Chapter 2 Momentum an Energy In this chapter we present some funamental results of continuum mechanics. The formulation is base on the principles of conservation of mass, momentum, angular momentum, an energy. It shoul be note that we use the wor principle to inicate those concepts that are use as primitive (e.g., principle of conservation of mass), whereas we use the wor law to inicate those concepts that are erive from the principles : this is one to emphasize the ifferent roles in the formulation even if sometimes this may soun awkwar (e.g., law of conservation of mechanical energy, or virtual work law ). We will assume as primitive the concepts of time, space, mass, force, internal energy, heat flux, an heat generate per unit volume. We will enote the time with t, the ensity with ρ, the internal energy with e, the force per unit volume with f, the stress vector (or force per unit area) with t, heat flux with h, an the heat generate per unit volume with r. 2.1 Conservation Principles Consier a material volume, i.e., a volume that is compose of the same particles at all times. We enote this volume with, an its bounary with. The principle of conservation of mass states that, for any arbitrary material volume,, within a continuum, the mass of this material volume remains constant in time, i.e., ρ = 0 (2.1) t The principle of conservation of momentum states that, for any arbitrary material volume,, within a continuum, the time erivative of the momentum in is equal to the resultant of all the forces acting of the continuum in, i.e., ρv = t 21 ρf + t (2.2)
22 The principle of conservation of angular momentum states that, for any arbitrary material volume,, within a continuum, the time erivative of the angular momentum in is equal to the resultant of all the moments acting on, i.e., ρx v = ρx f + x t (2.3) t The principle of conservation of energy states that, for any arbitrary material volume,, within a continuum, the sum of the time erivatives of the kinetic an internal energies is equal to the work per unit time one by the all the forces acting on, plus the heat generate within the volume, minus the heat flux through the bounary surface, i.e., ( ) v 2 ρ t 2 + e = ρf v + t v + r h (2.4) As we will see, these principles, with the aition of the constitutive relations (which are subject to the constraints erive from the secon principle of thermoynamics) are the basis for all the material covere in this volume. 2.2 Dynamics The results of this section are base on the principles of conservation of mass, momentum, an angular momentum. 2.2.1 tress Tensor In Eqs. 2.2, 2.3, an 2.4 we introuce the stress vector (or force per unit area), t, acting on the surface,. ince is the surface of an arbitrary control volume,, the stress vector at a given point is, in general, a function of the normal to the surface at that point. In this ection we want to show that the stress vector t, at a given point, is a homogeneous linear function of the components of the normal to at that point. In other wors, the components of t are relate to those of n, through a matrix multiplication. The matrix of the coefficients form a tensor calle the stress tensor. In orer to accomplish this consier the principle of conservation of momentum, an apply the secon Reynols transport theorem to obtain ρ Dv Dt = Next note that, as 0, ρ Dv Dt = O(l3 ) ρf + t (2.5) ρf = O(l 3 ) (2.6)
23 where l 3 is a characteristic length of. Therefore, Eq. 2.5 implies t = O(l 3 ) (2.7) Note that in general the integral is of the orer of l 2. Then Eq. 2.7 implies that the terms of orer l 2 are equal to zero. This may be exploite in orer to obtain the esire relationship between t an n. In orer to o this consier the Cauchy tetraheron, i.e., a tetraheron with three faces normal to the coorinates axes, x i. Figure 2.1: Cauchy Tetraheron If A i is the area of the surface normal to the x i -axis, an A is the area of the fourth surface (Fig. 2.1), Eq. 2.7 yiels, applying the mean value theorem, t 1 A 1 + t 2 A 2 + t 3 A 3 + ta = O(l 3 ) (2.8) or, iviing by A, an noting that A i = An i (i = 1, 2, 3), where n i are the components of n, t + t 1 n 1 + t 2 n 2 + t 3 n 3 = O(l) (2.9) By taking the limit as l goes to zero, one obtains t i = τ ji n j (2.10)
24 where τ ji = (t j ) i (2.11) The matrix of the coefficients τ ji form a tensor calle the stress tensor. In vector notations, Eq. 2.10 may be written as t = Tn (2.12) where T = e i e j τ ji (2.13) is the stress tensor. 2.2.2 Cauchy Equations of Motion Using 2.10, the principle of conservation of momentum, Eq. 2.5, may be written as ρ Dv Dt = ρf + Tn = ρf + iv T (2.14) where iv T = τ ji x j e i (2.15) Therefore, taking into account the arbitrariness of, we obtain the Cauchy equations of motion ρ Dv Dt 2.2.3 ymmetry of tress Tensor = ρf + iv T (2.16) In this ection we will show that the principle of conservation of angular momentum is equivalent to the symmetry of the stress tensor, T. Combining Eq. 2.3 with the secon Reynols transport theorem, Eq. 1.38, one obtains (noting that v v = 0) ρ D Dt (v x) = ρ Dv Dt x = ρf x + t x (2.17) In inicial notations, expressing t i in terms of the stress tensor τ ji, an using Gauss theorem, Dv j ρe ijk Dt x k = ρe ijk f j x k + e ijk t j x k = ρe ijk f j x k + (e ijk τ lj x k ) (2.18) x l
