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{{Banking}}
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'''Fractional-reserve banking''' is the practice whereby a [[bank]] retains [[bank reserves|reserves]] in an amount equal to only a portion of the amount of its customers' [[demand deposit|deposits]] to satisfy potential demands for withdrawals. Reserves are held at the bank as currency, or as deposits reflected in the bank's accounts at the [[central bank]]. The remainder of customer-deposited funds is used to fund investments or loans that the bank makes to other customers. {{Citation needed|date=September 2013}} Most of these loaned funds are later redeposited into other banks, allowing further lending. Because [[bank deposit]]s are usually considered money in their own right, fractional-reserve banking permits the [[money supply]] to grow to a multiple (called the [[money multiplier]]) of the underlying reserves of [[base money]] originally created by the central bank.<ref name="AbelAndrew">{{Cite book |last=Abel |first=Andrew |last2=Bernanke |first2=Ben |authorlink2=Ben Bernanke |title=Macroeconomics |publisher=Pearson |year=2005 |edition=5th|pages=522–532 |chapter=14.1 |postscript={{inconsistent citations}}}}</ref><ref name="Mankiw">{{Cite book |last=Mankiw |first=N. Gregory |title=Macroeconomics |publisher=Worth |year=2002 |edition=5th |pages=482–489 |chapter=Chapter 18: Money Supply and Money Demand |postscript={{inconsistent citations}} }}</ref>

To mitigate the risks of [[bank run]]s (when a large proportion of depositors seek withdrawal of their demand deposits at the same time) or, when problems are extreme and widespread, [[systemic risk|systemic crises]], the governments of most countries [[Bank regulation|regulate and oversee]] commercial banks, provide [[deposit insurance]] and act as [[lender of last resort]] to commercial banks.<ref name="AbelAndrew" /><ref name="Mankiw"/> In most countries, the central bank (or other monetary authority) regulates bank credit creation, imposing [[reserve requirements]] and other [[capital adequacy]] ratios. This limits the amount of [[money creation]] that occurs in the commercial banking system, and helps ensure that banks have enough funds to meet the demand for withdrawals.<ref name="Mankiw"/>

Fractional-reserve banking is the current form of banking in all countries worldwide.<ref>Frederic S. Mishkin, Economics of Money, Banking and Financial Markets, 10th Edition. Prentice Hall 2012</ref>

==History==
Fractional-reserve banking predates the existence of governmental monetary authorities and originated many centuries ago in bankers' realization that generally not all depositors demand payment at the same time.<ref>[http://mises.org/Books/Mengerprinciples.pdf Carl Menger: Principles of Economics]</ref>

Savers looking to keep their valuables in safekeeping depositories deposited [[gold]] and [[silver]] at [[goldsmith]]s, receiving in exchange a [[Promissory note|note]] for their [[deposit account|deposit]] (''see [[Bank of Amsterdam]]''). These notes gained acceptance as a [[medium of exchange]] for commercial transactions and thus became as an early form of circulating [[paper money]].<ref name="moneyfacts">{{cite book |last=United States. Congress. House. Banking and Currency Committee. |title=Money facts; 169 questions and answers on money – a supplement to A Primer on Money, with index, Subcommittee on Domestic Finance ... 1964. |location=Washington D.C. |year=1964 |url=http://www.baldwinlivingtrust.com/pdfs/AllAboutMoney.pdf}}</ref>

As the notes were used directly in [[trade]], the goldsmiths observed that people would not usually redeem all their notes at the same time, and they saw the opportunity to invest their coin reserves in interest-bearing loans and bills. This generated [[income]] for the goldsmiths but left them with more notes on issue than reserves with which to pay them. A process was started that altered the role of the goldsmiths from passive guardians of [[bullion]], charging fees for safe storage, to interest-paying and interest-earning banks. Thus fractional-reserve banking was born.

However, if [[creditor]]s (note holders of gold originally deposited) lost faith in the ability of a bank to pay their notes, many would try to redeem their notes at the same time. If in response a bank could not raise enough funds by calling in loans or selling bills, it either went into [[insolvency]] or defaulted on its notes. Such a situation is called a [[bank run]] and caused the demise of many early banks.<ref name="moneyfacts"/>

Starting in the late 1600s nations began to establish central banks which were given the legal power to set [[reserve requirement]]s and to issue the reserve assets, or [[monetary base]], in which form such reserves are required to be held.<ref>Charles P. Kindleberger, A Financial History of Western Europe. Routledge 2007</ref> The reciprocal of the reserve requirement, called the [[money multiplier]], limits the size to which the transactions in money supply may grow for a given level of reserves in the banking system. In order to mitigate the impact of bank failures and financial crises, governments created central banks – public (or semi-public) institutions that have the authority to centralize the storage of precious metal bullion amongst private banks to allow transfer of gold in case of bank runs, regulate commercial banks, impose reserve requirements, and act as lender-of-last-resort if any bank faced a bank run. The emergence of central banks reduced the risk of bank runs inherent in fractional-reserve banking and allowed the practice to continue as it does today.<ref name="Mankiw">{{Cite book |last=Mankiw |first=N. Gregory |title=Macroeconomics |publisher=Worth |year=2002 |edition=5th |pages=482–489 |chapter=Chapter 18: Money Supply and Money Demand}}</ref><ref name="paf">The Federal Reserve in Plain English – An easy-to-read guide to the structure and functions of the Federal Reserve System. See page 5 of the document for the purposes and functions: http://www.frbsf.org/publications/education/plainenglish/index.html</ref>

Over time, economists, central banks, and governments have changed their views as to the policy variables which should be targeted by monetary authorities. These have included interest rates, reserve requirements, and various measures of the money supply and [[monetary base]].

==How it works==
In most legal systems, a bank deposit is not a [[bailment]]. In other words, the funds deposited are no longer the property of the customer. The funds become the property of the bank, and the customer in turn receives an asset called a [[deposit account]] (a [[checking account|checking]] or [[savings account]]). That deposit account is a ''liability'' of the bank on the bank's books and on its [[balance sheet]]. Because the bank is authorized by law to create credit up to an amount equal to a multiple of the amount of its reserves, the bank's reserves on hand to satisfy payment of deposit liabilities amount to only a fraction of the total amount which the bank is obligated to pay in satisfaction of its demand deposits.

Fractional-reserve banking ordinarily functions smoothly. Relatively few depositors demand payment at any given time, and banks maintain a buffer of reserves to cover depositors' cash withdrawals and other demands for funds. However, during a bank run or a generalized [[financial crisis]], demands for withdrawal can exceed the bank's funding buffer, and the bank will be forced to raise additional reserves to avoid defaulting on its obligations. A bank can raise funds from additional borrowings (e.g., by borrowing in the [[interbank lending market]] or from the central bank), by selling assets, or by calling in short-term loans. If creditors are afraid that the bank is running out of reserves or is insolvent, they have an incentive to redeem their deposits as soon as possible before other depositors access the remaining reserves. Thus the fear of a bank run can actually precipitate the crisis.

Many of the practices of contemporary bank regulation and [[central banking]], including centralized [[Clearing (finance)|clearing]] of payments, central bank lending to member banks, regulatory auditing, and government-administered [[deposit insurance]], are designed to prevent the occurrence of such bank runs.

==Economic function==
Fractional-reserve banking allows banks to create credit in the form of bank deposits, which represent immediate liquidity to depositors. The banks also provide longer-term loans to borrowers, and act as [[financial intermediaries]] for those funds.<ref name="Mankiw"/><ref name="Abel_Bernanke">{{Cite book |last=Abel |first=Andrew |last2=Bernanke |first2=Ben |authorlink2=Ben Bernanke |title=Macroeconomics |publisher=Pearson |year=2005 |edition=5th|pages=266–269 |chapter=7}}</ref> Less liquid forms of deposit (such as [[time deposits]]) or riskier classes of financial assets (such as equities or long-term bonds) may lock up a depositor's wealth for a period of time, making it unavailable for use on demand. This "borrowing short, lending long," or [[maturity transformation]] function of fractional-reserve banking is a role that many economists consider to be an important function of the commercial banking system.<ref>[http://delong.typepad.com/sdj/2010/03/the-maturity-transformation-and-liquidity-transformation-and-safety-transformation-industtry.html Maturity Transformation] Brad DeLong</ref>

Additionally, according to [[macroeconomics|macroeconomic]] theory, a well-regulated fractional-reserve bank system also benefits the economy by providing regulators with powerful tools for influencing the [[money supply]] and interest rates. Many economists believe that these should be adjusted by the government to promote [[Economic stability|macroeconomic stability]].<ref name="Mankiw_Ch9">{{Cite book |last=Mankiw |first=N. Gregory |title=Macroeconomics |publisher=Worth |year=2002 |edition=5th |pages=238–255 |chapter=9}}</ref>

Modern central banking allows banks to practice fractional-reserve banking with inter-bank business transactions with a reduced risk of bankruptcy. The process of fractional-reserve banking expands the money supply of the economy but also increases the risk that a bank cannot meet its depositor withdrawals.<ref name="purpose">Page 57 of 'The FED today', a publication on an educational site affiliated with the Federal Reserve Bank of Kansas City, designed to educate people on the history and purpose of the United States Federal Reserve system. [http://www.philadelphiafed.org/publications/economic-education/fed-today/fed-today_lesson-6.pdf The FED today Lesson 6]</ref><ref>{{cite web |url=http://www.bankofengland.co.uk/publications/speeches/2009/speech381.pdf
|title=Mervyn King, Finance: A Return from Risk
|publisher=Bank of England|quote= Banks are dangerous institutions. They borrow short and lend long. They create liabilities which promise to be liquid and hold few liquid assets themselves. That though is hugely valuable for the rest of the economy. Household savings can be channelled to finance illiquid investment projects while providing access to liquidity for those savers who may need it.... If a large number of depositors want liquidity at the same time, banks are forced into early liquidation of assets – lowering their value ...'}}</ref>

