CAPÍTULO IV 1

CAPÍTULO IV

ÍNDICE

4.1 Introducción IV-1 4.2 Formulación en desplazamientos. Ecuaciones de Navier IV-3 4.4 Problema de contorno. Condiciones de contorno IV-11 4.4.1 Condiciones de contorno en tensiones IV-13 4.4.3 Condiciones de contorno mixtas IV-17 4.4.4 Condiciones de contorno para cargas puntuales IV-19 4.4.5 Condiciones de contorno considerando la simetría IV-20 4.4.6 Condiciones de contorno considerando la antisimetría IV-22 4.6 Consideración del efecto de la temperatura IV-27 4.6.1 Ecuaciones de Navier termoelásticas IV-32 4.6.3 Analogía de Neumann - Duhamel IV-33

CAPÍTULO IV 2

CAPÍTULO IV

PLANTEAMIENTO GENERAL

DEL PROBLEMA ELÁSTICO

4. 1 Introducción En la formulación general de la Mecánica de los Medios Continuos se establece que cualquier medio debe satisfacer las siguientes leyes: - Conservación de la masa - Principio de Momentum - Principio de Momento de Momentum - Primera y segunda ley de la termodinámica Aparte de estas leyes, que son básicas para un Medio Continuo, existen unas particulares de la Elasticidad, que son: - Desplazamiento, deformación y compatibilidad de deformaciones - Leyes constitutivas o de comportamiento del material De la leyes enunciadas, el principio de conservación de la masa se cumple si:

( ) ( ) ndeformacióladedespuésndeformacióladeantes dvdv ⋅=⋅ ρρ

donde ρ es la densidad y dv es el diferencial de volumen. Sea un sistema de coordenadas (x1,x2,x3) y un sólido cuyo volumen es B(t) en un instante de tiempo determinado. Sea r el vector posición de una partícula del sólido respecto a un origen de coordenadas. Sea V el vector velocidad de una partícula de coordenadas ( )ppp xxx 321 ,, . Se define Momentum o Momentum lineal de un cuerpo en la configuración B(t) a:

∫=Μ)(tB

vdρV

y Momentum del Momentum:

∫ ×=ΜΜ)(tB

vdρVr

CAPÍTULO IV 3

En ambas expresiones ρ = densidad del material y la integración es sobre el volumen B(t). La aplicación de la 3º ley de Newton ( F = m a ) conduce a:

∫∫∫∫= +≡==∂∂

)()()(

.

tBStBtBdvdsdvdv

tM vFTFaV ρρ

Es decir la variación del Momentum es igual a la fuerza total F aplicada sobre el cuerpo. Como puede observarse el principio de Momentum está implícito en las ecuaciones de equilibrio interno, por lo que su cumplimiento conlleva la aceptación del Principio. Por otra parte la variación del Momentum del Momentum es igual al momento torsor aplicado:

∫∫∫∫ ×+×≡=×=∂∂

×=)()()(

.

tBST

tBtBdvdsMdvdv

tMM vFrTrFrVr ρ

El momento torsor se toma con respecto a un eje que pasa por el mismo punto origen que el vector posición. El principio del momento de Momentum está implícito en el hecho de que el tensor de tensiones sea simétrico. Efectivamente, como se recordará en el capítulo I se demostró que el equilibrio de momentos respecto a un eje implicaba la simetría del tensor de tensiones. La primera ley de la Termodinámica establece, en su forma habitual, que la variación de energía interna es igual al trabajo realizado por las tensiones.

ijijU εσ21

=

La segunda ley de la termodinámica establece la producción de entropía en un sistema elástico. Al igual que la primera ley se emplea con profusión, la segunda se usa ocasionalmente e implícitamente, como por ejemplo para asegurar que el trabajo neto realizado durante el proceso de carga y descarga es positivo. Una vez establecido el marco de trabajo, se repasan ciertas conclusiones obtenidas en capítulos anteriores, que son relevantes para el objetivo de este apartado. - Relación desplazamiento-deformación:

( )ijjiij uu ,,21

+=ε 4-1

- Ley de comportamiento:

ijkkijij Ev

Ev δσσε −

+=

1 4-2

CAPÍTULO IV 4

ijkkijij G δλεεσ += 2 4-3 - Ecuaciones de equilibrio: ( ) 0, =+ ivjij Fσ 4-4 Estas expresiones representan un conjunto de 15 ecuaciones con 15 incógnitas. Las ecuaciones son: - 6 ecuaciones en deformaciones - 6 ecuaciones en tensiones - 3 ecuaciones de equilibrio y las incógnitas: - 6 deformaciones ( ε ), 6 tensiones (σ ), y tres desplazamientos. El problema elástico está pues formulado y además es consistente, en el sentido de que cumple todas las leyes primordiales de la Mecánica del Medio Continuo. La resolución de estas ecuaciones implica una integración, y por tanto, unas constantes de integración que se calculan forzando el cumplimiento de las Condiciones de Contorno. Sabido pues que se cumplen todas las restricciones de la Mecánica del Medio Continuo es necesario recordar las hipótesis simplificativas realizadas que son: - Pequeñas deformaciones y pequeños desplazamientos - No se han considerado los efectos de inercia ni del tiempo - El material es homogéneo y de comportamiento isótropo y lineal. Por tanto la solución será válida si el sólido es sometido a un estado de cargas tal que se pueda considerar un régimen de pequeñas deformaciones y desplazamientos, además la aplicación de las cargas sea lo suficientemente lenta como para no contemplar efectos dinámicos, y el material constitutivo sea isótropo, homogéneo y lineal. 4.2 Formulación en desplazamientos. Ecuaciones de Navier Consiste en eliminar las tensiones y deformaciones y establecer unas ecuaciones donde solamente figuren como incógnitas los desplazamientos. Para lograr el objetivo se parte de las ecuaciones anteriores y sustituir:

( )ijjiij uu ,,21

+=ε 4-5

en: ijkkijij G δλεεσ += 2 4-6

CAPÍTULO IV 5

Obteniendo:

( ) ( ) ijijjiij uuuuuG δλσ 3,32,21,1,,212 ++++= 4-7

Sustituyendo esta expresión en las ecuaciones de equilibrio (4-4) se obtiene: ( ) ( )[ ] ( ) 0

,3,32,21,1,, =+++++ ivjijijji FuuuuuG δλ 4-8

Que simplificando resulta: ( ) ( )[ ] ( ) 0

,,,, =+++ ivjijkkijji FuuuG δλ 4-9

Teniendo en cuenta que: ( )[ ] ( ) ijjkikikkjijkk uuuu ,,,,,, ===δ 4-10

resulta finalmente: ( ) ( ) 0,, =+++ ivijjjji FuGGu λ 4-11 Que son las denominadas ecuaciones de Navier. El desarrollo de estas ecuaciones se escribe a continuación:

( ) ( )

( ) ( )

