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ejercicio mecanica del medio continuo
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  • Course
  • Electromagnetismo
  • 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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  • September 10, 2023
  • 1
  • 2023/2024
  • Exam (elaborations)
  • Only questions

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  • mecanica del medio continuo

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  • Universidad Autónoma de Madrid
  • Licenciatura En Fisica
  • Electromagnetismo
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Ejercicio 2

Para un determinado instante, el movimiento de un medio continuo viene
de
nido por:
x1 = X1 − AX3 (1)
x2 = X2 − AX3 (2)
x3 = −AX1 − AX2 + X3 (3)

Obtener el tensor gradiente material de deformaci on F (X) en dicho instante.
A partir de las ecuaciones de movimiento inversas obtener el tensor gradiente
espacial de la deformacion F −1 (X). Con los resultados obtenidos comprobar
que F −1 F = 1.
a)- Encontrar el tensor gradiente de deformacion.
 
X1 − AX3
⃗ =  ∂ ∂ ∂
 
F = ⃗x ⊗ ▽ X2 − AX3 ∂x1 ∂x2 ∂x3
−AX1 + AX2 + X3
realizando la multiplicacion de matrices y su derivacion da como resultado
 
1 0 −A
=  0 1 −A
−A A 1

b)- encontrar las ecuaciones del moviemineto inverso

X1 = (1 + A2 )x1 − A2 x2 + Ax3
⃗ x, t) = X2 = A2 x1 + (1 − A2 )x2 + Ax3
X(⃗
X3 = Ax1 − Ax2 + x3


c)-encontar el tensor gradiente espacial de deformacion

realizamos el producto de las ecuaciones de movimiento con las derivadas
parciales

 
(1 + A2 )x1 −A2 x2 Ax3 
F −1 (1 − A2 )x2 Ax3  ∂x∂ 1 ∂ ∂

= F = ⃗x ⊗ ▽ =  A2 x1 ∂x2 ∂x3
Ax1 Ax2 x3
 
1 + A2 −A2 A
=  A2 1 − A2 A
A A 1
d)- realizar la comporbacion del tensor gradiente de deformacion con el
tensor espacial de deformacion
  
1 0 −A 1 + A2 −A2 A
F · F −1 =  0 1 −A  A2 1 − A2 A
−A A 1 A A 1
obteniendo la matriz identidad

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