (9) is a reasonable approximation to Uimagt. The dotted line Fig. l b schematically shows Uimag~ as a function of position in the multilayer. The extrema in Ui,,,~ge occur at the midpoints of each layer, where Ui(1) "" _

/~1b2 -tc--~rln(hl/4ro),

Ui(2) _~ + t ~

b2 ln(h2/4r0).

(10)

Compared to the difference in energy between an isolated dislocation in material 1 and material 2, the energy difference in a multilayer geometry is substantially less, due to the factor to, which modifies the multilayer solution. Each of the contributions to dislocation resistance discussed can be comparable to or larger than that produced by lattice Peierls barriers. The activation barrier associated with biaxial stresses increases with the difference, la2 - al h and with M1 hi, M2 ha. Energy minima and maxima occur at alternating interfaces. The corresponding activation barrier produced by image effects increases with layer thicknesses, hi and h2, relative to r0, as well to. Here, energy minima and maxima occur at alternating midpoints of layers, rather than interfaces. See Fig. lb. This additional lattice resistance suggests some unique aspects of crack-dislocation interactions, and corresponding implications for ductile-brittle response of crack tips. The Rice-Thomson model and extensions to it [9] distinguish between ductile and brittle response of a crack tip by whether a loaded, sharp crack tip is capable of first emitting a dislocation, or instead, cleaving. This approach, although dependent on inherently nonlinear features embodied in the core cutoff parameter, represents the competition process between dislocation emission and cleavage in terms of critical local stress intensity factors at the crack tip. Figure 2b shows a planar emission surface[8] with intercepts given by kit, klIe, and kIllt, based on the emission criterion that the incipient dislocation at a core distance from the crack tip is repelled from the crack. The cleavage surface is given by a critical value of kz = kle, based on bond breaking [8]. The outcome of the competition for a given loading path in local k-space is determined by whether the dislocation emission or cleavage surface is reached first. The local k description remains a valid approach when existing dislocations or other defects which shield the crack tip are at large distances from the crack tip compared to the dimensions over which the dislocation emission or fracture processes occur. This condition would appear to be satisfied for emission in homogeneous materials, where a dislocation emerging from the crack tip continues to glide through a dislocation-free zone,

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interactions are also decreased by the finite thickness of the film. On the other hand, the finite thickness increases the stress intensity, K , of the crack, for a given load and crack length, and thereby increases the forces on dislocations due to the crack. When the dislocation is emitted at an angle 0 # 0 in Fig. 2a, emission is easier on the soft side, as expected. 3. M o d u l a t e d R e s i s t a n c e to D i s l o c a t i o n M o t i o n

There are several distinct features of layered structures which, in principle, permit control of dislocation mobility by the choice of thickness, elastic moduli, and crystal orientation of the discrete layers. The net effect is to produce a periodic resistance to dislocation motion across layers, on a length scale equal to the individual layer dimensions in the structure. This resistance stems from (i) the periodic fluctuation of in-plane stress state produced by coherency or thermal stresses, (ii) the variation in dislocation self energy with position in the multilayer due to moduli differences, and (iii) activation barriers to slip across interfaces. To evaluate the first two effects, the multilayer is assumed to consist of a periodic arrangement of N distinct types of isotropic layers, each described by a thickness hi, elastic modulus Ej, and Poisson's ratio vj. The multilayer is assumed to have two equivalent, orthogonal, in-plane directions along which there is equal extension or contraction, but no shear. Thus an equi-biaxial stress state exists in each layer. The magnitude of stress in each layer is estimated as ~J = a - a__.____tj,

"i = Miti,

aj

(2)

where Mj = E J ( 1 - vj) is the biaxial modulus[5a], a is an arbitrary dimension along one of the in-plane directions, and aj is the corresponding value of a in layer j when stress-free, i.e., when detached from the multilayer. In the case of coherent interfaces in a cube--on-cube epitaxy, for example, the aj are conveniently chosen to correspond to the stress-free lattice parameters in each layer. Applying the condition that no macroscopic in-plane loading exists, N

. , h , = o,

¢a)

i=l

permits solution for a, and the resulting biaxial stress state in layer j is given by

a.....Li= Mj

N

h

Y~.i=l[Mi il - 1. N aj Ek=1[Mkh~lak]

(4)

For the case, see Fig. la, where only two layer types exist (N = 2), the biaxial stress in layer 1 is given by ~I =

M1

a2 - al

1Jr -M~a2hl

al

- - ,

M2alh2

(5)

and an appropriate exchange of indices (1,2) yields the corresponding biaxial stress in layer 2. For application to a so called "misfit" dislocation at position tt = c, with in-plane Burgers vector b = be_z and dislocation line sense _~= _% corresponding to an extra half plane of atoms for y < c, the y-component of the energetic force acting per unit length of dislocation line is f,=+bcrzz.

