Part of this text was originally published on the Icarus Interstellar Website

Introduction

Materials are the bedrock upon which starship designs are built. There is no use postulating a starship drive that requires materials that do not, or cannot, exist. At the same time, a review of material properties shows a steady advance in certain fields(TBC find a reference). Therefore, some extrapolation from presently known properties is legitimate.

Designers are looking for the lightest, the strongest, the toughest, the most corrosion, heat and radiation resistant. Plus some contradictory requirements according to application: high or low thermal resistivity, high or low transparency.

Strength of materials

There is a huge difference between the theoretical ideal strength of materials and their actual ones for practical applications. Aluminium, for example, has a theoretical yield strength of 6,900 MPa₍₁₎. But for existing aluminium products, the actual yield strength varies from 35 to 500 MPa. The strongest developed aluminium alloys have about 7% of their theoretical yield strength, and most of them have much less. This holds true for pretty well all materials, from metals to ceramics. Their actual strength properties are usually between 3% and 10% of their ideal ones. So it is important to take the values published for relatively new materials such as graphene and carbon nanotubes with a skeptical attitude. Graphene may have a theoretical yield strength of 130,000 MPa, but planning for anything more than 13,000 MPa in real products is probably optimistic.

Rigidity numbers such as Young’s modulus, however, are fairly exact, even though most materials are not pure, but alloys, and are therefore a mix of the properties of the components.

A few definitions, because these terms are often confusing:

Tensile strength: The maximum stress that a material can withstand without breaking. Sometimes called ultimate tensile strength or ultimate strength. This is an experimental value.

Yield strength: The stress that causes permanent deformation. Metals will stretch and yield without breaking, ceramics will often have practically the same yield strength as tensile strength, and break without any stretching. Also an experimental value.

Allowable stress: The maximum stress recommended for practical use. For most metals, it is about a quarter of the tensile strength.

Creep, deformation, fatigue, yield and failure

The tables of materials yield points and temperature don’t tell the whole story. Before they break down, materials will stretch. As time passes, cyclical loading will reduce their strength. As the temperature goes up, under the influence of the stresses upon them, they will flow, or creep; stretching. Until, for example, a turbine blade tip hits the turbine walls and self destructs. Or a pipe wall starts to thin, balloons out and fails catastrophically. Because of this, loads need to be reduced significantly in order to keep strain considerably below the tabulated values.

For example, a fairly common stainless steel pipe, ASTM 304, will have a tensile strength of 525 MPa, but will have a yield strength of 205 MPa. However, it will never be used at more than 125 MPa of stress, even at lower temperatures, to ensure good long term performance. As the usage temperature gets higher, the allowable stress goes down. At 800 K, the allowable stress is down under 100 MPa, and it then declines rapidly, down to 25 MPa at 980K. And we are still a long way from the melting point of steel, at 1773 K ().

Corrosion

Looking at table 1, we can see that for a thickness of a few mm, Tungsten of Zirconium Diboride do not increase thermal resistance by more than a few dozen degrees. So their use as corrosion protection from liquid metals or hot hydrogen for a carbon composite radiator piping system might be possible. On the other hand, in a turbine or pump blade application, we would want maximum thermal resistance to reduce the temperature inside the turbine blade.

Heat transfer

Heat flow through materials depends on thermal conductivity

Applications

Rotating machines

Turbines used for pumps and power generation are among the most extreme applications for materials on a starship, as on Earth. High stress from rotation, high temperatures, cyclical loading and fatigue, corrosion and erosion from high fluid velocities all take their toll. One of the good points of using helium gas as a coolant is that it removes corrosion from the parameter set, possibly allowing for higher operating temperatures.

Metals do not behave very well at very high temperatures and under high stress, but at lower temperatures, they remain very strong. Ceramics, on the other hand, are not very tough, and can break easily. But they keep their properties at much higher temperatures than metals. So one of the solution found by engine makers has been to blend the two together: a metal blade with a ceramic coating. The solution needs to be complemented with cooling for the metal on the interior of the blade using either a gas or cooling fluids. The heat flow out of the blade is equal to the heat flow into the blade through the coating.

