Coupled Problem of Thermoelasticity: Solution in a Series of Functions Form

Abstract

hyperbolic-parabolic type. And this is the theory of which one problem we will solve in this encyclopedia entry. Coupled Thermoelasticity The governing equations of coupled thermoelasticity for a linear homogeneous isotro-pic material are: The equation of motion div S þ b ¼ r€ u; S ¼ S T ð1Þ The strain-displacement relation E ¼ 1 2 ðHu þ Hu T Þ ð 2Þ Hooke's law S ¼ 2mE þ lðtr EÞ1 À gT1; T ¼ y À y 0 ð3Þ The energy equation H 2 T À 1 k _ T À gy 0 k tr _ E ¼ À Q k ð4Þ Here 1 is the unit second-order tensor, y is the temperature, y 0 is the reference temperature , and T is temperature change. From the combination of the first three equations, the displacement-temperature equation of motion is obtained: mH 2 u þ ðl þ mÞHðdiv uÞ À gHT þ b ¼ r€ u ð5Þ Equations (4) and (5) represent the displacement temperature equations of coupled thermoelasticity for a solid elastic body. The full description of the problem requires the energy equation and the equation of motion complemented by appropriate initial and boundary conditions for thermal and mechanical loads.