Uniqueness (Kirchhoff's Theorem)

If two displacement fields ${\displaystyle \textstyle \mathbf {u} }$ and ${\displaystyle \textstyle \mathbf {u} ^{'}}$ correspond to the same strain field, then ${\displaystyle \textstyle \mathbf {u} =\mathbf {u} ^{'}+\mathbf {w} }$, where ${\displaystyle \textstyle \mathbf {w} }$ is a rigid displacement field.

Existence (Compatibility Theorem)

The strain field ${\displaystyle \textstyle {\boldsymbol {\varepsilon }}}$ corresponding to a ${\displaystyle \textstyle C^{2}}$ continuous displacement field satisfies the compatibility equation

${\displaystyle {\boldsymbol {\nabla }}\times {{\boldsymbol {\nabla }}\times {\boldsymbol {\varepsilon }}}=0}$

In index notation

${\displaystyle e_{ijk}e_{lmn}{\cfrac {\partial ^{2}\varepsilon _{jm}}{\partial X_{k}\partial X_{n}}}=0}$

The converse also holds if the body is simply connected. The compatibility equations can also be written as

${\displaystyle {\begin{aligned}2{\cfrac {\partial ^{2}\varepsilon _{12}}{\partial X_{1}\partial X_{2}}}&={\cfrac {\partial ^{2}\varepsilon _{11}}{\partial X_{2}^{2}}}+{\cfrac {\partial ^{2}\varepsilon _{22}}{\partial X_{1}^{2}}}\\2{\cfrac {\partial ^{2}\varepsilon _{23}}{\partial X_{2}\partial X_{3}}}&={\cfrac {\partial ^{2}\varepsilon _{22}}{\partial X_{3}^{2}}}+{\cfrac {\partial ^{2}\varepsilon _{33}}{\partial X_{2}^{2}}}\\2{\cfrac {\partial ^{2}\varepsilon _{31}}{\partial X_{3}\partial X_{1}}}&={\cfrac {\partial ^{2}\varepsilon _{33}}{\partial X_{1}^{2}}}+{\cfrac {\partial ^{2}\varepsilon _{11}}{\partial X_{3}^{2}}}\\{\cfrac {\partial ^{2}\varepsilon _{11}}{\partial X_{2}\partial X_{3}}}&={\cfrac {\partial ^{2}\varepsilon _{12}}{\partial X_{3}\partial X_{1}}}-{\cfrac {\partial ^{2}\varepsilon _{23}}{\partial X_{1}^{2}}}+{\cfrac {\partial ^{2}\varepsilon _{31}}{\partial X_{1}\partial X_{2}}}\\{\cfrac {\partial ^{2}\varepsilon _{22}}{\partial X_{3}\partial X_{1}}}&={\cfrac {\partial ^{2}\varepsilon _{23}}{\partial X_{1}\partial X_{2}}}-{\cfrac {\partial ^{2}\varepsilon _{31}}{\partial X_{2}^{2}}}+{\cfrac {\partial ^{2}\varepsilon _{12}}{\partial X_{2}\partial X_{3}}}\\{\cfrac {\partial ^{2}\varepsilon _{33}}{\partial X_{1}\partial X_{2}}}&={\cfrac {\partial ^{2}\varepsilon _{31}}{\partial X_{2}\partial X_{3}}}-{\cfrac {\partial ^{2}\varepsilon _{12}}{\partial X_{3}^{2}}}+{\cfrac {\partial ^{2}\varepsilon _{23}}{\partial X_{3}\partial X_{1}}}\end{aligned}}}$

The compatibility condition also implies the following relationship between the infinitesimal strain tensor and the axial vector corresponding to the infinitesimal rotation tensor:

${\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {\varepsilon }}={\boldsymbol {\nabla }}{\boldsymbol {\omega }}}$

Problem 1

Show that the compatibility relation for plane stress is satisfied by unrestrained thermal expansion (${\displaystyle \varepsilon _{11}=\varepsilon _{22}=\alpha T}$, ${\displaystyle \varepsilon _{12}=0}$), where ${\displaystyle \alpha }$ is the coefficient of thermal expansion and ${\displaystyle T}$ is the temperature, provided that the temperature is a two-dimensional harmonic function, i.e.,

${\displaystyle {\frac {\partial ^{2}T}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}T}{\partial x_{2}^{2}}}=0}$

Hence deduce that, subject to certain restrictions which you should explicitly specify, no thermal stresses will be induced in a thin body with a steady-state, two-dimensional temperature distribution and no surface tractions.

Solution

The plane stress compatibility equation is

${\displaystyle \varepsilon _{11,22}+\varepsilon _{22,11}-2\varepsilon _{12,12}=0}$

Plugging in the expressions for strain,

${\displaystyle \alpha T_{,22}+\alpha T_{,11}=0\,}$

or,

${\displaystyle {\frac {\partial ^{2}T}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}T}{\partial x_{2}^{2}}}=0}$

The above equation is the steady-state heat conduction equation without any internal sources.

If there are no surface tractions, the state ${\displaystyle \sigma _{11}=\sigma _{22}=\sigma _{12}}$ satisfies the BCs. Since the steady-state heat conduction equation is also the compatibility equation, compatibility is automatically satisfied by the above stress state. Therefore, no thermal stresses are induced in this situation. However, extra conditions need to be applied if the body is multiply-connected.