Athermal bond thickness for
axisymmetric optical elements
Tutorial by Eric Frater
Introduction
• Motivations
– Survival of optics
– Survival of bond
– Performance of optics
• Concerns
– Thermal stress
• Radial stress
• Shear stresses
– Glass distortion
r
r
𝜃
z
Example design
• Cell: Aluminum 6061-T6
• Optic: Schott N-BK7
• Adhesives:
– MG chemicals RTV 566
– 3M 2216 B/A (gray)
• Design:
– Bond provides constraint
– Uniform and continuous bond-line
– Zero-strain in materials at nominal
bonding temp
Material constants
Subscript notation:
“c”: cell
“b”: bond
“o”: optic
Required:
αb > αc > αo
or
αb = αc = αo
MATERIAL
N-BK7
6061-T6
2216 B/A (gray)[1]
RTV 566
αc
rc
αb
αo
ro
α(ppm/°C) Poisson ratio, ν
7.1
24
102
200
.21
.33
~.43
~.499
E (Gpa)
82
69
69
~.003
Quick note: 2216 B/A and RTV 566 very different adhesives. As seen in the table,
RTV compliance highly dependent on aspect ratio of bond.
[1] Yoder, Paul R. Mounting Optics in Optical Instruments
Bayar equation
• Consider positive ΔT in example design
• Assume bond only radially constrained
• Require:
∆ℎ = ∆𝑟𝑐 − ∆𝑟𝑜
ℎ𝛼𝑏 ΔT=(𝑟𝑜 +ℎ)𝛼𝑐 ΔT−𝑟𝑜 𝛼𝑜 ΔT
ΔT
𝜃
r
z
r
ΔT
𝑟𝑜 (𝛼𝑐 − 𝛼𝑜 )
ℎ=
𝛼 𝑏 − 𝛼𝑐
(Bayar equation[2])
Note: This vastly over-predicts thickness, neglects ν
Example: h= 2.75mm (2216 B/A), h= 1.22mm (RTV 566)
[2] Bayar, Mete. “Lens Barrel Optomechanical
Design Principles”
Radial strain and Hooke’s Law
• From Bayar equation (valid in all cases):
𝛿ℎ
𝑟𝑜
𝜀𝑟 =
= ΔT 𝛼𝑏 − 𝛼𝑐 − (𝛼𝑐 − 𝛼𝑜 )
ℎ
ℎ
• Radial stress from Hooke’s Law:
𝐸
𝜎𝑟 =
1 + 𝜈 1 − 2𝜈
1 − 𝜈 𝜀𝑟 + 𝜈 𝜀𝑧 + 𝜀𝜃
Athermalizing:
– Define εr and εθ
– Set radial stress equal to zero (pre-factor drops out)
– Solve for athermal bond thickness h
Van Bezooijen equation
• Assume bond is perfectly constrained in
r, z, θ
𝜃
r
z
r
𝛼𝑜 + 𝛼𝑐
𝜀𝑧 = 𝜀𝜃 = ∆𝑇 𝛼𝑏 −
2
• Solving for σr=0,
ℎ=
𝑟𝑜 (𝛼𝑐 −𝛼𝑜 )
2𝜈
𝛼𝑏 −𝛼𝑐 +1−𝜈
𝛼 +𝛼
𝛼𝑏 − 𝑜 2 𝑐
.
ΔT
ΔT
(van Bezooijen equation[3])
Note: This under-predicts thickness, neglects axial bulging of bond
Example: h= 1.03mm (2216 B/A), h= 0.40mm (RTV 566)
[3] Van Bezooijen, Roel. “Soft Retained AST Optics”
Modified van Bezooijen equation
• Assume bond is perfectly constrained in
r, θ and unconstrained in z
𝛼𝑜 + 𝛼𝑐
𝜀𝜃 = ∆𝑇 𝛼𝑏 −
2
𝜀𝑧 = 0
• Solving for σr=0,
ℎ=
𝑟𝑜 (𝛼𝑐 −𝛼𝑜 )
𝜈
𝛼𝑏 −𝛼𝑐 +
1−𝜈
𝛼 +𝛼
𝛼𝑏 − 𝑜 𝑐
2
.
