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Cocoa bean

The cocoa bean (technically cocoa seed) or simply cocoa, also called cacao , is the dried and fully fermented seed of Theobroma cacao, the cacao tree, from which cocoa solids (a mixture of nonfat substances) and cocoa butter (the fat) can be extracted. Cocoa trees are native to the Amazon rainforest. They are the basis of chocolate and Mesoamerican foods including tejate, an indigenous Mexican drink.

The cacao tree was first domesticated at least 5,300 years ago by the Mayo-Chinchipe culture in South America, before it was introduced in Mesoamerica. Cacao was consumed by pre-Hispanic cultures in spiritual ceremonies and its beans were a common currency in Mesoamerica. The cacao tree grows in a limited geographical zone; today, West Africa produces nearly 81% of the world's crop. The three main varieties of cocoa plant are Forastero, Criollo, and Trinitario; Forastero is the most widely used.

In 2020, global cocoa bean production reached 5.8 million tonnes, with Ivory Coast leading at 38% of the total, followed by Ghana and Indonesia. Cocoa beans, cocoa butter, and cocoa powder are traded on futures markets, with London focusing on West African cocoa and New York on Southeast Asian cocoa. Various international and national initiatives aim to support sustainable cocoa production, including the Swiss Platform for Sustainable Cocoa (SWISSCO), the German Initiative on Sustainable Cocoa (GISCO), and Belgium's Beyond Chocolate. At least 29% of global cocoa production was compliant with voluntary sustainability standards in 2016. Deforestation due to cocoa production remains a concern, especially in West Africa. Sustainable agricultural practices, such as agroforestry, can support cocoa production while conserving biodiversity. Cocoa contributes significantly to economies such as Nigeria's, and demand for cocoa products continues to grow steadily at over 3% annually since 2008.

To produce 1 kg of chocolate, around 300 to 600 cocoa beans are processed. The beans are roasted, cracked, and deshelled, resulting in pieces called nibs, which are ground into a thick paste known as chocolate liquor or cocoa paste. The liquor is processed into chocolate by adding cocoa butter, sugar, and sometimes vanilla and lecithin. Alternatively, cocoa powder and cocoa butter can be separated using a hydraulic press or the Broma process. Treating cocoa with an alkali produces Dutch process cocoa, which has a different flavor profile than untreated cocoa. Roasting can also be done on the whole bean or nib, affecting the final flavor. Cocoa contains phytochemicals like flavanols, procyanidins, and other flavonoids, and flavanol-rich chocolate and cocoa products may have a small blood pressure lowering effect. The beans also contain theobromine and a small amount of caffeine. The tree takes five years to grow and has a typical lifespan of 100 years.

Etymology
The word cocoa comes from the Spanish word cacao, which is derived from the Nahuatl word cacauatl. The Nahuatl word, in turn, ultimately derives from the reconstructed Proto-Mixe–Zoquean word kakawa.

(The reason for the inversion of the letters "o" and "a" in the borrowing from the Spanish "cacao" to the English "cocoa" is unknown, but etymological dictionaries speculate that it originated with a simple isolated arbitrary spelling confusion during the borrowing, possibly due to influence from the unrelated Spanish word "coco", but does not appear to represent any larger linguistic pattern or have any evident explanation beyond that, neither semantic nor euphonic. )

Used on its own, the term cocoa may also mean:

 * Hot cocoa, the drink more commonly known as hot chocolate

Terms derived from cocoa include:

 * Cocoa paste, ground cocoa beans: the mass is melted and separated into:
 * Cocoa butter, a pale, yellow, edible fat
 * Cocoa solids, the dark, bitter mass that contain most of cacao's notable phytochemicals, including caffeine and theobromine
 * Cocoa powder, a powder made by removing most of the cocoa butter from ground cacao seeds, or its derivatives, such as Dutch process cocoa

Cocoa beans are technically not beans or legumes, but seeds.

History
The cacao tree is native to the Amazon rainforest. It was first domesticated at least 5,300 years ago, in equatorial South America from the Santa Ana-La Florida (SALF) site in what is present-day southeast Ecuador (Zamora-Chinchipe Province) by the Mayo-Chinchipe culture, before being introduced in Mesoamerica.

More than 4,000 years ago, it was consumed by pre-Hispanic cultures along the Yucatán, including the Maya, and as far back as Olmeca civilization in spiritual ceremonies. It also grows in the foothills of the Andes in the Amazon and Orinoco basins of South America, in Colombia and Venezuela. Wild cacao still grows there. Its range may have been larger in the past; evidence of its wild range may be obscured by cultivation of the tree in these areas since long before the Spanish arrived.

As of November 2018, evidence suggests that cacao was first domesticated in equatorial South America, before being domesticated in Central America roughly 1,500 years later. Artifacts found at Santa-Ana-La Florida, in Ecuador, indicate that the Mayo-Chinchipe people were cultivating cacao as long as 5,300 years ago. Chemical analysis of residue extracted from pottery excavated at an archaeological site at Puerto Escondido, in Honduras, indicates that cocoa products were first consumed there sometime between 1500 and 1400 BC. Evidence also indicates that, long before the flavor of the cacao seed (or bean) became popular, the sweet pulp of the chocolate fruit, used in making a fermented (5.34% alcohol) beverage, first drew attention to the plant in the Americas. The cocoa bean was a common currency throughout Mesoamerica before the Spanish conquest.

Cacao trees grow in a limited geographical zone, of about 20° to the north and south of the Equator. Nearly 70% of the world crop today is grown in West Africa. The cacao plant was first given its botanical name by Swedish natural scientist Carl Linnaeus in his original classification of the plant kingdom, where he called it Theobroma ("food of the gods") cacao.

Cocoa was an important commodity in pre-Columbian Mesoamerica. A Spanish soldier who was on Hernan Cortés' side during the conquest of the Aztec Empire tells that when Moctezuma II, emperor of the Aztecs, dined, he took no other beverage than chocolate, served in a golden goblet. Flavored with vanilla or other spices, his chocolate was whipped into a froth that dissolved in the mouth. No fewer than 60 portions each day reportedly may have been consumed by Moctezuma II, and 2,000 more by the nobles of his court.

Chocolate was introduced to Europe by the Spaniards, and became a popular beverage by the mid-17th century. Spaniards also introduced the cacao tree into the West Indies and the Philippines. It was also introduced into the rest of Asia, South Asia and into West Africa by Europeans. In the Gold Coast, modern Ghana, cacao was introduced by a Ghanaian, Tetteh Quarshie.

Varieties
The three main varieties of cocoa plant are Forastero, Criollo, and Trinitario. The first is the most widely used, comprising 80–90% of the world production of cocoa. Cocoa beans of the Criollo variety are rarer and considered a delicacy. Criollo also tends to be less resistant to several diseases that attack the cocoa plant, hence very few countries still produce it. One of the largest producers of Criollo beans is Venezuela (Chuao and Porcelana). Trinitario (from Trinidad) is a hybrid between Criollo and Forastero varieties. It is considered to be of much higher quality than Forastero, has higher yields, and is more resistant to disease than Criollo.

Criollo
Representing only 5% of all cocoa beans grown, Criollo is the rarest and most expensive cocoa on the market, and is native to Central America, the Caribbean islands and the northern tier of South American states. The genetic purity of cocoas sold today as Criollo is disputed, as most populations have been exposed to the genetic influence of other varieties.

Criollo is particularly difficult to grow, as they are vulnerable to a variety of environmental threats and produce low yields of cocoa per tree. The flavor of Criollo is described as delicate yet complex, low in classic chocolate flavor, but rich in "secondary" notes of long duration.

Forastero
The most commonly grown bean is Forastero, a large group of wild and cultivated cocoas, most likely native to the Amazon Basin. The African cocoa crop is entirely made up of Forastero. They are significantly hardier and of higher yield than Criollo. The source of most chocolate marketed, Forastero cocoas are typically strong in classic "chocolate" flavor, but have a short duration and are unsupported by secondary flavors, producing "quite bland" chocolate. Forastero is particularly tannic and is therefore more astringent and bitter than the other varieties of cocoa.

Nacional
The Nacional is a rare variety of cocoa bean found in areas of South America such as Ecuador and Peru. Some experts in the 21st century had formerly considered the Nacional bean to be extinct after an abrupt end in 1916, when an outbreak of witch's broom disease devastated the Nacional variety throughout these countries. Pure genotypes of the bean are rare because most Nacional varieties have been interbred with other cocoa bean varieties. Ecuadorian Nacional traces its genetic lineage as far back as 5,300 years, to the earliest-known cacao trees domesticated by humanity. In the 18th and 19th centuries, Nacional was considered by many European chocolatiers to be the most coveted source of cacao in the world due to its floral aroma and complex flavor profile.

Trinitario
Trinitario is a natural hybrid of Criollo and Forastero. Trinitario originated in Trinidad after an introduction of Forastero to the local Criollo crop. Nearly all cocoa produced over the past five decades is of the Forastero or lower-grade Trinitario varieties.

Kurdicus
Theobroma cacao 'Kurdicus'human-made hybrid It is Turkey's first local cocoa type, produced by the tropical plant collector Sadık Ay in Turkey, a country where cocoa cultivation is not carried out, as a result of the three-parent hybridization of PA7xNA37 cocoa hybrids with another Cultivar hybrid. Studies continue on this new hybrid species, which shows rapid development and has the ability to be immune to diseases. The species carries Forastero and Trinitario genetics, but also has 30% Criollo character.

Cultivation
A cocoa pod (fruit) is about 17 to 20 cm long and has a rough, leathery rind about 2 to 3 cm thick (this varies with the origin and variety of pod) filled with sweet, mucilaginous pulp (called baba de cacao in South America) with a lemonade-like taste enclosing 30 to 50 large seeds that are fairly soft and a pale lavender to dark brownish purple color.

