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120-441: The decline in the cost of carbon fibre during the 1970s allowed sailplane designers to design large components from carbon-fibre reinforced plastic (CFRP). The use of carbon-fibre lay-ups, designed to reduce aero-elasticity of wings, has allowed CFRP gliders to be designed to cruise at much higher speeds than those with wings made from fibreglass (GFRP) or wood, as well as take advantage of thinner aerofoil sections. The Ventus has

240-420: A flow of viscous liquid , the force F may not be perpendicular to S ; hence the stress across a surface must be regarded a vector quantity, not a scalar. Moreover, the direction and magnitude generally depend on the orientation of S . Thus the stress state of the material must be described by a tensor , called the (Cauchy) stress tensor ; which is a linear function that relates the normal vector n of

360-438: A vacuum bag can be used. A fiberglass, carbon fiber, or aluminum mold is polished and waxed, and has a release agent applied before the fabric and resin are applied, and the vacuum is pulled and set aside to allow the piece to cure (harden). There are three ways to apply the resin to the fabric in a vacuum mold. The first method is manual and called a wet layup, where the two-part resin is mixed and applied before being laid in

480-439: A "particle" as being an infinitesimal patch of the plate's surface, so that the boundary between adjacent particles becomes an infinitesimal line element; both are implicitly extended in the third dimension, normal to (straight through) the plate. "Stress" is then redefined as being a measure of the internal forces between two adjacent "particles" across their common line element, divided by the length of that line. Some components of

600-560: A 15-metre wingspan, using CFRP in its structure, complying with 15 m Racing Class specifications. Extended wingtips can be fitted to increase the span to 16.6 metres for competition in the Open Class. The fuselage was also produced in two sizes to suit pilots of different heights – the Ventus A for short pilots and Ventus B with a longer and wider cockpit. Both Ventus A and Ventus B have a complex flap/brake arrangement similar to that of

720-401: A 2/2 weave. The process by which most CFRPs are made varies, depending on the piece being created, the finish (outside gloss) required, and how many of the piece will be produced. In addition, the choice of matrix can have a profound effect on the properties of the finished composite. Many CFRP parts are created with a single layer of carbon fabric that is backed with fiberglass. A tool called

840-437: A central wing-box made of CFRP; it is the first to have a smoothly contoured wing cross-section instead of the wings being partitioned span-wise into sections. This flowing, continuous cross section optimises aerodynamic efficiency. Moreover, the trailing edge, along with the rear bulkhead, empennage , and un-pressurised fuselage are made of CFRP. However, many delays have pushed order delivery dates back because of problems with

960-447: A chopper gun is used to quickly create these composite parts. Once a thin shell is created out of carbon fiber, the chopper gun cuts rolls of fiberglass into short lengths and sprays resin at the same time, so that the fiberglass and resin are mixed on the spot. The resin is either external mix, wherein the hardener and resin are sprayed separately, or internal mixed, which requires cleaning after every use. Manufacturing methods may include

1080-392: A coordinate system with axes e 1 , e 2 , e 3 {\displaystyle e_{1},e_{2},e_{3}} , the stress tensor is a diagonal matrix, and has only the three normal components λ 1 , λ 2 , λ 3 {\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}

1200-466: A core. Applications for CFRPs include the following: The Airbus A350 XWB is built of 53% CFRP including wing spars and fuselage components, overtaking the Boeing 787 Dreamliner , for the aircraft with the highest weight ratio for CFRP, which is 50%. This was one of the first commercial aircraft to have wing spars made from composites. The Airbus A380 was one of the first commercial airliners to have

1320-457: A cylindrical bar such as a shaft is subjected to opposite torques at its ends. In that case, the shear stress on each cross-section is parallel to the cross-section, but oriented tangentially relative to the axis, and increases with distance from the axis. Significant shear stress occurs in the middle plate (the "web") of I-beams under bending loads, due to the web constraining the end plates ("flanges"). Another simple type of stress occurs when

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1440-454: A differential formula for friction forces (shear stress) in parallel laminar flow . Stress is defined as the force across a small boundary per unit area of that boundary, for all orientations of the boundary. Derived from a fundamental physical quantity (force) and a purely geometrical quantity (area), stress is also a fundamental quantity, like velocity, torque or energy , that can be quantified and analyzed without explicit consideration of

1560-466: A material may arise by various mechanisms, such as stress as applied by external forces to the bulk material (like gravity ) or to its surface (like contact forces , external pressure, or friction ). Any strain (deformation) of a solid material generates an internal elastic stress , analogous to the reaction force of a spring , that tends to restore the material to its original non-deformed state. In liquids and gases , only deformations that change

1680-505: A mold, with epoxy either pre-impregnated into the fibers (also known as pre-preg ) or "painted" over it. High-performance parts using single molds are often vacuum-bagged and/or autoclave -cured, because even small air bubbles in the material will reduce strength. An alternative to the autoclave method is to use internal pressure via inflatable air bladders or EPS foam inside the non-cured laid-up carbon fiber. For simple pieces of which relatively few copies are needed (one or two per day),

1800-432: A number of field applications strengthening concrete, masonry, steel, cast iron, and timber structures. Their use in industry can be either for retrofitting to strengthen an existing structure or as an alternative reinforcing (or prestressing) material instead of steel from the outset of a project. Retrofitting has become the increasingly dominant use of the material in civil engineering, and applications include increasing

1920-422: A refinery in a one-step process. Capture and reuse of the carbon and monomers is then possible. CFRPs can also be milled or shredded at low temperature to reclaim the carbon fiber; however, this process shortens the fibers dramatically. Just as with downcycled paper, the shortened fibers cause the recycled material to be weaker than the original material. There are still many industrial applications that do not need