==Money creation process==
{{main|Money creation}}

There are two types of money in a fractional-reserve banking system operating with a central bank:<ref name="bis">Bank for International Settlements – The Role of Central Bank Money in Payment Systems. See page 9, titled, "The coexistence of central and commercial bank monies: multiple issuers, one currency": http://www.bis.org/publ/cpss55.pdf
A quick quotation in reference to the 2 different types of money is listed on page 3. It is the first sentence of the document:
:"Contemporary monetary systems are based on the mutually reinforcing roles of central bank money and commercial bank monies."</ref><ref name="ecb">European Central Bank – Domestic payments in Euroland: commercial and central bank money:
http://www.ecb.int/press/key/date/2000/html/sp001109_2.en.html One quotation from the article referencing the two types of money:
:"At the beginning of the 20th almost the totality of retail payments were made in central bank money. Over time, this monopoly came to be shared with commercial banks, when deposits and their transfer via cheques and giros became widely accepted. Banknotes and commercial bank money became fully interchangeable payment media that customers could use according to their needs. While transaction costs in commercial bank money were shrinking, cashless payment instruments became increasingly used, at the expense of banknotes"</ref><ref>Macmillan report 1931 account of how fractional banking works http://books.google.ca/books?hl=en&id=EkUTaZofJYEC&dq=British+Parliamentary+reports+on+international+finance&printsec=frontcover&source=web&ots=kHxssmPNow&sig=UyopnsiJSHwk152davCIyQAMVdw&sa=X&oi=book_result&resnum=1&ct=result#PPA34,M1</ref>
# '''Central bank money:''' money created or adopted by the central bank regardless of its form – precious metals, commodity certificates, banknotes, coins, electronic money loaned to commercial banks, or anything else the central bank chooses as its form of money
# '''Commercial bank money:''' demand deposits in the commercial banking system; sometimes referred to as "chequebook money"
When a deposit of central bank money is made at a commercial bank, the central bank money is removed from circulation and added to the commercial banks' reserves (it is no longer counted as part of [[Money supply#Empirical measures|M1 money supply]]). Simultaneously, an equal amount of new commercial bank money is created in the form of bank deposits. When a loan is made by the commercial bank (which keeps only a fraction of the central bank money as reserves), using the central bank money from the commercial bank's reserves, the m1 money supply expands by the size of the loan.<ref name="Mankiw"/> This process is called "deposit multiplication".

===Example of deposit multiplication===
The table below displays the relending model of how loans are funded and how the money supply is affected. It also shows how central bank money is used to create commercial bank money from an initial deposit of $100 of central bank money. In the example, the initial deposit is lent out 10 times with a fractional-reserve rate of 20% to ultimately create $500 of commercial bank money (it is important to note that the 20% reserve rate used here is for ease of illustration, actual [[reserve requirement]]s are usually a lot ''lower'', for example around 3% in the USA and UK). Each successive bank involved in this process creates new commercial bank money on a diminishing portion of the original deposit of central bank money. This is because banks only lend out a portion of the central bank money deposited, in order to fulfill reserve requirements and to ensure that they always have enough reserves on hand to meet normal transaction demands.

The relending model begins when an initial $100 deposit of central bank money is made into Bank A. Bank A takes 20 percent of it, or $20, and sets it aside as reserves, and then loans out the remaining 80 percent, or $80. At this point, the money supply actually totals $180, not $100, because the bank has loaned out $80 of the central bank money, kept $20 of central bank money in reserve (not part of the money supply), and substituted a newly created $100 IOU claim for the depositor that ''acts equivalently to and can be implicitly redeemed for'' central bank money (the depositor can transfer it to another account, write a check on it, demand his cash back, etc.). These claims by depositors on banks are termed ''demand deposits'' or ''commercial bank money'' and are simply recorded in a bank's accounts as a liability (specifically, an IOU to the depositor). From a depositor's perspective, commercial bank money is equivalent to central bank money – it is impossible to tell the two forms of money apart unless a bank run occurs.<ref name="Mankiw"/>

At this point in the relending model, Bank A now only has $20 of central bank money on its books. The loan recipient is holding $80 in central bank money, but he soon spends the $80. The receiver of that $80 then deposits it into Bank B. Bank B is now in the same situation as Bank A started with, except it has a deposit of $80 of central bank money instead of $100. Similar to Bank A, Bank B sets aside 20 percent of that $80, or $16, as reserves and lends out the remaining $64, increasing money supply by $64. As the process continues, more commercial bank money is created. To simplify the table, a different bank is used for each deposit. In the real world, the money a bank lends may end up in the same bank so that it then has more money to lend out.

[[File:Fractional-reserve-banking base100 0.8reserve rate.svg|thumb|360px|right|The expansion of $100 of central bank money through fractional-reserve lending with a 20% reserve rate. $400 of commercial bank money is created virtually through loans.]]
{| class="wikitable"
|+ Table Sources:
! Individual Bank
! Amount Deposited
! Lent Out
! Reserves
|-
| height="16" align="CENTER" | A
| bgcolor="#00AA00" align="CENTER" | 100
| align="CENTER" | 80
| bgcolor="#FF0000" align="CENTER" | 20
|-
| height="16" align="CENTER" | B
| align="CENTER" | 80
| align="CENTER" | 64
| bgcolor="#FF0000" align="CENTER" | 16
|-
| height="16" align="CENTER" | C
| align="CENTER" | 64
| align="CENTER" | 51.20
| bgcolor="#FF0000" align="CENTER" | 12.80
|-
| height="16" align="CENTER" | D
| align="CENTER" | 51.20
| align="CENTER" | 40.96
| bgcolor="#FF0000" align="CENTER" | 10.24
|-
| height="16" align="CENTER" | E
| align="CENTER" | 40.96
| align="CENTER" | 32.77
| bgcolor="#FF0000" align="CENTER" | 8.19
|-
| height="16" align="CENTER" | F
| align="CENTER" | 32.77
| align="CENTER" | 26.21
| bgcolor="#FF0000" align="CENTER" | 6.55
|-
| height="16" align="CENTER" | G
| align="CENTER" | 26.21
| align="CENTER" | 20.97
| bgcolor="#FF0000" align="CENTER" | 5.24
|-
| height="16" align="CENTER" | H
| align="CENTER" | 20.97
| align="CENTER" | 16.78
| bgcolor="#FF0000" align="CENTER" | 4.19
|-
| height="16" align="CENTER" | I
| align="CENTER" | 16.78
| align="CENTER" | 13.42
| bgcolor="#FF0000" align="CENTER" | 3.36
|-
| height="16" align="CENTER" | J
| align="CENTER" | 13.42
| align="CENTER" | 10.74
| bgcolor="#FF0000" align="CENTER" | 2.68
|-
| height="16" align="CENTER" | K
| align="CENTER" | 10.74
| align="CENTER" | 0.0
| bgcolor="#FF0000" align="CENTER" | 10.74
|-
| height="16" align="CENTER" |
| align="CENTER" |
| align="CENTER" |
| align="CENTER" |
|-
|
| align="CENTER" | '''Total Amount of Deposits:'''
| align="CENTER" | '''Total Amount Lent Out:'''
| align="CENTER" | '''Total Reserves:'''
|-
| height="16" align="CENTER" |
| bgcolor="#99CCFF" align="CENTER" | 457.05
| bgcolor="#99CCFF" align="CENTER" | 357.05
| bgcolor="#00AA00" align="CENTER" | 100
|-
|}
Although no new money was physically created in addition to the initial $100 deposit, new commercial bank money is created through loans. The boxes marked in red show the location of the original $100 deposit throughout the entire process. The total reserves will always equal the original amount, which in this case is $100. As this process continues, more commercial bank money is created. The amounts in each step decrease towards a limit. If a graph is made showing the accumulation of deposits, one can see that the graph is curved and approaches a limit. This limit is the maximum amount of money that can be created with a given reserve rate. When the reserve rate is 20%, as in the example above, the maximum amount of total deposits that can be created is $500 and the maximum increase in the money supply is $400.

For an individual bank, the deposit is considered a ''[[Liability (accounting)|liability]]'' whereas the loan it gives out and the reserves are considered ''[[assets]]''. Deposits will always be equal to loans plus a bank's reserves, since loans and reserves are created from deposits. This is the basis for a bank's ''[[balance sheet]]''. Fractional-reserve banking allows the money supply to expand or contract. Generally the expansion or contraction of the money supply is dictated by the balance between the rate of new loans being created and the rate of existing loans being repaid or defaulted on. The balance between these two rates can be influenced to ''some degree'' by actions of the central bank. However, the central bank has no direct control over the amount of money created by commercial (or high street) banks.

===Money multiplier===
{{main|Money multiplier}}
[[File:Fractional-reserve banking with varying reserve requirements.gif|thumb|right|360px|The expansion of $100 through fractional-reserve banking with varying reserve requirements. Each curve approaches a limit. This limit is the value that the "money multiplier'" calculates.]]
The most common mechanism used to measure this increase in the money supply is typically called the "money multiplier". It calculates the maximum amount of money that an initial deposit can be expanded to with a given reserve ratio.

====Formula====
The money multiplier, ''m'', is the inverse of the reserve requirement, ''R'':<ref>http://www.mhhe.com/economics/mcconnell15e/graphics/mcconnell15eco/common/dothemath/moneymultiplier.html</ref>
:<math>m=\frac1R</math>

;Example

For example, with the reserve ratio of 20 percent, this reserve ratio, ''R'', can also be expressed as a fraction:
:<math>R=\tfrac15</math>
So then the money multiplier, ''m'', will be calculated as:
:<math>m=\frac{1}{1/5}=5</math>
This number is multiplied by the initial deposit to show the maximum amount of money it can be expanded to.

