1.
Bourdon tube
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Many techniques have been developed for the measurement of pressure and vacuum. Instruments used to measure and display pressure in a unit are called pressure gauges or vacuum gauges. A manometer is an example as it uses a column of liquid to both measure and indicate pressure. Likewise the widely used Bourdon gauge is a device which both measures and indicates, and is probably the best known type of gauge. A vacuum gauge is a pressure gauge used to measure the pressures lower than the ambient atmospheric pressure. Other methods of pressure measurement involve sensors which can transmit the pressure reading to an indicator or control system. Everyday pressure measurements, such as for vehicle tire pressure, are made relative to ambient air pressure. In other cases measurements are made relative to a vacuum or to other specific reference. Gauge pressure is zero-referenced against ambient air pressure, so it is equal to absolute pressure minus atmospheric pressure, to distinguish a negative pressure, the value may be appended with the word vacuum or the gauge may be labeled a vacuum gauge. These are further divided into two subcategories, high and low vacuum, the applicable pressure ranges of many of the techniques used to measure vacuums have an overlap. Hence, by combining different types of gauge, it is possible to measure system pressure continuously from 10 mbar down to 10−11 mbar. Differential pressure is the difference in pressure between two points, the zero reference in use is usually implied by context, and these words are added only when clarification is needed. Tire pressure and blood pressure are gauge pressures by convention, while atmospheric pressures, deep vacuum pressures, for most working fluids where a fluid exists in a closed system, gauge pressure measurement prevails. Pressure instruments connected to the system will indicate pressures relative to the current atmospheric pressure, the situation changes when extreme vacuum pressures are measured, absolute pressures are typically used instead. Differential pressures are used in industrial process systems. Differential pressure gauges have two ports, each connected to one of the volumes whose pressure is to be monitored. Moderate vacuum pressure readings can be ambiguous without the proper context, thus a vacuum of 26 inHg gauge is equivalent to an absolute pressure of 30 inHg −26 inHg =4 inHg. Atmospheric pressure is typically about 100 kPa at sea level, but is variable with altitude, if the absolute pressure of a fluid stays constant, the gauge pressure of the same fluid will vary as atmospheric pressure changes
2.
Pressure
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Pressure is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure is the relative to the ambient pressure. Various units are used to express pressure, Pressure may also be expressed in terms of standard atmospheric pressure, the atmosphere is equal to this pressure and the torr is defined as 1⁄760 of this. Manometric units such as the centimetre of water, millimetre of mercury, Pressure is the amount of force acting per unit area. The symbol for it is p or P, the IUPAC recommendation for pressure is a lower-case p. However, upper-case P is widely used. The usage of P vs p depends upon the field in one is working, on the nearby presence of other symbols for quantities such as power and momentum. Mathematically, p = F A where, p is the pressure, F is the normal force and it relates the vector surface element with the normal force acting on it. It is incorrect to say the pressure is directed in such or such direction, the pressure, as a scalar, has no direction. The force given by the relationship to the quantity has a direction. If we change the orientation of the element, the direction of the normal force changes accordingly. Pressure is distributed to solid boundaries or across arbitrary sections of normal to these boundaries or sections at every point. It is a parameter in thermodynamics, and it is conjugate to volume. The SI unit for pressure is the pascal, equal to one newton per square metre and this name for the unit was added in 1971, before that, pressure in SI was expressed simply in newtons per square metre. Other units of pressure, such as pounds per square inch, the CGS unit of pressure is the barye, equal to 1 dyn·cm−2 or 0.1 Pa. Pressure is sometimes expressed in grams-force or kilograms-force per square centimetre, but using the names kilogram, gram, kilogram-force, or gram-force as units of force is expressly forbidden in SI. The technical atmosphere is 1 kgf/cm2, since a system under pressure has potential to perform work on its surroundings, pressure is a measure of potential energy stored per unit volume. It is therefore related to density and may be expressed in units such as joules per cubic metre. Similar pressures are given in kilopascals in most other fields, where the prefix is rarely used
3.
Balance wheel
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A balance wheel, or balance, is the timekeeping device used in mechanical watches and some clocks, analogous to the pendulum in a pendulum clock. It is a wheel that rotates back and forth, being returned toward its center position by a spiral torsion spring. It is driven by the escapement, which transforms the motion of the watch gear train into impulses delivered to the balance wheel. Each swing of the wheel allows the train to advance a set amount. From its invention in the 14th century until tuning fork and quartz movements became available in the 1960s, virtually every portable timekeeping device used some form of balance wheel. Modern watch balance wheels are made of Glucydur, a low thermal expansion alloy of beryllium, copper and iron. The two alloys are matched so their residual temperature responses cancel out, resulting in lower temperature error. The wheels are smooth, to air friction, and the pivots are supported on precision jewel bearings. Older balance wheels used weight screws around the rim to adjust the poise, Balance wheels rotate about 1½ turns with each swing, that is, about 270° to each side of their center equilibrium position. The rate of the wheel is adjusted with the regulator. This holds the part of the spring behind the slit stationary, moving the lever slides the slit up and down the balance spring, changing its effective length, and thus the resonant vibration rate of the balance. Since the regulator interferes with the action, chronometers and some precision watches have ‘free sprung’ balances with no regulator. Their rate is adjusted by weight screws on the balance rim, a balances vibration rate is traditionally measured in beats per hour, or BPH, although beats per second and Hz are also used. The length of a beat is one swing of the balance wheel, balances in precision watches are designed with faster beats, because they are less affected by motions of the wrist. Alarm clocks and kitchen timers often have a rate of 4 beats per second, Watches made prior to the 1970s usually had a rate of 5 beats per second. Current watches have rates of 6,8 and a few have 10 beats per second, during WWII, Elgin produced a very precise stopwatch that ran at 40 beats per second, earning it the nickname Jitterbug. Audemars Piguet currently produces a movement that allows for a high balance vibration of 12 beats/s. The precision of the best balance wheel watches on the wrist is around a few seconds per day, the most accurate balance wheel timepieces made were marine chronometers, which by WWII had achieved accuracies of 0.1 second per day
4.
Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
5.
Force
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In physics, a force is any interaction that, when unopposed, will change the motion of an object. In other words, a force can cause an object with mass to change its velocity, force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity and it is measured in the SI unit of newtons and represented by the symbol F. The original form of Newtons second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. In an extended body, each part usually applies forces on the adjacent parts, such internal mechanical stresses cause no accelation of that body as the forces balance one another. Pressure, the distribution of small forces applied over an area of a body, is a simple type of stress that if unbalanced can cause the body to accelerate. Stress usually causes deformation of materials, or flow in fluids. In part this was due to an understanding of the sometimes non-obvious force of friction. A fundamental error was the belief that a force is required to maintain motion, most of the previous misunderstandings about motion and force were eventually corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Sir Isaac Newton formulated laws of motion that were not improved-on for nearly three hundred years, the Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known, in order of decreasing strength, they are, strong, electromagnetic, weak, high-energy particle physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction. Since antiquity the concept of force has been recognized as integral to the functioning of each of the simple machines. The mechanical advantage given by a machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount of work. Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was famous for formulating a treatment of buoyant forces inherent in fluids. Aristotle provided a discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotles view, the sphere contained four elements that come to rest at different natural places therein. Aristotle believed that objects on Earth, those composed mostly of the elements earth and water, to be in their natural place on the ground. He distinguished between the tendency of objects to find their natural place, which led to natural motion, and unnatural or forced motion
6.
Spring (device)
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A spring is an elastic object used to store mechanical energy. Springs are usually out of spring steel. There are a number of spring designs, in everyday usage the term often refers to coil springs. When a spring is compressed or stretched from its resting position, the rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. That is, it is the gradient of the force versus deflection curve, an extension or compression springs rate is expressed in units of force divided by distance, for example lbf/in or N/m. A torsion spring is a spring that works by twisting, when it is twisted about its axis by an angle, a torsion springs rate is in units of torque divided by angle, such as N·m/rad or ft·lbf/degree. The inverse of spring rate is compliance, that is, if a spring has a rate of 10 N/mm, the stiffness of springs in parallel is additive, as is the compliance of springs in series. Springs are made from a variety of materials, the most common being spring steel. Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel, some non-ferrous metals are also used including phosphor bronze and titanium for parts requiring corrosion resistance and beryllium copper for springs carrying electrical current. Simple non-coiled springs were used throughout history, e. g. the bow. In the Bronze Age more sophisticated spring devices were used, as shown by the spread of tweezers in many cultures, coiled springs appeared early in the 15th century, in door locks. The first spring powered-clocks appeared in that century and evolved into the first large watches by the 16th century, in 1676 British physicist Robert Hooke discovered Hookes law which states that the force a spring exerts is proportional to its extension. Compression spring – is designed to operate with a compression load, flat spring – this type is made of a flat spring steel. Machined spring – this type of spring is manufactured by machining bar stock with a lathe and/or milling operation rather than a coiling operation, since it is machined, the spring may incorporate features in addition to the elastic element. Machined springs can be made in the load cases of compression/extension, torsion. Serpentine spring - a zig-zag of thick wire - often used in modern upholstery/furniture, the most common types of spring are, Cantilever spring – a spring which is fixed only at one end. Coil spring or helical spring – a spring is of two types, Tension or extension springs are designed to become longer under load and their turns are normally touching in the unloaded position, and they have a hook, eye or some other means of attachment at each end. Compression springs are designed to become shorter when loaded and their turns are not touching in the unloaded position, and they need no attachment points
7.
