Abstract

We have developed a method of computational simulation and estimation of thermo-optical distortions in solid-state laser rods, electro-optic shutters, and modulators, considering real heat-transfer conditions, anisotropic material properties, and an arbitrary cross-section shape of the element. Numerical investigations for Nd:YAG laser rods and potassium dihydrogen phosphate electro-optic shutters have been carried out, new results have been obtained, and known analytical solutions have been corrected.

© 1993 Optical Society of America

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References

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  1. W. Koechner, Solid State Laser Engineering (Springer-Verlag, New York, 1976).
  2. N. Gopi, T. P. S. Nathan, B. K. Sinha, “Experimental studies of transient, thermal depolarization in a Nd:glass laser rod,” Appl. Opt. 29, 2259–2265 (1990).
    [CrossRef] [PubMed]
  3. J. F. Nye, Physical Properties of Crystals (Clarendon, Oxford, 1964).
  4. I. P. Kaminow, “Strain effects in electrooptical light modulators,” Appl. Opt. 3, 511–515 (1964).
    [CrossRef]
  5. D. Eimerl, “High average power harmonic generator,” IEEE J. Quantum Electron. QE-23, 525–592 (1987).
  6. E. K. Malbutis, J. J. Reksnys, S. V. Sakalauskas, “Contribution of thermoeleastic stresses into dn/dT for crystals of hexagonal and trigonal symmetry heated by laser radiation,” Kvantovaya Elektron. (Moscow) 2, 2493–2498 (1975).
  7. W. Koechner, D. K. Rice, “Effect of birefringence on the performance of linearly polarized Nd:YAG lasers,” IEEE J. Quantum Electron. QE-7, 557–566 (1970).
    [CrossRef]
  8. M. A. Karr, “Nd:YAG laser due to internal Brewster polarizer,” Appl. Opt. 10, 893–895 (1971).
    [CrossRef] [PubMed]
  9. W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44, 3162–3170 (1973).
    [CrossRef]
  10. L. N. Soms, A. A. Tarasov, V. V. Shashkin, “On the problem of linearly polarized light by YAG:Nd3+ laser rod under conditions of thermally induced birefringence,” Kvantovaya Elektron. (Moscow) 7, 619–620 (1980).
  11. G. N. Dulnev, A. E. Michailov, V. G. Parfenov, “The modeling of the thermal regimes of the pump systems,” J. Eng. Phys. Engl. Transl. 53, 107–113 (1987).
  12. D. Skinner, I. Tregellas-Williams, “Total energy and energy distribution in a laser crystal due to optical pumping, as calculated by Monte-Carlo method,” Aust. J. Phys. 19, 7–17 (1966).
  13. A. V. Lykov, Heat-Mass Transfer (Energiya, Moscow, 1972).
  14. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972).
  15. O. C. Zienkiewicz, The Finite Element Method in Engineering Science (McGraw-Hill, New York, 1977).
  16. D. H. Anderson, J. C. Tannenhill, R. H. Pletcher, Computational Mechanics and Heat Transfer (Hemisphere, New York, 1984), Vol. 1.
  17. S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, New York, 1970).
  18. B. N. Grechushnikova, D. Brodovskiy, “Thermal stresses in cubic crystals,” Kristallographiya 1, 597–599 (1956).

1990 (1)

1987 (2)

D. Eimerl, “High average power harmonic generator,” IEEE J. Quantum Electron. QE-23, 525–592 (1987).

G. N. Dulnev, A. E. Michailov, V. G. Parfenov, “The modeling of the thermal regimes of the pump systems,” J. Eng. Phys. Engl. Transl. 53, 107–113 (1987).

1980 (1)

L. N. Soms, A. A. Tarasov, V. V. Shashkin, “On the problem of linearly polarized light by YAG:Nd3+ laser rod under conditions of thermally induced birefringence,” Kvantovaya Elektron. (Moscow) 7, 619–620 (1980).

1975 (1)

E. K. Malbutis, J. J. Reksnys, S. V. Sakalauskas, “Contribution of thermoeleastic stresses into dn/dT for crystals of hexagonal and trigonal symmetry heated by laser radiation,” Kvantovaya Elektron. (Moscow) 2, 2493–2498 (1975).

