advanced mechanics of silids worksheet#02

8/16/2019 Advanced Mechanics of Silids Worksheet#02

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01: Sutherland’s viscosity law is used to represent the variation of gas viscosity with

temperature,

µ

µ˳ =

T

T ¿3

2(T ˳ S)

¿¿¿

where μ is the dynamic viscosity, T is the absolute temperature, μo

and To

are reference values

of μ and T, respectively, and S is a constant with units of temperature!that depends on the gas"

#or air, To

= $%& ', μo

=1 "%1 (10)*

+g m -s!, and S =110 ". '

/rite a script m)file that evaluates μair

at &0o" n your e2pression for μ, use variables, not the

numerical constants, for , T , and S"μ ˳

Solution:

T=&03T = T4$%&3

To = $%&3

uo= 1"%151e)*3S = 110".3

u= TTo6&$!5To5S!!T4S!!5uo


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$" onvert the script written in the preceding e2ercise to a function" The function should have one

input parameter, T, and one output parameter, μ"

Solution:function u = viscosityt!T = T4$%&3

To = $%&3uo = 1"%151e)*3

S = 110".3

u = TTo!6&$!5To5S!!T4S!!5uo3

&" 7odify the function from the preceding e2ercise so that it accepts a vector of T values as well

as a scalar T values"


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Solution:

function u = viscosityt!T = T4$%&3

To = $%&3

uo =1"%151e)*3T =110".3

u = T"To!"6&$!"5To5S!!"T4S!!"5uo3


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4. Write a function that evaluates μair

in the range 0 ≤ T ≤ 230o

C using the

function written in the preceding exercise. Plot the utherland !odel data

against the shown "elow.

0 1"%$0e)*

$0 1"81%e)*.0 1"911e)*

0 $"00$e)*

80 $"091e)*100 $"1%%e)*

1$% $"$9.e)*

1%% $".9&e)* $$% $"%01e)*

Solution:

function u = plotviscosityT,Tv,uv!T = T4$%&3

To =$%&3

uo =1"%151e)*3

S =110".3u = T"To!"6&$!"5To5S!!"T4S!!"5uo3

plot T,u,;));!3hold on3

plotTv,uv,;);!3


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*" ? that can be used to:

a! #ind the principal stresses for the given state of stress"

b! #ind absolute ma2imum shear stress for the given state of stress"

c! #ind octahedral shear stress for the given state of stress"

d! #ind the principal strains for the given state of stress"e!


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% exy,eyz,ezx are s$earin& strains

ex = (sx-(v.sy)-(v.sz))E'

ey = (sy-(v.sx)-(v.sz))E'

ez = (sz-(v.sx)-(v.sy))E'

exy = sxy(2"/.4)'

eyz = syz(2"/.4)'

ezx = szx(2"/.4)'

es = zeros(3,3)'

% es is t$e strain tensor

es(1,1) = ex'

es(2,2) = ey'

es(3,3) = ez'

es(1,2) = exy'

es(2,1) = exy'

es(2,3) = eyz'

es(3,2) = eyz'

es(1,3) = ezx'

es(3,1) = ezx'

% e is t$e Ei&en strain vector

% e1, e2, e3 as t$e rincia! strains

e = ei&(es)'e1 = max(e)'

e3 = min(e)'

% f condition is used to find t$e e2

if (e(1)e3 ** e(1)+e1)

e2 = e(1)'

e!seif (e(2)e3 ** e(2)+e1)

e2 = e(2)'

e!se

e2 = e(3)'

end

% 678 989:E 68 #;8E##% r12 = radius of circ!e


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!ot(x2,y2)'

x!aorma! stress)

y!a


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s1 =*0

s$ =$*s& = )%*

tma2 = $"*000

toct =*."00$e1 = 0"00&0

e$ = 0"001*

e& = )0"00.*

advanced mechanics of silids worksheet#02

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