3.2 The Cauchy stress tensor 71 Fig. 3.3. Tangent surfaces at P have the same traction vector 3.2.1 Symmetry of the Cauchy stress tensor (Cauc hy Theorem) In Sect. 4.4.2 we will prove that from the equilibrium equations of a nonpolar continuum we get w = w W > (3.8) that is to say, the Cauchy stress tensor is symmetric. Example 3.1. JJJJJ In the following figure, at the point S , inside the w-configuration of a contin- uum body, the stress tensor is w . Cutting the continuum with the surface w V 1 ,wegetatS the traction vector w t 1 = w n 1 · w > if we cut with w V 2 , the traction vector at S is: w t 2 = w n 2 · w = 72 Nonlinear continua Secant surfaces at a point inside a continuum In an arbitrary coordinate system { w { l } we can write w t 1 · w n 2 = w q 1 l w lm w q 2 m > w t 2 · w n 1 = w q 2 l w lm w q 1 m > and since from Eq. (3.8) w lm = w ml we get w t 1 · w n 2 = w t 2 · w n 1 = The above result, a direct consequence of the Cauchy Theorem, is known as the projection theorem or reciprocal theorem of Cauchy (Malvern 1969).JJJJJ 3.3 Conjugate stress/strain rate measures Let us assume, at an instant (load level) w,acontinuumbodyB in equilibrium under the action of external body forces w b and external surface forces w t. Assuming a velocity field w v( w x) on B ,thepower provided by the external forces is: w S h{w = Z w Y w w b · w v w dY + Z w V w t · w v w dV= (3.9a) Using Eq. (3.7) we can rewrite Eq.(3.9a) as 3.3 Conjugate stress/strain rate measures 73 w S h{w = Z w Y w w b · w v w dY + Z w V ¡ w n · w ¢ · w v w dV= (3.9b) From the Divergence Theorem (Hildebrand 1976), w S h{w = Z w Y £ w w b · w v + u · ¡ w · w v ¢¤ w dY (3.9c) in troducing Eq. (2.110a-2.110c) and after some algebra, w S h{w = Z w Y £ w : w l + ¡ w w b + u · w ¢ · w v ¤ w dY= (3.9d) Since the w-configuration is an equilibrium configuration, the following equation (to be proved in Chap. 4, Eq.(4.27b)) holds w w b + u · w = w G w v Gw > (3.9e) and an obvious result that we also need is: G w v Gw · w v = G Gw 1 2 w v · w v ¸ = (3.9f) The kinetic energy of the body B,attheinstantw,isdefined as w N = Z w Y w 2 w v · w v w dY = Z p 1 2 w v · w v dp= (3.9g) In the second integral of the above equation w e integrate over the mass of the body B. Since the mass of the body is invariant, G w N Gw = Z p G w v Gw · w v dp= (3.9h) Finally, using the decomposition of the velocity gradient tensor in Eq. (2.112a) and considering that since ( w ) is a symmetric tensor and ( w $) is a skew-symmetric one, w : w $ =0 (3.9i) we get w S h{w = G w N Gw + Z w Y w : w d w dY= (3.9j) We define w S = Z w Y w : w d w dY (3.10) as the stresses power. Ob viously w S is the fraction of w S h{w that is not trans- formed into kinetic energy and that is either stored in the body material or dissipated by the body material, depending on its prope rties (see Chapter 5). From Eq. (3.10) we define the spatial tensors w and w d to be ener gy conjugate (Atluri 1984). In what follows we will define other pairs of energy conjugate stress/strain rate measures. 74 Nonlinear continua 3.3.1 The Kirchho stress tensor From Eqs. (3.10) and (2.34d) and the mass-conservation principle (to be dis- cussed in Chapter 4, Eq.