Peter Hertel. University of Osnabrück, Germany. Lecture presented at APS, Nankai University, China.

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1 Balance University of Osnabrück, Germany Lecture presented at APS, Nankai University, China Spring 2012

2 Linear and angular momentum and First and Second Law

3 point A material point is a piece of matter which is tiny from an engineer s point of view but huge from a physicist s point of view one mm 3 of an ideal gas under normal conditions still contains particles ideal gas: N 2 N 2 = N i.e. relative fluctuation is δn N the material point is always in thermodynamic equilibrium we usually suppress the t, x arguments

4 Content and flow think of an additive and transportable quantity Y mass, charge, momentum, energy etc. the content of Y in a region V is Q(Y ; t, V) = dv ϱ(y ; t, x) V the density ϱ(y ) = ϱ(y ; t, x) is a field the flow of Y through an area A is I(Y ; t, A) = da j(y ; t, x) A the current density j(y ) = j(y ; t, x) has a strength and a direction

5 Production the rate of production for Y in a region V is Π(Y ; t, V) = dv π(y ; t, x) V the volumetric production rate π(y ) = π(y ; t, x) describes: how much quantity Y is produced (or vanishes) per unit time and unit volume e.g. particles are produced in chemical reactions momentum is produced by external forces internal energy may be produced by friction entropy S is produced by irreversible effects π(s) 0 is the of thermodynamics

6 content of Y in V changes because Y is redistributed i.e. flows through the surface V or because Y is produced inside V as an d Q(Y ; t, V) = I(Y ; t, V) + Π(Y ; t, V) dt Gauss theorem dv f = da f V V generic ϱ(y ) + j(y ) = π(y ) must hold true for all times t and at all locations x

7 Number of particles Y = N a denotes the number of particles of species a = 1, 2,... particle density n a = ϱ(n a ) particle current density j a = j(n a ) velocities defined by j a = n a v a vanish or are generated in chemical reactions there are Γ r chemical reactions of type r per unit time and unit volume volumetric production rates are π a = Γ r ν ra r the ν ra are stoichiometric coefficients ν ra particles of species a are produced in a reaction of type r particle t n a + i n a vi a = Γ r ν ra r

8 Mass M particles of species a carry a mass m a mass density ϱ = ϱ(m) = a ma n a in each reaction, mass is conserved meaning Γ ra m a = 0 a consequently π(m) = Γ ra m a = 0 a r mass current density can be written as j(m) = ϱ(m)v mass continuity reads t ϱ + i ϱv i = 0

9 Charge Q e particles of species a carry a charge q a charge is conserved in each chemical reaction consequently π(q e ) = 0 charge density ϱ e = q a n a a current is split into convection and conduction part j(q e ) = j e = ϱ e v + J e charge continuity reads t ϱ e + i {ϱ e v i + J e i } = 0 without proof: only conduction contributions J are proper vector fields e.g. Ohm s refers to J e however, j e appears in Maxwell s s

10 P k is an additive transportable quantity its density has three components ϱ(p k ) = ϱv k mass times velocity per unit volume current densities for each component j i (P k ) = ϱv k v i T ki conduction part T ki is stress tensor minus sign is a convention volumetric momentum production rate is force f k per unit volume f k = ϱg k + ϱ e E k momentum t ϱv k + i {ϱv k v i T ki } = f k

11 Substantial time derivative time change as noted by a co-moving observer (D t f)(t, x) = i.e. D t f = t f + v f f(t + dt, x + dt v) f(t, x) dt specific quantities σ(y ) defined by ϱ(y ) = ϱ σ(y ) eqations may also be written as ϱ D t σ(y ) = i J i (Y ) + π(y ) in particular momentum ϱ D t v k = i T ki + f k

12 ctnd. angular momentum requires T ki = T ik at each location x there is a coordinate system such that the stress tensor is orthogonal: T T ki = T ik = 0 T T 3 a particular material can support only stress (positive T ) and pressure (negative T ) within certain limits the field of structural mechanics : work out the stress tensor, diagonalize it locally, and check whether its eigenvalues are within the allowed limits before building the bridge

13 Kinetic energy the kinetic energy density is ϱ(e k ) = ϱ 2 v2 it follow that ϱ 2 D t σ(e k ) = i J i (E k ) + π(e k ) where J i (E k ) = v k T ki and π(e k ) = T ik G ik + v k f k the volicity gradient is G ik = iv k + k v i 2

14 Potential energy quasistatic gravitational and electric potentials f k = ϱ k Φ g ϱ e k Φ e density of potential energy ϱ(e p ) = ϱφ g + ϱ e Φ e it follow that j i (E p ) = ϱj i (M) + ϱ e j i (Q) with j i (M) = ϱv i and j i (Q) = ϱ e + J e i π(e p ) = v i f i J e i E i

15 E i = U J e i E i T ij G ij E p v i f i E k Volumetric production rates for kinetic, potential and internal energy. Arrows pointing towards an energy form indicate a plus sign.

16 Internal energy total energy is the sum of kinetic, potential, and internal energy E i = U total energy is conserved pi(e) = π(e k + E p + U) = π(e k ) + π(e p ) + π(u) = 0 internal energy production rate per unit volume π(u) = T ik G ik + J e i E i specific internal energy is denotet by u conduction current density for internal energy... is the heat current density J u i stress tensor consists of the reversible part T ik and an irreversible part T ik likewise J e i and J e i

17 the for internal energy is... the of thermodynamics ϱ D t u = heat conduction i J u i deformation work +T ik G ik internal friction +T ik G ik polarization work +J e i E i and Joule s heat +J e i E i

18 each matierial point is always in equilibrium it undergoes a reversable change entropy S and temperature T introduced by du = T ds + dw were dw is the work done on the system entropy ϱ D t s = i J s i + π(s) entropy current j s = ϱsv + 1 T J u a µ a T J a chemical potentials µ a diffusion current densities J a

19 the second main of thermodynamics says 0 π(s) = heat conduction, Ji u 1 i T diffusion J a µ a i i T a friction +T ik i v k + k v i 2 Joule s heat 1 T J e i i Φ e chemical reactions + 1 Γ r A r T r

20 chemical affinity of reaction r is defined by A r = ν ra µ a a all contributions are of type flux times non-equilibrium indicator which would vanish in global equilibrium. there are five and only five contributions to entropy production First and are partial differential s there are much more fields than s. therefore: additional s required

ME338A CONTINUUM MECHANICS

ME338A CONTINUUM MECHANICS lecture notes 10 thursday, february 4th, 2010 Classical continuum mechanics of closed systems in classical closed system continuum mechanics (here), r = 0 and R = 0, such that