Discussion
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?
Answer: B
There is a constant body force per unit volume in the y-direction.
The equilibrium equations are \(\partial\sigma_{ij}/\partial x_{j} + b_{i} = 0\). With all shear stresses zero and the normal stresses depending only on y:
\[ \begin{aligned} x:\ \frac{\partial \sigma_{xx}}{\partial x} + b_{x} = 0 &\Rightarrow b_{x} = 0 \\ y:\ \frac{\partial \sigma_{yy}}{\partial y} + b_{y} = 0 &\Rightarrow b_{y} = -C_{2} \\ z:\ \frac{\partial \sigma_{zz}}{\partial z} + b_{z} = 0 &\Rightarrow b_{z} = 0 \end{aligned} \]
The body force is \(-C_{2}\) in the y-direction, constant and non-zero because \(C_{2} \neq 0\). It does not vary with y, and its direction is set by the sign of \(C_{2}\), not \(C_{1}\), since \(\sigma_{xx}\) and \(\sigma_{zz}\) vary only with y and contribute nothing to the x and z equations.
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