Question

A stress field is given by \( \sigma_{xx} = \sigma_{zz} = C_1 y \); \( \sigma_{yy} = C_2 y \); \( \tau_{xy} = \tau_{yz} = \tau_{zx} = 0 \), where \( C_1 \) and \( C_2 \) are non-zero constants. If the stress field satisfies equilibrium, which one of the following options is correct?

• A. There is no body force per unit volume.
• B. There is a constant body force per unit volume in the y-direction.
• C. The body force per unit volume varies linearly in the y-direction.
• D. The direction of the body force per unit volume depends on the value of \( C_1 \).

Hint:

For stress fields in equilibrium, the body force per unit volume can be computed from the variation of stress with respect to position. In this case, the body force is constant in the \( y \)-direction.

Answer 1

The Correct Option is B

To satisfy equilibrium in the absence of external forces, the body force per unit volume \( \mathbf{f} \) must be related to the spatial variation of the stress field. The equilibrium equations for a stress field in 3D are given by: \[ \frac{\partial \sigma_{ij}}{\partial x_j} + f_i = 0 \] In this case, the stress components are:
\( \sigma_{xx} = C_1 y \)
\( \sigma_{yy} = C_2 y \)
\( \sigma_{zz} = C_1 y \)
Shear stresses \( \tau_{xy} = \tau_{yz} = \tau_{zx} = 0 \)
We need to focus on the equilibrium in the \( y \)-direction. The equilibrium equation for the \( y \)-direction (\( f_y \)) is: \[ \frac{\partial \sigma_{yy}}{\partial y} + f_y = 0 \] Since \( \sigma_{yy} = C_2 y \), we have: \[ \frac{\partial}{\partial y} (C_2 y) + f_y = 0 \quad \Rightarrow \quad C_2 + f_y = 0 \quad \Rightarrow \quad f_y = -C_2 \] Thus, the body force per unit volume in the \( y \)-direction is a constant value of \( -C_2 \), which confirms that there is a constant body force in the \( y \)-direction.