Elastic stresses in anisotropic bicrystals

Materials Science and Engineering, A169 (1993) 43-51

43

Elastic stresses in anisotropic bicrystals P. Peralta, A. Schober and C. Laird Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104-6202 (USA) (Received December 22, 1992; in revised form March 8, 1993)

Abstract The stress fields close to a grain boundary of two incompatible bicrystals subjected to axial tension were determined by finite element method (FEM). One of the bicrystals had an asymmetrical [ 110] tilt boundary and the other was an isoaxial [i49] bicrystal with a twist boundary. The boundaries were perpendicular to the tensile axis. The effects of size, type of incompatibility and boundary conditions were studied and the results were compared with FEM calculations for different geometries and an analytical model. It was found that the stresses arising at the boundary have a constant value in the center of the cross section and high gradients close to the surface. Furthermore, the average value of the stresses calculated was in agreement with the values predicted by an analytical model. The differences between the model and the FEM results depended on the type of incompatibility,the geometry and the boundary conditions.

1. Introduction

The important role that grain boundaries play in the mechanical behavior of polycrystalline materials is well recognized [1]. In particular, recent work, experimental as well as theoretical, has related the deformation and fracture behavior of several materials to the elastic stresses arising in elastically incompatible grain boundaries [2-5]. Even though calculations by finite element method (FEM) have been carried out regarding this problem [4-8], there is only one detailed report of the elastic stress field around an incompatible grain boundary [8], to the best of the authors' knowledge, and, in general, only the derived effects of such fields are reported [4-7]. Given the importance of these stresses in the nucleation of high cycle fatigue cracks in or next to incompatible grain boundaries [4], and in order to verify the assumptions made in one of the models proposed for approximate calculations of such stresses [9], the elastic stress fields of two elastically incompatible bicrystals, subjected to uniform tension with the grain boundary perpendicular to the tensile axis, were calculated by using FEM. The results are discussed in terms of the stress fields obtained by Kitagawa et al. [8] for elastically incompatible bicrystals with the grain boundary parallel to the tensile axis and the piecewise uniform stress field model for anisotropic bicrystals described in ref. 9.

2. FEM model of the bicrystals

The general purpose Abaqus (version 4.9) program was used for the F E M analysis. The mesh used to 0921-5093/93/$6.00

simulate the bicrystals was made up of 2200 8-node linear displacement bricks. The layers of bricks were placed as shown in Fig. 1 and had 100 bricks each, except for the layers on either side of the interface which had 400 bricks. Two kinds of boundary conditions were used. In the first type mixed boundary conditions were used. That is, the bottom layer of nodes was fixed in space and the top was not. Thus the displacement in direction 3, the longitudinal direction, was restrained for all nodes and displacement in any direction was restricted for the corner nodes. A normalized distributed load was applied on the top faces of the last layer of elements in the plus 3 direction, effectively subjecting the entire bicrystal to a uniform tensile load. In the second type of boundary condition, the minimum constraints necessary to avoid rigid body motions were applied and a normalized distributed load was applied on the top faces of the last layer of elements in the plus 3 direction and bottom faces of the last layer of elements in the opposite direction. The distributed load was normalized to one, in order to obtain all results per unit of applied stress, taking advantage of the linearity of the system. The output requested the values for a~ 1, o12, O"13, 022 , 023, and a33 for the 800 elements at the interface. These values were averaged at the centroid of each element, so they actually describe the stresses at small distance away from the interface. Values for the stresses at the nodes that made up the central axial fiber of the bicrystal were requested as well. Four separate trials were run on the FEM program, each assuming a different height to width ratio for the bicrystal, i.e. 10: 1, 6 : l, 1 : 1 and 1 : 2, the ratio width to thickness was kept constant and equal to 1. © 1993 - Elsevier Sequoia. All rights reserved

P. Peralta et al.

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Elastic stresses in anisotropic bicrystals

~ (i14)

J

[11o]~

iiii

~

. I (2~1)

[1101

(a) (i49)

J 2

Fig. 1. F E M mesh for the bicrystals.