Stiffness
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Stiffness is the rigidity of an object — the extent to which it resists deformation in response to an applied force. The complementary concept is flexibility or pliability, the more flexible an object is, the stiffness, k, of a body is a measure of the resistance offered by an elastic body to deformation. In Imperial units, stiffness is typically measured in pounds per inch, generally speaking, deflections of an infinitesimal element in an elastic body can occur along multiple DOF. For example, a point on a beam can undergo both a vertical displacement and a rotation relative to its undeformed axis. When there are M degrees of freedom a M x M matrix must be used to describe the stiffness at the point, in industry, the term influence coefficient is sometimes used to refer to the coupling stiffness. For a body with multiple DOF, in order to calculate a particular direct-related stiffness, under such a condition, the above equation can be used to obtain the direct-related stiffness for the degree of freedom which is unconstrained. The ratios between the forces and the produced deflection are the coupling stiffnesses. A description including all possible stretch and shear parameters is given by the elasticity tensor, the inverse of stiffness is flexibility or compliance, typically measured in units of metres per newton. In rheology it may be defined as the ratio of strain to stress, in the SAE system, rotational stiffness is typically measured in inch-pounds per degree. For example, for an element in tension or compression, the stiffness is k = A E L where A is the cross-sectional area, E is the elastic modulus. For the special case of unconstrained uniaxial tension or compression, Youngs modulus can be thought of as a measure of the stiffness of a structure. The stiffness of a structure is of importance in many engineering applications. A high modulus of elasticity is sought when deflection is undesirable, while a low modulus of elasticity is required when flexibility is needed, in biology, the stiffness of the extracellular matrix is important for guiding the migration of cells in a phenomenon called durotaxis. Another application of stiffness finds itself in skin biology, the skin maintains its structure due to its intrinsic tension, contributed to by collagen, an extracellular protein which accounts for approximately 75% of its dry weight. The pliability of skin is a parameter of interest that represents its firmness and extensibility, encompassing characteristics such as elasticity, stiffness and these factors are of functional significance to patients. This is of significance to patients with injuries to the skin, whereby the pliability can be reduced due to the formation. This can be evaluated both subjectively, or objectively using a device such as the Cutometer, the Cutometer applies a vacuum to the skin and measures the extent to which it can be vertically distended
8.
Robert Hooke
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Robert Hooke FRS was an English natural philosopher, architect and polymath. Allan Chapman has characterised him as Englands Leonardo, Robert Gunthers Early Science in Oxford, a history of science in Oxford during the Protectorate, Restoration and Age of Enlightenment, devotes five of its fourteen volumes to Hooke. Hooke studied at Wadham College, Oxford during the Protectorate where he became one of a tightly knit group of ardent Royalists led by John Wilkins. Here he was employed as an assistant to Thomas Willis and to Robert Boyle and he built some of the earliest Gregorian telescopes and observed the rotations of Mars and Jupiter. In 1665 he inspired the use of microscopes for scientific exploration with his book, based on his microscopic observations of fossils, Hooke was an early proponent of biological evolution. Much of Hookes scientific work was conducted in his capacity as curator of experiments of the Royal Society, much of what is known of Hookes early life comes from an autobiography that he commenced in 1696 but never completed. Richard Waller mentions it in his introduction to The Posthumous Works of Robert Hooke, the work of Waller, along with John Wards Lives of the Gresham Professors and John Aubreys Brief Lives, form the major near-contemporaneous biographical accounts of Hooke. Robert Hooke was born in 1635 in Freshwater on the Isle of Wight to John Hooke, Robert was the last of four children, two boys and two girls, and there was an age difference of seven years between him and the next youngest. Their father John was a Church of England priest, the curate of Freshwaters Church of All Saints, Robert Hooke was expected to succeed in his education and join the Church. John Hooke also was in charge of a school, and so was able to teach Robert. He was a Royalist and almost certainly a member of a group who went to pay their respects to Charles I when he escaped to the Isle of Wight, Robert, too, grew up to be a staunch monarchist. As a youth, Robert Hooke was fascinated by observation, mechanical works and he dismantled a brass clock and built a wooden replica that, by all accounts, worked well enough, and he learned to draw, making his own materials from coal, chalk and ruddle. Hooke quickly mastered Latin and Greek, made study of Hebrew. Here, too, he embarked on his study of mechanics. It appears that Hooke was one of a group of students whom Busby educated in parallel to the work of the school. Contemporary accounts say he was not much seen in the school, in 1653, Hooke secured a choristers place at Christ Church, Oxford. He was employed as an assistant to Dr Thomas Willis. There he met the natural philosopher Robert Boyle, and gained employment as his assistant from about 1655 to 1662, constructing, operating and he did not take his Master of Arts until 1662 or 1663
9.
Latin
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Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages, such as Italian, Portuguese, Spanish, French, and Romanian. Latin, Italian and French have contributed many words to the English language, Latin and Ancient Greek roots are used in theology, biology, and medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin, Vulgar Latin was the colloquial form spoken during the same time and attested in inscriptions and the works of comic playwrights like Plautus and Terence. Late Latin is the language from the 3rd century. Later, Early Modern Latin and Modern Latin evolved, Latin was used as the language of international communication, scholarship, and science until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the language of the Holy See and the Roman Rite of the Catholic Church. Today, many students, scholars and members of the Catholic clergy speak Latin fluently and it is taught in primary, secondary and postsecondary educational institutions around the world. The language has been passed down through various forms, some inscriptions have been published in an internationally agreed, monumental, multivolume series, the Corpus Inscriptionum Latinarum. Authors and publishers vary, but the format is about the same, volumes detailing inscriptions with a critical apparatus stating the provenance, the reading and interpretation of these inscriptions is the subject matter of the field of epigraphy. The works of several hundred ancient authors who wrote in Latin have survived in whole or in part and they are in part the subject matter of the field of classics. The Cat in the Hat, and a book of fairy tales, additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissners Latin Phrasebook. The Latin influence in English has been significant at all stages of its insular development. From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed inkhorn terms, as if they had spilled from a pot of ink. Many of these words were used once by the author and then forgotten, many of the most common polysyllabic English words are of Latin origin through the medium of Old French. Romance words make respectively 59%, 20% and 14% of English, German and those figures can rise dramatically when only non-compound and non-derived words are included. Accordingly, Romance words make roughly 35% of the vocabulary of Dutch, Roman engineering had the same effect on scientific terminology as a whole
10.
Anagram
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The original word or phrase is known as the subject of the anagram. Any word or phrase that exactly reproduces the letters in order is an anagram. Someone who creates anagrams may be called an anagrammatist, and the goal of a serious or skilled anagrammatist is to produce anagrams that in some way reflect or comment on their subject, Anagrams are often used as a form of mnemonic device. Such an anagram may be a synonym or antonym of its subject, a parody and it sometimes changes a proper noun or personal name into a sentence, such as with William Shakespeare = I am a weakish speller or Madam Curie = Radium came. It can change parts of speech, such as the adjective silent to the verb listen, Anagrams itself can be anagrammatized as Ars magna. Anagrams can be traced back to the time of Moses, as Themuru or changing and they were popular throughout Europe during the Middle Ages, for example with the poet and composer Guillaume de Machaut. They are said to go back at least to the Greek poet Lycophron, in the third century BCE, Anagrams in Latin were considered witty over many centuries. Est vir qui adest, explained below, was cited as the example in Samuel Johnsons A Dictionary of the English Language, any historical material on anagrams must always be interpreted in terms of the assumptions and spellings that were current for the language in question. In particular, spelling in English only slowly became fixed, there were attempts to regulate anagram formation, an important one in English being that of George Puttenhams Of the Anagram or Posy Transposed in The Art of English Poesie. The origins of these are not documented, Latin continued to influence letter values. There was a tradition of allowing anagrams to be perfect if the letters were all used once. This can be seen in a popular Latin anagram against the Jesuits, Societas Jesu turned into Vitiosa seces, the rules were not completely fixed in the 17th century. When it comes to the 17th century and anagrams in English or other languages, the lawyer Thomas Egerton was praised through the anagram gestat honorem, the physician George Ent took the anagrammatic motto genio surget, which requires his first name as Georgius. James Is courtiers discovered in James Stuart a just master, walter Quin, tutor to the future Charles I, worked hard on multilingual anagrams on the name of father James. Dryden in MacFlecknoe disdainfully called the pastime the torturing of one poor word ten thousand ways, an example from France was a flattering anagram for Cardinal Richelieu, comparing him to Hercules or at least one of his hands, where Armand de Richelieu became Ardue main dHercule. Examples from the century are the transposition of Horatio Nelson into Honor est a Nilo. With the advent of surrealism as a movement, anagrams regained the artistic respect they had had in the Baroque period. The German poet Unica Zürn, who made use of anagram techniques, came to regard obsession with anagrams as a dangerous fever
11.
Elasticity (physics)
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In physics, elasticity is the ability of a body to resist a distorting influence or deforming force and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate forces are applied on them, if the material is elastic, the object will return to its initial shape and size when these forces are removed. The physical reasons for elastic behavior can be different for different materials. In metals, the atomic lattice changes size and shape when forces are applied, when forces are removed, the lattice goes back to the original lower energy state. For rubbers and other polymers, elasticity is caused by the stretching of polymer chains when forces are applied, perfect elasticity is an approximation of the real world. The most elastic body in modern science found is Quartz fibre which is not even a perfect elastic body, so perfect elastic body is an ideal concept only. Most materials which possess elasticity in practice remain purely elastic only up to very small deformations. In engineering, the amount of elasticity of a material is determined by two types of material parameter, the first type of material parameter is called a modulus, which measures the amount of force per unit area needed to achieve a given amount of deformation. The SI unit of modulus is the pascal, a higher modulus typically indicates that the material is harder to deform. The second type of measures the elastic limit, the maximum stress that can arise in a material before the onset of permanent deformation. Its SI unit is also pascal, when describing the relative elasticities of two materials, both the modulus and the elastic limit have to be considered. Rubbers typically have a low modulus and tend to stretch a lot, of two rubber materials with the same elastic limit, the one with a lower modulus will appear to be more elastic, which is however not correct. When an elastic material is deformed due to a force, it experiences internal resistance to the deformation. The various moduli apply to different kinds of deformation, for instance, Youngs modulus applies to extension/compression of a body, whereas the shear modulus applies to its shear. The elasticity of materials is described by a curve, which shows the relation between stress and strain. The curve is nonlinear, but it can be approximated as linear for sufficiently small deformations. For even higher stresses, materials exhibit behavior, that is, they deform irreversibly. Elasticity is not exhibited only by solids, non-Newtonian fluids, such as viscoelastic fluids, in response to a small, rapidly applied and removed strain, these fluids may deform and then return to their original shape. Under larger strains, or strains applied for longer periods of time, because the elasticity of a material is described in terms of a stress-strain relation, it is essential that the terms stress and strain be defined without ambiguity
12.
