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CAPTULO VI - 0

CAPTULO VI

NDICE

6.1 Introduccin VI-1

6.2 Consideraciones previas VI-2

6.3 Resultados experimentales VI-11

6.4 Criterios de plastificacin VI-14

6.4.1 Criterio de Rankine - Lam VI-14

6.4.2 Criterio de Saint-Venant - Poncelet VI-156.4.3 Criterio de Tresca - Guest VI-16

6.4.4 Criterio de Von Mises - Hencky VI-21

6.5 Consideraciones finales VI-31


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CAPTULO VI - 1

CAPTULO VI

CRITERIOS DE PLASTIFICACIN

6 - 1 Introduccin.

De acuerdo con el desarrollo de la programacin de este curso, en el primer captulo seestudi el estado tensional de los puntos de un slido. En el segundo captulo lasdeformaciones que sufra pero con la particularidad de que ambos estados se analizaronde forma independiente. Posteriormente, en el tercer captulo, se estudi la ley decomportamiento que une ambos estados. Este proceso de estudio puede ser representadoen el siguiente diagrama:

ij

uT

ij

Lema de Cauchy Relacin desplazamiento-Deformacin

Ley de Hooke

Ecc. De Lam

Objetivo de laElasticidad

Figura 6-1

Es decir el vector tensin T se relaciona con el tensor de tensiones ij a travs dellema de Cauchy. Luego el tensor de tensiones y el de deformaciones ij se relacionanentre si a travs de la ley de comportamiento, y por ltimo el vector desplazamiento use relaciona con el tensor de deformaciones mediante la relacin desplazamiento-deformacin. Como puede observarse el objetivo final de la Elasticidad es relacionar el

vector Tensin con los desplazamientos (o lo que es lo mismo carga-desplazamiento) ,sin embargo esto no puede hacerse directamente sino que es necesario idear unasvariables intermedias, tensiones y deformaciones, y realizar un recorrido a travs deellas para alcanzar el fin propuesto. Este modelo de la Teora de la Elasticidad se puedeafirmar que est parcialmente completo a falta, nicamente de aadir algn sistema queindique cundo ciertos puntos del slido abandonan el comportamiento elstico y entranen fase plstica.

Como se recordar del captulo III, un slido entra en fase plstica cuando algun/os desu/s punto/s superan el Lmite Elstico del material. Esta determinacin es sencilla derealizar si el slido tuviese una geometra muy simple y estuviese sometido a un estadode cargas igual que al del Ensayo de Traccin. En este caso simplemente con hacer:


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CAPTULO VI - 2

E

se asegurara que ningn punto plastificar. En la expresin anterior E es el LmiteElstico del material y I la nica tensin existente en el interior del slido.

Sin embargo lo usual es que un cuerpo tenga una geometra general y est sometido aun estado de cargas cualesquiera siendo el caso ms general cuando el tensor detensiones est completo. Ante tal situacin de cargas cabra preguntarse Cundoalcanza un punto del slido el Lmite Elstico del material ? o bien Cul es lacombinacin de tensiones ms desfavorable que hace que se alcance antes el LmiteElstico ?.

Para responder a estas preguntas se desarrolla este captulo y lo que se denominaCriterios de Plastificacin o herramientas que permiten predecir si un punto de un slidova a alcanzar la plastificacin ante un determinado estado de cargas. A tal fin el captulose articula de la siguiente forma: En primer lugar se hacen unas consideraciones previas,luego se desarrolla una pequea revisin de los criterios de plastificacin msrelevantes, y por ltimo se hace especial hincapi en los Criterios de Von Mises y deTresca aceptados hoy en da por la Instruccin espaola.

7-2 Consideraciones previas.

Con el fin de sistematizar el establecimiento de un Criterio de Plastificacin esnecesario enunciar una serie de postulados. Tales son:

1 Postulado

Si un punto de un slido elstico plastifica lo hace solo y exclusivamente en funcin delas tensiones que actan sobre l y no de la forma en que las tensiones han alcanzado losvalores crticos. Esta hiptesis permite suponer la existencia de una cierta funcin

ijf (funcin de plastificacin) tal que:

El material se comporta elsticamente si 0


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CAPTULO VI - 3

Ahora bien si el material es istropo implica que cualquiera que sea el sistema dereferencia del tensor de tensiones siempre tendr las mismas tensiones principales. Portanto la funcin f puede ser expresada haciendo uso de las tensiones principales as:

),,( IIIIIIff = 6-6

Ahondando an ms en la idea, la funcinf incluso podra ser expresada, de una formams general, en funcin de los invariantes:

),,( 321 IIIff = 6-7

Esta ltima expresin indica que la funcin de plastificacin ni siquiera dependedirectamente de las tensiones principales sino de una combinacin de ellas.

2 Postulado

La funcin f no debera cambiar si los ejes se intercambian, es decir si el eje 2 pase aser el eje 1, por ejemplo, la funcin de plastificacin debera ser la misma. Este hechoconduce al siguiente resultado:

etcffff IIIIIIIIIIII ),,(),,(),,( ===

Una conclusin importante de que la funcin de plastificacin tenga el mismo valorcualquiera que sea la rotacin de ejes es: la funcin de plastificacin f es una funcinsimtrica de las tensiones principales.

Vamos a extendernos un poco mas y exponer algn ejemplo que ayude a entender estepostulado. A tal fin supngase el polinomio caracterstico que proporciona las races otensiones principales:

0322

13 =+ III

Hasta ahora, y por convenio, se ha denominado I a la mayor de las tres races

cumplindose que: IIII >> . Sin embargo una funcin de plastificacin nopuede depender de este convenio, sino que tiene que ser independiente de la forma enque se denomine a cada raz o tensin principal. Supngase dos casos de asignacin deraces:

Caso 1 Caso 2

321 ;; === IIIIII 231 ;; === IIIIII

Est claro que se debe cumplirse que: ),,(),,( IIIIIII fff == .

Si se hace una representacin isomtrica de las tensiones principales, resulta:

CASO 1 CASO 2


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CAPTULO VI - 4

I II

III

I III

II

f (I, II, III ) f (I, III, II )

III

III

I III

II

Figura 6-2

La consecuencia de que ),,(),,( IIIIIIIII ff = es que f debe ser simtricaalrededor del eje I, para que al intercambiarII porIII la funcin de plastificacin tengael mismo valor. Si se repite este razonamiento para los otros dos ejes resulta por

conclusin que la funcin de plastificacin debe ser simtrica respecto a los tres ejes.3 Postulado

La funcin de plastificacin f toma valores iguales cuando los signos de las tensionescambian:

f fij ij( ) ( ) = 6-8

Esto quiere decir que el material se comportar de igual forma tanto si trabaja a traccincomo a compresin, y por tanto la funcin de plastificacin tambin ha de ser simtrica

respecto a los ejes I , II, III, pero en su parte negativa. Es ms, si se requiere que lacondicin se mantenga despus de que el material haya sido sometido a unadeformacin plstica inicial, entonces ser necesario no considerar el efecto Baushinger.

Efecto Baushinger:Sea una probeta de un acero que se somete al siguiente ensayo. Primero se carga y sesobrepasa el lmite elstico (figura 6-2) del material (tramo A B C). Luego se descarga yse llega a cero. A continuacin se invierte la carga y se somete la probeta a compresinhasta que alcance el lmite elstico pero negativo es decir -E ( tramo C D H). Seobserva que el nuevo lmite elstico negativo no coincide con el de traccin ( es menor)y el comportamiento del material es elstico pero no lineal (efecto Baushinger).

A

BC

D

H

- E

Figura 6-3


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CAPTULO VI - 5

4 Postulado

Este ltimo postulado establece que la plastificacin slo depende de la parte desviadoradel tensor de tensiones. Este muy importante postulado es consecuencia de la

observacin del hecho experimental (corroborada por los ensayos de Bridgman amediados del siglo XX) de que un estado triaxial no provoca plastificacin.Nota:Cuando una probeta de un material dctil es sometida a un estado de traccin ocompresin triaxial (un estado triaxial de cargas es tal que todas las presiones sobre un

punto de un slido son iguales en magnitud y sentido, por ejemplo el estado hidrostticode la figura 7-3) el slido no plastifica y la rotura se produce como si el material fuesefrgil (cuando un material se comporta de forma frgil no aparece en ningn instante la

plastificacin).

