Exam (elaborations)

ejercicio mecanica del medio continuo

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Course
Mecanica del medio continuo

Institution
Universidad Autónoma De Madrid

Estos ejercicios son como ventanas a un mundo de conocimientos que te ayudarán a comprender mejor cómo se comportan los materiales y las estructuras en situaciones reales. Cada ejercicio viene con una explicación detallada y paso a paso de cómo abordar y resolver los desafíos, lo que te permit...

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Uploaded on September 10, 2023
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mecanica del medio continuo

Institution
Universidad Autónoma de Madrid
Education
Licenciatura En Fisica
Course
Mecanica del medio continuo

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Ejercicio resuelto

referencia de ejercicio: J.N Reddy, An introduction to continum mechanics,
pag 205, ejercicio 4.8.

4.8 Las componentes del tensor de tensiones de Cauchy en un punto P en el
cuerpo deformado con respecto al sistema de coordenadas estan (x1 , x2 , x3 )
dadas por:
 
1 4 −2
[σ] =  4 0 0  M P a
−2 0 3
a) Determinar el vector de tensiones de Cauchy tn̂ en el punto P de un plano
que pasa por el punto y paralelo al plano 2x1 + 3x2 + x3 = 4
b) Encuentre la longitud de tn̂ y el angulo entre tn̂ y el vector normal al
plano.
c)Determine las componentes del tensor de tensiones de Cauchy en un sistema
de coordenadas rectangulares (x1 , x2 , x3 ) cuyos vectores base ortonormales ēˆi
estan dados en terminos de los vectores base êi del sistema de coordenadas
(x1 , x2 , x3 )
1 1
ê2 = √ (ê1 − ê3 ), ēˆ3 = (2ê1 − ê2 − ê3 )
2 3

a) el vector normal al plano 2x1 + 3x2 + x3 = 4

donde la normal esta dada por
▽f
n̂ =
| ▽ f|
1
n̂ = √ (2ê1 + 3ê2 + ê3 )
12
    
1 4 −2 2 12
1 1  
t = √ =  4 0 0  3 = √ 8 MP a
14 −2 0 3 1 14 −1
 

el vector de tensiones resultantes es:

1
t(n̂) = √ (12ê1 + 8ê2 − ê3 )M P a
14
b) Encuentre la longitud de tn̂ y el angulo entre tn̂ y el vector normal al plano

la longitud del vector t(n̂)
s
12 8 1
t(n̂) = ( √ )2 + ( √ )2 + ( √ )2
14 14 14

t = 3,86M P a

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