Chapter 7, Problem 7.3P

Interpretation Introduction

Interpretation:

Axial strain rate for flow in one direction is to be shown to be given by $\partial v_x/\partial x$ . Also, the rate of volume change is to be calculated. Rate of volume change by generalizing the equation for three-dimensional element is to be determined.

Concept Introduction:

The change in the strain of a material with respect to the time is known as the rate of strain. It accounts for both the rate of expansion or rate of compression, and the rate of deformation of the material due to progressive rate of shear. Strain is represented by $\delta$, thus, the formula for strain rate to be usedis:

  $\frac{\partial \delta }{\partial t}=\underset{\Delta x\to 0}{\lim }\left(\frac{v\left(x+\Delta x\right)\Delta t-v\left(x\right)\Delta t}{\Delta x\Delta t}\right)$  ........(1)

Similarly, the rate of volume change is:

  $\frac{\partial V}{\partial t}=\underset{\Delta x\to 0}{\lim }\left(\frac{v\left(x+\Delta x\right)\Delta t-v\left(x\right)\Delta t}{\Delta x\Delta t}\right)$  ........(2)

Here, $x$ is the distance and $v$ is the velocity of the element.