2040-720: A single carbon fiber with diameter of 5–7 μm is sealed in a glass capillary. At the tip the capillary is either sealed with epoxy and polished to make carbon-fiber disk microelectrode or the fiber is cut to a length of 75–150 μm to make carbon-fiber cylinder electrode. Carbon-fiber microelectrodes are used either in amperometry or fast-scan cyclic voltammetry for detection of biochemical signalling. CFRPs are now widely used in sports equipment such as in squash, tennis, and badminton racquets, sport kite spars, high-quality arrow shafts, hockey sticks, fishing rods, surfboards , high end swim fins, and rowing shells . Amputee athletes such as Jonnie Peacock use carbon fiber blades for running. It

2160-426: A small sustaining engine ("turbo") and are designated with a T while some are equipped with a more powerful engine and are self-launching and are designated with an M . The best measured glide ratio of a Ventus at 15 metres wingspan is 44:1, improving to 46:1 with the 16.6 metre tips, giving a creditable performance for the 1980s and 1990s. Ventus sailplanes won two World Gliding Championships , competing against

2280-463: A stretched elastic band, is subject to tensile stress and may undergo elongation . An object being pushed together, such as a crumpled sponge, is subject to compressive stress and may undergo shortening. The greater the force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Stress has dimension of force per area, with SI units of newtons per square meter (N/m ) or pascal (Pa). Stress expresses

2400-452: A surface S to the traction vector T across S . With respect to any chosen coordinate system , the Cauchy stress tensor can be represented as a symmetric matrix of 3×3 real numbers. Even within a homogeneous body, the stress tensor may vary from place to place, and may change over time; therefore, the stress within a material is, in general, a time-varying tensor field . In general,

2520-1007: A surface will always be a linear function of the surface's normal vector n {\displaystyle n} , the unit-length vector that is perpendicular to it. That is, T = σ ( n ) {\displaystyle T={\boldsymbol {\sigma }}(n)} , where the function σ {\displaystyle {\boldsymbol {\sigma }}} satisfies σ ( α u + β v ) = α σ ( u ) + β σ ( v ) {\displaystyle {\boldsymbol {\sigma }}(\alpha u+\beta v)=\alpha {\boldsymbol {\sigma }}(u)+\beta {\boldsymbol {\sigma }}(v)} for any vectors u , v {\displaystyle u,v} and any real numbers α , β {\displaystyle \alpha ,\beta } . The function σ {\displaystyle {\boldsymbol {\sigma }}} , now called

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2640-434: A surface with normal vector n {\displaystyle n} (which is covariant - "row; horizontal" - vector) with coordinates n 1 , n 2 , n 3 {\displaystyle n_{1},n_{2},n_{3}} is then a matrix product T = n ⋅ σ {\displaystyle T=n\cdot {\boldsymbol {\sigma }}} (where T in upper index

2760-413: A system must be balanced by internal reaction forces, which are almost always surface contact forces between adjacent particles — that is, as stress. Since every particle needs to be in equilibrium, this reaction stress will generally propagate from particle to particle, creating a stress distribution throughout the body. The typical problem in stress analysis is to determine these internal stresses, given

2880-434: A system of partial differential equations involving the stress tensor field and the strain tensor field, as unknown functions to be determined. The external body forces appear as the independent ("right-hand side") term in the differential equations, while the concentrated forces appear as boundary conditions. The basic stress analysis problem is therefore a boundary-value problem . Stress analysis for elastic structures

3000-489: A two-dimensional one, and/or replace the general stress and strain tensors by simpler models like uniaxial tension/compression, simple shear, etc. Still, for two- or three-dimensional cases one must solve a partial differential equation problem. Analytical or closed-form solutions to the differential equations can be obtained when the geometry, constitutive relations, and boundary conditions are simple enough. Otherwise one must generally resort to numerical approximations such as

3120-1092: Is transposition , and as a result we get covariant (row) vector) (look on Cauchy stress tensor ), that is [ T 1 T 2 T 3 ] = [ n 1 n 2 n 3 ] ⋅ [ σ 11 σ 21 σ 31 σ 12 σ 22 σ 32 σ 13 σ 23 σ 33 ] {\displaystyle {\begin{bmatrix}T_{1}&T_{2}&T_{3}\end{bmatrix}}={\begin{bmatrix}n_{1}&n_{2}&n_{3}\end{bmatrix}}\cdot {\begin{bmatrix}\sigma _{11}&\sigma _{21}&\sigma _{31}\\\sigma _{12}&\sigma _{22}&\sigma _{32}\\\sigma _{13}&\sigma _{23}&\sigma _{33}\end{bmatrix}}} The linear relation between T {\displaystyle T} and n {\displaystyle n} follows from

3240-410: Is actually the average of a very large number of atomic forces between their molecules; and physical quantities like mass, velocity, and forces that act through the bulk of three-dimensional bodies, like gravity, are assumed to be smoothly distributed over them. Depending on the context, one may also assume that the particles are large enough to allow the averaging out of other microscopic features, like

3360-583: Is an essential tool in engineering for the study and design of structures such as tunnels, dams, mechanical parts, and structural frames, under prescribed or expected loads. It is also important in many other disciplines; for example, in geology, to study phenomena like plate tectonics , vulcanism and avalanches ; and in biology, to understand the anatomy of living beings. Stress analysis is generally concerned with objects and structures that can be assumed to be in macroscopic static equilibrium . By Newton's laws of motion , any external forces being applied to such

3480-406: Is assumed fixed, the normal component can be expressed by a single number, the dot product T · n . This number will be positive if P is "pulling" on Q (tensile stress), and negative if P is "pushing" against Q (compressive stress). The shear component is then the vector T − ( T · n ) n . The dimension of stress is that of pressure , and therefore its coordinates are measured in

