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unit4_notes - 16.21 Techniques of Structural Analysis and...

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16.21 Techniques of Structural Analysis and Design Spring 2005 Unit #4 - Thermodynamics Principles Ra´ul Radovitzky February 22, 2005 First Law of Thermodynamics d K + U = P + H (1) dt where: K : kinetic energy U : internal energy P : Power of external forces H : hear exchange per unit time 1 u u 1 ∂u i ∂u i K = ρ dV = ρ dV (2) 2 V ∂t · ∂t 2 V ∂t ∂t U = ρ U 0 dV = U 0 dV (3) V V 1

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where U 0 , U 0 are the internal energy densities per unit mass and per unit volume, respectively. u u P = f dV + t dS (4) · ∂t S · ∂t V In components: ∂u i ∂u i P = f i dV + t i dS (5) ∂t S ∂t V Replacing t i = n j σji in this expression: ∂u i ∂u i P = f i dV + n j σji dS (6) ∂t S ∂t V Using Gauss’ Theorem: ∂u i ∂u i P = f i dV + σ ji dV V ∂t V ∂x j ∂t �� ∂σ ji ∂u i ∂u i = + f i + σ ji dV ∂x j ∂t ∂x j ∂t V �� �� 2 u i ∂u i ρ (why?) (7) ∂t 2 ∂t ∂x j 2
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unit4_notes - 16.21 Techniques of Structural Analysis and...

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