Question

___________ is a compatibility equation.

(a) \(\frac{∂^2 ε_y}{∂y^2} +\frac{∂^2 ε_y}{∂x^2} = \frac{∂^2 Γ_{xy}}{∂x∂y}\)

(b) \(\frac{∂^2 ε_x}{∂y^2} +\frac{∂^2 ε_y}{∂x^2} = \frac{∂^2 Γ_{xy}}{∂x∂y}\)

(c) \(\frac{∂^2 ε_x}{∂y^2} +\frac{∂^2 ε_y}{∂x^2} = \frac{∂^2 Γ_{xy}}{∂x∂y}\)

(d) \(\frac{∂^2 ε_x}{∂y^2} +\frac{∂^2 ε_y}{∂x^2} = \frac{∂^2 Γ_{xy}}{∂x∂y}\)

This question was posed to me in final exam.

My question is from Strain Components and Compatibility Equations topic in chapter Elements of Elasticity of Geotechnical Engineering I

Answer 1

The correct answer is (b) \(\frac{∂^2 ε_x}{∂y^2} +\frac{∂^2 ε_y}{∂x^2} = \frac{∂^2 Γ_{xy}}{∂x∂y}\)

The explanation is: The strain equations in terms of displacements is given by,

\(ε_X=\frac{∂u}{∂x} —————-(1)\)

\(ε_Y=\frac{∂v}{∂y} —————- (2)\)

\(Γ_{xy}=\frac{∂v}{∂x}+\frac{∂u}{∂y} ———(3)\)

Differentiating (1) twice with respect to y, (2) twice with respect to x and (3) once with respect to x and then y,

\(\frac{∂^2 ε_x}{∂y^2} =\frac{∂^3 u}{dx∂y^2} ————-(4)\)

\(\frac{∂^2 ε_y}{∂x^2} =\frac{∂^3 v}{dy∂x^2} ————-(5)\)

\(\frac{∂^2 Γ_{xy}}{∂x∂y}=\frac{∂^3 u}{dx∂y^2}+\frac{∂^3 v}{dy∂x^2} ——-(6)\)

∴ from (4), (5) and (6), we get,

\(\frac{∂^2 ε_x}{∂y^2} +\frac{∂^2 ε_y}{∂x^2} =\frac{∂^2 Γ_{xy}}{∂x∂y}\) which is a compatibility equation.