The money creation process is also affected by the currency drain ratio (the propensity of the public to hold banknotes rather than deposit them with a commercial bank), and the safety reserve ratio ([[excess reserves]] beyond the legal requirement that commercial banks voluntarily hold – usually a small amount). Data for "excess" reserves and vault cash are published regularly by the [[Federal Reserve in the United States]].<ref>http://www.federalreserve.gov/releases/h3/Current/ Federal Reserve Board, "AGGREGATE RESERVES OF DEPOSITORY INSTITUTIONS AND THE MONETARY BASE" (Updated weekly).</ref> In practice, the actual money multiplier varies over time, and may be substantially lower than the theoretical maximum.<ref>http://books.google.com/books?id=FdrbugYfKNwC&pg=PA169&lpg=PA169&dq=united+states+money+multiplier&source=web&ots=C_Hw1u82xe&sig=m7g0bMz167DijFsOCbn5f4aWAOU#PPA170,M1 Bruce Champ & Scott Freeman, Modeling Monetary Economies, p. 170 (Figure 9.1).</ref>

==Money supplies around the world==
[[File:Components of the United States money supply2.svg|thumb|400px|Components of US money supply (currency, [[Money supply#Empirical measures|M1, M2, and M3]]) since 1959. In January 2007, the amount of "central bank money" was $750.5 billion while the amount of "commercial bank money" (in the M2 supply) was $6.33 trillion. M1 is currency plus demand deposits; M2 is M1 plus time deposits, savings deposits, and some money-market funds; and M3 is M2 plus large time deposits and other forms of money.
The M3 data ends in 2006 because [http://www.federalreserve.gov/releases/h6/discm3.htm the federal reserve ceased reporting it].]]
[[File:Euro money supply Sept 1998 - Oct 2007.jpg|thumb|right|400px|Components of the euro money supply 1998–2007]]
{{see also|Money supply}}
Fractional-reserve banking determines the relationship between the amount of "central bank money" in the official money supply statistics and the total money supply. Most of the money in these systems is "commercial bank money". Fractional-reserve banking allows the creation of commercial bank money, which increases the money supply through the [[deposit creation multiplier]]. The issue of money through the banking system is a mechanism of monetary transmission, which a [[central bank]] can influence only indirectly by raising or lowering [[interest rate]]s (although banking regulations may also be adjusted to influence the money supply, depending on the circumstances).

This table gives an outline of the makeup of money supplies worldwide. Most of the money in any given money supply consists of commercial bank money.<ref name="bis"/> The value of commercial bank money is based on the fact that it can be exchanged freely at a bank for central bank money.<ref name="bis"/><ref name="ecb"/>

The actual increase in the money supply through this process may be lower, as (at each step) banks may choose to hold [[excess reserves|reserves in excess]] of the statutory minimum, borrowers may let some funds sit idle, and some members of the public may choose to hold cash, and there also may be delays or frictions in the lending process.<ref>http://books.google.com/books?id=I-49pxHxMh8C&pg=PA303&dq=deposit+reserves&lr=&sig=hMQtESrWP6IBRYiiaZgKwIoDWVk#PPA295,M1 William MacEachern, Macroeconomics: A Contemporary Introduction, p. 295</ref> Government regulations may also be used to limit the money creation process by preventing banks from giving out loans even though the reserve requirements have been fulfilled.<ref>ebook: The Federal Reserve – Purposes and Functions:http://www.federalreserve.gov/pf/pf.htm
:see pages 13 and 14 of the pdf version for information on government regulations and supervision over banks</ref>

==Regulation==
Because the nature of fractional-reserve banking involves the possibility of [[bank run]]s, central banks have been created throughout the world to address these problems.<ref name="paf"/><ref name="rbievolution">Reserve Bank of India – Report on Currency and Finance 2004–05 (See page 71 of the full report or just download the section ''Functional Evolution of Central Banking''): http://www.rbi.org.in/scripts/AnnualPublications.aspx?head=Report%20on%20Currency%20and%20Finance&fromdate=03/17/06&todate=03/19/06: The monopoly power to issue currency is delegated to a central bank in full or sometimes in part. The practice regarding the currency issue is governed more by convention than by any particular theory. It is well known that the basic concept of currency evolved in order to facilitate exchange. The primitive currency note was in reality a promissory note to pay back to its bearer the original precious metals. With greater acceptability of these promissory notes, these began to move across the country and the banks that issued the promissory notes soon learnt that they could issue more receipts than the gold reserves held by them. This led to the evolution of the fractional-reserve system. It also led to repeated bank failures and brought forth the need to have an independent authority to act as lender-of-the-last-resort. Even after the emergence of central banks, the concerned governments continued to decide asset backing for issue of coins and notes. The asset backing took various forms including gold coins, bullion, foreign exchange reserves and foreign securities. With the emergence of a fractional-reserve system, this reserve backing (gold, currency assets, etc.) came down to a fraction of total currency put in circulation.</ref>

===Central banks===
{{main|Central bank}}
Government controls and [[bank regulation]]s related to fractional-reserve banking have generally been used to impose restrictive requirements on note issue and deposit taking on the one hand, and to provide relief from bankruptcy and creditor claims, and/or protect creditors with government funds, when banks defaulted on the other hand. Such measures have included:
# Minimum [[Reserve requirement|required reserve ratios]] (RRRs)
# Minimum [[capital ratio]]s
# Government bond deposit requirements for note issue
# 100% Marginal Reserve requirements for note issue, such as the [[Bank Charter Act 1844]] (UK)
# Sanction on bank defaults and protection from creditors for many months or even years, and
# Central bank support for distressed banks, and government guarantee funds for notes and deposits, both to counteract bank runs and to protect bank creditors.

===Reserve requirements===
The currently prevailing view of [[reserve requirement]]s is that they are intended to prevent banks from:
# generating too much money by making too many loans against the narrow money deposit base;
# having a shortage of cash when large deposits are withdrawn (although the reserve is thought to be a legal minimum, it is understood that in a crisis or [[bank run]], reserves may be made available on a temporary basis).

In practice, some central banks do not require reserves to be held, and in some countries that do, such as the USA and the EU they are not required to be held during the day when the banks are lending, and banks can borrow from other banks at near the central bank policy rate to ensure they have the necessary amount of required reserves by the close of business. ''Required'' reserves are therefore considered by some central bankers, monetary economists and textbooks to only play a very small role in limiting money creation in these countries. Most commentators agree however, that they help the banks have sufficient supplies of highly liquid assets, so that the system operates in an orderly fashion and maintains public confidence. The UK for example, which does not have required reserves, does have requirements that the banks keep a certain amount of cash, and in Australia while there are no reserve requirements, there ''are'' a variety of requirements to ensure the banks have a stabilising ratio of liquid assets, such as deposits held with local banks. Individual countries adhere to varying [[reserve requirement#Required reserves|required reserve ratios]] which have changed over time.

In addition to reserve requirements, there are other required [[financial ratio]]s that affect the amount of loans that a bank can fund. The [[Capital requirement|capital requirement ratio]] is perhaps the most important of these other required ratios. When there are [[Reserve requirement#Reserve ratios|no mandatory reserve requirements]], which are considered by some economists to restrict lending, the capital requirement ratio acts to prevent an infinite amount of bank lending.

===Liquidity and capital management for a bank===
{{main|Capital requirement|Market liquidity}}
To avoid defaulting on its obligations, the bank must maintain a minimal reserve ratio that it fixes in accordance with, notably, regulations and its liabilities. In practice this means that the bank sets a reserve ratio target and responds when the actual ratio falls below the target. Such response can be, for instance:
# Selling or redeeming other assets, or [[securitization]] of illiquid assets,
# Restricting investment in new loans,
# Borrowing funds (whether repayable on demand or at a fixed maturity),
# Issuing additional [[Capital requirement#Regulatory capital|capital instruments]], or
# Reducing [[dividends]].{{Citation needed|date=February 2011}}

Because different funding options have different costs, and differ in reliability, banks maintain a stock of low cost and reliable sources of liquidity such as:
# Demand deposits with other banks
# High quality marketable debt securities
# Committed lines of credit with other banks{{Citation needed|date=February 2011}}

As with reserves, other sources of liquidity are managed with targets.

The ability of the bank to borrow money reliably and economically is crucial, which is why confidence in the bank's creditworthiness is important to its liquidity. This means that the bank needs to maintain adequate capitalisation and to effectively control its exposures to risk in order to continue its operations. If creditors doubt the bank's assets are worth more than its liabilities, all demand creditors have an incentive to demand payment immediately, causing a bank run to occur.{{Citation needed|date=February 2011}}

Contemporary bank management methods for liquidity are based on maturity analysis of all the bank's assets and liabilities (off balance sheet exposures may also be included). Assets and liabilities are put into residual contractual maturity buckets such as 'on demand', 'less than 1 month', '2–3 months' etc. These residual contractual maturities may be adjusted to account for expected counter party behaviour such as early loan repayments due to borrowers refinancing and expected renewals of term deposits to give forecast cash flows. This analysis highlights any large future net outflows of cash and enables the bank to respond before they occur. Scenario analysis may also be conducted, depicting scenarios including stress scenarios such as a bank-specific crisis.{{Citation needed|date=February 2011}}

==Hypothetical example of a bank balance sheet and financial ratios==
An example of fractional-reserve banking, and the calculation of the "reserve ratio" is shown in the balance sheet below:
{|class="wikitable"
|-
! colspan=4 | Example 2: ANZ National Bank Limited Balance Sheet as at 30 September 2007{{Citation needed|date=January 2008}}
|-
|'''Assets'''
| align = "center" | '''NZ$m'''
|'''Liabilities'''
| align = "center" | '''NZ$m'''
|-
|Cash
| align = "right" | 201
|Demand deposits
| align = "right" | 25,482
|-
|Balance with Central Bank
| align = "right" | 2,809
|Term deposits and other borrowings
| align = "right" | 35,231
|-
|Other liquid assets
| align = "right" | 1,797
|Due to other financial institutions
| align = "right" | 3,170
|-
|Due from other financial institutions
| align = "right" | 3,563
|Derivative financial instruments
| align = "right" | 4,924
|-
|Trading securities
| align = "right" | 1,887
|Payables and other liabilities
| align = "right" | 1,351
|-
|Derivative financial instruments
| align = "right" | 4,771
|Provisions
| align = "right" | 165
|-
|Available for sale assets
| align = "right" | 48
|Bonds and notes
| align = "right" | 14,607
|-
|Net loans and advances
| align = "right" | 87,878
|Related party funding
| align = "right" | 2,775
|-
|Shares in controlled entities
| align = "right" | 206
|[subordinated] Loan capital
| align = "right" | 2,062
|-
|Current tax assets
| align = "right" | 112
|'''Total Liabilities'''
| align = "right" | '''99,084'''
|-
|Other assets
| align = "right" | 1,045
|Share capital
| align = "right" | 5,943
|-
|Deferred tax assets
| align = "right" | 11
|[revaluation] Reserves
| align = "right" | 83
|-
|Premises and equipment
| align = "right" | 232
|Retained profits
| align = "right" | 2,667
|-
|Goodwill and other intangibles
| align = "right" | 3,297
|'''Total Equity'''
| align = "right" | '''8,703'''
|-
|'''Total Assets'''
| align = "right" | '''107,787'''
|'''Total Liabilities plus Net Worth'''
| align = "right" | '''107,787'''
|}

In this example the cash reserves held by the bank is NZ$3,010m (NZ$201m Cash + NZ$2,809m Balance at Central Bank) and the Demand Deposits (liabilities) of the bank are NZ$25,482m, for a cash reserve ratio of 11.81%.