(4.20d)) we get w S = Z w Y w : w d w dY = Z Y w w : w d dY= (3.11) The Kirchho stre ss tensor is defined as w = w w (3.12) where, : density in the reference configuration and Y : volume of the refer- ence configuration. It is important to note that although the Kirchho stress tensor w a s in- troduced by calculating w S via an integral defined over the reference volume, Eq. (3.12) clearly shows that w is definedinthesamespacewhere w is defined: the spatial configuration. Hence, using in the w-configuration an arbi- trary curvilinear coordinate system { w { d } with covariant base vectors w g d we obtain w = w de w g d w g e (3.13a) w = w de w g d w g e (3.13b) and w de = w w de = (3.13c) 3.3.2 The first Piola-Kirchho stress tensor From Eqs. (3.11) and (3.9i) we can write w S = Z Y w : w l dY= (3.14) In Chap. 2 we learned how to derive representations in the reference config- uration of tensors defined in the spatial configuration via pull-back operations. We will now obtain a representation of the Kirchho stress tensor in the form of a two-point tensor. In the reference configuration we define an arbitrary coordinate system { { D } with covariant base vectors g D ;andinthespatialconfiguration a system { w { d } with covariant base vectors w g d .Wealsodefine a c onvected system { l } with covariant base vectors in the reference configuration e g l and covariant base vectors in the spatial configuration w e g l . In the spatial configuration we can write the Kirchho stress tensor as 3.3 Conjugate stress/strain rate measures 75 w = w ˜ lm w e g l w e g m (3.15a) a pull-back of the above tensor to the reference configuration is: w T ` = w ˜ lm e g l e g m = h w T ` i LM g L g M = (3.15b) We define a two-point representation of w as w P = w ˜ lm e g l w e g m = £ w P ¤ Lm g L w g m = (3.15c) After some algebra, w S Lm = w ˜ op C { L C o C w { m C p = " w st C o C w { s C p C w { t # C { L C o C w { m C p = w sm ¡ w [ 1 ¢ L s = (3.15d) The second-order two-point tensor w P is the first Piola-Kirchho stress tensor.ItisapparentfromEq.(3.15d)thatitisanon-symmetric tensor. We can write, due to the symmetry of the Kirchho stress tensor: S = Z Y w de w o de dY = Z Y w ed w o de dY= (3.16a) Hence, using Eq.(3.15d): S = Z Y w S Ed w [ e E w o de dY= (3.16b) Using Eq. (2.111a) we have S = Z Y w S Ed w ˙ [ dE dY = Z Y w P ·· w ˙ X dY= (3.17) We can also write the above as (Malvern 1969): S = Z Y w P W : w ˙ X dY= (3. 18) The above equation defines the two-point tensors w P W and w ˙ X as energy conjugates . We will not try to force a so-called “physical interpretation” of the first Piola-Kirchho stress tensor; instead we will regard it only as a useful math- ematical tool. 76 Nonlinear continua 3.3.3 The second Piola-Kirchho stress tensor The pull-back configuration of w to the reference configuration is: w T ` = h w lm ¡ w [ 1 ¢ L l ¡ w [ 1 ¢ M m i g L g M = (3.19) The tensor w T ` ,defined by the above equation, is the se cond Piola- Kirchho stress tensor and it is a symmetric tensor. Using Bathe’s notation (Bathe 1996), we identify the second Piola-Kirchho stress tensor, corresponding to the w-configuration and referred to the config- uration in w =0as w S. Example 3.2. JJJJJ From Eqs. (3.15d) and (3.19) we get w V LM = ¡ w [ 1 ¢ M m w S Lm that is to say, w V LM = £ w ! ¡ w S Lm ¢¤ LM = JJJJJ From Example 2.16: w ˙% DE = £ w ! ¡ w g de ¢¤ DE (3.20a) and since from Eq. (3.19), w V DE = £ w ! ¡ w de ¢¤ DE (3.20b) using Eqs. (3.11) and (2.88), we get w S = Z Y w S : w ˙% dY= (3.20c) From Eq. (3.20c) we define the tensors w S and w ˙% to be energy conjugate (Atluri 1984). Here, we will also not try to force a “physical interpretation” of the second Piola-Kirchho stress tensor. An important point to be analyzed is the transformation of w S under rigid- body rotations. • Let us first consider the w-configuration of a certain body B. At an arbitrary point S the Cauchy stress tensor is w . 