Figura 6-4

Antes de continuar es necesario recordar la descomposicin de un tensor en su parteesfrica y desviadora

+

=+=

300

030

003

300

030

003

1

1

1

1

1

1

I

I

I

I

I

I

III

II

I

desvij

esf

ijij

6-9

Donde octTI

=31 , y como puede observarse el tensor esfrico crea un estado triaxial de

cargas y por tanto no produce plastificacin. Consecuentemente el responsable de laplastificacin debera ser la parte desviadora del tensor de tensiones tal y como sepropuso al principio de este postulado.


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CAPTULO VI - 6

Representacin en el espacio de las tensiones principales

Es una representacin que ayuda a entender la superficie de plastificacin y consiste endefinir un sistema de referencia cuyos ejes sean las tensiones principales (es importantellamar la atencin que este sistema de referencia no es un sistema cartesiano ni

constituye un espacio por lo que es simplemente una representacin). En talrepresentacin la diagonal principal es una recta en la que se cumple que: I = II = III(estado hidroesttico) y por tanto es el lugar de los puntos de la parte esfrica de untensor de tensiones.

I

II

III

ij

ijesf

ijdesv I = II = III

Figura 6-5

Tambin en dicha representacin el tensor de tensiones es un punto o un pseudovectoral igual que el tensor esfrico y desviador con la particularidad de que son

perpendiculares. La comprobacin de tal perpendicularidad consiste en realizar elproducto escalar siguiente:

( )

( ) 03333

3

33,

3,

33,

3,

3

11

1111

1111111

=

=

++

=

II

IIII

IIIIIIIIIIIIIIIIIII

6-10

y ver que efectivamente es nulo

Realizadas las consideraciones anteriores, el siguiente paso es definir la superficie deplastificacin como un cilindro cuyo eje es concntrico a la diagonal principal delpseudosistema mencionado. A esta conclusin se llega observando que la plastificacin

no depende de la posicin de sino de y por tanto cualquier seccin

realizada por un plano perpendicular a la diagonal principal debe tener la misma formatal y como muestra la figura siguiente:

esfij

desvij


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CAPTULO VI - 7

I

II

III

ijesf

ijdesv

f ( I , II ,III )

Figura 6-6

Representacin de Haig Westergaard

Consiste en realizar una interseccin a la superficie f por un plano perpendicular a ladiagonal principal y que pase por el origen. El resultado es una representacin plana delos infinitos pseudovectores desviadores (hay que recalcar que la interseccin esindependiente del punto sobre el que se realice ya que la superficie f es un cilindro). Sise elige un sistema isomtrico para representar el resultado de la interseccin se obtienela figura siguiente:

III

III

Superficie fdePlastificacinintersectada por unplano

figura 6-7

La aplicacin del segundo postulado a la funcin de plastificacin conduce a que debeser simtrica alrededor de los ejes (I ,II ,III), y si no se considera el efecto Baushingertambin debera ser simtrica alrededor de los ejes ( I, II, III) pero en su parte negativa.Esta condicin obliga a que la funcin de plastificacin sea simtrica respecto a 6segmentos de rea de 60 cada uno, tal y como se muestra en la figura 6-8 (las distanciasOM y OQ son constantes).


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CAPTULO VI - 9

I I I

I I I

I I

I I

I

I

Figura 6-10

Otro comportamiento tambin vlido, y muy parecido al real, sera como muestra la

figura 6-11

I I I

I I I

I I

I I

I

I

Figura 6-11

Por ltimo hay que aadir que existen infinitas posibilidades para descomponer untensor en uno esfrico y otro desviador. Sin embargo slo existe una que hace que los

pseudovectores sean perpendiculares y es tomar el esfrico de tal forma que sus trestensiones principales sean precisamente las tensiones octadricas.

6-3 Resultados experimentales

Cualquier criterio de plastificacin que se obtenga de una forma ms o menos tericadeber estar de acuerdo, o bien sus predicciones deberan acercarse o estar prximas, alos resultados obtenidos de forma experimental. Lo ideal sera que en una situacincompleja de carga y geometra se pudiese realizar un ensayo que proporcionase elverdadero comportamiento y Lmite Elstico del caso que est estudiando.

La realidad, sin embargo, es que los ensayos se tienen que realizar con slidos degeometra sencilla y estados de carga simples y luego extrapolar las consecuencias asituaciones ms complejas a travs de los criterios de Plastificacin.


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CAPTULO VI - 10

Los ensayos que normalmente se emplean, aparte del ensayo de traccin, tratan desimular situaciones complejas, para ello se usan tubos y se les somete a cargas axilesde traccin, torsin, y presin. Combinando estos tres tipos de carga se consiguen unaserie de ensayos donde aparecen tensiones normales y tangenciales, dos tensionesnormales, etc. Por ejemplo en la figura 6-12 se muestra un tubo de pared delgada y los

resultados tericos que se obtienen cuando es sometido a axil, torsor, y presin interna:

F

P

T

T 1

R

e

z

Figura 6-12

Las tensiones que provocan este tipo de esfuerzos son:

- Debido al axil F :er

F

21=

- Debido al Momento Torsor T: er

Tz 22

=

- Debido a una presin interna p suponiendo tapaderas mviles:e

Rp=

Por tanto se dispone de dos tensiones normales: 11 y y de una tensin

tangencial: 1 que se pueden combinar produciendo estados de carga cuasi-planos.

Los casos extremos son: que slo exista axil, o que slo exista torsin:

II = 0III =0

II = 0

F 0 ; T=0 ; p = 0 F = 0 ; T 0 ; p = 0

T T

T T

I = 11I = z

I = - z

Figura 6-13


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CAPTULO VI - 11

y el objetivo de los ensayos es obtener resultados experimentales de las combinacionesde ambos estados de carga.

Entre los mas conocidos estn los realizados por Lode publicados en 1926, Ros yEichinger publicados en 1926 y los de Taylor y Quinney publicados en 1931. Los

ensayos tenan por objetivo corroborar los criterios de plastificacin tericos con losobtenidos experimentalmente utilizando para ello tres tipos de materiales: acero, nquel,y cobre. Los resultados mostraron que exista una buena correspondencia entre losvalores tericos y los obtenidos experimentalmente si se empleaba el criterio de VonMises- Hencky y bastantes prximos si se empleaba el criterio de Tresca, Los resultadosde los autores Taylor y Quinney usando tubos de cobre, aluminio y acero ysometindolos a carga axil y momento torsor se representan en la figura 6-14 quemuestra la variacin del Lmite Elstico en funcin de las tensiones normales ytangenciales.

Figura 6-14

Como puede observarse los resultados se expresan en unos ejes adimensionales. Ellosignifica que, tericamente, cualquier material debera plastificar siempre en el mismo

punto para una determinada combinacin de axil y cortante, o lo que es lo mismo quelos smbolos crculo, tringulo y cuadrado de la figura 6-14 deberan coincidir paracualquier punto. Lgicamente la realidad no es as, pero puede observarse que es

bastante parecida.

Los extremos de la batera de ensayos son los casos de axil puro

= 1E

T

y el ensayo

de torsin pura

= 1

E

T

. Segn muestran los resultados en un caso de torsin pura

(slo existira tensin tangencial) se obtiene:

56.0=E

T

6-11

Este resultado es importante porque va a servir de test o prueba para comprobar laexactitud o inexactitud de un criterio de plastificacin cuando se particularice para elcaso de torsin pura.


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CAPTULO VI - 12

6-4 Criterios de Plastificacin

El objetivo fundamental de un criterio de plastificacin es proveer al diseador de unaley o regla que le informe, de una forma sencilla, si las tensiones en un punto de unslido van a alcanzar un lmite establecido. En nuestro caso es el Lmite Elstico del

material. Histricamente se trataba de ligar directamente los resultados del ensayo detraccin con los que se obtena del clculo. As surgieron criterios como los Rankine ySaint-Venant, que siendo lgicos para su poca hoy en da son de poco uso. Sinembargo parece instructivo exponerlos, y as se tendr una idea de como ha sido laevolucin del conocimiento en este campo.

6-4-1 Criterio de Rankine-Lam (1858 y 1852)

Este criterio es el ms inmediato, y establece que la plastificacin se producir cuandolas tensiones existentes en algn punto del slido alcancen el Lmite Elstico E delmaterial. Es decir:

EIIIEIIEI ;; 6-12

Este criterio se representa en el espacio de las tensiones principales como un cubo delado 2 E centrado en el origen, figura 6-15-a, y en el espacio de Haig-Westergaard,figura 6-15-b, sera la interseccin del cubo por un plano perpendicular a la diagonal

principal pasando por el origen.

