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Table of contents

1 Preliminaries 1

1.1 Continuum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Questions of notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Coordinate and component transformations . . . . . . . . . . . . . . . . . . 3

1.4 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Tensor analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Time derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Conservation laws and stress 12

2.1 Mass density and equation of continuity . . . . . . . . . . . . . . . . . . . . 13

2.2 Forces and conservation of linear momentum . . . . . . . . . . . . . . . . . 15

2.3 Stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Center of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Conservation of angular momentum . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.8 Eigenvalue problem for symmetric tensors . . . . . . . . . . . . . . . . . . . 23

2.9 Normal and shear stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Deformation 293.1 Finite deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Infinitesimal deformation or small strain . . . . . . . . . . . . . . . . . . . . 34

3.3 Geometric interpretation of small strain . . . . . . . . . . . . . . . . . . . . 36

3.4 Rate of deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5 Principal strains and special deformations . . . . . . . . . . . . . . . . . . . 40

3.6 Compatibility conditions for small strain . . . . . . . . . . . . . . . . . . . . 42

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4 Constitutive relations 44

4.1 Youngs experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Frame-indifference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 Isotropy and Reiner-Rivlin fluids . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 Linear isotropic media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.5 Viscoelastic media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.6 Kramers-Kronigs relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 Ideal fluids 63

5.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 Simplifying hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 Bernoullis law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4 Solar wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.5 Hydrostatic equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.6 Sound waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.7 Linear waves in magnetized plasmas . . . . . . . . . . . . . . . . . . . . . . 81

5.8 Large amplitude ion-acoustic solitons . . . . . . . . . . . . . . . . . . . . . 88

5.9 Large amplitude circularly polarized waves in plasmas . . . . . . . . . . . . 935.10 Newtonian cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.11 Complex potential theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6 Viscous fluids 120

6.1 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.3 Turbulence : Reynolds equations . . . . . . . . . . . . . . . . . . . . . . . . 128

6.4 Turbulence : Kolmogorovs law . . . . . . . . . . . . . . . . . . . . . . . . . 1326.5 Matched asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 136

6.6 Boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7 Linear elasticity 145

7.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.2 Elastostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.3 Elastic waves in general linear media . . . . . . . . . . . . . . . . . . . . . . 149

7.4 Elastic waves in linear isotropic media . . . . . . . . . . . . . . . . . . . . . 151


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8 Epilogue: Kinetic theory 155

8.1 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

8.2 Macroscopic averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.3 Macroscopic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159


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List of figures

2.1 Sequence of domains enclosing P. . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Surface elements around P. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Comparison of initial and final states. . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Decomposition of neighbouring displacements. . . . . . . . . . . . . . . . . . 35

3.3 Deformation of angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Velocities of neighbouring particles. . . . . . . . . . . . . . . . . . . . . . . . 38

4.1 Youngs experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Contour of integration in the complex -plane. . . . . . . . . . . . . . . . . . 60

5.1 Pressure-density relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2 Streamlines and the velocity field. . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3 Steady outflow of a tank. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.4 Hydrostatic equilibrium of a star. . . . . . . . . . . . . . . . . . . . . . . . . 77

5.5 Propagation of radio signals through the ionosphere. . . . . . . . . . . . . . . 87

5.6 Profile of a solitary wave joining constant states at = and localized in . 935.7 Two observers O and O and a distant galaxy P. . . . . . . . . . . . . . . . . 98

5.8 Flow of a source or sink in the origin. . . . . . . . . . . . . . . . . . . . . . . 108

5.9 Flow around a point vortex in the origin. . . . . . . . . . . . . . . . . . . . . 109

5.10 Flow around a circle or cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . 110

5. 11 Fl ow al ong an edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.12 F low in a corner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.1 Reynolds experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.2 Velocity diagram in turbulent flow. . . . . . . . . . . . . . . . . . . . . . . . 129

6.3 Comparison between turbulent and Hagen-Poiseuille velocity profiles. . . . . 133

6.4 Energy spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.5 Power law in the inertial range. . . . . . . . . . . . . . . . . . . . . . . . . . 136

