Hybrids A-B

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Acta Materialia 51 (2003) 58015821 www.actamat-journals.com

Designing hybrid materials

M.F. Ashby a,, Y.J.M. Brechet b

a Engineering Department, University of Cambridge, Trumpington Street, CB2 1PZ Cambridge, UKb L.T.P.C.M., Domaine Universitaire de Grenoble, BP75, 38402 Saint Martin dHeres Cedex, France

Accepted 31 August 2003

Abstract

The properties of engineering materials can be mapped, displaying the ranges of mechanical, thermal, electrical andoptical behavior they offer. These maps reveal that there are holes: some areas of property-space are occupied andothers are empty. The holes can sometimes be filled and the occupied areas extended by making hybrids of two ormore materials or of material and space. Particulate and fibrous composites are examples of one type of hybrid, butthere are many others: sandwich structures, foams, lattice structures and more. Here we explore ways of designinghybrid materials, emphasizing the choice of components, their shape and their scale. The new variables expand thedesign space, allowing the creation of new materials with specific property profiles. 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Designing materials; Hybrid materials; Composites; Sandwich structures; Foams

1. Introduction: hybrid materials

1.1. Extending material-property space

Fig. 1 is an example of a material-propertychart. It shows the thermal conductivities of some2300 different materials, plotted against their

Youngs moduli. It is one of many, each a slicethrough material-property space; the assembly ofall the slices can be thought of as a map of thisspace [1,2]. All the charts have one thing in com-mon: parts of them are populated with materials

Corresponding author. Tel.: +44-01223-332-635; fax: +44-01223-332-662. The Golden Jubilee IssueSelected topics in Materials

Science and Engineering: Past, Present and Future, edited byS. Suresh.

1359-6454/$30.00 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/S1359-6454(03)00441-5

and parts are not. Some parts are inaccessible forfundamental reasons that relate to the size of atomsand the nature of the forces that bind their atomstogether. But other parts are empty even though,in principle, they are accessible. If they wereaccessed, the new materials that are there couldallow novel design possibilities.

One approach to thisthe traditional oneisthat of developing new metal alloys, new polymerchemistries and new compositions of glass and cer-amics so as to extend the populated areas of theproperty charts, but this can be an expensive anduncertain process. An alternative is to combine twoor more existing materials so as to allow a super-position of their propertiesin short, to create ahybrid (Fig. 2). The spectacular success of carbonand glass-fiber reinforced composites at oneextreme, and of foamed materials at another


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5802 M.F. Ashby, Y.J.M. Brechet / Acta Materialia 51 (2003) 58015821

Fig. 1. A material-property chart of thermal conductivity and Youngs modulus for 2300 materials. Each small circle is a plot ofthese properties for a real material. The large ellipses enclose, approximately, the circles for a given family of materials. A large

area of the chart is empty: there are no materials with high conductivity and low modulus. The challenge is to create hybrids that

fill the hole. (This and the other charts were created using the CES 4 software system, Ref. [42].)

Fig. 2. Hybrid materials combine the properties of two (or

more) monolithic materials, or of one material and space. They

include fibrous and particulate composites, foams and lattices,sandwiches and almost all natural materials. One might imagine

two further dimension: those of shape and scale.

(hybrids of material and space) in filling previouslyempty areas of the property charts is encourage-

ment enough to explore ways in which such

hybrids might be designed. What is the best wayto go about doing so?

1.2. What might we hope to achieve?

Fig. 3 shows schematically the fields occupiedby two families of materials, plotted on a chart

with properties P1 and P2 as axes. Within each fielda single member of that family is identified(materials M1 and M2). What might be achieved

by making a hybrid of the two? The figure showsfour scenarios, each typical of a different class of

hybrid. We consider the case when large values

of P1 and P2 are desirable, low values not. Then,depending on the shapes of the materials and the

way they are combined, we may find any one ofthe following.

The best of both scenario (Point A). The ideal,often, is the creation of a hybrid with the best

properties of both components. There are


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5803M.F. Ashby, Y.J.M. Brechet / Acta Materialia 51 (2003) 58015821

Fig. 3. The possibilities of hybridization. The properties of the

hybrid reflect those of its component materials, combined inone of several possible ways.

examples, most commonly when a bulk pro-

perty of one material is combined with the sur-

face properties of another. Zinc coated steel hasthe strength and toughness of steel with the cor-rosion resistance of zinc. Glazed pottery

exploits the formability and low cost of clay

with the impermeability and durability of glass. The rule of mixtures scenario (Point B). When

bulk properties are combined in a hybrid, as in

structural composites, the best that can be

obtained is often the arithmetic average of theproperties of the components, weighted by their

volume fractions. Thus unidirectional fiber com-posites have an axial modulus (the one parallelto the fibers) that lies close to the rule of mix-tures.

The weaker link dominates scenario (Point

C). Sometimes we have to live with a lessercompromise, typified by the stiffness of particu-late composites, in which the hybrid propertiesfall below those of a rule of mixtures, lying

closer to the harmonic than the arithmetic mean

of the properties. Although the gains are lessspectacular, they can still be useful.

The worst of both scenario (point D)notsomething we want.

These set certain fixed points, but the list is not

exhaustive. Other combinations are possible, some

relying on the physics of percolation, others on

atomistic effects. These will emerge below.

