? ? ? 16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #4 - Thermodynamics Principles First Law of Thermodynamics d dt ? K + U ? = P + H (1) 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 UdV, U : Internal energy density (3)U = ρ ? ? V ? ? ?u ?u P = f · dV + t · dS (4) V ?t S ?t 1 ˙ ˙ ? ? ? ? ? ? ?? ? In components: ?u i ?u i P = f i dV + t i dS (5) V ?t S ?t Replacing t i = n j σji in this expression: ?u i ?u i P = f i dV + n j σji dS (6) V ?t S ?t Using Gauss’ Theorem: ?u i ? ? ? ?u i ? P = f i dV + σ ji dV ?t V ?x j ?t ? V ?? ?σ ji ? ?u i ? ?u i ? = + f i + σ ji dV V ?t ?x j ?t ? ?x j ?? ? ? ?? ? ? 2 u i ? ?u i ρ ?t 2 (why?) ?t ?x j (7) ? ? ? 2 u i ?u i ? ?u i ? = ρ V ? ?t 2 ?? ?t ? + σ ji ?t ?x j dV 1 ? ? ?u i ? 2 ? σ ji ? ji 2 ?t ?t ?t Notation: Time derivatives: ?( ) ?t = ( ˙ ) Examples: ?u i ? ?t = u˙ i , ?u = u ?t ? 2 u i ? ?t 2 = ¨u i ?? ij = ? ij ? ?t Spatial derivatives: ?( ) = ( ) ,i ?x i Examples: 2 ˙ ˙ ˙ ˙ ? ? ?? ? ? ? ? ? ? σ x j j i = σ ji,j With this notation, the power of the external forces can be rewritten as: d 1 ?u i ?u i P = ρ dV + σ ji ? ji dV dt ? V 2 ?t ?t ? ? V ?? ? (8) K deformation power where the “ρdV ” inside the ?rst integral was included inside the time deriva- tive since it is a constant due to conservation of mass. We conclude that part of the power of the external forces goes into changing the kinetic energy of the material and the rest into deforming the material. We call the latter the deformation power and it represents the rate at which the stresses do work on the deforming material. Replacing in the ?rst law, equation (1): d ? ? d ? ? K + U = K + σ ji ? ji dV + H (9) dt dt V After canceling the kinetic energy from both sides, the ?rst law expresses the fact that the internal energy of a deforming material can be changed either by heating or by deforming the material: dU dt = d dt ? V ρ ? U dV = ? V σ ji ˙? ji dV + H (10) In the isothermal case (H = 0): ? ? ? ? ? U ρ ? σ ij ? ij dV = 0 (11) V ?t or, in local form: ? ? U ρ = σ ij ? ij (12) ?t In ideal elasticity, we assume that all the work of deformation is converted into internal energy, i.e., the internal energy density is a state function of the deformation: U = U (? ij ) (13) 3 ˙ ˙ ˙ Then: ? ? ? ? U U = ? ij (14) ?t ?? ij Replace in ?rst law, equation (12: ? ? U ρ ? ij = σ ij ? ij ? (15) ?? ij ? ? U ρ = σ ij (16) ?? ij i.e., the stresses derive from a potential. 4