(fi)

The energy of the dislocated multilayer, as a function of dislocation position c, is given by the work done on the body in moving the dislocation from the origin, at which the energy is arbitrarily set to zero, to y = c.

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of dimension equal to several hundred b, until arrested by lattice resistance. When the multilayer length scale becomes considerably smaller than the dislocation-free zone, existing dislocations are suppressed from moving away from the crack tip, and departures from the local k approach occur. Specifically, the stress field in the crack tip region over which the dislocation emission and cleavage processes occur is no longer uniquely characterized by the local stress intensity factors, due to the singular stress fields of nearby dislocations. (See §4 below.) On the basis of linear elastic theory, the "nonlocal" shielding leads to a relative displacement of both surfaces. For the case of an existing dislocation at position (R, 0), with the same sign as the incipient dislocation to be emitted, the emission surface displaces by [10]

kit

= fi(O)

+ gi(0)

,

(11)

where i = I, I I , I I I denotes the mode of loading and r0 is the core size used in the emission criterion. For the special ease to be discussed in §4, where the slip plane is 0 = 0, the first dislocation is emitted at a critical Mode II loading, #b kilt =

2(1 - ~) 2,/~-ff'

(12)

and the increase, A k l t , , for emission of the second dislocation is given by Eqn. (11), with fli(O = O) = 0 and gzi(O = 0) = 2. The corresponding displacement in the cleavage surface intercept is too complicated to go into here, but is smaller than Eqn. (11). Thus, existing dislocations are predicted to suppress continued emission on the same slip system considerably more than cleavage. Further, the increase Aki~ can be a considerable fraction of ki~ when existing dislocations are within 10b - 20b of the crack tip. An enhanced nonlocal effect in multilayers suggests that the fundamental outcome of dislocation emission/cleavage processes is changed. In simulations of dislocation emission from crack tips using elastic crack and dislocation theory, a transition from an emitting crack to a cleaving crack is observed when available slip systems at the crack tip become saturated with emitted dislocations. The implication is that emission to cleavage transitions in multilayers may be controlled by the coherency and image effects which control dislocation motion. 4. A t o m i c M o d e l s

We have developed a lattice Green's function technique for the calculation of the structure of equilibrium cracks on interfaces under mixed tensile/shear load, during dislocation emission[ll], and have applied it, in work still incomplete, to the hexagonal lattice. The results, so far, of that work indicate that several new phenomena associated with dislocation emission may be important in determining the toughness of multilayers. In an atomic model, there are more elastic parameters than in elastic theory[12], because the spring constant between two lay,~rs may be different from that of either bulk phase. When the interfacial spring constants are weaker than either bulk spring constant, we find that the lattice crack extension force can differ considerably from the elastic prediction. This elastic effect at the interface is independent of the bond strength between the layers. The criterion for emitting a dislocation on the cleavage plane (0 = 0 in Fig. 2a) also depends sensitively on the spring constant between the layers, as well as on the strength of the bond. We find that dislocation cores are generally much broader on the interface (for typical values of the parameters) than in either bulk phase, and this means that emission will be easier on the interface. Thus in the language of §2, the emission criterion in the lattice is considerably more complicated than the interfacial Mode I / I I continuum prediction of Eqn. (1). But we do find that the emission does reflect, other things being equal, the magnitude of the appropriate effective modulus for Mode I/II, analogous to that given in Eqn. (1) for Mode III. One of our primary results is that, in the absence of misfit stress, the Mode II loading for emission is very asymmetric in s i g n - - t h a t is, it is hard to emit dislocations for Mode II shear load of one sign and easy for the other. This finding is expected from the continuum theory, and is, in part, a consequence of the shear stresses at the crack tip generated by the elastic mismatch at the interface. More interesting is the fact that cleavage is quite sensitive to the sign of the Mode II loading. By contrast, in studies of cleavage under mixed