String (music)
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A string is the vibrating element that produces sound in string instruments such as the guitar, harp, piano, and members of the violin family. Strings are lengths of a material that a musical instrument holds under tension so that they can vibrate freely. Wound strings have a core of one material, with an overwinding of other materials and this is to make the string vibrate at the desired pitch, while maintaining a low profile and sufficient flexibility for playability. This enabled stringed instruments to be made with less thick bass strings, on string instruments that the player plucks or bows directly, this enabled instrument makers to use thinner strings for the lowest-pitched strings, which made the lower-pitch strings easier to play. The end of the string that mounts to the tuning mechanism is usually plain. Depending on the instrument, the other, fixed end may have either a plain, loop. When a ball or loop is used with a guitar, this ensures that the string stays fixed in the bridge of the guitar, when a ball or loop is used with a violin-family instrument, this keeps the string end fixed in the tailpiece. Fender Bullet strings have a cylinder for more stable tuning on guitars equipped with synchronized tremolo systems. Strings for some instruments may be wrapped with silk at the ends to protect the string, the color and pattern of the silk often identifies attributes of the string, such as manufacturer, size, intended pitch, etc. There are several varieties of wound strings available, the simplest wound strings are roundwound—with round wire wrapped in a tight spiral around either a round or hexagonal core. Such strings are usually simple to manufacture and the least expensive and they have several drawbacks, however, Roundwound strings have a bumpy surface profile that produce friction on the players fingertips. This causes squeaking sounds when the fingers slide over the strings. Roundwound strings higher friction surface profile may hasten fingerboard and fret wear, when the core is round, the winding is less secure and may rotate freely around the core, especially if the winding is damaged after use. Flatwound strings also have either a round or hex core, however, the winding wire has a rounded square cross-section that has a shallower profile when tightly wound. This makes for more playing, and decreased wear for frets. Squeaking sounds due to fingers sliding along the strings are also decreased significantly, flatwound strings also have a longer playable life because of smaller grooves for dirt and oil to build up in. On the other hand, flatwound strings sound less bright than roundwounds, flatwounds also usually cost more than roundwounds because of less demand, less production, and higher overhead costs. Manufacturing is also difficult, as precise alignment of the flat sides of the winding must be maintained
13.
Toy balloon
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A toy balloon or party balloon, is a small balloon mostly used for decoration, advertising and child toy. Toy balloons are made of rubber or aluminized plastic. They come in a variety of sizes and shapes, but are most commonly 10 to 30 centimetres in diameter. Toy balloons are not considered to include sky lanterns, although these too are or were used as toys in some parts of the world. The Consumer Products Safety Commission found that children had inhaled latex balloons whole or choked on fragments of broken balloons, parents a monthly magazine about raising children advised parents to buy Mylar balloons instead of latex balloons. Early balloons were made from pig bladders and animal intestines, the Aztecs created the first balloon sculptures using cat intestines, which were then presented to the gods as a sacrifice. There are references to balloons made of whale intestine in Swiss Family Robinson, the first rubber balloons were made by Professor Michael Faraday in 1824 for use in his experiments with hydrogen at the Royal Institution in London. The caoutchouc is exceedingly elastic, he wrote in the Quarterly Journal of Science the same year, Faraday made his balloons by cutting round two sheets of rubber laid together and pressing the edges together. The tacky rubber welded automatically, and the inside of the balloon was rubbed with flour to prevent the opposing surfaces joining together. Toy balloons were introduced by pioneer rubber manufacturer Thomas Hancock the following year in the form of a kit consisting of a bottle of rubber solution. Vulcanized toy balloons, which unlike the kind were unaffected by changes in temperature, were first manufactured by J. G. Ingram of London in 1847 and can be regarded as the prototype of modern toy balloons, in the 1920s Neil Tillotson designed and produced a latex balloon with a cats face and ears from a cardboard form which he cut by hand with a pair of scissors. He managed to make his first sale of these balloons with an order of 15 gross to be delivered for the annual Patriots Day Parade on April 19,1931, the first colored balloons were sold at the 1933-34 Chicago Worlds Fair. Inflatable foil balloons are made from plastics, such as aluminized PET film, foil balloons are not elastic like rubber balloons, so that detailed and colorful pictures printed on their surfaces are not distorted when the balloon is inflated. When no longer required, it is recommended to cut the balloon to release the helium, every toy balloon has an opening through which gases are blown into it, followed by a connecting tube known as the neck. Balloons are usually filled by using ones breath, a pump, the opening can then be permanently tied off or clamped temporarily. By filling a balloon with a gas lighter than air, such as helium, helium is the preferred gas for floating balloons, because it is inert and will not catch fire or cause toxic effects when inhaled. Small, light objects are placed in balloons along with helium and released into the air and, when the balloon eventually falls
14.
Linear elasticity
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Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua, linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of continuum mechanics. The fundamental linearizing assumptions of linear elasticity are, infinitesimal strains or small deformations, in addition linear elasticity is valid only for stress states that do not produce yielding. These assumptions are reasonable for many engineering materials and engineering design scenarios, linear elasticity is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis. The system of equations is completed by a set of linear algebraic constitutive relations. For elastic materials, Hookes law represents the behavior and relates the unknown stresses. Note, the Einstein summation convention of summing on repeated indices is used below and these are 3 independent equations with 6 independent unknowns. Strain-displacement equations, ε i j =12 where ε i j = ε j i is the strain and these are 6 independent equations relating strains and displacements with 9 independent unknowns. The equation for Hookes law is, σ i j = C i j k l ε k l where C i j k l is the stiffness tensor and these are 6 independent equations relating stresses and strains. An elastostatic boundary value problem for a media is a system of 15 independent equations. Specifying the boundary conditions, the value problem is completely defined. To solve the two approaches can be taken according to boundary conditions of the boundary value problem, a displacement formulation. In isotropic media, the stiffness tensor gives the relationship between the stresses and the strains, for an isotropic medium, the stiffness tensor has no preferred direction, an applied force will give the same displacements no matter the direction in which the force is applied. If the medium is homogeneous, then the elastic moduli will be independent of the position in the medium, the constitutive equation may now be written as, σ i j = K δ i j ε k k +2 μ. This expression separates the stress into a part on the left which may be associated with a scalar pressure. A simpler expression is, σ i j = λ δ i j ε k k +2 μ ε i j where λ is Lamés first parameter. More simply, ε i j =12 μ σ i j − ν E δ i j σ k k =1 E where ν is Poissons ratio and E is Youngs modulus. Elastostatics is the study of linear elasticity under the conditions of equilibrium, in all forces on the elastic body sum to zero
15.
Taylor series
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In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the functions derivatives at a single point. The concept of a Taylor series was formulated by the Scottish mathematician James Gregory, a function can be approximated by using a finite number of terms of its Taylor series. Taylors theorem gives quantitative estimates on the error introduced by the use of such an approximation, the polynomial formed by taking some initial terms of the Taylor series is called a Taylor polynomial. The Taylor series of a function is the limit of that functions Taylor polynomials as the degree increases, a function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an interval is known as an analytic function in that interval. The Taylor series of a real or complex-valued function f that is differentiable at a real or complex number a is the power series f + f ′1. Which can be written in the more compact sigma notation as ∑ n =0 ∞ f n, N where n. denotes the factorial of n and f denotes the nth derivative of f evaluated at the point a. The derivative of order zero of f is defined to be f itself and 0 and 0. are both defined to be 1, when a =0, the series is also called a Maclaurin series. The Maclaurin series for any polynomial is the polynomial itself. The Maclaurin series for 1/1 − x is the geometric series 1 + x + x 2 + x 3 + ⋯ so the Taylor series for 1/x at a =1 is 1 − +2 −3 + ⋯. The Taylor series for the exponential function ex at a =0 is x 00, + ⋯ =1 + x + x 22 + x 36 + x 424 + x 5120 + ⋯ = ∑ n =0 ∞ x n n. The above expansion holds because the derivative of ex with respect to x is also ex and this leaves the terms n in the numerator and n. in the denominator for each term in the infinite sum. The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a result, but rejected it as an impossibility. It was through Archimedess method of exhaustion that a number of progressive subdivisions could be performed to achieve a finite result. Liu Hui independently employed a similar method a few centuries later, in the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama. The Kerala school of astronomy and mathematics further expanded his works with various series expansions, in the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a method for constructing these series for all functions for which they exist was finally provided by Brook Taylor. The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, if f is given by a convergent power series in an open disc centered at b in the complex plane, it is said to be analytic in this disc
16.
Elastic limit
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A yield strength or yield point is the material property defined as the stress at which a material begins to deform plastically. Prior to the point the material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed, some fraction of the deformation will be permanent, in the three-dimensional principal stresses, an infinite number of yield points form together a yield surface. The yield point determines the limits of performance for mechanical components, in structural engineering, this is a soft failure mode which does not normally cause catastrophic failure or ultimate failure unless it accelerates buckling. It is often difficult to define yielding due to the wide variety of stress–strain curves exhibited by real materials. In addition, there are possible ways to define yielding. This definition is used, since dislocations move at very low stresses. Elastic limit Beyond the elastic limit, permanent deformation will occur, the elastic limit is therefore the lowest stress at which permanent deformation can be measured. This requires a manual procedure, and the accuracy is critically dependent on the equipment used. For elastomers, such as rubber, the limit is much larger than the proportionality limit. Also, precise measurements have shown that plastic strain begins at low stresses. Yield point The point in the curve at which the curve levels off. Offset yield point When a yield point is not easily defined based on the shape of the curve an offset yield point is arbitrarily defined. The value for this is set at 0.1 or 0. 2% plastic strain. The offset value is given as a subscript, e. g. Rp0. 2=310 MPa, high strength steel and aluminum alloys do not exhibit a yield point, so this offset yield point is used on these materials. Upper and lower yield points Some metals, such as mild steel, the material response is linear up until the upper yield point, but the lower yield point is used in structural engineering as a conservative value. If a metal is only stressed to the yield point. A yield criterion, often expressed as yield surface, or yield locus, is a hypothesis concerning the limit of elasticity under any combination of stresses
17.