During harvest, the pods are opened, the seeds are kept, and the empty pods are discarded and the pulp made into juice. The seeds are placed where they can ferment. Due to heat buildup in the fermentation process, cacao beans lose most of the purplish hue and become mostly brown in color, with an adhered skin which includes the dried remains of the fruity pulp. This skin is released easily by winnowing after roasting. White seeds are found in some rare varieties, usually mixed with purples, and are considered of higher value.

Harvesting
Cocoa trees grow in hot, rainy tropical areas within 20° of latitude from the Equator. Cocoa harvest is not restricted to one period per year and a harvest typically occurs over several months. In fact, in many countries, cocoa can be harvested at any time of the year. Pesticides are often applied to the trees to combat capsid bugs, and fungicides to fight black pod disease.

Immature cocoa pods have a variety of colours, but most often are green, red, or purple, and as they mature, their colour tends towards yellow or orange, particularly in the creases. Unlike most fruiting trees, the cacao pod grows directly from the trunk or large branch of a tree rather than from the end of a branch, similar to jackfruit. This makes harvesting by hand easier as most of the pods will not be up in the higher branches. The pods on a tree do not ripen together; harvesting needs to be done periodically through the year. Harvesting occurs between three and four times weekly during the harvest season. The ripe and near-ripe pods, as judged by their colour, are harvested from the trunk and branches of the cocoa tree with a curved knife on a long pole. Care must be used when cutting the stem of the pod to avoid damaging the junction of the stem with the tree, as this is where future flowers and pods will emerge. One person can harvest an estimated 650 pods per day.

Harvest processing
The harvested pods are opened, typically with a machete, to expose the beans. The pulp and cocoa seeds are removed and the rind is discarded. The pulp and seeds are then piled in heaps, placed in bins, or laid out on grates for several days. During this time, the seeds and pulp undergo "sweating", where the thick pulp liquefies as it ferments. The fermented pulp trickles away, leaving cocoa seeds behind to be collected. Sweating is important for the quality of the beans, which originally have a strong, bitter taste. If sweating is interrupted, the resulting cocoa may be ruined; if underdone, the cocoa seed maintains a flavor similar to raw potatoes and becomes susceptible to mildew. Some cocoa-producing countries distill alcoholic spirits using the liquefied pulp.

A typical pod contains 30 to 40 beans and about 400 dried beans are required to make 1 lb of chocolate. Cocoa pods weigh an average of 400 g and each one yields 35 to 40 g dried beans; this yield is 9–10% of the total weight in the pod. One person can separate the beans from about 2000 pods per day.

The wet beans are then transported to a facility so they can be fermented and dried. The farmer removes the beans from the pods, packs them into boxes or heap them into piles, then covers them with mats or banana leaves for three to seven days. Finally, the beans are trodden and shuffled about (often using bare human feet) and sometimes, during this process, red clay mixed with water is sprinkled over the beans to obtain a finer color, polish, and protection against molds during shipment to factories in other countries. Drying in the sun is preferable to drying by artificial means, as no extraneous flavors such as smoke or oil are introduced which might otherwise taint the flavor.

The beans should be dry for shipment, which is usually by sea. Traditionally exported in jute bags, over the last decade, beans are increasingly shipped in "mega-bulk" parcels of several thousand tonnes at a time on ships, or standardized to 62.5 kg per bag and 200 (12.5 MT) or 240 (15 MT) bags per 20 ft container. Shipping in bulk significantly reduces handling costs. Shipment in bags, either in a ship's hold or in containers, is still common.

Throughout Mesoamerica where they are native, cocoa beans are used for a variety of foods. The harvested and fermented beans may be ground to order at tiendas de chocolate, or chocolate mills. At these mills, the cocoa can be mixed with a variety of ingredients such as cinnamon, chili peppers, almonds, vanilla, and other spices to create drinking chocolate. The ground cocoa is also an important ingredient in tejate.

Child slavery

The first allegations that child slavery is used in cocoa production appeared in 1998. In late 2000, a BBC documentary reported the use of enslaved children in the production of cocoa in West Africa. Other media followed by reporting widespread child slavery and child trafficking in the production of cocoa.

The cocoa industry was accused of profiting from child slavery and trafficking. The Harkin–Engel Protocol is an effort to end these practices. In 2001, it was signed and witnessed by the heads of eight major chocolate companies, US senators Tom Harkin and Herb Kohl, US Representative Eliot Engel, the ambassador of the Ivory Coast, the director of the International Programme on the Elimination of Child Labor, and others. It has, however, been criticized by some groups including the International Labor Rights Forum as an industry initiative which falls short, as the goal to eliminate the "worst forms of child labor" from cocoa production by 2005 was not reached. The deadline was extended multiple times and the goal changed to a 70% child labor reduction.

Child labour was growing in some West African countries in 2008–09 when it was estimated that 819,921 children worked on cocoa farms in Ivory Coast alone; by 2013–14, the number went up to 1,303,009. During the same period in Ghana, the estimated number of children working on cocoa farms was 957,398 children.

The 2010 documentary The Dark Side of Chocolate revealed that children smuggled from Mali to the Ivory Coast were forced to earn income for their parents, while others were sold as slaves for €230.

In 2010, the US Department of Labor formed the Child Labor Cocoa Coordinating Group as a public-private partnership with the governments of Ghana and Côte d'Ivoire to address child labor practices in the cocoa industry.

As of 2017, approximately 2.1 million children in Ghana and Côte d'Ivoire were involved in harvesting cocoa, carrying heavy loads, clearing forests, and being exposed to pesticides. According to Sona Ebai, the former secretary general of the Alliance of Cocoa Producing Countries: "I think child labor cannot be just the responsibility of industry to solve. I think it's the proverbial all-hands-on-deck: government, civil society, the private sector. And there, you really need leadership." As Reported in 2018, a 3-year pilot program, conducted by Nestlé with 26,000 farmers mostly located in Côte d'Ivoire, observed a 51% decrease in the number of children doing hazardous jobs in cocoa farming.

Lawsuits
In 2021, several companies were named in a class action lawsuit filed by eight former children from Mali who alleged that the companies aided and abetted their enslavement on cocoa plantations in Ivory Coast. The suit accused Barry Callebaut, Cargill, The Hershey Company, Mars, Mondelez, Nestlé, and Olam International, of knowingly engaging in forced labour, and the plaintiffs sought damages for unjust enrichment, negligent supervision, and intentional infliction of emotional distress.

Production
In 2020, world production of cocoa beans was 5.8 million tonnes, led by Ivory Coast with 38% of the total. Secondary producers were Ghana and Indonesia (each with about 14%).

Cocoa trading
Cocoa beans are traditionally shipped and stored in burlap sacks, in which the beans are susceptible to pest attacks. Fumigation with methyl bromide was to be phased out globally by 2015. Additional cocoa protection techniques for shipping and storage include the application of pyrenoids, as well as hermetic storage in sealed bags or containers with lowered oxygen concentrations. Safe long-term storage facilitates the trading of cocoa products at commodity exchanges.

Cocoa beans, cocoa butter and cocoa powder are traded on futures markets. The London market is based on West African cocoa and New York on cocoa predominantly from Southeast Asia. Cocoa is the world's smallest soft commodity market. The futures price of cocoa butter and cocoa powder is determined by multiplying the bean price by a ratio. The combined butter and powder ratio has tended to be around 3.5. If the combined ratio falls below 3.2 or so, production ceases to be economically viable and some factories cease extraction of butter and powder and trade exclusively in cocoa liquor.

Cocoa futures traded on the ICE Futures US Softs exchange, are valued at 10 Tonnes per contract with a tick size of 1 and tick value of US$10.

Sustainability
Multiple international and national initiatives collaborate to support sustainable cocoa production. These include the Swiss Platform for Sustainable Cocoa (SWISSCO), the German Initiative on Sustainable Cocoa (GISCO), and Beyond Chocolate, Belgium. A memorandum between these three initiatives was signed in 2020 to measure and address issues including child labor, living income, deforestation and supply chain transparency. Similar partnerships between cocoa producing and consuming countries are being developed, such as the cooperation between the International Cocoa Organization (ICCO) and the Ghanaian Cocoa Authority, who aim to increase the proportion of sustainable cocoa being imported from Ghana to Switzerland to 80% by 2025. The ICCO is engaged in projects around the world to support sustainable cocoa production and provide current information on the world cocoa market.

Voluntary sustainability standards
There are numerous voluntary certifications including Fairtrade and UTZ (now part of Rainforest Alliance) for cocoa which aim to differentiate between conventional cocoa production and that which is more sustainable in terms of social, economic and environmental concerns. As of 2016, at least 29% of global cocoa production was compliant with voluntary sustainability standards. However, among the different certifications there are significant differences in their goals and approaches, and a lack of data to show and compare the results on the farm level. While certifications can lead to increased farm income, the premium price paid for certified cocoa by consumers is not always reflected proportionally in the income for farmers. In 2012 the ICCO found that farm size mattered significantly when determining the benefits of certifications, and that farms an area less than 1ha were less likely to benefit from such programs, while those with slightly larger farms as well as access to member co-ops and the ability to improve productivity were most likely to benefit from certification. Certification often requires high up-front costs, which are a barrier to small farmers, and particularly, female farmers. The primary benefits to certification include improving conservation practices and reducing the use of agrochemicals, business support through cooperatives and resource sharing, and a higher price for cocoa beans which can improve the standard of living for farmers.

Fair trade cocoa producer groups are established in Belize, Bolivia, Cameroon, the Congo, Costa Rica, the Dominican Republic, Ecuador, Ghana, Haiti, India, Ivory Coast, Nicaragua, Panama, Paraguay, Peru, Sierra Leone, and São Tomé and Príncipe.