=== Other financial ratios ===
The key [[financial ratio]] used to analyze fractional-reserve banks is the [[cash reserve ratio]], which is the ratio of cash reserves to demand deposits. However, other important financial ratios are also used to analyze the bank's liquidity, financial strength, profitability etc.

For example the ANZ National Bank Limited balance sheet above gives the following financial ratios:
# The cash reserve ratio is $3,010m/$25,482m, i.e. 11.81%.
# The liquid assets reserve ratio is ($201m+$2,809m+$1,797m)/$25,482m, i.e. 18.86%.
# The equity capital ratio is $8,703m/107,787m, i.e. 8.07%.
# The tangible equity ratio is ($8,703m-$3,297m)/107,787m, i.e. 5.02%
# The total capital ratio is ($8,703m+$2,062m)/$107,787m, i.e. 9.99%.

It is very important how the term 'reserves' is defined for calculating the reserve ratio, as different definitions give different results. Other important financial ratios may require analysis of disclosures in other parts of the bank's financial statements. In particular, for [[liquidity risk]], disclosures are incorporated into a note to the financial statements that provides maturity analysis of the bank's assets and liabilities and an explanation of how the bank manages its liquidity.

=== How the example bank manages its liquidity ===
{{see also|Duration gap}}
The ANZ National Bank Limited explains its methods as:{{Citation needed|date=January 2008}}
{{quote|Liquidity risk is the risk that the Banking Group will encounter difficulties in meeting commitments associated with its financial liabilities, e.g. overnight deposits, current accounts, and maturing deposits; and future commitments e.g. loan draw-downs and guarantees. The Banking Group manages its exposure to liquidity risk by maintaining sufficient liquid funds to meet its commitments based on historical and forecast cash flow requirements.}}

{{quote|The following maturity analysis of assets and liabilities has been prepared on the basis of the remaining period to contractual maturity as at the balance date. The majority of longer term loans and advances are housing loans, which are likely to be repaid earlier than their contractual terms. Deposits include substantial customer deposits that are repayable on demand. However, historical experience has shown such balances provide a stable source of long term funding for the Banking Group. When managing liquidity risks, the Banking Group adjusts this contractual profile for expected customer behaviour.}}

{|class="wikitable"
|-
! colspan=7 | Example 2: ANZ National Bank Limited Maturity Analysis of Assets and Liabilities as at 30 September 2007{{Citation needed|date=January 2008}}
|-
|
| align = "center" | '''Total Carrying Value'''
| align = "center" | '''< 3 Months'''
| align = "center" | '''3–12 Months'''
| align = "center" | '''1–5 Years'''
| align = "center" | '''5+ Years'''
| align = "center" | '''No Specified Maturity'''
|-
|'''Assets'''
|
|
|
|
|
|
|-
|Liquid assets
| align = "right" | 4,807
| align = "right" | 4,807
|
|
|
|
|-
|Due from other financial institutions
| align = "right" | 3,563
| align = "right" | 2,650
| align = "right" | 440
| align = "right" | 187
| align = "right" | 286
|
|-
|Derivative financial instruments
| align = "right" | 4,711
|
|
|
|
| align = "right" | 4,711
|-
|Assets available for sale
| align = "right" | 48
| align = "right" | 33
| align = "right" | 1
| align = "right" | 13
|
| align = "right" | 1
|-
|Net loans and advances
| align = "right" | 87,878
| align = "right" | 9,276
| align = "right" | 9,906
| align = "right" | 24,142
| align = "right" | 44,905
|
|-
|Other assets
| align = "right" | 4,903
| align = "right" | 970
| align = "right" | 179
|
|
| align = "right" | 3,754
|-
|'''Total Assets'''
| align = "right" | '''107,787'''
| align = "right" | '''18,394'''
| align = "right" | '''10,922'''
| align = "right" | '''25,013'''
| align = "right" | '''45,343'''
| align = "right" | '''8,115'''
|-
|'''Liabilities'''
|
|
|
|
|
|
|-
|Due to other financial institutions
| align = "right" | 3,170
| align = "right" | 2,356
| align = "right" | 405
| align = "right" | 32
| align = "right" | 377
|
|-
|Deposits and other borrowings
| align = "right" | 70,030
| align = "right" | 53,059
| align = "right" | 14,726
| align = "right" | 2,245
|
|
|-
|Derivative financial instruments
| align = "right" | 4,932
|
|
|
|
| align = "right" | 4,932
|-
|Other liabilities
| align = "right" | 1,516
| align = "right" | 1,315
| align = "right" | 96
| align = "right" | 32
| align = "right" | 60
| align = "right" | 13
|-
|Bonds and notes
| align = "right" | 14,607
| align = "right" | 672
| align = "right" | 4,341
| align = "right" | 9,594
|
|
|-
|Related party funding
| align = "right" | 2,275
| align = "right" | 2,275
|
|
|
|
|-
|Loan capital
| align = "right" | 2,062
|
| align = "right" | 100
| align = "right" | 1,653
| align = "right" | 309
|
|-
|'''Total Liabilities'''
| align = "right" | '''99,084'''
| align = "right" | '''60,177'''
| align = "right" | '''19,668'''
| align = "right" | '''13,556'''
| align = "right" | '''746'''
| align = "right" | '''4,937'''
|-
|Net liquidity gap (Total Assets less Total Liabilities)
| align = "right" | 8,703
| align = "right" | (41,783)
| align = "right" | (8,746)
| align = "right" | 11,457
| align = "right" | 44,597
| align = "right" | 3,178
|-
|'''Net Liquidity Gap – Cumulative'''
| align = "right" | '''8,703'''
| align = "right" | '''(41,783)'''
| align = "right" | '''(50,529)'''
| align = "right" | '''(39,072)'''
| align = "right" | '''5,525'''
| align = "right" | '''8,703'''
|}

==Criticisms of standard textbook descriptions of fractional reserve banking==

Some economists, including a former governor of the Bank of England, dispute the standard textbook descriptions of fractional-reserve banking.<ref>{{cite web|last=Sheard|first=Paul|title=Repeat After Me: Banks Cannot And Do Not "Lend Out" Reserves|url=http://www.standardandpoors.com/spf/upload/Ratings_US/Repeat_After_Me_8_14_13.pdf|publisher=Standard and Poor's}}</ref><ref>{{cite web|last=King|first=Mervyn|title=The transmission mechanism of monetary policy|url=http://wenku.baidu.com/view/12767661783e0912a2162a61|publisher=Bank of England}}</ref><ref>{{cite web|last=Goodhart|first=Charles|title=Money, credit and bank behaviour: need for a new approach|url=http://www.thefreelibrary.com/Money,+credit+and+bank+behaviour%3A+need+for+a+new+approach.-a0250677146}}</ref><ref>{{cite web|last=Carpenter|first=Seth|title=Money, Reserves, and the Transmission of Monetary Policy: Does the Money Multiplier Exist?|url=http://www.federalreserve.gov/pubs/feds/2010/201041/201041pap.pdf}}</ref><ref>{{cite web|last=Tobin|first=James|title=Commercial Banks as Creators of "Money"|url=http://cowles.econ.yale.edu/P/cm/m21/m21-01.pdf}}</ref> For example, Lord [[Adair Turner]], formally the UK's chief financial regulator, said "Banks do not, as too many textbooks still suggest, take deposits of existing money from savers and lend it out to borrowers: they create credit and money [[ex nihilo]] – extending a loan to the borrower and simultaneously crediting the borrower’s money account".<ref>{{cite web|last=Turner|first=Adair|title=Credit Money and Leverage, what Wicksell, Hayek and Fisher knew and modern macroeconomics forgot|url=http://ineteconomics.org/sites/inet.civicactions.net/files/Adair%20Turner%20Stockholm%20School%20of%20Economics%20September%2012.pdf}}</ref>

==Criticisms of fractional reserve banking==
{{Expand section|date=October 2013}}

In 1935, economist [[Irving Fisher]] proposed a system of 100% reserve banking as a means of reversing the deflation of the [[Great depression]]. He wrote: "100 per cent banking [...] would give the Federal Reserve absolute control over the money supply. Recall that under the present fractional reserve system of
depository institutions, the money supply is determined in the short run by such non-policy variables as the currency/deposit ratio of the public and the excess
reserve ratio of depository institutions."<ref>{{Cite book
| title = 100% Money
| publisher = Pickering & Chatto Ltd
| first = Irving
| last = Fisher
|authorlink= Irving Fisher
| isbn = 978-1-85196-236-5
| year = 1997}}</ref>{{page?|date=October 2013}}

[[Milton Friedman]] said "Our present fractional reserve banking system has two major defects. First, it involves extensive governmental intervention into lending and investing activities that should preferably be left to the free market. Second, decisions by holders of money about the form in which they want to hold money and by banks about the structure of their assets tend to affect the amount available to lend. This has often been referred to as the 'inherent instability' of a fractional reserve system". In a book he wrote in which he proposed that fractional reserve banking should be abolished and replaced with [[full reserve banking]]. <ref>{{cite book|last=Friedman|first=Milton|title=A Program For Monetary Stability|year=1992|page=66}}</ref> In his review in the ''[[American Economic Review]]'', economist [[Lawrence Ritter]] wrote that Friedman's argument is unconvincing and his evidence unsatisfactory.<ref>Lawrence S. Ritter ''The American Economic Review'' Vol. 50, No. 4 (September, 1960) pp. 765-768</ref>

==See also==
* [[Endogenous money]]
* [[Basel II]]
* [[Basel III]]
* [[Asset liability management]]

==References==
{{reflist|30em}}

==Further reading==
* Crick, W.F. (1927), The genesis of bank deposits, ''Economica'', vol 7, 1927, pp 191–202.
* [[Milton Friedman|Friedman, Milton]] (1960), ''A Program for Monetary Stability'', New York, [[Fordham University Press]].
* Meigs, A.J. (1962), ''Free reserves and the money supply'', Chicago, University of Chicago, 1962.
* {{cite book |last=Paul |first=Ron |title=[[End the Fed]] |year=2009 |publisher=Grand Central Publishing |location=New York |isbn=978-0-446-54919-6 |url=http://mises.org/daily/3687 |authorlink=Ron Paul |chapter=2 The Origin and Nature of the Fed}}
* Philips, C.A. (1921), ''Bank Credit'', New York, Macmillan, chapters 1–4, 1921,
* Thomson, P. (1956), Variations on a theme by Philips, ''[[American Economic Review]]'' vol 46, December 1956, pp. 965–970.