25 where e ijk is the permutation symbol e ijk = 1 if i, j, k = cyclic permutation Noting the arbitrariness of the volume, we have or = 1 if i, j, k = anticyclic permutation = 0 otherwise (2.19) Dv j ρe ijk Dt x k = ρe ijk f j x k + (e ijk τ lj x k ) (2.20) x l ρe ijk Dv j Dt x k = ρe ijk f j x k + e ijk τ lj x l x k + e ijk τ kj (2.21) This is the the law of conservation of angular momentum, in ifferential form. Next, note that, taking the cross prouct of Cauchy equations of motion, Eq. 2.16 times x, we have Finally, subtracting 2.22 from 2.21 yiels which implies i.e., the stress tensor is symmetric: ρe ijk Dv j Dt x k = ρe ijk f j x k + e ijk τ lj x l x k (2.22) e ijk τ kj = 0 (2.23) τ jk = τ kj (2.24) T = T T (2.25) 2.2.4 Conservation of Mechanical Energy Taking the ot prouct of Eq. 2.16 with v an integrating one obtains ρ Dv Dt v = ρf v + v iv T (2.26) Noting that Dv Dt v = 1 Dv 2 2 Dt (2.27)
26 an one obtains t ρ v2 2 = v iv T = iv (Tv) T : gra v (2.28) ρf v + n Tv T : gra v (2.29) Equation 2.30 represents the law of conservation of mechanical energy. Because of the symmetry of the stress tensor, it may be rewritten as ρ v2 t 2 = ρf v + n Tv T : D (2.30) where D = 1 2 (v i,j + v j,i )e i e j (2.31) 2.3 Energy The results of this section are base on the principles of conservation of mass, momentum, angular momentum, an energy. 2.3.1 Conservation of Thermoynamic Energy ubtracting the law of conservation of mechanical energy, Eq. 2.30 from the principle of conservation of energy, Eq. 2.4 one obtains ρe = t It shoul be note that the term T : D + r h (2.32) T : D (2.33) provies the link between the laws of conservation of mechanical an the thermoynamic energy. As we will see in Chapter 3 (on the secon principle of thermoynamics), the work one by the internal stresses may broken into two parts, one reversible an one irreversible. 2.3.2 Heat Flux ector Next, following a process similar to that use to emonstrate the existence of a stress tensor, we want to show that the heat flux, h, is a linear function of the normal, n, to the surface.
27 In orer to show this, consier the Cauchy tetraheron (Fig. 2.2) introuce above, an note that, by the same reasoning, Eq. 2.32 yiels h = O(l 3 ) (2.34) Figure 2.2: Cauchy Tetraheron Applying the mean value theorem, we have h A + h 1 A 1 + h 2 A 2 + h 3 A 3 = O(l 3 ) (2.35) Diviing by A, an noting that A i = An i (i = 1, 2, 3), where n i are the components of n, we have h + h 1 n 1 + h 2 n 2 + h 3 n 3 = O(l) (2.36) or, taking the limit as l goes to zero, h = q n (2.37) where q k = h k. The coefficients q k form a vector, calle the heat-flux vector.
28 2.3.3 Conservation of Thermoynamic Energy Differential Form Using Eq. 2.37, an Gauss theorem, Eq. 2.32 yiels ρ De Dt = T : D + or, in the limit, consiering the arbitrariness of, ρ De Dt r iv q (2.38) = T : D + r iv q (2.39) 2.4 Alternative Axioms Galilean Relativity In this ection we will show that the axioms at the beginning of this ection, i.e., the classical conservation laws of continuum mechanics (consevation of mass, momentum, angular momentum, an energy) may be replace with the following assumptions: the principle of conservation of energy, its invariance to uniform rigi-boy translations (Galilean relativity), an symmetry of the stress tensor. The principle of Galilean relativity of the energy equation states that Eq. 2.4 is the same in all the frames of reference in uniform translation with respect to each other. In orer to enforce this principle, substitute, in Eq. 2.4, v with v 0 + v, with v 0 constant. This yiels ( ) v 2 ρ 0 t 2 + v 0 v + v2 2 + e = ρf v 0 + ρf v + t v 0 + t v + r h (2.40) ince the energy expression must be the same in all frames of reference, we have, comparing with Eq. 2.4, [ ] [ v 2 ρ 0 t 2 + ρv t ] ρf t v 0 = 0 (2.41) Finally, noting that v 2 0 an v 0 are linearly inepenent, we obtain the law of conservation of mass, Eq. 2.1, an the law of conservation of momentum, Eq. 2.2. The law of conservation of angular momentum may be obtaine from the symmetry of the stress tensor, by following, in reverse, the same proceure use to prove the symmetry of the stress tensor.
29 References Green, A. E., an Naghy, P. M., A General Theory of an Elastic Plastic Continuum, Archives of Rational Mechanics, ol. 18, pp. 251-281, 1964. Green an Rivlin, Archives of Rational Mechanics, ol. 17, 113 1964 errin, J., Mathematical Principles of Classical Flui Mechanics, in E.:. Fluegge, Encyclopeia of Physics, ol. III/1, pp.125 263, 1959.
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