ME338A CONTINUUM MECHANICS

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1 global vs local balance equations ME338A CONTINUUM MECHANICS lecture notes 11 tuesay, may 06, 2008 The balance equations of continuum mechanics serve as a basic set of equations require to solve an initial bounary value problem of thermomechanics for the primary variables. This section is evote to erivation of the funamental balance laws of continuum thermomechanics. In what follows, we consier a certain volume P close by the bounary P. For this part of the boy, we balance a volumetric source an a surface flux with the temporal change of the quantity for which the balance principle is constructe. Initially, the balance equation is erive on the entire subset P, i.e., in a global form. In orer to erive the local form of a balance law, we transform the surface flux terms into a volume term through Gauss integral theorem an localize the resulting expression to any arbitrary point x provie that the continuity conitions are met. X T N A P F F t t n a x S material vs spatial balance equations The global an local material balance equations vali on the reference configuration an at point X can be recast into their spatial format vali on the current configuration an at point x, respectively. 88

2 3 alance equations 3.3 alance of mass total mass m of a boy P m = ρ 0 V = ρ t v (3.3.1) in terms of material an spatial mass ensity ρ 0 an ρ t from transformation of volume elements v = J V with ρ P t v = J ρ S P t V = ρ P 0 V thus ρ 0 = J ρ t (3.3.2) mass exchange of boy P with its environment through mass flux across the surface P m sur = 0 (3.3.3) an through volume source within P m vol = 0 (3.3.4) global balance of mass / integral form The time rate of change of the total mass m of a boy P is balance with the mass exchange ue to the contact mass flux m sur an the at-a-istance mass exchange m vol. t m = msur + m vol (3.3.5) global balance of mass, material version, i.e., in terms of ρ 0 ρ 0 V = 0 (3.3.6) t global balance of mass, spatial version, i.e., in terms of ρ t t J ρ t V = 0 (3.3.7) 89 local balance of mass / ifferential form moification of rate terms,localization to any point insie P t ρ 0 V = t J ρ t V = 0 (3.3.8) local balance of mass, material version, in terms of ρ 0 t ρ 0 = 0 (3.3.9) conservation of mass The mass in a material boy oes not change. alternative statement from global spatial version t (J ρ t) = J t ρ t + ρ t t J [ ] = J t ρ t + ρ t iv(v) [ ] = J t ρ t + iv(ρ t v) = 0 with the following transformations J/t = J F t : Ḟ = J iv(v) ρ t /t = ρ t /t + x ρ t v ρ t iv(v)+ x ρ t v = iv(ρ t v) local balance of mass, spatial version, in terms of ρ t t ρ t + ρ t iv(v) =0 soli mechanics (3.3.10) t ρ t + iv(ρ t v)=0 (3.3.11) continuity equation Euler 1757 flui mechanics 90

3 3 alance equations remarks balance equations can be phrase in a global / integral or in a local / ifferential format, both combinations can either be formulate in terms of material quantities or in terms of spatial quantities, the particular choice of the balance equation epens on the application for escriptions in fixe omains, the conservation of mass ρ 0 /t = 0 is usually fulfille automatically, it consists of only one term, the time rate of change of the balance quantity within the fixe omain, these formulations are typical in soli mechanics for escriptions in moving omains, the conservation of mass ρ t / t + iv(ρ t v)=0 is referre to as the continuity equation, it nicely illustrates that for time erivatives in moving omains, we have a time evolution term insie the moving omain ρ t / t an a convective term iv(ρ t v) accounting for the in- or outflux through the moving bounary, these formulations are common in flui mechanics terms between the material an spatial formulation can be transforme into one another through the Piola transforms an Reynol s transport theorem balance equations have a somewhat hierarchical orer, lower orer balance equations can be use to simplify higher orer balance equations, e.g., we will see that the balance of mass can be use to simplify the balance of linear momentum summary of useful material vs spatial transformations F 1 { } N { } 0 A P N F F t { } n n { } t a material vs spatial volume terms Piola transform global comparison of volume terms { } 0 an { } t { } 0 V = { } t v (3.3.12) with volume transformation v = J V local comparison of volume terms { } 0 an { } t { } 0 = J { } t (3.3.13) material vs spatial surface terms S Piola transform global comparison of surface terms { } an { } { } A = { } a (3.3.14) with area transformation a = J F 1 A, Nanson s formula local comparison of surface terms { } an { } { } = J { } F t Div{ } = J iv{ } (3.3.15) 91 92