==External links==
* [http://www.ecfr.gov/cgi-bin/retrieveECFR?gp=&SID=fdc5c4af6a2bf9cdef6edbf498a643a7&n=12y2.0.1.1.5&r=PART&ty=HTML#12:2.0.1.1.5.0.2.6 Regulation D of the Federal Reserve Board of the U.S.]
* [http://www.novapoly.com/articles/finance/fractional-reserve-banking-model/ Interactive Fractional-Reserve Calculator] Calculator that details deposit multiplication for any reserve requirement.
* [http://www.federalreserveeducation.org/fed101_html/policy/frtoday_depositCreation.pdf Federalreserveeducation.org – The Principle of Multiple Deposit Creation]{{broken citation|date=January 2013}}
* [http://www.newyorkfed.org/aboutthefed/fedpoint/fed45.html Reserve Requirements – Fedpoints – Federal Reserve Bank of New York]{{broken citation|date=January 2013}}
* [http://www.bis.org/publ/cpss55.pdf Bank for International Settlements – The Role of Central Bank Money in Payment Systems]

{{Use dmy dates|date=June 2011}}

[[Category:Banking]]
[[Category:Central banks]]
[[Category:Criticisms of economics]]
[[Category:Heterodox economics]]
[[Category:Monetary economics]]
[[Category:Monetary policy]]
[[Category:Monetary reform]]
[[Category:Systemic risk]]

[[de:Mindestreserve]]

Doubly periodic function

2014-01-13T10:35:40Z

50.131.197.174:

A '''Van Hove singularity''' is a singularity (non-smooth point) in the [[density of states]] (DOS) of a crystalline [[solid]]. The [[wavevector]]s at which Van Hove singularities occur are often referred to as [[Critical point (mathematics)|critical points]] of the [[Brillouin zone]]. (The [[Critical point (physics)|critical point]] found in [[phase diagram]]s is a completely separate phenomenon.) For three-dimensional crystals, they take the form of kinks (where the density of states is not [[differentiable]]). The most common application of the Van Hove singularity concept comes in the analysis of [[optical absorption]] spectra. The occurrence of such singularities was first analyzed by the [[Belgium|Belgian]] physicist [[Léon Van Hove]] in 1953 for the case of [[phonon]] densities of states.<ref>L. Van Hove, [http://dx.doi.org/10.1103/PhysRev.89.1189 "The Occurrence of Singularities in the Elastic Frequency Distribution of a Crystal,"] Phys. Rev. 89, 1189–1193 (1953).</ref>

== Theory ==
Consider a one-dimensional lattice of ''N'' particles, with each particle separated by distance ''a'', for a total length of L = ''Na''. A standing wave in this lattice will have a [[wave number]] ''k'' of the form

:<math>k=\frac{2\pi}{\lambda}=n\frac{2\pi}{L}</math>

where <math>\lambda</math> is wavelength, and ''n'' is an integer. (Positive integers will denote forward waves, negative integers will denote reverse waves.) The smallest wavelength possible is ''2a'' which corresponds to the largest possible wave number <math>k_{max}=\pi/a</math> and which also corresponds to the maximum possible |n|: <math>n_{max}=L/2a</math>. We may define the density of states ''g(k)dk'' as the number of standing waves with wave vector ''k'' to ''k+dk'':<ref>*M. A. Parker(1997-2004)[http://www.ece.rutgers.edu/~maparker/classes/582-Chapters/Ch07-Sol-State-Carriers/Ch07S16DensityStates.pdf "Introduction to Density of States" ''Marcel-Dekker Publishing''] p.7.</ref>

:<math>g(k)dk = dn =\frac{L}{2\pi}\,dk</math>

Extending the analysis to [[wavevector]]s in three dimensions the density of states in a [[particle in a box|box]] will be

:<math>g(\vec{k})d^3k = d^3n =\frac{L^3}{(2\pi)^3}\,d^3k</math>

where <math>d^3k</math> is a volume element in ''k''-space, and which, for electrons, will need to be multiplied by a factor of 2 to account for the two possible [[Spin (physics)|spin]] orientations. By the [[chain rule]], the DOS in energy space can be expressed as

:<math>dE =
\frac{\partial E}{\partial k_x}dk_x +
\frac{\partial E}{\partial k_y}dk_y +
\frac{\partial E}{\partial k_z}dk_z =
\vec{\nabla}E \cdot d\vec{k}</math>

where <math>\vec{\nabla}</math> is the gradient in k-space.

The set of points in ''k''-space which correspond to a particular energy ''E'' form a surface in ''k''-space, and the gradient of ''E'' will be a vector perpendicular to this surface at every point.<ref>*{{cite book
| first = John | last = Ziman | authorlink = John Ziman | year = 1972
| title = Principles of the Theory of Solids | publisher = Cambridge University Press
| id = ISBN B0000EG9UB }}</ref> The density of states as a function of this energy ''E'' is:

:<math>g(E)dE = \iint_{\partial E}g(\vec{k})\,d^3k = \frac{L^3}{(2\pi)^3}\iint_{\partial E}dk_x\,dk_y\,dk_z</math>

where the integral is over the surface <math>\partial E</math> of constant ''E''. We can choose a new coordinate system <math>k'_x,k'_y,k'_z\,</math> such that <math>k'_z\,</math> is perpendicular to the surface and therefore parallel to the gradient of ''E''. If the coordinate system is just a rotation of the original coordinate system, then the volume element in k-prime space will be

:<math>dk'_x\,dk'_y\,dk'_z = dk_x\,dk_y\,dk_z</math>

We can then write ''dE'' as:

:<math>dE=|\vec{\nabla}E|\,dk'_z</math>

and, substituting into the expression for ''g(E)'' we have:

:<math>g(E)=\frac{L^3}{(2\pi)^3}\iint\frac{dk'_x\,dk'_y}{|\vec{\nabla}E|}</math>

where the <math>dk'_x\,dk'_y</math> term is an area element on the constant-''E'' surface. The clear implication of the equation for <math>g(E)</math> is that at the <math>k</math>-points where the [[dispersion relation]] <math>E(\vec{k})</math> has an extremum, the integrand in the DOS expression diverges. The Van Hove singularities are the features that occur in the DOS function at these <math>k</math>-points.

A detailed analysis<ref>*{{cite book | last=Bassani | first=F. | coauthors = Pastori Parravicini, G. | title=Electronic States and Optical Transitions in Solids | publisher=Pergamon Press | year=1975 | isbn=0-08-016846-9}} This book contains an extensive discussion of the types of Van Hove singularities in different dimensions and illustrates the concepts with detailed theoretical-versus-experimental comparisons for [[germanium|Ge]] and [[graphite]].</ref> shows that there are four types of Van Hove singularities in three-dimensional space, depending on whether the band structure goes through a [[local maximum]], a [[local minimum]] or a [[saddle point]]. In three dimensions, the DOS itself is not divergent although its derivative is. The function g(E) tends to have square-root singularities (see the Figure) since for a spherical [[free electron]] [[Fermi surface]]

:<math>E = \hbar^2 k^2/2m</math> so that <math>|\vec{\nabla}E| = \hbar^2 k/m = \hbar \sqrt{ \frac{2E}{m}}</math>.

In two dimensions the DOS is logarithmically divergent at a saddle point and in one dimension the DOS itself is infinite where <math>\vec{\nabla}E</math> is zero.

[[Image:NewvanHove.png|thumb|right|A sketch of the DOS g(E) versus energy E for a simulated three-dimensional solid. The Van Hove singularities occur where dg(E)/dE diverges.]]

== Experimental observation ==
The optical absorption spectrum of a solid is most straightforwardly calculated from the [[electronic band structure]] using [[Fermi's Golden Rule]] where the relevant [[Perturbation theory (quantum mechanics)|matrix element]] to be evaluated is the [[dipole operator]] <math>\vec{A} \cdot \vec{p}</math> where <math>\vec{A}</math> is the [[vector potential]] and <math>\vec{p}</math> is the [[momentum]] operator. The density of states which appears in the Fermi's Golden Rule expression is then the '''joint density of states''', which is the number of electronic states in the conduction and valence bands that are separated by a given photon energy. The optical absorption is then essentially the product of the dipole operator matrix element (also known as the '''oscillator strength''') and the JDOS.

The divergences in the two- and one-dimensional DOS might be expected to be a mathematical formality, but in fact they are readily observable. Highly anisotropic solids like [[graphite]] (quasi-2D) and [[Bechgaard salt]]s (quasi-1D) show anomalies in spectroscopic measurements that are attributable to the Van Hove singularities. Van Hove singularities play a significant role in understanding [[optical properties of carbon nanotubes|optical intensities in single-walled nanotubes]] (SWNTs) which are also quasi-1D systems. The Dirac point in [[graphene]] is a Van-Hove singularity that can be seen directly as a peak in electrical resistance, when the graphene is charge-neutral.

