Advanced Light Source - Users Handbook

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The Inverse Laplace Transformation of Eq. 95 results:   I1 pffiffiffiffi x 1 2 atierfc pffiffiffiffi À eÀdx Tðx; tÞ ¼ 2d k 2 at   pffiffiffiffi 1 ad2 tþdx x þ e erfc d at þ pffiffiffiffi 2d 2 at  ! p ffiffiffiffi 1 ad2 tÀdx x p ffiffiffiffi þ e erfc d at À 2d 2 at ð2:95Þ ð2:96Þ where erf is the error function, erfc is the complementary error function, and ierfc is the integral of complementary error function, which are: Z x 2 2 eÀv dv erf ðvÞ ¼ pffiffiffi p 0 ð2:97Þ erfcðvÞ ¼1 À erf ðvÞ 1 Àv2 ierfcðvÞ ¼ pffiffiffi e À verfcðvÞ p Introducing dimensionless quantities as: s ¼ ad2 t : x0 ¼ xd : T 0 ¼ kd T I1 Substituting the dimensionless quantities in Eq.

With respect to t, the Laplace transformation of Eq. 86 yields: Ã o2 T I1 d Àdx 1 Â e þ ¼ sT À Tðx; 0Þ 2 ox ks a ð2:89Þ where T ¼ Tðx; sÞ. Using the initial condition, Tðx; 0Þ ¼ 0; Eq. 87 yields: o2 T I1 d Àdx e À q2 T ¼ À ox2 ks ð2:90Þ where q2 ¼ as . 89 has a solution: T ¼ Aeqx þ BeÀqx þ I1 ad À Á eÀdx ks s À ad2 ð2:91Þ where A and B are the constants and they are calculated through the boundary conditions. Substituting boundary condition, ooxT ¼ 0 at the surface (x ¼ 0), it gives: 20 2 Equilibrium Laser Pulse Heating and Thermal Stress Analysis A¼Bþ I1 ad2 À Á qks s À ad2 ð2:92Þ The boundary condition, T ¼ 0 at x ¼ 1, results A ¼ 0 in Eq.

1 x 7 6  h2 t    4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À Á þ h e Erf h t À c 5 Á U½1Š    1 p t À x =c 1 ð2:287Þ 52 2 Equilibrium Laser Pulse Heating and Thermal Stress Analysis and À Á rx 212 ¼ À  ðh þ 1Þc3 1 hÀ Á     2 ðb þ 1Þðh þ b =t Þ c2 1 þ b =t 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffi     1 b b x 7 6 Àb þ   5 Á U½1Š  4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ À  eÀb Erf À Á t c1 t p t À x =c  =t 1 ð2:288Þ and  À Á ðh þ 1Þc3 1Àh Á À Á Á rx 312 ¼ À 2   h À c2 1 À c2 c2 1 1 1 þ b =t 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi!

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