1973 (1)

W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44, 3162–3170 (1973).
[CrossRef]

1971 (1)

1970 (1)

W. Koechner, D. K. Rice, “Effect of birefringence on the performance of linearly polarized Nd:YAG lasers,” IEEE J. Quantum Electron. QE-7, 557–566 (1970).
[CrossRef]

1966 (1)

D. Skinner, I. Tregellas-Williams, “Total energy and energy distribution in a laser crystal due to optical pumping, as calculated by Monte-Carlo method,” Aust. J. Phys. 19, 7–17 (1966).

1964 (1)

1956 (1)

B. N. Grechushnikova, D. Brodovskiy, “Thermal stresses in cubic crystals,” Kristallographiya 1, 597–599 (1956).

Anderson, D. H.

D. H. Anderson, J. C. Tannenhill, R. H. Pletcher, Computational Mechanics and Heat Transfer (Hemisphere, New York, 1984), Vol. 1.

Brodovskiy, D.

B. N. Grechushnikova, D. Brodovskiy, “Thermal stresses in cubic crystals,” Kristallographiya 1, 597–599 (1956).

Dulnev, G. N.

G. N. Dulnev, A. E. Michailov, V. G. Parfenov, “The modeling of the thermal regimes of the pump systems,” J. Eng. Phys. Engl. Transl. 53, 107–113 (1987).

Eimerl, D.

D. Eimerl, “High average power harmonic generator,” IEEE J. Quantum Electron. QE-23, 525–592 (1987).

Goodier, J. N.

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, New York, 1970).

Gopi, N.

Grechushnikova, B. N.

B. N. Grechushnikova, D. Brodovskiy, “Thermal stresses in cubic crystals,” Kristallographiya 1, 597–599 (1956).

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972).

Kaminow, I. P.

Karr, M. A.

Koechner, W.

W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44, 3162–3170 (1973).
[CrossRef]

W. Koechner, D. K. Rice, “Effect of birefringence on the performance of linearly polarized Nd:YAG lasers,” IEEE J. Quantum Electron. QE-7, 557–566 (1970).
[CrossRef]

W. Koechner, Solid State Laser Engineering (Springer-Verlag, New York, 1976).

Lykov, A. V.

A. V. Lykov, Heat-Mass Transfer (Energiya, Moscow, 1972).

Malbutis, E. K.

E. K. Malbutis, J. J. Reksnys, S. V. Sakalauskas, “Contribution of thermoeleastic stresses into dn/dT for crystals of hexagonal and trigonal symmetry heated by laser radiation,” Kvantovaya Elektron. (Moscow) 2, 2493–2498 (1975).

Michailov, A. E.

G. N. Dulnev, A. E. Michailov, V. G. Parfenov, “The modeling of the thermal regimes of the pump systems,” J. Eng. Phys. Engl. Transl. 53, 107–113 (1987).

Nathan, T. P. S.

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Clarendon, Oxford, 1964).

Parfenov, V. G.

G. N. Dulnev, A. E. Michailov, V. G. Parfenov, “The modeling of the thermal regimes of the pump systems,” J. Eng. Phys. Engl. Transl. 53, 107–113 (1987).

Pletcher, R. H.

D. H. Anderson, J. C. Tannenhill, R. H. Pletcher, Computational Mechanics and Heat Transfer (Hemisphere, New York, 1984), Vol. 1.

Reksnys, J. J.

E. K. Malbutis, J. J. Reksnys, S. V. Sakalauskas, “Contribution of thermoeleastic stresses into dn/dT for crystals of hexagonal and trigonal symmetry heated by laser radiation,” Kvantovaya Elektron. (Moscow) 2, 2493–2498 (1975).

Rice, D. K.

W. Koechner, D. K. Rice, “Effect of birefringence on the performance of linearly polarized Nd:YAG lasers,” IEEE J. Quantum Electron. QE-7, 557–566 (1970).
[CrossRef]

Sakalauskas, S. V.

E. K. Malbutis, J. J. Reksnys, S. V. Sakalauskas, “Contribution of thermoeleastic stresses into dn/dT for crystals of hexagonal and trigonal symmetry heated by laser radiation,” Kvantovaya Elektron. (Moscow) 2, 2493–2498 (1975).

Shashkin, V. V.

L. N. Soms, A. A. Tarasov, V. V. Shashkin, “On the problem of linearly polarized light by YAG:Nd3+ laser rod under conditions of thermally induced birefringence,” Kvantovaya Elektron. (Moscow) 7, 619–620 (1980).

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972).

Sinha, B. K.

Skinner, D.