1.3. When is a hybrid a material?

There is a certain duality about the way in which

hybrids are viewed and discussed. Some, like filledpolymers, composites or wood are treated as

materials in their own right, each characterized by

its own set of material properties. Otherslike gal-vanized steelare seen as one material (steel) towhich a coating of a second (zinc) has been

applied, even though this could be regarded as a

new material with the strength of steel but the sur-face properties of zinc (stinc, perhaps?). Sand-wich panels illustrate the duality, sometimesviewed as two sheets of face-material separated by

a core material, and sometimesto allow compari-son with bulk materialsas a material with theirown density, flexural stiffness and strength. To callany one of these a material and characterize itas such is a useful shorthand, allowing designersto use existing methods when designing with them.

But if we are to design the hybrid itself, we must

deconstruct it, and think of it as a combination of

materials (or of material and space) in a definedgeometry.

2. The method: A + B + shape + scale

First, a working definition: a hybrid material isa combination of two or more materials in a prede-termined geometry and scale, optimally serving a

specific engineering purpose [3], which we para-phrase as A + B + shape + scale. Here we allow

for the widest possible choice of A and B, includ-ing the possibility that one of them is a gas or sim-

ply space. These new variables expand the designspace, allowing an optimization of properties that

is not possible if choice is limited to single, mono-

lithic materials.The basic idea, illustrated in Fig. 4, is this.

Monolithic materials offer a certain portfolio of

properties on which much engineering design isbased. But if the design requirements are excep-

tionally demanding, no single material may be


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Fig. 4. The steps in designing a hybrid to meet given design requirements.

found that can meet them: the requirements lie ina hole in property space. Then the way forward is

to identify and separate the conflicting require-ments, seeking optimal material solutions for each,and then combine them in ways that retain the

desirable attributes of both. The best choice is the

one that ranks most highly when measured by the

performance metrics that motivate the design: min-imizing mass or cost, or maximizing some aspect

of performance (the criteria of excellence). Thealternative combinations are examined and

assessed, using the criteria of excellence to rank

them. The output is a specification of a hybrid interms of its component materials and geometry.

Consider as an example the design of a hybrid

material for long-span power cables. The objec-tives are to minimize the electrical resistance, but

at the same time to maximize the strength sincethis allows a greater span. In multi-objective opti-

mization, of which this problem is an example, itis conventional to express each objective such thata minimum is sought; we thus seek materials with

the lowest values of resistivity, R, and reciprocal

of tensile strength, 1/sts. Fig. 5 shows the result:materials that best meet the design requirements lie

near the bottom left. Those with the lowest resist-

ancecopper, aluminum, and some of theiralloysare not very strong, and the materials thatare strongestdrawn carbon and low-alloy steeldo not conduct very well. Now consider a cable

made by interweaving strands of copper and steelsuch that each occupies half the cross-section.

Assuming the steel carries no current and the cop-per no load (the most pessimistic scenario) the per-

formance of the cable will lie at the point shownon the figureit has twice the resistivity of thecopper and half the strength of the steel. It occupies

a part of property space that was previously empty,

offering performance that was not previously poss-ible.

But while some conflicting requirements can be

met in this way, others need a more inventiveapproach.1 So the question arises: are there general

ways in which material hybridization can be

explored systematically? It is unrealistic to supposethat one method and one tool can solve all such

problems. Instead we examine examples of hybrid

design and attempt to extract principles that could

help tackle other, as yet unformulated problems ofthis class.

3. A + B: selecting components for composites

Aircraft engineers, automobile makers, and

designers of sports equipment all have one thing in

common: they want materials that are stiff, strong,tough and light. The single-material choices that

best achieve this are the light alloys: alloys basedon magnesium, aluminum and titanium. Much

research aims at improving their properties. Butthey are not all that lightpolymers have muchlower densities. Nor are they all that stiffcer-amics are much stiffer and, especially in the form

of small particles or thin fibers, much stronger.These facts are exploited in the subset of hybrids

that we usually refer to as particulate and fibrous composites.

Any two materials can, in principle, be com-

bined to make a composite, and they can be mixed

in many geometries (Fig. 6). In this section, we

1 An interesting example is that of flexible ferromagnets.Monolithic ferromagnetic materials are stiff, metallic or cer-

amic, solids. Elastomeric ferromagnetic hybrids offer several

properties that these monolithic solids do not. The hybrids are

made by mixing up to 30% of sub-micron iron particles into an

elastomer resin before polymerising it. The result is a compliant

ferromagnetic material that has the property that it is mag-

netostrictive, and that its stiffness increases when placed in the

magnetic field because the magnetic dipoles that are induced inthe particles attract one another. The material has a fast (1 ms)

response time, making it suitable for vibration damping.


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Fig. 5. Designing a hybridhere, one with high strength and high electrical conductivity. The figure shows the resistivity andreciprocal of tensile strength for 1700 metals and alloys. We seek materials with the lowest values of both. The construction is for

a hybrid of hard-drawn OFHC copper and drawn low alloy steel, but the figure itself allows many hybrids to be investigated [42].