4 material vs spatial time erivative material time material position X = { } (3.3.16) t { } := t { } X fixe spatial time spatial position x t { } := t { } x fixe (3.3.17) Euler theorem local comparison of time erivatives t { } = t { } + x{ } v (3.3.18) Reynol s transport theorem global comparison of time erivatives { } V = t t { } v + { } v n a (3.3.19) local comparison of time erivatives [ ] t { } 0 = J t { } t + iv({ } t v) (3.3.20) 3 alance equations 3.4 alance of linear momentum total linear momentum p of a boy P p = ρ 0 v V = ρ t v v (3.4.1) momentum exchange of boy P with environment through contact forces f sur f sur = T A = t a (3.4.2) an at-a-istance forces f vol f vol = ρ 0 b V = ρ t b v (3.4.3) in terms of contact/surface forces T = P N an t = σ n an volume forces b X T A P N F F t t x n a S Reynol s transport theorem The rate of change of the quantity { } 0 in a fixe material volume equals the rate of change of the quantity { } t in a fixe spatial control volume plus the flux through the bounary of the control omain. 93 global balance of momentum / integral form The time rate of change of the total momentum p of a boy P is balance with the momentum exchange ue to contact momentum flux / surface force f sur an the at-a-istance momentum exchange / volume force f vol. t p = f sur + f vol (3.4.4) 94

5 global balance of momentum, material version, in ρ 0 & T ρ 0 v V = T A + ρ 0 b V (3.4.5) t global balance of momentum, spatial version, in ρ t & t J ρ t v V = t a + ρ t b v (3.4.6) t local balance of momentum / ifferential form moification of rate terms p /t t p = t (ρ 0 v) V = t (J ρ t v) V (3.4.7) moification of surface terms f sur f sur = T A Cauchy = P N A Gauss = Div(P t ) V thus = t a Cauchy = σ n a Gauss = t (ρ 0 v) V = Div(P t ) + ρ 0 b V t (J ρ t v) V = J iv(σ t )+Jρ t b V iv(σ t ) v (3.4.8) (3.4.9) local balance of momentum, material version, in ρ 0 & P t (ρ 0 v) =Div(P t )+ρ 0 b (3.4.10) reuction by subtracting weighte version of balance of mass v ρ 0 /t = 0 an with a = v /t 3 alance equations Cauchy s first law of motion, Cauchy [1827] ρ 0 a = Div(P t )+ρ 0 b (3.4.11) equilibrium equation soli mechanics Piola transforms where { } 0 = ρ 0 an { } t = ρ t,an{ } = P an { } = σ ρ 0 v = J ρ t v P = J σ F t Div(P t )=J iv(σ t ) ρ 0 b = J ρ t v Reynol s transport theorem where { } 0 = ρ 0 v an { } t = ρ t v t (ρ 0v) =J [ (ρ tv)+iv(ρ t v v) local balance of momentum, spatial version, in ρ t & σ ρ t a = iv(σ t )+ρ t b t (ρ t v) =iv(σ t ρ t v v)+ρ t b equilibrium equation ] (3.4.12) (3.4.13) soli & flui mechanics the balance of momentum is maybe the most important equation in soli an flui mechanics, again, it can be phrase globally or locally, on fixe an moving omains 95 96

6 from global to local (i) moification of rate terms transform all integral terms to fixe omain (ii) moification of surface terms transform all bounary flux terms on into volume terms in with Green/Gauss theorem (iii) localize the global version for each point in P from material to spatial (i) apply Piola transform for volume terms { } 0 = J { } t (ii) apply Piola transform for surface terms Div{ } = J iv{ } (iii) apply Reynol s transport theorem for time erivatives t { } 0 = J [ t { } t + iv({ } t v) ] for quasi-static problems, e.g., in the static analysis of structures or for material testing, the acceleration term is usually neglecte, i.e., a 0, such that the equilibrium equation woul reuce to Div(P t )+ρ 0 b = 0 iv(σ t )+ρ t b = 0 for vanishing volume forces as common to most applications, i.e., b = 0, the equilibrium equation woul then reuce to Div(P t )=0 iv(σ t )=0 a typical example of a volume force term woul be gravity, however, in most analyses this time is typically neglecte 97