( ) ( ) 0

0

0

33

3

2

2

1

1

323

32

22

32

21

32

23

3

2

2

1

1

223

22

22

22

21

22

13

3

2

2

1

1

123

12

22

12

21

12

=+

∂∂

+∂∂

+∂∂

∂∂

++

∂

∂+

∂

∂+

∂

∂

=+

∂∂

+∂∂

+∂∂

∂∂

++

∂

∂+

∂

∂+

∂

∂

=+

∂∂

+∂∂

+∂∂

∂∂

++

∂

∂+

∂

∂+

∂

∂

v

v

v

Fxu

xu

xu

xG

xu

xu

xuG

Fxu

xu

xu

xG

xu

xu

xuG

Fxu

xu

xu

xG

xu

xu

xuG

λ

λ

λ

4-12

que como puede observarse son 3 ecuaciones diferenciales, en derivadas parciales acopladas entre si, con tres incógnitas: las componentes del vector desplazamiento. Las ecuaciones de Navier también pueden ser escritas en notación vectorial de la siguiente forma: ( ) ( ) ( ) 02 =+⋅∇∇++∇ vFuGuG λ 4-13

CAPÍTULO IV 6

o bien:

( ) ( ) ( ) ( ) ( ) ( )( ) 0,,,,,, 321321

3212 =+⋅∇

∂∂

∂∂

∂∂

++∇ VVV FFFuxxx

Guuu λG

Otra expresión vectorial se logra haciendo uso de la descomposición de Helmhotz: ( ) ( ) uuu 2∇−⋅∇∇=×∇×∇ 4-14 que aplicada a 4-13 conduce a: ( ) ( ) ( ) ( ) 0=+×∇×∇−⋅∇∇+ vFuGuGλ 4-15 La solución de las tres ecuaciones diferenciales implica forzosamente una integración y por tanto determinar una serie de constantes mediante la aplicación de las condiciones de contorno. Es fácil intuir, a la vista de la forma de las ecuaciones, que no es sencillo hallar una solución y por tanto muy pocos problemas han podido ser resuelto. Entre los problemas factibles de resolución se encuentran los que presentan simetría de revolución (ya que en ellos el rotacional es nulo, y además el desplazamiento tiene una sola componente), y los problemas asociados a medios infinitos (pues no tienen condiciones de contorno). 4. 4 Problema de contorno. Condiciones de contorno Matemáticamente hablando existen dos grandes grupos de problemas: - Problemas de Contorno - Problemas de valor inicial Se denominan Problemas de Contorno aquellos en que, usando la misma pieza, la semejanza o disimilitud entre un problema u otro radica en: a) forma y lugar de sujeción de la pieza b) valor, tipología, y área en que actúan las cargas. Problemas de valor inicial se refieren a problemas evolutivos en que se alcanza una u otra solución dependiendo de los valores iniciales del problema. Lógicamente este no es el objetivo de la asignatura, por lo que no se añadirá nada más al respecto Dependiendo del tipo de condición de contorno que esté sometida una pieza recibe una denominación determinada así: - Condición de Neumann: Cuando las incógnitas son los desplazamientos (problema

de primer grado) - Condición de Dirichlet: Cuando la incógnita es el vector Tensión (problema de

segundo grado ya que en definitiva la condición de contorno se plantea sobre la variable derivada del desplazamiento)

CAPÍTULO IV 7

- Condición Mixta: Cuando en una parte del contorno se conocen los desplazamientos y en otra el vector tensión.

Sobre cada punto del contorno siempre se conocerá, al menos, una condición de contorno en cada una de las tres direcciones del espacio. Ese conocimiento será del desplazamiento, o del vector tensión pero no ambas a la vez. Es decir si el desplazamiento en dirección x1 es dato, significa que la componente x1 del vector tensión es incógnita y viceversa. Dicho esto a continuación se hace un repaso de las condiciones de contorno más interesantes y su expresión en un problema determinado. 4. 4. 1 Condiciones de contorno en tensiones: Antes de comenzar es necesario decir que las condiciones de contorno de expresan sobre el vector tensión no sobre el Tensor de Tensiones. Este asunto es importante remarcarlo porque la experiencia acumulada dice que este hecho suele llevar al alumno a confusión. Por tanto: la expresión de las condiciones de contorno se hace con el vector tensión, y la asignación de positivo o negativo depende exclusivamente si sigue o no a los ejes coordenados definidos sobre la pieza. Sea el siguiente ejemplo:

Q

Q

P P X1

X2

a b

c d

h

L

Figura 4-1 La expresión de las condiciones de contorno sobre las caras es: Cara ab: n= ( 0,1, 0 ) X2 = h / 2

−≡

=

=

==00

010

232

22

12

2/332331

322221

312111

22

QT

hXhXσσσ

σσσσσσσσσ

En que la asignación de signo negativo a Q se debe a que va en sentido contrario al eje X1

CAPÍTULO IV 8

Cara cd: n= ( 0,- 1, 0 ) X2 = - h / 2

−≡

−

−−=

−

=

−=−=00

01

0

232

22

12

2/332331

322221

312111

22

QT

hXhXσσσ

σσσσσσσσσ

Y de igual forma la asignación de signo negativo a Q se debe a que va en sentido contrario al eje X1 Cara ac: n= ( -1, 0 , 0) X1 = - L / 2

≡

−

−−=

−

=

−==00

001

13

12

11

2/2/332331

322221

312111

12

PT

LXhXσσσ

σσσσσσσσσ

Y la asignación de signo positivo a P se debe a que va en el mismo sentido que el eje X1. Es interesante observar que P es de compresión, aún así lleva signo positivo Cara bd: n= ( 1, 0 , 0) X1 = L / 2

≡

=

=

=00

001

13

12

11

2/332331

322221

312111

1

PT

LXσσσ

σσσσσσσσσ

Y la asignación de signo positivo a P se debe a que va en el mismo sentido que el eje X1. Condiciones de contorno en que la presión no es constante

P

A B

C D

E

F

G

X2

X1

2 H

2L

X3

Figura 4-2

CAPÍTULO IV 9

Sea el caso de la figura 4-2. Se trata de un cuerpo prismático sometido a una presión lineal sobre su cara superior de tal forma que en su extremo izquierdo toma valor P y en el derecho es nula: Las condiciones de contorno serán : Cara ABEG ( x3 = H ) los cosenos directores de la normal son ( 0 , 0, 1 ), y el vector tensión se expresa:

−−

=

=

⋅

12

00

100

33

23

13

333231

232221

131211

Lypσ

σσ

σσσσσσσσσ

4-16

Vector tensión

particularizado para x3 = H

= De igual forma para las caras laterales, por ejemplo la By la ecuación de dicha cara es y=L

=

⋅

010

23

22

12

333231

232221

131211

σσσ

σσσσσσσσσ

Vector tensión particularizado para x2 = L

4. 4.3 Condiciones de contorno en desplazam En primer lugar se verá el caso bidimensional y luego e - Apoyo sobre cojinetes:

X2

X1

A

D

Figura 4-3

Condición de contorno dato

EDF la normal es = (0, 1, 0) ;

=

000

Condición de contorno dato =

ientos

l tridimensional.