Seismology
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Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. A related field that uses geology to infer information regarding past earthquakes is paleoseismology, a recording of earth motion as a function of time is called a seismogram. A seismologist is a scientist who does research in seismology, scholarly interest in earthquakes can be traced back to antiquity. Early speculations on the causes of earthquakes were included in the writings of Thales of Miletus, Anaximenes of Miletus, Aristotle. In 132 CE, Zhang Heng of Chinas Han dynasty designed the first known seismoscope, in 1664, Athanasius Kircher argued that earthquakes were caused by the movement of fire within a system of channels inside the Earth. In 1703, Martin Lister and Nicolas Lemery proposed that earthquakes were caused by chemical explosions within the earth, the Lisbon earthquake of 1755, coinciding with the general flowering of science in Europe, set in motion intensified scientific attempts to understand the behaviour and causation of earthquakes. The earliest responses include work by John Bevis and John Michell, Michell determined that earthquakes originate within the Earth and were waves of movement caused by shifting masses of rock miles below the surface. From 1857, Robert Mallet laid the foundation of instrumental seismology and he is also responsible for coining the word seismology. In 1897, Emil Wiecherts theoretical calculations led him to conclude that the Earths interior consists of a mantle of silicates, surrounding a core of iron. In 1906 Richard Dixon Oldham identified the separate arrival of P-waves, S-waves and surface waves on seismograms, in 1910, after studying the 1906 San Francisco earthquake, Harry Fielding Reid put forward the elastic rebound theory which remains the foundation for modern tectonic studies. The development of this depended on the considerable progress of earlier independent streams of work on the behaviour of elastic materials. In 1926, Harold Jeffreys was the first to claim, based on his study of waves, that below the mantle. In 1937, Inge Lehmann determined that within the liquid outer core there is a solid inner core. By the 1960s, earth science had developed to the point where a comprehensive theory of the causation of seismic events had come together in the now well-established theory of plate tectonics, seismic waves are elastic waves that propagate in solid or fluid materials. There are two types of waves, Pressure waves or Primary waves and Shear or Secondary waves. S-waves are transverse waves that move perpendicular to the direction of propagation, therefore, they appear later than P-waves on a seismogram. Fluids cannot support perpendicular motion, so S-waves only travel in solids, the two main surface wave types are Rayleigh waves, which have some compressional motion, and Love waves, which do not. Rayleigh waves result from the interaction of vertically polarized P- and S-waves that satisfy the conditions on the surface
18.
Molecular mechanics
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Molecular mechanics uses classical mechanics to model molecular systems. The potential energy of all systems in mechanics is calculated using force fields. Molecular mechanics can be used to study molecule systems ranging in size, for accurate reproduction of vibrational spectra, the Morse potential can be used instead, at computational cost. The dihedral or torsional terms typically have multiple minima and thus cannot be modeled as harmonic oscillators and this class of terms may include improper dihedral terms, which function as correction factors for out-of-plane deviations. The non-bonded terms are more computationally costly to calculate in full, since a typical atom is bonded to only a few of its neighbors. Fortunately the van der Waals term falls off rapidly, the repulsive part r−12 is however unphysical, because repulsion increases exponentially. Description of van der Waals forces by the Lennard-Jones 6–12 potential introduces inaccuracies, generally a cutoff radius is used to speed up the calculation so that atom pairs which distances are greater than the cutoff have a van der Waals interaction energy of zero. The basic functional form is the Coulomb potential, which falls off as r−1. A variety of methods are used to address this problem, the simplest being a cutoff similar to that used for the van der Waals terms. However, this introduces a discontinuity between atoms inside and atoms outside the radius. Other more sophisticated but computationally intensive methods are particle mesh Ewald, in addition to the functional form of each energy term, a useful energy function must be assigned parameters for force constants, van der Waals multipliers, and other constant terms. These terms, together with the bond, angle, and dihedral values, partial charge values, atomic masses and radii. Parameterization is typically done through agreement with experimental values and theoretical calculations results, each force field is parameterized to be internally consistent, but the parameters are generally not transferable from one force field to another. The main use of mechanics is in the field of molecular dynamics. This uses the field to calculate the forces acting on each particle. Another application of mechanics is energy minimization, whereby the force field is used as an optimization criterion. This method uses an algorithm to find the molecular structure of a local energy minimum. These minima correspond to stable conformers of the molecule and molecular motion can be modelled as vibrations around and it is thus common to find local energy minimization methods combined with global energy optimization, to find the global energy minimum
19.
Acoustics
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Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of technology may be called an acoustical engineer. The application of acoustics is present in almost all aspects of society with the most obvious being the audio. Hearing is one of the most crucial means of survival in the animal world, accordingly, the science of acoustics spreads across many facets of human society—music, medicine, architecture, industrial production, warfare and more. Likewise, animal species such as songbirds and frogs use sound, art, craft, science and technology have provoked one another to advance the whole, as in many other fields of knowledge. Robert Bruce Lindsays Wheel of Acoustics is a well accepted overview of the fields in acoustics. The word acoustic is derived from the Greek word ἀκουστικός, meaning of or for hearing, ready to hear and that from ἀκουστός, heard, audible, which in turn derives from the verb ἀκούω, I hear. The Latin synonym is sonic, after which the term used to be a synonym for acoustics. Frequencies above and below the range are called ultrasonic and infrasonic. If, for example, a string of a length would sound particularly harmonious with a string of twice the length. In modern parlance, if a string sounds the note C when plucked, a string twice as long will sound a C an octave lower. In one system of tuning, the tones in between are then given by 16,9 for D,8,5 for E,3,2 for F,4,3 for G,6,5 for A. Aristotle understood that sound consisted of compressions and rarefactions of air which falls upon, a very good expression of the nature of wave motion. The physical understanding of acoustical processes advanced rapidly during and after the Scientific Revolution, mainly Galileo Galilei but also Marin Mersenne, independently, discovered the complete laws of vibrating strings. Experimental measurements of the speed of sound in air were carried out successfully between 1630 and 1680 by a number of investigators, prominently Mersenne, meanwhile, Newton derived the relationship for wave velocity in solids, a cornerstone of physical acoustics. The eighteenth century saw advances in acoustics as mathematicians applied the new techniques of calculus to elaborate theories of sound wave propagation. Also in the 19th century, Wheatstone, Ohm, and Henry developed the analogy between electricity and acoustics, the twentieth century saw a burgeoning of technological applications of the large body of scientific knowledge that was by then in place. The first such application was Sabine’s groundbreaking work in architectural acoustics, Underwater acoustics was used for detecting submarines in the first World War
20.
Spring scale
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A spring scale or spring balance or newton meter is a type of weighing scale. It consists of spring fixed at one end with a hook to attach an object at the other and it works by Hookes Law, which states that the force needed to extend a spring is proportional to the distance that spring is extended from its rest position. Therefore, the markings on the spring balance are equally spaced. A spring scale can not measure mass, only weight, also, the spring in the scale can permanently stretch with repeated use. A spring scale will only read correctly in a frame of reference where the acceleration in the axis is constant. This can be shown by taking a spring scale into an elevator, if two or more spring balances are hung one below the other in series, each of the scales will read approximately the same, the full weight of the body hung on the lower scale. The scale on top would read slightly heavier due to supporting the weight of the lower scale itself. Spring balances come in different sizes, generally, small scales that measure newtons will have a less firm spring than larger ones that measure tens, hundreds or thousands of newtons or even more depending on the scale of newtons used. The largest spring scale ranged in measurement from 5000-8000 newtons, a spring balance may be labeled in both units of force and mass. Strictly speaking, only the values are correctly labeled. Main uses of spring balances are industrial, especially related to weighing heavy loads such as trucks, storage silos and they are also common in science education as basic accelerators. They are used when the accuracy afforded by other types of scales can be sacrificed for simplicity, cheapness, a spring balance measures the weight of an object by opposing the force of gravity acting with the force of an extended spring. The first spring balance in Britain was made around 1770 by Richard Salter of Bilston and he and his nephews John & George founded the firm of George Salter & Co. still notable makers of scales and balances, who in 1838 patented the spring balance. They also applied the same spring balance principle to steam locomotive safety valves, weighing scale Media related to spring balance at Wikimedia Commons Media related to spring scales at Wikimedia Commons
21.
Pressure measurement
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Many techniques have been developed for the measurement of pressure and vacuum. Instruments used to measure and display pressure in a unit are called pressure gauges or vacuum gauges. A manometer is an example as it uses a column of liquid to both measure and indicate pressure. Likewise the widely used Bourdon gauge is a device which both measures and indicates, and is probably the best known type of gauge. A vacuum gauge is a pressure gauge used to measure the pressures lower than the ambient atmospheric pressure. Other methods of pressure measurement involve sensors which can transmit the pressure reading to an indicator or control system. Everyday pressure measurements, such as for vehicle tire pressure, are made relative to ambient air pressure. In other cases measurements are made relative to a vacuum or to other specific reference. Gauge pressure is zero-referenced against ambient air pressure, so it is equal to absolute pressure minus atmospheric pressure, to distinguish a negative pressure, the value may be appended with the word vacuum or the gauge may be labeled a vacuum gauge. These are further divided into two subcategories, high and low vacuum, the applicable pressure ranges of many of the techniques used to measure vacuums have an overlap. Hence, by combining different types of gauge, it is possible to measure system pressure continuously from 10 mbar down to 10−11 mbar. Differential pressure is the difference in pressure between two points, the zero reference in use is usually implied by context, and these words are added only when clarification is needed. Tire pressure and blood pressure are gauge pressures by convention, while atmospheric pressures, deep vacuum pressures, for most working fluids where a fluid exists in a closed system, gauge pressure measurement prevails. Pressure instruments connected to the system will indicate pressures relative to the current atmospheric pressure, the situation changes when extreme vacuum pressures are measured, absolute pressures are typically used instead. Differential pressures are used in industrial process systems. Differential pressure gauges have two ports, each connected to one of the volumes whose pressure is to be monitored. Moderate vacuum pressure readings can be ambiguous without the proper context, thus a vacuum of 26 inHg gauge is equivalent to an absolute pressure of 30 inHg −26 inHg =4 inHg. Atmospheric pressure is typically about 100 kPa at sea level, but is variable with altitude, if the absolute pressure of a fluid stays constant, the gauge pressure of the same fluid will vary as atmospheric pressure changes
22.