𝜃
r
z
r
ΔT
ΔT
(modified van Bezooijen equation[4])
Note: This over-predicts thickness, allows excessive axial bulging
Example: h= 1.50mm (2216 B/A), h= 0.60mm (RTV 566)
[4] Monti, Christpher L. “Athermal bonded mounts:
Incorporating aspect ratio into a closed-form solution”
Aspect ratio
• Aspect ratio and axial constraint:
– Part of bond expands freely in z
– Middle section is perfectly
constrained in z
– Modifies the axial strain 𝜀𝑧
ℎ
𝜀𝑧 = ∆𝑇 1 −
𝐿
Unconstrained in z
if h=L
𝛼𝑜 + 𝛼𝑐
𝛼𝑏 −
2
ℎ=
𝐿
𝑅𝑎𝑠𝑝𝑒𝑐𝑡 =
ℎ
z
r
𝑟𝑜 (𝛼𝑐 − 𝛼𝑜 )
.
𝛼𝑜 + 𝛼𝑐
𝜈
ℎ
𝛼𝑏 − 𝛼𝑐 + 1 − 𝜈 2 − 𝐿 𝛼𝑏 −
2
Varies from 1-2 between limits
of van Bezooijen eq.’s
Closed-form aspect ratio approximation
𝜃
−𝑏 + 𝑏2 − 4𝑎𝑐
ℎ=
2𝑎
𝑎=
r
z
r
−𝜈
𝛼𝑜 + 𝛼𝑐
𝛼𝑏 −
𝐿
2
𝛼𝑜 + 𝛼𝑐
2
𝑐 = −𝑟𝑜 (1 − 𝜈)(𝛼𝑐 − 𝛼𝑜 )
𝑏 = 1 − 𝜈 𝛼𝑏 − 𝛼𝑐 + 2𝜈 𝛼𝑏 −
ΔT
ΔT
(Aspect ratio approximation[4])
Note: Provides a best-guess for h in closed-form
Example: h= 1.13mm (2216 B/A), h= 0.41mm (RTV 566)
[4] Monti, Christpher L. “Athermal bonded mounts:
Incorporating aspect ratio into a closed-form solution”
Conclusions
• Bayar equation
– Good conceptual starting point
– Tends to vastly over-estimate h
– Applicable to highly segmented
bonds
THICKNESS EQUATION
2216 B/A
RTV 566
Bayar
2.75 mm
1.22 mm
van Bezooijen
1.03 mm
0.40 mm
Modified van Bezooijen
1.50 mm
0.60 mm
Aspect ratio approximation
1.13 mm
0.41 mm
• Van Bezooijen equation
– Takes all strains into account
– Much more accurate than Bayar eq.
– Under-predicts h due to bulk effects
• Aspect ratio approximation
– Approximates varying bulk effects due to aspect ratio of bond
– Matches empirical FEA-derived corrections to van Bezooijen eq. well for
𝑅𝑎𝑠𝑝𝑒𝑐𝑡>4 and ν ∈ (.45, . 5) [4,5]
[5] Michels, Gregory, and Keith Doyle. “Finite Element
Modeling of Nearly Incompressible Bonds”
References
1.
2.
3.
4.
5.
Yoder, Paul R. Mounting Optics in Optical Instruments, 2nd ed. SPIE Press
Monograph Vol. PM181 (2008), p. 732.
Bayar, Mete. “Lens Barrel Optomechanical Design Principles”, Optical
Engineering. Vol. 20 No. 2 (April 1981)
Van Bezooijen, Roel. “Soft Retained AST Optics” Lockheed Martin
Technical Memo
Monti, Christpher L. “Athermal bonded mounts: Incorporating aspect
ratio into a closed-form solution”, SPIE 6665, 666503 (2007)
Michels, Gregory, and Keith Doyle. “Finite Element Modeling of Nearly
Incompressible Bonds”, SPIE 4771, 287 (2002)