Thermal Stresses in a Bronze Sealed Sapphire Window

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In an annular housing/metal-seal/window arrangement, the stresses are as for thick-walled cylinders.

The stresses and displacements are:

radial stress

s_r = (-a² p (b² / r²-1)-b² q (1-a² / r²)) / (b²-a²)

hoop stress

s_q = (a² p (b² / r²+1)-b² q (1+a² / r²)) / (b²-a²)

radial displacement--

u = (r (1-m) (a² p-b² q)+(1+m) (p-q) a² b² / r) / (E (b²-a²))

u = (a² (b² (m+1) (p-q)-p r² (m-1))+b² q r² (m-1)) / (E r (b²-a²))

Substituting for the appropriate variables for the housing gives

s_rh = a_h² p_h (b_h²-r_h²) / (r_h² (a_h²-b_h²))

s_qh = a_h² p_h (b_h²+r_h²) / (r_h² (b_h²-a_h²))

u_h = a_h p_h (a_h² (m_h-1)-b_h² (m_h+1)) / (E_h (a_h²-b_h²))

and for the metal-seal

s_rg = (-a_g² p_g (b_g² / r_g²-1)-b_g² q_g (1-a_g² / r_g²)) / (b_g²-a_g²)

s_qg = (a_g² p_g (b_g² / r_g²+1)-b_g² q_g (1+a_g² / r_g²)) / (b_g²-a_g²)

u_g = (a_g² (b_g² (m_g+1) (p_g-q_g)-p_g r_g² (m_g-1))+b_g² q_g r_g² (m_g-1)) / (E_g r_g (b_g²-a_g²))

u_ga = a_g (a_g² p_g (m_g-1)+b_g² (2 q_g-p_g (m_g+1))) / (E_g (a_g²-b_g²))

u_gb = b_g (a_g² (2 p_g-q_g (m_g+1))+b_g² q_g (m_g-1)) / (E_g (b_g²-a_g²))

and finally for the window

s_rw = -q_w

s_qw = -q_w

u_w = q_w r_w (m_w-1) / E_w

At the interface between the housing and the glass seal

the radius is

a_h = 0.3125

b_h = 0.492

b_g = 0.3125

a_g = 0.2755

E_h = 29000000

E_g = 17000000

E_w = 50000000

The thermal interference between the housing and the sealing glass is

u_gh = a_h dt (a_h-a_g)

= 0.3125 (2441-530) (8 10^(-6)-9.8 10^(-6))

= -0.0010749375

u_gw = a_g dt (a_g-a_w)

= 0.2755 (2441-530) (9.8 10^(-6)-4.278 10^(-6))

= 0.002907225

To solve for the stresses at this interface, the pressures and displacements must be set equal for the respective touching parts

p_h = q_g

q_w = p_g

u_gh = u_h-u_g= 2 b_g^3 (p_g-q_g) / (E_g (a_g²-b_g²))-2 a_h b_h² p_h / (E_h (a_h²-b_h²))+a_h p_h (m_h-1) / E_h+b_g (2 p_g-q_g (m_g+1)) / E_g

u_gw = u_g-u_w= a_g (a_g² p_g (m_g-1)+b_g² (2 q_g-p_g (m_g+1))) / (E_g (a_g²-b_g²))-q_w r_w (m_w-1) / E_w

6.809938199 10^-4 = 2 0.3125³ (p_g-q_g) / (7440000 (0.2755²-0.3125²))-2 (0.3125) (0.492)² q_g / (29000000 ((0.3125)²-(0.492)²))+~

0.3125 q_g (0.29-1) / 29000000+0.3125 (2 p_g-q_g (0.25+1)) / 7440000

6.809938199 10^-4 = 3.530472224 10^-7 q_g-2.930699148 10^-7 p_g

1.17818126 10^-4 = 0.2755 ((0.2755)² p_g (0.25-1)+(0.3125)² (2 q_g-p_g (0.25+1))) / (7440000 ((0.2755)²-(0.3125)²))-p_g (0.2755) (-0.02-1)~

/ 50000000

1.17818126 10^-4 = 3.102775992 10^-7 p_g-3.324295767 10^-7 q_g

[6.809938199 10^-4 = 3.530472223 10^-7 q_g-2.930699147 10^-7 p_g,1.17818126 10^-4 = 3.102775992 10^-7 p_g-3.324295766 10^-7 q_g]

[p_g = 22115, q_g = 20287]

Hoop stress in the housing

s_qh = (0.3125)² (20286.80234) ((0.492)²+(0.3125)²) / ((0.3125)² ((0.492)²-(0.3125)²))

= 47725

Hoop stress in the seal

s_qg = ((0.2755)² (22114.87796) ((0.3125)² / (0.2755)²+1)-(0.3125)² (20286.80234) (1+(0.2755)² / (0.2755)²)) / ((0.3125)²-(0.2755)²)

= -5703.5

(This analysis was done quickly and accurately on my copy of the computer algebra system, Derive.)

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