Clock
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A clock is an instrument to measure, keep, and indicate time. The word clock is derived from the Celtic words clagan and clocca meaning bell, a silent instrument missing such a striking mechanism has traditionally been known as a timepiece. In general usage today a clock refers to any device for measuring and displaying the time, Watches and other timepieces that can be carried on ones person are often distinguished from clocks. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units, the day, the lunar month. Devices operating on several physical processes have been used over the millennia, a sundial shows the time by displaying the position of a shadow on a flat surface. There is a range of duration timers, an example being the hourglass. Water clocks, along with the sundials, are possibly the oldest time-measuring instruments, spring-driven clocks appeared during the 15th century. During the 15th and 16th centuries, clockmaking flourished, the next development in accuracy occurred after 1656 with the invention of the pendulum clock. A major stimulus to improving the accuracy and reliability of clocks was the importance of precise time-keeping for navigation, the electric clock was patented in 1840. The development of electronics in the 20th century led to clocks with no clockwork parts at all, the timekeeping element in every modern clock is a harmonic oscillator, a physical object that vibrates or oscillates at a particular frequency. This object can be a pendulum, a fork, a quartz crystal. Analog clocks usually indicate time using angles, Digital clocks display a numeric representation of time. Two numeric display formats are used on digital clocks, 24-hour notation. Most digital clocks use electronic mechanisms and LCD, LED, or VFD displays, for convenience, distance, telephony or blindness, auditory clocks present the time as sounds. There are also clocks for the blind that have displays that can be read by using the sense of touch, some of these are similar to normal analog displays, but are constructed so the hands can be felt without damaging them. The evolution of the technology of clocks continues today, the study of timekeeping is known as horology. The apparent position of the Sun in the sky moves over the course of a day, shadows cast by stationary objects move correspondingly, so their positions can be used to indicate the time of day. A sundial shows the time by displaying the position of a shadow on a flat surface, sundials can be horizontal, vertical, or in other orientations
23.
Solid mechanics
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Solid mechanics is fundamental for civil, aerospace, nuclear, and mechanical engineering, for geology, and for many branches of physics such as materials science. It has specific applications in other areas, such as understanding the anatomy of living beings. One of the most common applications of solid mechanics is the Euler-Bernoulli beam equation. Solid mechanics extensively uses tensors to describe stresses, strains, as shown in the following table, solid mechanics inhabits a central place within continuum mechanics. The field of rheology presents an overlap between solid and fluid mechanics, a material has a rest shape and its shape departs away from the rest shape due to stress. The amount of departure from rest shape is called deformation, the proportion of deformation to original size is called strain and this region of deformation is known as the linearly elastic region. It is most common for analysts in solid mechanics to use linear material models, however, real materials often exhibit non-linear behavior. As new materials are used and old ones are pushed to their limits, There are four basic models that describe how a solid responds to an applied stress, Elastically – When an applied stress is removed, the material returns to its undeformed state. Linearly elastic materials, those that deform proportionally to the applied load and this implies that the material response has time-dependence. Plastically – Materials that behave elastically generally do so when the stress is less than a yield value. When the stress is greater than the stress, the material behaves plastically. That is, deformation occurs after yield is permanent. Thermoelastically - There is coupling of mechanical with thermal responses, in general, thermoelasticity is concerned with elastic solids under conditions that are neither isothermal nor adiabatic. The simplest theory involves the Fouriers law of conduction, as opposed to advanced theories with physically more realistic models. This theorem includes the method of least work as a special case 1874,1922, Timoshenko corrects the Euler-Bernoulli beam equation 1936, Hardy Cross publication of the moment distribution method, an important innovation in the design of continuous frames. Martin, and L. J. Applied mechanics Materials science Continuum mechanics Fracture mechanics L. D, landau, E. M. Lifshitz, Course of Theoretical Physics, Theory of Elasticity Butterworth-Heinemann, ISBN 0-7506-2633-X J. E. Marsden, T. J. Hughes, Mathematical Foundations of Elasticity, Dover, ISBN 0-486-67865-2 P. C. Chou, N. J. Pagano, Elasticity, Tensor, Dyadic, goodier, Theory of elasticity, 3d ed
24.
Deformation (mechanics)
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Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body, a deformation may be caused by external loads, body forces, or changes in temperature, moisture content, or chemical reactions, etc. Strain is a description of deformation in terms of displacement of particles in the body that excludes rigid-body motions. In a continuous body, a deformation field results from a field induced by applied forces or is due to changes in the temperature field inside the body. The relation between stresses and induced strains is expressed by constitutive equations, e. g. Hookes law for linear elastic materials, deformations which are recovered after the stress field has been removed are called elastic deformations. In this case, the continuum completely recovers its original configuration, on the other hand, irreversible deformations remain even after stresses have been removed. Another type of deformation is viscous deformation, which is the irreversible part of viscoelastic deformation. In the case of elastic deformations, the response function linking strain to the stress is the compliance tensor of the material. Strain is a measure of deformation representing the displacement between particles in the relative to a reference length. A general deformation of a body can be expressed in the form x = F where X is the position of material points in the body. Such a measure does not distinguish between rigid body motions and changes in shape of the body, a deformation has units of length. We could, for example, define strain to be ε ≐ ∂ ∂ X = F ′ − I, hence strains are dimensionless and are usually expressed as a decimal fraction, a percentage or in parts-per notation. Strains measure how much a given deformation differs locally from a rigid-body deformation, a strain is in general a tensor quantity. Physical insight into strains can be gained by observing that a given strain can be decomposed into normal and this could be applied by elongation, shortening, or volume changes, or angular distortion. However, it is sufficient to know the normal and shear components of strain on a set of three perpendicular directions. In this case, the undeformed and deformed configurations of the continuum are significantly different and this is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue. Infinitesimal strain theory, also called small strain theory, small deformation theory, small displacement theory, in this case, the undeformed and deformed configurations of the body can be assumed identical. Large-displacement or large-rotation theory, which assumes small strains but large rotations, in each of these theories the strain is then defined differently
25.
Stress (mechanics)
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For example, when a solid vertical bar is supporting a weight, each particle in the bar pushes on the particles immediately below it. When a liquid is in a container under pressure, each particle gets pushed against by all the surrounding particles. The container walls and the pressure-inducing surface push against them in reaction and these macroscopic forces are actually the net result of a very large number of intermolecular forces and collisions between the particles in those molecules. Strain inside a material may arise by various mechanisms, such as stress as applied by external forces to the material or to its surface. Any strain of a material generates an internal elastic stress, analogous to the reaction force of a spring. In liquids and gases, only deformations that change the volume generate persistent elastic stress, however, if the deformation is gradually changing with time, even in fluids there will usually be some viscous stress, opposing that change. Elastic and viscous stresses are usually combined under the mechanical stress. Significant stress may exist even when deformation is negligible or non-existent, stress may exist in 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, stress that exceeds certain strength limits of the material will result in permanent deformation or even change its crystal structure and chemical composition. In some branches of engineering, the stress is occasionally used in a looser sense as a synonym of internal force. For example, in the analysis of trusses, it may refer to the total traction or compression force acting on a beam, since ancient times humans have been consciously aware of stress inside materials. Until the 17th century the understanding of stress was largely intuitive and empirical, with those tools, Augustin-Louis Cauchy was able to give the first rigorous and general mathematical model for stress in a homogeneous medium. Cauchy observed that the force across a surface was a linear function of its normal vector, and, moreover. The understanding of stress in liquids started with Newton, who provided a formula for friction forces in parallel laminar flow. Stress is defined as the force across a small boundary per unit area of that boundary, following the basic premises of continuum mechanics, stress is a macroscopic concept. In a fluid at rest the force is perpendicular to the surface, in a solid, or in 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 depend on the orientation of S. Thus the stress state of the material must be described by a tensor, called the stress tensor, with respect to any chosen coordinate system, the Cauchy stress tensor can be represented as a symmetric matrix of 3×3 real numbers
26.
Linear map
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In mathematics, a linear map is a mapping V → W between two modules that preserves the operations of addition and scalar multiplication. An important special case is when V = W, in case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the meaning as linear map. A linear map always maps linear subspaces onto linear subspaces, for instance it maps a plane through the origin to a plane, Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring, let V and W be vector spaces over the same field K. e. that for any vectors x1. Am ∈ K, the equality holds, f = a 1 f + ⋯ + a m f. It is then necessary to specify which of these fields is being used in the definition of linear. If V and W are considered as spaces over the field K as above, for example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. A linear map from V to K is called a linear functional and these statements generalize to any left-module RM over a ring R without modification, and to any right-module upon reversing of the scalar multiplication. The zero map between two left-modules over the ring is always linear. The identity map on any module is a linear operator, any homothecy centered in the origin of a vector space, v ↦ c v where c is a scalar, is a linear operator. This does not hold in general for modules, where such a map might only be semilinear, for real numbers, the map x ↦ x2 is not linear. Conversely, any map between finite-dimensional vector spaces can be represented in this manner, see the following section. Differentiation defines a map from the space of all differentiable functions to the space of all functions. It also defines an operator on the space of all smooth functions. If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f, V → W to dimF × dimF matrices in the way described in the sequel are themselves linear maps. The expected value of a variable is linear, as for random variables X and Y we have E = E + E and E = aE
27.
Tensor
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In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors. Elementary examples of such include the dot product, the cross product. Geometric vectors, often used in physics and engineering applications, given a coordinate basis or fixed frame of reference, a tensor can be represented as an organized multidimensional array of numerical values. The order of a tensor is the dimensionality of the array needed to represent it, or equivalently, for example, a linear map is represented by a matrix in a basis, and therefore is a 2nd-order tensor. A vector is represented as a 1-dimensional array in a basis, scalars are single numbers and are thus 0th-order tensors. Because they express a relationship between vectors, tensors themselves must be independent of a choice of coordinate system. The precise form of the transformation law determines the type of the tensor, the tensor type is a pair of natural numbers, where n is the number of contravariant indices and m is the number of covariant indices. The total order of a tensor is the sum of two numbers. The concept enabled an alternative formulation of the differential geometry of a manifold in the form of the Riemann curvature tensor. There are several approaches to defining tensors, although seemingly different, the approaches just describe the same geometric concept using different languages and at different levels of abstraction. For example, an operator is represented in a basis as a two-dimensional square n × n array. The numbers in the array are known as the scalar components of the tensor or simply its components. They are denoted by giving their position in the array, as subscripts and superscripts. For example, the components of an order 2 tensor T could be denoted Tij , whether an index is displayed as a superscript or subscript depends on the transformation properties of the tensor, described below. The total number of required to identify each component uniquely is equal to the dimension of the array. However, the term generally has another meaning in the context of matrices. Just as the components of a change when we change the basis of the vector space. Each tensor comes equipped with a law that details how the components of the tensor respond to a change of basis
28.
