# Optomechanics – SOLVED: Differential equations of elasticity for the maximum tensile stress in glass…

Colleagues:

Those of you who are familiar with my expression,

s(x)=-s(y)=pb(1-2v)/(a+b)^2,

for tensile stresses in a ring-mounted glass lens (SPIE, 7424-14, 2009) should be delighted to know I didn’t give up there.  I recently solved the differential equations of elasticity for the maximum tensile stress in the glass for a short line-contact load,

s(x)=-s(y)<P(1-2v)/(2pL^2).

In this expression s(x) is the tensile stress, s(y) is the compressive stress, P is the total load, v is Poisson’s ratio for the glass and L is the length of the line contact.

As in the prior case these stresses are much lower (by factors of perhaps 1/1,000 to 1/1,000,000 depending on the length L) than those predicted by the erroneous Delgado and Hallinan method used by optical engineers since the 1970s.  In the limits, where L approaches either 0.0 or infinity, my expression approaches those in the classic literature (ie., Timoshenko and Goodier), just as it should.

And it worked!  The low stress level opens up a whole new family of design options.  Just another one of those “small things” that brings joy to an optomechanical engineer’s heart.

Oh, in my new expression “<” means “less than” because I integrated only over the length, L, assuming the contact width was infinitesimal.  That suited my engineering (worst case) purpose.  There’s still time for some grad student to integrate over the finite width for the full solution.

We’re just keeping AEH’s tools sharp, too.

Oh, October… the Great Goblin was enchanting but the little kids were TRULY GREAT!!!

Al H.
11-2-15