3600-478: Is based on the theory of elasticity and infinitesimal strain theory . When the applied loads cause permanent deformation, one must use more complicated constitutive equations, that can account for the physical processes involved ( plastic flow , fracture , phase change , etc.). Engineered structures are usually designed so the maximum expected stresses are well within the range of linear elasticity (the generalization of Hooke's law for continuous media); that is,

3720-409: Is dependent on the fiber orientation and machining condition of the cutting process. To reduce tool wear various types of coated tools are used in machining CFRP and CFRP-metal stack. The primary element of CFRPs is a carbon filament ; this is produced from a precursor polymer such as polyacrylonitrile (PAN), rayon , or petroleum pitch . For synthetic polymers such as PAN or rayon, the precursor

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3840-438: Is first spun into filament yarns, using chemical and mechanical processes to initially align the polymer chains in a way to enhance the final physical properties of the completed carbon fiber. Precursor compositions and mechanical processes used during spinning filament yarns may vary among manufacturers. After drawing or spinning, the polymer filament yarns are then heated to drive off non-carbon atoms ( carbonization ), producing

3960-641: Is given in the article on viscosity . The same for normal viscous stresses can be found in Sharma (2019). The relation between stress and its effects and causes, including deformation and rate of change of deformation, can be quite complicated (although a linear approximation may be adequate in practice if the quantities are small enough). Stress that exceeds certain strength limits of the material will result in permanent deformation (such as plastic flow , fracture , cavitation ) or even change its crystal structure and chemical composition . In some situations,

4080-518: Is mitigated by the material's unsurpassed strength-to-weight ratio, and low weight is essential for high-performance automobile racing. Race-car manufacturers have also developed methods to give carbon fiber pieces strength in a certain direction, making it strong in a load-bearing direction, but weak in directions where little or no load would be placed on the member. Conversely, manufacturers developed omnidirectional carbon fiber weaves that apply strength in all directions. This type of carbon fiber assembly

4200-619: Is most widely used in the "safety cell" monocoque chassis assembly of high-performance race-cars. The first carbon fiber monocoque chassis was introduced in Formula One by McLaren in the 1981 season. It was designed by John Barnard and was widely copied in the following seasons by other F1 teams due to the extra rigidity provided to the chassis of the cars. Many supercars over the past few decades have incorporated CFRPs extensively in their manufacture, using it for their monocoque chassis as well as other components. As far back as 1971,

4320-413: Is much more difficult to process and more expensive. Despite their high initial strength-to-weight ratios, a design limitation of CFRPs are their lack of a definable fatigue limit . This means, theoretically, that stress cycle failure cannot be ruled out. While steel and many other structural metals and alloys do have estimable fatigue or endurance limits, the complex failure modes of composites mean that

4440-409: Is often a thermoset resin such as epoxy , but other thermoset or thermoplastic polymers, such as polyester , vinyl ester , or nylon, are sometimes used. The properties of the final CFRP product can be affected by the type of additives introduced to the binding matrix (resin). The most common additive is silica , but other additives such as rubber and carbon nanotubes can be used. Carbon fiber

4560-513: Is often used for safety certification and monitoring. Most stress is analysed by mathematical methods, especially during design. The basic stress analysis problem can be formulated by Euler's equations of motion for continuous bodies (which are consequences of Newton's laws for conservation of linear momentum and angular momentum ) and the Euler-Cauchy stress principle , together with the appropriate constitutive equations. Thus one obtains

4680-408: Is perpendicular to the layer, the net internal force across S , and hence the stress, will be zero. As in the case of an axially loaded bar, in practice the shear stress may not be uniformly distributed over the layer; so, as before, the ratio F / A will only be an average ("nominal", "engineering") stress. That average is often sufficient for practical purposes. Shear stress is observed also when

4800-553: Is several times stronger and tougher than typical CFRPs and is used in the Lockheed Martin F-35 Lightning II as a structural material for aircraft. CNRP still uses carbon fiber as the primary reinforcement, but the binding matrix is a carbon nanotube-filled epoxy. Stress (mechanics) In continuum mechanics , stress is a physical quantity that describes forces present during deformation . For example, an object being pulled apart, such as

4920-646: Is significantly improved if a thin layer of carbon fibers is moulded near the surface because a dense, compact layer of carbon fibers efficiently reflects heat. CFRPs are being used in an increasing number of high-end products that require stiffness and low weight, these include: CFRPs have a long service lifetime when protected from the sun. When it is time to decommission CFRPs, they cannot be melted down in air like many metals. When free of vinyl (PVC or polyvinyl chloride ) and other halogenated polymers, CFRPs can be thermally decomposed via thermal depolymerization in an oxygen-free environment. This can be accomplished in

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5040-442: Is sometimes referred to as graphite-reinforced polymer or graphite fiber-reinforced polymer ( GFRP is less common, as it clashes with glass-(fiber)-reinforced polymer ). CFRP are composite materials . In this case the composite consists of two parts: a matrix and a reinforcement. In CFRP the reinforcement is carbon fiber, which provides its strength. The matrix is usually a thermosetting plastic, such as polyester resin, to bind

5160-1089: Is still being flown at the World Championship level in 2017. data from:Schempp-Hirth Flugzeugbau GmbH website Data from Jane's All the World's Aircraft 1988–89 General characteristics Performance Related lists Carbon fiber Carbon fiber-reinforced polymers ( American English ), carbon-fibre-reinforced polymers ( Commonwealth English ), carbon-fiber-reinforced plastics , carbon-fiber reinforced-thermoplastic ( CFRP , CRP , CFRTP ), 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. The binding polymer