I

III

II

(-E , E , E )

(-E , E , -E )

(E , E , -E )

2E

Interseccin del cubocon un plano que pasa

por el origen

Figura 6-15

Este criterio no es vlido para materiales dctiles ya que para el ensayo siguiente:


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CAPTULO VI - 13

T

T- II I

Figura 6-16

predecira que la plastificacin se producir cuando la tensin tangencial T E == (

en realidad debera expresarse que 1=E

T

). Este resultado es casi el doble del

obtenido mediante los ensayos de Lode, o Quinney, que dice que la plastificacin se

producir cuando 56.0=E

T

. Consecuencia de ello es que este criterio no es vlido

para materiales dctiles. Sin embargo s es un criterio vlido para materiales frgiles,por ejemplo en rotura de cristales, ya que en este tipo de materiales si se cumple que

EIT == .

6-4-2 Criterio de Saint-Venant -Poncelet (1870 y 1839)

Este criterio, propuesto independientemente por ambos autores, establece que un puntode un slido plastificar cuando las deformaciones principales alcancen el valor de ladeformacin E (deformacin correspondiente al Lmite Elastico en el ensayo detraccin).

Puesto que:

E

EE

=

la plastificacin ocurrir (en un caso general) cuando:

EIIIEIIEI ;; 6-13

Si se expresa las deformaciones en funcin de las tensiones a travs de la ley de Hooke:

( )[ ]E

vE

EIIIIIII

=+=

16-14

o bien:


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CAPTULO VI - 14

( )[ ] EIIIIv =+ 6-15

y de igual forma para las otras direcciones:

( )[ ] EIIII v =+ 6-16( )[ ] EIII v =+ 6-17

La comprobacin de la bondad de los resultados se realiza para el caso de carga delcriterio anterior. Efectivamente =I 0== III y . Sustituyendo estosvalores en 6-15 ; 6-16 ; y 6-17 resulta:

( )[ ] ( ) EIIIII vv ++ 1( )[ ] EIIIII vv ++ )1(

( )[ ] EIIII v + 0

El ms desfavorable es:v

E

+

1

y debido a que ( ) =maxT resulta:

)1( vT E

+

6-18

Tomando un valor del mdulo de Poisson de 0.30 resultara:

77.0

3.1

1

1

1==

+

=v

T

E

6-19

Valor que, mejorando el obtenido con el criterio anterior sigue an estando alejado del

establecido experimentalmente (T

e

= 056. para un caso de cortadura pura).

En el espacio de la tensiones principales este criterio representara un romboedro, y lainterseccin con el plano perpendicular a la diagonal principal sera un hexgonoregular Figuras 6-17-a y 6-17-b.

Figura 6-17


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CAPTULO VI - 15

6-4-3 Criterio de Tresca - Guest (1868 y 1872)

Durante la realizacin del ensayo de traccin se observa que en las inmediaciones delLmite Elstico aparecen en el material unas lneas a 45, conocidas como lneas deLders, que hicieron pensar la posibilidad de que la plastificacin estuviese gobernada

por las tensiones tangenciales (como se sabe las tensiones tangenciales mximas seproducen en planos que forman 45 con la direccin principal I ).

El criterio de Tresca establece que: un punto de un slido plastificar cuando la tensin

tangencial mxima T alcance el valor de la tensin tangencial T . Siendo la

tensin tangencial que se produce en el ensayo de traccin cuando la probeta alcanza ellmite elstico del lmite elstico en el ensayo de traccin.

E

ET

El circulo de Mohr del ensayo de traccin es:

E

TE

Figura 6-18

Y por tanto2EET =

Para el caso general de un punto de un slido en tres dimensiones, los crculos de Mohrcorrespondientes seran:

III II I T

T

( T)max

Figura 6-19


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CAPTULO VI - 16

y la tensin tangencial mxima es:2

max IIIIT

=

Para un caso general, como el anterior, y no teniendo en cuenta el convenio de queI > II > III el criterio de Tresca se expresara de la siguiente forma:

22;

22;

22EIIIIIEIIIEIIII =

=

=

6-20

o bien:

EIIIEIIEIII === ;; 6-21

A continuacin es necesario hacerse la siguiente pregunta Cunto de bien, o cunto demal, funciona este criterio?

Para ello se retoma el ensayo de cortadura pura: 0; === IIII

Sustituyendo estos valores en 6-21 resulta:

EEII 6-22

EEII 2 6-23

EEIIII 6-24

De las desigualdades anteriores la mas desfavorable es 2E

. Por tanto el resultadoque predice el criterio de Tresca para este tipo de caso es que la plastificacin se

producir cuando la tensin tangencial alcance el valor de:

( )( )2maxET

= 6-25

y adimensionalizando:

5.0=E

T

6-26

este valor terico (0.5) es muy prximo al experimental (0.56) concluyndose que elcriterio de Tresca produce resultados bastante aceptables.

En el espacio de las tensiones principales cada una de las igualdades de la expresin 6-21 representa a dos planos paralelos a la diagonal principal de acuerdo con la figura 6-20:


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CAPTULO VI - 17

III

II

I Figura 6-20

No es inmediato vislumbrar geomtricamente tal conclusin. Para ello se considera unode los casos por ejemplo: EIII = y se supondr que el resultado es positivo.

Est claro que en el pseudoespacio de las tensiones principales EIII = es unplano tal que corta al eje I en E y al eje II en - E (puntos A y B de la figura 6-21) yes paralelo al eje III (misma figura).

Por otra parte se sabe que la diagonal principal forma ngulos iguales con los tres ejesprincipales de valor: arccos( 1 / 3 ) que a su vez se puede expresar como:

=

3

2

3

1arccos arcsen

Diagonal PrincipalPlano I - II = E

II

III

I

B

A

O

3

2

arcsen

figura 6-21

Si se hace el producto escalar del vector normal al plano EII = y el vectorcoincidente con la diagonal principal y el resultado fuese nulo significara que ambosson perpendiculares, o lo que es lo mismo que el plano EIII = es paralelo a la

diagonal principal. Para ello:


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CAPTULO VI - 18

Vector normal al plano EI = =>

0,

2

2,

2

2

Vector en la direccin de la diagonal principal:

3

1,

3

1,

3

1

realizando el producto escalar:

03

1,

3

1,

3

10,

2

2,

2

2=

Lo cual corroborara la aseveracin de que el plano EI = es paralelo a ladiagonal principal. De igual forma se podra hacer para el resto de los planos ycomprobar que efectivamente son todos paralelos a la diagonal principal y ratificar laveracidad de la figura 6-20.

Para hacer una representacin de Haig-Westergaard se intersecta el cilindro de Trescacon un plano perpendicular a la diagonal principal resultando un hexgono regular, que

se representa en la figura 6-22. Los cuyos ejes se han denominado ) ,, IIIIII parano confundirlos con los principales y se han situado de acuerdo con una representacinisomtrica.

I - II = -EI - II = E

II - III = E

II - III = -E

III - I = -E

III - I = E

O

A

III

EOAOA

3

2

3

2 ==

III

Figura 6-22

Es interesante remarcar que en la figura 6-22 la distancia O A no coincide con ladistancia OA del espacio de las tensiones principales puesto que son representaciones

diferentes y para demostrar que OA3

2 =OA se hace uso de la figura 6-23. En ella

se representa un plano perpendicular a la diagonal principal sobre el que se proyecta elsegmento OA resultando:


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CAPTULO VI - 19

Planoperpendicular ala diagonal

principal

O

A

A = 90

= arcsen(2 / 3)

II

II

II

III

IIIDiagonal

Principal

Figura 6-23

El segmento OAes la proyeccin del segmento OA sobre el eje I y se cumple que:

= 90

Por tanto: OA= OA cos = OA sen = OA2

3tal y como se quera demostrar.

Un tema interesante es que el criterio de Tresca puede ser escrito en funcin de losinvariantes de la siguiente forma:

064

6416

964

3627464

2

222

23

32 =+

EEe IIII

La importancia de esta expresin radica en poner de manifiesto dos cosas:

a) que el criterio de Tresca es el primero en exponer que es independiente del primerinvariante y por consiguiente independiente del tensor esfrico.

b) Por otra parte, al ser expuesto en forma de invariantes significa que es independientede la orientacin del sistema de referencia y de la numeracin de este.