Fig. 6. Schematic of hybrids of the composite type: unidirec-

tional fibrous, laminated fiber, chopped fiber and particulatecomposites. Bounds and limits, described in the text, bracket

the properties of all of these.

restrict the discussion to fully dense, strongly

bonded, composites such that there is no tendency

for the components to separate at their interfaceswhen the composite is loaded, and to those in

which the scale of the reinforcement is large com-pared to that of the atom or molecule size and the

dislocation spacing, allowing the use of con-

tinuum methods.On a macroscopic scaleone which is large

compared to that of the componentsa compositebehaves like a homogeneous solid with its own set

of thermo-mechanical properties. Calculating theseprecisely can be done, but it is difficult. It is mucheasier to bracket them by bounds or limits: upperand lower values between which the properties lie

[47]. The term bound will be used to describea rigorous boundary, one which the value of theproperty cannotsubject to certain assumptionsexceed or fall below. It is not always possible to

derive bounds; then the best that can be done is toderive limits outside which it is unlikely that thevalue of the property will lie. The important point


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is that the bounds or limits bracket the properties

of all arrangements of matrix and reinforcement

shown in Fig. 6; by using them we avoid the need

to model individual geometries.

3.1. Density

When a volume fraction f of a reinforcement r(density rr) is mixed with a volume fraction (1

f) of a matrix m (density rm) to form a compositewith no residual porosity, the composite density is

given exactly by a rule of mixtures (an arithmetic

mean, weighted by volume fraction)

r frr (1f)rm. (1)

The geometry or shape of the reinforcement does

not matter except in determining the maximumpacking-fraction of reinforcement and thus the

upper limit for f.

3.2. Modulus

The modulus of a composite is bracketed by the

well-known Voigt and Reuss bounds. The upper

bound, Eu, is obtained by postulating that on load-ing the two components suffer the same strain; the

stress is then the volume-average of the localstresses and the composite modulus follows a ruleof mixtures:

Eu f Er (1f)Em. (2)

Here Er is the Youngs modulus of the reinforce-ment and Em that of the matrix. The lower bound,

El, is found by postulating instead that the twocomponents carry the same stress; the strain is the

volume-average of the local strains and the com-

posite modulus is

E1 EmEr

f Em (1f)Er(3)

More precise bounds are possible [8,9], but thesimple ones are adequate to illustrate the method.

3.3. Hybrid design for stiffness at minimumweight

We need a criterion of excellence to assess the

merit of any given hybrid. Here our criterion is

Table 1

Criteria of excellence for minimum weight design

Mode of loading and Stiffness at minimum weight

geometry

Tensile loading of ties E/rBending of beams E1/ 2/rBending of plates E1/ 3/r

stiffness per unit mass, measured by the indices

listed in the table (for derivations see Ref. [2]). If

a possible hybrid has a value of any one of thesethat exceed those of the light alloys, it achievesour goal.

Consider, as an illustration of the method, thedesign of a composite for a light, stiff beam of

fixed section-shape, to be loaded in bending. Theefficiency is measured by the index E1 /2/r shownin Table 1. Imagine, as an example, that the beam

is at present made of an aluminum alloy. Berylliumis both lighter and stiffer than aluminum; ceramics

are stiffer, but not all are lighter. What can these

hybrids offer?Fig. 7 is a small part of the Er property chart.

Fig. 7. Part of the Er property chart, showing aluminiumalloys, beryllium and alumina (Al203). Bounds for the moduli

of hybrids made by mixing them are shown. The diagonal con-

tours plot the criterion of excellence, E1/ 2/r.


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Three groups of materials are shown: aluminum

and its alloys, alumina (Al2O3) and beryllium (Be).

Composites made by mixing them have densities

given exactly by Eq. (1) and moduli that are brack-eted by the bounds of Eqs. (2) and (3). Both of

these moduli depend on volume fraction ofreinforcement, and through this, on density. Upperand lower bounds for the modulusdensityrelationship can thus be plotted onto the Er chartusing volume fraction f as a parameter, as shown

in Fig. 7. Any composite made by combiningaluminum with alumina will have a modulus con-

tained in the envelope for AlAl2O3; the same forAlBe. Fibrous reinforcement gives a longitudinal

modulus (parallel to the fibers) near the upperbound; particulate reinforcement or transverselyloaded fibers give moduli near the lower one.

Superimposed on Fig. 7 is a grid showing the

criterion of excellence E1/ 2/r. The bound-envelopefor Alberyllium composites extends almost nor-mal to the grid, while that for AlAl2O3 lies at ashallow angle to it. Beryllium fibers improve per-formance (as measured by E1 /2/r) roughly fourtimes as much as alumina fibers do, for the samevolume fraction. The difference for particulate

reinforcement is even more dramatic. The lower

bound for AlBe lies normal to the contours: 30%of particulate beryllium increases E1/2/r by a fac-tor of 1.5. The lower bound for AlAl2O3 is,initially, parallel to the E1 /2/r grid: 30% of par-ticulate Al2O3 gives almost no gain. The underly-ing reason is clear: both beryllium and Al2O3increases the modulus, but only beryllium

decreases the density; the criterion of excellence ismore sensitive to density than to modulus.

In Fig. 8 we return to the big picture. It shows

the moduli and densities of metals and polymers,

and, encircled by a broken ellipse, those of highperformance carbon, aramid, PE and glass fibers.The construction illustrated in Fig. 7 leads to famil-ies of polymermatrix composites that lie in theshaded ellipse with that name, and to families ofmetalmatrix composites that lie in the ellipseabove it. Both ellipses occupy areas of property

space that were previously unoccupied by bulk

materials, and it is an important one, enabling thedesign of new lightweight mechanical structures.

Similar methods can be used to select materials

optimum strength, and for tailored values of ther-

mal conductivity, expansion coefficient and spe-cific heat [7]. The properties of specific composites

can, of course, be computed in conventional ways.The advantage of this graphical approach is the

breadth and freedom of conceptual thinking that itallows and the ease of comparison of possible newhybrids with the population of existing materials.

