3 Balance equations global vs local balance equations The balance equations of continuum mechanics serve as a basic set of equations required to solve an initial bound- ary value problem of thermomechanics for the primary vari- ables. This section is devoted to derivation of the fundamen- tal balance laws of continuum thermomechanics. In what P ME338A follows, we consider a certain volume closed by the bound- ary ∂P. For this part of the body, we balance a volumetric CONTINUUM MECHANICS source and a surface flux with the temporal change of the quantity for which the balance principle is constructed. Ini- tially, the balance equation is derived on the entire subset P, i.e., in a global form. In order to derive the local form of a lecture notes 11 balance law, we transform the surface flux terms into a vol- tuesday, may 06, 2008 ume term through Gauss’ integral theorem and localize the resulting expression to any arbitrary point x ∈PS provided that the continuity conditions are met. T˜ n N t da X dA F x − PB t F PS ∂P ∂PB S B S material vs spatial balance equations The global and local material balance equations valid on the reference configuration PB and at point X ∈PB can be re- cast into their spatial format valid on the current configura- tion PS and at point x ∈PS, respectively. 88 3 Balance equations 3 Balance equations 3.3 Balance of mass local balance of mass / differential form modification of rate terms,localization to any point inside P total mass m of a body P d d ρ0 dV = J ρt dV = 0 (3.3.8) m = ρ0 dV = ρt dv (3.3.1) PB dt PB dt PB PS local balance of mass, material version, in terms of ρ0 in terms of material and spatial mass density ρ0 and ρt from transformation of volume elements dv = J dV with d ρ dv = J ρ dV = ρ dV thus ρ = 0 (3.3.9) PS t PB t PB 0 dt 0 ρ0 = J ρt (3.3.2) conservation of mass solid mechanics mass exchange of body P with its environment through mass flux across the surface ∂P “The mass in a material body does not change.” sur alternative statement from global spatial version m = 0 (3.3.3) d ( ρ )= d ρ + ρ d and through volume source within P J t J t t J dt dt dt mvol = 0 (3.3.4) d = J ρt + ρtdiv(v) (3.3.10) dt global balance of mass / integral form ∂ = ρ + (ρ ) = J ∂ t div t v 0 “The time rate of change of the total mass m of a body P is t balanced with the mass exchange due to the contact mass with the following transformations sur vol − flux m and the at-a-distance mass exchange m .” dJ/dt = J F t : F˙ = J div(v) d ρ = ∂ρ ∂ + ∇ ρ · m = msur + mvol (3.3.5) d t/dt t/ t x t v dt ρtdiv(v)+∇xρt · v = div(ρt v) global balance of mass, material version, i.e., in terms of ρ0 local balance of mass, spatial version, in terms of ρt d ρ0 dV = 0 (3.3.6) dt PB d ∂ ρ ρ + ρ div(v)=0 ρ + div(ρ v)=0 (3.3.11) global balance of mass, spatial version, i.e., in terms of t t t ∂ t t dt t d ρ = J t dV 0 (3.3.7) continuity equation dt PB Euler 1757 fluid mechanics 89 90 3 Balance equations 3 Balance equations remarks summary of useful material vs spatial transformations ◦ balance equations can be phrased in a global / integral or −1 ·{}· F N n in a local / differential format, both combinations can either N {♦}·n be formulated in terms of material quantities or in terms of da spatial quantities, the particular choice of the balance equa- {◦} {◦} 0 dA F t − tion depends on the application PB t F PS ∂P ∂PB S ◦ for descriptions in fixed domains PB, the conservation of B S mass dρ0 /dt = 0 is usually fulfilled automatically, it con- sists of only one term, the time rate of change of the balance material vs spatial volume terms Piola transform quantity within the fixed domain PB, these formulations are {◦} {◦} global comparison of volume terms 0 and t typical in solid mechanics {◦}0 dV = {◦}t dv (3.3.12) PB PS ◦ for descriptions in moving domains PS, the conservation with volume transformation dv = J dV of mass ∂ρt / ∂t + div(ρt v)=0 is referred to as the conti- local comparison of volume terms {◦}0 and {◦}t nuity equation, it nicely illustrates that for time derivatives in moving domains, we have a time evolution term inside {◦} = {◦} the moving domain ∂ρt / ∂t and a convective term div(ρt v) 0 J t (3.3.13) accounting for the in- or outflux through the moving bound- ary, these formulations are common in fluid mechanics material vs spatial surface terms Piola transform ◦ {} {♦} terms between the material and spatial formulation can global comparison of surface terms and Piola transforms be