En los casos bidimensionales en que II es nula el criterio se expresa particularizandola expresin 6-10 para: I I III III II= = =; ; 0 . Resultando:

22;

22;

22EIIIEIEIIII ===

6-27

La representacin de estas condiciones en el plano ( I , III ) es:


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CAPTULO VI - 20

I

III

I - III = E

III - I = E

III = E

III = - E

I = E

I = - E

Figura 6-24

7-4-3 Criterio de Von Mises - Hencky (1913 y 1924)

Tratando de suavizar la superficie angulosa de plastificacin que prev el criterio deTresca, Von Mises propuso, de forma heurstica, que la representacin en el espacio delas tensiones principales de un criterio de plastificacin debera ser un cilindro de

seccin circular de radio: E3

2en vez de un cilindro de seccin hexagonal como en

el caso de Tresca. Tal representacin sera:

Figura 6-25

Y la representacin de Haig-Westergaard:


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CAPTULO VI - 21

O

A

Von Mises

Tresca

EOA 3

2 =

III

III

Figura 6-26

La demostracin terica del criterio fue realizada posteriormente por Hencky. Porltimo Naday en la dcada de los aos 30 puso de manifiesto la coincidencia con lastensiones tangenciales octadricas.

El criterio de Von Mises - Hencky se puede expresar de la siguiente forma: Un punto deun slido plastificar cuando la densidad de energa de distorsin (densidad de energaasociada al tensor desviador) alcance el valor de la densidad de energa de distorsinque tiene una probeta en un ensayo de traccin en el instante en que la tensin llegue allmite elstico.

La Densidad de Energa de Deformacin (energa por unidad de volumen) almacenadadurante el proceso de carga de un slido cualquiera es:

ijijU 2

1=

Sustituyendo las deformaciones en funcin de las tensiones, se obtiene:

( ) ( )( ) ( )

++++++++=

313

223

212

3311332222211333

222

211

122

21

v

vE

U 6-28

y en el caso de que el tensor est expresado en el sistema principal sera:

( ) ( )[ ]IIIIIIIIIIIIIIIIII vE

U ++++= 22

1 322 6-29

Por otra parte un tensor de tensiones expresado en el sistema principal y descompuesto

en su parte esfrica y desviadora es:


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CAPTULO VI - 22

+

=+=

300

03

0

003

300

03

0

003

1

1

1

1

1

1

I

I

I

I

I

I

III

II

I

desvij

esfijij

La Densidad de Energa de Deformacin sera la suma de la Densidad de Energa deDeformacin de la parte esfrica, ms la correspondiente a la parte desviadora, as:

desvesf UUU += 6-30

Puesto que en la plastificacin slo interviene la parte desviadora se halla acontinuacin la Densidad de Energa de Deformacin correspondiente al tensor

desviador. Su obtencin es simplemente sustituyendo en 6-29:

3

;3

;3

111 IIIIIIIIIIIIIII

resultando:

+

+

+

+

=

333333

3332

1

111111

21

21

21

IIIIII

E

v

III

EU

IIIIIIIIIIII

IIIIIIdesv

6-31

Operando:

( ) ( )

( ) ( ) ( )

++

+++++

++

+++++=

IIIIIIIIIIII

IIIIIIIIIIII

IIIIIIIIIIII

desv

IIIIEv

EU

2

1111

22222

33333

33

2

2

1

6-32

Sumando y restando a cada parntesis del segundo corchete lo que le falta paracompletar el primer invariante, se obtiene:


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CAPTULO VI - 23

( )

( )

+++++

++

++=

IIIIIIIIIIIIIII

IIIIIIIII

desv

Ev

EU

3

32

1

2

2222

6-33

Multiplicando y dividiendo por 3 y operando se obtiene:

[ ]

[ ]IIIIIIIIIIIIIIIIII

IIIIIIIIIIIIIIIIIIdesv

E

v

EU

2222226

2222226

1

222

222

++

+++=6-34

expresin que representada en su forma habitual es:

( ) ( ) ([ ]2226

1IIIIIIIIIIII

desv )E

vU ++

+= 6-35

o bien:

( ) ( ) ([ ]22212

1IIIIIIIIIIII

desv

GU ++= ) 6-36

La Densidad de Energa de Deformacin de la parte desviadora del tensor de tensionesen el ensayo de traccin cuando se alcanza el lmite elstico se obtiene particularizandola expresin anterior para: I = E ; II = 0 ; III = 0 obtenindose:

( ) ( )[ ]2212

1EE

desvtraccens

GU += 6-37

Igualando se obtiene el conocido criterio de Von Mises:

( ) ( ) ( ) 2222 2 EIIIIIIII =++ 6-38

o bien:

( ) ( ) ( )E

IIIIIIIIIIII

++

2

222

6-39

Como puede observarse, el criterio de Von Mises representa claramente un cilindro deseccin circular cuyo eje es la diagonal principal tal y como se representa en la figura 6-25. La interseccin con el plano III = 0 es una elipse que si se proyecta en el plano

perpendicular a la diagonal principal es un cilindro de radio

2

3E de acuerdo con lafiguras 6-25 y 6-26.


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CAPTULO VI - 24

Existen varias formas de obtener el mismo resultado, por ejemplo:

- Limitando el segundo invariante del tensor desviador. Por tanto el criteriotambin podra definirse como: La plastificacin de un punto de un slido comenzarcuando el segundo invariante del tensor desviador alcance el valor del correspondiente

al del ensayo de traccin.

- Limitar la tensin intrnseca tangencial octadrica del tensor desviador. Por tantotambin podra definirse como: La plastificacin de un punto de un slido elsticocomenzar cuando la tensin intrnseca tangencial octadrica del tensor desviadoralcance la tensin intrnseca tangencial octadrica del tensor desviador correspondienteal ensayo de traccin .

En el caso bidimensional ( II = 0 ) el criterio de Von Mises adopta la siguienteexpresin:

( ) ( ) ( )E

IIIIIIII

++

2

2226-40

o bien:

222EIIIII + 6-41

que es la ecuacin de una elipse. La figura siguiente representa la elipse del criterio de

Von Mises superpuesta al hexgono irregular del criterio de Tresca:

I

III

Figura 6-27

Por ltimo la comprobacin de la fiabilidad de este criterio frente a un caso real serealiza a travs del ensayo de cortadura. Es decir

0;; === IIIIII 6-42

sustituyendo se obtiene:


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CAPTULO VI - 25

( ) ( )3

32

22 222

EEE

+6-43

Como en este ensayo ( ) =maxT resulta: ( ) ( ) 577.03max

max E

E TT

Valor muy parecido al obtenido experimentalmente ( 0.56) , concluyndose que estecriterio es el que proporciona unos resultados tericos ms prximos a losexperimentales.

Para terminar es interesante expresar el criterio de Von Mises desde el punto de vista dela Resistencia de Materiales. En esta disciplina la forma usual del tensor de tensioneses:

6-44

=

00

00

13

12

131211

ij

ya que supone que las tensiones 22 , 33 y 23 tienen valores despreciables. Si sehallan las tensiones principales se obtiene:

( ) ( )2

40

2

4 213212

21111

213

212

21111

++

==+++

= IIIIII

Y sustituyendo estos valores en el criterio de Von Mises se obtendra:

( ) E ++ 213212211 3 6-45

y en el caso muy comn de que slo existe una tensin normal ( ) y una tensintangencial ( ) resulta:

( )E

+ 22 3 6-46

A continuacin se muestra un grfico de los resultados de los ensayos experimentalesobtenidos por Taylor y Quinney y se comparan con los que proporcionara el criterio deTresca y de Von Mises. El ensayo consisti en someter a un tubo (de pared delgada) auna combinacin de dos acciones: un esfuerzo axil de traccin y un momento torsor. Elesfuerzo axil produce un tensor de tensiones donde slo existe I mientras que el

momento torsor produce un tensor de tensiones donde slo existe 12 . Por tanto se esten un caso que corresponde a la ltima expresin. Combinando las acciones se produceuna nube de puntos y la conclusin es evidente: el criterio de Von Mises proporcionamejor aproximacin a los resultados del ensayo que cualquier otro.


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CAPTULO VI - 26

Figura 6-28

7-5 Consideraciones finales.

Hasta ahora se ha hablado de superficies de plastificacin, pero no de superficies deroturas. El tema tiene su lgica ya que un material deja de comportarse elsticamentecuando alcanza el lmite elstico y dejan de ser vlidas las ecuaciones de la elasticidadcuando lo supera de ah la importancia de su determinacin. Por otra parte ladeterminacin de la superficies de plastificacin solo tiene sentido en materialesdctiles pero no en materiales frgiles.