== Notes ==
<references/>

[[Category:Condensed matter physics]]

Magnetic helicity

2014-01-09T01:03:08Z

50.131.141.62: /* External links */ replaced dead link to Mitch Berger publications with live one

{{merge from|Glutamate dehydrogenase 1|date=October 2011}}
{{enzyme
| Name = glutamate dehydrogenase (GLDH)
| EC_number = 1.4.1.2
| CAS_number = 9001-46-1
| IUBMB_EC_number = 1/4/1/2
| GO_code = 0004352
| image =
| width =
| caption =
}}
{{enzyme
| Name = glutamate dehydrogenase [NAD(P)+]
| EC_number = 1.4.1.3
| CAS_number = 9029-12-3
| IUBMB_EC_number = 1/4/1/3
| GO_code = 0004353
| image =
| width =
| caption =
}}
{{enzyme
| Name = glutamate dehydrogenase (NADP+)
| EC_number = 1.4.1.4
| CAS_number = 9029-11-2
| IUBMB_EC_number = 1/4/1/4
| GO_code = 0004354
| image =
| width =
| caption =
}}

'''Glutamate dehydrogenase''' (GLDH) is an [[enzyme]], present in most microbes and the [[mitochondria]] of [[eukaryotes]], as are some of the other enzymes required for [[urea]] synthesis, that converts [[glutamate]] to [[alpha-Ketoglutaric acid|α-ketoglutarate]], and vice versa. In animals, the produced ammonia is usually used as a substrate in the [[urea cycle]]. Typically, the α-ketoglutarate to glutamate reaction does not occur in mammals, as glutamate dehydrogenase equilibrium favours the production of ammonia and α-ketoglutarate. Glutamate dehydrogenase also has a very low affinity for ammonia (high [[Michaelis constant]] <math>K_m</math> of about 1 mM), and therefore toxic levels of ammonia would have to be present in the body for the reverse reaction to proceed (that is, α-ketoglutarate and ammonia to glutamate and NAD(P)+). In bacteria, the ammonia is assimilated to amino acids via glutamate and aminotransferases.<ref name="Lightfoot_1988">{{cite journal | author = Lightfoot DA, Baron AJ, Wootton JC | year = 1988 | title = Expression of the Escherichia coli glutamate dehydrogenase gene in the cyanobacterium Synechococcus PCC6301 causes ammonium tolerance | journal = Plant Molecular Biology | volume = 11 | issue = 3 | pages = 335–344 | doi = 10.1007/BF00027390 }}</ref> In plants, the enzyme can work in either direction depending on environment and stress.<ref name="pmid16046826">{{cite journal | author = Mungur R, Glass AD, Goodenow DB, Lightfoot DA | title = Metabolite Fingerprinting in Transgenic Nicotiana tabacum Altered by the Escherichia coli Glutamate Dehydrogenase Gene | journal = J. Biomed. Biotechnol. | volume = 2005 | issue = 2 | pages = 198–214 |date=June 2005 | pmid = 16046826 | pmc = 1184043 | doi = 10.1155/JBB.2005.198 | url = }}</ref><ref = <ref name = "Grabowska_2011"/> Transgenic plants expressing microbial GLDHs are improved in tolerance to herbicide, water deficit, and pathogen infections.<ref name="Lightfoot_2007">{{cite journal | author =Lightfoot DA, Bernhardt K, Mungur R, Nolte S, Ameziane R, Colter A, Jones K, Iqbal MJ, Varsa E, Young B | year = 2007 | title = Improved drought tolerance of transgenic Zea mays plants that express the glutamate dehydrogenase gene (gdhA) of E. coli | journal=Euphytica | volume=156 | issue = 1–2 | pages = 103–116 | doi = 10.1007/s10681-007-9357-y}}</ref> They are more nutritionally valuable.<ref name="isbn0-8138-1502-9">{{cite book | editor = Wood, Andrew; Matthew A. Jenks | authorlink = | author = Lightfoot DA | chapter = Genes for use in improving nitrogen use efficiency in crops | title = Genes for Plant Abiotic Stress | edition = | language = | publisher = Wiley-Blackwell | location = | year = 2009 | origyear = | pages = 167–182 | quote = | isbn = 0-8138-1502-9 | oclc = | doi = | url = | accessdate = }}</ref>

<gallery>
Image:Glutaminsäure - Glutamic acid.svg|[[Glutamate]]
Image:Alpha-ketoglutaric acid.png |[[alpha-Ketoglutaric acid|α-Ketoglutarate]]
</gallery>

The enzyme represents a key link between [[catabolic]] and [[metabolic pathways]], and is, therefore, ubiquitous in eukaryotes.

==Clinical application==

GLDH can be measured in a [[medical laboratory]] to evaluate the liver function. Elevated [[blood serum]] GLDH levels indicate liver damage and GLDH plays an important role in the differential diagnosis of liver disease, especially in combination with [[aminotransferases]]. GLDH is localised in [[mitochondria]], therefore practically none is liberated in generalised inflammatory diseases of the liver such as viral hepatitides. Liver diseases in which necrosis of hepatocytes is the predominant event, such as toxic liver damage or hypoxic liver disease, are characterised by high serum GLDH levels. GLDH is important for distinguishing between acute viral hepatitis and acute toxic liver necrosis or acute hypoxic liver disease, particularly in the case of liver damage with very high aminotransferases. In [[clinical trials]], GLDH can serve as a measurement for the safety of a drug.

==Cofactors==
[[Nicotinamide adenine dinucleotide|NAD]]+(or [[NADP]]+) is a [[Cofactor (biochemistry)|cofactor]] for the glutamate dehydrogenase reaction, producing α-ketoglutarate and [[ammonium]] as a byproduct.<ref name = "Grabowska_2011">{{cite journal | author = Grabowska A, Nowicki M, Kwinta J | title = Glutamate dehydrogenase of the germinating triticale seeds: gene expression, activity distribution and kinetic characteristics | journal = Acta Phys. Plant. | volume = 33 | doi = 10.1007/s11738-011-0801-1 | url = http://www.springerlink.com/content/m05q717303575376/ | year = 2011 | issue = 5 | pages = 1981–90 }}</ref>

Based on which cofactor is used, glutamate dehydrogenase enzymes are divided into the following three classes:
* EC 1.4.1.2: L-glutamate + H2O + NAD+ <math>\rightleftharpoons</math> 2-oxoglutarate + NH3 + NADH + H+
* EC 1.4.1.3: L-glutamate + H2O + NAD(P)+ <math>\rightleftharpoons</math> 2-oxoglutarate + NH3 + NAD(P)H + H+
* EC 1.4.1.4: L-glutamate + H2O + NADP+ <math>\rightleftharpoons</math> 2-oxoglutarate + NH3 + NADPH + H+

==Role in flow of nitrogen==
Ammonia incorporation in animals and microbes occurs through the actions of glutamate dehydrogenase and [[glutamine synthetase]]. Glutamate plays the central role in [[mammalian]] and microbe nitrogen flow, serving as both a nitrogen donor and a nitrogen acceptor.

б==Regulation of glutamate dehydrogenase==
In humans, the activity of glutamate dehydrogenase is controlled through [[ADP-ribose|ADP-ribosylation]], a covalent modification carried out by the gene [[sirt4]]. This regulation is relaxed in response to [[caloric restriction]] and low [[blood glucose]]. Under these circumstances, glutamate dehydrogenase activity is raised in order to increase the amount of α-ketoglutarate produced, which can be used to provide energy by being used in the [[citric acid cycle]] to ultimately produce [[adenosine triphosphate|ATP]].

In microbes, the activity is controlled by the concentration of ammonium and or the like-sized rubidium ion, which binds to an allosteric site on GDH and changes the Km ([[Michaelis constant]]) of the enzyme.<ref name="pmid6221721">{{cite journal | author = Wootton JC | title = Re-assessment of ammonium-ion affinities of NADP-specific glutamate dehydrogenases. Activation of the Neurospora crassa enzyme by ammonium and rubidium ions | journal = Biochem. J. | volume = 209 | issue = 2 | pages = 527–31 |date=February 1983 | pmid = 6221721 | pmc = 1154121 | doi = }}</ref>

The control of GDH through ADP-ribosylation is particularly important in [[insulin]]-producing [[β cells]]. Beta cells secrete insulin in response to an increase in the ATP:[[adenosine diphosphate|ADP]] ratio, and, as amino acids are broken down by GDH into α-ketoglutarate, this ratio rises and more insulin is secreted. SIRT4 is necessary to regulate the metabolism of amino acids as a method of controlling insulin secretion and regulating blood [[glucose]] levels.

Mutations alter the allosteric binding site of GTP cause permanent activation of glutamate dehydrogenase lead to disorder known as hyperinsulinism-hyperammonemia.

==Regulation==
[[Allosteric regulation]]:

This protein may use the [[morpheein]] model of [[allosteric regulation]].<ref name=pmid22182754>{{cite journal | author = T. Selwood and E. K. Jaffe. | title = Dynamic dissociating homo-oligomers and the control of protein function. | journal = Arch. Biochem. Biophys. | volume = 519| issue = 2| pages = 131–43| year = 2011 | pmid = 22182754 | url = http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Citation&list_uids=22182754 | doi=10.1016/j.abb.2011.11.020 | pmc=3298769}}</ref>

Allosteric inhibitors:
*[[Adenosine triphosphate]] (ATP)
*[[Guanosine triphosphate]] (GTP)
Activators:
*[[Adenosine diphosphate]] (ADP)
*[[Guanosine diphosphate]] (GDP)

== Isozymes ==

Humans express the following glutamate dehydrogenase [[isozyme]]s:

{|
|{{infobox protein
|Name= [[GLUD1|glutamate dehydrogenase 1]]
|caption=
|image=
|width=
|HGNCid=4335
|Symbol=[[GLUD1]]
|AltSymbols=GLUD
|EntrezGene=2746
|OMIM=138130
|RefSeq=NM_005271
|UniProt=P00367
|PDB=
|ECnumber=1.4.1.3
|Chromosome=10
|Arm=q
|Band=21.1
|LocusSupplementaryData=-24.3
}}
|{{infobox protein
|Name=[[GLUD2|glutamate dehydrogenase 2]]
|caption=
|image=
|width=
|HGNCid=4336
|Symbol=[[GLUD2]]
|AltSymbols=GLUDP1
|EntrezGene=2747
|OMIM=300144
|RefSeq=NM_012084
|UniProt=P49448
|PDB=
|ECnumber=1.4.1.3
|Chromosome=X
|Arm=q2
|Band=5
|LocusSupplementaryData=
}}
|}

==See also==
* [[Anaplerotic reactions]]

== References ==
{{Reflist}}

[http://users.comcen.com.au/~journals/glutamateabs2012.htm Amino acid sequence analysis of glutamate dehydrogenase from different source organisms]==External links==
* {{MeshName|Glutamate+dehydrogenase}}

{{CH-NH2 oxidoreductases}}
{{Mitochondrial enzymes}}
{{Citric acid cycle enzymes}}
{{Amino acid metabolism enzymes}}

[[Category:EC 1.4.1]]

Ice-sheet dynamics

2013-12-13T05:38:10Z

50.131.42.209: /* Flow dynamics */

[[Rodney Hill]] has developed several yield criteria for anisotropic plastic deformations. The earliest version was a straightforward extension of the [[von Mises yield criterion]] and had a quadratic form. This model was later generalized by allowing for an exponent ''m''. Variations of these criteria are in wide use for metals, polymers, and certain composites.