D. Skinner, I. Tregellas-Williams, “Total energy and energy distribution in a laser crystal due to optical pumping, as calculated by Monte-Carlo method,” Aust. J. Phys. 19, 7–17 (1966).

Soms, L. N.

L. N. Soms, A. A. Tarasov, V. V. Shashkin, “On the problem of linearly polarized light by YAG:Nd3+ laser rod under conditions of thermally induced birefringence,” Kvantovaya Elektron. (Moscow) 7, 619–620 (1980).

Tannenhill, J. C.

D. H. Anderson, J. C. Tannenhill, R. H. Pletcher, Computational Mechanics and Heat Transfer (Hemisphere, New York, 1984), Vol. 1.

Tarasov, A. A.

L. N. Soms, A. A. Tarasov, V. V. Shashkin, “On the problem of linearly polarized light by YAG:Nd3+ laser rod under conditions of thermally induced birefringence,” Kvantovaya Elektron. (Moscow) 7, 619–620 (1980).

Timoshenko, S. P.

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, New York, 1970).

Tregellas-Williams, I.

D. Skinner, I. Tregellas-Williams, “Total energy and energy distribution in a laser crystal due to optical pumping, as calculated by Monte-Carlo method,” Aust. J. Phys. 19, 7–17 (1966).

Zienkiewicz, O. C.

O. C. Zienkiewicz, The Finite Element Method in Engineering Science (McGraw-Hill, New York, 1977).

Appl. Opt. (3)

Aust. J. Phys. (1)

D. Skinner, I. Tregellas-Williams, “Total energy and energy distribution in a laser crystal due to optical pumping, as calculated by Monte-Carlo method,” Aust. J. Phys. 19, 7–17 (1966).

IEEE J. Quantum Electron. (2)

W. Koechner, D. K. Rice, “Effect of birefringence on the performance of linearly polarized Nd:YAG lasers,” IEEE J. Quantum Electron. QE-7, 557–566 (1970).
[CrossRef]

D. Eimerl, “High average power harmonic generator,” IEEE J. Quantum Electron. QE-23, 525–592 (1987).

J. Appl. Phys. (1)

W. Koechner, “Transient thermal profile in optically pumped laser rods,” J. Appl. Phys. 44, 3162–3170 (1973).
[CrossRef]

J. Eng. Phys. Engl. Transl. (1)

G. N. Dulnev, A. E. Michailov, V. G. Parfenov, “The modeling of the thermal regimes of the pump systems,” J. Eng. Phys. Engl. Transl. 53, 107–113 (1987).

Kristallographiya (1)

B. N. Grechushnikova, D. Brodovskiy, “Thermal stresses in cubic crystals,” Kristallographiya 1, 597–599 (1956).

Kvantovaya Elektron. (Moscow) (2)

L. N. Soms, A. A. Tarasov, V. V. Shashkin, “On the problem of linearly polarized light by YAG:Nd3+ laser rod under conditions of thermally induced birefringence,” Kvantovaya Elektron. (Moscow) 7, 619–620 (1980).

E. K. Malbutis, J. J. Reksnys, S. V. Sakalauskas, “Contribution of thermoeleastic stresses into dn/dT for crystals of hexagonal and trigonal symmetry heated by laser radiation,” Kvantovaya Elektron. (Moscow) 2, 2493–2498 (1975).

Other (7)

J. F. Nye, Physical Properties of Crystals (Clarendon, Oxford, 1964).

W. Koechner, Solid State Laser Engineering (Springer-Verlag, New York, 1976).

A. V. Lykov, Heat-Mass Transfer (Energiya, Moscow, 1972).

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972).

O. C. Zienkiewicz, The Finite Element Method in Engineering Science (McGraw-Hill, New York, 1977).

D. H. Anderson, J. C. Tannenhill, R. H. Pletcher, Computational Mechanics and Heat Transfer (Hemisphere, New York, 1984), Vol. 1.

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, New York, 1970).

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Figures (12)

Fig. 1
Fig. 1

Optical element and Cartesian coordinate system.

Fig. 2
Fig. 2

Temperature field calculation grid for an actual laser head.

Fig. 3
Fig. 3

Finite elements for calculation: (a) translation components, (b) temperature field.

Fig. 4
Fig. 4

Calculation grid for (a) circular and (b) rectangular cross section.

Fig. 5
Fig. 5

Steady-state change of transmission of KDP shutter with circular cross section (at spatially uniform light beam). A, analytical solution from Ref. 3; B, corrected solution according to the formula of Eqs. (27).