Inmediatamente surge la cuestin: Cundo un material es dctil o frgil? . Estapregunta trat de ser respondida en el captulo concerniente al estudio de la ley decomportamiento. Sin embargo hay que tener presente que no siempre un material dctil

se comporta como tal ya que si trabajara en forma triaxial tendra un comportamientofrgil, no siendo de aplicacin lo visto en este captulo. Por ello es muy importante quesituaciones de carga que conducen a estados triaxiales sean, en la medida de lo posible,soslayados mediante disposiciones constructivas adecuadas. Es el caso que se presentaa continuacin:

2

1

3

Cordones desoldadura

Figura 6-29

En esta figura se representan tres piezas de acero unidas por tres cordones de soldaduraformando ngulos de 90 entre s. La realizacin de un cordn de soldadura genera en elmaterial unas tensiones (de origen trmico y producidas por el calor generado en el

proceso de soldeo), que permanecen en l y se denominan tensiones residuales. As:

El cordn de soldadura sobre el eje 1 producir 11


28/43

CAPTULO VI - 27



Consecuentemente el origen estar sometido a un estado triaxial de cargas no deseable,por ello es necesario buscar alternativas constructivas que eviten estas situaciones,siendo una de ellas ejecutar los cordones de la siguiente forma:

2

1

3

Cordones desoldadura Hueco

Figura 6-30

De esta forma el origen estar libre de tensiones y no se producir un estado triaxial decargas.


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CAPTULO VI - 28

Trained as a civil engineer, William Rankine was appointed to the chair of civilengineering and mechanics at Glasgow in 1855. He developed methods to solve theforce distribution in frame structures.He worked on heat, and attempted to derive Sadi Carnot's law from his own hypothesis.His work was extended by Maxwell. Rankine also wrote on fatigue in the metal ofrailway axles, on Earth pressures in soil mechanics and the stability of walls. He waselected a Fellow of the Royal Society in 1853.Among his most important works are Manual of Applied Mechanics (1858), Manual ofthe Steam Engine and Other Prime Movers (1859) and On the Thermodynamic Theoryof Waves of Finite Longitudinal Disturbance.
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Carnot_Sadi.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Carnot_Sadi.html


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George Biddell Airy

Born: 27 July 1801 in Alnwick, Northumberland, EnglandDied: 2 Jan 1892 in Greenwich, England

George Airy's father was William Airy while his mother was Ann Biddell. William Airy

was from Lincolnshire and Ann was the daughter of a farmer from Suffolk. OriginallyWilliam had been a farmer too, but he had educated himself and risen to the position oftax inspector. When George was born his parents were living in Northumberland whereWilliam was a collector of excise, but in the following year the family moved toHereford when William was transferred there.George attended Byatt Walker's school in Colchester and at the age of ten he took first

place at the end of his primary school career. He had learnt some useful skills at theschool such as arithmetic, double-entry book keeping and how to use a slide rule. Hehad probably learned more, however, from studying his father's books. He wrote in hisautobiography that (see [3]):-... he was not a favourite with his school mates.Eggen writes [1]:-An introverted but not shy child, Airy was, even for the time and especially for hiscircumstances, a young snob. Nevertheless, he overcame some of the dislike of hisschoolmates by his great skill and inventiveness in the construction of peashooters andother such devices.Before Airy left Byatt Walker's school his father had transferred again, this time toEssex. From 1812 Airy spent his summers with his uncle, Arthur Biddell, who had afarm near Ipswich. Clearly Airy was not too happy at home because he asked his uncleif he could live with him rather than with his own family. Things had taken a turn forthe worse at home since his father lost his tax collectors job in 1813 and the family

were, from that time, living in poverty. Because of the financial circumstances thefamily seem to have been quite glad that Airy's uncle had almost taken over the role ofhis father.The fact that Airy spent about half his time with his uncle over the next five years wasimportant for him. Arthur Biddell was a man of learning who had a fine librarycontaining books on chemistry, optics and mechanics which Airy avidly studied, and inaddition he had many leading scientists as his friends. Their influence on the youngAiry was marked and was a major factor in his seeking an academic career.During these five years, 1814 to 1819, Airy attended Colchester Grammar School wherehe was [7]:-... noted for his memory, repeating in one examination 2394 lines of Latin verse.

Airy entered Trinity College, Cambridge in 1819 as a sizar, meaning that he paid areduced fee but essentially worked as a servant to make good the fee reduction.


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CAPTULO VI - 30

However it was only because his uncle provided financial support that he was able toundertake university studies at all. To supplement his income Airy took private pupilsand this, of course, gave him less time for his own studies. Despite this his performancewas outstanding and he graduated as Senior Wrangler (the top First Class student) in1823 and was a Smith's prizeman. Woodhouse, who had left the Lucasian chair in 1822

to become Plumian Professor of Astronomy, was one of Airy's examiners for theSmith's prize, the other being Thomas Turton who had succeeded Woodhouse to theLucasian chair. In the following year Airy was awarded a fellowship at Trinity Collegeand began his academic career.We should comment on why Airy did so well in the Tripos examinations, being farahead of the next best student. The Tripos examinations at that time were less a test ofmathematical ability and more a test of the candidates ability to learn vast amounts ofmaterial and methods. At this Airy proved exceptionally good, partly because of hisexcellent memory, but also because of his remarkable organisational abilities. As anundergraduate he kept paper beside him to record every thought he had. Latereverything was transferred to the books and diaries which he kept. He maintained this

routine throughout his life and this record, almost of his every thought, still exists toprovide remarkable evidence of the period [3]:-The ruling feature of his character was order. From the time he went up to Cambridge tothe end of his life his system of order was strictly maintained.Clerke writes in [7]:-He never destroyed a document, but devised an ingenious plan of easy reference to thehuge bulk of his papers.In 1824 Airy met Richarda Smith while on a walking holiday. He proposed two daysafter they first met but her father, Richard Smith, the vicar of a church near Chatsworth,refused to allow the marriage on the grounds that Airy could not support his daughterfinancially. This made Airy determined to obtain a position with the financial statuswhich would allow him to marry.Only three years after graduating from Cambridge, he was appointed Lucasian Professorof Mathematics at Cambridge. It is rather surprising that the Lucasian Professor onlyreceived 99 per year while Airy was already receiving 150 as an assistant tutor. Airywondered whether he could afford to compete for the chair when he was advised in1826 that Turton was leaving, but Peacock persuaded him that the status was moreimportant than the money. He became one of three candidates, French and Babbage

being the other two. When Babbage stated that he was about to start legal proceedingsover the election, French withdrew. Airy triumphed and a rivalry with Babbage whichwas to last for many years began.

In addition to the Lucasian Chair, Airy was appointed a member of the Board ofLongitude which gave him another 100 per year provided he attended four meetings.He explained his actions (see [3]):-My prospects in the law or other profession might have been good if I could havewaited but marriage would have been out of the question and I much preferred amoderate income in no long time. I had now in some measure taken science as my line(but not irrevocably) and I thought it best to work it well for a time at least and wait foraccidents.These, of course, are not the words of a man driven by a love of his subject. Hecertainly still did not have the financial position to allow him to marry Richarda so hetried for other posts. His attempt to secure the vacant post of Astronomer Royal for

Ireland failed in 1827.
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Woodhouse.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Woodhouse.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Woodhouse.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Woodhouse.html


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and a Fellow of the Royal Society of London in 1836, having received the Society'sCopley Medal in 1831. He gave the Bakerian lecture to the Society entitled On thetheoretical explanation of an apparent new polarity of light in 1840. He received theSociety's Royal Medal in 1845 for a paper on the Irish tides.The Royal Astronomical Society elected Airy to be their President in 1845. Then, in

1851, Airy was elected President of the British Association, and in 1871 he was electedPresident of the Royal Society of London holding the post for two years. The Institut deFrance elected him to membership to fill the position which became vacant on the deathof John Herschel in 1872 and in the same year he accepted a knighthood havingdeclined it on three previous occasions on the grounds that he could not afford the fees.Soon after this, in 1874, he organised an expedition to observe the transit of Venus.Outside his professional scientific interests, Airy was a man of broad tastes. He liked

poetry, history, theology, antiquities, architecture, engineering, and geology. He evenpublished papers on his other interests including one which tried to identify the exactplace where Julius Caesar landed in Britain and the exact place from which he left. Inaddition he published a number of papers on religious matters.