3.4. Percolation: properties that switch on and

off

Fig. 9, a chart of electrical resistivity against

elastic stiffness (here measured by Youngs

modulus), has an enormous hole. Materials thatconduct well are stiff; those that are flexible areinsulators. Consider designing materials to fill thehole; to be more specific, consider designing onethat has low modulus, can be molded like a poly-

mer, and is a good electrical conductor. Suchmaterials find application in anti-static clothing andmats, as pressure sensing elements, even as solder-

less connections.Metals, carbon and some carbides and intermet-

allics are good conductors, but they are stiff andcannot be molded. Thermoplastic and thermoset-

ting elastomers can be molded but do not conduct.How are they to be combined? Metal coating ofpolymers is workable if the product is to be used in

a protected environment, but the coating is easily

damaged. If a robust, flexible, product is needed,bulk rather than surface conduction is essential.

This can be achieved by mixing conducting par-

ticles into the polymer.To understand how to optimize this we need the

concept of percolation. Percolation problems are

easy to define, but not easy to solve. Research

since 1960 has provided approximate solutions tomost of the percolation problems associated with

the design of hybrids (see Ref. [10] for a review).Think of mixing conducting and insulating spheres

of the same size to give a large array. If there are

too few conducting spheres for them to touch, thearray is obviously an insulator. If each conductingsphere contacts just one other, there is still no con-

necting path. If, on average, each touched two,there is still no path. Adding more spheres gives

larger clusters, but they can be large yet still dis-


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Fig. 8. Youngs modulus and density for 1850 polymers, metals and fibers (broken ellipse). Combining these to create polymer andmetal matrix composites fills a previously empty hole in material-property space [42].

Fig. 9. When conducting, particles or fibers are mixed into an insulating elastomer, a hole in material-property space is filled.Carbon-filled butyl rubbers lie in this part of the space [42].


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crete. The array first becomes a conductor when asingle trail of contacts links one surface to the

other, that is, when the fraction p of conducting

spheres reaches the percolation threshold, pc. Forsimple cubic packing pc = 0.248, for close packing

pc = 0.180. For a random array it is somewhere inbetweenapproximately 0.2.2

Make the spheres smaller and the transition issmeared out. The percolation threshold is still 0.2,

but the first connecting path is now thin andextremely deviousit is the only one, out of thevast number of almost complete paths, that actually

connects. Increase the volume fraction and the

number of conducting paths increases initially as

(p

pc)

2

, then linearly, reverting to a rule of mix-tures [13]. If the particles are very small, as much

as 40% may be needed to give good conduction.But a loading of 40% seriously degrades the mold-

ability and compliance of the polymer.

Shape gives a way out. If the spheres arereplaced by fibers, they touch more easily and thepercolation threshold falls. If their aspect ratio is

f = L/d (where L is the fiber length, d thediameter) thento an adequate approximationempirically, the percolation threshold falls from fcto roughly fc/f

1/2 [1418]. Fig. 9 shows the area

of the property chart where these hybrids lie. Withsufficient aspect ratio the percolation thresholdfalls to a few percentage points.

The concept of percolation is a necessary tool

in designing hybrids. Electrical conductivity worksthat way; so too does the passage of liquids

through foams or porous mediano connectedpaths, and no fluid flows; just one (out of a millionpossibilities) and there is a leak. Add a few more

connections and there is a flood. Percolation ideasare particularly important in understanding the

transport properties of hybrids: properties thatdetermine the flow of electricity or heat, of fluid,or of flow by diffusion, specially when the differ-ences in properties of the components are extreme.

Most polymers differ from metals in their electricalconductivity by a factor of about 1020. The dif-

2 These results are for infinite, or at least very large, arrays.Experiments [11,12] generally give values in the range 0.190.22, with some variability because of the finite size of thesamples.

fusion of water through solids differs from the flowrate of water through channels by a similar factor.

It is then that single connections really matter.

Percolation influences mechanical propertiestoo, particularly when mechanical connection is

important, as in arrays of loose powders or fibers.If there are no bonds between the particles or fib-ers, the array has no tensile stiffness or strength.If each particle is bonded to another, or to several

forming discrete clusters, there is still no stiffness

or strength. These only appear when there are con-nected paths running completely through the array.

The plasticity of 2-phase hybrids, too, can be

viewed as a percolation problem. Plasticity may

start in one phase at a low stress, allowing patchesof slip to form, but full plasticity requires that the

slip patches link to give connected paths throughthe entire cross-section of the sample. Mechanical

and electrical percolation can be combined,

exploiting scale. A latex reinforced with a smallvolume fraction of cellulose fibers coated withpolypyrrole to make them conducting, gives a

material with a shear modulus two orders of mag-nitude higher than the rule of mixture predicts

combined with good conduction because of thehigh aspect ratio of the fibers [19]. Here we are

escaping the continuum boundsa topic we returnto later.

