Question 2.5: Derive expressions for the effective Young’s modulus and Poi......

Let x = [112]. The axes y and z must be normal to x and each other; for example, y = [11\overline{1]} and z = [1\overline{1}0 ]. The direction cosines between these axes and the cubic axes of the crystal can be found by dot products; for example, l_{1x} = [100]·[112] = (1·1 + 0·1 + 0·2)/(1² + 0² + 0²)(1² + 1² + 2²) = 1/ \sqrt{6} . Forming a table of direction cosines:

For tension along [112], \sigma_{y} =\sigma_{z} = \sigma_{yz} = \sigma_{zx}=\sigma_{xy}=0 , so the only finite stress on the x, y, z axis system is \sigma_{x}. Transforming to the 1, 2, and 3 axes using Equation (1.8),

\sigma_{x^{\prime} x } =l_{x^{\prime} x } l_{x^{\prime} x } \sigma_{x x }+l_{x^{\prime} y } l_{x^{\prime} x } \sigma_{yx}+l_{x^{\prime} z} l_{x^{\prime} x } \sigma_{zx }
+l_{x^{\prime} x } l_{x^{\prime} y } \sigma_{xy }+l_{x^{\prime} y } l_{x^{\prime} y } \sigma_{yy }+l_{x^{\prime} z} l_{x^{\prime} y} \sigma_{zy }
+l_{x^{\prime} x } l_{x^{\prime} z } \sigma_{xz }+l_{x^{\prime} y} l_{x^{\prime} z } \sigma_{yz }+l_{x^{\prime} z} l_{x^{\prime} z } \sigma_{zz}       (1.8a)

\sigma_{x^{\prime} y^{\prime} } =l_{x^{\prime} x } l_{y^{\prime} x } \sigma_{x x }+l_{x^{\prime} y } l_{y^{\prime} x } \sigma_{yx}+l_{x^{\prime} z} l_{y^{\prime} x } \sigma_{zx }
+l_{x^{\prime} x } l_{y^{\prime} y } \sigma_{xy }+l_{x^{\prime} y } l_{y^{\prime} y } \sigma_{yy }+l_{x^{\prime} z} l_{y^{\prime} y} \sigma_{zy }
+l_{x^{\prime} x } l_{y^{\prime} z } \sigma_{xz }+l_{x^{\prime} y} l_{y^{\prime} z } \sigma_{yz }+l_{x^{\prime} z} l_{y^{\prime} z } \sigma_{zz}.       (1.8b)

\sigma_{1} =(1/6)\sigma_{x} \ \ \ \ \ \ \sigma_{23} =(1/3)\sigma_{x}
\sigma_{2} =(1/6)\sigma_{x} \ \ \ \ \ \ \sigma_{31} =(1/3)\sigma_{x}
\sigma_{3} =(2/3)\sigma_{x} \ \ \ \ \ \ \sigma_{12} =(1/6)\sigma_{x}.

From Equation (2.13) for a cubic crystal, the resulting elastic strains are

e_{11}=s_{11}\sigma_{11} +s_{12}\sigma_{22} +s_{13}\sigma_{33}
e_{22}=s_{12}\sigma_{11} +s_{11}\sigma_{22} +s_{12}\sigma_{33}
e_{33}=s_{12}\sigma_{11} +s_{12}\sigma_{22} +s_{11}\sigma_{33}
\gamma_{23} =s_{44} \sigma_{23}
\gamma_{31} =s_{44} \sigma_{31}
\gamma_{12} =s_{44} \sigma_{12}.       (2.13)

e_{1}/ \sigma_{x} =(1/6)s_{11} +(1/6)s_{12}+(2/3)s_{12}=(1/6)s_{11}+(5/6)s_{12},

e_{2}/ \sigma_{x} =(1/6)s_{12} +(1/6)s_{11}+(2/3)s_{12}=(1/6)s_{11}+(5/6)s_{12},

e_{3}/ \sigma_{x} =(1/6)s_{12} +(1/6)s_{12}+(2/3)s_{11}=(2/3)s_{11}+(1/3)s_{12} ,

Using Equation (1.27), transform these strains onto the x, y, and z axes,

\varepsilon_{x^{\prime} x } =l_{x^{\prime} x^{\prime} } l_{x^{\prime} x^{\prime} } \varepsilon_{x x }+l_{x^{\prime} y } l_{x^{\prime} x^{\prime} } \varepsilon_{yx}+l_{x^{\prime} z} l_{x^{\prime} x^{\prime} } \varepsilon_{zx }
+l_{x^{\prime} x^{\prime} } l_{x^{\prime} y } \varepsilon_{xy }+l_{x^{\prime} y } l_{x^{\prime} y } \varepsilon_{yy }+l_{x^{\prime} z} l_{x^{\prime} y}\varepsilon_{zy }
+l_{x^{\prime} x^{\prime} } l_{x^{\prime} z }\varepsilon_{xz }+l_{x^{\prime} y} l_{x^{\prime} z }\varepsilon_{yz }+l_{x^{\prime} z} l_{x^{\prime} z }\varepsilon_{zz}       (1.27a)

\varepsilon_{x^{\prime} y^{\prime}} =l_{x^{\prime} x^{\prime} } l_{y^{\prime} x^{\prime} } \varepsilon_{x x }+l_{x^{\prime} y } l_{y^{\prime} x } \varepsilon_{yx}+l_{x^{\prime} z} l_{y^{\prime} x } \varepsilon_{zx }
+l_{x^{\prime} x^{\prime} } l_{x^{\prime} y } \varepsilon_{xy }+l_{x^{\prime} y } l_{y^{\prime} y } \varepsilon_{yy }+l_{x^{\prime} z} l_{y^{\prime} y}\varepsilon_{zy }
+l_{x^{\prime} x^{\prime} } l_{y^{\prime} z }\varepsilon_{xz }+l_{x^{\prime} y} l_{y^{\prime} z }\varepsilon_{yz }+l_{x^{\prime} z} l_{y^{\prime} z }\varepsilon_{zz}.       (1.27b)

e_{x} = (1/6)e_{1}+(1/6)e_{2}+(2/3)e_{3}+(1/3)\gamma_{23}+(1/3)\gamma_{31}+(1/6)\gamma_{12}
=[1/6)s_{11}+(1/2)s_{12}+(1/4)s_{44}]/\sigma_{x},

e_{y} = (1/3)e_{1}+(1/3)e_{2}+(1/3)e_{3}-(1/3)\gamma_{23}-(1/3)\gamma_{31}+(1/3)\gamma_{12}
=[(1/3)s_{11}+(2/3)s_{12}+(1/6)s_{44}]/\sigma_{x},

Young’s modulus is E_{[112]} = (e_{x}/\sigma_{x} )^{-1}=\left[(1/6) s_{11} + (1/2) s_{12}+ (1/4) s_{44}\right]^{-1}. There are two values of Poisson’s ratio: \upsilon_{y} = -e_{y}/ e_{x} and \upsilon_{z} = -e_{z}/ e_{x} .