5.2 Transient thermal distortions

In the quasistatic regime, the time scale for temperature evolution is clearly too long to generate inertial effects in the material. Thus, the elastodynamic equations reduce to elastic. The equilibrium equations are therefore unchanged with respect to Equation (3.91View Equation). The displacement vector is unchanged in form with respect to the static case, except that the time enters as an evolution parameter through temperature. The temperature field being, as usual,
∑ T (t,r,z) = Ts(t,z)J0(ksr). (5.39 ) s

The generic thermoelastic longitudinal displacement is of the form

∑ uz(t,r,z) = Bs (t,z)J0(ζsr∕a ). (5.40 ) s
We are interested in the displacement of the reflecting surface, and the general solution gives
{ cosh γ Bs (t,− h ∕2) = 2α(1 + σ) ------s[e′s(t) sinh γs − ksoscosh γs] Γ ′′s
sinh γ } ---′-s-[o′s(t)cosh γs − ksessinh γs] , (5.41 ) Γs
where the function 𝒯s(t,z) is a special solution of
2 2 (∂z − ks)𝒯s(t,z) = Ts(t,z) (5.42 )
and where the notation of Equation (3.148View Equation) has been employed.

5.2.1 Case of coating absorption

In the case of a heat source on the coating, we have found the temperature field from Equation (5.33View Equation), so that we have

[ ( ) 𝜖P h3p ∑ cos uj 1 − exp (− t∕τ′ ) 𝒯s(t,z) = − ------s2 -------------------2-s,j-2-2-cos(κjz ) 8πKa j (1 + sin(2uj)∕2uj)(γs + uj) ( ′′ ) ] --sinvj--1 −-exp(−-t∕τs,j)-- − (1 − sin(2vj)∕2vj)(γ2s + v2j)2 sin(κjz) (5.43 )
and consequently
𝜖h3ps- kses(t) = − 8Ka ζsUs,j(t) (5.44 )
𝜖h3ps ksos(t) = ------ζsVs,j(t) (5.45 ) 8Ka
′ 𝜖h3ps- es(t) = − 8Ka χVs,j(t) (5.46 )
3 e′(t) = 𝜖h--psχU (t) (5.47 ) s 8Ka s,j
with
( ) cos2uj 1 − exp (− t∕τs′,j Us,j(t) = --------------------2----22- (5.48 ) (1 + sin(2uj)∕2uj )(γ s + u j)
( ) --sin2vj--1 −-exp(−-t∕-τ′′s,j-- Vs,j(t) = (1 − sin(2v )∕2v )(γ2+ v2)2, (5.49 ) j j s j
so that finally
3 ∑ [ ] B (t,− h ∕2) = − α(1-+-σ)𝜖P-h-p d1,ssinh-γsUs,j(t)-+ d2,scoshγsVs,j(t)- . (5.50 ) s 4πKa3 s Γ ′s Γ ′s′ j
The contribution to curvature of the Saint-Venant correction reduces to
δu (t,r,− h ∕2) = c(t)r2 (5.51 ) z
with the time dependent curvature
∑ c(t) = 3χP-α-𝜖h psJ0-(ζs)-[(χ + 2a∕h)(1 + γ2∕v2) 2πKa3 ζ2s s j s,j ′′ +2 σd2,s(sinh γs − γscosh γs)∕ γsΓs]Vs,j(t). (5.52 )
See in Figures 44View Image, 45View Image, and 46View Image, the time evolution of the surface deformation under a constant power flux. The dashed curves corresponding to the steady state are computed with Equations (3.120View Equation) and (3.122View Equation). Using our averaging technique, we can compute the transient curvature radius for the three considered examples (Figures 47View Image, 48View Image, and 49View Image).
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Figure 44: Coating absorption: time evolution of the reflecting surface caused by thermal expansion. Mode LG0,0, w = 2 cm
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Figure 45: Coating absorption: time evolution of the reflecting surface caused by thermal expansion. Flat mode, b = 9.1 cm
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Figure 46: Coating absorption: time evolution of the reflecting surface caused by thermal expansion. Mode LG5,5, w = 3.5 cm
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Figure 47: Coating absorption: time evolution of the curvature radius of the thermal lens caused by thermal expansion. Mode LG0,0, w = 2 cm
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Figure 48: Coating absorption: time evolution of the curvature radius of the thermal lens caused by thermal expansion. Flat mode, b = 9.1 cm
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Figure 49: Coating absorption: time evolution of the curvature radius of the thermal lens caused by thermal expansion. Mode LG5,5, w = 3.5 cm

5.2.2 Case of bulk absorption

The temperature field defined by Equation (5.37View Equation) gives

βP h4p ∑ sin u ∕u 𝒯s (t,z) = − -------s --------------j--j-------2--[1 − exp(− t∕τs,j)]cos(κjz ) (5.53 ) 8πKa2 j (1 + sin (2uj )∕2uj)(γ2s + uj)2
so that
βP h4 kses(t) = − ------3psζsWs,j(t) (5.54 ) 8πKa
′ βP h3 o(t) = ------3psχWs,j (t) (5.55 ) 8πKa
with
( ′ ) Ws,j(t) = sin(2uj-)∕2uj--1-−-exp-(−-t∕τs,j)-- (5.56 ) (1 + sin (2uj)∕2uj)(γ2s + u2j)2
yielding, finally,
α (1 + σ)βP h4 ∑ p d sinh γ Bs(t,− h∕2) = − ----------3--- -s-1,s-′----sWs,j(t). (5.57 ) 4πKa j Γ s
Due to the symmetry of the temperature field in z and to the resulting symmetry of the stress field Θrr (a,z), the contribution to curvature of the Saint-Venant correction (mean torque) is zero. See in Figures 50View Image, 51View Image, and 52View Image the time evolution of the distorted surface for our three examples. The curvature evolution is plotted in Figures 53View Image,54View Image, and 55View Image.
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Figure 50: Bulk absorption: time evolution of the reflecting surface caused by thermal expansion. Mode LG0,0, w = 2 cm
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Figure 51: Bulk absorption: time evolution of the reflecting surface caused by thermal expansion. Flat mode, b = 9.1 cm
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Figure 52: Bulk absorption: time evolution of the reflecting surface caused by thermal expansion. Mode LG5,5, w = 3.5 cm
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Figure 53: Bulk absorption: time evolution of the curvature radius caused by thermal expansion. Mode LG0,0, w = 2 cm
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Figure 54: Bulk absorption: time evolution of the curvature radius caused by thermal expansion. Flat mode, b = 9.1 cm
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Figure 55: Bulk absorption: time evolution of the curvature radius caused by thermal expansion. Mode LG5,5, w = 3.5 cm

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