In 2018, the Beyond Chocolate partnership was created between multiple stakeholders in the global cocoa industry to decrease deforestation and provide a living income for cocoa farmers. The many international companies are currently participating in this agreement and the following voluntary certification programs are also partners in the Beyond Chocolate initiative: Rainforest Alliance, Fairtrade, ISEAL, BioForum Vlaanderen.

Many major chocolate production companies around the world have started to prioritize buying fair trade cocoa by investing in fair trade cocoa production, improving fair trade cocoa supply chains and setting purchasing goals to increase the proportion of fair trade chocolate available in the global market.

The Rainforest Alliance lists the following goals as part of their certification program:
 * Forest protection and sustainable land management
 * Improve rural livelihoods to reduce poverty
 * Address human rights issues such as child labor, gender inequality and indigenous land rights

The UTZ Certified-program (now part of Rainforest Alliance) included counteracting against child labor and exploitation of cocoa workers, requiring a code of conduct in relation to social and environmentally friendly factors, and improvement of farming methods to increase profits and salaries of farmers and distributors.

Environmental impact
The relative poverty of many cocoa farmers means that environmental consequences such as deforestation are given little significance. For decades, cocoa farmers have encroached on virgin forest, mostly after the felling of trees by logging companies. This trend has decreased as many governments and communities are beginning to protect their remaining forested zones. However, deforestation due to cocoa production is still a major concern in parts of West Africa. In Côte d'Ivoire and Ghana, barriers to land ownership have led migrant workers and farmers without financial resources to buy land to illegally expand their cocoa farming in protected forests. Many cocoa farmers in this region continue to prioritize expansion of their cocoa production, which often leads to deforestation.

Sustainable agricultural practices such as utilizing cover crops to prepare the soil before planting and intercropping cocoa seedlings with companion plants can support cocoa production and benefit the farm ecosystem. Prior to planting cocoa, leguminous cover crops can improve the soil nutrients and structure, which are important in areas where cocoa is produced due to high heat and rainfall which can diminish soil quality. Plantains are often intercropped with cocoa to provide shade to young seedlings and improve drought resilience of the soil. If the soil lacks essential nutrients, compost or animal manure can improve soil fertility and help with water retention.

In general, the use of chemical fertilizers and pesticides by cocoa farmers is limited. When cocoa bean prices are high, farmers may invest in their crops, leading to higher yields which, in turn tends to result in lower market prices and a renewed period of lower investment.

While governments and NGOs have made efforts to help cocoa farmers in Ghana and Côte d'Ivoire sustainably improve crop yields, many of the educational and financial resources provided are more readily available to male farmers versus female farmers. Access to credit is important for cocoa farmers, as it allows them to implement sustainable practices, such as agroforestry, and provide a financial buffer in case disasters like pest or weather patterns decrease crop yield.

Cocoa production is likely to be affected in various ways by the expected effects of global warming. Specific concerns have been raised concerning its future as a cash crop in West Africa, the current centre of global cocoa production. If temperatures continue to rise, West Africa could simply become unfit to grow the beans. The International Center for Tropical Agriculture warned in a paper published in 2013 that Ghana and Côte d'Ivoire, the world's two top cocoa growers, will experience a decline in suitable areas for cocoa production as global temperatures rise by up to 2 °C by 2050.

Climate change, coupled with pests, poor soil health, and the demand for sustainable cocoa, has led to a rapid decline in cocoa productivity, resulting in reduced income for smallholder cocoa farmers. Severe droughts have led to soil fertility decline, causing a decrease in yields, and resulting in some farmers abandoning cocoa production.

Cocoa beans also have a potential to be used as a bedding material in farms for cows. Using cocoa bean husks in bedding material for cows may contribute to udder health (less bacterial growth) and ammonia levels (lower ammonia levels on bedding).

Agroforestry
Cocoa beans may be cultivated under shade, as done in agroforestry. Agroforestry can reduce the pressure on existing protected forests for resources, such as firewood, and conserve biodiversity. Integrating shade trees with cocoa plants reduces risk of soil erosion and evaporation, and protects young cocoa plants from extreme heat. Agroforests act as buffers to formally protected forests and biodiversity island refuges in an open, human-dominated landscape. Research of their shade-grown coffee counterparts has shown that greater canopy cover in plots is significantly associated with greater mammal species diversity. The amount of diversity in tree species is fairly comparable between shade-grown cocoa plots and primary forests.

Economic effects
Cocoa contributes significantly to Nigerian economic activity, comprising the largest part of the country's foreign exchange, and providing income for farmers. Farmers can grow a variety of fruit-bearing shade trees to supplement their income to help cope with the volatile cocoa prices. Although cocoa has been adapted to grow under a dense rainforest canopy, agroforestry does not significantly further enhance cocoa productivity. However, while growing cocoa in full sun without incorporating shade plants can temporarily increase cocoa yields, it will eventually decrease the quality of the soil due to nutrient loss, desertification and erosion, leading to unsustainable yields and dependency on inorganic fertilizers. Agroforestry practices stabilize and improve soil quality, which can sustain cocoa production in the long term.

Over time, cocoa agroforestry systems become more similar to forest, although they never fully recover the original forest community within the life cycle of a productive cocoa plantation (approximately 25 years). Thus, although cocoa agroforests cannot replace natural forests, they are a valuable tool for conserving and protecting biodiversity while maintaining high levels of productivity in agricultural landscapes.

In West Africa, where about 70% of global cocoa supply originates from smallholder farmers, recent public–private initiatives such as the Cocoa Forest Initiatives in Ghana and Côte d'Ivoire (World Cocoa Foundation, 2017) and the Green Cocoa Landscape Programme in Cameroon (IDH, 2019) aim to support the sustainable intensification and climate resilience of cocoa production, the prevention of further deforestation and the restoration of degraded forests. They often align with national REDD+ policies and plans.

Consumption
People around the world enjoy cocoa in many different forms, consuming more than 3 million tons of cocoa beans yearly. Once the cocoa beans have been harvested, fermented, dried and transported they are processed in several components. Processor grindings serve as the main metric for market analysis. Processing is the last phase in which consumption of the cocoa bean can be equitably compared to supply. After this step all the different components are sold across industries to many manufacturers of different types of products.

Global market share for processing has remained stable, even as grindings increase to meet demand. One of the largest processing countries by volume is the Netherlands, handling around 13% of global grindings. Europe and Russia as a whole handle about 38% of the processing market. Average year after year demand growth has been just over 3% since 2008. While Europe and North America are relatively stable markets, increasing household income in developing countries is the main reason of the stable demand growth. As demand is awaited to keep growing, supply growth may slow down due to changing weather conditions in the largest cocoa production areas.

Chocolate production
To make 1 kg (2.2 lb) of chocolate, about 300 to 600 beans are processed, depending on the desired cocoa content. In a factory, the beans are roasted. Next, they are cracked and then deshelled by a "winnower". The resulting pieces of beans are called nibs. They are sometimes sold in small packages at specialty stores and markets to be used in cooking, snacking, and chocolate dishes. Since nibs are directly from the cocoa tree, they contain high amounts of theobromine. Most nibs are ground, using various methods, into a thick, creamy paste, known as chocolate liquor or cocoa paste. This "liquor" is then further processed into chocolate by mixing in (more) cocoa butter and sugar (and sometimes vanilla and lecithin as an emulsifier), and then refined, conched and tempered. Alternatively, it can be separated into cocoa powder and cocoa butter using a hydraulic press or the Broma process. This process produces around 50% cocoa butter and 50% cocoa powder. Cocoa powder may have a fat content of about 12%, but this varies significantly. Cocoa butter is used in chocolate bar manufacture, other confectionery, soaps, and cosmetics.

Treating with an alkali produces Dutch process cocoa, which is less acidic, darker, and more mellow in flavor than untreated cocoa. Regular (non-alkalized) cocoa is acidic, so when cocoa is treated with an alkaline ingredient, generally potassium carbonate, the pH increases. This process can be done at various stages during manufacturing, including during nib treatment, liquor treatment, or press cake treatment.

Another process that helps develop the flavor is roasting, which can be done on the whole bean before shelling or on the nib after shelling. The time and temperature of the roast affect the result: A "low roast" produces a more acid, aromatic flavor, while a high roast gives a more intense, bitter flavor lacking complex flavor notes.

Phytochemicals and research
Cocoa contains various phytochemicals, such as flavanols (including epicatechin), procyanidins, and other flavonoids. A systematic review presented moderate evidence that the use of flavanol-rich chocolate and cocoa products causes a small (2 mmHg) blood pressure lowering effect in healthy adults—mostly in the short term.

The highest levels of cocoa flavanols are found in raw cocoa and to a lesser extent, dark chocolate, since flavonoids degrade during cooking used to make chocolate.

Methylxanthines

The beans contain theobromine, and between 0.1% and 0.7% caffeine, whereas dry coffee beans are about 1.2% caffeine.

Theobromine found in the cocoa solids is fat soluble.

title: Cocoa bean


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463

Year 463 (CDLXIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Basilius and Vivianus (or, less frequently, year 1216 Ab urbe condita). The denomination 463 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Europe

 * Childeric I, king of the Salian Franks, allies with the Roman general Aegidius. During a battle near Orléans, the Visigoths under King Theodoric II are defeated by the Franks while crossing the Loire River.
 * The Suebi live under a diarchy, and fight a civil war over the kingship in Galicia (Northern Spain).

Asia

 * The Kibi Clan Rebellion against the Yamato state (Japan) in the Korean Peninsula begins.

Births

 * Houfei Di, emperor of the Liu Song Dynasty (d. 477)

Deaths

 * Frumar, Suevic king of Galicia (approximate date)
 * Richimund, Suevic king of Galicia (approximate date)
 * Romanus of Condat, hermit and saint (approximate date)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: 463


text:
467

Year 467 (CDLXVII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Pusaeus and Iohannes (or, less frequently, year 1220 Ab urbe condita). The denomination 467 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Roman Empire

 * April 12 – Emperor Leo I has his general Anthemius elected emperor of the Western Roman Empire. He allies himself with Ricimer, de facto ruler of Rome, and marries Anthemius's daughter Alypia to him, to strengthen the relationship and end the hostilities between the Eastern and Western Empire.
 * Summer – Vandal War (461-468): King Genseric extends his pirate raids in the Mediterranean Sea; the Vandals sack and enslave the people living in Illyricum, the Peloponnese and other parts of Greece. Leo I joins forces with the Western Empire.

