A Introduction to tensor analysis In this Appendix, assuming that the reader is acquainted with vector analy- sis, we present a short introduction to tensor analysis. However, since tensor analysis is a fundamental tool for understanding continuum mechanics, we strongly recommend a deeper study of this subject. Some of the books that can be used for that purpose are: (Synge & Schild 1949, McConnell 1957, Santaló 1961, Aris 1962, Sokolniko 1964, Fung 1965, Green & Zerna 1968, Flügge 1972, Chapelle & Bathe 2003). A.1 Coordinates transformation Let us assume that in a three-dimensional space ( 3) we can define a system of Cartesian coordinates: we call this space the Euclidean< space. In this Appendix we will restrict our presentation to the case of the Eu- clidean space. In the 3 spacewedefine a system of Cartesian coordinates } >= 1> 2> 3 , and< an arbitrary system of curvilinear coordinates l >l{=1> 2> 3 . The following} relations hold: { } l = l(} >=1> 2> 3) >l=1> 2> 3 = (A.1) The above functions are single-valued, continuous and with continuous first derivatives. We call M the Jacobian of the coordinates transformation defined by Eq. (A.1). Hence Cl = (A.2) M = "C} # An admissible transformation is one in which det M =0,thatistosay, a transformation where a region of nonzero volume in one6 system does not collapse into a point in the other system and vice versa. 214 Nonlinear continua A proper transformation is an admissible transformation in which det MA 0. A.1.1 Contravariant transformation rule From Eq. (A.1) we obtain Cl dl = d} = (A.3) C} When the coordinates system is changed, the mathematical entities dl at a certain point of 3 that transform following the same rule as does the coordinate dierentials< (Eq. (A.3)) are said to transform according to a con- travariant transformation rule. We indicate these mathematical entities using upper indices. l Now we consider two systems of curvilinear coordinates l and ˆ , related by the following equations: { } © ª l l ˆ = ˆ (m >m=1> 2> 3) >l=1> 2> 3 (A.4a) and o n = n(ˆ >o=1> 2> 3) >n=1> 2> 3 = (A.4b) We can write the coordinate dierentials as: l l C ˆm d = m d (A.4c) Cˆ l l Cˆ dˆ = dm = (A.4d) Cm In the same way, a contravariant mathematical entity can be defined in either of the two systems dl = dl(m >m=1> 2> 3) >l=1> 2> 3 (A.4e) m dˆl =ˆdl(ˆ >m=1> 2> 3) >l=1> 2> 3 (A.4f) and we transform it from one curvilinear system to the other following a trans- formation rule similar to the transformation rule followed by the coordinate dierentials: l l C m d = m dˆ >l=1> 2> 3 (A.4g) Cˆ l Cˆ dˆl = dm >l=1> 2> 3 = (A.4h) Cm A.2 Vectors 215 Although the contravariant transformation rule applies to dl and not to l, using a notation abuse, we follow the convention of using upper indices for the coordinates. A.1.2 Covariant transformation rule Given an arbitrary continuous and dierentiable function i(1>2>3) and using the chain rule, we write Ci Ci Cm l = m l >l=1> 2> 3 = (A.5a) Cˆ C Cˆ We define Ci dm = >m=1> 2> 3 = (A.5b) Cm l In the ˆ coordinate system we define { } Ci dˆm = m >m=1> 2> 3 (A.5c) Cˆ Cm dˆl = dm l >l=1> 2> 3 (A.5d) Cˆ m Cˆ dl =ˆdm >l=1> 2> 3 = (A.5e) Cl When the coordinates system is changed, the mathematical entities dl at a certain point of 3that transform following the same rule as does the derivatives of a scalar< function (Eqs. (A.5d) and (A.5e)) are said to transform accordingtoacovariant transformation rule. We indicate those mathematical entities using lower indices. A.2 Vectors There are some physical properties like mass, temperature, concentration of a given substance, etc., whose values do not change when the coordinate system used to describe the problem is changed. These variables are referred to as scalars. On the other hand, there are other physical variables like velocity, accel- eration, force, etc. that do not change their intensity and direction when the coordinate system used to describe the problem is changed. They are called vectors. In what follows, we will make use of the above intuitive definition of scalars and vectors. However, in Sect. A.4 we will see that they represent two partic- ular kinds of tensors (order 0 and 1, respectively). 