3.5. Creating anisotropy

The elastic and plastic properties of bulk mono-

lithic solids are frequently anisotropic, but weakly

sothe properties do not depend strongly on direc-tion. Hybridization gives a way of creating and

controlling anisotropy, and it can be large. We

have already seen an example in Fig. 7, which

shows the upper and lower bounds for the moduliof composites. The longitudinal properties of

unidirectional long-fiber composites lie near theupper bound, the transverse properties near the

lower one. The vertical width of the band in

between them measures the anisotropy.Consider a second example, that of creating

hybrids with anisotropic thermal conductivity. A

saucepan made from a single material when heatedon an open flame, develops hot spots that canlocally burn its contents. That is because the sauce-


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pan is thin, and heat is transmitted through the

thickness more quickly than it can be spread trans-

versely to bring the entire pan surface to a uniform

temperature. The metals of which saucepans areusually madecast iron, or aluminum or copperhave a isotropic thermal conductivities whereaswhat we clearly want is a thermal conductivity thatis higher in the transverse direction than in thethrough-thickness direction. A bi-layer (or multi-

layer) hybrid can achieve this.

Heat transmitted transversely in a bi-layer sheethas two parallel paths; the total heat transmitted is

a sum of that in each of the paths. If it is made of

a layer of material 1 with thickness t1 and conduc-

tivity l1, bonded to a layer of material 2 with thick-ness t2 and conductivity l2, the conductivity paral-lel to the layers is

lII fl1 (1f)l2 (4)

(a the rule of mixtures), where f= t1/ (t1 + t2). Per-pendicular to the layers the conductivity is

1

l

f

l1

(1f)

l2(5)

(the harmonic mean). Fig. 10 shows l and l plot-ted against f for a bi-layer of copper (l=390W/m K) and cast iron (l = 30 W/m K). For singlematerials the two are equal; layering them gives

Fig. 10. Creating anisotropy. The thermal conductivities of

copper and cast iron are isotropic. Anisotropy is created by

combining them as a bi-layer.

trajectories for l and l that separate. Themaximum separation occurs broadly where each

occupies about half the thickness, where the ratio

of the conductivities (the anisotropy ratio) is 3.8.Mechanical anisotropy is most easily created

and managed through shape. This is the topic ofthe next section.

4. Shape: structures, sandwiches and

segmented assemblies

The shape and configuration of components Aand B of a hybrid play a key role in determiningits properties. Shape can be used to enhance or

diminish stiffness and strength, to impart damagetolerance, andas we have already seentomanipulate the percolation limit.

4.1. Shape efficiency and shape factors

Beams with hollow-box or I-sections are stiffer

and stronger in bending than solid sections of the

same cross-sectional area; so, too, are panels withribs or waffle stiffeners, or those with an expandedcore to create a sandwich (Fig. 11). These are

Fig. 11. Making high-efficiency structures. Shape gives thesections a greater flexural stiffness and strength per unit massthan the solid section from which they are made.


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examples of the use of shape to increase structural

efficiency. To characterize this we need a metrica way of measuring the structural efficiency of a

section shape, independent of the material of whichit is made. An obvious one is that given by the

ratio j of the stiffness or strength of the shapedsection to that of a neutral reference shape. Fora beam we take the reference shape to be that ofa solid square section with the same cross-sectional

area, and thus the same mass per unit length, as

the shaped section. For a panel, we take it to be asolid, plain panel with the same mass per unit area

as the shaped section, as shown in the figure. Wecall j the shape factor [2,20] and define that for

stiffness je as

je Flexural stiffness of shaped section

Flexural stiffness of reference section(6)

and that for strength jf as

jf Flexural strength of shaped section

Flexural strength of reference section(7)

Shape can be used to reduce flexural stiffnessand strength as well as increase them. Springs, sus-pensions, flexible cables and other structures that

must flex yet have high tensile strength, use shapeto give a low bending stiffness. Low shapeefficiency is achieved by forming the material intostrands or leaves, as suggested Fig. 12. Values of

j for the stiffness of structural sections can be as

Fig. 12. Making low-efficiency structures. Shape gives thesections a lower flexural stiffness and strength per unit massthan the solid section from which they are made.

high as 50; for multi-strand or multi-leaf structures

as low as 0.01.

Note the origins of efficiency. The flanges of the

I-section or the faces of the sandwich panels lie farfrom the neutral axis; they stretch when the section

is loaded in bending. Subdivision, as in Fig. 11,lowers efficiency because the slender strands orleaves bendeasily, but do not stretch when the sec-tion is bent: an n-strand cable is less stiff by a fac-

tor of 3/n than the solid reference section; an n-leaf panel by a factor 1/n2. There is an underlyingprinciple here: stretch dominated structures have

high structural efficiency; bending dominated

structures have low.

4.2. Shape on a micro-scale

The sections of Fig. 11 achieve efficiencythrough their macroscopic shape. Structural

efficiency can be manipulated in another way:through shape on a small scale; microscopic ormicro-structural shape. Wood is an example. Thesolid component of wood (a composite of cellu-lose, lignin and other polymers) is shaped into pris-

matic cells, each cell like the hollow tube of Fig.11. The effect is to disperse the solid component

further from the axis of bending or twisting of thebranch or trunk of the tree, increasing its flexuralstiffness and strength. This is not the only possi-

bility; low efficiency structures give materials withlow stiffness and strength, desirable in cushioningand packaging.

4.3. Ultra-light, low stiffness hybrids

The point has been made that stretching is a stiff

mode of loading, bending is a compliant one. A

material that responds, at the micro-structurallevel, by bending no matter how it is loaded

remotely is much less stiff than one that respondsby stretching. Material made by foaming have

structures that respond in this way.