5280-412: Is subjected to tension by opposite forces of magnitude F {\displaystyle F} along its axis. If the system is in equilibrium and not changing with time, and the weight of the bar can be neglected, then through each transversal section of the bar the top part must pull on the bottom part with the same force, F with continuity through the full cross-sectional area , A . Therefore,

5400-401: Is the total composite modulus, V m {\displaystyle V_{m}} and V f {\displaystyle V_{f}} are the volume fractions of the matrix and fiber respectively in the composite, and E m {\displaystyle E_{m}} and E f {\displaystyle E_{f}} are the elastic moduli of

5520-437: Is then reduced to a scalar (tension or compression of the bar), but one must take into account also a bending stress (that tries to change the bar's curvature, in some direction perpendicular to the axis) and a torsional stress (that tries to twist or un-twist it about its axis). Stress analysis is a branch of applied physics that covers the determination of the internal distribution of internal forces in solid objects. It

5640-576: Is too small to be detected. In a solid material, such strain will in turn generate an internal elastic stress, analogous to the reaction force of a stretched spring , tending to restore the material to its original undeformed state. Fluid materials (liquids, gases and plasmas ) by definition can only oppose deformations that would change their volume. If the deformation changes with time, even in fluids there will usually be some viscous stress, opposing that change. Such stresses can be either shear or normal in nature. Molecular origin of shear stresses in fluids

5760-436: Is typical). As a consequence, only small cross-sectional areas of the material are used. Small areas of very high strength but moderate stiffness material will significantly increase strength, but not stiffness. CFRPs can also be used to enhance shear strength of reinforced concrete by wrapping fabrics or fibers around the section to be strengthened. Wrapping around sections (such as bridge or building columns) can also enhance

5880-412: Is used as a shank plate in some basketball sneakers to keep the foot stable, usually running the length of the shoe just above the sole and left exposed in some areas, usually in the arch. Controversially, in 2006, cricket bats with a thin carbon-fiber layer on the back were introduced and used in competitive matches by high-profile players including Ricky Ponting and Michael Hussey . The carbon fiber

6000-505: The (Cauchy) stress tensor , completely describes the stress state of a uniformly stressed body. (Today, any linear connection between two physical vector quantities is called a tensor , reflecting Cauchy's original use to describe the "tensions" (stresses) in a material.) In tensor calculus , σ {\displaystyle {\boldsymbol {\sigma }}} is classified as a second-order tensor of type (0,2) or (1,1) depending on convention. Like any linear map between vectors,

6120-566: The ASW 20 and later the LS6 in the 15 m class. Whilst the Ventus A and B acquired reputations for unparalleled high-speed glide performance, the Ventus C introduced improvements in handling. About 613 Ventus sailplanes were built before being replaced by the Schempp-Hirth Ventus 2 in 1995. The Ventus 1 remains a competitive glider winning national level competitions. The Ventus 1A

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6240-639: The Citroën SM offered optional lightweight carbon fiber wheels. Use of the material has been more readily adopted by low-volume manufacturers who used it primarily for creating body-panels for some of their high-end cars due to its increased strength and decreased weight compared with the glass-reinforced polymer they used for the majority of their products. CFRPs have become a notable material in structural engineering applications. Studied in an academic context as to their potential benefits in construction, CFRPs have also proved themselves cost-effective in

6360-573: The Glasflügel Mosquito and the Mini-Nimbus , but the Ventus C reverted to conventional upper-surface air-brakes that are separate from the trailing-edge flaps. Wingtip extensions could be fitted to the Ventus A and B to increase the wingspan to 16.6 m. Longer extensions could be fitted to the Ventus C to increase the span to 17.6 m. Modified extensions with winglets are also available. Some Ventus Bs and Cs are equipped with

6480-591: The brittle nature of CFRPs, in contrast to the ductility of steel. Though design codes have been drawn up by institutions such as the American Concrete Institute , there remains some hesitation among the engineering community about implementing these alternative materials. In part, this is due to a lack of standardization and the proprietary nature of the fiber and resin combinations on the market. Carbon fibers are used for fabrication of carbon-fiber microelectrodes . In this application typically

6600-610: The capitals , arches , cupolas , trusses and the flying buttresses of Gothic cathedrals . Ancient and medieval architects did develop some geometrical methods and simple formulas to compute the proper sizes of pillars and beams, but the scientific understanding of stress became possible only after the necessary tools were invented in the 17th and 18th centuries: Galileo Galilei 's rigorous experimental method , René Descartes 's coordinates and analytic geometry , and Newton 's laws of motion and equilibrium and calculus of infinitesimals . With those tools, Augustin-Louis Cauchy

6720-478: The ductility of the section, greatly increasing the resistance to collapse under dynamic loading. Such 'seismic retrofit' is the major application in earthquake-prone areas, since it is much more economic than alternative methods. If a column is circular (or nearly so) an increase in axial capacity is also achieved by wrapping. In this application, the confinement of the CFRP wrap enhances the compressive strength of

6840-993: The orthogonal shear stresses . The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the Mohr's circle of stress distribution. As a symmetric 3×3 real matrix, the stress tensor σ {\displaystyle {\boldsymbol {\sigma }}} has three mutually orthogonal unit-length eigenvectors e 1 , e 2 , e 3 {\displaystyle e_{1},e_{2},e_{3}} and three real eigenvalues λ 1 , λ 2 , λ 3 {\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}} , such that σ e i = λ i e i {\displaystyle {\boldsymbol {\sigma }}e_{i}=\lambda _{i}e_{i}} . Therefore, in

6960-457: The principal stresses . If the three eigenvalues are equal, the stress is an isotropic compression or tension, always perpendicular to any surface, there is no shear stress, and the tensor is a diagonal matrix in any coordinate frame. In general, stress is not uniformly distributed over a material body, and may vary with time. Therefore, the stress tensor must be defined for each point and each moment, by considering an infinitesimal particle of