B

C

CAPÍTULO IV 10

En este caso las condiciones de contorno sobre la cara DC son las siguientes:

00 21 == uT Ya que el cuerpo puede deslizarse sin rozamiento en dirección X1, y el desplazamiento X2 está impedido. Una ampliación de este caso es el siguiente:

X2

X1

A B

C D

Figura 4-4

Cara DC: T 00 21 == u Cara AD: T 00 12 == u Un caso análogo es el siguiente:

X1

X2

A B

C D

Figura 4-5 En la cara DC:

00 12 == uT

CAPÍTULO IV 11

4. 4. 4 Condiciones de contorno para cargas puntuales. En el caso de cargas puntuales: Fuerzas o Momentos, la aplicación de las condiciones de contorno no puede hacerse directamente como en los casos anteriores. El problema es conceptual en dos sentidos. Por un lado es que las condiciones de contorno se expresan en presiones no en fuerzas, y por otro que una fuerza aplicada sobre un punto

provoca una tensión infinita (AF

=σ si el área A es infinitamente pequeña σ tiende a

infinito). Si se observa microscópicamente, puede considerarse que la fuerza está aplicada sobre un área muy pequeña pero de dimensión finita, consecuentemente la tensión tendrá valores muy altos. Esto trae como consecuencia que en una pequeña zona alrededor de la aplicación de la carga puntual se sobrepasa el límite elástico del material, (lo que implica que esa pequeña zona está plastificada) y por tanto no es de aplicación la teoría de la Elasticidad la pieza. Para obviar este inconveniente se aísla una zona que rodea a la carga puntual y se aplica equilibrio de fuerzas y momentos, así:

P

O O

A A

B B

P

TX1

TX2

X2

X1

Figura 4-6

4. 4. 5 Condiciones de contorno considerando la simetría. Es interesante notar que en muchos problemas se pueden presentar condiciones de simetría o antimetría. Se dice que un cuerpo tiene un eje (un plano en el caso de tres dimensiones) de simetría cuando la porción del cuerpo, cargas y condiciones de apoyo, que queda a un lado del eje (plano) es la imagen especular de la otra porción que está situada al otro lado del eje (plano), tal y como se muestra en la figura siguiente.

CAPÍTULO IV 12

Eje de Simetría

A B

Figura 4-7

Si la pieza es simétrica en geometría y carga, lo lógico es que alguna componente del desplazamiento sea también simétrico. Para ver ello obsérvese los puntos A y B de la figura anterior. Ambos están simétricamente situados; si se admite que la componente horizontal del desplazamiento del punto A es hacia la derecha, entonces la simetría obligaría a que la componente horizontal del desplazamiento del punto B sea el mismo pero hacia la izquierda. Esto significa que justamente los puntos que están situados sobre el eje de simetría han de tener desplazamiento horizontal nulo. Por tanto y como primera conclusión: Si un cuerpo tiene un eje de simetría implica que en sus puntos la componente del desplazamiento perpendicular a dicho eje es nula. A continuación se estudia lo que ocurre con el estado tensional. Para ello se vuelve a los dos puntos anteriores A y B simétricamente situados (figura siguiente). Las tensiones normales en el punto A son especularmente iguales a las del punto B y no cambian de signo. Sin embargo las tensiones tangenciales (se han dibujado separadas de las normales por claridad) de los puntos C y D cambian de signo al realizar una imagen especular. Es decir si las tensiones tangenciales en el punto C son positivas implica que las del punto D serán negativas. Por tanto sobre el eje de simetría no pueden existir tensiones tangenciales. Conclusión para los puntos situados sobre el eje de simetría: a) las tensiones normales no son nulas. b) Las tensiones tangenciales son nulas. c) El desplazamiento normal al eje de simetría es nulo.

CAPÍTULO IV 13

Eje de Simetría

A B

C D

Situación imposible

Figura 4-8

4. 4.6 Condiciones de contorno considerando la antisimetría. Una situación antisimétrica puede generarse a partir de una simétrica simplemente cambiando el signo a las variables simétricas en uno de los lados. Un caso de antisimetría es el que se presenta a continuación:

Eje de Antisimetría

A B

Figura 4-9 Igual que el caso anterior consideremos dos puntos A y B simétricamente situados. Siguiendo la idea expresada inicialmente se dibuja la situación simétrica y luego se invierte. Las conclusiones que se obtienen respecto a los desplazamientos son:

A B

Eje de Simetría

A B

Eje de Antimetría

Figura 4-10

CAPÍTULO IV 14

El desplazamiento vertical ha de ser nulo en el eje de antisimetría y el horizontal no. Para el caso de las tensiones:

Eje de Simetría Eje de Antimetría

A A B B

Figura 4-11 A la vista de la figura es fácil predecir que las tensiones tangenciales van ser distinta de cero en los puntos sobre el eje de antisimetría (pues no cambian de signo al pasar de un lado al otro del eje), y las tensiones normales van a ser nulas en el eje de antisimetría pues si cambian de signo. Conclusión para los puntos situados sobre el eje de antisimetría: a) El desplazamiento en la dirección del eje de antimetría es nulo. b) Las tensiones normales al eje de antimetría son nulas. c) las tensiones tangenciales no son nulas. 4. 6 Consideración del efecto de la Temperatura. En el análisis realizado hasta el momento se ha supuesto que la temperatura se mantenía constante, cuando ello no es el caso se formula lo que se conoce como Termoelasticidad, que podría definirse como la parte de la Elasticidad que estudia el comportamiento de los cuerpos cuando se encuentran sometidos a un campo de temperatura. Cuando el campo de temperatura es variable en el tiempo, es necesario recurrir a la ley de Fourier y formular la Termoelastodinámica. Sin embargo el planteamiento que se realiza a continuación es más sencillo y consiste en suponer un incremento de temperatura y analizar el estado inicial (antes de variar la temperatura) y final (cuando la temperatura se ha estabilizado), siendo las ecuaciones constitutivas dependientes de la cantidad en que ha variado la temperatura, es decir de ∆ T (variable temperatura), pero no del camino seguido. Nota: La ley de Fourier establece que: la cantidad de calor que fluye por unidad de tiempo a través de un elemento de superficie de área unidad es proporcional al gradiente de temperatura en la dirección de la

normal al elemento: nTq∂∂

−= κ donde la constante κ es la conductividad térmica y el signo menos es

debido a que cuanto mayor sea el flujo de calor más desciende la temperatura en el cuerpo.

CAPÍTULO IV 15

En un problema termoelástico la relación entre las tensiones, deformaciones, y temperatura es bastante complicada y, al igual que anteriormente ocurría con las ecuaciones de Navier estáticas, la solución exacta presenta enormes dificultades. Antes de continuar es necesario poner de manifiesto las hipótesis simplificativas que se realizan en el planteamiento de la termoelasticidad: - Suponer que los cambios de temperatura provocan pequeñas deformaciones, consecuentemente las ecuaciones básicas, deformaciones y tensiones, están relacionadas de forma lineal. - El calor producido en el interior del cuerpo en el proceso de deformación puede ser despreciado. - Los cambios de temperatura en el interior del cuerpo, en general, se pueden considerar como lentos, consecuentemente las deformaciones provocadas también serán lentas y los efectos de inercia no se considerarán. - Las constantes elásticas permanecen constantes en todo el proceso. Con estas simplificaciones se obtiene lo que se conoce como Termoelasticidad de pequeñas deformaciones y de pequeños y lentos cambios de temperatura. Históricamente hablando la termoelasticidad comenzó en 1837 cuando Duhamel publicó su famosa " Memoire sur les Phenoménes Thermo-mécaniques", aunque su verdadero desarrollo no llegó hasta la década de 1960 con el desarrollo de los ordenadores y de los métodos aproximados de cálculo que se aplicaron a la resolución de problemas termoelásticos complicados. La teoría de Duhamel se basa en la realidad experimental de que cuando un cuerpo se caliente o enfría se produce una dilatación o contracción, y que cuando se vuelve a la temperatura original el cuerpo recupera su forma inicial (siempre y cuando no se superen unos márgenes establecidos de temperatura). Sea el siguiente sencillo experimento: someter una varilla de longitud L y sección A a un aumento de temperatura. Se parte pues de que se conoce la temperatura inicial (la ambiente) las dimensiones iniciales de la varilla, y el incremento de temperatura a que se la somete.

