1.1 The Reference State

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Continuum mechanics deals with the change of field variables due to external actions.

Examples of field variables are displacements, stresses, temperatures and magnetic induction.

Actions include mechanical forces, heating, and so on. In general, a reference state is

chosen with respect to which the change of field variables is measured. Let the fields of

interest be defined in the reference state in a set of points, the so-called material points,

occupying a volume V0 with a surface A0 in Eucledian space R3 (Figure 1.1). Assume that

the reference space is described by a set of curvilinear coordinates _XK_K=1,2,3 related to

a rectangular system {ZK}K=1,2,3 by

ZK = ZK(X1,X2,X3). (1.1)

Coordinates in the reference state are also called material coordinates. Consider an infinitesimal

vector dX. One can write

dX = ∂X

∂ZK

dZK (1.2)

(summation over repeated indices).

IK = ∂X

∂ZK

(1.3)

is a set of basis vectors in the rectangular system. Accordingly, IK,K = 1, 2, 3 do not

depend on ZK. In an analogous way, one can write

dX = ∂X

∂XK

dXK. (1.4)

The Finite Element Method for Three-dimensional Thermomechanical Applications Guido Dhondt

2004 John Wiley & Sons, Ltd ISBN: 0-470-85752-8

2 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

X1

X2

X3

Z1

Z2

Z3 G1

G3 G2

I 1

I 2 I 3

X

dX

A0

V0

Figure 1.1 Material coordinate systems

The vectors

GK = ∂X/∂XK (1.5)

constitute a basis in the curvilinear coordinate system. One can write (compare Equation (1.2)

with Equation (1.4))

GK dXK = IL dZL (1.6)

or

GK = ∂ZL

∂XK

IL. (1.7)

The size dS of a vector dX is defined as

dS2 := dX · dX (1.8)

where the “·” denotes the inner product of two vectors (also called the dot product or the

contraction of two vectors). In rectangular coordinates, one finds (substitute Equation (1.2)

into Equation (1.8))

dS2 = IK dZK · IL dZL

= dZK dZLIK · IL

=: dZK dZLIKL. (1.9)

The metric tensor IKL takes the value 1 for K = L and 0 for K _= L. In curvilinear

coordinates, one obtains (substitute Equation (1.4) into Equation (1.8)),

dS2 = GK dXK · GL dXL

= dXK dXLGK · GL

=: dXK dXLGKL (1.10)

DISPLACEMENTS, STRAIN, STRESS AND ENERGY 3

GKL is called the metric tensor for the coordinate system {XK}. In general, GKL _= 0

for K _= L, and GKL _= 1 for K = L. Thus, the basis vectors GK are not necessarily

orthonormal. Using the set {GK}, one can define another set {GL} through the relations

GK · GL = δ L

K (1.11)

where δ L

K

= 0 for K _= 0 and δ L

K

= 1 for K = L. In modern Riemannian geometry, {GL}

are called one-forms (or covariant tensors of rank 1 ). They map the vectors {GK} (which

are also called contravariant tensors of rank 1 ) into a scalar by Equation (1.11). {GL}

forms a basis for the vector space of one-forms and is also called the dual basis of {Gk}.

If α is a one-form, one writes

α = αKGL. (1.12)

The dot product of a vector V and a one-form α is defined by

V · α = V KGK · αLGL = V KαK (1.13)

through Equation (1.11). In the same way, the dot product of two vectors and two one-forms

yields

V ·W = V KGK · WLGL = V KWLGKL (1.14)

α · β = αKGK · βLGL = αKβLGKL (1.15)

where GKL is defined by

GKL := GK · GL. (1.16)

Notice that in Equations (1.13), (1.14) and (1.15) the same symbol is used for the dot

product. The context shows whether a (covariant or contravariant) metric tensor is needed.

Multiplying a vector V with the one-form GL yields

V · GL = V KGK · GL = V Kδ L

K

= V L. (1.17)

Thus, the components V L of V can be obtained by taking the scalar product of V with the

basis one-form GL. Hence,

V = (V · GL)GL. (1.18)

Similar statements to Equation (1.17) and Equation (1.18) can be made on the basis of

one-forms:

α · GL = αL (1.19)

α = (α · GL)GL. (1.20)

Although the separation of tensors of rank one into vectors and one-forms is instructive

from a theoretical point of view, there is no reason why a vector cannot be written in terms

4 DISPLACEMENTS, STRAIN, STRESS AND ENERGY

of a contravariant basis or a one-form in terms of a covariant basis. Substituting GK in

Equation (1.18) and GK in Equation (1.19), one obtains

GK = GKLGL (1.21)

GK = GKLGL. (1.22)

The operation in Equation (1.21) and in Equation (1.22) is called raising and lowering of

the index respectively. As we will see later on, some fields are naturally represented by

covariant tensors (such as the Lagrangian strain and normals on a plane), whereas others are

predestinate for a contravariant representation (such as stresses and normals in a direction).

They can be viewed as dual fields.

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