7080-442: The strain rate can be quite complicated, although a linear approximation may be adequate in practice if the quantities are sufficiently small. Stress that exceeds certain strength limits of the material will result in permanent deformation (such as plastic flow , fracture , cavitation ) or even change its crystal structure and chemical composition . Humans have known about stress inside materials since ancient times. Until

7200-445: The 17th century, this understanding was largely intuitive and empirical, though this did not prevent the development of relatively advanced technologies like the composite bow and glass blowing . Over several millennia, architects and builders in particular, learned how to put together carefully shaped wood beams and stone blocks to withstand, transmit, and distribute stress in the most effective manner, with ingenious devices such as

7320-430: The absence of external forces; such built-in stress is important, for example, in prestressed concrete and tempered glass . Stress may also be imposed on a material without the application of net forces , for example by changes in temperature or chemical composition, or by external electromagnetic fields (as in piezoelectric and magnetostrictive materials). The relation between mechanical stress, strain, and

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7440-426: The brittle fracture mechanics presents unique challenges to engineers in failure detection since failure occurs catastrophically. As such, recent efforts to toughen CFRPs include modifying the existing epoxy material and finding alternative polymer matrix. One such material with high promise is PEEK , which exhibits an order of magnitude greater toughness with similar elastic modulus and tensile strength. However, PEEK

7560-449: The bulk of the material, varying continuously with position and time. Other agents (like external loads and friction, ambient pressure, and contact forces) may create stresses and forces that are concentrated on certain surfaces, lines or points; and possibly also on very short time intervals (as in the impulses due to collisions). In active matter , self-propulsion of microscopic particles generates macroscopic stress profiles. In general,

7680-407: The concrete. However, although large increases are achieved in the ultimate collapse load, the concrete will crack at only slightly enhanced load, meaning that this application is only occasionally used. Specialist ultra-high modulus CFRP (with tensile modulus of 420 GPa or more) is one of the few practical methods of strengthening cast iron beams. In typical use, it is bonded to the tensile flange of

7800-498: The cross-section), but will vary over the cross section: the outer part will be under tensile stress, while the inner part will be compressed. Another variant of normal stress is the hoop stress that occurs on the walls of a cylindrical pipe or vessel filled with pressurized fluid. Another simple type of stress occurs when a uniformly thick layer of elastic material like glue or rubber is firmly attached to two stiff bodies that are pulled in opposite directions by forces parallel to

7920-402: The deformations caused by internal stresses are linearly related to them. In this case the differential equations that define the stress tensor are linear, and the problem becomes much easier. For one thing, the stress at any point will be a linear function of the loads, too. For small enough stresses, even non-linear systems can usually be assumed to be linear. Stress analysis is simplified when

8040-409: The desired component. The benefit is the speed of the entire process. Some car manufacturers, such as BMW, claimed to be able to cycle a new part every 80 seconds. However, this technique has a very high initial cost since the molds require CNC machining of very high precision. For difficult or convoluted shapes, a filament winder can be used to make CFRP parts by winding filaments around a mandrel or

8160-709: The effect of gravity and other external forces can be neglected. In these situations, the stress across any imaginary internal surface turns out to be equal in magnitude and always directed perpendicularly to the surface independently of the surface's orientation. This type of stress may be called isotropic normal or just isotropic ; if it is compressive, it is called hydrostatic pressure or just pressure . Gases by definition cannot withstand tensile stresses, but some liquids may withstand very large amounts of isotropic tensile stress under some circumstances. see Z-tube . Parts with rotational symmetry , such as wheels, axles, pipes, and pillars, are very common in engineering. Often

8280-434: The elements σ x , σ y , σ z {\displaystyle \sigma _{x},\sigma _{y},\sigma _{z}} are called the orthogonal normal stresses (relative to the chosen coordinate system), and τ x y , τ x z , τ y z {\displaystyle \tau _{xy},\tau _{xz},\tau _{yz}}

8400-424: The external forces that are acting on the system. The latter may be body forces (such as gravity or magnetic attraction), that act throughout the volume of a material; or concentrated loads (such as friction between an axle and a bearing , or the weight of a train wheel on a rail), that are imagined to act over a two-dimensional area, or along a line, or at single point. In stress analysis one normally disregards

8520-550: The fatigue failure properties of CFRPs are difficult to predict and design against; however emerging research has shed light on the effects of low velocity impacts on composites. Low velocity impacts can make carbon fibre polymers susceptible to damage. As a result, when using CFRPs for critical cyclic-loading applications, engineers may need to design in considerable strength safety margins to provide suitable component reliability over its service life. Environmental effects such as temperature and humidity can have profound effects on

8640-401: The final carbon fiber. The carbon fibers filament yarns may be further treated to improve handling qualities, then wound onto bobbins . From these fibers, a unidirectional sheet is created. These sheets are layered onto each other in a quasi-isotropic layup, e.g. 0°, +60°, or −60° relative to each other. From the elementary fiber, a bidirectional woven sheet can be created, i.e. a twill with

8760-495: The following: One method of producing CFRP parts is by layering sheets of carbon fiber cloth into a mold in the shape of the final product. The alignment and weave of the cloth fibers is chosen to optimize the strength and stiffness properties of the resulting material. The mold is then filled with epoxy and is heated or air-cured. The resulting part is very corrosion-resistant, stiff, and strong for its weight. Parts used in less critical areas are manufactured by draping cloth over

8880-415: The form of hydrogen embrittlement has been blamed for the gradual deterioration of the prestressing wires in many PCCP lines. Over the past decade, CFRPs have been used to internally line PCCP, resulting in a fully structural strengthening system. Inside a PCCP line, the CFRP liner acts as a barrier that controls the level of strain experienced by the steel cylinder in the host pipe. The composite liner enables