L0

T0

∆ T

L

Figura 4-12

CAPÍTULO IV 16

Si la longitud final después de elevar la temperatura un ∆ T es L se cumple que:

TLLL ∆+= α00 4-17 Donde α es el Coeficiente de Dilatación Térmica, definido por Duhamel en 1837, y es un parámetro característico de cada material. Operando en la expresión 4-17 :

TLLTLLLL ∆==

∆⇒∆=∆=− αεα

000 4-18

Por tanto una consecuencia de aumentar la temperatura es que se produce una deformación normal de valor α ∆T (el aumento o disminución de temperatura no provoca deformaciones tangenciales). Generalizando a los tres ejes se obtiene:

ijij Tδαε ∆= 4-19 Esta relación, obtenida por Duhamel, es fundamental en la termoelasticidad. A continuación se expone, a modo de ejemplo, los valores del coeficiente α para diversos materiales

MATERIAL α ⋅ 10-6 / º CAcero 11.7

Aluminio 23.4 Hormigón 11.2 Ladrillo 9

Evidentemente si la varilla está sometida a un cambio de temperatura uniforme de tal forma que pueda alargarse o acortarse libremente sin impedimento alguno (figura 4-17) se producirán deformaciones normales pero no aparecerá tensión alguna. De igual forma, si se eleva la temperatura de la varilla pero se impide el desplazamiento en alguna dirección entonces existirá una tensión en esa dirección cuyo valor es (figura 4-18):

σ ε= −E

∆ T

Figura 4-13

A continuación se analiza con detenimiento la afirmación de que la temperatura deba ser uniforme o constante.

CAPÍTULO IV 17

- En primer lugar si un cuerpo es sometido a un incremento de temperatura lineal o

constante y no existe impedimento alguno a su expansión se producirá un estado de deformaciones tal como se muestra en 4-19:

- Una distribución cualquiera de temperatura no crea un estado de deformaciones

como 4-19. Para la demostración de esta afirmación se parte de las ecuaciones de compatibilidad y de la ecuación 4-19. Efectivamente sustituyendo 4-19 en las tres primeras ecuaciones de compatibilidad se obtiene:

( ) ( )

( ) ( )

( ) ( ) 0

0

0

23

2

22

2

23

2

21

2

22

2

21

2

=∂

∆∂+

∂

∆∂

=∂

∆∂+

∂

∆∂

=∂

∆∂+

∂

∆∂

xT

xT

xT

xT

xT

xT

αα

αα

αα

4-20

Y sustituyendo en las tres siguientes se obtiene:

( ) ( ) ( ) 0;0;0

31

2

32

2

21

2=

∂∂

∆∂=

∂∂

∆∂=

∂∂

∆∂

xxT

xxT

xxT ααα 4-21

De 4-20 se obtiene que:

( ) ( ) ( ) 02

3

2

22

2

21

2=

∂

∆∂=

∂

∆∂=

∂

∆∂

xT

xT

xT ααα

que en unión de 4-21 conduce a que:

321 DxCxBxAT +++=∆ 4-22

o lo que es lo mismo: Para que se cumplan las ecuaciones de compatibilidad la distribución de temperatura debe ser lineal o constante. Lógicamente surge de inmediato la siguiente pregunta: ¿ Qué ocurre cuando el incremento de temperatura no es lineal ni constante sino cualquiera? La respuesta a esta pregunta es que el campo de deformaciones total será suma de dos campos de deformaciones: Uno tal como 4-19 (que no cumple compatibilidad) más otro (en principio incógnita y que tampoco cumple por si solo compatibilidad) pero tal que sumado a 4-19 produce un campo de deformaciones total que cumple las ecuaciones de compatibilidad, es decir:

CAPÍTULO IV 18

( ) ( ) ( )incognitaijeccijTotalij εεε +=

−− 194

Neumann propuso en 1841 que el campo de deformaciones incógnita fuese precisamente la ley de Hooke o ley de comportamiento. Es decir:

( ) ijkkijincognitaij Ev

Ev δσσε −

+

=1 4-23

por tanto el campo de deformaciones total será:

( ) ijijkkijTotalij TEv

Ev δαδσσε ∆+−

+

=1 4-24

Así 4-24 representa una ley de comportamiento termoelástica, que se puede invertir y obtener las ecuaciones de Lamé termoelásticas. Para ello despejando σ ij de 4-24 se obtiene:

∆−+

+

= ijijkkijij TEv

vE δαδσεσ

1 4-25

Y para expresar σ kk en función de las deformaciones se suman las tres ecuaciones de 4-25 relativas a las deformaciones normales (es lo mismo que hacer que en 4-25 los índices coincidan i = j = k) obteniendo:

∆−+

+

= TEv

vE

kkkkkk ασεσ 331

4-26

y despejando de esta última kkσ se obtiene:

[ Tv

Ekkkk ∆−

−

= αεσ 321

] 4-27

Sustituyendo en 4-25 y teniendo en cuenta que:

( ) ( ) λ=−⋅+

=+ vv

EvGv

E211

;21

se obtiene finalmente:

ijijkkijij Tv

EG δαδλεεσ ∆−

−+=21

2 4-28

Esta última expresión se denomina ley de Neumann-Duhamel, y es interesante notar ciertos aspectos: - Para ∆T = 0 se obtienen las ecuaciones de Lamé elásticas.

CAPÍTULO IV 19

- La ley de comportamiento termoelástica contempla dos sumandos: Uno corresponde a la ley de comportamiento elástica como si no existiese temperatura, y el otro corresponde a la ley de comportamiento térmica que tiene en cuenta la existencia de temperatura.

- Es inmediato ver que una solicitación térmica cualquiera lleva asociado la aparición de un campo de tensiones (4-28). Al contrario: es posible contemplar una solicitación térmica ( ∆ T Fexternas≠ =0 ;

externas

0 ) superpuesta a un campo de tensiones (∆ T F= ≠0 0; ) de tal forma que la ley de comportamiento final sea 4-28 es decir que el campo total de tensiones sea la suma del elástico ( ∆T = 0) más el térmico correspondiente.