3.3 Conjugate stress/strain rate measures 77 • Let us now assume that we evolve from the w-configuration to a (w + w)- configuration imposing on B and on its external loads a rigid body rotation. At the point S we can write: w+w w X = w+w w R > (3.21a) and therefore, w+w dx = w+w w R · w dx = (3.21b) For the external loads, w+w t = w+w w R · w t > (3.22a) w+w b = w+w w R · w b = (3.22b) For a velocity vector, w+w v = w+w w R · w v = (3.22c) For the external normal vector, w+w n = w+w w R · w n = (3.22d) Example 3.3. JJJJJ For an arbitrary force vector w f (it can be a force per unit surface, per unit volume, etc.) and considering the evolution described above, w+w f · w+w v = w+w w R · w f · w+w v = w f · w+w w R W · w+w v = w f · w+w w R W · w+w w R · w v and since the rotation tensor is orthogonal, w+w f · w+w v = w f · w v = The above equation states the intuitive notion that a rigid-body rotation cannot aect the value of the deformation power performed by the external forces. JJJJJ 78 Nonlinear continua At w we can write w t = w n · w (3.23a) and at (w + w), w+w t = w+w n · w+w = (3.23b) In troducing Eq. (3.22d) in the above, w+w t = ¡ w+w w R · w n ¢ · w+w = (3.23c) And with Eq. (2.28a) and (3.22a), we finally have w t = w n · h w+w w R W · w+w · w+w w R i = (3.23d) For deriving the above equation, we used that w+w w R · w t = w t · w+w w R W = Hence, w+w = w+w w R · w · w+w w R W = (3.23e) The above equation indicates that the Cauchy stress t ensor fulfills the cri- terion for objectivity under isometric transformations, established for Eulerian tensors in Sect. 2.12.2. We define an arbitrary system { w { d 0 } in the w-configuration and a system { w+w { d } in the (w + w)-configuration. Hence, from Eq. (3.23e), w+w d e = w+w w U d f 0 w f 0 g 0 ¡ w+w w U W ¢ g 0 e (3.24a) and using Eq. (2.28c), we get w+w d e = w+w w U d f 0 w f 0 g 0 w+w w U o p 0 w+w j oe w j p 0 g 0 (3.24b) therefore, w+w do = w f 0 p 0 w+w w U d f 0 w+w w U o p 0 = (3.24c) It is easy to show that for the Kirchho stress tensor we can also write w+w = w+w w R · w · w+w w R W = (3.25) From Eq. (3.19), we obtain w+w V LM = w+w lm ¡ w+w [ 1 ¢ L l ¡ w+w [ 1 ¢ M m (3.26a) but since, w+w [ d D = w+w w U d d 0 w [ d 0 D (3.26b) ¡ w+w [ 1 ¢ D d = ¡ w [ 1 ¢ D d 0 ¡ w+w w U W ¢ d 0 d (3.26c) using Eqs. (3.24c) and (3.26c) in Eq. (3.26a), we get 3.3 Conjugate stress/strain rate measures 79 w+w V LM = w V LM (3.27a) therefore, w+w S = w S = (3.27b) The above equation indicates that the second Piola-Kirchho stress ten- sor fulfills the criterion for objectivity under isometric transformations, estab- lished for Lagrangian tensors in Sect. 2.12.2. 