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# 2.6 The displacement field is given by $u_{x}=k\left(x^{2}+2 z\right), \quad u_{y}=k\left(4 x+2 y^{2}+z\right), \quad u_{z}=4 k z^{2}$ $k$ is a very small constant. What are the strains at $(2,2,3)$ in directions (a) $n_{x}=0, n_{y}=1 / \sqrt{2}, n_{z}=1 / \sqrt{2}$ (b) $n_{x}=1, n_{y}=n_{z}=0$ (c) $n_{x}=0.6, n_{y}=0, n_{z}=0.8$ [Ans. (a) $\frac{33}{2} k$, (b) $4 k$, (c) $17.76 k]$ 2.7 For the displacement field given in Problem 2.6, with $k=0.001$, determine the change in angle between two line segments $P Q$ and $P R$ at $P(2,2,3)$ having direction cosines before deformation as (a) $\begin{array}{l} \text { PQ: } \quad n_{x 1}=0, n_{y 1}=n_{z 1}=\frac{1}{\sqrt{2}} \\ \text { PR: } \quad n_{x 2}=1, n_{y 2}=n_{z 2}=0 \end{array}$ (b) $\begin{array}{l} \text { PQ: } \quad n_{x 1}=0, n_{y 1}=n_{z 1}=\frac{1}{\sqrt{2}} \\ \text { PR: } \quad n_{x 2}=0.6, n_{y 2}=0, n_{z 2}=0.8 \end{array}$2.7 For the displacement field given in Problem 2.6, with $k=0.001$, determine the change in angle between two line segments $P Q$ and $P R$ at $P(2,2,3)$ having direction cosines before deformation as (a) $\begin{array}{l} P Q: \quad n_{x 1}=0, n_{y 1}=n_{z 1}=\frac{1}{\sqrt{2}} \\ \text { PR: } \quad n_{x 2}=1, n_{y 2}=n_{z 2}=0 \\ \end{array}$ (b) $\begin{array}{l} P Q: \quad n_{x 1}=0, n_{y 1}=n_{z 1}=\frac{1}{\sqrt{2}} \\ P R: \quad n_{x 2}=0.6, n_{y 2}=0, n_{z 2}=0.8 \end{array}$ $\left[\begin{array}{l} \text { Ans. (a) } 90^{\circ}-89.8^{\circ}=0.2^{\circ} \\ \text { (b) } 55.5^{\circ}-50.7^{\circ}=4.8^{\circ} \end{array}\right]$  