Calculations of breaking length
where is the length, is the tensile strength, is the density and is the acceleration due to gravity ( m/s)
The specific strength is a material's (or muscle's) strength (force per unit area at failure) divided by its density. It is also known as the strength-to-weight ratio or strength/weight ratio or strength-to-mass ratio. In fiber or textile applications, tenacity is the usual measure of specific strength. The SI unit for specific strength is Pa⋅m 3/kg, or N⋅m/kg, which is dimensionally equivalent to m2/s2, though the latter form is rarely used. Specific strength has the same units as specific energy, and is related to the maximum specific energy of rotation that an object can have without flying apart due to centrifugal force.
Another way to describe specific strength is breaking length, also known as self support length: the maximum length of a vertical column of the material (assuming a fixed cross-section) that could suspend its own weight when supported only at the top. For this measurement, the definition of weight is the force of gravity at the Earth's surface (standard gravity, 9.80665 m/s2) applying to the entire length of the material, not diminishing with height. This usage is more common with certain specialty fiber or textile applications.
The materials with the highest specific strengths are typically fibers such as carbon fiber, glass fiber and various polymers, and these are frequently used to make composite materials (e.g. carbon fiber-epoxy). These materials and others such as titanium, aluminium, magnesium and high strength steel alloys are widely used in aerospace and other applications where weight savings are worth the higher material cost.
Note that strength and stiffness are distinct. Both are important in design of efficient and safe structures.
where is the length, is the tensile strength, is the density and is the acceleration due to gravity ( m/s)
| Material | Tensile strength (MPa) | Density (g/cm3) | Specific strength (kN·m/kg) | Breaking length (km) | Source |
|---|---|---|---|---|---|
| Concrete | 2–5 | 2.30 | 5.22 | 0.44 | [ citation needed ] |
| Polyoxymethylene; POM | 69 | 1.42 | 49 | 4.95 | [1] |
| Rubber | 15 | 0.92 | 16.3 | 1.66 | [ citation needed ] |
| Copper | 220 | 8.92 | 24.7 | 2.51 | [ citation needed ] |
| Polypropylene; PP | 25–40 | 0.90 | 28–44 | 2.8–4.5 | [2] |
| (Poly)acrylonitrile-butadiene-styrene; ABS | 41–45 | 1.05 | 39–43 | [3] | |
| Polyethylene terephthalate; polyester; PET | 80 | 1.3–1.4 | 57–62 | [4] | |
| Piano wire; ASTM 228 Steel | 1590–3340 | 7.8 | 204–428 | [5] | |
| Polylactic acid; polylactide; PLA | 53 | 1.24 | 43 | [6] | |
| Low carbon steel (AISI 1010) | 365 | 7.87 | 46.4 | 4.73 | [7] |
| Stainless steel (304) | 505 | 8.00 | 63.1 | 6.4 | [8] |
| Maraging steel (18Ni(350)) | 2450 | 8.2 | 298.78 | 29.7 | [9] |
| Brass | 580 | 8.55 | 67.8 | 6.91 | [10] |
| Nylon | 78 | 1.13 | 69.0 | 7.04 | [11] |
| Titanium | 344 | 4.51 | 76 | 7.75 | [12] |
| CrMo Steel (4130) | 560–670 | 7.85 | 71–85 | 7.27–8.70 | [13] [14] |
| Aluminium alloy (6061-T6) | 310 | 2.70 | 115 | 11.70 | [15] |
| Oak | 90 | 0.78–0.69 | 115–130 | 12–13 | [16] |
| Inconel (X-750) | 1250 | 8.28 | 151 | 15.4 | [17] |
| Magnesium alloy | 275 | 1.74 | 158 | 16.1 | [18] |
| Aluminium alloy (7075-T6) | 572 | 2.81 | 204 | 20.8 | [19] |
| Pine wood (American eastern white) | 78 | 0.35 | 223 | 22.7 | [20] |
| Titanium alloy (Beta C) | 1250 | 4.81 | 260 | 26.5 | [21] |
| Bainite | 2500 | 7.87 | 321 | 32.4 | [22] |
| Reversibly Assembled Cellular Composite Materials | 0.073 | 0.0072 | 10,139 | 1035 | [23] [24] |
| Self-Reprogrammable Mechanical Metamaterials | 0.01117 | 0.0103 | 1,084 | 111 | [25] |
| Balsa | 73 | 0.14 | 521 | 53.2 | [26] |
| Carbon–epoxy composite | 1240 | 1.58 | 785 | 80.0 | [27] |
| Spider silk | 1400 | 1.31 | 1,069 | 109 | [ citation needed ] |
| Silicon carbide fiber | 3440 | 3.16 | 1,088 | 110 | [28] |
| Miralon carbon nanotube yarn C-series | 1375 | 0.7–0.9 | 1,100 | 112 | [29] |
| Glass fiber | 3400 | 2.60 | 1,307 | 133 | [30] |
| Basalt fiber | 4840 | 2.70 | 1,790 | 183 | [31] |
| 1 μm iron whiskers | 14000 | 7.87 | 1,800 | 183 | [22] |
| Vectran | 2900 | 1.40 | 2,071 | 211 | [30] |
| Carbon fiber (AS4) | 4300 | 1.75 | 2,457 | 250 | [30] |
| Kevlar | 3620 | 1.44 | 2,514 | 256 | [32] |
| Dyneema (UHMWPE) | 3600 | 0.97 | 3,711 | 378 | [33] |
| Zylon | 5800 | 1.54 | 3,766 | 384 | [34] |
| Carbon fiber (Toray T1100G) | 7000 | 1.79 | 3,911 | 399 | [35] |
| Carbon nanotube (see note below) | 62000 | 0.037–1.34 | 46,268–N/A | 4716–N/A | [36] [37] |
| Colossal carbon tube | 6900 | 0.116 | 59,483 | 6066 | [38] |
| Graphene | 130500 | 2.090 | 62,453 | 6366 | [39] |
| Fundamental limit | 9×1013 | 9.2×1012 | [40] |
The data of this table is from best cases, and has been established for giving a rough figure.
