Question

For two dimensional case, for both plane stress as well as plain strain case the compatibility equation is _______

(a) \(\frac{∂^2 ε_x}{∂z^2} +\frac{∂^2 ε_z}{∂x^2} =\frac{∂^2 Γ_{xz}}{∂x∂z}\)

(b) \(\frac{∂^2 ε_z}{∂z^2} +\frac{∂^2 ε_y}{∂x^2} =\frac{∂^2 Γ_{zy}}{∂z∂y}\)

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

(d) \(\frac{∂^2 ε_z}{∂z^2} +\frac{∂^2 ε_y}{∂x^2} =0\)

This question was addressed to me in examination.

This interesting question is from Elasticity Elements topic in section Elements of Elasticity of Geotechnical Engineering I

Answer 1

Correct answer is (a) \(\frac{∂^2 ε_x}{∂z^2} +\frac{∂^2 ε_z}{∂x^2} =\frac{∂^2 Γ_{xz}}{∂x∂z}\)

To elaborate: For two dimensional case, the six compatibility equations are evidently reduced to one single equation;

\(\frac{∂^2 ε_x}{∂z^2} +\frac{∂^2 ε_z}{∂x^2} =\frac{∂^2 Γ_{xz}}{∂x∂z}.\)  

This is because, in the plain strain case, one dimension (y) is very large in comparison to the other two directions. So, the strain components in this direction are zero. Also in plain stress condition, the stresses in y-direction are considered as zero.