9000-412: The fundamental laws of conservation of linear momentum and static equilibrium of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations ( Cauchy's equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that

9120-461: The grains of a metal rod or the fibers of a piece of wood . Quantitatively, the stress is expressed by the Cauchy traction vector T defined as the traction force F between adjacent parts of the material across an imaginary separating surface S , divided by the area of S . In a fluid at rest the force is perpendicular to the surface, and is the familiar pressure . In a solid , or in

9240-423: The internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the relative deformation of the material. For example, when a solid vertical bar is supporting an overhead weight , each particle in the bar pushes on the particles immediately below it. When a liquid is in a closed container under pressure , each particle gets pushed against by all

9360-408: The layer; or a section of a soft metal bar that is being cut by the jaws of a scissors-like tool . Let F be the magnitude of those forces, and M be the midplane of that layer. Just as in the normal stress case, the part of the layer on one side of M must pull the other part with the same force F . Assuming that the direction of the forces is known, the stress across M can be expressed simply by

9480-440: The load capacity of old structures (such as bridges, beams, ceilings, columns and walls) that were designed to tolerate far lower service loads than they are experiencing today, seismic retrofitting, and repair of damaged structures. Retrofitting is popular in many instances as the cost of replacing the deficient structure can greatly exceed the cost of strengthening using CFRP. Applied to reinforced concrete structures for flexure,

9600-419: The manufacture of these parts. Many aircraft that use CFRPs have experienced delays with delivery dates due to the relatively new processes used to make CFRP components, whereas metallic structures have been studied and used on airframes for decades, and the processes are relatively well understood. A recurrent problem is the monitoring of structural ageing, for which new methods are constantly investigated, due to

9720-429: The material body is under equal compression or tension in all directions. This is the case, for example, in a portion of liquid or gas at rest, whether enclosed in some container or as part of a larger mass of fluid; or inside a cube of elastic material that is being pressed or pulled on all six faces by equal perpendicular forces — provided, in both cases, that the material is homogeneous, without built-in stress, and that

9840-568: The matrix and fibers respectively. The other extreme case of the elastic modulus of the composite with the fibers oriented transverse to the applied load can be found using the equation: The fracture toughness of carbon fiber reinforced plastics is governed by the mechanisms: 1) debonding between the carbon fiber and polymer matrix, 2) fiber pull-out, and 3) delamination between the CFRP sheets. Typical epoxy-based CFRPs exhibit virtually no plasticity, with less than 0.5% strain to failure. Although CFRPs with epoxy have high strength and elastic modulus,

9960-623: The matrix in CFRPs such as compressive, interlaminar shear, and impact properties. The epoxy matrix used for engine fan blades is designed to be impervious against jet fuel, lubrication, and rain water, and external paint on the composites parts is applied to minimize damage from ultraviolet light. Carbon fibers can cause galvanic corrosion when CRP parts are attached to aluminum or mild steel but not to stainless steel or titanium. Carbon Fiber Reinforced Plastics are very hard to machine, and cause significant tool wear. The tool wear in CFRP machining

10080-519: The medium surrounding that point, and taking the average stresses in that particle as being the stresses at the point. Human-made objects are often made from stock plates of various materials by operations that do not change their essentially two-dimensional character, like cutting, drilling, gentle bending and welding along the edges. The description of stress in such bodies can be simplified by modeling those parts as two-dimensional surfaces rather than three-dimensional bodies. In that view, one redefines

10200-433: The mold and placed in the bag. The other one is done by infusion, where the dry fabric and mold are placed inside the bag while the vacuum pulls the resin through a small tube into the bag, then through a tube with holes or something similar to evenly spread the resin throughout the fabric. Wire loom works perfectly for a tube that requires holes inside the bag. Both of these methods of applying resin require hand work to spread

10320-448: The most general case, called triaxial stress , the stress is nonzero across every surface element. Combined stresses cannot be described by a single vector. Even if the material is stressed in the same way throughout the volume of the body, the stress across any imaginary surface will depend on the orientation of that surface, in a non-trivial way. Cauchy observed that the stress vector T {\displaystyle T} across

10440-420: The nature of the material or of its physical causes. Following the basic premises of continuum mechanics, stress is a macroscopic concept. Namely, the particles considered in its definition and analysis should be just small enough to be treated as homogeneous in composition and state, but still large enough to ignore quantum effects and the detailed motions of molecules. Thus, the force between two particles

10560-647: The need to re-true a wheel and the reduced mass reduces the moment of inertia of the wheel. CFRP spokes are rare and most carbon wheelsets retain traditional stainless steel spokes. CFRPs also appear increasingly in other components such as derailleur parts, brake and shifter levers and bodies, cassette sprocket carriers, suspension linkages, disc brake rotors, pedals, shoe soles, and saddle rails. Although strong and light, impact, over-torquing, or improper installation of CFRP components has resulted in cracking and failures, which may be difficult or impossible to repair. The fire resistance of polymers and thermo-set composites

10680-452: The physical causes of the forces or the precise nature of the materials. Instead, one assumes that the stresses are related to deformation (and, in non-static problems, to the rate of deformation) of the material by known constitutive equations . Stress analysis may be carried out experimentally, by applying loads to the actual artifact or to scale model, and measuring the resulting stresses, by any of several available methods. This approach

10800-424: The physical dimensions and the distribution of loads allow the structure to be treated as one- or two-dimensional. In the analysis of trusses, for example, the stress field may be assumed to be uniform and uniaxial over each member. Then the differential equations reduce to a finite set of equations (usually linear) with finitely many unknowns. In other contexts one may be able to reduce the three-dimensional problem to

