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Finite Element Analysis - Thermomechanics of Solids Part 7

Tham khảo tài liệu 'finite element analysis - thermomechanics of solids part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | y Thermal and Thermomechanical Response 7.1 BALANCE OF ENERGY AND PRODUCTION OF ENTROPY 7.1.1 Balance of Energy 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 K E w Q 7.1 in which E is the internal energy with density ệ E J pệdV 7.2a w is the rate of mechanical work satisfying w J u T T dS 7.2b Q is the rate of heat input with heat production h and heat flux q satisfying Q J phdV - J nTqdS and A . . . K is the rate of increase in the kinetic energy K i puT du dV. 1 dt 7.2c 7.2d It has been tacitly assumed that all work is done on S and that body forces do no work. 107 2003 by CRC CRC Press LLC 108 Finite Element Analysis Thermomechanics of Solids Invoking the divergence theorem and balance of linear momentum furnishes Jlx u T Ip d-v T V L ut tr TD ph VTq dV 0. 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 _ __T pệ tr T D VTq ph. 7.4 To convert to undeformed coordinates note that J nTqdS J qT JF Tn0 dS0 J qTn0dS0 q0 JF q. 7.5 In undeformed coordinates Equation 7.3 is rewritten as J P0 ề tr SE Poh VTq 0 dV0 0 furnishing the local form p ị tr SE P0h vTq 0 0. 7.6a 7.6b 7.1.2 Entropy Production Inequality Following the thermodynamics of ideal and non-ideal gases the entropy production inequality is introduced as follows see Callen 1985 H J pndV J h dV J T 1 dS 7.7a in which H is the total entropy n is the specific entropy per unit mass and T is the absolute temperature. This relation provides a framework for describing the irreversible 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 pTn h VTq .