2.7 The p o lar decomposition 31 ( w dV) 2 =( dV) 2 £ n · w C · n ¤ and therefore, in the fixed Cartesian system that we are using, w dV dV = £ [ q] W [ w F][ q] ¤ 1@2 and using the numerical values calculated in the previous example, w dV dV =1=03528 = The reader can check the above numerical values using very simple geometrical considerations. In the case of n = l ,itiseasytoshowthat w dV dV = l = For [ p] W = £ 1=00=00=0 ¤ > we get w dV dV =1=0 = The v ectors m and n, which are orthogonal in the reference confi gu- ration, form an angle w in the spatial configuration, w dx p = w dV p w m ; k w mk =1 w dx q = w dV q w n ; k w nk =1 w dx p · w dx q =( w dV p )( w dV q )cos w =( dV p )( dV q )[ p] W [ w F][ q] µ w dV dV ¶ p µ w dV dV ¶ q cos w =[ p] W [ w F][ q] hence, w =cos 1 " [ p] W [ w F][ q] ¡ w dV dV ¢ p ¡ w dV dV ¢ q # and using the calculated numerical values, we get w =75 = Once again, it is very simple for the reader to check the above numerical result. JJJJJ In the above examples we have numerically calculated the eigenvalues and eigenvectors of the tensor w C ; however, in some problems it is necessary to 32 Nonlinear continua dierentiate those eigenvalues and eigenvectors and it is therefore necessary to use an analytical expression of them. As is wellknown the eigenvalues of w C are given by the roots o f the follow- ing polynomial (Strang 1980), s( 2 l )= 6 l + L F 1 4 l L F 2 2 l + L F 3 =0 (l =1> 2> 3) (2.59a) where, (McConnell 1957) L F 1 = wu( w C) (2.59b) L F 2 = 1 2 £ (L F 1 ) 2 wu( w C 2 ) ¤ (2.59c) L F 3 = ghw( w C) > (2.59d) and it is easy to verify the Serrin representation (Simo & Taylor 1991): w * d w * d = w 2 d w b (L F 1 2 d ) w g + L F 3 2 d w b 1 2 4 d L F 1 2 d + L F 3 2 d > (2. 60a) D D = w 2 D w C (L F 1 2 D ) g + L F 3 2 D w C 1 2 4 D L F 1 2 D + L F 3 2 D > (2.60b) with no addition in “a”or “A”in the above equations. According to what we showed in Sect. 2.7.5, d = D for d = D . Example 2.9. JJJJJ To verify Eq. (2.60a)we start from, w b = 2 l w * l w * l w b 1 = 2 l w * l w * l L F 1 = 2 1 + 2 2 + 2 3 L F 2 = 2 1 2 2 + 2 1 2 3 + 2 2 2 3 L F 3 = 2 1 2 2 2 3 = For d =1 w b (L F 1 2 1 ) w g + L F 3 2 1 w b 1 = 2 4 1 L F 1 2 1 +L F 3 2 1 2 1 w * 1 w * 1 = The above verifies Eq. (2.60a) for the case d =1; the demonstrations for d =2> 3 are identical. 2.8 Strain measures 33 To verify Eq. (2.60b) we start from, w C = L 2 L L w C 1 = L 2 L L and proceed as before. JJJJJ It is important to note that Eqs. (2.60a-2.60b) are only valid if the denom- inator of the r.h.s. is not zero. The denominator is zero if we ha ve repeated eigenvalues. 2.8 Strain measures Intheliteraturewecanfind a large num ber of strain measures that have been proposed to characterize a deformation process. There are dierent approac hes for analyzing the deformation of continuum bodies and we usually find that, for a given approach, one particular strain measure may be more suitable than others. In this section we will present a number of these strain measures without making any claim of completeness. 2.8.1 The Green deformation tensor We ha ve already presen ted the Green deformation tensor in Sect. 2.7.1. It is important to remember that this second-order t ensor is defined in the reference configuration and that for two vectors dx 1 and dx 2 defined at a point " in the reference configuration, the corresponding vectors in the spatial configuration satisfy the relation, w dx 1 · w dx 2 = dx 1 · w C · dx 2 = (2.61) 2.8.2 The Finger deformation tensor We have already presented the Finger deformation tensor in Sect. 2.7.3. It is important to remember that this second-order tensor is definedinthespatial configuration. Using Eq. (2.45a), w b 1 = w X W · w X 1 > (2.62) and for two vectors w dx 1 and w dx 2 defined at a point w x in the spatial configuration we can write, 34 Nonlinear continua w dx 1 · w b 1 · w dx 2 = w dx 1 · w X W · dx 2 (2.63a) using Eq. (2.28a), we get w dx 1 · w b 1 · w dx 2 = dx 1 · dx 2 = (2.63b) 2.8.3 The Green-Lagrange deformation tensor From Eq.