216 Nonlinear continua A.2.1 Base vectors Asetofn linearly independent vectors is a basis of the space q and any othervectorin q can be constructed as a linear combination of< those base vectors. < Let us consider the three linearly independent vectors g (l =1> 2> 3) in l 3. Any vector v inthesamespacecanbewrittenas: < v = yl g = (A.6) l The mathematical entities yl (l =1> 2> 3) are the components of v in the basis g (l =1> 2> 3). l Example A.1. JJJJJ In a Cartesian system } >=1> 2> 3 thebasevectorsare { } e1 =(1> 0> 0) > e2 =(0> 1> 0) > e3 =(0> 0> 1) > where we have indicated the projection of the base vectors on the Cartesian axes. The position vector r of a point S in 3 is < r = } e = Hence, dr = d} e > but also, Cr dr = d} = C} Therefore, we get Cr e = >=1> 2> 3 = C} JJJJJ A.2.2 Covariant base vectors In the arbitrary curvilinear system l >l=1> 2> 3 we can write, at any point S of the space, { } Cr dr = dl = (A.7a) Cl Since A.2 Vectors 217 Fig. A.1. Covariant base vectors at a point P dr = dlg (A.7b) l we obtain Cr g = >=1> 2> 3 = (A.7c) l Cl The vectors g ,defined with the above equation, are the covariant base l vectors of the curvilinear coordinate system l at the point S . From its definition, the vector g is tangent{ to} the line, = ( ) and 1 2 2 S 3 = 3 (S ) . Similar conclusions can be reached for the covariant base vectors g and 2 g . 3 In a Cartesian system, we can write Eq. (A.7c) as: C} g = e >l=1> 2> 3 = (A.8) l Cl l In a second curvilinear system ˆ >l=1> 2> 3 , { } 218 Nonlinear continua m l Cˆ dr = dˆ gˆ = dl gˆ = (A.9a) l l C m Hence, we have m Cˆ g = gˆ >l=1> 2> 3 = (A.9b) l Cl m Due to the similarity between Eqs. (A.9b) and Eq. (A.5e) the base vectors g are called covariant base vectors. l A.2.3 Contravariant base vectors In an arbitrary curvilinear coordinate system l >l=1> 2> 3 we define the contravariant base vectors (dual basis) (gl >l{=1> 2> 3) with the} equation gl g = l > (A.10) · m m where the dot indicates a scalar product (“dot product”) between two vectors l l l and m is the Kronecker delta (m =1for l = m and m =0for l = m ). l 6 Defining in 3 two curvilinear systems l and ˆ and using Eq. (A.9b), we obtain < { } { } Cp gˆl gˆ = gˆl g = l = (A.11a) m m p m · · Cˆ Hence, using Eq. (A.10), we obtain Cp gl g = gˆl g = (A.11b) m m p · · Cˆ If we define l Cˆ gˆl = go >l=1> 2> 3 (A.11c) Co from Eq. (A.11b), we obtain l Cˆ Cp gl g = go g (A.11d) m o m p · C Cˆ · and l l Cˆ Cp Cˆ gl g = o = = l > (A.11e) m o m p m m · C Cˆ Cˆ where we can see that the relation (A.11a) is satisfied. Therefore, Eq.(A.11c) can be considered the transformation rule for the contravariant base vectors. Due to the similarity between Eqs. (A.11c) and (A.4h) the base vectors gl are called contravariant base vectors. A.3 Metric of a coordinates system 219 Fig. A.2. Covariant and contravariant base vectors In Fig.A.2 we represent in a space 2,atapointS ,thecovariantand contravariant base vectors, in order to provide< the reader with a useful geo- metrical insight. It is important to note that in a Cartesian system the covariant and contravariant base vectors are coincident. A.3 Metric of a coordinates system If a position vector r defines a point S in 3 and a position vector (r +dr) < defines a neighboring point S 0, the distance between these points is given by dv = dr dr = (A.12) · p A.3.1 Cartesian coordinates In a Cartesian system in 3,wehave < dv2 =d} d} (e e ) = (A.13a) · 220 Nonlinear continua If we call e e = > then · 2 dv =d} d} = (A.13b) We call the nine number the Cartesian components of the metric ten- sor at S (this notation will be clarified in Sect. A.4.3). From its definition, it is obvious that =1if = and =0 if = . 6 A.3.2 Curvilinear coordinates. Covariant metric components In an arbitrary curvilinear system l >l=1> 2> 3 the distance between S { } and S 0 is given by dv2 =dr dr =dl dm (g g ) = (A.14a) · l · m We call j = j = g g (A.14b) lm ml l · m the covariant components of the metric tensor at S in the curvilinear system l (this notation will be clarified in Section A.4.3.).