4. 6. 1 Ecuaciones de Navier Termoelásticas. Siguiendo los mismos pasos que en el caso elástico se obtiene: - Ecuaciones de equilibrio: ( ) 0, =+ ivjij Fσ

- Ley de Comportamiento: ijijkkijij Tv

EG δαδλεεσ ∆−

−+=21

2

- Relación Desplazamiento-deformación: ( )ijjiij uu ,,21

+=ε

Introduciendo la tercera en la segunda y el resultado en la primera se obtiene:

( ) ( ) 021 ,,,, =+∆

−−++ iviijkkijji FT

vEuuuG αδλ

Agrupando términos:

( ) ( ) 021 ,,, =+∆

−−++ iviijjjji FT

vEuGuG αλ 4-29

que son las conocidas ecuaciones de Navier en Termoelasticidad. 4. 6. 3 Analogía Neumann - Duhamel. Observando las ecuaciones planteadas es fácil darse cuenta que todas ellas contienen dos partes bien diferenciadas: una es como si no existiese temperatura y otra tiene en cuenta la influencia de ésta. Tal particularidad sugirió la idea siguiente ( propuesta por Neumann y Duhamel): un problema termoelástico puede ser resuelto como uno elástico sin temperatura. Para ello no hay más que observar que las ecuaciones de Navier:

( ) ( ) 021 ,,, =+∆

−−++ iviijjjji FT

vEuGuG αλ

CAPÍTULO IV 20

pueden ser escritas como:

( ) ( ) 021 ,,, =

+∆

−−+++ iviijjjji FT

vEuGuG αλ

y llamando a: ( ) ( )*,21 iVivi FFTv

E=

+∆

−− α se obtiene

( ) ( ) 0*

,, =+++ iVijjjji FuGuG λ De igual forma el vector tensión puede ser escrito como:

jijijkkijjiji nTv

EGnT

∆

−−+== δαδλεεσ

212

Pasando al primer miembro el último sumando del paréntesis se obtiene:

[ ] jijkkijiiji nGnTv

ET δελεδα +=∆−

+ 221

Que puede ser escrita como:

[ ] jijkkiji nGT δλεε += 2*

Donde : *21 iiiji TnTv

E=∆

−+ δαT

Por tanto un problema termoelástico puede ser resuelto como uno elástico siempre que se modifiquen las fuerzas de volumen y de contorno de la siguiente forma:

( ) ( ) ( ) ( ) iiViVVV Tv

EFFquetalFF ∆−

−=→ α21

**

iii nTv

ETTquetalTT ∆−

+=→ α21

**

Sea el siguiente ejemplo Una placa cargada como indica la figura siguiente y se incrementa la temperatura un ∆ T. Hallar el tensor de tensiones

CAPÍTULO IV 21

P

P

∆ T

FV =0

Figura 4-14

Si se aplica la analogía de Neumann-Duhamel debe resolverse el problema elástico equivalente

P

P

Tv

E∆

−α

21Tv

E∆

−α

21 iiV Tv

EF ,21)( ∆

−−= α

Figura 4-15

Por último es necesario tener en cuenta que el vector desplazamiento de ambos problemas es idéntico, al igual que el tensor de deformaciones. Sin embargo el tensor de tensiones calculado a partir del de deformaciones (ecuaciones de Lamé) no es idéntico al verdadero tensor de tensiones. Ello es debido a que el tensor de tensiones en el problema termoelástico, obedece a una ley de comportamiento siguiente:

∆

−−+= ijijkkijij T

vEG δαδλεεσ21

2 4-30

Y en el caso elástico obedece a la ley siguiente:

[ ]ijkkijij G δλεεσ += 2 4-31

CAPÍTULO IV 22

Claude Louis Marie Henri Navier

Born: 10 Feb 1785 in Dijon, France

Died: 21 Aug 1836 in Paris, France

Claude-Louis Navier's father was a lawyer who was a member of the National Assembly in Paris during the time of the French Revolution. However Navier's father died in 1793 when Navier was only eight years old. At this time the family were living in Paris but after Navier's father died, his mother returned to her home town of Chalon-sur-Saône and left Navier in Paris to be cared for by her uncle Emiland Gauthey. Emiland Gauthey was a civil engineer who worked at the Corps des Ponts et Chaussées in Paris. He was considered the leading civil engineer in France and he certainly gave Navier an interest in engineering. Despite encouraging Navier to enter the École Polytechnique, Gauthey seems not to have been that successful in teaching Navier, who may just have been a late developer, for he only just scraped into to École Polytechnique in 1802. However, from almost bottom place on entry, Navier made such progress in his first year at the École Polytechnique that he was one of the top ten students at the end of the year and chosen for special field work in Boulogne in his second year. During this first year at the École Polytechnique, Navier was taught analysis by Fourier who had a remarkable influence on the young man. Fourier became a life-long friend of Navier as well as his teacher, and he took an active interest in Navier's career from that time on. In 1804 Navier entered the École des Ponts et Chaussées and graduated as one of the top students in the school two years later. It was not long after Navier's graduation that his granduncle Emiland Gauthey died and Navier, who had left Paris to undertake field work, returned to Paris, at the request of the Corps des Ponts et Chaussées, to take on the task of editing Gauthey's works. Anderson writes in [3]:- Over the next 13 years, Navier became recognised as a scholar of engineering science. He edited the works of his granduncle, which represented the traditional empirical approach to numerous applications in civil engineering. In that process, on the basis of his own research in theoretical mechanics, Navier added a somewhat analytical flavour to the works of Gauthey. That, in combination with textbooks that Navier wrote for practicing engineers, introduced the basic principles of engineering science to a field that previously had been almost completely empirical. Navier took charge of the applied mechanics courses at the École des Ponts et Chaussées in 1819, being named as professor there in 1830. He did not just carry on the traditional teaching in the school, but rather he changed the syllabus to put much more emphasis on physics and on mathematical analysis. In addition, he replaced Cauchy as professor at the École Polytechnique from 1831. His ideas for teaching were not shared

CAPÍTULO IV 23

by all, however, and soon after his appointment to the professorship at the École Polytechnique Navier became involved in a dispute with Poisson over the teaching of Fourier's theory of heat. A specialist in road and bridge building, he was the first to develop a theory of suspension bridges which before then had been built to empirical principles. His major project to build a suspension bridge over the Seine was, however, to end in failure. The real reason that the project ran into difficulties was that the Municipal Council never supported it. Despite this it went ahead but, when the bridge was almost complete, a sewer ruptured at one end causing a movement of one of the bridge supports. The problem was not considered a major one by the Corps des Ponts et Chaussées who reported that repairs were straightforward, but the Municipal Council were looking for an excuse to stop the project and they had the bridge dismantled. Navier is remembered today, not as the famous builder of bridges for which he was known in his own day, but rather for the Navier-Stokes equations of fluid dynamics. He worked on applied mathematics topics such as engineering, elasticity and fluid mechanics and, in addition, he made contributions to Fourier series and their application to physical problems. He gave the well known Navier-Stokes equations for an incompressible fluid in 1821 while in 1822 he gave equations for viscous fluids. We should note, however, that Navier derived the Navier-Stokes equations despite not fully understanding the physics of the situation which he was modelling. He did not understand about shear stress in a fluid, but rather he based his work on modifying Euler's equations to take into account forces between the molecules in the fluid. Although his reasoning is unacceptable today, as Anderson writes in [3]:- The irony is that although Navier had no conception of shear stress and did not set out to obtain equations that would describe motion involving friction, he nevertheless arrived at the proper form for such equations. Navier received many honours, perhaps the most important of which was election to the Académie des Sciences in Paris in 1824. He became Chevalier of the Legion of Honour in 1831. Finally we should say a little of Navier's political position. Of course he lived through a period when there was great political movements throughout Europe and in France in particular. The two men who had the most influence on Navier's political thinking were Auguste Comte, the French philosopher known as the founder of sociology and of positivism, and Henri de Saint-Simon who started the Saint-Simonian movement which proposed a socialist ideology based on society taking advantage of science and technology. Comte had been educated at the École Polytechnique, entering in 1814, where he had studied mathematics. Navier appointed him as one of his assistants at the École Polytechnique and this connection was to see Navier become an ardent supporter of the ideas of Comte and Saint-Simon. Navier believed in an industrialised world in which science and technology would solve most of the problems. He also took a stand against war and against the bloodletting of the French Revolution and the military aggression of Napoleon. From 1830 Navier was employed as a consultant by the government to advise on how science and technology could be used to better the country. He advised on policies of road transport, the construction of both roads and railways. His many reports show both his remarkable abilities as an engineer coupled with his strong political views on building an industrialised society for the advantage of all.