There were certainly sides to his character which made him unpopular with thosearound him. We have already mentioned how he was a snob at school. In later life hewas sarcastic and enforced a rigid discipline on his staff at the Royal Observatory. In hisdefence we would have to note that he enforced such a rigid discipline on himself that itmust have seemed natural to him to expect the same from others.An illustration of Airy's personality is shown from his long running disagreements withBabbage. They had a dispute over the quality of a telescope in 1832, he stated thatBabbage's calculating engine was "worthless" ten years later and effectively stoppedgovernment funding of the project, and in 1854 he took the side of the narrow gauge forrailways while Babbage supported the wide gauge. In all these disputes Airy came outthe winner, but it is far from clear that he took the "right" side in the arguments.We should end with a few words on Airy's importance as a scientist. His own wordscertainly show that he had a realistic view of himself (see for example [1]):-... in those parts of astronomy which ... [require] only method and judgement, with verylittle science in the persons employed, we have done much; while in those whichdepend exclusively on individual effort we have done little ... our principal progress has

been made in the lower branches of astronomy while to the higher branches of sciencewe have not added anything.Eggen writes in [1]:-Airy was not a great scientist, but he made great science possible.However, others have a higher opinion of Airy's achievements. Chapman [6] believes

that:-Airy has been unfairly relegated to the scientific sidelines ...His son summed up Airy's life as follows [3]:-The life of Airy was essentially that of a hard-working business man, and differed fromthat of other hard-working people only in the quality and variety of his work. It was notan exciting life, but it was full of interest ...
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Babbage.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Babbage.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Babbage.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Babbage.html


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Richard von MisesBorn: 19 April 1883 in Lemberg, Austria (now Lvov, Ukraine)Died: 14 July 1953 in Boston, Massachusetts, USA

Richard von Mises was born in Lvov when it was under Austrian control and known asLemberg. His father, Arthur Edler von Mises, worked for the Austrian State Railwaysas a technical expert and his mother was Adele von Landau. Richard was the second sonof the family, the elder son being Ludwig von Mises who went on to become as famousas Richard. Ludwig, who was about eighteen months older than Richard, went on to

become an economist who contributed to liberalism in economic theory and made hisbelief in consumer power an important part of that theory. Richard also had a youngerbrother, who died as an infant.It was on the technical course that von Mises embarked, studying mathematics, physicsand engineering at the Technische Hochschule in Vienna. After graduating he wasappointed as Georg Hamel's assistant in Brnn. The city of Brnn is today called Brnoin the south-eastern Czech Republic. Up to World War II the inhabitants were

predominantly German, although today they are now mainly Czech. Von Mises wasawarded a doctorate from Vienna in 1907 and the following year he was awarded hishabilitation from Brnn, becoming qualified to lecture on engineering and machineconstruction.He was professor of applied mathematics at Strasburg from 1909 until 1918, althoughthis was a period which was interrupted by World War I. Even before the outbreak ofthe war von Mises had qualified as a pilot and he gave the first university course on

powered flight in 1913. When war broke out von Mises joined the Austro-Hungarianarmy and piloted aircraft. He had lectured on the design of aircraft before the war and

he now put this into practice leading a team which constructed a 600-horsepower planefor the Austrian army in 1915.After the end of the war von Mises was appointed to a new chair of hydrodynamics andaerodynamics at the Technische Hochschule in Dresden. Appointed in 1919 he soonmoved again, this time to the University of Berlin to become the director of the newInstitute of Applied Mathematics which had been set up there. Schmidt had argued forthe setting up of the Institute in 1918:-The pervasion of practical life by mathematical methods, as a result of the developmentof technology before the war and, above all, the unexpected need for ... mathematiciansduring the war make it an undeniable necessity to install applied mathematics at thelargest Prussian university... Among university students, however, one frequently finds

the opinion that applied mathematics is a subject of inferior importance, which does not
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Schmidt.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Schmidt.html


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require one's full attention. to create a new tradition, it needs an important personality ofapproved name. Such a personality can only be attracted by a full professorship.Theses words by Schmidt show great wisdom, and the authorities did indeed create thefull professorship and made an inspired choice for the first holder in von Mises.Ostrowski wrote in a 1965 lecture (see for example [16]):-

Only with the appointment of Richard von Mises to the University of Berlin did the firstserious German school of applied mathematics with a broad sphere of influence comeinto existence. Von Mises was an incredibly dynamic person and at the same timeamazingly versatile like Runge. He was especially well versed in the realm oftechnology.The Institute of Applied Mathematics flourished under his control. In 1921 he foundedthe journal Zeitschrift fr Angewandte Mathematik und Mechanik and he became theeditor of the journal. In the first edition he wrote an introduction stressing the widerange of applied mathematics and also the fact that the line between pure and appliedmathematics is not a fixed one, but one which changes over time as different areas of"pure mathematics" find applications in practical situations.

He set up a new curriculum for applied mathematics at the university which spread oversix semesters and included applications of mathematics to astronomy, geodesy andtechnology. It was not a "soft option" and von Mises went out of his way to stress thatapplied mathematics was every bit as rigorous as pure mathematics requiring [13]:-... a mathematical model of the widest possible generality, where the argument could bemade with clarity, elegance, and rigour.His Institute rapidly became a centre for research into areas such as probability,statistics, numerical solutions of differential equations, elasticity and aerodynamics.Von Mises was also an excellent lecturer. Collatz, one of his students, wrote:-I was enrolled in Berlin in 1930. ... Professor Dr Richard von Mises [gave] excellent,very clear and stimulating lectures on applied analysis ....The paper [8], written by Collatz, discusses von Mises' work on numerical mathematics,discusses his founding of the journal Zeitschrift fr Angewandte Mathematik undMechanik and looks at the difficulties faced by von Mises in bringing up the status ofapplied mathematics during the 1920s and early 1930s.On 30 January 1933, however, Hitler came to power and on 7 April 1933 the CivilService Law provided the means of removing Jewish teachers from the universities, andof course also to remove those of Jewish descent from other roles. All civil servantswho were not of Aryan descent (having one grandparent of the Jewish religion madesomeone non-Aryan) were to be retired. Von Mises in one sense was not Jewish for hewas a Roman Catholic by religion. He still fell under the non-Aryan definition of the act

but there was an exemption clause which exempted non-Aryans who had fought inWorld War I. Von Mises certainly qualified under this clause and it would have allowedhim to keep his chair in Berlin in 1933. He realised, quite correctly, that the exemptionclause would not save him for long. On the 10 June 1933 he wrote to von Krmn abouta young German, Walter Tollmien, who was looking for a position:-I have to advise you that the irrevocable prerequisite for any kind of employment orscholarship or suchlike is to make a statement on his word of honour that his fourgrandparents are Aryan and in particular are of non-Jewish descent. ... I believe that in afavourable case the prospects are not quite so bad as indeed a large part of all the

previous candidates can be omitted under the present law.Von Krmn forwarded the letter to Tollmien, writing "Indeed a document of our time"

on the back!
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Ostrowski.htmlhttp://./javascript:win1('./Glossary/probability_theory',350,200)http://./javascript:win1('./Glossary/differential_equation',350,200)http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Karman.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Karman.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Karman.htmlhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Karman.htmlhttp://./javascript:win1('./Glossary/differential_equation',350,200)http://./javascript:win1('./Glossary/probability_theory',350,200)http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Ostrowski.html


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Von Mises saw an offer of a chair in Turkey as a way out of his predicament inGermany but he tried to ensure that his pension rights were preserved. On 12 October1933 he wrote to the ministry explaining that it would benefit Germany if he accepted a

post in Turkey and that he should be allowed to retain his pension rights for his 24 yearsof service. He received the reply that he would have to relinquish all rights of a salary, a

pension or support for his dependants. He protested in a further letter to the Ministrythat he was legally entitled to rights that he was not prepared to relinquish. The NaziTheodor Vahlen wanted to take over as director of the Institute despite poor academicqualities. He promised von Mises that if he would support him to succeed as Director ofthe Institute then he would ensure that von Mises would not lose his pension rights.In October 1933 von Mises wrote his letter to support Vahlen as his successor. Collatz,von Mises' student, wrote:-I took my Staatsexamen in November 1933, and Professor von Mises examined me onthe day before his departure. The same day, he talked to me for about one hour, givingadvice for my further research ...Vahlen was appointed Director of the Institute in December 1933. Having taken up the

new chair in Istanbul, von Mises received a letter in January 1934 denying him anyrights at all. It was something that von Mises continued to feel extremely aggrievedabout, writing to the Ministry in 1953, shortly before his death, still trying to restore hisrights.The mathematician Hilda Geiringer followed him to Istanbul in 1934. There she wasappointed as professor of mathematics.In 1938 Kemal Atatrk died and those in Turkey who had fled from the Nazis fearedthat their safe haven would become unsafe. In 1939 von Mises left Turkey for theUnited States. He became professor at Harvard University and in 1944 he wasappointed Gordon-McKay Professor of Aerodynamics and Applied Mathematics there.Geiringerfollowed him to the USA and they were married in 1943.Von Mises worked on fluid mechanics, aerodynamics, aeronautics, statistics and

probability theory. He classified his own work, not long before his death, into eightareas: practical analysis, integral and differential equations, mechanics, hydrodynamicsand aerodynamics, constructive geometry, probability calculus, statistics and

philosophy. He introduced a stress tensor which was used in the study of the strength ofmaterials. His studies of wing theory for aircraft led him to investigate turbulence.Much of his work involved numerical methods and this led him to develop newtechniques in numerical analysis. His most famous, and at the same time mostcontroversial, work was in probability theory.He made considerable progress in the area of frequency analysis which was started by

