Abstract

The tensor equations of elasticity including temperature terms in nonorthogonal coordinates are presented in a form suitable for the method of dynamic relaxation. A nonorthogonal coordinate system that includes a wide class of circular solid mirrors or lenses is described. The equations may be solved for any prescribed temperature distribution. Computed results are given for several cases.

© 1970 Optical Society of America

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References

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  1. A. J. Malvick, Appl. Opt. 7, 2117 (1968).
    [CrossRef] [PubMed]
  2. A. J. Malvick, E. T. Pearson, Appl. Opt. 7, 1207 (1968).
    [CrossRef] [PubMed]
  3. A. S. Day, Engineer 219, 218 (1965).
  4. J. R. H. Otter, A. C. Cassell, R. E. Hobbs, Proc. Inst. Civil Eng. 35, 633 (1966).
    [CrossRef]
  5. R. E. Hobbs, “Dynamic relaxation and model techniques applied to arch dams,” Ph.D. dissertation, London University (1967).
  6. A. E. Green, W. Zerna, Theoretical Elasticity (Oxford U.P., New York, 1963).

1968 (2)

1966 (1)

J. R. H. Otter, A. C. Cassell, R. E. Hobbs, Proc. Inst. Civil Eng. 35, 633 (1966).
[CrossRef]

1965 (1)

A. S. Day, Engineer 219, 218 (1965).

Cassell, A. C.

J. R. H. Otter, A. C. Cassell, R. E. Hobbs, Proc. Inst. Civil Eng. 35, 633 (1966).
[CrossRef]

Day, A. S.

A. S. Day, Engineer 219, 218 (1965).

Green, A. E.

A. E. Green, W. Zerna, Theoretical Elasticity (Oxford U.P., New York, 1963).

Hobbs, R. E.

J. R. H. Otter, A. C. Cassell, R. E. Hobbs, Proc. Inst. Civil Eng. 35, 633 (1966).
[CrossRef]

R. E. Hobbs, “Dynamic relaxation and model techniques applied to arch dams,” Ph.D. dissertation, London University (1967).

Malvick, A. J.

Otter, J. R. H.

J. R. H. Otter, A. C. Cassell, R. E. Hobbs, Proc. Inst. Civil Eng. 35, 633 (1966).
[CrossRef]

Pearson, E. T.

Zerna, W.

A. E. Green, W. Zerna, Theoretical Elasticity (Oxford U.P., New York, 1963).

Appl. Opt. (2)

Engineer (1)

A. S. Day, Engineer 219, 218 (1965).

Proc. Inst. Civil Eng. (1)

J. R. H. Otter, A. C. Cassell, R. E. Hobbs, Proc. Inst. Civil Eng. 35, 633 (1966).
[CrossRef]

Other (2)

R. E. Hobbs, “Dynamic relaxation and model techniques applied to arch dams,” Ph.D. dissertation, London University (1967).

A. E. Green, W. Zerna, Theoretical Elasticity (Oxford U.P., New York, 1963).

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Figures (4)

Fig. 1
Fig. 1

Coordinate system for a typical mirror, showing nonorthogonal curvilinear elements.

Fig. 2
Fig. 2

Displacements of Steward Observatory 90-in. (2.3-m) mirror due to three rotationally symmetric temperature distributions: (a) T = 1°C everywhere; (b) linear variation from T = 0°C on back to T = 1°C on front, (c) linear variation from T = 0°C on inside edge to T = 1°C on outside edge.

Fig. 3
Fig. 3

Warm spot at r = 76 cm on the Steward Observatory 90-in. (2.3-m) mirror surface with a thermal energy of 2190°C cm3.

Fig. 4
Fig. 4

The Steward Observatory 90-in. (2.3-m) mirror simulated in a transient condition attributable to a distant source lying in the plane of the mirror. Temperature T = [(rri)/(rori)]2 cosx3, for the half mirror below the horizontal diameter. In the upper half, T = 0°C.

