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.
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 flywheel is a mechanical device which 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 aluminium 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 amount of energy stored in a given system or region of space per unit volume. It is sometimes confused with energy per unit mass which is properly 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.
The pound per square inch or, more accurately, pound-force per square inch, is a unit of measurement of pressure or of stress based on avoirdupois units. It is the pressure resulting from a force with magnitude of one pound-force applied to an area of one square inch. In SI units, 1 psi is approximately 6,895 pascals.
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.
7005 is an aluminium wrought alloy used in bicycle frames. Due to its relative ease of welding, it does not require expensive heat treating. It is, however, harder to form, making manufacture more challenging. It has an Ultimate Tensile Strength of 350 MPa, a Fatigue Strength of 150 MPa and a density of 2.78 g/cm3. It does not need to be precipitation hardened, but can be cooled in air.
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.
Ti-6Al-4V, also sometimes called TC4, Ti64, or ASTM Grade 5, is an alpha-beta titanium alloy with a high specific strength and excellent corrosion resistance. It is one of the most commonly used titanium alloys and is applied in a wide range of applications where low density and excellent corrosion resistance are necessary such as e.g. aerospace industry and biomechanical applications.
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.
In materials science, reinforcement is a constituent of a composite material which increases the composite's stiffness and tensile strength.
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