Materials for high radiation areas

If deuterium-deuterium fusion is retained as the final power source for the Icarus propulsion system, there will necessarily be a large radiation flux coming from the fusion area in the drive, mostly in the form of high energy neutrons and x-rays. Radiation flux damages materials, but, surprisingly, not necessarily in a negative way. Radiation annealing can harden some materials, while the same process can make some materials brittle. The heat from the absorbed radiation needs to be managed as well. In some cases, we will want our materials to be as transparent to radiation as possible, in other cases, such as in radiation shields, we will want the opposite, maximum opacity.

Most of the information on this comes from classical fission reactors. So there is quite a lot of experimental data.

The secret to radiation resistance is twofold: transparency and controlled defects. Close to the fusion drive, transparency will have to be at a maximum, since the neutron and x-ray flux will be very high. For a design of a Z-pinch or magnetic confinement type of drive, materials are required right next to the fusion reaction, to carry the required currents and create the magnetic fields, as in figure 8, on the left. For pellet explosion driven ships, as on the right in figure 8, the requirements are slightly less extreme since there are no physical components near the explosion. However, the radiation flux will still be very high. Regarding the physical strength of materials near the fusion reactions, it is interesting to note that they do not need to be intrinsically high. The actual forces from the ship’s thrust are quite modest. For Icarus, about 30 times lower than those from a Saturn 5 rocket, or about 1000 kPa (100 tonnes) versus 30 000 kPa (3000 tonnes). The strength requirements mostly come about from the desire to limit the ship’s weight. So we will want very transparent materials, but they will not necessarily have to be all that strong.

To qualify the radiative environment expected the Firefly Icarus drive can be used as an extreme case. The Deuterium-Deuterium fusion reaction used by Firefly produces radiation mainly as fast neutrons in the 14 MeVolt energy bracket, and a wide range of x-rays energies. The total neutron power is about 15,000 gigawatts, or 15,000 gigajoules per second, and the X-ray power is about 7,000 gigawatts. Dividing the neutron power by the neutron energy gives us 6.7 x10²⁴ neutrons per second coming from the drive. The source is a long thin line. For a pulse propulsion ICF drive, the explosion is a point source. The radiation flux goes down as an inverse square law.

Inverse square law for radiation

Pf= power or neutron flux (W/m²) or (neutrons/m2)or x-ray flux (W/m²)

Pf=P/4*pi*r²

For the Z-pinch drive the equation is a little more complex, since we have a line source rather than a point source: Pf=P/(4*pi*r² + l*2*r*pi) Where ‘l’ is the length of the line, the fusion area of the pinch. Past a certain distance, the results are almost identical.

P=Power (Watts) or Neutrons

r=distance from the source (m)

As a comparison basis, the neutron flux through the zirconium cladding of fuel pellets in a nuclear reactor can reach 1E+21 n/cm2, or 1E+25 neutrons/m2 ₍₄₎. Which is ten times the radiation level near the fusion reaction as seen in the above table, but for much slower neutrons. So the radiation levels seem comparable to one we are already familiar with, and for which existing materials can provide adequate resistance and durability.

Most materials are quite transparent to neutrons and x rays. But not perfectly. If the material is transparent enough, and thin enough, most high energy radiation will go right through. The behavior of this at the atomic level is complex, but fortunately, at the macroscopic level, for high energy neutrons and x-rays it can be simplified down to a simple equation, the Beer-Lambert law, where the energy absorbed is a factor of thickness and an attenuation coefficient ‘μ’ , an experimental value that varies from material to material and according to the type of radiation.

Transparency

Most materials are quite transparent to neutrons and x rays. But not perfectly. If the material is transparent enough, and thin enough, most high energy radiation will go right through. The behavior of this at the atomic level is complex, but fortunately, at the macroscopic level, for high energy neutrons and x-rays it can be simplified down to a simple equation, the Beer-Lambert law, where the energy absorbed is a factor of thickness and an attenuation coefficient ‘μ’ , an experimental value that varies from material to material and according to the type of radiation.