Note: Multiwalled carbon nanotubes have the highest tensile strength of any material yet measured, with labs producing them at a tensile strength of 63 GPa, [36] still well below their theoretical limit of 300 GPa. The first nanotube ropes (20 mm long) whose tensile strength was published (in 2000) had a strength of 3.6 GPa, still well below their theoretical limit. [41] The density is different depending on the manufacturing method, and the lowest value is 0.037 or 0.55 (solid). [37]
The International Space Elevator Consortium uses the "Yuri" as a name for the SI units describing specific strength. Specific strength is of fundamental importance in the description of space elevator cable materials. One Yuri is conceived to be the SI unit for yield stress (or breaking stress) per unit of density of a material under tension. One Yuri equals 1 Pa⋅m3/kg or 1 N⋅m/kg, which is the breaking/yielding force per linear density of the cable under tension. [42] [43] A functional Earth space elevator would require a tether of 30–80 megaYuri (corresponding to 3100–8200 km of breaking length). [44]
The null energy condition places a fundamental limit on the specific strength of any material. [40] The specific strength is bounded to be no greater than c2 ≈ 9×1013 kN⋅m/kg , where c is the speed of light. This limit is achieved by electric and magnetic field lines, QCD flux tubes, and the fundamental strings hypothesized by string theory.[ citation needed ]
Tenacity is the customary measure of strength of a fiber or yarn. It is usually defined as the ultimate (breaking) force of the fiber (in gram-force units) divided by the denier. Because denier is a measure of the linear density, the tenacity works out to be not a measure of force per unit area, but rather a quasi-dimensionless measure analogous to specific strength. [45] A tenacity of corresponds to:[ citation needed ] Mostly Tenacity expressed in report as cN/tex.
A carbon nanotube (CNT) is a tube made of carbon with a diameter in the nanometre range (nanoscale). They are one of the allotropes of carbon. Two broad classes of carbon nanotubes are recognized:
Kevlar (para-aramid) is a strong, heat-resistant synthetic fiber, related to other aramids such as Nomex and Technora. Developed by Stephanie Kwolek at DuPont in 1965, the high-strength material was first used commercially in the early 1970s as a replacement for steel in racing tires. It is typically spun into ropes or fabric sheets that can be used as such, or as an ingredient in composite material components.
A space elevator, also referred to as a space bridge, star ladder, and orbital lift, is a proposed type of planet-to-space transportation system, often depicted in science fiction. The main component would be a cable anchored to the surface and extending into space. An Earth-based space elevator would consist of a cable with one end attached to the surface near the equator and the other end attached to a counterweight in space beyond geostationary orbit. The competing forces of gravity, which is stronger at the lower end, and the upward centrifugal force, which is stronger at the upper end, would result in the cable being held up, under tension, and stationary over a single position on Earth. With the tether deployed, climbers (crawlers) could repeatedly climb up and down the tether by mechanical means, releasing their cargo to and from orbit. The design would permit vehicles to travel directly between a planetary surface, such as the Earth's, and orbit, without the use of large rockets.
A flywheel is a mechanical device that uses the conservation of angular momentum to store rotational energy, a form of kinetic energy proportional to the product of its moment of inertia and the square of its rotational speed. In particular, assuming the flywheel's moment of inertia is constant then the stored (rotational) energy is directly associated with the square of its rotational speed.
Glass fiber is a material consisting of numerous extremely fine fibers of glass.
