Relation between Curvature and Misfit Strain
Referring to the figure in the Force Balance section, the forces P and –P generate a moment, given by
\[M = P\left( {\frac{{h + H}}{2}} \right)\]
where h and H are deposit and substrate thicknesses respectively. Since the curvature, κ, (through-thickness strain gradient) is given by the ratio of moment, M, to beam stiffness, Σ
\[\kappa = \frac{M}{{\rm{\Sigma }}}\]
P can be expressed as
\[P = \frac{{2\kappa {\rm{\Sigma }}}}{{h + H}} (i)\]
The beam stiffness is given by
\[ {\rm{\Sigma }} = b\mathop \smallint \limits_{ - H - \delta }^{h - \delta } E\left( {{y_c}} \right)y_c^2 d{y_c} = b{E_\rm{d}}h\left( {\frac{{{h^2}}}{3} - h\delta + {\delta ^2}} \right) + b{E_\rm{s}}H\left( {\frac{{{H^2}}}{3} + H\delta + {\delta ^2}} \right) (ii) \]
where δ (the distance between the neutral axis (yc = 0) and the interface (y = 0)) is given by
\[ \delta = \frac{{\left( {{h^2}{E_\rm{d}} - {H^2}{E_\rm{s}}} \right)}}{{2\left( {h{E_\rm{d}} + H{E_\rm{s}}} \right)}} (iii)\]
The magnitude of P is found by expressing the misfit strain as the difference between the strains resulting from application of the P forces (ie by writing a strain balance):
\[\Delta ε = {ε_\rm{s}} - {ε_\rm{d}} = \frac{P}{{Hb{E_\rm{s}}}} + \frac{P}{{hb{E_\rm{d}}}}\]
\[ \frac{P}{b} = Δ ε \left(\frac{{ {h{E_\rm{d}}H{E_\rm{s}}} }}{{h{E_\rm{d}} + H{E_\rm{s}}}}\right) (iv) \]
Combination of this with Eqns.(i)-(iii) gives a general expression for the curvature, κ, arising from imposition of a uniform misfit strain, \(\Delta ε \)
\[ \kappa = \frac{{6{E_\rm{d}}{E_\rm{s}}\left( {h + H} \right)hH Δ ε }}{{{E_\rm{d}}^2{h^4} + 4{E_\rm{d}}{E_\rm{s}}{h^3}H + 6{E_\rm{d}}{E_\rm{s}}{h^2}{H^2} + 4{E_\rm{d}}{E_\rm{s}}h{H^3} + {E_\rm{s}}^2{H^4}}} (v)\]
The stress at the interface (y = 0), and at the free surfaces (y = h or -H), can be written in terms of the base level in each constituent (arising from the force balance) and the change due to the stress gradient (= curvature × stiffness):
\[ \left. \sigma{_d} \right|_{y=h} = \frac{{-P}} {b h} + {E_\rm{d}}\kappa \left(h - \delta \right) \]
\[\left. \sigma{_d} \right|_{y=0} = \frac{{-P}} {b h} − {E_\rm{d}}\kappa{\delta} \]
\[ \left. \sigma{_s} \right|_{y=-H} = \frac{{P}} {b h} − {E_\rm{s}}\kappa \left(H + \delta \right) \]
\[ \left. \sigma{_s} \right|_{y=0} = \frac{{P}} {b h} − {E_\rm{s}}\kappa{\delta} \]
Since the gradient is linear in each constituent, this gives the complete stress profile.