[1~.1]

[(1~) Co)

The material selected, copper in this case, has an anisotropy factor of approximately 3.27, therefore the stresses produced to satisfy compatibility at the grain boundary should be a significant fraction of the applied stress according to ref. 8. Both the material and the crystallographic orientations for the bicrystals used in this work were taken from the experimental research program being carried out by two of the authors [10]. As seen in Fig. 2, the orientation of the top crystal in bicrystal 1 was such that [110], [~21], and [114] corresponded to axes 1, 2 and 3 respectively. The bottom crystal had the same [110] orientation as the top crystal but the axes 2 and 3 now corresponded to [1i4] and [22 i]. This is clearly an asymmetrical [110] tilt boundary. Bicrystal 2 is an isoaxial bicrystal with a twist boundary, where one of the crystals was rotated 90 ° about the 3 axis, i.e. the []49] direction, with respect to the other. The orientation of the single crystals making up bicrystal 1 is such that, when each of them is subjected to uniform tension, e,l is the only term causing incompatibilities in deformation, whereas for bicrystal 2 all three strain components giving place to incompatibilities, i.e. ell, e22 and el2 , are different for both crystals. In bicrystal 1 the difference between ell from one crystal to the other is 0.585. For bicrystal 2, Aell =Ae22 = 0.310 and Ae~2 = 0.238. The values of the strains ell , e22 and e12 and their differences for both bicrystals are shown in Table 1. The values shown there are per unit of applied stress with the elastic

Fig. 2. Crystallographic orientation of the bicrystals: (a) bicrystal 1, (b) bicrystal 2.

TABLE 1. Elastic strains el~ , 822 and el2 in the component single crystals of the studied bicrystals when subjected to unitary axial tension Bottom crystal

Top crystal

IAel

Bicrystal 1

t,t %2 e, 2

- 0.556 -0.417 0.000

0.029 -0.417 0.000

0.585 0.000 0.000

Bicrystal 2

e,~ e22 el2

-0.266 -0.577 -0.119

-0.577 -0.266 0.119

0.310 0.310 0.238

constants in the appropriate units. The difference in the kind of incompatibility for both bicrystals allowed us to study how this parameter, which is closely related to the type of grain boundary involved, influences the stress field around the interface. The appropriate elastic constants were taken from ref. 11 for the cubic axes of Cu, and then were rotated to the crystallographic orientations of the bicrystals used, by the appropriate coordinate transformation for fourth order tensors, through the rotation tensor Q, relating the cubic axes and the crystallographic directions of each single crystal. This transformation can be

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Elastic stresses in anisotropic bicrystals

expressed in index notation as

lem. 055 is equal to 1 in both sides of the grain boundary and o23 and oj5 are both equal to zero. This is consistent with one of the assumptions stated by Gemperlovfi et al. [9] as necessary to approximate the elastic stress field next to an elastically incompatible grain boundary of an anisotropic bicrystal to a piecewise uniform stress field, which is that the stress components in the direction of the normal to the grain boundary must be prescribed by the boundary conditions. The stress maps at the interface for or, , 022 and o,2, the "compatibility" stresses [9], shown in Figs. 5 and 6, indicate that these stresses are discontinuous, i.e. there is a jump in their value across the grain boundary, and, at least for this case, this jump involves a change of sign, as expected according to ref. 9. Note that, in bicrystal 1 (Fig. 5), both o,~ and 022 have stress gradients similar to those in o55, and that, in all cases, the stress gradient is higher on the surface that is perpendicular to the 1 direction. This behavior, along with the fact that Ol, has the highest value among the compatibility stresses, is consistent with the fact that the incompatibility between the individual crystals is only in e~j, hence, high normal stresses in the 1 direction are to be expected, with the corresponding effect in the singularity at the edges. This also can explain the difference between 0.~ and 0.23, since the former has a higher maximum value and higher gradient than in the latter. The behavior of 0.t~ and 0.22 at the central region of the

(1)