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loading in homogeneous materials, we found that the cleavage condition depen~led only very weakly on Mode II loading. The second primary result is associated with the nonlocal embrittling effect of nearby dislocations, introduced in §3. Fig. 3 shows the structure of a crack emitting a dislocation in the presence of a prior shielding dislocation in its vicinity. From computed values of p, t, and k1ie for the first dislocation, Eqn. (12) infers that rQ = 1.lb. From the computed values of kIle to emit a second dislocation, with an existing dislocation at R --- 16b, Akile/kli~ = 0.17. This "measured" value from the atomic simulations is much larger than the value, AkllJklIe = 0.03, calculated from the elastic estimate in Eqn. (11). Thus the nonlocal effect in the lattice is predicted to be much larger than the elastic estimate, at least in Mode II emission. 5. C o n c l u s i o n s

Preliminary findings for fracture of multilayers suggest that these materials should be very interesting. The materials are known to have high yield strength relative to their bulk constituents, and in the case of Cu/Ni, appear to cleave on interfaces. The high yield strength may occur from lack of dislocation sources in carefully prepared material, but also from resistance to dislocation motion produced by coherency and thermal stresses, image forces from nearby interfaces, and activation barriers to slip across interfaces. These effects are controlled by the choice of layer thickness, constituent modulus, and epitaxy. In the context of the Rice-Thomson model, dislocations emitted from an interfacial crack in a multilayer are not expected to glide as far from the crack tip as compared to a bilayer, for example. One implication for fracture toughness is that nearby dislocations will more effectively shield the crack. However, elastic theory and observations of emission in a nonlinear atomistic model both suggest that the close proximity of emitted dislocations suppresses continued emission and favors a transition to brittle, cleavage behavior. The final outcome of these competing effects, in terms of an increase or decrease in fracture toughness, requires further analysis beyond the single dislocation studies presented here. References 1. J. P. Hirth and A. G. Evans, J. Appl. Phys., 60, 2372 (1986). 2. D. S. Lashmore and R. Thomson, J. Matl. Rsh., in press. 3. G. C. Sih and E. P. Chen, J. Franklin Inst., 290, 25 (1970). 4a. S. Chu, J. Appl. Phys., 53, 3019 (1982). 4b. Z. Suo and J. W. Hutchinson, Int. J. Fract., 43, 1 (1990) 5a. M. Ovecoglu, M. Doerner and W. Nix, Acta Met., 35, 2947 (1987). 5b. A. K. Head, Phil. Mag., 44, 92 (1953). 6. S. Shive, C. Hu and S. Lee, Marls. Sc. Eng., A l l 2 , 59 (1989). 7. S. V. Kamat, J. Hirth and B. Carnahan, Scripta Met., 21, 1587 (1987). 8. I.-H. Lin and R. Thomson, Acta Met., 34, 187 (1986). 9. J. R. Rice, J. Mech. Phys. Sol., 40, 239 (1992). 10. P. Anderson, S. J. Zhou, and R. Thomson, to be published. 11. R. Thomson, S. Zhou, A. Carlsson and V. Tewary, to be published. 12. J. R. Rice and G. Sih, Trans. ASME, 87, 418 (1965).

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h2

1

U

hz

7-I

h2

=i (a) (i)

27, No. 6

U

Z

2

Vol.

Y (m

Ul

(b)

Fig. 1. (a) Multilayer geometry. (b) Potential energy of a dislocation as a function of position, c, due to in-plane stresses (full line), and (ii) image effects (dotted Une) generated by the interfaces.

ki (a)

(~)

Fig. 2: (a) Crack tip geometry. The actual dislocation appears at radius R, while the effective distance is Rell. (h) emission and cleavage surfaces in local k-space. Emission intercepts occur at ki,, kIle and kIlI,, respectively, and the cleavage intercept is at kic.

Fig. 3. Structure of the cohesive zone of a crack in a homogeneous hexagonal lattice under mixed Mode I and II loading. (This example is not for the more complicated interracial case, which we do not have the space to discuss.) The crack is in static equilibrium, and is on the point of emitting a dislocation onto the cleavage plane ahead of the crack (Mode II emission) in the presence of a pinned dislocation on the same slip plane. The first dislocation is at the atom whose color is black near the right end of the cohesive zone. The force exerted across the cleavage plane by the atoms is shown by the degree of lightness of the filled circles. (Zero force is black.) The filled circles give the radius of the atoms, the open circles show the range of the nonlinear force law.