Venn. He combined the idea of a Venn limit and a random sequence of events.Ostrowski in the same lecture which we quoted from above wrote (see for example[16]):-Because of his dynamic personality his occasional major blunders were somehowtolerated. One has even forgiven him his theory of probability.This judgement by Ostrowski is rather harsh, however, and many consider von Mises'theories to be important in the development of the subject. After the measure theoryapproach by Kolmogorov had become favoured by almost all statisticians over vonMises' limiting frequency theory approach, there was a return to von Mises ideas andthere was an attempt to incorporate them into the measure theoretic approach ofKolmogorov who wrote himself in 1963:-

... that the basis for the applicability of the results of the mathematical theory ofprobability to real 'random phenomena' must depend on some form of the frequency
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concept of probability, the unavoidable nature of which has been established by vonMises in a spirited manner.In paper [18] discusses:-... von Mises' notion of a random sequence in the context of his approach to probabilitytheory. [The author claims] that the acceptance of Kolmogorov's rival axiomatisation

was due to a different intuition about probability getting the upper hand, as illustratedby the notion of a martingale.Phillip Frank, writing in [4] says:-... in looking over the work of von Mises ... we cannot fail to recognise a wholespectrum of research, extending from the philosophical meaning of science to practicalmethods of numerical computation. ... von Mises has always been a truly broad-mindedman ... notwithstanding the wide range of his topics, his work shows a great intrinsicunity: starting from a definite center, it branches out in systematic investigations of agreat diversity of problems. ... von Mises chose the topics according to a very definiteview-point, determined by his ideas about the essence and method of every thoroughlyscientific research.

In von Mises' book Positivism: A Study in Human Understanding (1951) he expressedhis views on science and life. He subscribes to a doctrine of positivism in the booksaying:-Positivism does not claim that all questions can be answered rationally, just as medicineis not based on the premise that all diseases are curable, or physics does not start outwith the postulate that all phenomena are explicable. But the mere possibility that theremay be no answers to some questions is no sufficient reason for not looking foranswers, or for not using those which are attainable.This interest in philosophy was only one of von Mises' interests outside the realm ofmathematics. Another was the fact that he was an international authority on the Austrian

poet Rainer Maria Rilke (1875-1926).In 1950 von Mises was offered honorary membership of the East German Academy ofScience. This was difficult for von Mises, particularly in McCarthy era America whereany link with communism would have been viewed as a serious offence. He sadlydeclined in a letter written on 15 September 1950:-I would very gladly accept the nomination in remembrance of my teaching activities inBerlin and thus re-establish the bond which connected me for a long time to the Germanscientific life. Unfortunately the present circumstances in Germany as well as those inthis country are such that the acceptance of such a distinction could be interpreted as a

political demonstration on my part. ... I only relinquish acceptance of this nominationunder the pressure of outward circumstances, a nomination which I regard as a great

honour in every respect.
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Jean Victor PonceletBorn: 1 July 1788 in Metz, Lorraine, FranceDied: 22 Dec 1867 in Paris, France

Jean-Victor Poncelet was a pupil of Monge. His development of the pole and polar linesassociated with conics led to the principle of duality.Poncelet took part in Napoleon's 1812 Russian campaign as an engineer. He was left fordead at Krasnoy and imprisoned until 1814 when he returned to France. During hisimprisonment he studied projective geometry. He also wrote a treatise on analyticgeometry Applications d'analyse et de gomtrie based on what he had learnt at thecole Polytechnique but it was only published 50 years later.From 1815 to 1825 he was a military engineer at Metz and from 1825 to 1835 professorof mechanics there. He applied mechanics to improve turbines and waterwheels morethan doubling the efficiency of the waterwheel.Poncelet was one of the founders of modern projective geometry simultaneouslydiscovered by Joseph Gergonne and Poncelet. His development of the pole and polarlines associated with conics led to the principle of duality. He also discovered circular

points at infinity.He published Trait des proprits projectives des figures in 1822 which is a study ofthose properties which remain invariant under projection. This work containsfundamental ideas of projective geometry such as the cross-ratio, perspective, involutionand the circular points at infinity. While writing this book he consulted with Servois.Poncelet published Applications d'analyse et de gomtrie in two volumes: 1862 and1864.
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George Stokes' father, Gabriel Stokes, was the Protestant minister of the parish ofSkreen in County Sligo. His mother, Elizabeth Haughton, was the daughter of a ministerof the church, so George Stokes' upbringing was a very religious one. He was theyoungest of six children and every one of his three older brothers went on to become a

priest. As the priest of the church in Cambridge which Stokes later attended wrote (see[15]):-Though he was never narrow in his faith and religious sympathies, he always held fast

by the simple evangelical truths he learnt from his father...In [4] the atmosphere in which George grew up is described in words which are morecolourful than those which might be used today:-The home-life in the Rectory at Skreen was very happy, and the children grew up in thefresh sea-air with well-knit frames and active minds. Great economy was required tomeet the educational needs of the large family...It was not only religious teaching, but a wider introduction to education, which GabrielStokes was able to give his children. In particular, having studied at Trinity CollegeDublin, he was able to teach George Latin grammar. Before going to school George wasalso taught by the clerk in his father's parish in Skreen. Leaving Skreen in 1832, Georgeattended school in Dublin. He spent three years at the Rev R H Wall's school in Hume

Street, Dublin, but he was not a boarder at the school, living for these three years withhis uncle John Stokes. In fact the family finances would not have allowed him a moreexpensive education, but at this school [4]:-He pursued the usual school studies, and attracted the attention of the mathematicalmaster by his solution of geometrical problems.It was during George's three years in Dublin that his father died and this event had, asone would expect, a major effect on the young man.In 1835, at the age of 16, George Stokes moved to England and entered Bristol Collegein Bristol. The two years which Stokes spent in Bristol at this College were importantones in preparing him for his studies at Cambridge. The Principal of the College, DrJerrard, was an Irishman who had attended Cambridge University with William Stokes,

one of George's elder brothers. Dr Jerrard was himself a mathematician but Stokes wastaught mathematics at Bristol College by Francis Newman (who was the brother of John


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Henry Newman, later Cardinal Newman, who became the leader of the OxfordMovement in the Church of England which was founded in 1833). Clearly Stokes talentfor mathematics was shown during his studies at Bristol College, for he wonmathematics prizes and Dr Jerrard wrote to him (see [4]):-I have strongly advised your brother to enter you at Trinity, as I feel convinced that you

will in all human probability succeed in obtaining a Fellowship at that College.It was not Trinity, rather Pembroke College, Cambridge, which Stokes entered in 1837.There are slight inconsistencies in what his mathematical background was on enteringCambridge. In the course at Bristol College (according to the College literature) (see[5]):-... a student was to become acquainted with the differential and integral calculus and togo on to statics, dynamics, conic sections and the first three sections ofNewton'sPrincipia...However, Stokes himself wrote in 1901 (see for example [4]):-I entered Pembroke College, Cambridge in 1837. In those days boys coming to theUniversity had not in general read so far in mathematics as is the custom at present; and