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6.6 Graphs of the exact solution and the inner and outer approximations . . . . 140

6.7 Velocity profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.1 P- and S-earthquake waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8.1 Description in phase space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156


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Chapter 1

Preliminaries

In this chapter we

give a short introduction to continuum mechanics recall concepts from coordinate transformations, vector and tensor algebra

and analysis

finally address time derivatives of material quantities, both in punctual asin integral form

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1.1 Continuum mechanics

The basic idea behind continuum mechanics is to describe the kinematical and dynamicalproperties of certain classes of material as if the material under consideration were fillingpart of physical space in a continuous way and had characteristics which depend continuouslyupon space and time. Of course, this is a model, an idealization if one thinks of the molecularor (sub)atomic structure of all matter. On the microscopic scale much more discontinuitythan continuity is to be found. Nevertheless, experience has learned that many macroscopicproperties can be described with an amazing degree of precision by indeed pretending thatcertain materials are continua, in the mathematical sense of the word. To quote just oneexample at this stage, for the flow of water through a pipe the knowledge of the molecularstructure of water will learn strictly nothing as far as the flow speed is concerned, if onlybecause of the vast number of molecules to be accounted for.

Treating matter as a continuum can be done for matter in the solid state, where it comple-ments the notion of rigid bodies and leads to the theory of elasticity, and also for matterin the liquid or gas states, where one gets hydro- and aerodynamics which are sometimesgrouped together into fluid dynamics. So it seems that matter under certain circumstancespermits us to use the mathematical concept of a continuum, and the whole question thenrevolves around knowing the limits of accuracy and applicability of this continuum conceptin practical cases. This is a rather difficult problem, because it stands almost totally di-vorced from the mathematical modelling which can proceed once basic concepts and lawsare defined and postulated, as will be done in the next chapter.

1.2 Questions of notation

As is the case with most branches of physics, continuum mechanics deals with certain phys-ical quantities which exist and can be talked of independently of a particular coordinatesystem. In mathematical terms this means that such physical quantities are represented bytensors and that the physical laws are expressible as tensor equations. Of course, these samephysical quantities will be measured and attributed numbers or values with respect to anappropriate coordinate system. As soon as a coordinate system is chosen, a tensor, althougha mathematical entity on its own, is completely specified through its components. These

components change when one passes from one coordinate system to another according tospecific rules.

This dual way of thinking of a tensor, i.e as a mathematical entity on its own and throughits coordinate representation, will reveal itself also in the notation. We write T for a certaintensor, or else we will use its components Tij... when referred to a coordinate system. Theindices i , j , . . . run from 1 to 3, in physical space that is, for one could as easily definetensors in n-dimensional space and the range of the indices would then be from 1 to n.Some of the mathematical properties recalled in the following paragraphs are indeed validfor general tensors in n-dimensional space. The coordinate systems used henceforth willalways be orthogonal Cartesian coordinate systems. This restriction avoids the complication

of having to distinguish between covariant and contravariant tensors, but still allows enough


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interesting properties and applications for a first course in continuum mechanics. The wordtensor will then mean Cartesian tensor.

A scalar is a tensor of order zero and has one, invariant component in each coordinatesystem. A vector is a tensor of order one and will be written in boldface (e.g. a, v ...). Itsthree components will be denoted by the same letter subscripted once (e.g. ai, vi ...).

Finally, a tensor of order two will be called a tensor, in short, and is denoted e.g. by T. Ithas nine components which are written Tij for i, j = 1, 2, 3. The unit tensor will be writtenas 1, with components ij, the familiar Kronecker deltas.

When a certain expression is written in index notation, use will be made of the Einsteinsummation convention in order to save the writing of summation signs with the appropriatesummation ranges. The summation convention implies that, whenever an index appears

twice in a product, the index or subscript takes the values 1,2,3 consecutively and the threeterms thus formed are summed. In the few cases where the summation convention cannotbe followed, this will be mentioned explicitly.