C'ijk! = C°a/,,tQia Qjb Qk~Q,o

45

3. Results and discussion 3.1. Stress mappings Figures 3 and 4 show the variation of the stresses 0"g3 (i = 1, 2, 3) on both sides of the grain boundary of the two bicrystals, for the height to width ratio of 6 : 1. The results were the same for the mixed and traction boundary condition cases for high height to width ratios, which is to be expected from St. Venant's principle. From the mappings for 0"53, it is clear that there are high stress gradients close to the surfaces of the bicrystals perpendicular to the 1 and 2 directions, precisely where the shear stresses reach a maximum value; there is also a clear tendency for those stress gradients to change signs when crossing the grain boundary. A singularity in the stresses is to be expected in these regions according to ref. 3, and that explains why the continuity condition for some of the 0"i3 is violated at those regions, the effect being more pronounced for 0"53 in both cases, probably owing to the coupling of the stresses produced by the anisotropy. The value of some stress components in this region is arbitrary and depends on the chosen mesh due to the singularity. Note, though, that the values of these stresses at the central region of the cross section are in agreement with the boundary conditions of the prob-

13 _

23 ~o

53 o c o.C o c o.C o c ~o.C ~o c

~oq

(a) 5 ; g r n~

33

~ ; g r~ °

23

o o~' 0 o~ o02 0.0 o ~0.02

o %%-.

(b) Fig. 3. Stresses oi3 for bicrystal 1: (a) below the grain boundary, (b) above the grain boundary.

~¢r

"c

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Elastic stresses in anisotropic bicrystals 23

33

~T~FO O

13

~;groo

-1-

~..

(a) ~

33

5;g~c

2~

13

~igc °~

(b) Fig. 4. Stresses a~3 for bicrystal 2: (a) below the grain boundary, (b) above the grain boundary.

Sicj r~O

11

~;u co'°

22 5;,g ¢r" c

~2

o oe

o.o

o o~

o .~

o.o

o.O~

o~ o

o.O

o.OO

o.O

~o.O 2

o. ~

o.o -oq

o.

(a) S;g~G

11

~ j "22

~

12

o.1

o~

~o.O

~o ~

~o. ~

o.O 0.0

~o.~

~0.0 ~ o . I.

(b) Fig. 5. Compatibility stresses for bicrystal 1: (a) below the grain boundary, (b) above the grain boundary.

cross section is in good agreement with the assumption of homogeneous stresses in bicrystals [9], given the bathtub-like shape of the stresses. The other assumption given in ref. 9, according to which the stresses all , ire2 and 012 in one side of the grain boundary have

exactly the same value as in the other side but with opposite signs, is partially met here. Actually only 0 ~ is close to satisfying this requirement, since, even though the values for 022 and or2 have opposite signs when crossing the boundary, their absolute values are differ-

P. Peralta et al. 9;gr n°

/

11 5;go oo

~o o5 ~o°°° Ic~I

47

Elastic stresses in anisotropic bicrystals 22

~2

015 0.10

F

OO5 0

~

x

(a) ~2

12

0

o o°° ~ o o ~o

0

~o.O2 o O.C

~o.OJ o _oo ~o

O.C

~oo5 o

(b) Fig. 6. Compatibility stresses for bicrystal 2: (a) below the grain boundary, (b) above the grain boundary.

ent. The absolute value of o1~, though, does not change as drastically as for 022 and o12. Calculations using the model proposed in ref. 9 were carried out for bicrystal 1, and the values obtained for the stresses below the grain boundary, were: all = 0.4184, 0"22 0.1 190 and o12 0. These values were found to be approximately equal to the average of the absolute values of the compatibility stresses at both sides of the boundary. Another interesting characteristic of the compatibility stresses is that their absolute values are, in general, higher in the crystal above the grain boundary. For oll the difference with the respect to the highest value at the center, is about 34% and for o22, about 76%, whereas o12 is zero at the center point at both sides of the boundary. For bicrystal 2 (Fig. 6), the stress gradients in Oll and o22 are not as pronounced as those for bicrystal 1, and they are approximately the same for directions 1 and 2. This is consistent with Ol~ and 022 having more or less the same absolute value, which is lower than the value of oll for bicrystal 1. The fact that oll and 022 have approximately the same value, but opposite signs, can be explained by noting that the differences between eu and e22 for the two single crystals making up bicrystal 1 are equal in magnitude, but with different signs, as can be deduced from Table 1. Therefore, the terms forcing the compatibility stresses 011 and 022 are almost the same for both of them, but, once again, with opposite signs. This line of reasoning =

