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DISORDER AND SCALING IN REGULAR AND HIERARCHICAL COMPOSITES

P.M. Dept.

Duxbury,

of Physics and Astronomy and

Center for Fundamental Materials Research, Michigan State University.

ABSTRACT We first

summarize the scaling behavior

of breakdown

strengths and

transport and elastic moduli of random two phase composites. sider the effect of disorder on two hierarchical

structures.

We then conThe first,

a

tree-like structure illustrates the fact that disordered trees, such as occur in many natural circulatory systems, conductance (CFC),

and current

flow.

leads us to suggest that,

hierarchical

show large fluctuations

in

their

The second, a continuous fiber composite due to its

greater flaw tolerance,

microstructural design may improve

the longitudinal

a

and

transverse toughness of CFC's.

I.

INTRODUCTION Disorder is

ubiquitous in synthetic and natural materials.

examples include fiber reinforced composites,

porous rock) and synthetic (e.g. catalyst carriers, cellular materials and alloys.

bone,

packaging materials)

Understanding of the effect of disorder on

elastic and transport properties is

quite sophisticated.

Although under-

standing of the effect of disorder on fracture and breakdown is advanced,

Important

natural (e.g. wood,

there has been considerable recent progress.

some previous work on disordered structures,

less well

After summarizing

we consider the effect of dis-

order on the properties of two broad classes of hierarchical structures. The first

are tree structures,

ring in

many natural systems.

electrical

which are typical of the geometries occurWe describe

the effect of disorder on

current flow in such structures as a paradigm for the effect of

flow properties in general.

Secondly, we consider the effect of disorder

on a synthetic continuous fiber composite gineering community. certain advantages

(CFC)

We argue that a hierarchical

of interest to the enconstruction

may have

over standard CFC microstructures due to its greater

flaw tolerance. The effect of a disordered microstructure on the moduli and strength of random systems can be calculated using one or more of the following 1 2 methods: rigorous bounds; effective medium theories; numerical Mat. Res. Soc. Symp. Proc. Vol. 255. 01992 Materials Research Society

322

simulations3 and scaling theories.4

Illustrative numerical

results for

diluted 2-dimensional networks are presented in Figure 1. 1 0

•

,'

+

0.8 + +

0.6

0.4

+

V v

+

V

0.2 GO

V

0.60

0.80

0.70

1.00

0.90

P

71 °F° --

0.8i-

0,40

00

0

0,%

0,0a

0.0

Figure 1.

(a)

b (fV)/jb(0) works. (b) b(P)/O00()

0.0

0.1

The conductance

0.3

O(f v)/GO0(+)

and breakdown

current

(V) of diluted 10000 node square lattice random resistor netThe elastic modulus B(p)/E 0 (0) and tensile strength of diluted 2500 node triangular lattice,

central

force spring

networks (p=l-f V).

In both lattice

and continuum two phase composites,

the effective

transport and elastic moduli may be calculated perturbatively in the dilute incl