1.3 Coordinate and component transformations

Physical space is seen as a three dimensional affine Euclidean space, modelled on IR3. Areference frame in space is given by the choice of an origin and of a positive (i.e. right-handed)orthonormal basis e(i) (i = 1, 2, 3). The position of a point P is given through its positionvector x = OP or, equivalently, through its Cartesian coordinates, which will be denotedxi for i = 1, 2, 3 (rather than through x, y, z). The triplet (e(1), e(2), e(3)) determines theorientation of the physical space. Once a base of a coordinate system is known, the relationbetween the position vector and the coordinates of a point P is simply

x = xi e(i).

By a coordinate system we will always mean a Cartesian coordinate system and we willdenote it by Ox1x2x3.

The next step is the transformation of one coordinate system Ox1x2x3, with unit vectors e(i),

to another one Ox1x2x

3, with unit vectors e

(j). Both coordinate systems have for simplicity

the same origin. The unit vectors then transform according to

e(i)

= Aij e(j),

where Aij are the direction cosines of the new axes referred to the old ones,

Aij = e(i) e(j) = cos

e(i)

, e(j)

.

Since we are dealing with orthogonal coordinate systems, this yields two sets of orthonor-mality relations:

AijAik = jk , AijAkj = ik,


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i.e. the matrix [Aij] is an orthogonal matrix. In addition, the transformation is orientationpreserving, i.e.: det[Aij] = 1. Thus the required coordinate transformations between the old

coordinates xi and the new ones xj of a point P is

xi = Aijxj.

The inverse transformation isxi = Ajix

j,

as can easily be checked,Ajix

j = AjiAjk xk = ikxk = xi.

Defining an arbitrary vector v in space as the difference between the position vectors of itstwo endpoints P and Q, allows us to deduce the transformation laws of the components of

v, which are denoted by vi in Ox1x2x3 and by vj in Ox1x2x3,

vi = Aij vj

We could also turn the argument around and postulate that a vector is defined in such away that its components obey this transformation law. It is easy to check and left to thereader to verify that vectors thus defined are the elements of a vector space.

Next come the transformation laws for the components of a tensor of order two or higher.For a tensor of order two these are

Tij = AipAjq Tpq

and for more general tensors

T ijk... = AipAjq Akr T pqr... .We will say that a tensor is isotropic if it is invariant under any orthogonal coordinatetransformation, i.e. Tijk... = AipAjq Akr T pqr... , for every orthogonal matrix A.Each component of a tensor of order n has n indices, and there are just n factors Aip in thetransformation equation. If one thinks of a tensor of rank two T as a linear transformation

which maps a vector u into a vector v,

v = T u,or in index notation

vi = Tijuj,

then the transformation laws for a tensor of order two can be deduced from those for avector.

Tensors of the same order can again be proved elements of a vector space. More precisely,if S and T are tensors of order two, say, and is an arbitrary real number, then S and

S + T are again tensors of order two, and all the traditional properties of a vector space are


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satisfied. Let us denote the space of second order tensors by V. For each element S of Vthere exists an inverse in V, denoted S, and the following distributive properties hold:

(S + T) = S + T, ( + )T = T + T,

where and are scalars.

Finally, lets go back to the transformation rule for the components of a general tensorTijk... = AipAjq Akr T pqr.... These hold not only for transformations from one right-handedbasis to another right-handed basis, but for any transformation of the basis. Quantities forwhich the transformation rule only holds for an orthogonal transformation with det[Aij] = 1are called pseudo-tensors. Stated otherwise, if the transformation equations for vector ortensor components only hold when det[Aij] = +1, then we are dealing with pseudo-tensorsrather than with tensors in the proper sense. An often-used pseudo-tensor of order three is

the alternating pseudo-tensor of Levi-Civita ijk with components

ijk

= +1 ifi,j,k are an even permutation of 1,2,3,= 1 if i,j,k are an odd permutation of 1,2,3,= 0 otherwise, when two or three indices are equal.