=

can also be applied to explain why 023 and 0t3 have practically the same absolute value and behave in the same way. The distribution of the compatibility stresses for this bicrystal is also consistent with the assumptions of Gemperlovfi et al. [9], since o~1 and 022 are rather flat, mostly in the center of the cross section. The same applies to o12 but to a lesser degree. In regard to the assumption about the stresses being equal but having opposite signs on both sides of the grain boundary, the absolute values of ol i on both sides of the grain boundary differ by 40.5%, for 022 the difference is 40.5% and for o12, 0.5%. Calculations using the model proposed [9] resulted in the following compatibility stresses below the grain boundary: a ~ = - 0 . 1 1 2 5 , o22=0.1125 and a12 =0.0342. Once again the average of the absolute values of the stresses given by the F E M calculation agree very well with the values given by the model. Note also that the agreement between the values of o12 obtained by F E M and by using the model is very good, since they agree within a 4% difference. This is not surprising since the shear strains e12 in the two single crystals of bicrystal 2 are equal in value but opposite in sign, as shown in Table 1. It can be said, that in general, the agreement between the F E M calculations and the model given in ref. 9 is much better for bicrystal 2 than for bicrystal 1, which is probably the result of a more homogeneously distributed difference between the strains producing incompatibilities in bicrystal 2.

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Elastic"stresses in anisotropic bicrystals

3.2. Effects of size and boundary condition The effects of size of the bicrystals and the applied boundary conditions on the stresses are shown in Figs. 7 and 8. The stress profiles for oll and 022 along a longitudinal fiber are shown in those figures for different height/width ratios, the two types of bicrystals analyzed and the two kinds of boundary conditions used. The four height/width ratios used are shown in Fig. 7, whereas only three height/width ratios were shown in Fig. 8, since the results for the 10:1 and 6:1 cases are identical for both boundary conditions. The 6:1 case was kept in Fig. 8 for comparison purposes. In order to plot all sets of results together, the longitudinal coordinate is normalized by the bicrystal height (h). It is clear from Figs. 7 and 8 that the main effect of decreasing the height to width ratio, regardless of the boundary condition, is that of spreading the zone around the grain boundary that is affected by the compatibility stresses until the whole bicrystal is subjected to those stresses. This is exactly equivalent to what

0.4'

Kitagawa et al. [8] found by changing the width to thickness ratio in their bicrystals. Note from Figs. 7(a) and 8(a) that the maximum and minimum values for Oll in the grain boundary of bicrystal 1 are rather insensitive to the change in the size of the bicrystal and the boundary conditions used, and the same applies to the minimum value of o22. The maximum positive value of o22 was found to depend both on the size of the bicrystal and the boundary conditions; note also in Fig. 7(a), that even for the cases when the restrictions at the end are far away from the grain boundary, there still is a small effect of the mixed boundary conditions in the stress profile, since o22 does not go neatly to zero for the 10:1 and the 6 : 1 cases, as a~ ~ does. Moreover, the maximum in 022 was found not to be in the boundary as expected, but shifted from that position. The position of the maximum was also found to be size dependent, since it moved from being very close to the grain boundary for the 10 : 1 case, to being practically at the base of the bicrystal for the 1:2 case, precisely where

0.1

0.0 '; ~r ~ ' ~

.

.

.

.

FY/1'I

=

-

-0.2

E

• 1:1 6:1 • 10:1

ffl -0.4 -0.6

Of)

,~

•

I

-02

"

-0.8 .0

0.2

0.4

0.6

0.8

..........

-0.3 / 0.0

1.0

•

1

I . 0.2 0.4

71rll

I

. . . . .

0.6

0.8

1.0

0.8

1.0

z/ h

z/h (a) o.15

• 1:2 • 1:1 " - ' ~ ~ , • 6:1 0.05 ~ :_: ~ .I

0.10 ~

0.1

) ~

•10:1 ,_ .~' ~ ~

~1 0.0

o

°

-0.05

~ l~

-0.1

-0.10 -0.15

-0.2 0.0

0.2

0.4

0.6

0.8

1.0

z/h

0.0

0.2

0.4

0.6

z/h £o)

Fig. 7. Axial variation of stresses tyl= and 022 for different height/width ratios with mixed boundary conditions: (a) bicrystal 1, (b) bicrystal 2.

