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4.1 Reynolds’ transp ort theorem 91 Fig. 4.2. Discontinuity surface • w w on w V 12 and using the generalized Reynolds’ transport theorem we obtain, G w w Gw Z w Y w # w dY = Z w Y C w # Cw w dY + Z w V w n · w v w # w dV + Z w V 12 w n 12 · w w w # w dV= (4.12a) Using the generalized Gauss’ theorem (Eq.(4.5)), Z w Y u · ( w v w #) w dY = Z w V w n · w v w # w dV + Z w V 12 w n 12 · w v w # w dV (4.12b) hence, G w w Gw Z w Y w # w dY = Z w Y C w # Cw + u · ( w v w #) ¸ w dY (4.12c) Z w V 12 w # w n 12 · ( w v w w) w dV= For the region on the positive side of w n 12 , in the same way, we get G w w Gw Z w Y + w # w dY = Z w Y + C w # Cw + u · ( w v w #) ¸ w dY (4.12d) + Z w V 12 w # + w n 12 · ( w v + w w) w dV= 92 Nonlinear continua The velocity field of the fictitious particles is coinciden t with the velocity field of the actual particles everywhere except on w V 12 . Therefore: G Gw Z w Y w # w dY = G w w Gw Z w Y w # w dY + G w w Gw Z w Y + w # w dY= (4.12e) From Eqs.(4.12c) to (4.12e) we get, G Gw Z w Y w # w dY = Z w Y C w # Cw + u · ( w v w #) ¸ w dY + Z w V 12 [[ w # ( w y q w z q )]] w dV (4.13) where, [[ w # ( w y q w z q )]] = w # + ( w y + q w z q ) w # ( w y q w z q ) > w y q = w n 12 · w v > w z q = w n 12 · w w = In order to obtain a localized version of Reynolds’ transport theorem at the discontinuity surface we consider the arbitrary material volume enclosed b y the dashed line in Fig. 4.3. Fig. 4.3. Derivation of the jump discontinuity condition w = w + w 3 For the enclosed material volume, using Eq.(4.13) and the generalized Gauss’ theorem, we write: G Gw Z w Y w # w dY = Z w Y C w # Cw w dY + Z w V w n · w v w # w dV (4.14) + Z w V + w n · w v w # w dV + Z w V 12 [[ w # ( w y q w z q )]] w dV> 4.2 Mass-conse rvation principle 93 when w dV + $ w V 12 and w dV $ w V 12 ; w + $ 0 and w $ 0 we get from Eq.(4.14): Z w V 12 ¡ [[ w # w y q ]] + [[ w # ( w y q w z q )]] ¢ w dV =0= (4.15) Therefore, i n order for the above integral equation to be valid for any arbitrary part of the discontinuity surface, we must fulfill [[ w # w y q ]] + [[ w # ( w y q w z q )]] = 0 (4.16a) at every point on w V 12 . Equation (4.16a) is known as the jump discontinuity condition. If we call w U = w z q w y q the discontinuity’s propagation speed, we can write [[ w # w U]] = [[ w # w y q ]] = (4.16b) The above equation is known as Kotchine’s theorem (Truesdell & Toupin 1960). 4.2 Mass-conservation principle In Sect. 2.2, Eq.(2.6) introduced the concept of mass of a contin uum body B. In the study of continuum media, under the assumptions of Newtonian mechanics, it is postulated that the mass of a continuum is conserved. Hence, G Gw Z w Y w w dY =0 (4.17) where w = w ( w { l >w). 4.2.1 Eulerian (spatial) formulation of the mass-conservation principle Using in Eq.(4.17) the expression of Reynolds’ transport theorem given in Eq.(4.4) we obtain: G Gw Z w Y w w dY = Z w Y C w Cw + u · ( w w v) ¸ w dY =0= (4.18) Since the above equation has to be fulfilled for any control volume inside the continuum, we can write for any point inside the spatial configuration: C w Cw + u · ( w w v)=0= (4.19) The above partial dierential equation is the localized spatial form of the mass-conservation principle in a Eulerian formulation anditiscalledthe continuity equation. 94 Nonlinear continua Example 4.3. JJJJJ Using components in a general curvilinear spatial coordinate system, the con- tinuit y equation is written as C w Cw + w y d C w C w { d + w w y d | d =0= JJJJJ Example 4.4. JJJJJ For an incompressible material G w Gw =0; hence the continuity equation is: u · w v =0> or in components, w y d | d =0= JJJJJ Example 4.5. JJJJJ Let a fluid of density w = w ( w { l >w) have a velocity field w v. Let us consider in the spatial configuration a volume (w) bounded b y a surface (w) that moves with an arbitrary velocity field w w. Following (Thorpe 1962), we first calculate the fluid mass instantaneously inside the volume (w): P = Z (w) w w dY and using the expression of the generalized Reynolds’ transport theorem in Eq.