Question

2. In the absence of body forces, do the following stress components satisfy the equations of equilibrium? 011 = a [až +v (zł – za)], 022 = a [xí +v (zá – z)] , 033 1,033 = av (2ỉ + x3) -2av2122,013 = 031 = 0,023 = 032 = 0 012 = 021 =

Answer 1

Solution:

Given that:

011 = a 22 [23 +v(21 - 23)

022 = a 2 [T++ (- x))

ძვ = av (1) (1 + 12)

$\sigma _{12}=\sigma _{21}=-2\alpha \nu x_{1}x_{2}$

13 = 31 — 0

م = 60 = 0

0 o = Try Tr: Try Oy Tys [TT: Tg:

Equlibrium equation:

дох Әт ху дTr: + + = 0 - 1 д1 Әx2 Әr3

д1 ху до у дту- + + = 0 -2 Әr1 Әx2 дх3

$\frac{\partial \tau _{xz} }{\partial x_{1}}+\frac{\partial \tau _{yz} }{\partial x_{2}}+\frac{\partial \sigma _{z}}{\partial x_{3}}=0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \rightarrow 3$

0 = a [23 +v (21-23)] -2avr 1.22 0 -2avc1.22 a[ri+v(.az - x)] 0 0 0 av(x + )

Equation 1 will be:

$2\alpha\nu x_{1}-2\alpha\nu x_{1}=0$

Equation 2 will be:

$-2\alpha\nu x_{2}+2\alpha\nu x_{2}=0$

Equation 3 will be:

$0+0+0=0$

Hence stress tensor satisfying the equilibrium equations.