Britannia

 * Ancient Hillforts in Britain are re-fortified, and the Wansdyke is built (approximate date).

Asia

 * Emperor Skandagupta dies after a 12-year reign, as Huns consolidate their conquests in western India. He is succeeded by his half-brother Purugupta.

Births

 * October 13 – Emperor Xiaowen of Northern Wei, emperor of Northern Wei (d. 499)
 * Cerdic, first king of Anglo-Saxon Wessex (approximate date)
 * Leo II, Byzantine emperor (d. 474)
 * Emperor Shun of Liu Song, Chinese emperor of Liu Song (d. 479)

Deaths

 * Benignus of Armagh, Irish bishop (approximate date)
 * Skandagupta, ruler of the Gupta Empire (India)

title: 467


text:
Reinforced concrete

Reinforced concrete, also called ferroconcrete, is a composite material in which concrete's relatively low tensile strength and ductility are compensated for by the inclusion of reinforcement having higher tensile strength or ductility. The reinforcement is usually, though not necessarily, steel bars (rebar) and is usually embedded passively in the concrete before the concrete sets. However, post-tensioning is also employed as a technique to reinforce the concrete. In terms of volume used annually, it is one of the most common engineering materials. In corrosion engineering terms, when designed correctly, the alkalinity of the concrete protects the steel rebar from corrosion.

Description
Reinforcing schemes are generally designed to resist tensile stresses in particular regions of the concrete that might cause unacceptable cracking and/or structural failure. Modern reinforced concrete can contain varied reinforcing materials made of steel, polymers or alternate composite material in conjunction with rebar or not. Reinforced concrete may also be permanently stressed (concrete in compression, reinforcement in tension), so as to improve the behavior of the final structure under working loads. In the United States, the most common methods of doing this are known as pre-tensioning and post-tensioning. For a strong, ductile and durable construction the reinforcement needs to have the following properties at least:
 * High relative strength
 * High toleration of tensile strain
 * Good bond to the concrete, irrespective of pH, moisture, and similar factors
 * Thermal compatibility, not causing unacceptable stresses (such as expansion or contraction) in response to changing temperatures.
 * Durability in the concrete environment, irrespective of corrosion or sustained stress for example.

History

François Coignet used iron-reinforced concrete as a technique for constructing building structures. In 1853, Coignet built the first iron reinforced concrete structure, a four-story house at 72 rue Charles Michels in the suburbs of Paris. Coignet's descriptions of reinforcing concrete suggests that he did not do it for means of adding strength to the concrete but for keeping walls in monolithic construction from overturning. The Pippen building in Brooklyn stands as a testament to his technique. In 1854, English builder William B. Wilkinson reinforced the concrete roof and floors in the two-story house he was constructing. His positioning of the reinforcement demonstrated that, unlike his predecessors, he had knowledge of tensile stresses.

Joseph Monier, a 19th-century French gardener, was a pioneer in the development of structural, prefabricated and reinforced concrete, having been dissatisfied with the existing materials available for making durable flowerpots. He was granted a patent for reinforcing concrete flowerpots by means of mixing a wire mesh and a mortar shell. In 1877, Monier was granted another patent for a more advanced technique of reinforcing concrete columns and girders, using iron rods placed in a grid pattern. Though Monier undoubtedly knew that reinforcing concrete would improve its inner cohesion, it is not clear whether he even knew how much the tensile strength of concrete was improved by the reinforcing.

Before the 1870s, the use of concrete construction, though dating back to the Roman Empire, and having been reintroduced in the early 19th century, was not yet a proven scientific technology. Thaddeus Hyatt, published a report entitled An Account of Some Experiments with Portland-Cement-Concrete Combined with Iron as a Building Material, with Reference to Economy of Metal in Construction and for Security against Fire in the Making of Roofs, Floors, and Walking Surfaces, in which he reported his experiments on the behavior of reinforced concrete. His work played a major role in the evolution of concrete construction as a proven and studied science. Without Hyatt's work, more dangerous trial and error methods might have been depended on for the advancement in the technology.

Ernest L. Ransome, an English-born engineer, was an early innovator of reinforced concrete techniques at the end of the 19th century. Using the knowledge of reinforced concrete developed during the previous 50 years, Ransome improved nearly all the styles and techniques of the earlier inventors of reinforced concrete. Ransome's key innovation was to twist the reinforcing steel bar, thereby improving its bond with the concrete. Gaining increasing fame from his concrete constructed buildings, Ransome was able to build two of the first reinforced concrete bridges in North America. One of his bridges still stands on Shelter Island in New Yorks East End, One of the first concrete buildings constructed in the United States was a private home designed by William Ward, completed in 1876. The home was particularly designed to be fireproof.

G. A. Wayss was a German civil engineer and a pioneer of the iron and steel concrete construction. In 1879, Wayss bought the German rights to Monier's patents and, in 1884, his firm, Wayss & Freytag, made the first commercial use of reinforced concrete. Up until the 1890s, Wayss and his firm greatly contributed to the advancement of Monier's system of reinforcing, established it as a well-developed scientific technology.

One of the first skyscrapers made with reinforced concrete was the 16-story Ingalls Building in Cincinnati, constructed in 1904.

The first reinforced concrete building in Southern California was the Laughlin Annex in downtown Los Angeles, constructed in 1905. In 1906, 16 building permits were reportedly issued for reinforced concrete buildings in the City of Los Angeles, including the Temple Auditorium and 8-story Hayward Hotel.

In 1906, a partial collapse of the Bixby Hotel in Long Beach killed 10 workers during construction when shoring was removed prematurely. That event spurred a scrutiny of concrete erection practices and building inspections. The structure was constructed of reinforced concrete frames with hollow clay tile ribbed flooring and hollow clay tile infill walls. That practice was strongly questioned by experts and recommendations for "pure" concrete construction were made, using reinforced concrete for the floors and walls as well as the frames.

In April 1904, Julia Morgan, an American architect and engineer, who pioneered the aesthetic use of reinforced concrete, completed her first reinforced concrete structure, El Campanil, a 72 ft bell tower at Mills College, which is located across the bay from San Francisco. Two years later, El Campanil survived the 1906 San Francisco earthquake without any damage, which helped build her reputation and launch her prolific career. The 1906 earthquake also changed the public's initial resistance to reinforced concrete as a building material, which had been criticized for its perceived dullness. In 1908, the San Francisco Board of Supervisors changed the city's building codes to allow wider use of reinforced concrete.

In 1906, the National Association of Cement Users (NACU) published Standard No. 1 and, in 1910, the Standard Building Regulations for the Use of Reinforced Concrete.

Use in construction

Many different types of structures and components of structures can be built using reinforced concrete elements including slabs, walls, beams, columns, foundations, frames and more.

Reinforced concrete can be classified as precast or cast-in-place concrete.

Designing and implementing the most efficient floor system is key to creating optimal building structures. Small changes in the design of a floor system can have significant impact on material costs, construction schedule, ultimate strength, operating costs, occupancy levels and end use of a building.

Without reinforcement, constructing modern structures with concrete material would not be possible.

Reinforced concrete elements
When reinforced concrete elements are used in construction, these reinforced concrete elements exhibit basic behavior when subjected to external loads. Reinforced concrete elements may be subject to tension, compression, bending, shear, and/or torsion.

Materials
Concrete is a mixture of coarse (stone or brick chips) and fine (generally sand and/or crushed stone) aggregates with a paste of binder material (usually Portland cement) and water. When cement is mixed with a small amount of water, it hydrates to form microscopic opaque crystal lattices encapsulating and locking the aggregate into a rigid shape. The aggregates used for making concrete should be free from harmful substances like organic impurities, silt, clay, lignite, etc. Typical concrete mixes have high resistance to compressive stresses (about 4000 psi); however, any appreciable tension (e.g., due to bending) will break the microscopic rigid lattice, resulting in cracking and separation of the concrete. For this reason, typical non-reinforced concrete must be well supported to prevent the development of tension.

If a material with high strength in tension, such as steel, is placed in concrete, then the composite material, reinforced concrete, resists not only compression but also bending and other direct tensile actions. A composite section where the concrete resists compression and reinforcement "rebar" resists tension can be made into almost any shape and size for the construction industry.

Key characteristics
Three physical characteristics give reinforced concrete its special properties:

 * 1) The coefficient of thermal expansion of concrete is similar to that of steel, eliminating large internal stresses due to differences in thermal expansion or contraction.
 * 2) When the cement paste within the concrete hardens, this conforms to the surface details of the steel, permitting any stress to be transmitted efficiently between the different materials. Usually steel bars are roughened or corrugated to further improve the bond or cohesion between the concrete and steel.
 * 3) The alkaline chemical environment provided by the alkali reserve (KOH, NaOH) and the portlandite (calcium hydroxide) contained in the hardened cement paste causes a passivating film to form on the surface of the steel, making it much more resistant to corrosion than it would be in neutral or acidic conditions. When the cement paste is exposed to the air and meteoric water reacts with the atmospheric CO2, portlandite and the calcium silicate hydrate (CSH) of the hardened cement paste become progressively carbonated and the high pH gradually decreases from 13.5 – 12.5 to 8.5, the pH of water in equilibrium with calcite (calcium carbonate) and the steel is no longer passivated.

As a rule of thumb, only to give an idea on orders of magnitude, steel is protected at pH above ~11 but starts to corrode below ~10 depending on steel characteristics and local physico-chemical conditions when concrete becomes carbonated. Carbonation of concrete along with chloride ingress are amongst the chief reasons for the failure of reinforcement bars in concrete.

