## 165 Applications

Two examples of the method follow, each exploring the viability of metal foams in a particular application. In the first, metal foams prove to be nonviable. In the second, despite their present high cost, they prove to be viable. The examples are deliberately simplified to bring out the method. The principles remain the same when further detail is added.

### Simple trade-off between two performance indices

Consider selection of a material for a design in which it is desired, for reasons of vibration control, to maximize the specific modulus E/p (E is Young's modulus and p is the density) and the damping, measured by the loss coefficient v. We identify two performance metrics, Pi and P2, defined such that minima are sought for both:

Figure 16.9 shows the trade-off plot. Each bubble on the figure describes a material; the dimensions of the bubble show the ranges spanned by these property groups for each material. Materials with high values of P1 have low values of P2, and vice versa, so a compromise must be sought. The optimum trade-off surface, suggested by the shaded band, identifies a subset of materials with good values of both performance metrics. If high E/p (low P1) is of predominant importance, then aluminum and titanium alloys are a good choice; if greater damping (lower P2) is required, magnesium alloys or cast irons are a better choice; and if high damping is the over-riding concern, tin or lead alloys and a range of polymers become attractive candidates. It is sometimes possible to use judgement to identify the best position on the tradeoff surface (strategy 1, above). Alternatively (strategy 2) a limit can be set for one metric, allowing an optimum for the other to be read off. Setting a limit of V > 0.1, meaning P2 < 10, immediately identifies pure lead and polyethylenes as the best choices in Figure 16.9. Finally, and preferably (strategy 3), a value function can be determined:

E v seeking materials which minimize V. Contours of constant V, like those of Figure 16.7, have slope

100000.

lonomer (IO) LDPE

= Density/modulus (Mg/m3/GPa)

100000.

lonomer (IO) LDPE

= Density/modulus (Mg/m3/GPa)

Figure 16.9 A trade-off plot for the performance metrics P1 = p/E and P2 = 1 /V- Each bubble refers to a material class. The metal foams are distinguished by filled ellipses (all other materials are fully dense). The shaded band show the optimum trade-off surface. Materials that lie on or near this surface most nearly optimize both performance metrics

The point at which one contour is tangent to the trade-off surface identifies the best choice of material. Implementation of this strategy requires values for the ratio a1/a2 which measures the relative importance of stiffness and damping in suppressing vibration. This requires estimates of the influence of each on overall performance, and can, in technical systems, be modeled. Here, however, it is unnecessary. The positions of three classes of metal foams are shown as black ovals. None lie on or near the trade-off surface; all are subdominant solutions to this particular problem. Metal foams, in this application, are non-viable.

### Co-minimizing mass and cost

One of the commonest trade-offs is that between mass and cost. Consider, as an example, co-minimizing the mass and cost of the panel of specified bending stiffness analysed in Chapter 5, Section 5.3. The mass of the panel is given by equation (5.4) which we rearrange to define the performance metric P1:

with fi given by fi is a constant for a given design, and does not influence the optimization. The geometric constant, C1, defined in equation (5.4), depends only on the distribution of loads on the panel and does not influence the subsequent argument. The cost, Cp, of the panel is simply the material cost per kg, C (from equation (16.4)), times the mass m, giving the second performance metric P2:

Figure 16.10 shows the trade-off plot. The horizontal axis, P1, is the material index p/E1/3. The vertical axis, correspondingly, is the index Cp/El/3.

Figure 16.10 shows the trade-off plot. The horizontal axis, P1, is the material index p/E1/3. The vertical axis, correspondingly, is the index Cp/El/3.

Aluminum alloys \

Polystyrene)

Performance metrics for panel of specified stiffness

CFRPs Alulight

Ai-SiC composites Alcan

GFRPs Magnesium alloys PP

Aluminum alloys \

Polystyrene)

Performance metrics for panel of specified stiffness

UPVC, Rigid "

Low alloy steels

Grey cast irons

Figure 16.10 A trade-off plot for the performance metrics (measured by material indices) for cost and mass of a panel of specified bending stiffness. Each bubble refers to a material class. The metal foams are distinguished by filled ellipses (all other materials are fully dense). The trade-offfront, constructed for non-foamed materials, separates the populated section of the figure from that which is unpopulated. Metal foams lie in the unpopulated sector

Conventional alloys (cast irons, steels, aluminum alloys) lie in the lower part of the diagram. Beryllium alloys, CFRPs and Al-based MMCs lie in the central and upper parts.

The trade-off surface for conventional, fully dense, materials is shown by the shaded band. Metal foams lie in the unpopulated sector of the diagram - all three classes of foam offer non-dominated solutions to this problem. But even so, they are viable only if the mass/value exchange constant lies in the right range. To explore this question for the panel of specified stiffness we define the value function

(since a2, relating value to cost, is unity). Values of ai, relating value to mass, are listed in Table 16.2. The equation is evaluated in Table 16.3 for two extreme values of a1 for a set of materials including cast irons, steels, aluminum alloys, titanium, beryllium and three metal foams. When a1 has the low value of 0.5 £/kg, nodular cast irons are the best choice. But if a1 is as

 Material Mg/m3 E GPa £/kg Pi P2 V ai = £0.5/kg V ai = £500/kg Cast iron, 7.30 175 0.25 1.31 0.33 0.99 655 nodular Low-alloy 7.85 210 0.45 1.32 0.59 1.25 660 steel (4340) A1-6061-T6 2.85 70 0.95 0.69 0.66 1.01 345 Al-061-20% 2.77 102 25 0.59 14.8 15.1 309 SiC, PM Ti-6-4 B265 4.43 115 20 0.91 18.2 18.7 473 grade 5 Beryllium 1.84 305 250 0.27 67.5 67.6 202 SR-200 Alporasa 0.25 1.0 40 0.23 10.0 10.1 125 Alulighta 0.30 0.8 16 0.3 5.2 5.4 155 Alcana 0.25 0.4 5.8 0.34 2.0 2.2 172

a All three types of metal foam are made in a range of densities, with a corresponding range of properties. These three examples are taken from the middle of the ranges. The costs are estimates only, broadly typical of current prices, but certain to change in the future. It is anticipated that large-scale production could lead to substantially lower costs.

a All three types of metal foam are made in a range of densities, with a corresponding range of properties. These three examples are taken from the middle of the ranges. The costs are estimates only, broadly typical of current prices, but certain to change in the future. It is anticipated that large-scale production could lead to substantially lower costs.

high as 500 £/kg, all three of the foams offer better value V than any of the competing materials. Alporas, using present data, is the best choice, meaning that it has the best combination of performance and cost.

### Multifunctionality

Metal foams appear to be most attractive when used in a multifunctional role. Multifunctionality exploits combinations of the potential applications listed in Table 1.2. The most promising of these are

• Energy absorption/blast mitigation

• Acoustic damping

• Thermal stand-off (firewalls)

• Low-cost assembly of large structures (exploiting low weight)

• Strain isolation (as a buffer between a stiff structure and a fluctuating temperature field, for instance).

The method described above allows optimization of material choice for any combination of these.

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