CAPÍTULO IV 24

Jean Baptiste Joseph Fourier

Born: 21 March 1768 in Auxerre, Bourgogne, France

Died: 16 May 1830 in Paris, France

Joseph Fourier's father was a tailor in Auxerre. After the death of his first wife, with whom he had three children, he remarried and Joseph was the ninth of the twelve children of this second marriage. Joseph's mother died went he was nine years old and his father died the following year. His first schooling was at Pallais's school, run by the music master from the cathedral. There Joseph studied Latin and French and showed great promise. He proceeded in 1780 to the École Royale Militaire of Auxerre where at first he showed talents for literature but very soon, by the age of thirteen, mathematics became his real interest. By the age of 14 he had completed a study of the six volumes of Bézout's Cours de mathematique. In 1783 he received the first prize for his study of Bossut's Méchanique en général. In 1787 Fourier decided to train for the priesthood and entered the Benedictine abbey of St Benoit-sur-Loire. His interest in mathematics continued, however, and he corresponded with C L Bonard, the professor of mathematics at Auxerre. Fourier was unsure if he was making the right decision in training for the priesthood. He submitted a paper on algebra to Montucla in Paris and his letters to Bonard suggest that he really wanted to make a major impact in mathematics. In one letter Fourier wrote Yesterday was my 21st birthday, at that age Newton and Pascal had already acquired many claims to immortality. Fourier did not take his religious vows. Having left St Benoit in 1789, he visited Paris and read a paper on algebraic equations at the Académie Royale des Sciences. In 1790 he became a teacher at the Benedictine college, École Royale Militaire of Auxerre, where he had studied. Up until this time there had been a conflict inside Fourier about whether he should follow a religious life or one of mathematical research. However in 1793 a third element was added to this conflict when he became involved in politics and joined the local Revolutionary Committee. As he wrote:- As the natural ideas of equality developed it was possible to conceive the sublime hope of establishing among us a free government exempt from kings and priests, and to free from this double yoke the long-usurped soil of Europe. I readily became enamoured of this cause, in my opinion the greatest and most beautiful which any nation has ever undertaken. Certainly Fourier was unhappy about the Terror which resulted from the French Revolution and he attempted to resign from the committee. However this proved impossible and Fourier was now firmly entangled with the Revolution and unable to

CAPÍTULO IV 25

withdraw. The revolution was a complicated affair with many factions, with broadly similar aims, violently opposed to each other. Fourier defended members of one faction while in Orléans. A letter describing events relates:- Citizen Fourier, a young man full of intelligence, eloquence and zeal, was sent to Loiret. ... It seems that Fourier ... got up on certain popular platforms. He can talk very well and if he put forward the views of the Society of Auxerre he has done nothing blameworthy... This incident was to have serious consequences but after it Fourier returned to Auxerre and continued to work on the revolutionary committee and continued to teach at the College. In July 1794 he was arrested, the charges relating to the Orléans incident, and he was imprisoned. Fourier feared the he would go to the guillotine but, after Robespierre himself went to the guillotine, political changes resulted in Fourier being freed. Later in 1794 Fourier was nominated to study at the École Normale in Paris. This institution had been set up for training teachers and it was intended to serve as a model for other teacher-training schools. The school opened in January 1795 and Fourier was certainly the most able of the pupils whose abilities ranged widely. He was taught by Lagrange, who Fourier described as the first among European men of science, and also by Laplace, who Fourier rated less highly, and by Monge who Fourier described as having a loud voice and is active, ingenious and very learned. Fourier began teaching at the Collège de France and, having excellent relations with Lagrange, Laplace and Monge, began further mathematical research. He was appointed to a position at the École Centrale des Travaux Publiques, the school being under the direction of Lazare Carnot and Gaspard Monge, which was soon to be renamed École Polytechnique. However, repercussions of his earlier arrest remained and he was arrested again imprisoned. His release has been put down to a variety of different causes, pleas by his pupils, pleas by Lagrange, Laplace or Monge or a change in the political climate. In fact all three may have played a part. By 1 September 1795 Fourier was back teaching at the École Polytechnique. In 1797 he succeeded Lagrange in being appointed to the chair of analysis and mechanics. He was renowned as an outstanding lecturer but he does not appear to have undertaken original research during this time. In 1798 Fourier joined Napoleon's army in its invasion of Egypt as scientific adviser. Monge and Malus were also part of the expeditionary force. The expedition was at first a great success. Malta was occupied on 10 June 1798, Alexandria taken by storm on 1 July, and the delta of the Nile quickly taken. However, on 1 August 1798 the French fleet was completely destroyed by Nelson's fleet in the Battle of the Nile, so that Napoleon found himself confined to the land that he was occupying. Fourier acted as an administrator as French type political institutions and administration was set up. In particular he helped establish educational facilities in Egypt and carried out archaeological explorations. While in Cairo Fourier helped found the Cairo Institute and was one of the twelve members of the mathematics division, the others included Monge, Malus and Napoleon Bonaparte. Fourier was elected secretary to the Institute, a position he continued to hold during the entire French occupation of Egypt. Fourier was also put in charge of collating the scientific and literary discoveries made during the time in Egypt. Napoleon abandoned his army and returned to Paris in 1799, he soon held absolute power in France. Fourier returned to France in 1801 with the remains of the