Fig. 6
Fig. 6

Change of transmission of KDP shutter: (a) steady state; (b) transient case (total internal heating power 1 W). A, circular cross section, spatially uniform light beam; B, square-shaped cross section, spatially uniform light beam; C, circular cross section, Gaussian light beam; D, square-shaped cross section, Gaussian light beam.

Fig. 7
Fig. 7

Change of transmission of KDP shutter versus ΔT0L in steady state. Conditions are the same as in Fig. 6(a).

Fig. 8
Fig. 8

High-power depolarization at uniform internal heating power distribution A d versus direction of initial light polarization γ in a Nd:YAG laser rod cut along the [001] (curve A) and the [111] (curve B) crystallographic direction.

Fig. 9
Fig. 9

Cross section of pumping cavity configuration.1, flash lamp; 2, housing; 3, reflector; 4, laser rod. D1 = 3 mm, D2 = 6 mm, D = 6.3 mm. R0 = 4.65 mm, δ = 0.5 mm, Δ = 2 mm for liquid-cooled systems and R0 = 4.35 mm, δ = 0.2 mm, Δ = 1 mm for air-cooled systems. Total length L = 65 mm.

Fig. 10
Fig. 10

Depolarization loss in Nd:YAG laser rod cut along (a) [001]; (b) [111] crystallographic direction in liquid-cooled laser head. A, B maximum and minimum depolarization obtained under assumption of uniform inner heat sources distribution in the rod; C, D, the same but considering actual heat transfer conditions in laser head. Steady state.

Fig. 11
Fig. 11

Nonuniformity of pump power and thermal distribution in Nd:YAG laser rod in air-cooled laser head at 40-W pump power: (a) isolines of relative internal heating power qv/qvmax; (b) isotherms of temperature changes (TT0), K(τ = 50 s).

Fig. 12
Fig. 12

Transient depolarization in Nd:YAG laser rod cut along the [001] crystallographic direction in air-cooled system. A, B, maximum and minimum depolarization obtained under assumption of uniform inner heat sources distribution in the rod; C, D, the same but considering actual heat transfer conditions in laser head. Pump power is 40 W.

Equations (53)