(4.11), we get dP dw = G w w Gw Z (w) w w dY = Z (w) C w Cw w dY + Z (w) w n · w w w w dV where w n is the external normal of the surface (w). Using Eq.(4=5) (generalized Gauss’ theorem), we get Z (w) u · ¡ w w v ¢ w dY = Z (w) w n · ( w w v) w dV and subtracting the above equation from the previous one, dP dw = Z (w) C w Cw + u · ( w w v) ¸ w dY + Z (w) w w n · ( w w w v) w dV= 4.3 Balance of momentum principle (Equilibrium) 95 Using Eq.(4.19), we see that the first integral on the r.h.s. is zero; hence, dP dw = Z (w) w w n · ( w w w v) w dV= The above equation is an integral equation of continuity for a control volume in motion in the fluid velocity field (Thorpe 1962). JJJJJ 4.2.2 Lagrangian (material) formulation of the mass conservation principle Equation (4.17) implies that, Z Y ( { D ) dY = Z w Y w ( w { d >w) w dY (4.20a) where ( > Y ) correspond to the reference configuration and ( w > w Y ) to the spatial configuration. Using Eq.(2.31) in the r.h.s. of Eq.(4.20a) and changing variables in the expression of w we obtain, Z Y ( { D ) dY = Z Y w ( { D >w) w M dY> (4.20b) hence, Z Y ( w w M) dY =0= (4.20c) Since the above equation has to be fulfilled for any control volume that we define inside the continuum, we can write for any point inside the reference configuration: = w w M> (4.20d) and therefore, G Gw ( w w M)=0= (4.20e) The above equation is the localized material form of the continuity equa- tion. 4.3 Balance of m omentum principle (Equilibrium) The principle of balance of momentum is the expression of Newton’s Second Law for continuum bodies. Quoting (Malvern 1969): “The momentum principle for a collection of particles states that the time rate of change of the total momentum of a given set of particles equals the vector sum of all the external forces acting on the particles of the set, provided Newton’s Third Law of action and reaction governs the internal forces. The continuum form of this principle is a basic po stulate of continuum mechanics”. 96 Nonlinear continua 4.3.1 Eulerian (spatial) formulation of the balance of momentum principle For a b ody B in the w-configuration we define its momentum as, Z w Y w w v w dY (4.21a) the resultant of the external forces acting on the elements of mass inside the body are, from Eq.(3.2): Z w Y w w b w dY> (4.21b) and the resultant of the external forces acting on the elements of the body’s surface are, from Eq.(3.4): Z w V w t w dV= (4.21c) Using Eqs.(4.21a-4.21c), w e can state Newton’s Second Law for the body B as, G Gw Z w Y w w v w dY = Z w Y w w b w dY + Z w V w t w dV= (4.22) Using the condition of equivalence bet ween external forces and Cauchy stresses inside a continuum, defined in Eq.(3.7), we get: G Gw Z w Y w w v w dY = Z w Y w w b w dY + Z w V w n · w w dV= (4.23) Using in the above the expression of Reynolds’ transport theorem given in Eq.(4.4), we get Z w Y C( w w v) Cw + u · ( w w v w v) ¸ w dY = Z w Y w w b w dY + Z w V w n · w w dV=(4.24) From Example 4.2, we obtain u · ( w w v w v)= w v · £ u ( w w v) ¤ + w w v (u · w v) (4.25a) also, from Eq.(2.20b), we get G ( w w v) Gw = C ( w w v) Cw + w v · £ u ( w w v) ¤ > (4.25b) and, from Eq.(4.5) (Generalized Gauss’ Theorem), we get 4.3 Balance of momentum principle (Equilibrium) 97 Z w V w n ã w w dV = Z w Y u ã ( w ) w dY= (4.25c) Using Eqs.(4.25a-4.25c) in Eq.(4.24) we arrive at the integral form of the Eulerian formulation of the balance of momentum principle: Z w Y G Gw ( w w v)+ w w v (u ã w v) á w dY = Z w Y Ê w w b + u ã w Ô w dY= (4.26) Since the above equation has to be fullled for any control volume that we dene inside the contin uum, w e can write for any point inside the spatial conguration: G Gw ( w w v)+ w w v (u ã w v)= w w b + u ã w (4.27a) and using in the above the continuity equation, we have w G w v Gw = w w b + u ã w = (4.27b) The above equation is the localized form of the balance of momentum principle in an Eulerian formulation and it is known as the equilibrium equa- tion. Example 4.6. JJJJJ Using Eq.