3.3.4 A stress tensor energy conjugate to the time derivative of the Henc ky strain tensor In Sect. 2.8.5 we defined the logarithmic or Hencky strain tensor. Let us now define, via a pull-back operation, the following stress tensor: w = w U ( w ) = (3.28) With the notation w U (·) we define the pull-back of the components of the tensor (·) using the tensor w R (Simo & Marsden 1984), that is to say, w is an unrotated representation of w . Fromthesymmetryof w ,theabovedefinition implies the symmetry of w = We will now demonstrate, following (Atluri 1984), that for an isotropic material w and w ˙ H are energy conjugate. We can write Eq. (3.14) as w S = Z Y w DE £ w U ¡ w g de ¢¤ DE dY (3.29a) therefore, w S = Z Y w DE w U d D w g de w U e E dY= (3.29b) From Eq. (2.28c), w U d D = ¡ w U W ¢ O o w j do j DO (3.29c) and using the above, the integrand in Eq. (3.29b) is: w DE ¡ w U W ¢ O o w j do j DO w g de w U e E = (3.29d) It is also easy to show that w R W · w d · w R = h ¡ w U W ¢ O o w j do j DO w g de w U e E i g D g E = (3.29e) Hence, using Eq. (2.118b), w S = Z Y 1 2 w : ³ w ˙ U · w U 1 + w U 1 · w ˙ U ´ dY= (3.29f) 80 Nonlinear continua In order to simplif y the algebra, in what follows w e will work in a Cartesian system; [D] willbethematrixformedwiththeCartesiancomponentsofa second-order tensor A . From Eqs. (2.122a-2.122d), we get [ w ˙ X][ w X] 1 =[ w U O ][ w ˙ ][ w ] 1 [ w U O ] W +[ w O ] (3.30a) [ w U O ][ w ][ w U O ] W [ w O ][ w U O ][ w ] 1 [ w U O ] W and [ w X] 1 [ w ˙ X]=[ w U O ][ w ] 1 [ w ˙ ][ w U O ] W (3.30b) +[ w U O ][ w ] 1 [ w U O ] W [ w O ][ w U O ][ w ][ w U O ] W [ w O ] it follows from the above two equations that 1 2 n [ w ˙ X][ w X] 1 +[ w X] 1 [ w ˙ X] o =[ w U O ][ w ] 1 [ w ˙ ][ w U O ] W (3.30c) + 1 2 [ w U O ][ w ] 1 [ w U O ] W [ w O ][ w U O ][ w ][ w U O ] W 1 2 [ w U O ][ w ][ w U O ] W [ w O ][ w U O ][ w ] 1 [ w U O ] W > and using once again Eqs. (2.122a-2.122d), we get 1 2 n [ w ˙ X][ w X] 1 +[ w X] 1 [ w ˙ X] o =[ w U O ][ w ] 1 [ w ˙ ][ w U O ] W (3.30d) + 1 2 [ w X] 1 [ w O ][ w X] 1 2 [ w X][ w O ][ w X] 1 = Using the result in Example 2.17, we can write 1 2 n [ w ˙ X][ w X] 1 +[ w X] 1 [ w ˙ X] o =[ w ˙ K] [ w O ][oq w X] (3.30e) +[oq w X][ w O ]+ 1 2 [ w X] 1 [ w O ][ w X] 1 2 [ w X][ w O ][ w X] 1 = Using the above in Eq. (3.29f) and w orking with the matrix components, w S = Z Y [ w ] n [ w ˙ K] [ w O ] [oq w X] (3.31a) +[oq w X] [ w O ] + 1 2 [ w X 1 ] [ w O ] [ w X] 1 2 [ w X] [ w O ] [ w X 1 ] ¾ dY= Since [ w ] and [ w X] are symmetric and [ w O ] is skew-symmetric, we can rewrite the above equation as [...]... change of the total amount of the tensorial property t carried by the particles that at time are inside the volume Using Eq.(2.31) we can write, 86 Nonlinear continua Z hence, Z ( ( ) d d = d ) d Z = d Z ( ( d ) ) d + d Z (4.2a) ( ) d d d (4.2b) JJJJJ Example 4.1 Working in Cartesian coordinates we can write, from Eq (2.34e) (Fung 19 65) : = 1 2 3 Using Eqs.(2.111a-2.111b) we can calculate the time rate... material velocity v of the particle instantaneously at - The velocity w of the surface The condition that the point (not the particle instantaneously at ) remains on = 0 when the surface moves is given by, = 0 + (4.9a) From geometrical considerations, and we can write, n = q w = g g (4.9b) (4.9c) Hence, = w · n = q (4.9d) and using in the above Eq.(4.9a), we get = q (4.9e) 90 Nonlinear continua We can also... in the form of partial di erential equations (localized form) On presenting the basic principles we are going to use both, the Eulerian (spatial) and Lagrangian (material) descriptions of motion Some reference books for this chapter are: (Truesdell & Toupin 1960, Fung 19 65, Eringen 1967, Malvern 1969, Slaterry 1972, Oden & Reddy 1976, Marsden & Hughes 1983, Panton 1984, Lubliner 19 85, Fung & Tong 2001)... adequate tool for deriving objective rates of Eulerian tensors is the Lie derivative The Lie derivative of the Cauchy stress tensor is: ³ ´ = £ v( ) ¤ = ˙ (3. 