Matrix (mathematics)
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In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 ×3, the individual items in an m × n matrix A, often denoted by ai, j, where max i = m and max j = n, are called its elements or entries. Provided that they have the size, two matrices can be added or subtracted element by element. The rule for multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field, a major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f = 4x. The product of two matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations, if the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a transformation is obtainable from the matrixs eigenvalues. Applications of matrices are found in most scientific fields, in computer graphics, they are used to manipulate 3D models and project them onto a 2-dimensional screen. Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions, Matrices are used in economics to describe systems of economic relationships. A major branch of analysis is devoted to the development of efficient algorithms for matrix computations. Matrix decomposition methods simplify computations, both theoretically and practically, algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory, a simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function. A matrix is an array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is an array of scalars each of which is a member of F. Most of this focuses on real and complex matrices, that is, matrices whose elements are real numbers or complex numbers. More general types of entries are discussed below, for instance, this is a real matrix, A =
29.
Homogeneity and heterogeneity
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Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity in a substance or organism. A material or image that is homogeneous is uniform in composition or character, the alternate spellings homogenous and heterogenous are commonly, but incorrectly, used. Heterogenous meanwhile is a term primarily confined to pathology which refers to the property of an object in the body having its origin outside the body, the concepts are the same to every level of complexity, from atoms to populations of animals or people, and galaxies. Hence, an element may be homogeneous on a larger scale and this is known as an effective medium approach, or effective medium approximations. Various disciplines understand heterogeneity, or being heterogeneous, in different ways, heterogeneous solids, liquids, and gases may be made homogeneous by melting, stirring, or by allowing time to pass for diffusion to distribute the molecules evenly. For example, adding dye to water will create a solution at first. Entropy allows for heterogeneous substances to become homogeneous over time, a heterogeneous mixture is a mixture of two or more compounds. Examples are, mixtures of sand and water or sand and iron filings, a rock, water and oil, a salad, trail mix. A mixture can be determined to be homogeneous when everything is settled and equal, various models have been proposed to model the concentrations in different phases. The phenomena to be considered are mass rates and reaction, homogeneous reactions are chemical reactions in which the reactants and products are in the same phase, while heterogeneous reactions have reactants in two or more phases. Reactions that take place on the surface of a catalyst of a different phase are also heterogeneous, a reaction between two gases or two miscible liquids is homogeneous. A reaction between a gas and a liquid, a gas and a solid or a liquid and a solid is heterogeneous, earth is a heterogeneous substance in many aspects. E. g. rocks are inherently heterogeneous, usually occurring at the micro-scale and mini-scale, in algebra, homogeneous polynomials have the same number of factors of a given kind. This can cause problems in attempts to summarize the meaning of the studies, in medicine and genetics, a genetic or allelic heterogeneous condition is one where the same disease or condition can be caused, or contributed to, by several factors. In the case of genetics, varying different genes or alleles, in cancer research, cancer cell heterogeneity is thought to be one of the underlying reasons that make treatment of cancer difficult. In physics, heterogeneous is understood to mean having physical properties that vary within the medium, in sociology, heterogeneous may refer to a society or group that includes individuals of differing ethnicities, cultural backgrounds, sexes, or ages. Academic Press Dictionary of Science and Technology
30.
Cross section (geometry)
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In geometry and science, a cross section is the intersection of a body in three-dimensional space with a plane, or the analog in higher-dimensional space. Cutting an object into slices creates many parallel cross sections, conic sections – circles, ellipses, parabolas, and hyperbolas – are formed by cross-sections of a cone at various different angles, as seen in the diagram at left. Any planar cross-section passing through the center of an ellipsoid forms an ellipse on its surface, a cross-section of a cylinder is a circle if the cross-section is parallel to the cylinders base, or an ellipse with non-zero eccentricity if it is neither parallel nor perpendicular to the base. If the cross-section is perpendicular to the base it consists of two line segments unless it is just tangent to the cylinder, in which case it is a single line segment. A cross section of a polyhedron is a polygon, if instead the cross section is taken for a fixed value of the density, the result is an iso-density contour. For the normal distribution, these contours are ellipses, a cross section can be used to visualize the partial derivative of a function with respect to one of its arguments, as shown at left. In economics, a function f specifies the output that can be produced by various quantities x and y of inputs, typically labor. The production function of a firm or a society can be plotted in three-dimensional space, also in economics, a cardinal or ordinal utility function u gives the degree of satisfaction of a consumer obtained by consuming quantities w and v of two goods. Cross sections are used in anatomy to illustrate the inner structure of an organ. A cross section of a trunk, as shown at left, reveals growth rings that can be used to find the age of the tree. Cavalieris principle states that solids with corresponding sections of equal areas have equal volumes. The cross-sectional area of an object when viewed from an angle is the total area of the orthographic projection of the object from that angle. For example, a cylinder of height h and radius r has A ′ = π r 2 when viewed along its central axis, a sphere of radius r has A ′ = π r 2 when viewed from any angle. For a convex body, each ray through the object from the viewers perspective crosses just two surfaces, descriptive geometry Exploded view drawing Graphical projection Plans
31.
Helix
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A helix is a type of smooth space curve, i. e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a line called the axis. Examples of helices are coil springs and the handrails of spiral staircases, a filled-in helix – for example, a spiral ramp – is called a helicoid. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices, the word helix comes from the Greek word ἕλιξ, twisted, curved. Helices can be either right-handed or left-handed, handedness is a property of the helix, not of the perspective, a right-handed helix cannot be turned to look like a left-handed one unless it is viewed in a mirror, and vice versa. Most hardware screw threads are right-handed helices, the alpha helix in biology as well as the A and B forms of DNA are also right-handed helices. The Z form of DNA is left-handed, the pitch of a helix is the height of one complete helix turn, measured parallel to the axis of the helix. A double helix consists of two helices with the axis, differing by a translation along the axis. A conic helix may be defined as a spiral on a conic surface, an example is the Corkscrew roller coaster at Cedar Point amusement park. A circular helix, has constant band curvature and constant torsion, a curve is called a general helix or cylindrical helix if its tangent makes a constant angle with a fixed line in space. A curve is a general helix if and only if the ratio of curvature to torsion is constant, a curve is called a slant helix if its principal normal makes a constant angle with a fixed line in space. It can be constructed by applying a transformation to the frame of a general helix. Some curves found in nature consist of multiple helices of different handedness joined together by transitions known as tendril perversions, in mathematics, a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a particular helix, Probably the simplest equations for one is x = cos , y = sin , z = t. As the parameter t increases, the point traces a right-handed helix of pitch 2θ and radius 1 about the z-axis, in cylindrical coordinates, the same helix is parametrised by, r =1, θ = t, h = t. A circular helix of radius a and slope b/a is described by the following parametrisation, another way of mathematically constructing a helix is to plot the complex-valued function exi as a function of the real number x. The value of x and the real and imaginary parts of the function value give this plot three real dimensions, except for rotations, translations, and changes of scale, all right-handed helices are equivalent to the helix defined above. The equivalent left-handed helix can be constructed in a number of ways, in music, pitch space is often modeled with helices or double helices, most often extending out of a circle such as the circle of fifths, so as to represent octave equivalency
32.
Mechanical equilibrium
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In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is zero. In addition to defining mechanical equilibrium in terms of force, there are alternative definitions for mechanical equilibrium which are all mathematically equivalent. In terms of momentum, a system is in if the momentum of its parts is all constant. In terms of velocity, the system is in equilibrium if velocity is constant, in a rotational mechanical equilibrium the angular momentum of the object is conserved and the net torque is zero. More generally in conservative systems, equilibrium is established at a point in space where the gradient of the potential energy with respect to the generalized coordinates is zero. If a particle in equilibrium has zero velocity, that particle is in static equilibrium, since all particles in equilibrium have constant velocity, it is always possible to find an inertial reference frame in which the particle is stationary with respect to the frame. An important property of systems at mechanical equilibrium is their stability, if we have a function which describes the systems potential energy, we can determine the systems equilibria using calculus. A system is in equilibrium at the critical points of the function describing the systems potential energy. We can locate these points using the fact that the derivative of the function is zero at these points, if the system is displaced an arbitrarily small distance from the equilibrium state, the forces of the system cause it to move even farther away. Second derivative >0, The potential energy is at a local minimum, the response to a small perturbation is forces that tend to restore the equilibrium. If more than one equilibrium state is possible for a system. Second derivative =0 or does not exist, The state is neutral to the lowest order, to investigate the precise stability of the system, higher order derivatives must be examined. In a truly neutral state the energy does not vary and the state of equilibrium has a finite width and this is sometimes referred to as state that is marginally stable or in a state of indifference. Generally an equilibrium is only referred to as if it is stable in all directions. Sometimes there is not enough information about the acting on a body to determine if it is in equilibrium or not. This makes it an indeterminate system. The special case of mechanical equilibrium of an object is static equilibrium
33.
Graph of a function
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In mathematics, the graph of a function f is the collection of all ordered pairs. If the function x is a scalar, the graph is a two-dimensional graph. If the function x is an ordered pair of real numbers, the graph is the collection of all ordered triples. Graphing on a Cartesian plane is referred to as curve sketching. The graph of a function on real numbers may be mapped directly to the representation of the function. The concept of the graph of a function is generalized to the graph of a relation, note that although a function is always identified with its graph, they are not the same because it will happen that two functions with different codomain could have the same graph. For example, the cubic polynomial mentioned below is a surjection if its codomain is the real numbers, to test whether a graph of a curve is a function of x, one uses the vertical line test. To test whether a graph of a curve is a function of y, if the function has an inverse, the graph of the inverse can be found by reflecting the graph of the original function over the line y = x. In science, engineering, technology, finance, and other areas, in the simplest case one variable is plotted as a function of another, typically using rectangular axes, see Plot for details. In the modern foundation of mathematics known as set theory, a function, F = { a, if x =1, d, if x =2, c, if x =3, is. The graph of the polynomial on the real line f = x 3 −9 x is. If this set is plotted on a Cartesian plane, the result is a curve, the graph of the trigonometric function f = sin cos is. If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface, oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function, f = −2 Given a function f of n variables, x 1, …, x n, the normal to the graph is. This is seen by considering the graph as a set of the function g = f − z. The graph of a function is contained in a Cartesian product of sets, fibre bundles arent cartesian products, but appear to be up close. There is a notion of a graph on a fibre bundle called a section
34.
Cartesian coordinates
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
35.
