9780203009512.ch7

# 9780203009512.ch7 - 0749_Frame_C07 Page 107 Wednesday,...

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107 Thermal and Thermomechanical Response 7.1 BALANCE OF ENERGY AND PRODUCTION OF ENTROPY 7.1.1 B ALANCE OF E NERGY The total energy increase in a body, including internal energy and kinetic energy, is equal to the corresponding work done on the body and the heat added to the body. In rate form, (7.1) in which: Ξ is the internal energy with density ξ (7.2a) is the rate of mechanical work, satisfying (7.2b) is the rate of heat input, with heat production h and heat ﬂux q , satisfying (7.2c) is the rate of increase in the kinetic energy, (7.2d) It has been tacitly assumed that all work is done on S , and that body forces do no work. 7 ˙ ˙ ˙ ˙ , K += + Ξ WQ ˙ ˙ ; Ξ= ρξ dV ˙ W ˙ ˙ , Wd S = u T τ ˙ Q ˙ , Q hdV dS =− ∫∫ ρ nq T and ˙ K ˙ ˙ ˙ . K = u u T d dt dV © 2003 by CRC CRC Press LLC

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108 Finite Element Analysis: Thermomechanics of Solids Invoking the divergence theorem and balance of linear momentum furnishes (7.3) The inner bracketed term inside the integrand vanishes by virtue of the balance of linear momentum. The relation holds for arbitrary volumes, from which the local form of balance of energy, referred to undeformed coordinates, is obtained as (7.4) To convert to undeformed coordinates, note that (7.5) In undeformed coordinates, Equation 7.3 is rewritten as (7.6a) furnishing the local form (7.6b) 7.1.2 E NTROPY P RODUCTION I NEQUALITY Following the thermodynamics of ideal and non-ideal gases, the entropy production inequality is introduced as follows (see Callen, 1985): (7.7a) in which H is the total entropy, η is the speciﬁc entropy per unit mass, and T is the absolute temperature. This relation provides a “framework” for describing the irre- versible nature of dissipative processes, such as heat ﬂow and plastic deformation. We apply the divergence theorem to the surface integral and obtain the local form of the entropy production inequality: (7.7b) ρξ ρ ˙ ˙ ˙ () . +− −− + = u u Dq TT T d dt tr h dV 0 ˙ . =− + tr T T h nq q F n qn q F q T T dS J dS dS J ∫∫ = == 00 00 0 0 1 . 0 0 0 0 ˙ ( ˙ , + [] = tr dV SE )h T q 0 0 0 ˙ ( ˙ ). + = tr SE h T q ˙ ˙ , H =≥ ρη dV h T dV T dS T Th T / T ˙ . ≥− ∇ + ∇ qq © 2003 by CRC CRC Press LLC
Thermal and Thermomechanical Response 109 The corresponding relation in undeformed coordinates is (7.7c) 7.1.3 T HERMODYNAMIC P OTENTIALS The Balance of Energy introduces the internal energy Ξ , which is an extensive variable—its value accumulates over the domain. The mass and volume averages of extensive variables are also referred to as extensive variables. This contrasts with intensive, or pointwise, variables, such as the stresses and the temperature. Another extensive variable is the entropy H . In reversible elastic systems, the heat ﬂux is completely converted into entropy according to (7.8) (We shall consider several irreversible effects, such as plasticity, viscosity, and heat conduction.) In undeformed coordinates, the balance of energy for reversible pro- cesses can be written as (7.9) We call this equation the thermal equilibrium equation. It is assumed to be

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## This note was uploaded on 05/18/2011 for the course MAE 269A taught by Professor Ju during the Spring '11 term at UCLA.

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9780203009512.ch7 - 0749_Frame_C07 Page 107 Wednesday,...

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