35) the above stress rate is known as Oldroyd stress rate (Marsden & Hughes 1983) 82 Nonlinear continua Example 3.4 JJJJJ To derive the expression of Oldroyd’s stress rate we start from Eq (2.129), ¡ v ¢ = ¡ + ¡ From Eq (2.109a) ˙ X = 1 ¢ ¸ + g 1 ¢... as, v = g therefore we can write, using Eq.(A .59 ), ¡ t ¢ v· = g · = } | g g g | g g g g g g Also, using Eq.(A.64), t ( · v) = | g g g g Finally, using Eq (A.62b), · (v )= = ³ ´ ³ | g g | + g | g ´ g g g g From the above equations we get, v · ( t ) + t ( · v) = · (v ) JJJJJ Using the above result we can express Reynolds’ transport theorem as, 88 Nonlinear continua Z ( ) d = Z t ( ) · (v + ( )) ¸ The... (3.37b) ª © ª ©d d Taking into account that d [ ] we get = d [ ] d £ d ( t) ¤ = [ ] [ ˙] [ ] + [ ˙] [ ] [ ] + [ ] [ ] [ ˙ ] (3.37c) and using Eqs (2.115a) and (2.116a) we get d £ d ( t) ¤ = [ +[ ] [ ˙] [ ] [ ][ ] [ ][ ] [ ] ][ ][ ] (3.37d) 84 Nonlinear continua Since, we finally arrive at h h R( R( t) i t) = [ i d £ d ] = [ ˙] [ ¤ ( t) [ ] ][ ] + [ ][ (3.37e) ] (3.37f) that in an arbitrary spatial coordinate... (4.9d) and using in the above Eq.(4.9a), we get = q (4.9e) 90 Nonlinear continua We can also define, using the material velocity of the particle instantaneously at , = v · n = q (4.9f) Using Eqs (4.9e), (4.9f) and the first equality in Eq.(4.8), we obtain r ˙ = ( ) (4.9g) If the particle instantaneously at remains on the surface during the motion, from obvious geometrical considerations = (4.9h) and using... discontinuity surface we define its normal ( n12 ) and its displacement velocity ( w), not necessarily coincident with the material velocity of the particle instantaneously at that point ( v) Considering the region on the negative side of n12 , we can define fictitious particles with the following velocity field: • v on ... result we can express Reynolds’ transport theorem as, 88 Nonlinear continua Z ( ) d = Z t ( ) · (v + ( )) ¸ The generalized Gauss’ theorem can be stated as (Malvern 1969): Z Z · t d = n · t d (4.4) (4 .5) where is the closed surface that bounds the volume and n is the surface’s outer normal vector Using Gauss’ theorem in Eq (4.4) and rearranging terms, we get the following expression of Reynolds’ transport... an arbitrary volume ( ) bounded by a surface ( ) that moves with an arbitrary velocity field w We can define inside the -configuration of the body B a surface, ( ) = 0 (4.7) If the surface moves with the particles instantaneously on it, we say that it is a material surface The Lagrange criterion states (Truesdell & Toupin 1960) that the necessary and su cient condition for the above-defined surface to be . measures 75 w = w ˜ lm w e g l w e g m (3.15a) a pull-back of the above tensor to the reference configuration is: w T ` = w ˜ lm e g l e g m = h w T ` i LM g L g M = (3.15b) We define. is: ³ w ´ de = £ O w v ( w ) ¤ de = w ˙ de w fe w o d f w df w o e f (3. 35) the above stress rate is known as Oldroyd stress rate (Marsden & Hughes 1983). 82 Nonlinear continua Example 3.4. JJJJJ To derive the expression. Eqs. (2.115a) and (2.116a) we get d dw £ w U ( w t) ¤ =[ w U] W [ w ˙ w][ w U] [ w U] W [ w U ][ w w][ w U] +[ w U] W [ w w][ w U ][ w U] = (3.37d) 84 Nonlinear continua Since, h O w R ( w t) i =[ w U] d dw £ w U ( w t) ¤ [ w U] W (3.37e) we