Slope
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In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. The direction of a line is increasing, decreasing, horizontal or vertical. A line is increasing if it goes up from left to right, the slope is positive, i. e. m >0. A line is decreasing if it goes down from left to right, the slope is negative, i. e. m <0. If a line is horizontal the slope is zero, if a line is vertical the slope is undefined. The steepness, incline, or grade of a line is measured by the value of the slope. A slope with an absolute value indicates a steeper line Slope is calculated by finding the ratio of the vertical change to the horizontal change between two distinct points on a line. Sometimes the ratio is expressed as a quotient, giving the number for every two distinct points on the same line. A line that is decreasing has a negative rise, the line may be practical - as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan. The rise of a road between two points is the difference between the altitude of the road at two points, say y1 and y2, or in other words, the rise is = Δy. Here the slope of the road between the two points is described as the ratio of the altitude change to the horizontal distance between any two points on the line. In mathematical language, the m of the line is m = y 2 − y 1 x 2 − x 1. The concept of slope applies directly to grades or gradients in geography, as a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point. When the curve given by a series of points in a diagram or in a list of the coordinates of points, thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment. This is described by the equation, m = Δ y Δ x = vertical change horizontal change = rise run. Given two points and, the change in x from one to the other is x2 − x1, substituting both quantities into the above equation generates the formula, m = y 2 − y 1 x 2 − x 1. The formula fails for a line, parallel to the y axis. Suppose a line runs through two points, P = and Q =, since the slope is positive, the direction of the line is increasing
36.
Torque
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Torque, moment, or moment of force is rotational force. Just as a force is a push or a pull. Loosely speaking, torque is a measure of the force on an object such as a bolt or a flywheel. For example, pushing or pulling the handle of a wrench connected to a nut or bolt produces a torque that loosens or tightens the nut or bolt, the symbol for torque is typically τ, the lowercase Greek letter tau. When it is called moment of force, it is denoted by M. The SI unit for torque is the newton metre, for more on the units of torque, see Units. This article follows US physics terminology in its use of the word torque, in the UK and in US mechanical engineering, this is called moment of force, usually shortened to moment. In US physics and UK physics terminology these terms are interchangeable, unlike in US mechanical engineering, Torque is defined mathematically as the rate of change of angular momentum of an object. The definition of states that one or both of the angular velocity or the moment of inertia of an object are changing. Moment is the term used for the tendency of one or more applied forces to rotate an object about an axis. For example, a force applied to a shaft causing acceleration, such as a drill bit accelerating from rest. By contrast, a force on a beam produces a moment, but since the angular momentum of the beam is not changing. Similarly with any force couple on an object that has no change to its angular momentum and this article follows the US physics terminology by calling all moments by the term torque, whether or not they cause the angular momentum of an object to change. The concept of torque, also called moment or couple, originated with the studies of Archimedes on levers, the term torque was apparently introduced into English scientific literature by James Thomson, the brother of Lord Kelvin, in 1884. A force applied at an angle to a lever multiplied by its distance from the levers fulcrum is its torque. A force of three newtons applied two metres from the fulcrum, for example, exerts the same torque as a force of one newton applied six metres from the fulcrum. More generally, the torque on a particle can be defined as the product, τ = r × F, where r is the particles position vector relative to the fulcrum. Alternatively, τ = r F ⊥, where F⊥ is the amount of force directed perpendicularly to the position of the particle, any force directed parallel to the particles position vector does not produce a torque
37.
Axis (mathematics)
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
38.
Vector (mathematics)
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A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers. The operations of addition and scalar multiplication must satisfy certain requirements, called axioms. Euclidean vectors are an example of a vector space and they represent physical quantities such as forces, any two forces can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces and these vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are commonly used. This is particularly the case of Banach spaces and Hilbert spaces, historically, the first ideas leading to vector spaces can be traced back as far as the 17th centurys analytic geometry, matrices, systems of linear equations, and Euclidean vectors. Today, vector spaces are applied throughout mathematics, science and engineering, furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques, Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra. The concept of space will first be explained by describing two particular examples, The first example of a vector space consists of arrows in a fixed plane. This is used in physics to describe forces or velocities, given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w, when a is negative, av is defined as the arrow pointing in the opposite direction, instead. Such a pair is written as, the sum of two such pairs and multiplication of a pair with a number is defined as follows, + = and a =. The first example above reduces to one if the arrows are represented by the pair of Cartesian coordinates of their end points. A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below, elements of V are commonly called vectors. Elements of F are commonly called scalars, the second operation, called scalar multiplication takes any scalar a and any vector v and gives another vector av. In this article, vectors are represented in boldface to distinguish them from scalars
39.
Equation
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In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the make the equality true. Variables are also called unknowns and the values of the unknowns which satisfy the equality are called solutions of the equation, there are two kinds of equations, identity equations and conditional equations. An identity equation is true for all values of the variable, a conditional equation is true for only particular values of the variables. Each side of an equation is called a member of the equation, each member will contain one or more terms. The equation, A x 2 + B x + C = y has two members, A x 2 + B x + C and y, the left member has three terms and the right member one term. The variables are x and y and the parameters are A, B, an equation is analogous to a scale into which weights are placed. When equal weights of something are place into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, likewise, to keep an equation in balance, the same operations of addition, subtraction, multiplication and division must be performed on both sides of an equation for it to remain an equality. In geometry, equations are used to describe geometric figures and this is the starting idea of algebraic geometry, an important area of mathematics. Algebra studies two main families of equations, polynomial equations and, among them the case of linear equations. Polynomial equations have the form P =0, where P is a polynomial, linear equations have the form ax + b =0, where a and b are parameters. To solve equations from either family, one uses algorithmic or geometric techniques, algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory and these equations are difficult in general, one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions. Differential equations are equations that involve one or more functions and their derivatives and they are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in such as physics, chemistry, biology. The = symbol, which appears in equation, was invented in 1557 by Robert Recorde. An equation is analogous to a scale, balance, or seesaw
40.
Displacement (vector)
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A displacement is a vector that is the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a line from the initial position to the final position of the point. The velocity then is distinct from the speed which is the time rate of change of the distance traveled along a specific path. The velocity may be defined as the time rate of change of the position vector. For motion over an interval of time, the displacement divided by the length of the time interval defines the average velocity. In dealing with the motion of a body, the term displacement may also include the rotations of the body. In this case, the displacement of a particle of the body is called linear displacement, for a position vector s that is a function of time t, the derivatives can be computed with respect to t. These derivatives have common utility in the study of kinematics, control theory, vibration sensing and other sciences, by extension, the higher order derivatives can be computed in a similar fashion. Study of these higher order derivatives can improve approximations of the displacement function. Such higher-order terms are required in order to represent the displacement function as a sum of an infinite series, enabling several analytical techniques in engineering. The fourth order derivative is called jounce
41.
Scalar (mathematics)
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A scalar is an element of a field which is used to define a vector space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector, more generally, a vector space may be defined by using any field instead of real numbers, such as complex numbers. Then the scalars of that space will be the elements of the associated field. A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, a vector space equipped with a scalar product is called an inner product space. The real component of a quaternion is also called its scalar part, the term is also sometimes used informally to mean a vector, matrix, tensor, or other usually compound value that is actually reduced to a single component. Thus, for example, the product of a 1×n matrix and an n×1 matrix, the term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix. The word scalar derives from the Latin word scalaris, a form of scala. The English word scale also comes from scala, according to a citation in the Oxford English Dictionary the first recorded usage of the term scalar in English came with W. R. A vector space is defined as a set of vectors, a set of scalars, and a multiplication operation that takes a scalar k. For example, in a space, the scalar multiplication k yields. In a function space, kƒ is the function x ↦ k, the scalars can be taken from any field, including the rational, algebraic, real, and complex numbers, as well as finite fields. According to a theorem of linear algebra, every vector space has a basis. It follows that every vector space over a scalar field K is isomorphic to a vector space where the coordinates are elements of K. For example, every vector space of dimension n is isomorphic to n-dimensional real space Rn. Alternatively, a vector space V can be equipped with a function that assigns to every vector v in V a scalar ||v||. By definition. If ||v|| is interpreted as the length of v, this operation can be described as scaling the length of v by k, a vector space equipped with a norm is called a normed vector space. The norm is defined to be an element of Vs scalar field K. Moreover, if V has dimension 2 or more, K must be closed under square root, as well as the four operations, thus the rational numbers Q are excluded
42.
Function (mathematics)
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In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that each real number x to its square x2. The output of a function f corresponding to a x is denoted by f. In this example, if the input is −3, then the output is 9, likewise, if the input is 3, then the output is also 9, and we may write f =9. The input variable are sometimes referred to as the argument of the function, Functions of various kinds are the central objects of investigation in most fields of modern mathematics. There are many ways to describe or represent a function, some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function, in science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation, sometimes the codomain is called the functions range, but more commonly the word range is used to mean, instead, specifically the set of outputs. For example, we could define a function using the rule f = x2 by saying that the domain and codomain are the numbers. The image of this function is the set of real numbers. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, another important operation defined on functions is function composition, where the output from one function becomes the input to another function. Linking each shape to its color is a function from X to Y, each shape is linked to a color, there is no shape that lacks a color and no shape that has more than one color. This function will be referred to as the color-of-the-shape function, the input to a function is called the argument and the output is called the value. The set of all permitted inputs to a function is called the domain of the function. Thus, the domain of the function is the set of the four shapes. The concept of a function does not require that every possible output is the value of some argument, a second example of a function is the following, the domain is chosen to be the set of natural numbers, and the codomain is the set of integers. The function associates to any number n the number 4−n. For example, to 1 it associates 3 and to 10 it associates −6, a third example of a function has the set of polygons as domain and the set of natural numbers as codomain
43.