The relative cross-sectional area of steel required for typical reinforced concrete is usually quite small and varies from 1% for most beams and slabs to 6% for some columns. Reinforcing bars are normally round in cross-section and vary in diameter. Reinforced concrete structures sometimes have provisions such as ventilated hollow cores to control their moisture & humidity.

Distribution of concrete (in spite of reinforcement) strength characteristics along the cross-section of vertical reinforced concrete elements is inhomogeneous.

Mechanism of composite action of reinforcement and concrete
The reinforcement in a RC structure, such as a steel bar, has to undergo the same strain or deformation as the surrounding concrete in order to prevent discontinuity, slip or separation of the two materials under load. Maintaining composite action requires transfer of load between the concrete and steel. The direct stress is transferred from the concrete to the bar interface so as to change the tensile stress in the reinforcing bar along its length. This load transfer is achieved by means of bond (anchorage) and is idealized as a continuous stress field that develops in the vicinity of the steel-concrete interface. The reasons that the two different material components concrete and steel can work together are as follows: (1) Reinforcement can be well bonded to the concrete, thus they can jointly resist external loads and deform. (2) The thermal expansion coefficients of concrete and steel are so close ($1$ to $1.5$ for concrete and $1.2$ for steel) that the thermal stress-induced damage to the bond between the two components can be prevented. (3) Concrete can protect the embedded steel from corrosion and high-temperature induced softening.

Anchorage (bond) in concrete: Codes of specifications
Because the actual bond stress varies along the length of a bar anchored in a zone of tension, current international codes of specifications use the concept of development length rather than bond stress. The main requirement for safety against bond failure is to provide a sufficient extension of the length of the bar beyond the point where the steel is required to develop its yield stress and this length must be at least equal to its development length. However, if the actual available length is inadequate for full development, special anchorages must be provided, such as cogs or hooks or mechanical end plates. The same concept applies to lap splice length mentioned in the codes where splices (overlapping) provided between two adjacent bars in order to maintain the required continuity of stress in the splice zone.

Anticorrosion measures
In wet and cold climates, reinforced concrete for roads, bridges, parking structures and other structures that may be exposed to deicing salt may benefit from use of corrosion-resistant reinforcement such as uncoated, low carbon/chromium (micro composite), epoxy-coated, hot dip galvanized or stainless steel rebar. Good design and a well-chosen concrete mix will provide additional protection for many applications.

Uncoated, low carbon/chromium rebar looks similar to standard carbon steel rebar due to its lack of a coating; its highly corrosion-resistant features are inherent in the steel microstructure. It can be identified by the unique ASTM specified mill marking on its smooth, dark charcoal finish. Epoxy-coated rebar can easily be identified by the light green color of its epoxy coating. Hot dip galvanized rebar may be bright or dull gray depending on length of exposure, and stainless rebar exhibits a typical white metallic sheen that is readily distinguishable from carbon steel reinforcing bar. Reference ASTM standard specifications A1035/A1035M Standard Specification for Deformed and Plain Low-carbon, Chromium, Steel Bars for Concrete Reinforcement, A767 Standard Specification for Hot Dip Galvanized Reinforcing Bars, A775 Standard Specification for Epoxy Coated Steel Reinforcing Bars and A955 Standard Specification for Deformed and Plain Stainless Bars for Concrete Reinforcement.

Another, cheaper way of protecting rebars is coating them with zinc phosphate. Zinc phosphate slowly reacts with calcium cations and the hydroxyl anions present in the cement pore water and forms a stable hydroxyapatite layer.

Penetrating sealants typically must be applied some time after curing. Sealants include paint, plastic foams, films and aluminum foil, felts or fabric mats sealed with tar, and layers of bentonite clay, sometimes used to seal roadbeds.

Corrosion inhibitors, such as calcium nitrite [Ca(NO2)2], can also be added to the water mix before pouring concrete. Generally, 1–2 wt. % of [Ca(NO2)2] with respect to cement weight is needed to prevent corrosion of the rebars. The nitrite anion is a mild oxidizer that oxidizes the soluble and mobile ferrous ions (Fe2+) present at the surface of the corroding steel and causes them to precipitate as an insoluble ferric hydroxide (Fe(OH)3). This causes the passivation of steel at the anodic oxidation sites. Nitrite is a much more active corrosion inhibitor than nitrate, which is a less powerful oxidizer of the divalent iron.

Reinforcement and terminology of beams

A beam bends under bending moment, resulting in a small curvature. At the outer face (tensile face) of the curvature the concrete experiences tensile stress, while at the inner face (compressive face) it experiences compressive stress.

A singly reinforced beam is one in which the concrete element is only reinforced near the tensile face and the reinforcement, called tension steel, is designed to resist the tension.

A doubly reinforced beam is the section in which besides the tensile reinforcement the concrete element is also reinforced near the compressive face to help the concrete resist compression and take stresses. The latter reinforcement is called compression steel. When the compression zone of a concrete is inadequate to resist the compressive moment (positive moment), extra reinforcement has to be provided if the architect limits the dimensions of the section.

An under-reinforced beam is one in which the tension capacity of the tensile reinforcement is smaller than the combined compression capacity of the concrete and the compression steel (under-reinforced at tensile face). When the reinforced concrete element is subject to increasing bending moment, the tension steel yields while the concrete does not reach its ultimate failure condition. As the tension steel yields and stretches, an "under-reinforced" concrete also yields in a ductile manner, exhibiting a large deformation and warning before its ultimate failure. In this case the yield stress of the steel governs the design.

An over-reinforced beam is one in which the tension capacity of the tension steel is greater than the combined compression capacity of the concrete and the compression steel (over-reinforced at tensile face). So the "over-reinforced concrete" beam fails by crushing of the compressive-zone concrete and before the tension zone steel yields, which does not provide any warning before failure as the failure is instantaneous.

A balanced-reinforced beam is one in which both the compressive and tensile zones reach yielding at the same imposed load on the beam, and the concrete will crush and the tensile steel will yield at the same time. This design criterion is however as risky as over-reinforced concrete, because failure is sudden as the concrete crushes at the same time of the tensile steel yields, which gives a very little warning of distress in tension failure.

Steel-reinforced concrete moment-carrying elements should normally be designed to be under-reinforced so that users of the structure will receive warning of impending collapse.

The characteristic strength is the strength of a material where less than 5% of the specimen shows lower strength.

The design strength or nominal strength is the strength of a material, including a material-safety factor. The value of the safety factor generally ranges from 0.75 to 0.85 in Permissible stress design.

The ultimate limit state is the theoretical failure point with a certain probability. It is stated under factored loads and factored resistances.

Reinforced concrete structures are normally designed according to rules and regulations or recommendation of a code such as ACI-318, CEB, Eurocode 2 or the like. WSD, USD or LRFD methods are used in design of RC structural members. Analysis and design of RC members can be carried out by using linear or non-linear approaches. When applying safety factors, building codes normally propose linear approaches, but for some cases non-linear approaches. To see the examples of a non-linear numerical simulation and calculation visit the references:

Prestressed concrete
Prestressing concrete is a technique that greatly increases the load-bearing strength of concrete beams. The reinforcing steel in the bottom part of the beam, which will be subjected to tensile forces when in service, is placed in tension before the concrete is poured around it. Once the concrete has hardened, the tension on the reinforcing steel is released, placing a built-in compressive force on the concrete. When loads are applied, the reinforcing steel takes on more stress and the compressive force in the concrete is reduced, but does not become a tensile force. Since the concrete is always under compression, it is less subject to cracking and failure.

Common failure modes of steel reinforced concrete
Reinforced concrete can fail due to inadequate strength, leading to mechanical failure, or due to a reduction in its durability. Corrosion and freeze/thaw cycles may damage poorly designed or constructed reinforced concrete. When rebar corrodes, the oxidation products (rust) expand and tends to flake, cracking the concrete and unbonding the rebar from the concrete. Typical mechanisms leading to durability problems are discussed below.

Mechanical failure
Cracking of the concrete section is nearly impossible to prevent; however, the size and location of cracks can be limited and controlled by appropriate reinforcement, control joints, curing methodology and concrete mix design. Cracking can allow moisture to penetrate and corrode the reinforcement. This is a serviceability failure in limit state design. Cracking is normally the result of an inadequate quantity of rebar, or rebar spaced at too great a distance. The concrete cracks either under excess loading, or due to internal effects such as early thermal shrinkage while it cures.

Ultimate failure leading to collapse can be caused by crushing the concrete, which occurs when compressive stresses exceed its strength, by yielding or failure of the rebar when bending or shear stresses exceed the strength of the reinforcement, or by bond failure between the concrete and the rebar.

Carbonation

Carbonation, or neutralisation, is a chemical reaction between carbon dioxide in the air and calcium hydroxide and hydrated calcium silicate in the concrete.

When a concrete structure is designed, it is usual to specify the concrete cover for the rebar (the depth of the rebar within the object). The minimum concrete cover is normally regulated by design or building codes. If the reinforcement is too close to the surface, early failure due to corrosion may occur. The concrete cover depth can be measured with a cover meter. However, carbonated concrete incurs a durability problem only when there is also sufficient moisture and oxygen to cause electropotential corrosion of the reinforcing steel.

One method of testing a structure for carbonatation is to drill a fresh hole in the surface and then treat the cut surface with phenolphthalein indicator solution. This solution turns pink when in contact with alkaline concrete, making it possible to see the depth of carbonation. Using an existing hole does not suffice because the exposed surface will already be carbonated.

Chlorides
Chlorides can promote the corrosion of embedded rebar if present in sufficiently high concentration. Chloride anions induce both localized corrosion (pitting corrosion) and generalized corrosion of steel reinforcements. For this reason, one should only use fresh raw water or potable water for mixing concrete, ensure that the coarse and fine aggregates do not contain chlorides, rather than admixtures which might contain chlorides.