10920-445: The plate). The analysis of stress can be considerably simplified also for thin bars, beams or wires of uniform (or smoothly varying) composition and cross-section that are subjected to moderate bending and twisting. For those bodies, one may consider only cross-sections that are perpendicular to the bar's axis, and redefine a "particle" as being a piece of wire with infinitesimal length between two such cross sections. The ordinary stress

11040-494: The polymer-based composites, including most CFRPs. While CFRPs demonstrate excellent corrosion resistance, the effect of moisture at wide ranges of temperatures can lead to degradation of the mechanical properties of CFRPs, particularly at the matrix-fiber interface. While the carbon fibers themselves are not affected by the moisture diffusing into the material, the moisture plasticizes the polymer matrix. This leads to significant changes in properties that are dominantly influenced by

11160-436: The proportion of the carbon fibers relative to the polymer. The two different equations governing the net elastic modulus of composite materials using the properties of the carbon fibers and the polymer matrix can also be applied to carbon fiber reinforced plastics. The equation: is valid for composite materials with the fibers oriented in the direction of the applied load. E c {\displaystyle E_{c}}

11280-414: The reinforcements together. Because CFRPs consist of two distinct elements, the material properties depend on these two elements. Reinforcement gives CFRPs their strength and rigidity, measured by stress and elastic modulus respectively. Unlike isotropic materials like steel and aluminum, CFRPs have directional strength properties. The properties of a CFRP depend on the layouts of the carbon fiber and

11400-649: The resin evenly for a glossy finish with very small pin-holes. A third method of constructing composite materials is known as a dry layup. Here, the carbon fiber material is already impregnated with resin (pre-preg) and is applied to the mold in a similar fashion to adhesive film. The assembly is then placed in a vacuum to cure. The dry layup method has the least amount of resin waste and can achieve lighter constructions than wet layup. Also, because larger amounts of resin are more difficult to bleed out with wet layup methods, pre-preg parts generally have fewer pinholes. Pinhole elimination with minimal resin amounts generally require

11520-671: The same units as pressure: namely, pascals (Pa, that is, newtons per square metre ) in the International System , or pounds per square inch (psi) in the Imperial system . Because mechanical stresses easily exceed a million Pascals, MPa, which stands for megapascal, is a common unit of stress. Stress in a material body may be due to multiple physical causes, including external influences and internal physical processes. Some of these agents (like gravity, changes in temperature and phase , and electromagnetic fields) act on

11640-553: The section, both increasing the stiffness of the section and lowering the neutral axis , thus greatly reducing the maximum tensile stress in the cast iron. In the United States, prestressed concrete cylinder pipes (PCCP) account for a vast majority of water transmission mains. Due to their large diameters, failures of PCCP are usually catastrophic and affect large populations. Approximately 19,000 miles (31,000 km) of PCCP were installed between 1940 and 2006. Corrosion in

11760-424: The single number τ {\displaystyle \tau } , calculated simply with the magnitude of those forces, F and the cross sectional area, A . τ = F A {\displaystyle \tau ={\frac {F}{A}}} Unlike normal stress, this simple shear stress is directed parallel to the cross-section considered, rather than perpendicular to it. For any plane S that

11880-709: The steel cylinder to perform within its elastic range, to ensure the pipeline's long-term performance is maintained. CFRP liner designs are based on strain compatibility between the liner and host pipe. CFRPs are more costly materials than commonly used their counterparts in the construction industry, glass fiber-reinforced polymers (GFRPs) and aramid fiber-reinforced polymers (AFRPs), though CFRPs are, in general, regarded as having superior properties. Much research continues to be done on using CFRPs both for retrofitting and as an alternative to steel as reinforcing or prestressing materials. Cost remains an issue and long-term durability questions still remain. Some are concerned about

12000-429: The strength of full-length carbon fiber reinforcement. For example, chopped reclaimed carbon fiber can be used in consumer electronics, such as laptops. It provides excellent reinforcement of the polymers used even if it lacks the strength-to-weight ratio of an aerospace component. In 2009, Zyvex Technologies introduced carbon nanotube-reinforced epoxy and carbon pre-pregs . Carbon nanotube reinforced polymer (CNRP)

12120-407: The stress T that a particle P applies on another particle Q across a surface S can have any direction relative to S . The vector T may be regarded as the sum of two components: the normal stress ( compression or tension ) perpendicular to the surface, and the shear stress that is parallel to the surface. If the normal unit vector n of the surface (pointing from Q towards P )

12240-507: The stress can be assumed to be uniformly distributed over any cross-section that is more than a few times D from both ends. (This observation is known as the Saint-Venant's principle ). Normal stress occurs in many other situations besides axial tension and compression. If an elastic bar with uniform and symmetric cross-section is bent in one of its planes of symmetry, the resulting bending stress will still be normal (perpendicular to

12360-411: The stress distribution in a body is expressed as a piecewise continuous function of space and time. Conversely, stress is usually correlated with various effects on the material, possibly including changes in physical properties like birefringence , polarization , and permeability . The imposition of stress by an external agent usually creates some strain (deformation) in the material, even if it

12480-475: The stress is evenly distributed over the entire cross-section. In practice, depending on how the bar is attached at the ends and how it was manufactured, this assumption may not be valid. In that case, the value σ {\displaystyle \sigma } = F / A will be only the average stress, called engineering stress or nominal stress . If the bar's length L is many times its diameter D , and it has no gross defects or built-in stress , then

12600-424: The stress is maximum for surfaces that are perpendicular to a certain direction d {\displaystyle d} , and zero across any surfaces that are parallel to d {\displaystyle d} . When the shear stress is zero only across surfaces that are perpendicular to one particular direction, the stress is called biaxial , and can be viewed as the sum of two normal or shear stresses. In