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A d = S I ( x , y ) sin 2 2 [ θ ( x , y ) - γ ] sin 2 [ δ ( x , y ) / 2 ] d S S I ( x , y ) d S .
c i ( T i ) ρ i T i τ = div [ k i ( T i ) grad ( T i ) ] + q v i ( x ¯ i , τ ) ,
c k ( T k ) ρ k [ T k τ + v k grad ( T k ) ] = div [ k k ( T k ) grad ( T k ) ] + q v k ( x ¯ k , τ ) ,
l = 1 N b S l α i l b · g ( x ¯ , T i , T l ) ( T i - T l ) d S l = 0 .
T i τ = 0 = T 0 i ( x ¯ ) ,             T j τ = 0 = T 0 k ( x ¯ ) ,
- k i ( T i ) T i n = j = 1 j i N b α i j b · b ( x ¯ , T i , T j ) ( T i - T j ) + k = 1 N f α i k b · f ( x ¯ , T i , T k ) ( T i - T k ) + l = 1 N g α i l b · g ( x ¯ , T i , T l ) ( T i - T l ) + m = 1 N a α i m b · a ( x ¯ , T i , T m ) ( T i - T m ) + q s i ( x ¯ , τ ) + e j = 1 N b S j q j r cos [ ϕ i ( x ¯ i , x ¯ j ) ] cos [ ϕ j ( x ¯ i , x ¯ j ) ] π r i j 2 ( x ¯ i , x ¯ j ) × d S j - i σ T i 4 .
q i r ( x ¯ ) - ( 1 - i ) j = 1 N b S j q j r cos [ ϕ i ( x ¯ i , x ¯ j ) ] cos [ ϕ j ( x ¯ i , x ¯ j ) ] π r i j 2 ( x ¯ i , x ¯ j ) d S j = i σ T i 4 .
- k k ( T k ) T k n = i = 1 N b α i k b · j ( x ¯ , T i , T k ) ( T i - T k ) .
c ( T ) ρ T τ = div [ k ( T ) grad ( T ) ] + q v ( x , y , τ )
- k ( T ) T n = i = 1 N α ( x , y , T , T i ) ( T - T i ) + q s ( x , y , τ )
T τ = 0 = T 0 ( x , y ) ,
σ 11 x 1 + σ 12 x 2 = 0 , σ 12 x 1 + σ 22 x 2 = 0.
{ σ } = [ D ] { } - [ D ] { 0 } ,
{ } T = ( 11 , 22 , 2 12 ) ,             { σ } T = [ σ 11 , σ 22 , σ 12 ] ,
{ 0 } T = { α ˜ } T ( T - T 0 ) .
11 = U 1 x 1 ;             12 = 1 2 ( U 1 x 2 + U 2 x 1 ) ;             22 = U 2 x 2 .
σ 11 n 1 + σ 12 n 2 = 0 ;             σ 12 n 1 + σ 22 n 2 = 0 ,
σ 33 = 0 ;             σ 23 = 0 ;             σ 13 = 0.
1 S S σ i 3 d S = 0 ;     1 S S x σ i 3 d S = 0 ;     1 S S y σ i 3 d S = 0 ,
σ i 3 = σ ˜ i 3 - 1 S S σ ˜ i 3 d S - x S x σ ˜ i 3 d S S x 2 d S - y S y σ ˜ i 3 d S S y 2 d S ,
B i j x i x j = 1.
B i i = 1 / [ n i ( T 0 ) + β i ( T - T 0 ) ] 2 + Δ B i i σ + Δ B i i E , B i j = Δ B i j σ + Δ B i j E ,             i j ,
δ ( x , y ) = π λ ( L + U 3 ) n 3 ( T 0 ) [ ( Δ B 11 - Δ B 22 ) 2 + 4 Δ B 12 2 ] 1 / 2 ,
θ ( x , y ) = arctan ( B 11 * / B 22 * ) ,
det [ B - ( 1 / n 2 ) I 3 ] = 0 ,
K o = A d 0 A d E 3 = 0 , K c = 1 - A d c 1 - A d E 3 0 .
σ 12 = q v 2 k α 11 - ( S 13 / S 33 ) α 33 6 ( S 11 - S 13 2 / S 33 ) + 2 ( S 12 - S 13 2 / S 33 ) + S 66 × r 2 sin ( 2 φ ) ,
σ 12 = q v 8 k ( S 11 - S 12 ) ( S 11 - S 13 2 / S 33 ) S 66 × [ α 11 - ( S 13 / S 33 ) α 33 ] r 2 sin ( 2 ϕ ) .
Δ K = [ 1 - H ( G ) ] / 2 , H ( G ) = 1 G 0 G J 0 ( x ) d x , G = 1 2 k λ n 3 ( T 0 ) π 66 × α 11 - ( S 13 / S 33 ) α 33 6 ( S 11 - S 13 2 / S 33 ) + 2 ( S 12 - S 13 2 / S 33 ) + S 66 P ,
I ( r ) = I 0 exp ( - 2 r 2 / R 2 ) ,
A d = 4 π - π / 4 π / 4 [ ξ tan ( 2 φ ) cos ( 2 γ ) - sin ( 2 γ ) ] 2 1 + ξ 2 tan 2 ( 2 φ ) × ( 1 - sin ( 2 G ) 2 G ) d φ , ξ = π 66 / ( π 11 - π 12 ) , G = B [ 1 + ξ 2 tan 2 ( 2 φ ) 1 + tan 2 ( 2 φ ) ] 1 / 2 , B = π L n 0 3 4 λ α E 1 - ν ( π 11 - π 12 ) Δ T 0 ,