(2.61), w dx 1 · w dx 2 dx 1 · dx 2 =2 dx 1 · 1 2 ( w C g) ¸ · dx 2 = (2.64) The Green-Lagrange strain tensor is defined in the reference configuration as w % = 1 2 ( w C g) = (2.65) The second order tensor w % describes the deformation corresponding to the w-configuration (spatial configuration) referred to the configuration at w =0 (reference configuration). Example 2.10. JJJJJ Considering a convected coordinate system © l ª with covariant base vectors w e g l in the spatial configuration and e g l in the reference one, we can write, w % = w e% op e g o e g p and it is easy to show that, w e% op = 1 2 h w e g o · w e g p e g o · e g p i = JJJJJ At the point " under study, we now evolve from the w-configuration to a w + w-configuration by means of a rotation w+w w R. Hence, w+w X = w+w w R · w X > (2.66a) w+w C = w X W · w+w w R W · w+w w R · w X > (2.66b) and taking into account that the rotation tensor is orthogonal, w e get 2.8 Strain measures 35 w+w C w C (2.66c) and therefore, w+w % w % = (2.66d) From the above, w e conclude that the Green deformation tensor and the Green -Lagrange strain tensor are not aected by rigid body rotations: that istosaytheyareindierent to rotations. 2.8.4 The Almansi deformation tensor From Eq. (2.63b), we get w dx 1 · w dx 2 dx 1 · dx 2 =2 w dx 1 · 1 2 ( w g w b 1 ) ¸ · w dx 2 = (2.67) The Almansi strain tensor is defined in the spatial configuration as w e = 1 2 ( w g w b 1 ) = (2.68) At the point " under study, we now evolve from the w-configuration to the w + w-configuration b y means of a rotation w+w w R. Hence, w+w X 1 = w X 1 · w+w w R W > (2.69a) and, w+w X W = w+w w R · w X W > (2.69b) using Eq. (2.62), w+w b 1 = w+w w R · w b 1 · w+w w R W > (2.69c) and therefore, w+w e = w+w w R · w e · w+w w R W = (2.69d) Hence, the Finger and Almansi tensors are aectedbyrigid-bodyrotations. 2.8.5 The Hencky deformation tensor The Hencky or logarithmic strain tensor is defined in the reference configura - tion as w H =ln w U = (2.70) When the problem is referred to a fixed Cartesian system using Eq. (2.58e), we get [ w K]=[ ] 5 7 ln 1 0=00=0 0=0ln 2 0=0 0=00=0ln 3 6 8 [] W = (2.71) Obviously, the Hencky deformation tensor is indierent to r otations,since from the polar decomposition, we can see that w U does not incorporate the eect of rigid-body rotations. 36 Nonlinear continua 2.9 Represen tation o f spatial tensors in the reference configuration (“ pull-back” ) For the regular motion depicted in Fig. 2.1, we can define: • An arbitrary curvilinear coordinate system { w { d } in the spatial configura- tion. At a point " ( w { d >d =1> 2> 3) we can determine the covariant base vectors w g d and the contravariant base vectors w g d . • An arbitrary curvilinear coordinate system { { D } in the reference config- uration. At the point " ( { D >D=1> 2> 3) we can determine the covariant base vectors g D and the contravariant base vectors g D . • A convected curvilinear coordinate system { l }.Atthepoint " in the spatial configuration we can determine the covariant base vectors w ˜ g d and the contrava riant base ve ctors w ˜ g d > while in the reference configuration we can determine the covariant base vectors ˜ g d and the contravariant base vectors ˜ g d 2.9.1 Pull-back of vector components Let us consider in the spatial configuration at the point " a vector, w b = w e lw g l = w e l w g l = w ˜ e lw e g l = w ˜ e l w e g l = (2.72) We define in the reference configuration the follow ing vectors (Dv orkin, Goldschmit, Pantuso & Repetto 1994): w B ` = w ˜ e l e g l =[ w B ` ] D g D (2.73a) w B ^ = w ˜ e l e g l =[ w B ^ ] D g D = (2.73b) After some algebra, [ w B ` ] D = w e m ( w [ 1 ) D m (2.74a) [ w B ^ ] D = w e m w [ m D = (2.74b) Adopting the notation used in manifolds analysis (Lang 1972, Marsden & Hughes 1983) we define the pull-back of the c ontravariant components w e m as £ w ! ( w e m ) ¤ D =[ w B ` ] D (2.75a) and the pull-back of the covariant components w e m as: £ w ! ( w e m ) ¤ D =[ w B ^ ] D = (2.75b) We can therefore rewrite Eqs. (2.73a-2.73b) and (2.74a-2.74b), 2.9 Representation of spatial tensors in the reference configuration (“pull-back”) 37 w B ` = w e m ( w [ 1 ) D m g D = w e m ( w [ 1 ) D m j DF g F (2.76a) w B ^ = w e m w [ m D g D = w e m w [ m D j DF g F = (2.76b) For two vectors w b and w w ,defined in the spatial configuration at " , using Eqs. (2.74a-2.74b), it is easy to show that: w B ` · w W ^ = w B ^ · w W ` = w b · w w = (2.77) Also, using Eqs.(2.74a-2.74b) and (2.36a-2.36b), we get [ w B ` ] Ew F DE =[ w B ^ ] D = (2.78) Using Eqs. (2.73a-2.73b), we can write ( w e g d ) ` = e g d (2.79a) ( w e g d ) ^ = e g d = (2.79b) Hence, we use the following notation (Moran, Ortiz & Shih 1990): w ! ( w e g d )=( w e g d ) ` = e g d (2.79c) w ! ( w e g d )=( w e g d ) ^ = e g d = (2.79d) From the above equations, we can get by inspection the geometrical inter- pretation of the vectors w B ` and w B ^ : • If w b,inthespatialconfiguration, is the tangent to a curve w c() at a point ",then w B ` is the tangent, in the reference configuration, to the curve C ()= w ! 1 [ w c()] ,atthepoint ". • In the transformation w B ` $ w b the modulus of the original vector gets stretched as the material fiber to which they are tangent. In convected coordinates we have w dx =d l w e g l (2.80a) dx =d l e g l (2.80b) that is to say, w dX ` = dx = (2.80c) • For two vectors w b and w w that are orthogonal in the spatial configuration it is obvious that: 38 Nonlinear continua Fig. 2.4. Mappings w B ` · w W ^ = w B ^ · w W ` = w b · w w =0= (2.81) Hence, the orthogonality of w b and w w implies the orthogonality of w B ` and w W ^ and the orthogonality of w B ^ and w W ` in the reference configuration. It is important to take into account that in Eqs. (2.74a-2.74b) the terms on the r.h.s. must be written as a function of the coordinates in the reference configuration. We can indicate this using a more formal nomenclature (e.g. Marsden & Hughes 1983), [ w B ` ] D = £ w ! ( w e d ) ¤ D = £ ( w [ 1 ) D d w ! ¤ ( w e d w !) (2.82a) [ w B ^ ] D = £ w ! ( w e d ) ¤ D = w [ d D ( w e d w !) = (2.82b) In order to understand the above equations we use Fig. 2.4 (Marsden & Hughes 1983). In this figure we indicate with “i j” the composition of the mapping “j” followed by the mapping “i”. Example 2.11. JJJJJ For a function i( w { d ) definedinthespatialconfiguration, we can write: 2.9 Representation of spatial tensors in the reference configuration (“pull-back”) 39 di = Ci C w { d w d{ d = In the above equation, we use a formal analogy with vector calculus in which (Marsden & Hughes 1983): di is a vector; Ci C w { d are its covariant components (di d )andg w { d are contravariant base vectors. Hence we can do a pull-back operation, £ w ! (di d ) ¤ D = w [ d D di d = C w { d C { D Ci C w { d = Ci C { D = Using a more formal nomencla ture and the mappings in Fig. 2.7, we get £ w ! (di d ) ¤ D = C(i w !) C { D = JJJJJ Example 2.12. JJJJJ Equation (2.11b) defines, in the spatial configuration, the velocity of a ma- terial point, w v = w y dw g d = Using the expressions for the pull-back of contravariant components, we write £ w ! ( w y d ) ¤ D =( w [ 1 ) D d w y d = Ifthecoordinatesystem{ { D },defined in the reference configuration, is a convected system with covariant base vectors, w ˜g D ,inthespatialconfigura- tion, we can write d{ Dw e g D = w d{ dw g d hence, w g d =( w [ 1 ) D d w e g D and therefore, w v = w y d ( w [ 1 ) D d w e g D = The components of the material