The mass energy attenuation coefficient ‘μ/ρ_en’ is a form used to express the attenuation coefficient when it is modified to take into account the generation of secondary radiation in the material. This coefficient has been tabulated by the NIST for X-rays₍₃₎. For neutrons, the mass attenuation coefficient is available at energies of 8 MeV₍₄₎ these are close to or higher than the energies expected for fusion drives.

From table 3, we can see that X-rays rays are best stopped by dense materials since μ/ρ gets larger with denser materials, while neutrons are best stopped by light materials. In a very general way, x-rays interact with electrons, while neutrons interact with nuclei. So x-ray opaque materials must have the maximum number of electrons, while the best neutron absorbers must have the most tightly packed nuclei, and these are found in the lightest elements.

Controlled defects

The use of controlled defects is the other important aspect of radiation resistance. The way to use these is to choose materials that have a large number of defects in the first place, but in a well known and controlled way. So the basic properties of the material are known and the design can be done. Radiation then interacts with these defects and, if the materials are well chosen, these interactions can actually reduce the influence of the defects. Increasing, rather than decreasing, the strength of the materials. This process is called radiation annealing. So we can start with values that are perhaps 1 to 2% of the ideal values, and expect these to increase a little with time. Fortunately, for strong materials with radiation annealing properties, such as carbon-carbon composites_(2), 1% of 60000 kPa is still 600 kPa, a very respectable number.

In the alternative, if we started from perfect materials, the radiation would inevitably break down this perfection, making the materials weaker. The defects would be random, and might lead to catastrophic failure. So by controlling the flaws in our materials and in designing for them, we will have controlled and reliable properties: Reliable mid range materials, rather than unreliable high strength materials. The appropriate Superman Kryptonite metaphor is left as an exercise to the reader.

Radiation shields

The radiation shields are at the top of the heat chain in the cooling system of the ship. They are the hottest elements of the ship, except for the plasma in the fusion drive. In practice, they are very similar to a fission reactor’ nuclear cores, except that the radiation comes from the outside, from the fusion drive, rather than from the inside, from nuclear fuel. So the thermal and radiation environment will be similar to the one that exists today in fuel elements for nuclear reactors. The neutron flux will be lower, but the temperature will probably be more extreme.

For the radiation shield, we need materials that are as opaque to neutrons and-x rays as possible.

Shield design

From table 3, for neutrons, let’s use boron, in the form of the high temperature ceramic boron carbide, and for X-rays we can use tungsten. Note that the neutron capture layer of boron should be closer to the radiation source, because the capture of neutrons will create gamma rays, that have to stopped as well. Furthermore, we want to stop 99.9% of the particles. A bit of experimentation with equation 6) will give us a thickness of about 200 mm for the boron and 5mm for the tungsten.

If the shield is 50m from the fusion reaction, and is 15m in diameter, then the energy flux from the neutrons will be 341 MW/m2 (from table 2) and the energy flux from the X-rays will be about 71 MW/m2 for a total of 512 MW/m2

P= (Unit power from neutrons +unit power from x-rays)*area

Where:

Area = 177 (m2), a 15m diameter shield. The actual size depends on the geometrical arrangement of the ship elements. But the smaller the shield, the lower the weight.

P= 512 MW/m2 * 177 m2 = 90 000 MW = 90 GW. So we find here the 100 GW of heat from the radiation shield.

As far as the physical arrangement of the shield is concerned, the shield cannot be a flat thin plate as in the left side of figure 9. The heat cannot leave the shield by conduction from the sides, the rate is much too low and the middle would melt. It cannot leave by radiation alone either, since the surface temperature required is over 8 100 K. Therefore, ample space is required for a coolant to circulate in the shield, and the design should increase as much as possible the surface area of the neutron and x-ray capture materials in order to increase heat transfer to the coolant. Interestingly, if liquid metals such as aluminium or beryllium are used as coolants, these have good neutron capture properties, and it might be possible to simplify the shield into some kind of piping loop, or perhaps simply a tank, in which the coolant circulates and absorbs the heat generated by the radiation.

