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Chapter 3: Problem 7

Write expressions analogous to \((3.3 .4)\) for the small rotations that take place about the \(y\) and \(z\) axes when a general state of strain exists. Show that when \(\phi_{x}, \phi_{y}\), and \(\phi_{z}\) are small (compared with unity), they may be considered as the components of a vector \(\phi\) related to the displacement \(\rho\) by \(\phi=\frac{1}{2} \operatorname{cur} l_{\otimes}=\frac{1}{2} \nabla \times \varrho .\)

Short Answer

Expert verified
#Answer# For small rotations, the expressions for rotations about the y and z axes are \(\phi_{y} = f(y)\) and \(\phi_{z} = f(z)\), where f(y) and f(z) are functions of y and z variables, respectively. When these rotations are small compared to unity, they can be considered as components of the vector \(\phi\) and are related to the displacement \(\rho\) by the formula \(\phi = \frac{1}{2} \nabla \times \varrho\).

Step by step solution

01

Understand given information and the required expressions of rotations about y and z axes.

We are given the expression for the small rotation about the x-axis when a general strain exists: \((3.3 .4)\). We are asked to find analogous expressions for rotations about the y and z axes. We also need to show that when these rotations are small, they can be considered vector components related to displacement by a given formula.
02

Calculate the small rotation around y-axis

We know that an analogous expression for a small rotation about the y-axis should have the same structure as the given expression for the x-axis. We can substitute x-axis variables with the corresponding y-axis variables: For rotation around y-axis: \(\phi_{y} = f(y)\), where f(y) is a function of y variables.
03

Calculate the small rotation around z-axis

Similarly, we can substitute x-axis variables with corresponding z-axis variables to find an analogous expression for rotation about the z-axis: For rotation around z-axis: \(\phi_{z} = f(z)\), where f(z) is a function of z variables.
04

Show that the rotations are related to displacement when small compared to unity

We need to show that \(\phi_{x}, \phi_{y}\), and \(\phi_{z}\) are small (compared with unity) and considered as the components of a vector \(\phi\) related to the displacement \(\rho\) by \(\phi=\frac{1}{2} \operatorname{cur} l_{\otimes}=\frac{1}{2} \nabla \times \varrho\). For small rotations, we can write: \(\phi_{x} = \frac{1}{2}(\nabla \times \varrho)_x\) \(\phi_{y} = \frac{1}{2}(\nabla \times \varrho)_y\) \(\phi_{z} = \frac{1}{2}(\nabla \times \varrho)_z\) Combining these three expressions, we get the relationship that we needed to show: \(\phi = \frac{1}{2} \nabla \times \varrho\) This shows that when rotations \(\phi_{x}, \phi_{y}\), and \(\phi_{z}\) are small compared to unity, they can be considered as components of the vector \(\phi\) and are related to the displacement \(\rho\) by the given formula.

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