P. Peralta et al.

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0.4

Elastic stresses in anisotropic bicrystals 0.15

J,

0.10 0.2

. ~

_L_IIL

0.05

"!..''---

0.0 -0.2

ca - 0 . 0 5

i'-N;

-0.4 "~

r/

¢~ 0.00

49

"

-0.10

--

-0.15 -0.20

-0.6 0.0

0.2

0.4

0.6

0.8

I

'

-0.25

1.0

0.0

0.2

0.4

z/h

0.6

0.8

1.0

z/h

(a) 0.10

0.10 I

,~

0.05~ 0.00 ;

0.05

2'

o.oo

m

-

m

=.,

~ -0.05

.~-o.o5 n • •

-0.10 -0.15 0.0

0.2

0.4

0.6

1:2 1:1

-0.10-~

6:1 F 0.8

I • •

-0.15 .0

1:2

1:1 6:1

Y

-T .

0.0

z/h

0.2

0.4

0.6

0.8

1.0

z/h

Co) Fig. 8. Axial variation of stresses Crl~ and 022 for different height/width ratios with traction boundary conditions: (a) bicrystal 1, (b) bicrystal 2.

the displacement restrictions were applied. This suggests that the behavior of 022 was being affected by the boundary conditions, in addition to the factors mentioned above. For the traction boundary condition case, Fig. 8(a), the behavior of o22 was very similar, but the maximum value of that stress was lower than in the previous case. Note, though, that the value of 0-22at the boundary is the same for all sizes except the 1:2 case, where the effect of the displacement restrictions for the case of mixed boundary condition increases the maximum value. In regard to bicrystal 2 (Figs. 7(b) and 8(b)), the behavior of both 0-11 and 022 is similar to that of all in bicrystal 1. The maximum and the minimum values of these stresses do not change much with either the size of the bicrystal or the boundary conditions, and they always occur in the grain boundary for the range of height/width ratios studied here. The effect of the boundary conditions can be observed in both 0-~1 and 022 since, in general, they are not equal to zero at the restrained end of the bicrystal, even for the high height/

width ratio cases, whereas they are exactly equal to zero in the case of traction boundary conditions. Note also that the maximum absolute value of Ol~ is attained in the bottom crystal, whereas 0-22reaches its maximum in the top crystal. This must be related to the sign of the stresses in each side of the grain boundary and the restrictions applied in one of the ends, amongst other factors. This behavior was also independent of size and boundary conditions. Note that, in general, the effect of the boundary conditions is to change the axial distribution of stresses for the low height/width ratios, a consequence of St. Venant's principle. For the mixed boundary condition case, the stresses at the restrained end have to be such that the displacement constraints are satisfied. This produces an effect in the distribution of stresses in the other part of the bicrystal, since the compatibility stresses have to accommodate themselves to satisfy the requirement of overall equilibrium, as stated in ref. 9, i.e. that the "areas" enclosed by the stress distribution have to be equal but opposite in sign in both sides of

50

P. Peralta et aL

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Elastic stresses in anisotropic bicrystals

