Is it possible to prove Gauss-Bonnet Theorem by using physics (Mechanics of materials) models?
For example in mechanics could one consider static equilibrium by action of hydrostatic pressure normal to a patch, tension in the curved patch boundary line or using strain energy, or by considering tension forces and/or moments in all three directions?
Surface Tension property $N$ or force per unit arc length of a closed boundary (or membrane energy in creating a new unit area), $N$ can balance pressure on a spherical soap-film as $ p = 2 N \kappa_n , $ where $\kappa_n$ is normal curvature of the bubble or by mean curvature $ 2 H = p /N $ for a doubly curved patch.
EDIT1:
My attempt ... Gauss Bonnet theorem for a singly connected patch. Proposing here an elementary physics model :
$$ \int K dA + \int k_g ds = 2 \pi\tag{1} $$
Defining elemental force $dF$ linking the above solid and tangential membrane " rotations" as applicable for pressure and surface tension by means of their force component $ dF$:
$$ dF = p \,dA \cos \phi = N \,ds \sin \phi ;\ \tag{2}$$
we have $$ \int \left( \frac{K}{p \cos \phi} + \frac{k_g}{ N\sin \phi } \right) dF = 2 \pi\tag{3}$$
as a topological constant.. However getting a full physical sense is elusive, at time of posting but was soon changed as following.
EDIT2:
Answering own question at least partially ..
We can derive Gauss Bonnet by considering mechanical Force Equilibrium of a spherical segment of the soap bubble in the following manner taken after equilibrium considerations of spherical or cylindrical pressure vessels... the attempt appears to me this way now, but may still need revision :
Considering static force and pressure equilibria, net force in horizontal direction is zero.
$$ 2 \pi R \cdot p \Delta z + N\cdot 2 \pi r = 0 $$
$$ \frac{1}{2 \pi R \cdot p \Delta z} +\frac{1}{ N\cdot 2 \pi r } = 0 \tag{4}$$
Divide numerator and denominator of each term by $R$ and let
$$ 1/R^2 = K ;\, \Delta z/R = \cos \phi ;\, r/R = \sin \phi \,; 1/R = k_g\, ;\tag{5} $$
$$ \frac{ 1/R^2}{2 \pi p (\Delta z/R) } + \frac {1/R}{2 \pi N ( r/R) } \tag{6}$$
$$ \frac{ K } { 2 \pi p\cos \phi} + \frac { k_g}{ 2 \pi N \sin \phi} = 0 \tag {7} $$
which relation when multiplied by $dF$ and integrated we get the same form of of Gauss Bonnet theorem Equn (3) with $2 \pi$ now appearing as an integration constant which can be identified as a topological constant.
EDIT 3:
Essentially it is seen that physical model geometrization is made possible by looking at area as reciprocal of pressure $p$ and boundary length as reciprocal of surface tension $T.$
The insight applied so far is that in a soap film application of pressure increases its area and boundary line dilations occur by linear (uni-axial) tension. The correct model should include all doubly curved surfaces, not spherical the surfaces only .