CAPÍTULO IV 26

expeditionary force and resumed his post as Professor of Analysis at the École Polytechnique. However Napoleon had other ideas about how Fourier might serve him and wrote:- ... the Prefect of the Department of Isère having recently died, I would like to express my confidence in citizen Fourier by appointing him to this place. Fourier was not happy at the prospect of leaving the academic world and Paris but could not refuse Napoleon's request. He went to Grenoble where his duties as Prefect were many and varied. His two greatest achievements in this administrative position was overseeing the operation to drain the swamps of Bourgoin and to oversee the construction of a new highway from Grenoble to Turin. He also spent much time working on the Description of Egypt which was not completed until 1810 when Napoleon made changes, rewriting history in places, to it before publication. By the time a second edition appeared every reference to Napoleon would have been removed. It was during his time in Grenoble that Fourier did his important mathematical work on the theory of heat. His work on the topic began around 1804 and by 1807 he had completed his important memoir On the Propagation of Heat in Solid Bodies. The memoir was read to the Paris Institute on 21 December 1807 and a committee consisting of Lagrange, Laplace, Monge and Lacroix was set up to report on the work. Now this memoir is very highly regarded but at the time it caused controversy. There were two reasons for the committee to feel unhappy with the work. The first objection, made by Lagrange and Laplace in 1808, was to Fourier's expansions of functions as trigonometrical series, what we now call Fourier series. Further clarification by Fourier still failed to convince them. As is pointed out in [4]:- All these are written with such exemplary clarity - from a logical as opposed to calligraphic point of view - that their inability to persuade Laplace and Lagrange ... provides a good index of the originality of Fourier's views. The second objection was made by Biot against Fourier's derivation of the equations of transfer of heat. Fourier had not made reference to Biot's 1804 paper on this topic but Biot's paper is certainly incorrect. Laplace, and later Poisson, had similar objections. The Institute set as a prize competition subject the propagation of heat in solid bodies for the 1811 mathematics prize. Fourier submitted his 1807 memoir together with additional work on the cooling of infinite solids and terrestrial and radiant heat. Only one other entry was received and the committee set up to decide on the award of the prize, Lagrange, Laplace, Malus, Haüy and Legendre, awarded Fourier the prize. The report was not however completely favourable and states:- ... the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigour. With this rather mixed report there was no move in Paris to publish Fourier's work. When Napoleon was defeated and on his way to exile in Elba, his route should have been through Grenoble. Fourier managed to avoid this difficult confrontation by sending word that it would be dangerous for Napoleon. When he learnt of Napoleon's escape from Elba and that he was marching towards Grenoble with an army, Fourier was extremely worried. He tried to persuade the people of Grenoble to oppose Napoleon and give their allegiance to the King. However as Napoleon marched into the town Fourier left in haste. Napoleon was angry with Fourier who he had hoped would welcome his return. Fourier was able to talk his way into favour with both sides and Napoleon made him Prefect of the Rhône. However Fourier soon resigned on receiving orders, possibly from Carnot, that the was to remove all administrators with royalist sympathies. He could not have

CAPÍTULO IV 27

completely fallen out with Napoleon and Carnot, however, for on 10 June 1815, Napoleon awarded him a pension of 6000 francs, payable from 1 July. However Napoleon was defeated on 1 July and Fourier did not receive any money. He returned to Paris. Fourier was elected to the Académie des Sciences in 1817. In 1822 Delambre, who was the Secretary to the mathematical section of the Académie des Sciences, died and Fourier together with Biot and Arago applied for the post. After Arago withdrew the election gave Fourier an easy win. Shortly after Fourier became Secretary, the Academy published his prize winning essay Théorie analytique de la chaleur in 1822. This was not a piece of political manoeuvring by Fourier however since Delambre had arranged for the printing before he died. During Fourier's eight last years in Paris he resumed his mathematical researches and published a number of papers, some in pure mathematics while some were on applied mathematical topics. His life was not without problems however since his theory of heat still provoked controversy. Biot claimed priority over Fourier, a claim which Fourier had little difficulty showing to be false. Poisson, however, attacked both Fourier's mathematical techniques and also claimed to have an alternative theory. Fourier wrote Historical Précis as a reply to these claims but, although the work was shown to various mathematicians, it was never published. Fourier's views on the claims of Biot and Poisson are given in the following, see [4]:- Having contested the various results [Biot and Poisson] now recognise that they are exact but they protest that they have invented another method of expounding them and that this method is excellent and the true one. If they had illuminated this branch of physics by important and general views and had greatly perfected the analysis of partial differential equations, if they had established a principal element of the theory of heat by fine experiments ... they would have the right to judge my work and to correct it. I would submit with much pleasure .. But one does not extend the bounds of science by presenting, in a form said to be different, results which one has not found oneself and, above all, by forestalling the true author in publication. Fourier's work provided the impetus for later work on trigonometric series and the theory of functions of a real variable.

CAPÍTULO IV 28

Jean Marie Constant Duhamel

Born: 5 Feb 1797 in St Malo, France

Died: 29 April 1872 in Paris, France

Jean-Marie Duhamel was a student at the École Polytechnique and then he became professor there in 1830. He was highly thought of as a teacher of mathematics and is reported to have given very fine lectures. During the period 1848 until 1851 Duhamel was Director of Studies at the École Polytechnique. From 1851, he again filled the analysis chair at the École Polytechnique. Also from 1851 he was professor at the Faculté des Sciences in Paris. Duhamel worked on partial differential equations and applied his methods to the theory of heat, to rational mechanics and to acoustics. His acoustical studies involved vibrating strings and the vibration of air in cylindrical and conical pipes. His techniques in the theory of heat were mathematically similar to Fresnel's work in optics. His theory of the transmission of heat in crystal structures was based on work of Fourier and Poisson. 'Duhamel's principle' in partial differential equations arose from his work on the distribution of heat in a solid with a variable boundary temperature. Duhamel was elected to the Académie des Sciences in 1840.

CAPÍTULO IV 29

John[1] Louis von Neumann Born 28 December 1903, Budapest, Hungary; Died 8 February 1957, Washington DC; Brilliant mathematician, synthesizer, and promoter of the stored program concept, whose logical design of the IAS became the prototype of most of its successors - the von Neumann Architecture.

Educ: University of Budapest, 1921; University of Berlin, 1921-23; Chemical Engineering, Eidgenössische Technische Hochschule [ETH] (Swiss Federal Institute of Technology), 1923-25; Doctorate, Mathematics (with minors in experimental physics and chemistry), University of Budapest, 1926; Prof. Exp: Privatdozent, University of Berlin, 1927-30; Visiting Professor, Princeton University, 1930-53; Professor of Mathematics, Institute for Advanced Study, Princeton University, 1933-57; Honors and Awards: D.Sc. (Hon), Princeton University, 1947; Medal for Merit (Presidential Award), 1947; Distinguished Civilian Service Award, 1947; D.Sc. (Hon), University of Pennsylvania, 1950; D.Sc. (Hon), Harvard University, 1950; D.Sc. (Hon), University of Istanbul, 1952; D.Sc. (Hon), Case Institute of Technology, 1952; D.Sc. (Hon), University of Maryland, 1952; D.Sc. (Hon), Institute of Polytechnics, Munich, 1953; Medal of Freedom (Presidential Award), 1956; Albert Einstein Commemorative Award, 1956; Enrico Fermi Award, 1956; Member, American Academy of Arts and Sciences; Member, Academiz Nacional de Ciencias Exactas, Lima, Peru; Member, Acamedia Nazionale dei Lincei, Rome, Italy; Member, National Academy of Sciences; Member, Royal Netherlands Academy of Sciences and Letters, Amsterdam, Netherlands; Member, Information Processing Hall of Fame, Infomart, Dallas TX, 1985.