Fig. 13 shows an idealized cell of a low-densityfoam. It consists of solid cell walls or edges sur-

rounding a void space, each cell with an overall

space-filling shape. Cellular solids are charac-terized by their relative density, which for the

structure shown here (with t ) is


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Fig. 13. A cell in a low density foam. When the foam is loaded, the cell edges bend, giving a low-modulus structure.

r

rs t

2 (8)

where r is the density of the foam, rs is the den-sity of the solid of which it is made, is the cellsize, and t is the thickness of the cell edges. A

remote compressive stress s exerts a force F s2 on the cell edges, causing them to bend asshown in the figure, and leading to a bending

deflection d. For the open-celled structure shownin the figure, the bending deflection is given by

dFL3

EsI(9)

where Es is the modulus of the solid of which thefoam is made and I= t4 /12 is the second momentof area of the cell edge of square cross-section, t t. The compressive strain suffered by the cell asa whole is then e = 2d/. Assembling these resultsgives the modulus E = s/e of the foam as

E

Es rrs

2

(10)

(bending dominated behavior)

Since E = Es when r = rs, we expect the constantof proportionality to be close to unitya specu-lation confirmed both by experiment and bynumerical simulation.

A similar approach can be used to model non-

linear properties such as strength. The cell walls

yield when the force exerted on them exceeds their

fully plastic moment

Mf sst

3

4(11)

where ssis the yield strength of the solid of whichthe foam is made. This moment is related to the

remote stress by M

FL

sL3

. Assembling theseresults gives the failure strength s

s

ssrrs

3/2

(12)

(bending dominated behavior)

This behavior is not confined to open-cell foamswith the structure shown in Fig. 14. Most closed-

cell foams also follow these scaling laws. At firstsight an unexpected result because the cell faces

must carry membrane stresses when the foam isloaded, and these should lead to a linear depen-

dence of both stiffness and strength on relative

density. The explanation lies in the fact that the

cell faces are very thin; they buckle or rupture at

stresses so low that their contribution to stiffness

and strength is small, leaving the cell edges to

carry most of the load (for further details, see

Ref. [21]).


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Fig. 14. A micro-truss structure and its unit cell.

4.4. Ultra-light, high stiffness hybrids

If conventional foams have low stiffness

because other configuration of the cell edgesallows them to bend, might it not be possible to

devise other configurations in which the cell edgeswere made to stretch instead? This thinking leadsto the idea of micro-truss structures [22,23]. To

understand these we need the Maxwell stability cri-terion.

The condition that a pin-jointed frame of b strutsand j frictionless joints to be both statically andkinematically determined i.e. just rigid [24,25], in

2-dimensions, is:

M b2j 3 0 (13)

and in 3-dimensions is

M b3j 6 0 (14)

If M 0, the frame is a mechanism. It has no

stiffness or strength, but will collapse if loaded. If

its joints are locked (instead of pin-jointed) the barsof the frame bend when the structure is loaded. If,

instead, M 0 the frame ceases to be a mech-anism; its members carry tension or compression

when the frame is loaded, and it becomes a stretch-

dominated structure.These criteria give a basis for the design of

efficient micro-truss structures. For the cellularstructure of Fig. 13 M 0, and it is bending domi-nated. For the structure shown in Fig. 14, however,

M 0 and it behaves as an almost isotropic,

stretch-dominated structure. On average one thirdof its bars carry tension when the structure is

loaded in simple tension, regardless of its direc-

tion. Thus

E

Es

1

3rrs (15)

for isotropic stretch-dominated behavior and

s

ss

1

3r

rs (16)for isotropic stretch-dominated behavior.3

Prismatic structures do even better, provided

they are loaded parallel to the prism axis. Fig. 15shows four such structures which are common innature. It is helpful to think, as before, of the

expansion of a solid bar, shown in the center, togive the structures, with no change of massherethe solid black represents solid material, the dottedareas represent low density foam and the open

areas represent space. The expansion has moved

material away from the axis of bending, increasingthe second moment of area about any axis con-

tained in the plane of the cross-section, and

increasing efficiency in the sense of Eqs. (6) and(7). For these structures the axial modulus and

strength (measured parallel to the prism axis) fol-

3 This assumes that failure occurs by the axial yielding of a

bar. If the bars are slender or have low modulus they may fail

instead by buckling, giving a lower strength.


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Fig. 15. Four extensive micro-structured materials which are

mechanically efficient: (a) prismatic cells, (b) fibers embeddedin a foamed matrix, (c) concentric cylindrical shells with foam

between, and (d) parallel plates separated by foamed spacers.

lows Eqs. (15) and (16), but with a constant ofproportionality not of 1/3 but of unity:

E

Es

r

rs (17)for prismatic stretch-dominated behavior and

s

ssrrs (18)

for prismatic stretch-dominated behavior.3

Loaded transversely, however, they are bendingdominated and follow power laws like those of

Eqs. (10) and (12); they are thus exceedingly

anisotropic.

These results are summarized in Fig. 16, inwhich the modulus E is plotted against the den-

sity r. Stretch dominated, prismatic microstruc-tures like wood give moduli that scale as r/rs(slope 1); bending dominated, cellular, microstruc-tures like that of polystyrene foam give moduli thatscale as (r/rs)

2 (slope 2). Given that the density

can be varied through a wide range, this allows

great scope for material design. Note how the useof microscopic shape has expanded the occupied

area of Er space.