A composite material is a material which is produced from two or more constituent materials. These constituent materials have notably dissimilar chemical or physical properties and are merged to create a material with properties unlike the individual elements. Within the finished structure, the individual elements remain separate and distinct, distinguishing composites from mixtures and solid solutions. Composite materials with more than one distinct layer are called composite laminates.
Carbon fibers or carbon fibres are fibers about 5 to 10 micrometers (0.00020–0.00039 in) in diameter and composed mostly of carbon atoms. Carbon fibers have several advantages: high stiffness, high tensile strength, high strength to weight ratio, high chemical resistance, high-temperature tolerance, and low thermal expansion. These properties have made carbon fiber very popular in aerospace, civil engineering, military, motorsports, and other competition sports. However, they are relatively expensive compared to similar fibers, such as glass fiber, basalt fibers, or plastic fibers.
Young's modulus is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Young's modulus is defined as the ratio of the stress applied to the object and the resulting axial strain in the linear elastic region of the material.
Ultimate tensile strength is the maximum stress that a material can withstand while being stretched or pulled before breaking. In brittle materials, the ultimate tensile strength is close to the yield point, whereas in ductile materials, the ultimate tensile strength can be higher.
Inconel is a nickel-chromium-based superalloy often utilized in extreme environments where components are subjected to high temperature, pressure or mechanical loads. Inconel alloys are oxidation- and corrosion-resistant. When heated, Inconel forms a thick, stable, passivating oxide layer protecting the surface from further attack. Inconel retains strength over a wide temperature range, attractive for high-temperature applications where aluminum and steel would succumb to creep as a result of thermally-induced crystal vacancies. Inconel's high-temperature strength is developed by solid solution strengthening or precipitation hardening, depending on the alloy.
In physics, energy density is the quotient between the amount of energy stored in a given system or contained in a given region of space and the volume of the system or region considered. Often only the useful or extractable energy is measured. It is sometimes confused with stored energy per unit mass, which is called specific energy or gravimetric energy density.
Specific modulus is a materials property consisting of the elastic modulus per mass density of a material. It is also known as the stiffness to weight ratio or specific stiffness. High specific modulus materials find wide application in aerospace applications where minimum structural weight is required. The dimensional analysis yields units of distance squared per time squared. The equation can be written as:
6061 aluminium alloy is a precipitation-hardened aluminium alloy, containing magnesium and silicon as its major alloying elements. Originally called "Alloy 61S", it was developed in 1935. It has good mechanical properties, exhibits good weldability, and is very commonly extruded. It is one of the most common alloys of aluminium for general-purpose use.
7075 aluminium alloy (AA7075) is an aluminium alloy with zinc as the primary alloying element. It has excellent mechanical properties and exhibits good ductility, high strength, toughness, and good resistance to fatigue. It is more susceptible to embrittlement than many other aluminium alloys because of microsegregation, but has significantly better corrosion resistance than the alloys from the 2000 series. It is one of the most commonly used aluminium alloys for highly stressed structural applications and has been extensively used in aircraft structural parts.
Carbon nanotubes (CNTs) are cylinders of one or more layers of graphene (lattice). Diameters of single-walled carbon nanotubes (SWNTs) and multi-walled carbon nanotubes (MWNTs) are typically 0.8 to 2 nm and 5 to 20 nm, respectively, although MWNT diameters can exceed 100 nm. CNT lengths range from less than 100 nm to 0.5 m.
Carbon fiber-reinforced polymers, carbon-fibre-reinforced polymers, carbon-fiber-reinforced plastics, carbon-fiber reinforced-thermoplastic, also known as carbon fiber, carbon composite, or just carbon, are extremely strong and light fiber-reinforced plastics that contain carbon fibers. CFRPs can be expensive to produce, but are commonly used wherever high strength-to-weight ratio and stiffness (rigidity) are required, such as aerospace, superstructures of ships, automotive, civil engineering, sports equipment, and an increasing number of consumer and technical applications.
Space tethers are long cables which can be used for propulsion, momentum exchange, stabilization and attitude control, or maintaining the relative positions of the components of a large dispersed satellite/spacecraft sensor system. Depending on the mission objectives and altitude, spaceflight using this form of spacecraft propulsion is theorized to be significantly less expensive than spaceflight using rocket engines.
In materials science, a polymer matrix composite (PMC) is a composite material composed of a variety of short or continuous fibers bound together by a matrix of organic polymers. PMCs are designed to transfer loads between fibers of a matrix. Some of the advantages with PMCs include their light weight, high resistance to abrasion and corrosion, and high stiffness and strength along the direction of their reinforcements.
A graphene helix, similar to the carbon nanotube, is a structure consisting of a two-dimensional sheet of graphene wrapped into a helix. These graphene sheets can have multiple layers, called multi-walled carbon structures, that add to these helices thus increasing their tensile strength but increasing the difficulty of manufacturing. Using van der Waals interactions it can make structures within one another.
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