== Quadratic Hill yield criterion ==
The quadratic Hill yield criterion<ref>R. Hill. (1948). ''A theory of the yielding and plastic flow of anisotropic metals.'' Proc. Roy. Soc. London, 193:281–297</ref> has the form
:<math>
F(\sigma_{22}-\sigma_{33})^2 + G(\sigma_{33}-\sigma_{11})^2 + H(\sigma_{11}-\sigma_{22})^2 + 2L\sigma_{23}^2 + 2M\sigma_{31}^2 + 2N\sigma_{12}^2 = 1 ~.
</math>
Here ''F, G, H, L, M, N'' are constants that have to be determined experimentally and <math>\sigma_{ij}</math> are the stresses. The quadratic Hill yield criterion depends only on the deviatoric stresses and is pressure independent. It predicts the same yield stress in tension and in compression.

=== Expressions for F, G, H, L, M, N ===
If the axes of material anisotropy are assumed to be orthogonal, we can write
:<math>
(G + H)~(\sigma_1^y)^2 = 1 ~;~~ (F + H)~(\sigma_2^y)^2 = 1 ~;~~ (F + G)~(\sigma_3^y)^2 = 1
</math>
where <math>\sigma_1^y, \sigma_2^y, \sigma_3^y</math> are the normal yield stresses with respect to the axes of anisotropy. Therefore we have
:<math>
F = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_2^y)^2} + \cfrac{1}{(\sigma_3^y)^2} - \cfrac{1}{(\sigma_1^y)^2}\right]
</math>
:<math>
G = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_3^y)^2} + \cfrac{1}{(\sigma_1^y)^2} - \cfrac{1}{(\sigma_2^y)^2}\right]
</math>
:<math>
H = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_1^y)^2} + \cfrac{1}{(\sigma_2^y)^2} - \cfrac{1}{(\sigma_3^y)^2}\right]
</math>
Similarly, if <math>\tau_{12}^y, \tau_{23}^y, \tau_{31}^y</math> are the yield stresses in shear (with respect to the axes of anisotropy), we have
:<math>
L = \cfrac{1}{2~(\tau_{23}^y)^2} ~;~~ M = \cfrac{1}{2~(\tau_{31}^y)^2} ~;~~ N = \cfrac{1}{2~(\tau_{12}^y)^2}
</math>

=== Quadratic Hill yield criterion for plane stress ===
The quadratic Hill yield criterion for thin rolled plates (plane stress conditions) can be expressed as
:<math>
\sigma_1^2 + \cfrac{R_0~(1+R_{90})}{R_{90}~(1+R_0)}~\sigma_2^2 - \cfrac{2~R_0}{1+R_0}~\sigma_1\sigma_2 = (\sigma_1^y)^2
</math>
where the principal stresses <math>\sigma_1, \sigma_2</math> are assumed to be aligned with the axes of anisotropy with <math>\sigma_1</math> in the rolling direction and <math>\sigma_2</math> perpendicular to the rolling direction, <math>\sigma_3 = 0 </math>, <math>R_0</math> is the [[Lankford coefficient|R-value]] in the rolling direction, and <math>R_{90}</math> is the [[Lankford coefficient|R-value]] perpendicular to the rolling direction.

For the special case of transverse isotropy we have <math>R=R_0 = R_{90}</math> and we get
:<math>
\sigma_1^2 + \sigma_2^2 - \cfrac{2~R}{1+R}~\sigma_1\sigma_2 = (\sigma_1^y)^2
</math>

:{| class="toccolours jy
collapsible collapsed "width="80%" style="text-align:left"
!Derivation of Hill's criterion for plane stress
|-
| For the situation where the principal stresses are aligned with the directions of anisotropy we have
:<math>
f := F(\sigma_2-\sigma_3)^2 + G(\sigma_3-\sigma_1)^2 + H(\sigma_1-\sigma_2)^2 - 1 = 0 \,
</math>
where <math>\sigma_1, \sigma_2, \sigma_3</math> are the principal stresses. If we assume an associated flow rule we have
:<math>
\epsilon^p_i = \lambda~\cfrac{\partial f}{\partial \sigma_i} \qquad \implies \qquad
\cfrac{d\epsilon^p_i}{d\lambda} = \cfrac{\partial f}{\partial \sigma_i} ~.
</math>
This implies that
:<math>
\begin{align}
\cfrac{d\epsilon^p_1}{d\lambda} &= 2(G+H)\sigma_1 - 2H\sigma_2 - 2G\sigma_3 \\
\cfrac{d\epsilon^p_2}{d\lambda} &= 2(F+H)\sigma_2 - 2H\sigma_1 - 2F\sigma_3 \\
\cfrac{d\epsilon^p_3}{d\lambda} &= 2(F+G)\sigma_3 - 2G\sigma_1 - 2F\sigma_2 ~.
\end{align}
</math>
For plane stress <math>\sigma_3 = 0</math>, which gives
:<math>
\begin{align}
\cfrac{d\epsilon^p_1}{d\lambda} &= 2(G+H)\sigma_1 - 2H\sigma_2\\
\cfrac{d\epsilon^p_2}{d\lambda} &= 2(F+H)\sigma_2 - 2H\sigma_1\\
\cfrac{d\epsilon^p_3}{d\lambda} &= - 2G\sigma_1 - 2F\sigma_2 ~.
\end{align}
</math>
The [[Lankford coefficient|R-value]] <math>R_0</math> is defined as the ratio of the in-plane and out-of-plane plastic strains under uniaxial stress <math>\sigma_1</math>. The quantity <math>R_{90}</math> is the plastic strain ratio under uniaxial stress <math>\sigma_2</math>. Therefore, we have
:<math>
R_0 = \cfrac{d\epsilon^p_2}{d\epsilon^p_3} = \cfrac{H}{G} ~;~~
R_{90} = \cfrac{d\epsilon^p_1}{d\epsilon^p_3} = \cfrac{H}{F} ~.
</math>
Then, using <math>H=R_0 G</math> and <math>\sigma_3=0</math>, the yield condition can be written as
:<math>
f := F \sigma_2^2 + G \sigma_1^2 + R_0 G(\sigma_1-\sigma_2)^2 - 1 = 0 \,
</math>
which in turn may be expressed as
:<math>
\sigma_1^2 + \cfrac{F+R_0 G}{G(1+R_0)}~\sigma_2^2 - \cfrac{2R_0}{1+R_0}~\sigma_1\sigma_2 = \cfrac{1}{(1+R_0)G}~.
</math>
This is of the same form as the required expression. All we have to do is to express <math>F,G</math> in terms of <math>\sigma_1^y</math>. Recall that,
:<math>
\begin{align}
F & = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_2^y)^2} + \cfrac{1}{(\sigma_3^y)^2} - \cfrac{1}{(\sigma_1^y)^2}
\right] \\
G & = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_3^y)^2} + \cfrac{1}{(\sigma_1^y)^2} - \cfrac{1}{(\sigma_2^y)^2}
\right] \\
H & = \cfrac{1}{2}\left[\cfrac{1}{(\sigma_1^y)^2} + \cfrac{1}{(\sigma_2^y)^2} - \cfrac{1}{(\sigma_3^y)^2}
\right]
\end{align}
</math>
We can use these to obtain
:<math>
\begin{align}
R_0 = \cfrac{H}{G} & \implies
(1+R_0)\cfrac{1}{(\sigma_3^y)^2} - (1+R_0)\cfrac{1}{(\sigma_2^y)^2} = (1-R_0)\cfrac{1}{(\sigma_1^y)^2} \\
R_{90} = \cfrac{H}{F} & \implies
(1+R_{90})\cfrac{1}{(\sigma_3^y)^2} - (1-R_{90})\cfrac{1}{(\sigma_2^y)^2} = (1+R_{90})\cfrac{1}{(\sigma_1^y)^2}
\end{align}
</math>
Solving for <math>\cfrac{1}{(\sigma_3^y)^2}, \cfrac{1}{(\sigma_2^y)^2}</math> gives us
:<math>
\cfrac{1}{(\sigma_3^y)^2} = \cfrac{R_0+R_{90}}{(1+R_0)~R_{90}}~\cfrac{1}{(\sigma_1^y)^2} ~;~~
\cfrac{1}{(\sigma_2^y)^2} = \cfrac{R_0(1+R_{90})}{(1+R_0)~R_{90}}~\cfrac{1}{(\sigma_1^y)^2}
</math>
Plugging back into the expressions for <math>F,G</math> leads to
:<math>
F = \cfrac{R_0}{(1+R_0)~R_{90}}~\cfrac{1}{(\sigma_1^y)^2} ~;~~
G = \cfrac{1}{1+R_0}~\cfrac{1}{(\sigma_1^y)^2}
</math>
which implies that
:<math>
\cfrac{1}{G(1+R_0)} = (\sigma_1^y)^2 ~;~~ \cfrac{F+R_0 G}{G(1+R_0)} = \cfrac{R_0(1+R_{90})}{R_{90}(1+R_0)} ~.
</math>
Therefore the plane stress form of the quadratic Hill yield criterion can be expressed as
:<math>
\sigma_1^2 + \cfrac{R_0~(1+R_{90})}{R_{90}~(1+R_0)}~\sigma_2^2 - \cfrac{2~R_0}{1+R_0}~\sigma_1\sigma_2 = (\sigma_1^y)^2
</math>
|}

== Generalized Hill yield criterion ==
The generalized Hill yield criterion<ref>R. Hill. (1979). '' Theoretical plasticity of textured aggregates. '' Math. Proc. Camb. Phil. Soc., 85(1):179–191.</ref> has the form
:<math>
\begin{align}
F|\sigma_{2}-\sigma_{3}|^m & + G|\sigma_{3}-\sigma_{1}|^m + H|\sigma_{1}-\sigma_{2}|^m + L|2\sigma_1 - \sigma_2 - \sigma_3|^m \\
& + M|2\sigma_2 - \sigma_3 - \sigma_1|^m + N|2\sigma_3 - \sigma_1 - \sigma_2|^m = \sigma_y^m ~.
\end{align}
</math>
where <math>\sigma_i</math> are the principal stresses (which are aligned with the directions of anisotropy), <math>\sigma_y</math> is the yield stress, and ''F, G, H, L, M, N'' are constants. The value of ''m'' is determined by the degree of anisotropy of the material and must be greater than 1 to ensure convexity of the yield surface.