Piping and tanks

Piping and tanks take us to both extremes of material applications, nuclear thermal and cryogenics. Tanks themselves are really just big pipes with end caps, plus a lot of fiddly details where the pipes connect to the tanks. And from the materials point of view, for once we are not in new territory; cryogenics systems are common, often quite large and already used for space applications. The tanks in the Icarus starship should be subject to much lower stresses than those found in a liquid fuel rocket.

Composite fiber tanks may be used, probably with an inner metal liner, or possibly graphene sheets₍₈₎ to prevent leaks due to permeability of most materials to the small atoms of helium and hydrogen. This is a good application for low weight carbon reinforced plastics. The tanks must absolutely be shielded from the drive by a radiation shield, since liquid deuterium will readily stop neutrons, and the tank walls themselves can absorb large quantities of neutrons and x-rays even with very small thicknesses.

For example, using the Beer Lambert law,, for a tank with a thin 0.5mm liner of beryllium, located at 100m from a fusion drive with no shield:

I/I_o = e^-(μ/ρ)ρl= e^{-0.0117*1848*0.0005} = 0.001; therefore the beryllium has absorbed only 0,1% of the radiation intensity. However, the radiation flux at that distance is still 100 MW per m2, so the tank coating will absorb about 100 kW per m2. And that’s only from the neutrons. To put the 100 kW per m2 into perspective, a standard baseboard heater has a power of about 1 kW per meter of length. So each and every square meter of our tank exposed to the radiation from the drive gets as much heating as from 100 meters of electric baseboards. Not good for cryogenic liquids.

Hoop stress in pipes and tanks

Although a modern, detailed analysis of stresses in pipe walls requires sophisticated finite element modeling, it’s good to know that a mathematical model based on a beer keg can do almost as well(5).

The hoop stress is the force exerted on the pipe wall by the pressure in the pipe. The relationship between the two is the following equation:

Hoop stress

σ= Hoop stress (Pa)

Stress = Pressure x radius / thickness of wall

σ=Pr/t

Valid for thin walled cylinders, such as pipes and long cylinders. For a thick walled cylinder, a radial stress component is added. Since we are designing our ship to be as light as possible, thick walled cylinders are not a required refinement.

σ=Pr/2t for tank ends.

P= Internal pressure (Pa)

r= Interior radius(m)

t= Pipe wall thickness(m)

S= Material tensile strengh (Pa)

the relationship with the material of the pipe is :

S/SF=σ

SF= Safety Factor

σ = the allowable hoop stress (Pa)

So, choosing a middle of the range carbon carbon composite at 360 MPa, and a factor of safety of three, because... it divides nicely, we will find:

S/SF= σ= 360 MPa / 3 = 120 MPa or 120,000 kPa

For our big pipe, 1.1m in diameter with a 0.55 m radius, the wall thickness will be:

t= Pr/σ= 8536*0.55 /120,000= 0.04m, or about 4 cm thick

And for the small pipes:

t= Pr/σ= 8536*0,075 /120 000= 0.005m, 5 mm thick.

The volume of the big pipe will be about 0,04m x 1,1m x pi x 100 = 4,4m3

And the volume of the small pipes will be:

0.005m x 0.15m x pi x 50 x 200 = 23.6 m3 for a total of 28 m3.

For a density of 1,500 kg/m3, the weight will be 42 000 kg, or 42 tonnes.

Other stress and strain equations:

Normal stress

σ= Normal stress (Pa)

Stress = Force/Area where the force is normal to the area

σ=F/A

Valid for most application under tension or compression. Will be superseded under compression if the structural member in too thin, and part will fail by buckling

F= Normal force (N)

A= Area(m2)

S= Material tensile strength (Pa)

the relationship with the material of the structural element is :

S/SF=σ

SF= Safety Factor

σ = the allowable normal stress (Pa)

Bending stress

σ= Bending stress (Pa)

Stress = Bending moment*distance from neutral axis/area moment of inertia

σ=M*c/I

Moment, torque and couple are synonyms and all are a Force (N) * distance(m)

Area moments of inertia can be found in many tables() for standard beams and load situations.

