Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

tensor A in terms of the three Cartesian basis vectors, we write

(l.147)

in which Aijkl are the components of A.

One example for a fourth-order tensor is the product of the four vectors u, v, w, x,

With these properties we can show that the components Aiikl of A can be expressed as

(1.150)

Hence, the double contraction of A with a second-order tensor B is a second-order tensor, denoted by A : B. With representations (I. I47), (1.59), and properties (1.149), ( 1.21) and (1.61) we find that

Following the definitions above and by analogy with (1.53), (1.56) and (1.57), note the important rules

EXAMPLE 1.5 Show eq. (l.152) which is the analogue of eq. (l.53).

Solution. Express tensor A, B, C in terms of their basis vectors according to ( 1.59).

With properties (1.148), (1.149), (1.21) and (1.61) we find that

(A® B): C = AijBktCmn(ei ® ej) ® (ek ® e,): (em® en)

=AijBklCmn(ek · em)(e, · en)(ei ® ej)

=AiiBktCmn8km81nei ® ei

A further example for a fourth-order tensor is the tensor product D = A ® B, while

A and Bare second-order tensors with components Aii and Bii, respectively. In index notation we may write Dijkl = AiiBkt·

The unique transpose of a fourth-order tensor A denoted by AT is governed, by

hold. From the identity ( 1.157) we deduce the index relation, namely (AT) iJkl = AkliJ.

We now define fourth-order unit tensors Il and Il so that

A=Il:A, AT=Il:A' ( 1.160)

for any second-order tensor A. These transformations assign to any A the tensor A itself, and its transpose, respectively. The fourth-order unit tensors may be represented by

The deviatoric part of a second-order tensor A may be described by means of a fourth-order tensor. From relations ( 1.13 I), ( 1.132h and transformations ( 1. I 60) 1,

(1.152) we conclude that

or with the definition of a fourth-order projection tensor P

The components of devA and A are related through the expression elev Aii = Pijkl Akt, with Pijkl

(observe the similar structure to eq. ( 1.1I2)). Show that

which assign to any second-order tensor A its symmetric and skew parts, respectively.

1.4 Eigenvalues, Eigenvectors of Tensors

The scalars Ai characterize eigenvalues (or principal values) of a tensor A if there exist corresponding nonzero normalized eigenvectors Di (or principal directions or principal axes) of A, so that

To identify the eigenvectors of a tensor, we use subsequently a hat on the vector quantity concerned, for example, ft.

Thus, a set of homogeneous algebraic equations for the unknown eigenvalues ,.\,

Eigenvalues characterize the physical nature of a tensor. They do not depend on coordinates. For a positive definite symmetric tensor A, all eigenvalues ,.\i are (real and) positive since, using (1.165), we have,.\ =Di·Ani > 0, i = 1, 2, 3. Moreover, the set of eigenvectors of a symmetric tensor A form a mutually orthogonal (orthonormal) basis {ni} (the proof of this statement is omitted).

Principal scalar invariants. For the system ( 1.166) to have solutions ni # o the determinant of the system must vanish. Thus,

(1.167)

where

( 1.168)

This requires that we solve a cubic equation in ,.\, usually written as

( 1.169)

called the characteristic polynomial (or equation) for A, the solutions of which are the eigenvalues Ai, i = 1, 2, 3.

Here, Ii(A), i = 1, 2, 3, are the so-called principal scalar invariants of A. In

for any positive integer a. Using this relation and ( 1.169) multiplied by Di, we obtain the well-known

(1.174)

It states that every (second-order) tensor A satisfies its own characteristic equation.

Spectral decomposition of a tensor. Any symmetric tensor A may be represented by its eigenvalues ,.\i, i = 1, 2, 3, and the corresponding eigenvectors of A forming an

26 1 Introduction to Vectors and Tensors

orthonormal basis {Di}· Using the unittensor, by analogy with ( l.65h, i.e. I= Di 0Di, and relations (1.58), (1.165) and (l.55) we obtain an expression which is known as the spectral decomposition (or spectral representation) of A, i.e.

i=l

The components Aii of tensor A relative to a basis of principal directions follow with ( 1.59) by replacing ei with the three orthonormal basis vectors Di. With

eqs. (l.165) and (1.21) we obtain

where the diagonal elements are the eigenvalues of A. This result may be obtained directly from the spectral decomposition (1.175) of A.

For a pair of equal roots, i.e. ,.\ 1 = ,.\2 # ,.\3 ( ,.\ 1 = ,.\2 == ,.\), with one unique eigenvector D3 associated with ,.\3 , we deduce from (1.175) that

(l.178)

where P~3, Pt3 denote projection tensors introduced in (1.125) and (1.126). In (1.178)

where every direction is a principal direction and every set of mutually orthogonal basis denotes principal axes.

EXERCISES

1.Show that the quantities in eqs. ( 1.170)-( 1.172), i.e. / 1, / 2 , / 3 , are indeed principal invariants of a tensor A under an orthogonal transformation of coordinates.

(b)Derive the associated normalized eigenvectors Di, i = 1, 2, 3, which in matrix representation read

3.Let a non-symmetric tensor A be given by the tensor product e1 ® e:h with the property e 1 • e2 = 0. Determine the eigenvalues and eigenvectors of tensor A.

4.Let A be a given second-order tensor. Show that the invariants of its deviatoric part, according to ( 1.163), are

Hint: To derive eq. ( 1.181) apply the Cayley-Hamilto11 equation in the form of

(devA) 3 - lc1evA (devA) 2 + IldevA devA - Illc1evA I= 0.

1.5Transformation Laws for Basis Vectors and Components

Consider two sets of mutually orthogonal basis vectors which share a common origin. They correspond to a 'new' and an 'old' (original) Cartesian coordinate system which we assume to be right-handed characterized by the two sets {ei} and {ei} of basis vectors, respectively. Hence, the new coordinate system may be obtained from the original one by a rotation of the basis vectors ei about their origin (see Figure 1.7). We find from ( 1.10) that the cosine of the angle between ei and ei is represented by Qii = cosO(ei, ei) = ei · ei. Note that the first index on Qii indicates the 'old' components whereas the second index holds for the 'new' components.

'

'

O(c2, c:!)'\

'~

'

'

'~ ' -

~C2

Figure 1.7 Rotation of mutually orthogonal basis vectors ei into ei.

(_

It is worthwhile to mention that vectors and tensors themselves remain invariant upon a change of basis - they are said to be independent of any coordinate system. However, their respective components do depend upon the coordinate system introduced, which is arbitrary. The components change their magnitudes by a rotation

(and/or reflection) of the basis vectors, but are independent of any translation (for the

characterization of a translation, see p. 62).

We now set up the transformation laws for various components of vectors and

tensors under a change of basis. Following the considerations above we have

where Q denotes an orthogonal tensor (seep. 16), with components QiJ which are the same in either basis. The components describe the orientation of the two sets of basis