A d = 1 2 ( 1 + ζ ) [ ξ cos 2 ( 2 γ ) + sin 2 ( 2 γ ) ] .
A d min = 0.5 / ( 1 + ξ ) ,             A d max = 0.5 ξ / ( 1 + ξ ) .
K A = A d ( O z ) [ 111 ] A d min ( O z ) [ 001 ] = { ( 1 + ξ ) / 2 ξ > 1 ( 1 + ξ ) / 2 ξ ξ < 1 .
A d 0 = B 2 6 [ ξ 2 cos 2 ( 2 γ ) + sin 2 ( 2 γ ) ] ,
A d min 0 = B 2 / 6 ,             A d max 0 = B 2 ξ 2 / 6
( π 11 - π 12 ) ( O z ) [ 111 ] = ( π 11 - π 12 + 2 π 66 ) 3 ( O z ) [ 001 ] ,
K A 0 = A d 0 ( O z ) [ 111 ] A d min 0 ( O z ) [ 001 ] = { ( 1 + 2 ξ ) 2 / 9 ξ > 1 ( 1 + 2 ξ ) 2 / 9 ξ 2 ξ < 1 .
γ = γ min = { π / 4 ξ > 1 0 ξ < 1 ( minimum depolarization ) ,
γ = γ max = { 0 ξ > 1 π / 4 ξ < 1 ( maximum depolarization ) .
( σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 ) = [ C 11 C 12 C 13 C 14 C 15 C 16 C 21 C 22 C 23 C 24 C 25 C 26 C 31 C 32 C 33 C 34 C 35 C 36 C 41 C 42 C 43 C 44 C 45 C 46 C 51 C 52 C 53 C 54 C 55 C 56 C 61 C 62 C 63 C 64 C 65 C 66 ] × ( 11 - α 11 ( T - T 0 ) 22 - α 22 ( T - T 0 ) 33 - α 33 ( T - T 0 ) 2 23 - α 23 ( T - T 0 ) 2 13 - α 13 ( T - T 0 ) 2 12 - α 12 ( T - T 0 ) ) .
( 11 - α 11 ( T - T 0 ) 22 - α 22 ( T - T 0 ) 33 - α 33 ( T - T 0 ) 2 23 - α 23 ( T - T 0 ) 2 13 - α 13 ( T - T 0 ) 2 12 - α 12 ( T - T 0 ) ) = [ S 11 S 12 S 13 S 14 S 15 S 16 S 21 S 22 S 23 S 24 S 25 S 26 S 31 S 32 S 33 S 34 S 35 S 36 S 41 S 42 S 43 S 44 S 45 S 46 S 51 S 52 S 53 S 54 S 55 S 56 S 61 S 62 S 63 S 64 S 65 S 66 ] ( σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 ) .
{ α ˜ } = ( α 11 α 22 α 12 ) ,
[ D ] = [ C 11 C 12 C 16 C 21 C 22 C 26 C 61 C 62 C 66 ] - [ C 13 C 14 C 15 C 23 C 24 C 25 C 63 C 64 C 65 ] × [ C 33 C 34 C 35 C 43 C 44 C 45 C 53 C 54 C 55 ] - 1 [ C 31 C 32 C 36 C 41 C 42 C 46 C 51 C 52 C 56 ] .
( σ 33 σ 13 σ 23 ) = - [ S 33 S 34 S 35 S 43 S 44 S 45 S 53 S 54 S 55 ] - 1 × [ ( σ 33 σ 13 σ 23 ) ( T - T 0 ) + [ S 31 S 32 S 36 S 41 S 42 S 46 S 51 S 52 S 56 ] ( σ 11 σ 22 σ 12 ) ]
{ α ˜ } = ( α 11 α 22 α 12 ) - [ S 13 S 14 S 15 S 23 S 24 S 25 S 63 S 64 S 65 ] × [ S 33 S 34 S 35 S 43 S 44 S 45 S 53 S 54 S 55 ] - 1 ( α 33 α 23 α 13 ) ,
[ D ] = [ S 11 S 12 S 16 S 21 S 22 S 26 S 61 S 62 S 66 ] - [ S 13 S 14 S 15 S 23 S 24 S 25 S 63 S 64 S 65 ] × [ S 33 S 34 S 35 S 43 S 44 S 45 S 53 S 54 S 55 ] - 1 [ S 31 S 32 S 36 S 41 S 42 S 46 S 51 S 52 S 56 ] - 1 .
T ( r ) = T a + Δ T 0 ( 1 - r 2 R 2 ) , Δ T 0 = q v R 2 / 4 k = P / 4 π k L ,
2 11 x 1 2 + 2 22 x 2 2 = 2 2 12 x 1 x 2 .
( S 11 - S 13 2 S 33 ) ( 2 σ 11 x 1 2 + 2 σ 22 x 2 2 ) + ( S 12 - S 13 2 S 33 ) ( 2 σ 11 x 2 2 + 2 σ 22 x 1 2 ) + ( α 11 - S 13 S 33 α 33 ) 2 T = S 66 2 σ 12 x 1 x 2 .
σ 11 = σ max ( 1 - x 2 - 3 y 2 ) / 2 , σ 22 = σ max ( 1 - 3 x 2 - y 2 ) / 2 , σ 12 = σ max x y ,
σ max = q v k α 11 - ( S 13 / S 33 ) α 33 6 ( S 11 - S 13 2 / S 33 ) + 2 ( S 12 - S 13 2 / S 33 ) + S 66 .
σ max = q v 8 k α E 1 - ν .

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