Von Neumann was a child prodigy, born into a banking family is Budapest, Hungary. When only six years old he could divide eight-digit numbers in his head. He received his early education in Budapest, under the tutelage of M. Fekete, with whom he published his first paper at the age of 18. Entering the University of Budapest in 1921, he studied Chemistry, moving his base of studies to both Berlin and Zurich before receiving his diploma in 1925 in Chemical Engineering. He returned to his first love of mathematics in completing his doctoral degree in 1928. he quickly gained a reputation in set theory, algebra, and quantum mechanics. At a time of political unrest in central Europe, he was invited to visit Princeton University in 1930, and when the Institute for Advanced Studies was founded there in 1933, he was appointed to be one of the original six Professors of Mathematics, a position which he retained for the remainder of his life. At the instigation and sponsorship of Oskar Morganstern, von Neumann and Kurt Gödel became US citizens in time for their clearance for wartime work. There is an anecdote which tells of Morganstern driving them to their immigration interview, after having learned about the US Constitution and the history of the country. On the drive there Morganstern asked them if they had any questions which he could answer. Gödel replied that he had no questions but he had found some logical inconsistencies in the

CAPÍTULO IV 30

Constitution that he wanted to ask the Immigration officers about. Morganstern strongly recommended that he not ask questions, just answer them! During 1936 through 1938 Alan Turing was a graduate student in the Department of Mathematics at Princeton and did his dissertation under Alonzo Church. Von Neumann invited Turing to stay on at the Institute as his assistant but he preferred to return to Cambridge; a year later Turing was involved in war work at Bletchley Park. This visit occurred shortly after Turing's publication of his 1934 paper "On Computable Numbers with an Application to the Entscheidungs-problem" which involved the concepts of logical design and the universal machine. It must be concluded that von Neumann knew of Turing's ideas, though whether he applied them to the design of the IAS Machine ten years later is questionable. [5] Von Neumann's interest in computers differed from that of his peers by his quickly perceiving the application of computers to applied mathematics for specific problems, rather than their mere application to the development of tables. During the war, von Neumann's expertise in hydrodynamics, ballistics, meteorology, game theory, and statistics, was put to good use in several projects. This work led him to consider the use of mechanical devices for computation, and although the stories about von Neumann imply that his first computer encounter was with the ENIAC, in fact it was with Howard Aiken's Harvard Mark I (ASCC) calculator. His correspondence in 1944 shows his interest with the work of not only Aiken but also the electromechanical relay computers of George Stibitz, and the work by Jan Schilt at the Watson Scientific Computing Laboratory at Columbia University. By the latter years of World War II von Neumann was playing the part of an executive management consultant, serving on several national committees, applying his amazing ability to rapidly see through problems to their solutions. Through this means he was also a conduit between groups of scientists who were otherwise shielded from each other by the requirements of secrecy. He brought together the needs of the Los Alamos National Laboratory (and the Manhattan Project) with the capabilities of firstly the engineers at the Moore School of Electrical Engineering who were building the ENIAC, and later his own work on building the IAS machine. Several "supercomputers" were built by National Laboratories as copies of his machine. Postwar von Neumann concentrated on the development of the Institute for Advanced Studies (IAS) computer and its copies around the world. His work with the Los Alamos group continued and he continued to develop the synergism between computers capabilities and the needs for computational solutions to nuclear problems related to the hydrogen bomb. Any computer scientist who reviews the formal obituaries of John von Neumann of the period shortly after his death will be struck by the lack of recognition of his involvement in the field of computers and computing. His Academy of Sciences biography, written by Salomon Bochner [1958], for example, includes but a single, short paragraph in ten pages - "... in 1944 von Neumann's attention turned to computing machines and, somewhat surprisingly, he decided to build his own. As the years progressed, he appeared to thrive on the multitudinousness of his tasks. It has been stated that von Neumann's electronic computer hastened the hydrogen bomb explosion on November 1, 1952." Dieudonné [1981] is a little more generous with words but appears to confuse the concept of the stored program concept with the wiring of computers: "Dissatisfied with the computing machines available immediately after the war, he was led to examine from its foundations the optimal method that such machines should follow, and he introduced new procedures in the logical organization, the "codes" by which a fixed system of wiring could solve a great variety of problems."!

CAPÍTULO IV 31

From the point of view of von Neumann's contributions to the field of computing, including the application of his concepts of mathematics to computing, and the application of computing to his other interests such as mathematical physics and economics, perhaps the most comprehensive is by Herman Goldstine [1972]. There has been some criticism of Goldstine's perspective since he personally was intimately involved in von Neumann's computing activities from the time of their chance meeting on the railroad platform at Aberdeen in 1944 [2] through their joint activities at the Institute for Advanced Studies in developing the IAS machine. There is no doubt that his insights into the organization of machines led to the infrastructure which is now known as the "von Neumann Architecture". However, von Neumann's ideas were not along those lines originally; he recognized the need for parallelism in computers but equally well recognized the problems of construction and hence settled for a sequential system of implementation. Through the report entitled First Draft of a Report on the EDVAC [1945], authored solely by von Neumann, the basic elements of the stored program concept were introduced to the industry. A retrospective examination of the development [3] of this idea reveals that the concept was discussed by J. Presper Eckert, John Mauchly, Arthur Burks, and others in connection with their plans for a successor machine to the ENIAC. The "Draft Report"

was just that, a draft, and although written by von Neumann was intended to be the joint publication of the whole group. The EDVAC was intended to be the first stored program computer, but the summer school at the Moore School in 1946 there was so much emphasis in the EDVAC that Maurice Wilkes, Cambridge University Mathematical Laboratory, conceived his own design for the EDSAC, which became the world's first operational, production, stored-program computer. In the 1950's von Neumann was employed as a consultant to IBM to review proposed and ongoing advanced technology projects. One day a week, von Neumann "held court" at 590 Madison Avenue, New York. On one of these occasions in 1954 he was confronted with the FORTRAN

concept; John Backus remembered von Neumann being unimpressed and that he asked "why would you want more than machine language?" Frank Beckman, who was also present, recalled that von Neumann dismissed the whole development as "but an application of the idea of Turing's `short code'." Donald Gillies, one of von Neumann's students at Princeton, and later a faculty member at the University of Illinois, recalled in the mid-1970's that the graduates students were being "used" to hand assemble programs into binary for their early machine (probably the IAS machine). He took time out to build an assembler, but when von Neumann found out about he was very angry, saying (paraphrased), "It is a waste of a valuable scientific computing instrument to use it to do clerical work." One last anecdote about von Neumann's brilliant mathematical capabilities. The von Neumann household in Princeton was open to many social activities and on one such occasion someone posed the "fly and the train" problem [4] to von Neumann. Quickly von Neumann came up with the answer. Suspecting that he had seen through the problem to discover a simple solution, he was asked how he solved the problem. "Simple", he responded, "I summed the series!" [From Nick Metropolis] The Institute of Electrical and Electronics Engineers (IEEE) continues to honor John von Neumann through the presentation of an annual award in his name. The IEEE John von Neumann Medal was established by the Board of Directors in 1990 and may be presented annually "for outstanding achievements in computer-related science and

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technology." The achievements may be theoretical, technological, or entrepreneurial, and need not have been made immediately prior to the date of the award. Quotations If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. Anyone who considers arithmetical methods of producing random numbers is, of course, in a state of sin. (Quoted in Knuth, 1968, Vol. 2, also in Goldstine, 1972, p. 297.)