Matrix product
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In mathematics, matrix multiplication or the matrix product is a binary operation that produces a matrix from two matrices. The definition is motivated by linear equations and linear transformations on vectors, which have applications in applied mathematics, physics. When two linear transformations are represented by matrices, then the matrix represents the composition of the two transformations. The matrix product is not commutative in general, although it is associative and is distributive over matrix addition, the identity element of the matrix product is the identity matrix, and a square matrix may have an inverse matrix. Determinant multiplicativity applies to the matrix product, the matrix product is also important for matrix groups, and the theory of group representations and irreps. Computing matrix products is both an operation in many numerical algorithms and potentially time consuming, making it one of the most well-studied problems in numerical computing. Various algorithms have been devised for computing C = AB, especially for large matrices, index notation is often the clearest way to express definitions, and is used as standard in the literature. The i, j entry of matrix A is indicated by ij or Aij, whereas a numerical label on a collection of matrices is subscripted only, e. g. A1, A2, assume two matrices are to be multiplied. M, and summing the results over k, i j = ∑ k =1 m A i k B k j. Thus the product AB is defined if the number of columns in A is equal to the number of rows in B. Each entry may be computed one at a time, sometimes, the summation convention is used as it is understood to sum over the repeated index k. To prevent any ambiguity, this convention will not be used in the article, usually the entries are numbers or expressions, but can even be matrices themselves. The matrix product can still be calculated exactly the same way, see below for details on how the matrix product can be calculated in terms of blocks taking the forms of rows and columns. The figure to the right illustrates diagrammatically the product of two matrices A and B, showing how each intersection in the matrix corresponds to a row of A. Note AB and BA are two different matrices, the first is a 1 ×1 matrix while the second is a 3 ×3 matrix, if A =, B =, their matrix product is, A B = =, however BA is not defined. The product of a square matrix multiplied by a column matrix arises naturally in algebra, for solving linear equations. By choosing a, b, c, p, q, r, u, v, w in A appropriately, A can represent a variety of such as rotations, scaling and reflections, shears. If A =, B =, their products are, A B = =
44.
Continuum mechanics
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Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century, research in the area continues till today. Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies, Continuum mechanics deals with physical properties of solids and fluids which are independent of any particular coordinate system in which they are observed. These physical properties are represented by tensors, which are mathematical objects that have the required property of being independent of coordinate system. These tensors can be expressed in coordinate systems for computational convenience, Materials, such as solids, liquids and gases, are composed of molecules separated by space. On a microscopic scale, materials have cracks and discontinuities, a continuum is a body that can be continually sub-divided into infinitesimal elements with properties being those of the bulk material. More specifically, the continuum hypothesis/assumption hinges on the concepts of an elementary volume. This condition provides a link between an experimentalists and a viewpoint on constitutive equations as well as a way of spatial and statistical averaging of the microstructure. The latter then provide a basis for stochastic finite elements. The levels of SVE and RVE link continuum mechanics to statistical mechanics, the RVE may be assessed only in a limited way via experimental testing, when the constitutive response becomes spatially homogeneous. Specifically for fluids, the Knudsen number is used to assess to what extent the approximation of continuity can be made, consider car traffic on a highway---with just one lane for simplicity. Somewhat surprisingly, and in a tribute to its effectiveness, continuum mechanics effectively models the movement of cars via a differential equation for the density of cars. The familiarity of this situation empowers us to understand a little of the continuum-discrete dichotomy underlying continuum modelling in general. To start modelling define that, x measure distance along the highway, t is time, ρ is the density of cars on the highway, cars do not appear and disappear. Consider any group of cars, from the car at the back of the group located at x = a to the particular car at the front located at x = b. The total number of cars in this group N = ∫ a b ρ d x, since cars are conserved d N / d t =0. The only way an integral can be zero for all intervals is if the integrand is zero for all x, consequently, conservation derives the first order nonlinear conservation PDE ∂ ρ ∂ t + ∂ ∂ x =0 for all positions on the highway. This conservation PDE applies not only to car traffic but also to fluids, solids, crowds, animals, plants, bushfires, financial traders and this PDE is one equation with two unknowns, so another equation is needed to form a well posed problem
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Boiler
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A boiler is a closed vessel in which water or other fluid is heated. The fluid does not necessarily boil, the heated or vaporized fluid exits the boiler for use in various processes or heating applications, including water heating, central heating, boiler-based power generation, cooking, and sanitation. The pressure vessel of a boiler is usually made of steel, stainless steel, especially of the austenitic types, is not used in wetted parts of boilers due to corrosion and stress corrosion cracking. In live steam models, copper or brass is used because it is more easily fabricated in smaller size boilers. For much of the Victorian age of steam, the material used for boilermaking was the highest grade of wrought iron. In the 20th century, design practice instead moved towards the use of steel, which is stronger and cheaper, with welded construction, which is quicker and requires less labour. It should be noted, however, that wrought iron boilers corrode far slower than their steel counterparts. This makes the longevity of older wrought-iron boilers far superior to those of welded steel boilers, cast iron may be used for the heating vessel of domestic water heaters. Although such heaters are usually termed boilers in some countries, their purpose is usually to produce hot water, not steam, the brittleness of cast iron makes it impractical for high-pressure steam boilers. The source of heat for a boiler is combustion of any of several fuels, such as wood, coal, oil, electric steam boilers use resistance- or immersion-type heating elements. Nuclear fission is used as a heat source for generating steam, either directly or, in most cases. Heat recovery steam generators use the heat rejected from other such as gas turbine. 18th century Haycock boilers generally produced and stored large volumes of very low-pressure steam and these could burn wood or most often, coal. Flued boiler with one or two large flues—an early type or forerunner of fire-tube boiler, fire-tube boiler, Here, water partially fills a boiler barrel with a small volume left above to accommodate the steam. This is the type of boiler used in nearly all steam locomotives, the heat source is inside a furnace or firebox that has to be kept permanently surrounded by the water in order to maintain the temperature of the heating surface below the boiling point. Fire-tube boilers usually have a low rate of steam production. Fire-tube boilers mostly burn solid fuels, but are adaptable to those of the liquid or gas variety. Water-tube boiler, In this type, tubes filled with water are arranged inside a furnace in a number of possible configurations and this type generally gives high steam production rates, but less storage capacity than the above
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Infinitesimal strain theory
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With this assumption, the equations of continuum mechanics are considerably simplified. This approach may also be called small deformation theory, small displacement theory and it is contrasted with the finite strain theory where the opposite assumption is made. In such a linearization, the non-linear or second-order terms of the strain tensor are neglected. Therefore, the displacement gradient components and the spatial displacement gradient components are approximately equal. From the geometry of Figure 1 we have a b ¯ =2 +2 = d x 1 +2 ∂ u x ∂ x +2 +2 For very small displacement gradients, i. e. e. Therefore, the elements of the infinitesimal strain tensor are the normal strains in the coordinate directions. The results of operations are called strain invariants. Since there are no shear strain components in this coordinate system, an octahedral plane is one whose normal makes equal angles with the three principal directions. The engineering shear strain on a plane is called the octahedral shear strain and is given by γ o c t =232 +2 +2 where ε1, ε2, ε3 are the principal strains. Several definitions of equivalent strain can be found in the literature, thus, a solution does not generally exist for an arbitrary choice of strain components. Therefore, some restrictions, named compatibility equations, are imposed upon the strain components, with the addition of the three compatibility equations the number of independent equations are reduced to three, matching the number of unknown displacement components. These constraints on the strain tensor were discovered by Saint-Venant, and are called the Saint Venant compatibility equations, the compatibility functions serve to assure a single-valued continuous displacement function u i. The strains associated with length, i. e. the normal strain ε33, plane strain is then an acceptable approximation. The strain tensor for plane strain is written as, ε _ _ = in which the double underline indicates a second order tensor and this strain state is called plane strain. The corresponding stress tensor is, σ _ _ = in which the non-zero σ33 is needed to maintain the constraint ϵ33 =0. This stress term can be removed from the analysis to leave only the in-plane terms. Antiplane strain is another state of strain that can occur in a body. For infinitesimal deformations the scalar components of ω satisfy the condition | ω i j | ≪1, note that the displacement gradient is small only if both the strain tensor and the rotation tensor are infinitesimal
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Cauchy stress tensor
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In continuum mechanics, the Cauchy stress tensor σ, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components σ i j that completely define the state of stress at a point inside a material in the state, placement. A graphical representation of this law is the Mohrs circle for stress. The Cauchy stress tensor is used for analysis of material bodies experiencing small deformations. For large deformations, also called finite deformations, other measures of stress are required, such as the Piola–Kirchhoff stress tensor, the Biot stress tensor, and the Kirchhoff stress tensor. There are certain invariants associated with the tensor, whose values do not depend upon the coordinate system chosen. These are the three eigenvalues of the tensor, which are called the principal stresses. On an element of area Δ S containing P, with normal vector n, in particular, the contact force is given by Δ F = T Δ S where T is the mean surface traction. Cauchy’s stress principle asserts that as Δ S becomes very small and tends to zero the ratio Δ F / Δ S becomes d F / d S and the couple stress vector Δ M vanishes. The resultant vector d F / d S is defined as the traction, also called stress vector, traction. Given by T = T i e i at the point P associated with a plane with a normal vector n, T i = lim Δ S →0 Δ F i Δ S = d F i d S. This equation means that the vector depends on its location in the body. It is not a field because it depends not only on the position x of a particular material point. Depending on the orientation of the plane under consideration, the vector may not necessarily be perpendicular to that plane. The shear stress can be decomposed into two mutually perpendicular vectors. According to the Cauchy Postulate, the stress vector T remains unchanged for all passing through the point P. This means that the vector is a function of the normal vector n only. Cauchy’s fundamental lemma is equivalent to Newtons third law of motion of action and reaction, the state of stress at a point in the body is then defined by all the stress vectors T associated with all planes that pass through that point
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Stiffness tensor
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Hookes law is a principle of physics that states that the force needed to extend or compress a spring by some distance X is proportional to that distance. That is, F = kX, where k is a constant factor characteristic of the spring, its stiffness, the law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram and he published the solution of his anagram in 1678 as, ut tensio, sic vis. Hooke states in the 1678 work that he was aware of the law already in 1660, an elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean. Hookes law is only a linear approximation to the real response of springs. Many materials will deviate from Hookes law well before those elastic limits are reached. On the other hand, Hookes law is an approximation for most solid bodies, as long as the forces. For this reason, Hookes law is used in all branches of science and engineering. It is also the principle behind the spring scale, the manometer. The modern theory of elasticity generalizes Hookes law to say that the strain of an object or material is proportional to the stress applied to it. In this general form, Hookes law makes it possible to deduce the relation between strain and stress for complex objects in terms of properties of the materials it is made of. Consider a simple helical spring that has one end attached to some fixed object, suppose that the spring has reached a state of equilibrium, where its length is not changing anymore. Let X be the amount by which the end of the spring was displaced from its relaxed position. Hookes law states that F = k X or, equivalently, X = F k where k is a real number. Moreover, the formula holds when the spring is compressed. According to this formula, the graph of the applied force F as a function of the displacement X will be a line passing through the origin. Hookes law for a spring is often stated under the convention that F is the force exerted by the spring on whatever is pulling its free end. In that case, the equation becomes F = − k X since the direction of the force is opposite to that of the displacement