tailieunhanh - Thermodynamics Kinetics of Dynamic Systems Part 13

Tham khảo tài liệu 'thermodynamics kinetics of dynamic systems part 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Irreversible Thermodynamics and Modelling of Random Media 349 We have neglected in 33 the viscous friction in the fluid phase. The equations 32 and 33 need closure assumptions for the contact stress tensor and the Reynolds-like tensor before being useful. The fluid pressure can be calculated with the õp f -. n as - r pfgp 0 . ỠXp 1 momentum equation for the fluid phase often reduced simply The modelling of the lasting contact stress tensor ơ iơ iơ p has been studied in i 1 N details in Borghi Bonelli 2007 . We will only summarize their results in the next paragraph but we will develop the question how the irreversible thermodynamics can help this modelling. The kinetic contribution due to the velocities fluctuations can be modelled with the principles that have been described in the first part of this chapter and we will discuss this point in . The modelling of the mean Cauchy stress tensor of lasting contacts The approach of Borghi and Bonelli gives an evolution equation for ỡ g . The grains are assumed elastic with small strains but large possible displacements and the contacts are displaying partly a reversible behaviour as proposed by Cambou 1998 Emeriault Cambou 1996 and partly irreversible sliding. The obtained evolution equation displays the rotational objective derivative of Jaumann without any additional assumption Õ P JL pvy -JL ơ pp y ce PyS f5 Ẹl -DpyS -DfyS dt dxy g dxy cg g g dxy y -2 ơ yroyp - 2 c .Qf - 2 w wj 34 8 sym sym g sym The first term on the . is a classical dispersion term due to the fluctuations of the velocity and appears for any averaged quantity within a randomly moving medium. The appearance of the Jaumann objective derivative is justified by the last term QjP being the averaged rotation rate tensor of the medium mainly related to ỠO gvy Ixp . The tensor ce fourth order is an effective elastic stiffness tensor of the medium due to the elastic behaviour within the grains and also particularly of the lasting .