It was once common for calcium chloride to be used as an admixture to promote rapid set-up of the concrete. It was also mistakenly believed that it would prevent freezing. However, this practice fell into disfavor once the deleterious effects of chlorides became known. It should be avoided whenever possible.

The use of de-icing salts on roadways, used to lower the freezing point of water, is probably one of the primary causes of premature failure of reinforced or prestressed concrete bridge decks, roadways, and parking garages. The use of epoxy-coated reinforcing bars and the application of cathodic protection has mitigated this problem to some extent. Also FRP (fiber-reinforced polymer) rebars are known to be less susceptible to chlorides. Properly designed concrete mixtures that have been allowed to cure properly are effectively impervious to the effects of de-icers.

Another important source of chloride ions is sea water. Sea water contains by weight approximately 3.5% salts. These salts include sodium chloride, magnesium sulfate, calcium sulfate, and bicarbonates. In water these salts dissociate in free ions (Na+, Mg2+, Cl−,, ) and migrate with the water into the capillaries of the concrete. Chloride ions, which make up about 50% of these ions, are particularly aggressive as a cause of corrosion of carbon steel reinforcement bars.

In the 1960s and 1970s it was also relatively common for magnesite, a chloride rich carbonate mineral, to be used as a floor-topping material. This was done principally as a levelling and sound attenuating layer. However it is now known that when these materials come into contact with moisture they produce a weak solution of hydrochloric acid due to the presence of chlorides in the magnesite. Over a period of time (typically decades), the solution causes corrosion of the embedded rebars. This was most commonly found in wet areas or areas repeatedly exposed to moisture.

Alkali silica reaction
This a reaction of amorphous silica (chalcedony, chert, siliceous limestone) sometimes present in the aggregates with the hydroxyl ions (OH−) from the cement pore solution. Poorly crystallized silica (SiO2) dissolves and dissociates at high pH (12.5 - 13.5) in alkaline water. The soluble dissociated silicic acid reacts in the porewater with the calcium hydroxide (portlandite) present in the cement paste to form an expansive calcium silicate hydrate (CSH). The alkali–silica reaction (ASR) causes localised swelling responsible for tensile stress and cracking. The conditions required for alkali silica reaction are threefold: (1) aggregate containing an alkali-reactive constituent (amorphous silica), (2) sufficient availability of hydroxyl ions (OH−), and (3) sufficient moisture, above 75% relative humidity (RH) within the concrete. This phenomenon is sometimes popularly referred to as "concrete cancer". This reaction occurs independently of the presence of rebars; massive concrete structures such as dams can be affected.

Conversion of high alumina cement
Resistant to weak acids and especially sulfates, this cement cures quickly and has very high durability and strength. It was frequently used after World War II to make precast concrete objects. However, it can lose strength with heat or time (conversion), especially when not properly cured. After the collapse of three roofs made of prestressed concrete beams using high alumina cement, this cement was banned in the UK in 1976. Subsequent inquiries into the matter showed that the beams were improperly manufactured, but the ban remained.

Sulfates
Sulfates (SO4) in the soil or in groundwater, in sufficient concentration, can react with the Portland cement in concrete causing the formation of expansive products, e.g., ettringite or thaumasite, which can lead to early failure of the structure. The most typical attack of this type is on concrete slabs and foundation walls at grades where the sulfate ion, via alternate wetting and drying, can increase in concentration. As the concentration increases, the attack on the Portland cement can begin. For buried structures such as pipe, this type of attack is much rarer, especially in the eastern United States. The sulfate ion concentration increases much slower in the soil mass and is especially dependent upon the initial amount of sulfates in the native soil. A chemical analysis of soil borings to check for the presence of sulfates should be undertaken during the design phase of any project involving concrete in contact with the native soil. If the concentrations are found to be aggressive, various protective coatings can be applied. Also, in the US ASTM C150 Type 5 Portland cement can be used in the mix. This type of cement is designed to be particularly resistant to a sulfate attack.

Steel plate construction
In steel plate construction, stringers join parallel steel plates. The plate assemblies are fabricated off site, and welded together on-site to form steel walls connected by stringers. The walls become the form into which concrete is poured. Steel plate construction speeds reinforced concrete construction by cutting out the time-consuming on-site manual steps of tying rebar and building forms. The method results in excellent strength because the steel is on the outside, where tensile forces are often greatest.

Fiber-reinforced concrete
Fiber reinforcement is mainly used in shotcrete, but can also be used in normal concrete. Fiber-reinforced normal concrete is mostly used for on-ground floors and pavements, but can also be considered for a wide range of construction parts (beams, pillars, foundations, etc.), either alone or with hand-tied rebars.

Concrete reinforced with fibers (which are usually steel, glass, plastic fibers) or cellulose polymer fiber is less expensive than hand-tied rebar. The shape, dimension, and length of the fiber are important. A thin and short fiber, for example short, hair-shaped glass fiber, is only effective during the first hours after pouring the concrete (its function is to reduce cracking while the concrete is stiffening), but it will not increase the concrete tensile strength. A normal-size fiber for European shotcrete (1 mm diameter, 45 mm length—steel or plastic) will increase the concrete's tensile strength. Fiber reinforcement is most often used to supplement or partially replace primary rebar, and in some cases it can be designed to fully replace rebar.

Steel is the strongest commonly available fiber, and comes in different lengths (30 to 80 mm in Europe) and shapes (end-hooks). Steel fibers can only be used on surfaces that can tolerate or avoid corrosion and rust stains. In some cases, a steel-fiber surface is faced with other materials.

Glass fiber is inexpensive and corrosion-proof, but not as ductile as steel. Recently, spun basalt fiber, long available in Eastern Europe, has become available in the U.S. and Western Europe. Basalt fiber is stronger and less expensive than glass, but historically has not resisted the alkaline environment of Portland cement well enough to be used as direct reinforcement. New materials use plastic binders to isolate the basalt fiber from the cement.

The premium fibers are graphite-reinforced plastic fibers, which are nearly as strong as steel, lighter in weight, and corrosion-proof. Some experiments have had promising early results with carbon nanotubes, but the material is still far too expensive for any building.

Non-steel reinforcement
There is considerable overlap between the subjects of non-steel reinforcement and fiber-reinforcement of concrete. The introduction of non-steel reinforcement of concrete is relatively recent; it takes two major forms: non-metallic rebar rods, and non-steel (usually also non-metallic) fibers incorporated into the cement matrix. For example, there is increasing interest in glass fiber reinforced concrete (GFRC) and in various applications of polymer fibers incorporated into concrete. Although currently there is not much suggestion that such materials will replace metal rebar, some of them have major advantages in specific applications, and there also are new applications in which metal rebar simply is not an option. However, the design and application of non-steel reinforcing is fraught with challenges. For one thing, concrete is a highly alkaline environment, in which many materials, including most kinds of glass, have a poor service life. Also, the behavior of such reinforcing materials differs from the behavior of metals, for instance in terms of shear strength, creep and elasticity.

Fiber-reinforced plastic/polymer (FRP) and glass-reinforced plastic (GRP) consist of fibers of polymer, glass, carbon, aramid or other polymers or high-strength fibers set in a resin matrix to form a rebar rod, or grid, or fiber. These rebars are installed in much the same manner as steel rebars. The cost is higher but, suitably applied, the structures have advantages, in particular a dramatic reduction in problems related to corrosion, either by intrinsic concrete alkalinity or by external corrosive fluids that might penetrate the concrete. These structures can be significantly lighter and usually have a longer service life. The cost of these materials has dropped dramatically since their widespread adoption in the aerospace industry and by the military.

In particular, FRP rods are useful for structures where the presence of steel would not be acceptable. For example, MRI machines have huge magnets, and accordingly require non-magnetic buildings. Again, toll booths that read radio tags need reinforced concrete that is transparent to radio waves. Also, where the design life of the concrete structure is more important than its initial costs, non-steel reinforcing often has its advantages where corrosion of reinforcing steel is a major cause of failure. In such situations corrosion-proof reinforcing can extend a structure's life substantially, for example in the intertidal zone. FRP rods may also be useful in situations where it is likely that the concrete structure may be compromised in future years, for example the edges of balconies when balustrades are replaced, and bathroom floors in multi-story construction where the service life of the floor structure is likely to be many times the service life of the waterproofing building membrane.

Plastic reinforcement often is stronger, or at least has a better strength to weight ratio than reinforcing steels. Also, because it resists corrosion, it does not need a protective concrete cover as thick as steel reinforcement does (typically 30 to 50 mm or more). FRP-reinforced structures therefore can be lighter and last longer. Accordingly, for some applications the whole-life cost will be price-competitive with steel-reinforced concrete.

The material properties of FRP or GRP bars differ markedly from steel, so there are differences in the design considerations. FRP or GRP bars have relatively higher tensile strength but lower stiffness, so that deflections are likely to be higher than for equivalent steel-reinforced units. Structures with internal FRP reinforcement typically have an elastic deformability comparable to the plastic deformability (ductility) of steel reinforced structures. Failure in either case is more likely to occur by compression of the concrete than by rupture of the reinforcement. Deflection is always a major design consideration for reinforced concrete. Deflection limits are set to ensure that crack widths in steel-reinforced concrete are controlled to prevent water, air or other aggressive substances reaching the steel and causing corrosion. For FRP-reinforced concrete, aesthetics and possibly water-tightness will be the limiting criteria for crack width control. FRP rods also have relatively lower compressive strengths than steel rebar, and accordingly require different design approaches for reinforced concrete columns.