12720-399: The stress patterns that occur in such parts have rotational or even cylindrical symmetry . The analysis of such cylinder stresses can take advantage of the symmetry to reduce the dimension of the domain and/or of the stress tensor. Often, mechanical bodies experience more than one type of stress at the same time; this is called combined stress . In normal and shear stress, the magnitude of

12840-684: The stress state of the medium at any point and instant can be specified by only six independent parameters, rather than nine. These may be written [ σ x τ x y τ x z τ x y σ y τ y z τ x z τ y z σ z ] {\displaystyle {\begin{bmatrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{xy}&\sigma _{y}&\tau _{yz}\\\tau _{xz}&\tau _{yz}&\sigma _{z}\end{bmatrix}}} where

12960-411: The stress tensor can be ignored, but since particles are not infinitesimal in the third dimension one can no longer ignore the torque that a particle applies on its neighbors. That torque is modeled as a bending stress that tends to change the curvature of the plate. These simplifications may not hold at welds, at sharp bends and creases (where the radius of curvature is comparable to the thickness of

13080-1620: The stress tensor can be represented in any chosen Cartesian coordinate system by a 3×3 matrix of real numbers. Depending on whether the coordinates are numbered x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}} or named x , y , z {\displaystyle x,y,z} , the matrix may be written as [ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 ] {\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}} or [ σ x x σ x y σ x z σ y x σ y y σ y z σ z x σ z y σ z z ] {\displaystyle {\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\\\end{bmatrix}}} The stress vector T = σ ( n ) {\displaystyle T={\boldsymbol {\sigma }}(n)} across

13200-431: The stress tensor is symmetric , that is σ 12 = σ 21 {\displaystyle \sigma _{12}=\sigma _{21}} , σ 13 = σ 31 {\displaystyle \sigma _{13}=\sigma _{31}} , and σ 23 = σ 32 {\displaystyle \sigma _{23}=\sigma _{32}} . Therefore,

13320-423: The stress within a body may adequately be described by a single number, or by a single vector (a number and a direction). Three such simple stress situations, that are often encountered in engineering design, are the uniaxial normal stress , the simple shear stress , and the isotropic normal stress . A common situation with a simple stress pattern is when a straight rod, with uniform material and cross section,

13440-440: The stress σ throughout the bar, across any horizontal surface, can be expressed simply by the single number σ, calculated simply with the magnitude of those forces, F , and cross sectional area, A . σ = F A {\displaystyle \sigma ={\frac {F}{A}}} On the other hand, if one imagines the bar being cut along its length, parallel to the axis, there will be no force (hence no stress) between

13560-402: The surrounding particles. The container walls and the pressure -inducing surface (such as a piston) push against them in (Newtonian) reaction . These macroscopic forces are actually the net result of a very large number of intermolecular forces and collisions between the particles in those molecules . Stress is frequently represented by a lowercase Greek letter sigma ( σ ). Strain inside

13680-440: The two halves across the cut. This type of stress may be called (simple) normal stress or uniaxial stress; specifically, (uniaxial, simple, etc.) tensile stress. If the load is compression on the bar, rather than stretching it, the analysis is the same except that the force F and the stress σ {\displaystyle \sigma } change sign, and the stress is called compressive stress. This analysis assumes

13800-718: The unusual multi-material and anisotropic nature of CFRPs. In 1968 a Hyfil carbon-fiber fan assembly was in service on the Rolls-Royce Conways of the Vickers VC10s operated by BOAC . Specialist aircraft designers and manufacturers Scaled Composites have made extensive use of CFRPs throughout their design range, including the first private crewed spacecraft Spaceship One . CFRPs are widely used in micro air vehicles (MAVs) because of their high strength-to-weight ratio. CFRPs are extensively used in high-end automobile racing. The high cost of carbon fiber

13920-407: The use of autoclave pressures to purge the residual gases out. A quicker method uses a compression mold , also commonly known as carbon fiber forging. This is a two (male and female), or multi-piece mold, usually made out of aluminum or steel and more recently 3D printed plastic. The mold components are pressed together with the fabric and resin loaded into the inner cavity that ultimately becomes

14040-422: The use of CFRPs typically has a large impact on strength (doubling or more the strength of the section is not uncommon), but only moderately increases stiffness (as little as 10%). This is because the material used in such applications is typically very strong (e.g., 3 GPa ultimate tensile strength , more than 10 times mild steel) but not particularly stiff (150 to 250 GPa elastic modulus, a little less than steel,

14160-421: The volume generate persistent elastic stress. If the deformation changes gradually with time, even in fluids there will usually be some viscous stress , opposing that change. Elastic and viscous stresses are usually combined under the name mechanical stress . Significant stress may exist even when deformation is negligible or non-existent (a common assumption when modeling the flow of water). Stress may exist in

14280-400: Was able to give the first rigorous and general mathematical model of a deformed elastic body by introducing the notions of stress and strain. Cauchy observed that the force across an imaginary surface was a linear function of its normal vector; and, moreover, that it must be a symmetric function (with zero total momentum). The understanding of stress in liquids started with Newton, who provided

14400-958: Was claimed to merely increase the durability of the bats, but it was banned from all first-class matches by the ICC in 2007. A CFRP bicycle frame weighs less than one of steel, aluminum, or titanium having the same strength. The type and orientation of the carbon-fiber weave can be designed to maximize stiffness in required directions. Frames can be tuned to address different riding styles: sprint events require stiffer frames while endurance events may require more flexible frames for rider comfort over longer periods. The variety of shapes it can be built into has further increased stiffness and also allowed aerodynamic tube sections. CFRP forks including suspension fork crowns and steerers, handlebars , seatposts , and crank arms are becoming more common on medium as well as higher-priced bicycles. CFRP rims remain expensive but their stability compared to aluminium reduces

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