Statistical Continuum Mechanics: Course Held at the by Ekkehart Kröner (auth.)

By Ekkehart Kröner (auth.)

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Extra info for Statistical Continuum Mechanics: Course Held at the Department of General Mechanics, October 1971

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It is rather the average behaviour of these quantities, i. e. the behaviour on the macro-time scale which is of interest. In order to see what is needed for such an average description consider the random function f(t). e. macro-time, information about f(t). Such averages are the mean f(t 1 ) 2-point correlation function f(t 1)f(t 2 ) tion function f (t 1) f (t~_) ... , where t 1 , ta ... tn are continuous variables. The complete statistical information about r is, at the same time, the complete macro-time information a- bout f.

We first observe that dueto eq. 3) The equations of motion ••• 49 Having substituted this in eq. , tz) and take the average. Interchanging then averaging with differentiation and using now subscript notation in Cartesian coordinates we arrive at a~1 \)'~ ( ~1, t1) '\)'\( (;2, t,) + ~ \)'~ (~1, t1)ui. (~1 ,t1) tYk (~,, t,) + +.!. () Q ()x1~ p(~ 1 ,t 1 )trk(~ 2 ,t 2 )-vV~tr~('~ 1 ,t1)U'k(~ 2 ,t 2)=0. ). ' t 2) == o. 8) In addition to eqs. 9) must be satisfied. The important point is that 3rd-order functions appear in the equations for the 2nd-order correlationfunctions.

O. We have not substituted t 1 for t because we do not intend to form correlation functions in time but rather in space. In applying this restriction which serves to simplify the situation 57 Formulation of the basic equations ••• we follow many of the authorswho have been concerned with the present problems. Proceeding as in chapter III we multiply by 'lTk (;. 2 , t) take the ensemble average and interchange this wi th the differentiation. t)t1k(;2,t) - +! 1~ p(~1tt)vk(~z,t) }+{ ~1 ... 3) where {~1++ k2} denotes the expression obtained by the simulta- neaus substi tution L - k, 1 - 2.

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