Some useful identities involving the components of the pseudo-tensor of Levi-Civit a and ofthe unit tensor of Kronecker are

ijk im = jkm jmk,ijk ij = 2k,

ijk ijk = 6,

ii = 3.For a given tensor T, with components Tij, one can always construct the transposed tensor,

denoted by Tt or T, such that(Tt)ij = Tji.

A coordinate independent definition would be through

u Tt = T ufor all u. A tensor is called symmetric iff

Tij = Tji or Tt = T.

Any tensor T of order two can be split in a unique way into a symmetric and an antisymmetricpart,

T =1

2(T + Tt) +

1

2(T Tt),

or in index notation

Tij =1

2(Tij + Tji) +

1

2(Tij Tji ).

It is possible to generalize the concept of the transposed of a tensor to tensors of higher order,where such a transposed tensor is called an isomer, but extreme care has to be exercised asto which pair of indices is involved in such a permutation. The pseudo-tensor of Levi-Civit ais totally antisymmetric, because any permutation of two indices changes the sign of the

components.


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1.4 Tensor products

Starting with scalars and we have only one kind of product, the usual one, , in thesense of number theory.

For vectors u and v in Euclidean space, one is supposed to be familiar with the scalar orinner product

u v = uivi,which yields a scalar, and (if the dimension of the space is three) with the vector or crossproduct u v

(u v)i = ijk uj vk,which is really a pseudo-vector.

Next, we can define a product on vectors which yield a (second-order) tensor: the so-calledtensor or outer product

(uv)ij = uivj .

One can indeed verify that the object defined by the right-hand side is a second-order tensorby checking the transformation property.

The ideas of inner and outer products can be extended to tensors of order two by putting

(S T)ij = SikTkj

for the inner product and(S T)ijk = SijTk

for the outer product. In more general terms even, the inner product of a tensor of order mwith a tensor of order n yields a tensor of order m + n 2,

(S T)ijk = SijpTpk,whereas the outer (or tensor) product gives a tensor of order m + n,

(S T)ijk = SijTk.

If m + n 2 is 2 or more, the inner product or contraction can be repeated to yield thedouble inner product or the double contraction of two tensors,

(S: T)ijk = SijpqTqpk.In particular, the double inner product of two tensors of order two is a scalar,

S : T = SijTji

One sometimes finds the double inner product defined in another way,

S T = SijTij,


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which usually gives a different result. It is to be noted that both definitions coincide if atleast one of the two tensors is symmetric.

Remark. In mathematical textbooks one will usually adopt the notation for tensorproduct (e.g. u v, S T, . . .).One could also try to extend the idea of the cross product of two vectors, but it seems onlyuseful to do so for the cross product of a vector with a tensor of order two, in that order, inview of the introduction of the curl of a tensor in the next paragraph,

(v T)ij = ipqvpTqj .

This cross product is a pseudo-tensor, as was already the case with the cross product of two

vectors.

To round off this paragraph, some forms of

(T v)i = Tij vj and (v T)i = vjTjiare given in the special case when the tensor itself is the tensor product of two other vectors,

(uv) w = u(v w) = uv w,u (vw) = (u v)w = u vw.

This shows that a notation such as uv w is unambiguous, and there is no need to write anybrackets.

If one introduces the trace of a tensor of order two,

tr T = Tii,

which is invariant under orthogonal coordinate transformations (i.e. tr T = Tii), one findsthe following relations as special cases:

tr (uv) = uivi = u v,

tr (S T) = (S T)ii = SijTji = S : T.

1.5 Tensor analysis

In this paragraph the notions of gradient, divergence and curl, which are known from vectoranalysis, are recalled and extended to general tensors.

The usual gradient of a scalar function of x (and possibly t) is a vector with components

(grad )i = ()i =

xi

= i.


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The partial derivative operator with respect to the coordinate xi will henceforth be denotedi. The symbolic vector nabla, , has components i and could formally be regarded

as a vector obeying the transformation laws, although it retains its derivative operatorcharacteristics. Hence the (generalized) gradient of a tensorT of order n is a tensor of ordern + 1, constructed as the tensor product of the nabla vector with the said tensor,

(grad T)ij... = (T)ij... = iTj....

