Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

The triple vector product is, in general, not associative, i.e. (u x v) x w ¥= u x (v x w).

EXERCISES

1. Use the properties ( 1.5), ( 1.6), ( 1.8) and ( 1.9) to show that

1.2 Algebra of Tensors

A second-order tensor A, denoted by A, B, C ... (or in the literature sometimes written as A, B, C .. .), may be thought of as a linear operator that acts on a vector u generating a vector v. Thus, we may write

which defines a linear transformation that assigns a vector v to each vector u. Since A is linear we have

for all vectors u, v and all scalars a. Since most tensors used in this text are of order two, we shall often omit the adjective 'second-order'.

where u denotes an arbitrary vector. The important second-order unit (or identity) tensor I and the second-order zero tensor 0 are defined, respectively, by the relations

Io = ul = u and Ou = uO = o for all vectors u. Note that tensor 0 maps every u to the zero vector o.

Tensor product. The tensor (or direct or matrix) product or the dyad of the

vectors u and v is denoted by u ® v (some authors use the notation uv). It is a secondorder tensor which linearly tran~jorms a vector w into a vector with the direction of u following the rule

( 1.53)

The dyad, not to be confused with the dot or cross product, has the linearity property

A dyadic is a linear combination of dyads with scalar coefficients, for example, v) + (J(w 0 x). Note further that no tensor may be expressed as a single tensor

product, in general, A = u ® v + w ® x # y ® z.

Any second-order tensor may be expressed as a dyadic. As an example, the secondorder tensor A may be represented by a linear combination of dyads formed by the (Cartesian) basis {ei}, i.e.

( 1.59)

We call A, which is resolved along basis vectors that are orthonormal, a Cartesian tensor of order two (more general tensors will be considered in Section 1.6). The nine

Cartesian components of A with respect to {ei}, represented by Ai1, form the entries of the matrix [A], i.e.

A:11

where by analogy with ( 1.22h

( 1.61)

holds. Relation ( 1.60) is known as the matrix notation of tensor A.

EXAMPLE 1.2 Let A be a Cartesian tensor of order two. Show that the projection of A onto the orthonormal basis vectors ei is according to

( 1.62)

where Aii are the nine Cartesian components of tensor A.

Solution. Using representation ( 1.59) and properties (1.53h, (1.2 I) and ( 1.61 ), we find that

On taking the dot product of ( 1.63),1 with ei, the nine Cartesian components Aii are completely determined, namely

ei · Aei = ei · A11e1 = Atj(ei ·e,)

If the relation v · Av > 0 holds for all vectors v # o, then A is said to be a positive semi-definite tensor. If the stronger condition v · Av > 0 holds for all nonzero vectors v, then A is said to be a positive definite tensor. Tensor A is called negative semidefinite if v · Av < 0 and negative definite if v · Av < 0 for all vectors v # o,

respectively.

The Cartesian components of the unit tensor I form the Kronecker delta symbol introduced in eq. ( 1.21 ). Thus,

( 1.65)

The summation, multiplication by scalars and dot products of tensors are governed mainly by properties known from ordinary arithmetic, for example,

Note that, in general, the dot product of second-order tensors is not commutative, i.e.

AB # BA and also Au # uA. Moreover, relations AB = 0 and Au = o do not, in general, imply that A, B or u are zero.

The components of the dot product AB along an orthonormal basis {ei }, as intro-

duced in ( 1.18), read, by means of representation ( 1.62) and relations (I.70), (1.63 ).i,

or equivalently

( 1.80)

For convenience, we adopt the convention that a repetition of only one index between a tensor and a vector (see, for example, eq. (1.68) with the dummy index j) or between two tensors (see, for example, eq. ( 1.80) 1 with the dummy index k) will not be indicated by a dot when symbolic notation is applied. Specifically that means for v = Au (vi = AifUj) and A == BC (Aii = BikCki) we do not write v = A· u and A = B · C, respectively.

Transpose of a tensor. The unique transpose of a second-order tensor A denoted by AT is governed by the identity

for all vectors u and v. The useful properties

(l.82)

for notational convenience.

Orthogonal tensor. An orthogonal tensor Q is a linear transformation satisfying the condition

for all vectors u and v (see Figure 1.5). As can be seen, the dot product u · v is invariant during that transformation, which means that both the angle () between u and v and the lengths of the vectors, lnl, lvl, are preserved.

u

Qu

Q

lul

v

Qv

Figure 1.5 Orthogonal tensor.

Hence, an orthogonal tensor has the property QTQ = QQT = I, which means that

Q- 1 = QT. Another important property is that det(QTQ) = (detQ) 2 with detQ =

± 1. If detQ = +1 (-1), Q is said to be proper (improper) orthogonal corresponding to a rotation (reflection), respectively.

Symmetric and skew tensors. Any tensor

into a symmetric tensor, here denoted by S, and a skew (or antisymmetric) tensor, here denoted by W. Hence, A = S + W, while

(l.112)

In this text, we also use the notation symA for Sand skewA for W. Tensors Sand W are governed by properties such as

terizing the projection of o onto the line spanned by e and onto the plane normal to e, respectively.

u

where P~ and P~ are projection tensors of order two. The projection tensor P~ applied to any vector o maps o into the direction of e, while Pt applied to o gives the projection of o onto the plane normal to e (see Figure 1.6).