The biaxial modulus
Biaxial Stress States
So far, attention has been concentrated on a single (in-plane) direction. For conventional (in-plane) uniaxial loading of a bilayer sample, or for applying a bending moment to it, this is appropriate. It’s also possible to generate a misfit strain in a single (in-plane) direction. However, while this is possible, it’s actually rather unusual. More commonly, the same misfit strain is generated simultaneously in all in-plane directions, a state that can be represented by creating the strain in two arbitrary (in-plane) directions that are normal to each other. This will lead to an Equal Biaxial stress state (since all in-plane directions are clearly equivalent and the through-thickness stress, σy, is often taken to be zero - there is no normal stress at a free surface). Differential thermal contraction would normally have this effect. It’s also possible to create an Unequal Biaxial stress state. This would arise, for example, during differential thermal contraction with one or both of the layers exhibiting in-plane anisotropy in thermal expansivity, so that the misfit strain would be different in different in-plane directions. (Also, anisotropy in stiffness would lead to different stresses in different in-plane directions, even if the misfit strains were equal.) It is, however, common to at least assume that all in-plane directions are equivalent, in terms of both properties and misfit strains.
Poisson Effects
The main reason why the case of an equal biaxial misfit strain differs from that of a uniaxial one is related to Poisson effects. The strains arising in the selected in-plane direction (the x-direction) will be accompanied by Poisson strains in the other two (principal) directions. That in the through-thickness (y) direction is often of little consequence, but in the other in-plane (z) direction, it will need to be added to the outcome of the effects arising in that direction (from the misfit strain in that direction). The upshot of this is actually rather simple. By symmetry, the two in-plane stresses (and strains) must be equal - ie σx = σz. For isotropic elastic properties and no through-thickness stress (σy = 0), the strain in the x-direction can be written in terms of the three principal stresses:
\[{ε_x}E = {\sigma _x} - \nu \left( {{\sigma _y} + {\sigma _z}} \right) = {\sigma _x}\left( {1 - v} \right)\]
where ν is the Poisson ratio. The ratio of stress to strain in the x-direction (and all in-plane directions) can therefore be expressed
\[ \frac {{\sigma _x}}{\epsilon_x} = \frac {{E}}{ \left(1 - \nu \right)} = E{^{'}} \]
This modified form of the Young’s modulus, E’ (often termed the Biaxial Modulus), is applicable in expressions referring to substrate/coating systems having an equal biaxial stress state. The effective stiffness (stress/strain ratio) has been raised by this Poisson effect. This higher value should be used in place of E throughout the formulations in the preceding pages (when the misfit strain is generated in all in-plane directions).