M= Moment (Nm)

c= distance from neutral axis(m)

I= Area Moment of Inertia (m4)

Axis perpendicular to the plane

The relationship with the material of the structural element :

S/SF=σ

S= Material tensile strength (Pa)

SF= Safety Factor

σ = the allowable bending stress (Pa)

Shear stress

τ= Shear stress (Pa)

Shear = Force/Area where the force is perpendicular to the area

τ=F/A

Shear is usually combined with other strains.

F= Force (N)

A= Area(m2)

S= Material tensile strength (Pa)

the relationship with the material of the pipe is :

S/SF=σ

SF= Safety Factor

σ = the allowable normal stress (Pa)

Torsional stress (shear)

τ= Shear stress (Pa)

Shear = Torque*distance from axis/polar moment of inertia

τ=Tr/J

Torsion is a form of shear stress. Maximum shear occurs where r=radius of structural member (usually a circular shaft)

T= Torque (Nm)

r=distance from center(m)

J=area moment of area(m4)

Axis normal to the plane

S= Material tensile strength (Pa)

the relationship with the material of the pipe is :

S/SF=σ

SF= Safety Factor

σ = the allowable normal stress (Pa)

All of the above forces and strains can be combined in various load situations that require a more complex mathematical treatment.

The future of materials

The following table lists properties we might expect from materials available in 2100. To build the table, yield strengths were doubled, and maximum use temperatures were increased from 50% to 66% of the melting point.

The allowable stress at room temperature is about 50% of the yield strength, and goes down to about 25% of the yield strength at the maximum use temperature.

We should expect the practical strength of many materials to double from their present values in the next century. But they will still be far below their ideal theoretical strengths. Melting points will remain the same, but materials will be able to be used closer to them. The most promising radiation and heat resistant materials are carbon-carbon composites, and for most applications that do not require corrosion resistance they will not need to be much improved over today’s values. Ceramics, in particular those containing boron, may be the best radiation shields. Fiber and nanoparticle reinforced ceramics may have much better properties than what we have today.

The quantities required for Icarus are not large on an industrial scale, but for some of these materials they are orders of magnitudes larger than today’s production capacities. As requirements for aircraft parts, power turbines and nuclear reactors increase during the next century, we can expect that industry will greatly increase the productions rates, making today’s exotic materials much more commonplace and affordable.

Present day materials are not really suitable for operation beyond 1800K. Substantial increases in existing material capacities and availability will be required to build Icarus and operate at temperatures in the 2000 - 2500 K range. Diamond pipes and graphene reinforced mega materials are not required, however. But if they are around, they’ll certainly be appreciated!

Acknowledgements

Many thanks to David Weiss, VP Engineering/R&D Eck Industries, Inc. for his input to this section.

References

(1) Pokluda, Jaroslav, and Ján Micheľ–Marián Buršák. "THEORETICAL STRENGTH OF SOLIDS." Here is a table for some materials http://bit.ly/1lxe1Sq

(2) Juhasz, Albert J. NASA Glenn Research Center, OH, United States, High Conductivity Carbon-Carbon Heat Pipes for Light Weight Space Power System Radiators http://ntrs.nasa.gov/search.jsp?R=20080045532

(3)Mass energy attenuation coefficients at the NIST http://www.nist.gov/pml/data/xraycoef/index.cfm

(4) Chang, G.S., 2004, INEEL, Integrated fast neutron flux http://www.inl.gov/technicalpublications/documents/3310878.pdf

(5) Go T. Chapman, Co Lo Storrs, National Energy Comission, 1955, Effective neutron removal cross sections for shielding http://web.ornl.gov/info/reports/1955/3445603498123.pdf

(6) Gorilla glass competitor http://www.us.schott.com/xensation/english/products/xcover/technicaldata.htmlà

(7) Thermophysical Properties of Materials for Nuclear Engineering: A Tutorial and Collection of Data , INTERNATIONAL ATOMIC ENERGY AGENCY, VIENNA 2008

http://bit.ly/1gzzVO2

(8) Bunch, J. Scott, Scott S. Verbridge, Jonathan S. Alden, Arend M. Van Der Zande, Jeevak M. Parpia, Harold G. Craighead, and Paul L. McEuen. "Impermeable atomic membranes from graphene sheets." Nano letters 8, no. 8 (2008): 2458-2462.

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