One drawback to the use of FRP reinforcement is their limited fire resistance. Where fire safety is a consideration, structures employing FRP have to maintain their strength and the anchoring of the forces at temperatures to be expected in the event of fire. For purposes of fireproofing, an adequate thickness of cement concrete cover or protective cladding is necessary. The addition of 1 kg/m3 of polypropylene fibers to concrete has been shown to reduce spalling during a simulated fire. (The improvement is thought to be due to the formation of pathways out of the bulk of the concrete, allowing steam pressure to dissipate. )

Another problem is the effectiveness of shear reinforcement. FRP rebar stirrups formed by bending before hardening generally perform relatively poorly in comparison to steel stirrups or to structures with straight fibers. When strained, the zone between the straight and curved regions are subject to strong bending, shear, and longitudinal stresses. Special design techniques are necessary to deal with such problems.

There is growing interest in applying external reinforcement to existing structures using advanced materials such as composite (fiberglass, basalt, carbon) rebar, which can impart exceptional strength. Worldwide, there are a number of brands of composite rebar recognized by different countries, such as Aslan, DACOT, V-rod, and ComBar. The number of projects using composite rebar increases day by day around the world, in countries ranging from USA, Russia, and South Korea to Germany.

title: Reinforced concrete


text:
464

Year 464 (CDLXIV) was a leap year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rusticus and Olybrius (or, less frequently, year 1217 Ab urbe condita). The denomination 464 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Roman Empire

 * Olybrius is elected Roman consul by the Eastern court in Constantinople.

Europe

 * The Suevic nation in Galicia (Northern Spain) is unified under King Remismund.
 * King Theodoric II sends Remismund gifts (for recognizing his kingship), including weapons, and a Gothic princess for a wife.
 * Aegidius dies (possibly poisoned) and is succeeded by his son Syagrius, who becomes ruler of the Domain of Soissons (Gaul).

Births

 * Hashim ibn 'Abd Manaf, great-grandfather of Mohammed (approximate date)
 * Wu Di, Chinese emperor of the Liang Dynasty (d. 549)

Deaths

 * Aegidius, Roman general (magister militum) (approximate date)
 * Conall Gulban, king of Tir Chonaill (approximate date)
 * Wang Xianyuan, empress and wife of Xiao Wu Di (b. 427)
 * Xiao Wu Di, emperor of the Liu Song Dynasty (b. 430)

title: 464


text:
465

Year 465 (CDLXV) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Hermenericus and Basiliscus (or, less frequently, year 1218 Ab urbe condita). The denomination 465 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Roman Empire

 * Basiliscus, with the help of his sister Verina (wife of emperor Leo I), becomes a consul in the Eastern Roman Empire.
 * August 15 – Libius Severus, puppet emperor of the Western Roman Empire, dies after a 4-year reign.
 * September 2 – A fire begins in Constantinople and, over the next six days, destroys the buildings in eight of the 14 sections into which the Eastern Roman Imperial capital had been divided.
 * Ricimer, de facto ruler, establishes political control for 2 years from his residence in Rome.

Britannia

 * Battle of Wippedesfleot: The Saxons under command of Hengist and Aesc are defeated by the Britons near Ebbsfleet (Kent). During the battle 12 Welsh leaders are killed (according to the Anglo-Saxon Chronicle).

Europe

 * King Remismund establishes a policy of friendship with the Visigoths, and promotes the conversion of the Suebi into Arianism in Galicia (Northern Spain).

China

 * Qian Fei Di, then Ming Di, becomes ruler of the Liu Song Dynasty after his nephew is assassinated.

Religion

 * November 19 – Pope Hilarius convokes a synod at Rome's Church of Santa Maria Maggiore.
 * Peter the Fuller becomes patriarch of Antioch (approximate date).

Births

 * July 5 – Ahkal Mo' Naab' I, Maya ruler of Palenque (d. 524)
 * Dubricius, bishop and saint (approximate date)
 * Liberius, Roman aristocrat (approximate date)
 * Procopius of Gaza, Christian sophist and rhetorician (approximate date)
 * Severus, patriarch of Antioch (approximate date)
 * Zacharias Rhetor, bishop and ecclesiastic historian (approximate date)

Deaths

 * May 5 – Gerontius, Archbishop of Milan
 * June 20 – Wen Cheng Di, emperor of Northern Wei (b. 440)
 * August 15 – Libius Severus, emperor of the Western Roman Empire
 * Buliugu Li, official of the Northern Wei dynasty
 * Eógan mac Néill, king of Tír Eoghain (Ireland)
 * Liu Chuyu, princess of the Liu Song Dynasty

title: 465


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Transmigration

Transmigration may refer to:

 * Transmigration of the soul, a religious philosophical concept
 * Transmigration program, the program to move landless people from densely populated areas of Indonesia to less populous areas of the country
 * Transmigration (novel), a 1970 novel by J. T. McIntosh
 * Transmigration, a 1993 album by Crematory
 * Transmigration, a 1994 album by Osamu Sato
 * Transmigration, a 2006 album by Bakithi Kumalo
 * Transmigration, a 2009 album by Robert Spano and the Atlanta Symphony Orchestra, winner of a Grammy Award for Best Surround Sound Album

title: Transmigration


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Talk:Operation Enduring Freedom

Why is FRENCH Flag noted under belligerents??
This is not to diminish the contributions of any nation concerened, but on first glance, one might assume that FRANCE is contributing on a massive scale when they in fact, ARE NOT!!

As of July, 2009, France actually ranks about 7th or 8th of all the NATO nations and has sustained 28 casualties from the start of their partipation.

Canada, on the other hand, with a "tiny" military in relation to France has continually fielded between 2,500 and 3,000 troops from the very beginnning of this conflict. Frances current committment is about 1,000 and they are located just outside Kabul. Probably the LEAST dangerous sector in the entire country if any area can be described as such. France has also announced that they will be COMPLETELY withdrawing its committment. (and may in fact have aleady begun to do so??)

The Canadians are currently tasked with partolling and figthing in one of the most dangerous provinces in all of Southern Afghanistan (along with the British) and are subject to attacks on a daily basis. They were one of (if not the very first) contributor to actually field Main Battle Tanks in support of their and other NATO troops.

Tragically, Canada has up to this point sustained over 120 dead (The third highest casualty rate after the US & Great Britain!!!), to France's 28.

My point is this.... When one prominently notes the flag of a nation, that in reality is NOT doing the majority of the fighting and as a side note underneath states "and approximately 39 other nations", you have in effect, belittled and diminished the real sacrificies of others, and that is SHAMEFUL. —Preceding unsigned comment added by 99.239.34.128 (talk) 16:01, 12 July 2009 (UTC)

''Emotion should not be taken into account. Shame and sacrifice are irrelevant. Simply because France is not doing the "majority" of the fighting does not mean that their contribution is not worth mentioning in an encyclopedia.''
 * I second the comment immediately above this one. Beingsshepherd (talk) 15:46, 4 August 2013 (UTC)Beingsshepherd

The Taliban
I propose deleting the subjective words: 'extreme' and 'harsh' from this section. I did so myself and my changes were (as nearly always) reverted. Beingsshepherd (talk) 15:44, 4 August 2013 (UTC)Beingsshepherd

OIF?
I was in Iraq in 2003 as an US Army soldier in 286TH Sig Co. attached to an ADA unit. I noticed that Operation Iraqi Freedom is not mentioned in this article anywhere. As we were told OIF was a part of OEF. — Preceding unsigned comment added by 69.123.230.253 (talk) 13:16, 9 September 2013 (UTC)

Links/To do
Need to mention the security pact deliberations, perhaps as "result" + .> US halts army shipments out of Afghanistan>> Failure to sign pact could end Afghan mission>> Planes Parked in Weeds in Kabul After $486 Million Spent>> Australian troops leave Afghanistan>> Karzai: no security deal without peace talks>> Afghan war takes rising toll on civilians[http://www.aljazeera.com/indepth/opinion/2014/03/can-un-save-afghanistan-20143134452256298.html >> Can the UN save Afghanistan? ](Lihaas (talk) 15:34, 26 November 2013 (UTC)).

Is it over?
I'm confused as to whether or not OEF is over. Two DOD sources give conflicting info. This says it's still ongoing, yet this says it's over. Illegitimate Barrister 02:57, 20 March 2015 (UTC)
 * Better to error on the side of caution, if the campaign medal is still authorized, keep it open IMHO.--RightCowLeftCoast (talk) 03:01, 20 March 2015 (UTC)
 * Alrightey, thanks. Illegitimate Barrister 03:02, 20 March 2015 (UTC)

I've sent an e-mail to the DOD asking about this. I shall await an answer and report back. Illegitimate Barrister 06:57, 21 March 2015 (UTC)
 * Alright, just got a reply back from the DOD's AFPS. It says: "Good morning - Operation Enduring Freedom - Afghanistan ended Dec. 31, 2014. It was succeeded by Operation Freedom's Sentinel. Thank you for your query." This pretty much confirms my suspicions that OEF is still ongoing, it's just that OEF-A is over. Illegitimate Barrister 13:30, 25 March 2015 (UTC)

OEF is not "the official name" of the WoT
Absent an "official" statement from an authorized US gov. official that corroborates the claim made in the first sentence of this article, it is simply false. OEF originally pertained exclusively to Afghanistan in distinction from the Iraq operations, all of which were under the umbrella term War on Terror. Granted WoT is a meaningless term to the extent it is defined as "US efforts to harm, defeat and destroy any person, organization or state the US designates as terrorist", but that does not mean that OEF is synonymous with WoT, either officially or colloquially.

As this article notes, OEF "ended" and was replaced with OFS. As the WoT main article notes, Bush arguably coined the term WoT (which he used as a synonym for 'crusade') in 9/2001, and that term referred broadly to any fight against terrorists anywhere, not just Afghanistan. Obama even stopped using the term WoT altogether, preferring instead to use some other hopelessly ambiguous term - but not OEF.