Equations (21)

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y 1 = y 1 ( x 1 , x 2 , x 3 )             y 2 = y 2 ( x 1 , x 2 , x 3 )             y 3 = y 3 ( x 1 , x 2 , x 3 ) .
d s 2 = d y 1 2 + d y 2 2 + d y 3 2 = g i j d x i d x j ,
τ i j G = g i α U i , α + g i β U i , β + 2 ν 1 - 2 ν g i j U β , β - 2 ( 1 + ν ) 1 - 2 ν g i j α t T ,
u i = U i ( g [ i i ] ) 1 2 .
σ i j = ( g [ j j ] g [ i i ] ) 1 2 τ i j .
U i , α = U i x α + { i β α } U β .
τ i j , α = τ i j x α + { i α β } τ β j + { j α β } τ i β .
{ i j α } = 1 2 g i β ( g j β x α + g α β x j - g j α x β ) .
σ i j = G ( g [ j j ] g [ i i ] ) 1 2 { g j α [ x α ( u i g [ i i ] 1 2 ) + { i β α } ( u β g [ β β ] 1 2 ) ] + g i β [ x β ( u j g [ j j ] 1 2 ) + { j α β } ( u α g [ α α ] 1 2 ) ] + 2 ν 1 - 2 ν g i j [ x β ( u β g [ β β ] 1 2 ) + { β α β } ( u α g [ α α ] 1 2 ) ] - 2 ( 1 + ν ) 1 - 2 ν g i i α t T } .
τ i j , i + F i = ρ A j ,
f j = F j ( g [ j j ] ) 1 2 ,
a j = A j ( g [ j j ] ) 1 2 .
ρ a j = f j + g [ j j ] 1 2 { x i [ σ i j ( g [ i i ] g [ j j ] ) 1 2 ] + { i i α } σ α j ( g [ α α ] g [ j j ] ) 1 2 + { j i α } σ i α ( g [ i i ] g [ α α ] ) 1 2 } .
τ i j = ( g [ i i ] g [ j j ] ) 1 2 σ i j = ( g [ j j ] g [ i i ] ) 1 2 σ j i
σ j i = ( g [ i i ] g [ i i ] g [ j j ] g [ j j ] ) 1 2 σ i j .
y 1 = x 1 cos x 3 , y 2 = x 1 sin x 3 , y 3 = x 2 [ ( k - 1 ) x 1 2 / r 0 2 + 1 ] + b x 1 2 ,
g i j = [ ( 1 + E 2 ) D E 0 D E D 2 0 0 0 x 1 2 ] .
g i j = [ 1 - E / D 0 - E / D ( 1 + E 2 ) / D 2 0 0 0 1 / x 1 2 ] ,
{ 2 21 } = { 2 12 } = E - 2 b x 1 x 2 D { 1 33 } = - x 1 { 2 11 } = E x 1 D { 3 31 } = { 3 13 } = 1 x 1 { 2 33 } = E x 1 D .
σ 11 = 2 G ( g 11 g 11 ) 1 2 { [ 1 - ν 1 - 2 ν g 11 g 11 - 1 2 x 1 + g 12 g 11 - 1 2 x 2 + g 11 g 11 - 1 2 ( E - 2 b x 1 x 2 D + 1 x 1 ) ν 1 - 2 ν ] u 1 + 1 - ν 1 - 2 ν g 11 g 11 - 1 2 u 1 x 1 + g 12 g 11 - 1 2 u 1 x 2 + ν 1 - 2 ν g 11 g 22 - 1 2 u 2 x 2 + ν 1 - 2 ν g 11 g 33 - 1 2 u 3 x 3 - 1 + ν 1 - 2 ν g 11 α t T } .
ρ a 1 = f 1 + ( g 11 1 2 g 11 - 1 2 x 1 + E - 2 b x 1 x 2 D + 1 x 1 ) σ 11 - ( g 11 1 2 x 1 ) σ 33 + σ 11 x 1 + ( g 22 ) 1 2 σ 12 x 2 + ( g 11 1 2 x 1 ) σ 13 x 3 .

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