4.5. Ultimate efficiency: the sandwich

A sandwich panel epitomizes the concept of ahybrid. It combines two materials in a specifiedgeometry and scaleone forming the faces, theother the coreto give a structure of high stiffnessand strength at low weight (Figs. 11 and 17). Theseparation of the faces by the core increases the

moment of inertia I and the section modulus Z of

the panel with little increase in weight, producingan efficient structure for resisting bending andbuckling loads. Sandwiches are found where

weight-saving is critical: in aircraft, trains, trucksand cars, in portable structures, and in sports

equipment. Nature, too, makes use of sandwichdesigns: sections through the human skull, thewing of a bird and the stalk and leaves of many

plants show a low-density foam-like core separat-

ing solid faces. The faces carry most of the load,so they must be stiff and strong; and they form the

exterior surfaces of the panel so they must tolerate

the environment in which they operate. The coreoccupies most of the volume, it must be light, and

stiff and strong enough to carry the shear stressesnecessary to make the whole panel behave as a

load bearing unit, but if the core is much thickerthan the faces these stresses are small.

So far we have spoken of the sandwich as a

structure: faces of material A supported on a core

of material B, each with its own density and modu-lus. But we can also think of it as a material with

its own set of properties, and this is useful because

it allows comparison with more conventionalmaterials. To do so we must analyze sandwich per-

formance [21,2629]. We shall use, as a criterionof excellence, the bending stiffness per unit width,Sw divided by the mass per unit area, ma.

The bending stiffness of the panel per unit

width, Sw, is given by

Sw (EI)sand (19)

112

(d3 c3)Ef1

1 BEftc

2GcL2

where the dimensions, d, c, t and L are identified


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This, of course, is an idealization. The core

always shears somewhat, and it does have some

mass; a more precise analysis copes with this [29].

It leads to the performance shown in Fig. 18, whichhas been constructed in the same way as Fig. 7. It

shows the modulus and density of A + B hybrids.The shaded band is bounded by the upper andlower bounds of Eqs. (1)(3), describing particu-late and fibrous composites. The flexural perform-ance of the sandwich is shown as a dashed line; at

its mid-section it lies a factor of 3 above the upperbound rule of mixtures of Eq. (2). The criterion of

excellence for minimum weight design with pre-

scribed bending stiffness, listed in Table 1, is that

of maximizing E

1/3

/r. Contours of this criterionare plotted as diagonal lines on the figure, increas-ing towards the top left. The sandwich out-per-forms all alternative hybrids of A + B.

4.6. Subdivision as a design variable

We have already seen how subdivision can

reduce stiffness (Fig. 12). Here we examineanother way in which it can be used: to impart

damage tolerance. A glass window, hit by a pro-

Fig. 18. Sandwich panels (broken line) extend the range of

flexural modulus per unit mass into areas not occupied bymonolithic materials. Their flexural moduli lie above that pre-dicted by a rule of mixtures by a factor of approximately 3.

jectile, will shatter. One made of small glass

bricks, laid as bricks usually are, will lose a brick

or two but not shatter totally; it is damage-tolerant.

By sub-dividing and separating the material, acrack in one segment does not penetrate into its

neighbors, allowing local but not global failure.That is the principle of topological toughening.Builders in stone and brick have exploited the ideafor thousands of years: both materials are almost

as brittle as glass, but buildings made of themeven those made without cement (dry-stonebuilding)survive ground movement, even earth-quakes, through their ability to deform with some

local failure, but without total collapse.

Taking the simplest view, two things are neces-sary for topological damage tolerance: discreteness

of the structural units, and an interlocking of theunits in such a way that the array as a whole can

carry load. Brick-like arrangements (Fig. 19a) are

damage tolerant in compression and shear, but dis-integrate under tension. Strand and layer-like struc-

tures (shown earlier as Fig. 12) are damage-tolerant

in tension because if one strand fails the crack doesnot penetrate its neighborsthe principle of multi-strand ropes and cables. The jigsaw puzzle con-figuration (Fig. 19b) carries in-plane tension, com-

pression, and shear, but at the cost of introducinga stress concentration factor of about R/r, where

R is the approximate radius of a unit and r that

of the interlock. Dyskin et al. [3032] explore aparticular set of topologies that rely on compress-ive or rigid boundary conditions to create continu-

ous layers that tolerate out-of-plane forces and

bending moments, illustrated in Fig. 19c. This isdone by creating interlocking units with non-planar

surfaces that have curvature both in the plane of

the surface and normal to it. Provided the array is

constrained at its periphery, the nesting shapes lim-its the relative motion of the units, locking them

together. The bending stiffness of the array is pro-portional to the stiffness of the boundary con-

straint, falling to zero as the constraint is relaxed.Topological interlocking of this sort allows the for-mation of continuous layers that can be used for

ceramic claddings or linings to give surface protec-

tion.The damage tolerance can be understood in the

following way. We suppose that the units of the


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Fig. 19. Examples of topological interlocking: discrete, unbonded structures that carry load. (a) Brick-like assemblies of rectangular

blocks carry axial compression (syy), but not tension or shear. (b) The 2-dimensional interlocking of a jig-saw puzzle carries in-planeloads ( sxx, syy, sxy). (c) The units suggested by Dyskin et al., when assembled into a continuous layer and clamped within arigid boundary around its edge, can carry out-of-plane loads and bending moments ( sxz, syz, Mxz, Myz) (Fig. 15(c) derived fromRef. [30]).

structure are all identical, each with a volume Vs,

and that they are assembled into a body of volume

Vt; there are therefore n = Vt/Vs segments. Wedescribe the probability of failure of a segment

under a uniform tensile stress sby a Weibull prob-ability function:

Pf(V,s) 1exp VsmVos

m

o (24)

where m, Vo and so are constants [33,34]. If thebody were made of a single monolithic piece ofthe brittle solid, this equation, with V= Vt, woulddescribe the failure probability. To calculate the

design stress st we set an acceptable value for P,which we call P (say 106, meaning that it is

acceptable if one in a million fail) and invert theequation to give

st so VoVt

ln(1P)1/m (25)Now consider the segmented body. A remote

stress, if sufficiently large, causes some segmentsto fail. We refer to the fraction that has failed as

the damage, D. If loaded such that each segmentcarries a uniform stress s, the damage is simplyPf(Vs,s). If some segments fail, the body as awhole remains intact; global failure requires that afraction D , the critical damage, (say, 10%) must

fail. Inverting Eq. (24) with V= Vs gives the globalfailure stress ss of the segmented body:

ss so

Vo

Vsln(1D)

1/m

(26)

Thus segmentation increases the allowable design

stress from st to s

s , factor of

ssst

nln(1D)ln(1P)

1/mnDP

1/m (27)(expanding the logarithm as a series and retaining

the first terman acceptable approximation forsmall P and D). Both n and D/P are con-

siderably greater than 1, so the equation suggests


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sm(1f) of the continuum approximation. Fine-grained materials, particularly those with grains of

a few nanometers, exhibit strengths that exceed

those of large-grained bulk samples. Scale, then,can have a profound influence on mechanicalproperties.

Transport properties, too, can be scale depen-dent. The mean-free paths of electrons, phononsand of diffusing atoms and molecules are limited

by the scale of the microstructure when this is

smalla fact exploited in micro-cellular foams togive exceptional thermal insulation. Convection,

acoustic absorption and light scattering, too, are

directly linked to aspects of structural scale.

6. Summary and conclusions

The properties of engineering materials can be

thought of as defining the axes of a multi-dimen-sional space with each property as a dimension.

Sections through this space can be mapped. These

maps reveal that some areas of property-space areoccupied, others are emptythere are holes. Theholes can sometimes be filled by making hybrids:combinations of two (or more) materials, or of

material in space, in chosen configuration andscale. Here we have surveyed conceptual tools forsuggesting and assessing hybrids to fill specifiedneeds. A useful starting point is the concept of a

hybrid as A + B + shape + scale. Successfulhybrids, as a rule, exploit the first three of these;with micron and nanometer scale fabrication tech-

nologies now a reality, it becomes possible to addthe last, opening up wider horizons.

Continuum bounding methods give tools for

scanning the possibilities offered by a set of

hybrids, provided the scale is such that the con-tinuum approximation applies. Shapethe way Aand B are configuredcan extend the populatedareas of property-space in ways that complement

efforts to create new monolithic materials. Dis-

criminating choice of shape can enhance or dimin-ish physical, mechanical, thermal and electrical

properties. Scale introduces a new variable. In

hybrids with structural units that are sub-micron, anew length scale (basically that of the atom) makes

itself evident. Here the bounds break down, and

continuum methods must be replaced by statistical

or dislocation mechanics.

Much, in a short paper such as this, has been

ignored. Fabricating successful hybrids can be dif-ficult and expensive (but so, too, is the alternativeof seeking to develop a new monolithic material).

Part of the difficulty stems from the multitude ofpossible choices: choice of materials, choice of

process to combine them, and choice of the internal

geometry and topology of the constitutive

materials. Part derives from the need to make these

choices in such a way as to optimally meet a set

of design requirements. The hybrid must be both

feasible and optimal.

To explore this design space efficiently optimiz-ation tools are needed. The starting point is a sim-

ple screening of optionsthe obvious way for-ward when the selection space is discrete. Thus it

is possible to create a database of, say, composites

by computing the properties of virtual materials

with 5%, 10%, 15% of reinforcing fibers anddiscrete choices of lay-up; selection proceeds by

rejecting all the combinations that fail to meet the

design requirements. This method becomes

impractical when many material choices and lay-

ups are allowed. When the optimization variablesare continuous (such as the volume fraction of

reinforcement in a composite with pre-chosen

constituents), linear programming or steepest

descent methods can be efficient. When the poten-tial landscape is very rugged, simulated annealingcan offer a practical alternative. When the variables

are both discrete (such as the choice of materials)

and continuous (the thicknesses of face sheets and

core of a sandwich, for instance), a genetic algor-

ithm, which allows an unevenly populated space to

be explored, can be efficient. Refs. [3741]examples of their use in hybrid design.These tools allow promising candidates to be

identified. They need the back-up of expert toolsadvising on the compatibility of the constituents,

the practicality of processes to assemble them into

a hybrid, and the loss of properties (the knock-down factor relating real and ideal hybrids) fora given combination of the materials, architecture

and process.


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Acknowledgements

The ideas, methods and tools described here

have evolved over the past 15 years. Numerouscolleagues in many countries have (sometimes

unknowingly) stimulated or contributed to thisevolution. Among these we would particularly like

to recognize Profs. Mick Brown, Chris Calladine,

Norman Fleck and David Cebon (all of CambridgeUniversity), Dave Embury (McMaster University,

Canada), Tony Evans (UCSB), John Hutchinson

(Harvard University) and Haydn Wadley (UVA),and Dr. Luc Salvo (University of Grenoble).

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