transformed into one another through the {}·dA = {♦}·da (3.3.14) and Reynold’s transport theorem ∂PB ∂PS with area transformation da = J F−1 · dA, Nanson’s formula ◦ balance equations have a somewhat “hierarchical order”, local comparison of surface terms {} and {♦} “lower order” balance equations can be used to simplify “higher order” balance equations, e.g., we will see that the {} = {♦}· −t {} = {♦} balance of mass can be used to simplify the balance of linear J F Div J div (3.3.15) momentum 91 92 3 Balance equations 3 Balance equations material vs spatial time derivative 3.4 Balance of linear momentum material time derivative: @fixed material position X P ∂ total linear momentum p of a body d {◦} = {◦} = {◦}˙ : ∂ (3.3.16) = ρ = ρ dt t X fixed p 0 v dV t v dv (3.4.1) PB PS spatial time derivative: @fixed spatial position x P momentum exchange of body with environment through sur ∂ ∂ contact forces f {◦} := {◦} (3.3.17) ∂t ∂t sur x fixed f = T d A = t d a (3.4.2) ∂P ∂P Euler theorem B S f vol local comparison of time derivatives and at-a-distance forces vol d ∂ f = ρ0 b dV = ρt b dv (3.4.3) {◦} = {◦} + ∇ {◦} · v (3.3.18) PB PS dt ∂t x in terms of contact/surface forces T = P · N and t = σ · n Reynold’s transport theorem and volume forces b global comparison of time derivatives d ∂ {◦} = {◦} + {◦} ⊗ · T˜ n dV ∂ dv v n da (3.3.19) dt PS PS t ∂PS N t local comparison of time derivatives da X dA F x − ∂ PB t d F PS ∂P {◦} = J {◦} + div({◦} ⊗ v) (3.3.20) ∂PB S dt 0 ∂t t t B S Reynold’s transport theorem global balance of momentum / integral form “The rate of change of the quantity {◦}0 in a fixed material “The time rate of change of the total momentum p of a body volume PB equals the rate of change of the quantity {◦}t in P is balanced with the momentum exchange due to contact a fixed spatial control volume PS plus the flux through the momentum flux / surface force f sur and the at-a-distance boundary of the control domain ∂PS.” momentum exchange / volume force f vol.” d p = f sur + f vol (3.4.4) dt 93 94 3 Balance equations 3 Balance equations global balance of momentum, material version, in ρ0 & T Cauchy’s first law of motion, Cauchy [1827] d ρ0 v dV = T dA + ρ0 b dV (3.4.5) P ∂P P t dt B B B ρ0 a = Div(P )+ρ0 b (3.4.11) global balance of momentum, spatial version, in ρt & t d equilibrium equation solid mechanics J ρt v dV = t da + ρt b dv (3.4.6) dt PB ∂PS PS Piola transforms local balance of momentum / differential form where {◦}0 = ρ0 and {◦}t = ρt,and{} = P and {♦} = σ modification of rate terms dp /dt ρ v = J ρ v 0 t d d d = σ · −t ( t)= (σt) p = (ρ0 v) dV = (J ρt v) dV (3.4.7) P J F Div P J div dt PB dt PB dt ρ0 b = J ρt v modification of surface terms f sur Reynold’s transport theorem sur Cauchy Gauss t {◦} = ρ {◦} = ρ f = T dA = P · N dA = Div(P ) dV where 0 0v and t tv ∂PB ∂PB PB ∂ = Cauchy= σ · Gauss= (σt) d t da n da div dv (ρ0v)=J (ρtv)+div(ρtv ⊗ v) (3.4.12) ∂PS ∂PS PS dt ∂ (3.4.8) local balance of momentum, spatial version, in ρt & σ thus d t ρ = (σt)+ρ (ρ0 v) dV = Div(P )+ρ0 b dV t a div t b PB dt PB ∂ (3.4.13) (3.4.9) (ρ )= (σt − ρ ⊗ )+ρ d ( ρ ) = (σt)+ ρ t v div t v v t b J t v dV J div J t b dV ∂t PB dt PB equilibrium equation solid & fluid mechanics local balance of momentum, material version, in ρ0 & P d (ρ v)=Div(Pt)+ρ b (3.4.10) dt 0 0 the balance of momentum is maybe the most important equa- tion in solid and fluid mechanics, again, it can be phrased reduction by subtracting weighted version of balance of mass globally or locally, on fixed and moving domains v dρ0/dt = 0 and with a = dv /dt 95 96 3 Balance equations from global to local (i) modification of rate terms transform all integral terms to fixed domain PB (ii) modification of surface terms transform all boundary flux terms on ∂PB into volume terms in PB 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 div{♦} (iii) apply Reynold’s transport theorem for time derivatives d {◦} = ∂ {◦} + ({◦} ⊗ ) dt 0 J ∂t t div t v for quasi-static problems, e.g., in the static analysis of struc- tures or for material testing, the acceleration term is usually neglected, i.e., a ≈ 0, such that the equilibrium equation would reduce to t t Div(P )+ρ0 b = 0 div(σ )+ρt b = 0 for vanishing volume forces as common to most applications, i.e., b = 0, the equilibrium equation would then reduce to Div(Pt)=0 div(σt)=0 a typical example of a volume force term would be gravity, however, in most analyses this time is typically neglected 97.