velocity vector in the convected system { { D } are, w ˜y D = w y d ( w [ 1 ) D d and therefore £ w ! ( w y d ) ¤ D = w ˜y D = JJJJJ 40 Nonlinear continua 2.9.2 Pull-back of tensor components Let us consider in the spatial configuration at the point " ( w { d >d =1> 2> 3) a second-order tensor, w t = w w lm w g l w g m = w w lm w g lw g m = w w l m w g l w g m = w ˜ w lm w e g l w e g m = w ˜ w lm w e g l w e g m = w ˜ w l m w e g l w e g m = (2.83) We define in the reference configuration the following second-order tensors (Dvorkin, Goldschmit, Pantuso & Repetto 1994): w T ` = w ˜ w lm e g l e g m =[ w T ` ] DE g D g E > (2.84a) w T _ = w ˜ w l m e g l e g m =[ w T _ ] D E g D g E > (2.84b) w T ^ = w ˜ w lm e g l e g m =[ w T ^ ] DE g D g E = (2.84c) After some algebra we get [ w T ` ] DE = w w de ( w [ 1 ) D d ( w [ 1 ) E e > (2.85a) [ w T _ ] D E = w w d e ( w [ 1 ) D d ( w [) e E > (2.85b) [ w T ^ ] DE = w w de ( w [) d D ( w [) e E = (2.85c) We define: • Pull-back of the contravariant components of w t , £ w ! ( w w lm ) ¤ DE =[ w T ` ] DE = (2.86a) • Pull-back of the mixed components of w t , £ w ! ( w w l m ) ¤ D E =[ w T _ ] D E = (2.86b) • Pull-back of the covariant components of w t , £ w ! ( w w lm ) ¤ DE =[ w T ^ ] DE = (2.86c) From Eqs. (2.84a-2.84c) and (2.85a-2.85c) we can write, [...]... pull-backs of the spatial metric tensor are: £ ¤ ) ( £ g = g = C g ¤ ) ( g g = g = C (2.93a) 1 (2.93b) • The pull-back of the Almansi strain tensor is: £ ( ) ¤ g g = E = (2.94a) From Eqs (2.77) (2.80a-2.80c) and (2.91a-2.91c), we get dx · e · dx = dx · • The pull-back of the left stretch tensor is: · dx (2.94b) 44 Nonlinear continua V = (2.95) U • In many problems related to metallurgy (e.g metal-forming... (2.107a) ˆ (2.107b) ˆ (2.107c) ˆ ˆ = ˆ ˆ Between the spatial coordinates of the material particles at and the spatial coordinates of the same particles at ˆ we can define a mapping (ˆ ) (see Eq (2.4)) Hence, we can also define a deformation gradient tensor, ˆX , and we can rewrite Eqs (2.107a-2.107c) as 50 Nonlinear continua = ( = ˆ ˆ = ( ˆ 1 ) ( ˆ ˆ 1 ) 1 ) ˆ ˆ ˆ = ˆ ˆ ˆ h ˆ ˆ = = ˆ (ˆ h ˆ h ˆ ) ˆ ˆ (ˆ... Performing a right polar decomposition on both sides of Eq (2.102c), we get R · U = Q( ) · R · U (2.103a) Taking into account that: The inner (dot) product of two orthogonal second-order tensors is an orthogonal second order tensor The polar decomposition is unique We get R = Q( ) · U = R (2.103b) (2.103c) U Hence, the rotation tensor (two-point tensor) and the right stretch tensor (Lagrangian tensor)... t · w (2.90) in the reference configuration, we can easily verify that the following relations are fulfilled: B = T · W (2.91a) B = T · W (2.91b) B = T · W (2.91c) 42 Nonlinear continua JJJJJ , it is possible to use the compo- Example 2. 13 Instead of using the mixed components , hence nents t = g = ˜ g ˜ g ˜ g we can then define in the reference configuration the second order tensor, T = ˜ ˜ g ˜ g = [T... reference frame (Marsden & Hughes 19 83) , we will use the word covariance to refer to this concept 2.12 Objectivity 45 2.12.1 Reference frame and isometric transformations We call an event the pair { x } formed by a vector x that defines a point in the Euclidean space and a time A reference or observation frame is a way of relating the physical world to the points in an . 2 1 + 2 2 + 2 3 L F 2 = 2 1 2 2 + 2 1 2 3 + 2 2 2 3 L F 3 = 2 1 2 2 2 3 = For d =1 w b (L F 1 2 1 ) w g + L F 3 2 1 w b 1 = 2 4 1 L F 1 2 1 +L F 3 2 1 2 1 w * 1 w * 1 = The. we can write, 34 Nonlinear continua w dx 1 · w b 1 · w dx 2 = w dx 1 · w X W · dx 2 (2.63a) using Eq. (2.28a), we get w dx 1 · w b 1 · w dx 2 = dx 1 · dx 2 = (2.63b) 2.8 .3 The Green-Lagrange. 2 d ) w g + L F 3 2 d w b 1 2 4 d L F 1 2 d + L F 3 2 d > (2. 60a) D D = w 2 D w C (L F 1 2 D ) g + L F 3 2 D w C 1 2 4 D L F 1 2 D + L F 3 2 D >