Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

The enjoyment I experienced in writing this textbook would not have been the same without the moral support of numerous friends. My thanks belong to all of them who tolerated my absence when I disappeared for many evenings and weekends in order to bring the ideas of this fascinating field to you, the reader.

2 1 Introduction to Vectors and Tensors

by u, v, w, ... (or in other texts frequently designated by ''i',~' 1!!' ..., or fi, 'V, 'lV, ...), is a directed line element in space. It is a model for physical quantities having both direction and length, for example,force, velocity or acceleration. Two vectors that have the same direction and length are said to be equal.

The sum of vectors yields a new vector, based on the parallelogram law of addi-

tion. The following properties

hold, where o denotes the unique zero vector with unspecified direction and zero length.

Let u be a vector and a be a real number (a scalar). Then the scalar multiplication au produces a new vector with the same direction as u if a > 0 or with the opposite direction to u if n < 0.

Further properties are:

The quantity lul (or !lull) is called the length (or norm or magnitude) of a vector

u, which is a non-negative real number. It is defined by the square root of u · u, i.e.

Thus, using ( l .10) we find the projection of a vector u along the direction of a

u

\0 l

-~ -""------ : ~------·-

e

Figure 1.1 Projection of u along a unit vector e.

Index notation. So far algebra has been presented in symbolic (or direct or abso-

lute) notation exclusively employing bold-face letters. It represents a very convenient and concise tool to manipulate most of the relations used in continuum mechanics. It will be the preferred representation in this text. However, particularly in computational mechanics, it is essential to refer vector (and tensor) quantities to a basis. Additionally, to gain more insight in some quantities and to carry out mathematical operations among (higher-order) tensors more readily (see next section) it is often helpful to refer to components.

In order to present coordinate (or component) expressions relative to a righthanded (or dextral) and orthonormal system we introduce a fixed set of three basis vectors e1 , e2 , c:i (sometimes introduced as i,j, k), called a (Cartesian) basis, with properties

These vectors of unit length which are mutually orthogonal form a so-called orthonormal system.

4 1 Introduction to Vectors and Tensors

Then any vector u in the three-dimensional Euclidean space is represented uniquely by a linear combination of the basis vectors e1, e2 , e3 , i.e.

( 1.19)

where the three real numbers u1 , 'U , u3 are the uniquely determined Cartesian (or rect-

2

angular) components of vector u along the given directions e1 , e2 , e:1, respectively (see Figure 1.2). The components of e1 , e2 , e3 are (1, 0, 0), (0, 1, 0), (0, 0, 1), respectively.

u

'lL]

Figure 1.2 Vector u with its Cartesian components u 1 , u2 , u3 •

Using index (or subscript or suffix) notation, relation (l.19) can be written as u = E~=t uiei or, in an abbreviated form by leaving out the summation symbol, simply

where we have adopt the summation convention, invented by Einstein. The summation convention says that whenever an index is repeated (only once) in the same term, then, a summation over the range of this index is implied unless otherwise indicated. We consider only the three-dimensional Euclidean space, which we characterize by means of Latin indices i, j, k ... running over 1, 2, 3. We denote the basis vectors by

{ ei he{ 1,2,3} collectively. Subsequently, in this text, the braces {•} will stand for a fixed set of basis vectors and the symbol • for any tensor element.

The index i that is summed over is said to be a dummy (or summation) index, since a replacement by any other symbol does not affect the value of the sum. An index that is not summed over in a given term is called a free (or live) index. Note that in the same equation an index is either dummy or free. Thus, relations ( 1.18) can be

hold. Note that 8ii also serves as a replacement operator; for example, the index on

'Ui becomes an j when the components 'Ui are multiplied by oii·

The projection of a vector u = ·u.iei onto the basis vectors ei yields the j-th compo-

Thus, from eq. ( 1.20) and properties ( 1.13), ( 1.21) and ( l.22h we have

( 1.23) Taking the basis {ei} and eqs. (l.13), (l.20), (l.21) and (l.22)2, the component expres-

sion for the dot product (I. I 0) gives

which is commonly used as the definition of the dot product. Thus, we may derive the dot product of u and v without knowledge of the angle between u and v.

In an analogous manner, the component expression for the square of the length of u, i.e. ( 1.15), is

Note that in eqs. ( 1.24) and ( 1.25) one index is repeated, indicating summation over

1, 2, 3. In symbolic notation this is indicated by one dot.

Cross product. The cross (or vector) product of u and v, denoted by u xv (in the literature also u /\ v), produces a new vector. The cross product is not commutative. It is defined as

6 1 Introduction to Vectors and Tensors

If relation (1.27) holds with u and v assumed to be nonzero vectors, we say that vector u is parallel to vector v. From eq. (1.29) we learn that

It characterizes the area of a parallelogram spanned by the vectors u and v (see Figure 1.3). The right-handed cross product of u and v, i.e. the vector u x v, is perpendicular to the plane spanned by u and v (0 is the angle between u and v).

uxv

v

.~ Area= lox vi

I

I

I

I

I

I

: u

I

I

f

-u xv= v x u

Figure 1.3 Cross product of vectors u and v.

In order to express the cross product in terms of components we introduce the

permutation (or alternating or Levi-Civita e-) symbol C.ijk' which is defined as

or in a more convenient short-hand notation, with ( 1.33),

( 1.35)

With relations ( 1.2 I), ( 1.34), (1.35) it is easy to verify that EiJk may be expressed as the determinant of a matrix

( 1.36)

where we have introduced the square brackets [•] for a matrix. The product of the

permutation symbols is related to the Kronecker delta by the relation

( 1.37)

With (1.22) 1 and (l.22h we deduce from (l.37) the important relations

Eijk Eijk = 6 . ( 1.38)

EXAMPLE 1.1 Obtain the coordinate expression for the cross product w = u x v.

Solution. Taking advantage of eqs. (1.20) and ( 1.35) we find that

(l .39)

with the three components

( 1.40)

( l.41)

( 1.42)

Now, using expressions ( 1.36), ( 1.39). , the vector product u x v relative to {ek} 1

may be written as

8 1 Introduction to Vectors and Tensors

The triple scalar (or box) product (u xv)· w represents the volume F of a parallelepiped 'spanned' by u, v, w forming a right-handed triad (see Figure 1.4).

uxv

w

v

Volume= (u xv) ·w

u

Figure 1.4 Triple scalar product.

By recalling definitions ( 1.29), ( 1.10) and ( 1.32), we have

base area height

Using index notation, then, from eqs. ( 1.20) and (1.35) we find with ( 1.2 I) the volume

1·'"to be

l" = (u x v) · w = Eijk'lli'VfWk

Hence, the triple scalar product ( 1.45):1 can be written in the convenient determinant form

Note that the vectors u, v, w are linearly dependent if and only if their triple scalar product vanishes (the parallelepiped has no volume).

The product u x (v x w) is called the triple vector product and may be verified with ( 1.39).i, ( 1.38)1, ( 1.22h and representations ( 1.20) and ( 1.24)4