=== Generalized Hill yield criterion for plane stress ===
For transversely isotropic materials with <math>1-2</math> being the plane of symmetry, the generalized Hill yield criterion reduces to (with <math>F=G</math> and <math>L=M</math>)
:<math>
\begin{align}
f := & F|\sigma_2-\sigma_3|^m + F|\sigma_3-\sigma_1|^m + H|\sigma_1-\sigma_2|^m + L|2\sigma_1 - \sigma_2 - \sigma_3|^m \\
& + L|2\sigma_2-\sigma_3-\sigma_1|^m + N|2\sigma_3-\sigma_1-\sigma_2|^m - \sigma_y^m \le 0
\end{align}
</math>
The [[R-value (plasticity)|R-value]] or [[Lankford coefficient]] can be determined by considering the situation where <math>\sigma_1 > (\sigma_2 = \sigma_3 = 0)</math>. The R-value is then given by
:<math>
R = \cfrac{(2^{m-1}+2) L - N + H}{(2^{m-1} - 1) L + 2 N + F} ~.
</math>
Under [[plane stress]] conditions and with some assumptions, the generalized Hill criterion can take several forms.<ref>Chu, E. (1995). ''Generalization of Hill's 1979 anisotropic yield criteria''. Journal of Materials Processing Technology, vol. 50, pp. 207-215.</ref>
* '''Case 1:''' <math>L = 0, H = 0.</math>
:<math>
f:= \cfrac{1+2R}{1+R}(|\sigma_1|^m + |\sigma_2|^m) - \cfrac{R}{1+R} |\sigma_1 + \sigma_2|^m - \sigma_y^m \le 0
</math>
* '''Case 2:''' <math>N = 0, F = 0.</math>
:<math>
f:= \cfrac{2^{m-1}(1-R)+(R+2)}{(1-2^{m-1})(1+R)}|\sigma_1 -\sigma_2|^m - \cfrac{1}{(1-2^{m-1})(1+R)} (|2\sigma_1 - \sigma_2|^m + |2\sigma_2-\sigma_1|^m)- \sigma_y^m \le 0
</math>
* '''Case 3:''' <math>N = 0, H = 0.</math>
:<math>
f:= \cfrac{2^{m-1}(1-R)+(R+2)}{(2+2^{m-1})(1+R)}(|\sigma_1|^m -|\sigma_2|^m) + \cfrac{R}{(2+2^{m-1})(1+R)} (|2\sigma_1 - \sigma_2|^m + |2\sigma_2-\sigma_1|^m)- \sigma_y^m \le 0
</math>
* '''Case 4:''' <math>L = 0, F = 0.</math>
:<math>
f:= \cfrac{1+2R}{2(1+R)}|\sigma_1 - \sigma_2|^m + \cfrac{1}{2(1+R)} |\sigma_1 + \sigma_2|^m - \sigma_y^m \le 0
</math>
* '''Case 5:''' <math>L = 0, N = 0.</math>. This is the [[Hosford yield criterion]].
:<math>
f := \cfrac{1}{1+R}(|\sigma_1|^m + |\sigma_2|^m) + \cfrac{R}{1+R}|\sigma_1-\sigma_2|^m - \sigma_y^m \le 0
</math>
: ''Care must be exercised in using these forms of the generalized Hill yield criterion because the yield surfaces become concave (sometimes even unbounded) for certain combinations of'' <math>R</math> and <math>m</math>.<ref>Zhu, Y., Dodd, B., Caddell, R. M. and Hosford, W. F. (1987). ''Limitations of Hill's 1979 anisotropic yield criterion.'' International Journal of Mechanical Sciences, vol. 29, pp. 733.</ref>

== Hill 1993 yield criterion ==
In 1993, Hill proposed another yield criterion <ref>Hill. R. (1993). ''User-friendly theory of orthotropic plasticity in sheet metals.'' International Journal of Mechanical Sciences, vol. 35, no. 1, pp. 19–25.</ref> for plane stress problems with planar anisotropy. The Hill93 criterion has the form
:<math>
\left(\cfrac{\sigma_6}{\sigma_0}\right)^2 + \left(\cfrac{\sigma_2}{\sigma_{90}}\right)^2 + \left[ (p + q - c) - \cfrac{p\sigma_1+q\sigma_2}{\sigma_b}\right]\left(\cfrac{\sigma_1\sigma_2}{\sigma_0\sigma_{90}}\right) = 1
</math>
where <math>\sigma_0</math> is the uniaxial tensile yield stress in the rolling direction, <math>\sigma_{90}</math> is the uniaxial tensile yield stress in the direction normal to the rolling direction, <math>\sigma_b</math> is the yield stress under uniform biaxial tension, and <math>c, p, q</math> are parameters defined as
:<math>
\begin{align}
c & = \cfrac{\sigma_0}{\sigma_{90}} + \cfrac{\sigma_{90}}{\sigma_0} - \cfrac{\sigma_0\sigma_{90}}{\sigma_b^2} \\
\left(\cfrac{1}{\sigma_0}+\cfrac{1}{\sigma_{90}}-\cfrac{1}{\sigma_b}\right)~p & =
\cfrac{2 R_0 (\sigma_b-\sigma_{90})}{(1+R_0)\sigma_0^2} - \cfrac{2 R_{90} \sigma_b}{(1+R_{90})\sigma_{90}^2} + \cfrac{c}{\sigma_0} \\
\left(\cfrac{1}{\sigma_0}+\cfrac{1}{\sigma_{90}}-\cfrac{1}{\sigma_b}\right)~q & =
\cfrac{2 R_{90} (\sigma_b-\sigma_{0})}{(1+R_{90})\sigma_{90}^2} - \cfrac{2 R_{0} \sigma_b}{(1+R_{0})\sigma_{0}^2} + \cfrac{c}{\sigma_{90}}
\end{align}
</math>
and <math>R_0</math> is the R-value for uniaxial tension in the rolling direction, and <math>R_{90}</math> is the R-value for uniaxial tension in the in-plane direction perpendicular to the rolling direction.

== Extensions of Hill's yield criteria ==
The original versions of Hill's yield criteria were designed for material that did not have pressure-dependent yield surfaces which are needed to model [[polymer]]s and [[foam]]s.

=== The Caddell-Raghava-Atkins yield criterion ===
An extension that allows for pressure dependence is Caddell-Raghava-Atkins (CRA) model <ref>Caddell, R. M., Raghava, R. S. and Atkins, A. G., (1973), ''Yield criterion for anisotropic and pressure dependent solids such as oriented polymers.'' Journal of Materials Science, vol. 8, no. 11, pp. 1641-1646.</ref> which has the form
:<math>
F (\sigma_{22}-\sigma_{33})^2 + G (\sigma_{33}-\sigma_{11})^2 + H (\sigma_{11}-\sigma_{22})^2 + 2 L \sigma_{23}^2 + 2 M \sigma_{31}^2 + 2 N\sigma_{12}^2 + I \sigma_{11} + J \sigma_{22} + K \sigma_{33} = 1~.
</math>

=== The Deshpande-Fleck-Ashby yield criterion ===
Another pressure-dependent extension of Hill's quadratic yield criterion which has a form similar to the [[Bresler Pister yield criterion]] is the Deshpande, Fleck and Ashby (DFA) yield criterion <ref>Deshpande, V. S., Fleck, N. A. and [[M. F. Ashby|Ashby, M. F.]] (2001). '' Effective properties of the octet-truss lattice material.'' Journal of the Mechanics and Physics of Solids, vol. 49, no. 8, pp. 1747-1769.</ref> for [[honeycomb structures]] (used in [[Sandwich structured composite|sandwich composite]] construction). This yield criterion has the form
:<math>
F (\sigma_{22}-\sigma_{33})^2 + G (\sigma_{33}-\sigma_{11})^2 + H (\sigma_{11}-\sigma_{22})^2 + 2 L \sigma_{23}^2 + 2 M \sigma_{31}^2 + 2 N\sigma_{12}^2 + K (\sigma_{11} + \sigma_{22} + \sigma_{33})^2 = 1~.
</math>

== References ==
<references/>

== External links ==
* [http://aluminium.matter.org.uk/content/html/eng/default.asp?catid=183&pageid=2144416653 Yield criteria for aluminum]
* [http://www.tecnun.es/Asignaturas/estcompmec/documentos/thinsheets.pdf Yield criteria for thin metal sheets]

{{DEFAULTSORT:Hill Yield Criteria}}
[[Category:Plasticity]]
[[Category:Solid mechanics]]
[[Category:Yield criteria]]

Chinese remainder theorem

2012-08-18T03:51:39Z

50.131.180.143: /* Concrete example */ 4 was forgotten

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