(A.62b), in the general Eulerian curvilinear system { w { d },weget, u ã w = w de | d w g e hence, using Eq.(A.55b), u ã w = C w de C w { d + w ve w d vd + w dv w e vd á w g e = From the result in Example A.10, we can easily get w p lo = 1 2 w j pm C w j lm C w { o + C w j mo C w { l C w j ol C w { m á = Therefore, u ã w = C w de C w { d + 1 2 Ă w ve w j dm + w dv w j em  à C w j vm C w { d + C w j md C w { v C w j dv C w { m ảá w g e = JJJJJ 98 Nonlinear continua Example 4.7. JJJJJ A perfect fluid is defined as a con tinuum in which, at every point, and for any surface, w n · w = w w n > where w is a scalar (no shear stresses). Since w is a symmetric second order tensor (to be shown in Sect. 4.4), its eigenvalues are real and its eigenvectors are orthogonal (Appendix, A.4.1). Referring the problem to the Cartesian system defined by the normalized eigenvectors, w ˆe we can write, w n = w ˆq w ˆe w = 3 X =1 w ˆ w ˆe w ˆe = Then, for the perfect fluid, w ˆq w ˆ = w w ˆq ( =1> 2> 3)(qr dgglwlrq rq ) = The above set of equations is fulfilled only if the three eigenvalues w ˆ are equal (hydrostatic stress tensor). Hence, w = w s w ˆe w ˆe = It is easy to show that as the three eigenvalues of w are equal, the above equation is valid in any Cartesian system; hence we can write, w lm = w s lm > where w s is the pressure. Generalizing the above to any arbitrary coordinate system w = w s w g = Using Eq. (A.62b) and the result in Example A.11, the equilibrium equation, Eq. (4.27b), can be written as, w G w v Gw = w w b + Cs C w { l w j lm w g m = From Eq. (A.57) we identify the last term on the r.h.s. of the above equation as u w s, hence we can write the equilibrium equation for a perfect fluid as, w G w v Gw = w w b + u w s= The above equation is known as the Euler equation for perfect fluids.Many authors get a minus sign for the second term on the r.h.s. because they define w lm = w s lm = JJJJJ 4.3 Balance of momentum principle (Equilibrium) 99 Example 4.8. JJJJJ Following with the topic discussed in Example 4.5 we consider a uid, moving with a velocity eld w v, and a moving control volume, moving with a velocity eld w w. In this example, following (Thorpe 1962), we are going to analyze the momentum balance inside the moving control volume. Using the generalized Reynolds transport theorem (Eq.(4.11)) for the uid momentum, G w w Gw Z (w) w w v w dY = Z (w) C( w w v) Cw w dY + Z (w) w w v ( w n ã w w) w dV= From the generalized Gauss theorem (Eq.(4.5)), Z (w) u ã ( w w v w v) w dY = Z (w) w n ã ( w w v w v) w dV= Subtracting the above equation from the previous one, G w w Gw Z (w) w w v w dY = Z (w) C( w w v) Cw + u ã ( w w v w v) á w dY + Z (w) w w v Ê w n ã ( w w w v) Ô w dV= Using the result in Example 4.2 and Eq.(4.19) (continuity equation), G w w Gw Z (w) w w v w dY = Z (w) w à C w v Cw + w v ã u w v ả w dY + Z (w) w w v Ê w n ã ( w w w v) Ô w dV= Onther.h.s.oftheaboveequation,thetermbetweenthebracketsinthe rst integral is the uid particles material acceleration. We can state, using Newtons second law, that the external force instantaneously acting on the particles inside (w) is, w F = Z (w) w w a w dY= Hence, w F = G w Gw Z (w) w w v w dY + Z (w) w w v Ê w n ã Ă w v w w ÂÔ w dV= JJJJJ 100 Nonlinear continua Example 4.9. JJJJJ Let us consider the body B and the particle S on its external surface. We define at S a convected coordinate system l with covariant base vectors w e g l in the spatial configuration and e g l in the material configuration. The con- vected system is definedsoastohave w e g 1 and w e g 2 in the plane tangent to w V at w S ; and therefore e g 1 and e g 2 define the plane tangent to V at S . Material and spatial normal vectors (Nanson’s formula) The external unit normals at S are w n = w e g 1 × w e g 2 | w e g 1 × w e g 2 | > and, n = e g 1 × e g 2 | e g 1 × e g 2 | = Also, the surface-area dierentials are w dV w n =( w e g 1 × w e g 2 )d 1 d 2 (D) > dV n =( e g 1 × e g 2 )d 1 d 2 (E) = If we define, w t 1 =d 1 w e g 1 > w t 2 =d 2 w e g 2 > t 1 =d 1 e g 1 > t 2 =d 2 e g 2 > [...]