Question 8.4: The number of independent slip systems is the same as the nu......

Let the [100], [010], and [001] directions be designated as 1, 2, and 3 respectively and let the slip systems be designated as follows:
[011]( 01\overline{1}) and [01\overline{1}] (011) as Aa, [101]( 10\overline{1}) and [101]( 10\overline{1}) as Bb,
and [110]( 1\overline{1}0) and [1\overline{1}0] (110) as Cc.
The strains along the cubic axes can be written as
e_{1}=l_{1A}l_{1a}\gamma _{Aa}+l_{1B}l_{1b}\gamma _{Bb}+l_{1C}l_{1c}\gamma _{Cc}=0+(1/2) \gamma _{Bb} +(1/2)\gamma _{Cc}
e_{2}=l_{2A}l_{2a}\gamma _{Aa}+l_{2B}l_{2b}\gamma _{Bb}+l_{2C}l_{2c}\gamma _{Cc}=0+(1/2) \gamma _{Aa} +(1/2)\gamma _{Cc}
\gamma _{23}=(l_{2A}l_{3a}+l_{2a}l_{3A})\gamma _{Aa}+(l_{2B}l_{3b}+l_{2b}l_{3B})\gamma _{Bb}+(l_{2C}l_{3c}+l_{2c}l_{3C})\gamma _{Cc}
\gamma _{31}=(l_{3A}l_{1a}+l_{3a}l_{1A})\gamma _{Aa}+(l_{3B}l_{1b}+l_{3b}l_{1B})\gamma _{Bb}+(l_{3C}l_{1c}+l_{3c}l_{1C})\gamma _{Cc}
\gamma _{12}=(l_{1A}l_{2a}+l_{1a}l_{2A})\gamma _{Aa}+(l_{1B}l_{2b}+l_{1b}l_{2B})\gamma _{Bb}+(l_{1C}l_{2c}+l_{1c}l_{2C})\gamma _{Cc}.
Substituting gives
l_{1A}= [100]·[110] = 1/ √2 , l_{1a}= [100]· [1\overline{1}0] = 1/ √2, l_{1B} = [100]·[101]
=1/ √2 ,
l_{1b} = [100]· [10\overline{1}] = 1/ √2 , l_{1C} = [100]·[011] = 0, l_{1c} = [100]· [01\overline{1}] = 0,
l_{2A} = [010]·[110] = 1/ √2 , l_{2a} = [010]· [1\overline{1}0]= -/ √2 , l_{2B}= [010]·[101] = 0,
l_{2b} = [010]·[101] = 0, l_{2C} = [010]·[011] = 1/ √2 , l_{2c} = [010]· [01\overline{1}] = 1/ √2 ,
l_{3A} = [001]·[110] = 0,l_{3a} = [001]· [1\overline{1}0]= 0,  l_{3B}= [001]·[101] = 1/ √2 ,
l_{3b}= [001]· [10\overline{1}]= -1/ √2 , l_{3C} = [001]·[011] = 1/ √2 , l_{3c}= [001]· [01\overline{1}]
=-1/ √2 .

Substituting (l_{2A}l_{3a}+l_{2a}l_{3A})=(l_{2B}l_{3b}+ l_{2b}l_{3B}) =(l_{2C}l_{3c}+l_{2c}l_{3C})=0 etc., it can be concluded that \gamma _{23}=\gamma _{31}= \gamma_{12}=0 , so the shear strains \gamma _{23},\gamma _{31}, and \gamma _{12} cannot be satisfied. Therefore there are only two independent strains, \varepsilon _{1} and \varepsilon _{2} , and so there are only two slip systems that are independent.