In 2D plane stress elasticity with zero body forces, the Beltrami-Michell compatibility equation governing the sum of normal stresses $(\sigma_{xx} + \sigma_{yy})$ is:

(A) $\nabla^2(\sigma_{xx} + \sigma_{yy}) = \frac{\partial^2(\sigma_{xx} + \sigma_{yy})}{\partial x^2} + \frac{\partial^2(\sigma_{xx} + \sigma_{yy})}{\partial y^2} = 0$

(B) $\sigma_{xx} + \sigma_{yy} = \text{constant}$

(C) $\nabla^4(\sigma_{xx} + \sigma_{yy}) = 0$

(D) $\frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \sigma_{yy}}{\partial y} = 0$

Answer Key:

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### 1. Strategy & Theoretical Principles In 2D linear elasticity, the strain compatibility condition is: $$\frac{\partial^2 \epsilon_{xx}}{\partial y^2} + \frac{\partial^2 \epsilon_{yy}}{\partial x^2} = 2 \frac{\partial^2 \epsilon_{xy}}{\partial x \partial y}$$ Substituting Hooke's law for plane stress and using the differential equations of equilibrium ($\partial \sigma_{xx}/\partial x + \partial \tau_{xy}/\partial y = 0$ and $\partial \tau_{xy}/\partial x + \partial \sigma_{yy}/\partial y = 0$) directly produces the Beltrami-Michell compatibility equation: $$\nabla^2(\sigma_{xx} + \sigma_{yy}) = 0$$ This proves that the first stress invariant (the sum of normal stresses) is a harmonic function. ### 2. Distractor Forensics - **Option A:** Correct Beltrami-Michell relation. - **Option B:** Normal stresses sum to constant only in special cases like pure torsion or uniform hydrostatic stress. ### 3. Exam Trap & Standard Reference - **Trap Alert:** The Airy stress function satisfies $\nabla^4 \phi = 0$, but the sum of stresses satisfies the harmonic equation $\nabla^2(\sigma_{xx} + \sigma_{yy}) = 0$. - **Standard Reference:** Timoshenko, S.P. & Goodier, J.N., *Theory of Elasticity*, Chapter 2.

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