... d (4.36a) where r = x x ; and x ; x are the position vectors of an arbitrary point and of the point , respectively The resultant moment with respect to of the external forces acting on the elements of mass inside the body is, Z r × b d (4.36b) and the resultant moment with respect to of the external forces acting on the elements of the body’s surface is, Z r × t d (4.36c) Using Eqs.(4.36a-4.36c) we... and again using Eq.(A.37e), we get d Therefore, ( 1 ) = p | || 1 | p | | ( 1) ( 2) ( 2) 102 Nonlinear continua d 1 ( ) = d 1 | s | | | | | and using Eq.(2.34i), we get n d = n · 1 X d The above equation is called Nanson’s formula (Bathe 19 96) JJJJJ Example 4.10 For an Eulerian vector a we define, using Eq.(2.76a), JJJJJ A = h¡ 1 i ¢ g Using the generalized Gauss’ theorem (Eq.(4.5)), Z Z · a d = n · a... (Malvern 1 969 ) 4.5.1 Eulerian (spatial) formulation of the energy balance For a body B in the -configuration we define u as its internal energy per unit mass Being the kinetic energy of B defined by Eq.(3.9g), the total energy in the considered body, at the instant , is Z = + u d (4. 46) The external forces acting on the body provide a mechanical power input Using Eq.(3.9j), we write 110 Nonlinear continua. .. the first Piola-Kirchho stress tensor, we obtain Z Z Z v) d = b d + ( P) d (4.28b) ( In the above (Malvern 1 969 ), ( P) = = | g + + ¸ g Equation (4.28b) is an integral form of the equilibrium equations It is important to note that although the integrals are calculated on volumes defined 104 Nonlinear continua in the reference configuration, the equilibrium is established in the spatial configuration The corresponding... principle for the continuum body B: Z Z Z r× v d = r× b d + r× t d (4.37a) With the condition of equivalence between external forces and Cauchy stresses inside a continuum defined in Eq.(3.7), we get 1 06 Nonlinear continua Z r × v d = Z r × b d + Z r × (n · (4.37b) JJJJJ Example 4.13 The last integral on the r.h.s of Eq.(4.37b) can be written as, Z r × ¡ ¢ n · d Z = ¡ Z = n · · ¡ ) d ¢ × r d × r ¢ d where... momentum principle (Equilibrium) 105 4.4 Balance of moment of momentum principle (Equilibrium) Quoting (Malvern 1 969 ) again: “In a collection of particles whose interactions are equal, opposite, and collinear forces, the time rate of change of the total moment of momentum for a given collection of particles is equal to the vector sum of the moments of the external forces acting on the system In the absence... Symmetry of Eulerian and Lagrangian stress measures From the Eulerian localized form of the balance of moment of momentum principle, we will first derive the symmetry of the Cauchy stress tensor 108 Nonlinear continua Using an intermediate result that we got in Example 4.13, we can write the localized form of the balance of moment of momentum as, v r × r × b = ¡ · × r ¢ (4.42) JJJJJ Example 4.14 The divergence... equilibrium is established in the spatial configuration The corresponding localized form is, v b + = ( P) (4.29) JJJJJ Example 4.12 From Eqs.(4.27b) and (4.29) we get, · = ( P) The above equation is a particular application of the Piola Identity JJJJJ In order to write the equilibrium equations in terms of fully material tensors we have to pull-back Eq.(4.29) For the material velocity field: £ ¤ V = . (4.12d) + Z w V 12 w # + w n 12 · ( w v + w w) w dV= 92 Nonlinear continua The velocity field of the fictitious particles is coinciden t with the velocity field of the actual particles everywhere except on w V 12 × w b w dY> (4.36b) and the resultant moment with respect to R of the external forces acting on the elements of the body’s surface is, Z w V w r × w t w dV= (4.36c) Using Eqs.(4.36a-4.36c) we can. can write [[ w # w U]] = [[ w # w y q ]] = (4.16b) The above equation is known as Kotchine’s theorem (Truesdell & Toupin 1 960 ). 4.2 Mass-conservation principle In Sect. 2.2, Eq.(2 .6) introduced the concept of
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