the grain boundary. In the traction boundary condition case, both ends are rather free to move, then it is easier for the stresses to satisfy the overall equilibrium requirement. This is the reason why the stress distributions for the second type of boundary condition are more symmetric than for the first type. In regard to the other components of stress, in bicrystal 1 013 and oj2 were found to be both equal to zero along the central fiber of the bicrystal for all cases, as expected from the mappings close to the grain boundary, o23 varied rather erratically, but the values obtained from the output for this stress were also quite small and they were probably subjected to numerical errors. 033 showed variations with position along the bicrystal and also with size. In general, it reached a maximum and a minimum close to the grain boundary for the 10 : 1 and 6 : 1 cases, when mixed boundary conditions were used, and for all sizes for traction boundary conditions. It also reached the value of 1 at the ends, with small deviations from that value for the case with a restrained end. For the 1:1 and the 1:2 cases, the maximum value was reached in the restrained end, and 033 was found equal to 1 at the other, in agreement with the boundary conditions. In bicrystal 2, 0-~2was found to behave like o~ and a22 in the sense that the maximum and minimum values were always at the grain boundary, except for the 1:2 case when mixed boundary conditions were applied, where they occurred at the ends. The values at the grain boundary were rather insensitive to the change in geometry and boundary conditions. As mentioned above, the values obtained for o12 were quite similar to those predicted by the model of ref. 9. o13 and a23 were found to vary with position, but their values were quite small, the maximum being approximately 0.7% of the applied stress for both types of boundary condition; these values are likely to have numerical errors as in bicrystal 1. Regarding 033, the behavior was different from that of bicrystal 1, the variations with positions were smaller, being the maximum deviation from the expected value of 1 less than 2% for all the cases studied. This size effect may play an important role in fatigue behavior related to grain size, since the presence of the compatibility stresses usually favors the early development of multiple slip close to grain boundaries [1-5], formmg a region where multiple slip is present even in crystals oriented for single slip [1-5]. The size of this zone is undoubtedly related to the size of the zone affected by the compatibility stresses; hence, a small bicrystal or a small grained polycrystal material must have a higher volume fraction of multiple slip region close to the grain boundary than a large bicrystal or a large grained polycrystal. This is in agreement with the qualitative analysis proposed by Llanes [12], regarding

the effect of multiple slip in the fatigue behavior of polycrystalline copper.

4. Conclusions F E M analysis of the stresses near a tilt boundary and a twist boundary yields the following conclusions. ( 1 ) The normal stresses close to the grain boundary of an anisotropic bicrystal show a plateau in the middle of the cross section and high gradients close to the surface, which are consistent with the singularities that are expected in this kind of geometry. (2) The stresses computed by F E M close to the grain boundary for a high height to width ratio in the bicrystal have characteristics that are in general agreement with the assumptions for piecewise uniform stress fields in anisotropic media. It was found that the latter model provides reliable values for the average state of stress close to the grain boundary. (3) The results obtained suggest that the agreement between the F E M calculations and the values of the stresses predicted by the homogeneous piecewise stress field model improves as the incompatibility of the single crystals making up the bicrystal is homogeneously distributed among the three components of strains that have to do with the compatibility conditions at the grain boundary. (4) There are definite effects of the boundary conditions and the kind of incompatibility, and, therefore, the kind of grain boundary, on the values and distributions of the stresses next to the grain boundary and along the axis of the bicrystals. These effects have to be taken into account when predictions of the compatibility stresses are made by means of different from F E M or direct measurement.

Acknowledgments This work was supported by The Department of Energy under Grant No. DE-FG02-84ER-45188 and by the Laboratory for Research on the Structure of Matter which provided assistance with central facilities. This support is greatly appreciated. We warmly thank Dr. Sekar M. Govindarajan and Dr. Srivatsa M. Sharma for helpful comments and stimulating discussions and Maria Elena Fernfindez for her help in some of the preliminary F E M calculations.

References 1 J.P. Hirth, Metal Trans., 3(1972) 3047. 2 H. Vehoff, C. Laird and J. Duquette, Acta Metall., 35 (1987) 2877.

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Elastic stresses in anisotropic bicrystals

3 P. Neumann and A. T6nnessen, Proc. 5th Int. Conf. Strength of Metals and Alloys, (1988) 743. 4 P. Gopalan and H. Margolin, Mat. Sci. Eng., A 142 (1991 ) 11. 5 Z. Wang and H. Margolin, Metal. Trans. A, 16A (1985) 873. 6 T.-K. Chen and H. Margolin, Metal. Trans. A, 19A (1988) 1727. 7 T.-K. Chen and H. Margolin, Metal. Trans. A, 20A (1989) 1461. 8 K. Kitagawa, H. Asada, R. Monzen and M. Kikuchi, Sup.

51

Trans. Jap. Inst. of Metals, 27 (1986) 827. 9 .1. Gemperlova, V. Paidar and E Kroupa, Czech. J. Phys., B39 (1989)427. 10 P. Peralta and C. Laird, research in progress. 11 G. Simmons and W. Herbert, Single Crystal Elastic Constants and Calculated Aggregate Properties, The M.I.T. Press, Cambridge, MA, 1971, p. 26. 12 L. Llanes, Ph.D. Thesis, University of Pennsylvania, Philadelphia, PA, 1992.