I had not begun the differential calculus when I entered the College, and had onlyrecently read analytical sections.In Stokes' second year at Cambridge he began to be coached by William Hopkins, afamous Cambridge coach who played a more important role than the lecturers. Stokeswrote [4]:-In my second year I began to read with a private tutor, Mr Hopkins, who was celebratedfor the very large number of his pupils gaining high places in the Universityexaminations for mathematical honours...Hopkins was to exert a strong influence on the direction of Stokes' mathematicalinterests. Hopkins [5]:-... praised the study of physical astronomy and physical optics, for example, becausethey revealed mathematics to be 'the only instrument of investigation by which mancould possibly have attained to a knowledge of so much of what is perfect and beautifulin the structure of the material universe, and the laws that govern it'.In 1841 Stokes graduated as Senior Wrangler (the top First Class degree) in theMathematical Tripos and he was the first Smith's prizeman. Pembroke Collegeimmediately gave him a Fellowship. He wrote [4]:-After taking my degree I continued to reside in College and took private pupils. Ithought I would try my hand at original research....It was William Hopkins who advised Stokes to undertake research into hydrodynamicsand indeed this was the area in which Stokes began to work. In addition to Hopkins'

advice, Stokes was also inspired to enter this field by the recent work of George Green.Stokes published papers on the motion of incompressible fluids in 1842 and 1843, inparticular On the steady motion of incompressible fluids in 1842. After completing thisresearch Stokes discovered that Duhamel had already obtained similar results but, sinceDuhamel had been working on the distribution of heat in solids, Stokes decided that hisresults were obtained in a sufficiently different situation to justify him publishing.Stokes then continued his investigations, looking at the situation where he took intoaccount internal friction in fluids in motion. After he had deduced the correct equationsof motion Stokes discovered that again he was not the first to obtain the equations since

Navier, Poisson and Saint-Venant had already considered the problem. In fact thisduplication of results was not entirely an accident, but was rather brought about by the

lack of knowledge of the work of continental mathematicians at Cambridge at that time.Again Stokes decided that his results were obtained with sufficiently different
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assumptions to justify publication and he published On the theories of the internalfriction of fluids in motion in 1845. The work also discussed the equilibrium and motionof elastic solids and Stokes used a continuity argument to justify the same equation ofmotion for elastic solids as for viscous fluids.Perhaps the most important event in the recognition of Stokes as a leading

mathematician was his Report on recent researches in hydrodynamics presented to theBritish Association for the Advancement of Science in 1846. But a study of fluids wascertainly not the only area in which he was making major contributions at this time. In1845 Stokes had published an important work on the aberration of light, the first of anumber of important works on this topic. He also used his work on the motion of

pendulums in fluids to consider the variation of gravity at different points on the earth,publishing a work on geodesy of major importance On the variation of gravity at thesurface of the earth in 1849.In 1849 Stokes was appointed Lucasian Professor of Mathematics at Cambridge. In1851 Stokes was elected to the Royal Society, awarded the Rumford medal of thatSociety in 1852, and he was appointed secretary of the Society in 1854. The Lucasian

chair paid very poorly so Stokes needed to earn additional money and he did this byaccepting an additional position to the Lucasian chair, namely that of Professor ofPhysics at the Government School of Mines in London.Stokes' work on the motion of pendulums in fluids led to a fundamental paper onhydrodynamics in 1851 when he published his law of viscosity, describing the velocityof a small sphere through a viscous fluid. He published several important investigationsconcerning the wave theory of light, such as a paper on diffraction in 1849. This paperis discussed in detail in [11] in which the authors write:-...the results of Stokes are related to the elastic theory of light, and supplement andexpand a number of questions, previously studied for the most part in the works of ACauchy. Stokes's methods for solving diffraction problems, differing considerably fromthe methods employed by Cauchy, form the basis of the further studies of themathematical theory of the phenomenon of diffraction.Stokes named and explained the phenomenon of fluorescence in 1852. Hisinterpretation of this phenomenon, which results from absorption of ultraviolet light andemission of blue light, is based on an elastic aether which vibrates as a consequence ofthe illuminated molecules. The paper [9] discusses this in detail and is particularlyinteresting since the author makes full use of Stokes' unpublished notebooks.In 1854 Stokes theorised an explanation of the Fraunhofer lines in the solar spectrum.He suggested these were caused by atoms in the outer layers of the Sun absorbingcertain wavelengths. However when Kirchhoff later published this explanation, Stokes

disclaimed any prior discovery.Stokes' career certainly took a rather different tack in 1857 when he moved from hishighly active theoretical research period into one where he became more involved withadministration and experimental work. Certainly his marriage in 1857 was notunconnected with the change in tack and, particularly since it gives us an insight intoStokes' personality, we shall look at the events. Stokes became engaged to marry MarySusanna Robinson, the daughter of the astronomer at Armagh Observatory in Ireland. In[4] a number of letters from Stokes to Mary Susanna Robinson are given. On 21January 1857 he wrote of his feelings for her:-I was capable of being moved, mathematically, as it were, by the belief that a particularcourse was right; and I do believe that God put these views in my mind, working by

means of that which was in me to supply that which was wanting.Three days later he wrote that she had stopped him becoming an old bachelor:-
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I feel that perhaps my marriage with you would be even the turning-point of mysalvation.A further three days later he wrote:-You are quite right in saying that it is well not to go brooding over one's own thoughtsand feelings, and in a family that is easy, but you don't know what it is to live utterly

alone.On the 31 March 1857 he wrote again expressing his feelings in rather mathematicalterms:-I too feel that I have been thinking too much of late, but in a different way, my headrunning on divergent series, the discontinuity of arbitrary constants, ... I often thoughtthat you would do me good by keeping me from being too engrossed by those things.These letters clearly did not express the love that Mary hoped to find in them and whenStokes wrote her a 55 page letter (which was possibly deliberately destroyed) about theduty he felt towards her, she came close to calling off the wedding at the last moment.On receiving her letter showing that she was unhappy to go ahead with the marriageStokes replied:-

Then it is right that you should even now draw back, nor heed though I should go to thegrave a thinking machine unenlivened and uncheered and unwarmed by the happinessof domestic affection.The marriage did go ahead and Stokes certainly turned away from his life of intensemathematical research. It may appear from the above quotations that in fact Stokes wasreally looking for this change in his life and perhaps he sought marriage partly so thatthis change in his life-style could come about.

Now Fellows at Cambridge had to be unmarried, and so on his marriage in 1857 Stokeshad to give up his fellowship at Pembroke College. However, a change in the rules in1862 allowed married men to hold fellowships and he was able to take up the fellowshipat Pembroke again. Stokes continued as secretary of the Royal Society from hisappointment in 1854 until 1885 when he was elected President of the Society. He heldthe position of President until 1890. He was also president of the Victoria Institute from1886 until his death in 1903. There were other administrative tasks which he undertook.In 1859 he had written to Thomson saying:-I have another iron in the fire now: I have just been appointed an additional secretary ofthe Cambridge University Commission.P G Tait mentioned this in his criticism of the way that science was organised in Britain[16]:-What a comment on things as they are is furnished by the spectacle of genius like that ofStokes' wasted on the drudgery of Secretary to the Commissioners for the University of

Cambridge; or of a Lecturer in the School of Mines; or the exhausting labour and totallyinadequate remuneration of a Secretary to the Royal Society.Stokes received the Copley medal from the Royal Society of London in 1893 and hewas given the highest possible honour by his College when he served as Master ofPembroke College in 1902-3.Stokes' influence is summed up well by Parkinson in [1]:-... Stokes was a very important formative influence on subsequent generations ofCambridge men, including Maxwell. With Green, who in turn had influenced him,Stokes followed the work of the French, especially Lagrange, Laplace, Fourier, Poisson,and Cauchy. This is seen most clearly in his theoretical studies in optics andhydrodynamics; but it should also be noted that Stokes, even as an undergraduate,

experimented incessantly. Yet his interests and investigations extended beyond physics,
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for his knowledge of chemistry and botany was extensive, and often his work in opticsdrew him into those fields.One notable omission from his publication list was a treatise on light. This omissionwas in part due to the change in his research output after 1857 but it was also partly dueto not wishing to report upon speculative ideas in a field which was in a rapid state of

progress. Stokes' failure to publish a treatise on optics is discussed in detail in [7].However, he did lecture on optics in his Burnett lectures at the University of Aberdeenin 1891-93 and these lectures were published.Stokes' mathematical and physical papers were published in five volumes, the first threeof which Stokes edited himself in 1880, 1883 and 1891. The last two were edited by SirJoseph Larmorwith the work being completed in 1905.These comments about Stokes' character in [3] are interesting:-From 1887 to 1892 he was one of the members of Parliament for Cambridge University.In spite, or perhaps because of, his great and profound knowledge and remarkableability, he rarely spoke in the House of Commons, but was always listened to withattention. In private life his simplicity and modesty were as conspicuous as his great

attainments.
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