A thin plate with a thickness of 1 mm is subjected to mechanical loads, a change in temperature, and a change in moisture content. Strain gages are used to measure the surface strains induced in the plate. They are found to be
at z = −t/2 = −0.5 mm: ε_{xx} = 250 μm/m, ε_{yy }− 1500 μm/m, γ_{xy} = 1000 μrad
at z = +t/2 = +0.5 mm: ε_{xx} = −250 μm/m, ε_{yy} − 1100 μm/m, γ_{xy} = 800 μrad
What midplane strains and curvatures are induced in the plate?
To solve this problem, we simply apply Equation 6.12
\left \{ \begin{matrix}\varepsilon _{xx} \\ \varepsilon _{yy} \\ \gamma _{xy} \end{matrix} \right \}=\left \{ \begin{matrix} \varepsilon^\circ _{xx} \\ \varepsilon^\circ _{yy} \\ \gamma ^\circ _{xy} \end{matrix} \right \}+z\left \{ \begin{matrix}\kappa _{xx} \\ \kappa _{yy} \\ \kappa _{xy} \end{matrix} \right \} (6.12)
to both surfaces of the plate. For example, using the measured strains for ε_{xx} we have:
atz=-t/2=-0.0005 m: \varepsilon _{xx}=250\mu m/m = \varepsilon ^o_{xx}-(0.0005)\kappa _{xx}
atz=+ t/2=-0.0005 m: \varepsilon _{xx}=-250\mu m/m = \varepsilon ^o_{xx}+(0.0005)\kappa _{xx}
Solving simultaneously, we find:
\varepsilon ^\circ_{xx}=0 \mu m/m \kappa _{xx}= -50 rad/mUsing a similar approach using the measured values for ε_{yy} and γ_{xy}, we find:
\varepsilon ^\circ_{yy}=1300 \mu m/m \kappa _{yy}= 0.40 rad/m\gamma ^\circ_{xy}=900 \mu rad \kappa _{xy}= -0.20 rad/m