Question 6.1: A thin plate with a thickness of 1 mm is subjected to mechan...
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 Eq. (12) to both surfaces of the plate. For example, using the measured strains for ε_{xx}, we have:
\left\{\begin{matrix} \varepsilon _{xx} \\ \varepsilon _{yy} \\ \gamma _{xy} \end{matrix} \right\} = \left\{\begin{matrix} \varepsilon° _{xx} \\ \varepsilon° _{yy} \\ \gamma° _{xy} \end{matrix} \right\} + z \left\{\begin{matrix} \kappa _{xx} \\ \kappa _{yy} \\ \kappa _{xy} \end{matrix} \right\} (12)
at z = -t/2 = -0.0005 m : ε_{xx} = 250 μm/m = ε°_{xx} – (0.0005)\kappa _{xx}
at z = +t/2 = +0.0005 m : ε_{xx} = -250 μm/m = ε°_{xx} + (0.0005)\kappa _{xx}
Solving simultaneously, we find:
ε°_{xx} = 0 μm/m, \kappa _{xx} = -0.50 rad/m
Using a similar approach utilizing the measured values for ε_{yy} and γ_{xy}, we find:
ε°_{yy} = -1300 μm/m, \kappa _{xx} = 0.40 rad/m
γ°_{xx} = 900 μrad, \kappa _{xx} = -0.20 rad/m