Classic Plate Equation: Constitutive
Hooke's Law 
The constitutive equation describes how the stresses and straines are related within the plate (Hooke's law). In linear elasticity, the most generalized Hooke's law contains six (6) components of stresses that are linearly related to six (6) components of strains. Nevertheless, classical plate theory typically uses isotropic material in the Hooke's Equation. The compliance (stress to strain) form is
The stiffness (strain to stress) form is
Note that the stresses and strains are functions of x, y, and z throughout the entire plate. 
Constitutive Equation of Classical Plates Materials 
Based on the kinematics of classical plates, the stressstrain relations can be simplified to
alternatively, they can be written as
and
which yields

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