ME338A CONTINUUM MECHANICS

Transcription

1 ME338A CONTINUUM MECHANICS lecture notes 10 thursday, february 4th, 2010

2 Classical continuum mechanics of closed systems in classical closed system continuum mechanics (here), r = 0 and R = 0, such that the mass density ρ is constant in time, i.e. D t (ρ v) = ρ D t v and thus ρ D t v = div (σ t ) + b (3.3.12) the above equation is referred to as Cauchy s first equation of motion, Cauchy [1827] 3.4 Balance of angular momentum total angular momentum l of a body l := x D t u d m = x v d m = ρ x v dv (3.4.1) angular momentum exchange of body with environment through momentum from contact forces l sur and momentum from at-a-distance forces l vol l sur := x t σ d A l vol := x b d V (3.4.2) with momentum due to contact/surface forces x t σ = x σ t n and volume forces x b, assumption: no additional external torques Global form of balance of angular momentum The time rate of change of the total angular momentum l of a body is balanced with the angular momentum exchange 82

3 due to contact momentum flux / surface force l sur and due to the at-a-distance momentum exchange / volume force l vol. D t l = l sur + l vol (3.4.3) and thus D t ρ x v dv = x t σ d A + x b d V (3.4.4) Local form of balance of angular momentum modification of rate term D t p D t l = D t ρ x vdv fixed = modification of surface term l sur l sur = x t σ d A Cauchy = Gauss = div (x σ t ) d V = x div (σ t ) d V + and thus D t (ρ x v) dv = + D t (ρ x v)dv (3.4.5) x σ t n d A x σ t d V (3.4.6) x div (σ t ) d V I σ t d V + x b d V (3.4.7) for arbitrary bodies local form of ang. mom. balance D t (ρ x v) = x div (σ t ) + I σ t + x b (3.4.8) 83

4 3.4.3 Reduction with lower order balance equations modification with the help of lower order balance equations D t (ρ x v) = D t x (ρ v) + x D t (ρ v) = x div (σ t ) + I σ t + x b (3.4.9) position vector x fixed, thus D t x = 0, with balance of linear momentum multiplied by position x x D t (ρ v) = x div (σ t ) + x b (3.4.10) local angular momentum balance in reduced format I σ t = 3 e: σ = 2 axl(σ) = 0 σ = σ t (3.4.11) the above equation is referred to as Cauchy s second equation of motion, Cauchy [1827] in the absense of couple stresses, surface and body couples, the stress tensor is symmetric, σ = σ t, else: micropolar / Cosserat continua vector product of second order tensors A B = 3 e: [A B t ] 84

5 3.5 Balance of energy total energy E of a body as the sum of the kinetic energy K and the internal energy I E := K := I := e dv = k dv = i dv k + i dv = K + I 1 2 ρ v v dv (3.5.1) energy exchange of body with environment e sur and e vol E sur := v t σ d A q n d A (3.5.2) E vol := v b d V + Q d V with contact heat flux q n = r n and heat source Q Global form of balance of energy The time rate of change of the total energy E of a body is balanced with the energy exchange due to contact energy flux E sur and the at-a-distance energy exchange E vol. D t E = E sur + E vol (3.5.3) and thus D t e dv = v t σ q n da + v b + QdV (3.5.4) 85

6 3.5.2 Local form of balance of energy modification of rate term D t E D t E = D t e dv fixed = D t e dv (3.5.5) modification of surface term E sur E sur = v t σ q n d A Cauchy = v σ t n q n d A div (v σ t ) + div ( q) d V Gauss = and thus D t e dv = div (v σ t q) d V + (3.5.6) v b+qdv (3.5.7) for arbitrary bodies local form of energy balance D t e = div (v σ t q) + v b + Q (3.5.8) Reduction with lower order balance equations modification with the help of lower order balance equations with with D t e = D t ( k + i ) = v D t (ρ v) + D t i = div (v σ t q) + v b + Q (3.5.9) D t k = D t ( 1 2 ρv v) = 1 2 D t(ρv) v v D t(ρv) = v D t (ρv) 86

7 (3.5.10) with balance of momentum multiplied by velocity v v D t (ρ v) = v div (σ t ) + v b = div (v σ t ) v : σ t + v b (3.5.11) with stress power v : σ t = σ t : v = σ t : D t ϕ = σ t : D t ϕ = σ t : D t sym ϕ = σ t : D t ǫ (3.5.12) local energy balance in reduced format/balance of int. energy D t i = σ t : D t ǫ div (q) + Q (3.5.13) First law of thermodynamics alternative definitions p ext := div (v σ t )+v b... external mechanical power p int := σ t : v=σ t : D t ǫ... internal mechanical power q ext := div ( q)+q... external thermal power (3.5.14) Balance of total energy / first law of thermodynamics the rate of change of the total energy e, i.e. the sum of the kinetic energy k and the potential energy i is in equilibrium with the external mechanical power p ext and the external thermal power q ext 87

8 D t e = D t k + D t i = p ext + q ext (3.5.15) the above equation is typically referred to as principle of interconvertibility of heat and mechanical work, Carnot [1832], Joule [1843], Duhem [1892] first law of thermodynamics does not provide any information about the direction of a thermodynamic process balance of kinetic energy D t k = p ext p int (3.5.16) the balance of kinetic energy is an alternative statement of the balance of linear momentum balance of internal energy D t i = q ext + p int (3.5.17) kinetic energy k and internal energy i are no conservation properties, they exchange the internal mechanical energy p int 88