This may sound like pedantry but it is critical that terminology used as a euphemism for war has a reasonably objective definition. Sloppy language allows for things like calling the decades old US interventions in Central/South America OEF, and according to this article therefore part of the WoT, despite US involvement predating the WoT, despite the lack of belligerents on one side.

This entire article is full of dubious claims, but the first sentence is fairly outrageous.BTercero (talk) 05:01, 9 June 2016 (UTC)

 * Lead adjusted to reflect sourced article content. ( Hohum  @ ) 17:35, 9 June 2016 (UTC)
 * Then what is OEF-A, et al? My email to the DOD was met with a reply that said OEF-A ended. You say that OEF ended in 2014, but this DOD page from 2015 says it is still ongoing. – Illegitimate Barrister (talk • contribs), 17:21, 22 September 2016 (UTC)

Infobox "Result"
Please note that Template:Infobox military conflict states against "result" that "this parameter may use one of two standard terms: "X victory" or "Inconclusive"." The infobox has been amended to reflect this. Please read the template "result" guidance in full before amending or reverting. It would probably be best to discuss any proposed change here first to seek consensus. Thanks. Gog the Mild (talk) 20:21, 17 March 2024 (UTC)

title: Talk:Operation Enduring Freedom


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Singularity (mathematics)

In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity.

For example, the reciprocal function $$f(x) = 1/x$$ has a singularity at $$x = 0$$, where the value of the function is not defined, as involving a division by zero. The absolute value function $$g(x) = |x|$$ also has a singularity at $$x = 0$$, since it is not differentiable there.

The algebraic curve defined by $$\left\{ (x,y):y^3-x^2 = 0 \right\}$$ in the $$(x, y)$$ coordinate system has a singularity (called a cusp) at $$(0, 0)$$. For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory.

Real analysis
In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discontinuities: type I, which has two subtypes, and type II, which can also be divided into two subtypes (though usually is not).

To describe the way these two types of limits are being used, suppose that $$f(x)$$ is a function of a real argument $$x$$, and for any value of its argument, say $$c$$, then the left-handed limit, $$f(c^-)$$, and the right-handed limit, $$f(c^+)$$, are defined by:
 * $$f(c^-) = \lim_{x \to c}f(x)$$, constrained by $$x < c$$ and

 * $$f(c^+) = \lim_{x \to c}f(x)$$, constrained by $$x > c$$.

The value $$f(c^-)$$ is the value that the function $$f(x)$$ tends towards as the value $$x$$ approaches $$c$$ from below, and the value $$f(c^+)$$ is the value that the function $$f(x)$$ tends towards as the value $$x$$ approaches $$c$$ from above, regardless of the actual value the function has at the point where $$x = c$$.

There are some functions for which these limits do not exist at all. For example, the function
 * $$g(x) = \sin\left(\frac{1}{x}\right)$$

does not tend towards anything as $$x$$ approaches $$c = 0$$. The limits in this case are not infinite, but rather undefined: there is no value that $$g(x)$$ settles in on. Borrowing from complex analysis, this is sometimes called an essential singularity.

The possible cases at a given value $$c$$ for the argument are as follows.
 * A point of continuity is a value of $$c$$ for which $$f(c^-) = f(c) = f(c^+)$$, as one expects for a smooth function. All the values must be finite. If $$c$$ is not a point of continuity, then a discontinuity occurs at $$c$$.
 * A type I discontinuity occurs when both $$f(c^-)$$ and $$f(c^+)$$ exist and are finite, but at least one of the following three conditions also applies:
 * $$f(c^-) \neq f(c^+)$$;
 * $$f(x)$$ is not defined for the case of $$x = c$$; or
 * $$f(c)$$ has a defined value, which, however, does not match the value of the two limits.
 * Type I discontinuities can be further distinguished as being one of the following subtypes:
 * A jump discontinuity occurs when $$f(c^-) \neq f(c^+)$$, regardless of whether $$f(c)$$ is defined, and regardless of its value if it is defined.
 * A removable discontinuity occurs when $$f(c^-) = f(c^+)$$, also regardless of whether $$f(c)$$ is defined, and regardless of its value if it is defined (but which does not match that of the two limits).
 * A type II discontinuity occurs when either $$f(c^-)$$ or $$f(c^+)$$ does not exist (possibly both). This has two subtypes, which are usually not considered separately:
 * An infinite discontinuity is the special case when either the left hand or right hand limit does not exist, specifically because it is infinite, and the other limit is either also infinite, or is some well defined finite number. In other words, the function has an infinite discontinuity when its graph has a vertical asymptote.
 * An essential singularity is a term borrowed from complex analysis (see below). This is the case when either one or the other limits $$f(c^-)$$ or $$f(c^+)$$ does not exist, but not because it is an infinite discontinuity. Essential singularities approach no limit, not even if valid answers are extended to include $$\pm\infty$$.
 * An essential singularity is a term borrowed from complex analysis (see below). This is the case when either one or the other limits $$f(c^-)$$ or $$f(c^+)$$ does not exist, but not because it is an infinite discontinuity. Essential singularities approach no limit, not even if valid answers are extended to include $$\pm\infty$$.

In real analysis, a singularity or discontinuity is a property of a function alone. Any singularities that may exist in the derivative of a function are considered as belonging to the derivative, not to the original function.

Coordinate singularities
A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example of this is the apparent singularity at the 90 degree latitude in spherical coordinates. An object moving due north (for example, along the line 0 degrees longitude) on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole (in the case of the example, jumping from longitude 0 to longitude 180 degrees). This discontinuity, however, is only apparent; it is an artifact of the coordinate system chosen, which is singular at the poles. A different coordinate system would eliminate the apparent discontinuity (e.g., by replacing the latitude/longitude representation with an $n$-vector representation).

Complex analysis
In complex analysis, there are several classes of singularities. These include the isolated singularities, the nonisolated singularities, and the branch points.

Isolated singularities
Suppose that $$\ f\ $$ is a function that is complex differentiable in the complement of a point $$\ a\ $$ in an open subset $$\ U\ $$ of the complex numbers $$\ \mathbb{C} ~.$$ Then:
 * The point $$\ a\ $$ is a removable singularity of $$\ f\ $$ if there exists a holomorphic function $$\ g\ $$ defined on all of $$\ U\ $$ such that $$\ f(z) = g(z)\ $$ for all $$\ z\ $$ in $$\ U \smallsetminus \{ a \} ~.$$ The function $$\ g\ $$ is a continuous replacement for the function $$\ f ~.$$
 * The point $$\ a\ $$ is a pole or non-essential singularity of $$\ f\ $$ if there exists a holomorphic function $$\ g\ $$ defined on $$\ U\ $$ with $$\ g(a)\ $$ nonzero, and a natural number $$\ n\ $$ such that $$\ f(z) = \frac{ g(z) }{\ (z-a)^n\ }\ $$ for all $$\ z\ $$ in $$\ U \smallsetminus \{ a \} ~.$$ The least such number $$\ n\ $$ is called the order of the pole. The derivative at a non-essential singularity itself has a non-essential singularity, with $$\ n\ $$ increased by $1$ (except if $$\ n\ $$ is $0$ so that the singularity is removable).
 * The point $$\ a\ $$ is an essential singularity of $$\ f\ $$ if it is neither a removable singularity nor a pole. The point $$\ a\ $$ is an essential singularity if and only if the Laurent series has infinitely many powers of negative degree.

Nonisolated singularities
Other than isolated singularities, complex functions of one variable may exhibit other singular behaviour. These are termed nonisolated singularities, of which there are two types:

 * Cluster points: limit points of isolated singularities. If they are all poles, despite admitting Laurent series expansions on each of them, then no such expansion is possible at its limit.
 * Natural boundaries: any non-isolated set (e.g. a curve) on which functions cannot be analytically continued around (or outside them if they are closed curves in the Riemann sphere).

Branch points
Branch points are generally the result of a multi-valued function, such as $$\ \sqrt{z\ }\ $$ or $$\ \log(z)\ ,$$ which are defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function. When the cut is genuinely required, the function will have distinctly different values on each side of the branch cut. The shape of the branch cut is a matter of choice, even though it must connect two different branch points (such as $$\ z = 0\ $$ and $$\ z = \infty\ $$ for $$\ \log(z)\ $$) which are fixed in place.

Finite-time singularity

A finite-time singularity occurs when one input variable is time, and an output variable increases towards infinity at a finite time. These are important in kinematics and Partial Differential Equations – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically, the simplest finite-time singularities are power laws for various exponents of the form $$x^{-\alpha},$$ of which the simplest is hyperbolic growth, where the exponent is (negative) 1: $$x^{-1}.$$ More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses $$(t_0-t)^{-\alpha}$$ (using t for time, reversing direction to $$-t$$ so that time increases to infinity, and shifting the singularity forward from 0 to a fixed time $$t_0$$).

An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of kinetic energy is lost on each bounce, the frequency of bounces becomes infinite, as the ball comes to rest in a finite time. Other examples of finite-time singularities include the various forms of the Painlevé paradox (for example, the tendency of a chalk to skip when dragged across a blackboard), and how the precession rate of a coin spun on a flat surface accelerates towards infinite—before abruptly stopping (as studied using the Euler's Disk toy).

Hypothetical examples include Heinz von Foerster's facetious "Doomsday's equation" (simplistic models yield infinite human population in finite time).

Algebraic geometry and commutative algebra
In algebraic geometry, a singularity of an algebraic variety is a point of the variety where the tangent space may not be regularly defined. The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation $y2 − x3 = 0$ defines a curve that has a cusp at the origin $x = y = 0$. One could define the $x$-axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, in this case, the $x$-axis is a "double tangent."

For affine and projective varieties, the singularities are the points where the Jacobian matrix has a rank which is lower than at other points of the variety.

An equivalent definition in terms of commutative algebra may be given, which extends to abstract varieties and schemes: A point is singular if the local ring at this point is not a regular local ring.

title: Singularity (mathematics)