Special cases of this equation are the gradient of a vector,

(grad v)ij = (v)ij =vjxi

= ivj

which is a tensor of order two, and the gradient of a tensor T of order two,

(grad T)ijk = (T)ijk = iTjk ,

which is a tensor of order three.

On the other hand, the usual divergence of a vector is a scalar,

div v = v = ivi.

Hence we define the divergence of a general tensorT

of order n as a tensor of order n

1,constructed as the inner product of the nabla vector with T:

(div T)ij... = ( T)ij... = pTpij...This includes as a special case the divergence of a tensor (of order two),

(div T)i = ( T)i = j Tji

Finally, in vector analysis, the curl of a vector is a pseudo-vector,

(curl v)i = ( v)i = ijk jvk,

and, generalizing this concept, the curl of a tensor T (of order two) is a pseudo-tensor of thesame order,

(curl T)ij = ( T)ij = ipqpTqj .The gradient, divergence and curl operations can be repeated several times, if the resultingexpressions permit, as in the following example of the double gradient of a scalar function,which is a tensor of order two,

(

)ij =

2

xixj = ij.


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Once these operations are established, it becomes straightforward to write a Taylor expansionof a tensor around a point with position vector xo, or with coordinates xoi , either in index

notation, Tij...(x) = Tij...(xo) + (x xo )Tij...(xok) + . . .or in vector notation,

T(x) = T(xo) + (x xo) T(xo) + 12

(x xo)(x xo) :T(xo) + . . . .

Some useful identities involving the nabla operator are:

(u v) = u v + v u + u ( v) + v ( u),

(uv) = (

u)v + u

v,

(T v) = v : T + Tt :v, ( v) = v + v .

1.6 Time derivatives

In continuum mechanics we are interested, among others, in the time evolution of certainobjects. The partial time derivative of a tensor depending explicitly on t will be denoted by

tT =T

t

,

with

(tT)ij... =Tij...

t

If the continuum moves in space, such that position vector of a material point becomes adifferentiable function of time, the change of a tensor along a flow line is given by the totalor material time derivative (i.e. the change of the tensor when following the material in itsmotion, hence the name of material derivative):

T =d T

dt=

d

dtT(x(t), t)

=T

t+

d x

dtT = tT + v T.

The operator

d

dt= t + v = t + vii

is called the material time derivative. The material derivative, as calculated above, is forfunctions which are defined in each point of the material. Note, in particular, that thematerial derivative of the position vector x at a point P is the velocity of the material in

that point (or the velocity of P for short).


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Further on, the problem will also arise of finding the total time derivative of quantities whichare expressed as a time-dependent volume integral, when the volume under consideration

coincides at all times with a well-defined part of the material in motion. Something analogousarises when in a one-dimensional integral both the integrand and the integration limitsdepend on a parameter,

I() =

1()0()

f(, ) d,

and the derivative of the integral is sought with respect to this parameter,

dI

d=

1()0()

f

d + f(, )

=1()

d1d

f(, )

=0()

d0d

.

The first term on the right-hand side of this equation comes from the dependence of theintegrand on the parameter, the two remaining terms occur because the limits vary with thesame parameter.

Let (x, t) be a continuous function of coordinates and time and (t) the integral of overa certain material volume V, which evolves in time in such a way that the same materialpoints remain contained in V,

(t) =

V

(x, t) dV.

To compute the material derivative

(t) =

d

dt V (x, t) dVwe choose some time-independent reference configuration, in such a way that a particle whichwas at position a at a given (original and fixed) instant of time is at x at time t. In this waywe can express

x = x(a, t).

For this transformation the volume-elements dV = d3x and dV0 = d3a are related through

dV = J dV0,

where dV0 refers to the initial state, and J is the Jacobian of the transformation,

J = det

xiaj

.

Hence

d

dt

V

(x, t) dV =d

dt

V0

[x(a, t), t]J dV0

=

V0

d

dt{[x(a, t), t]J} dV0

= V0 J + J dV0,


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using the definition of the material time derivative of . To work out what J is, we alsoneed

ddt xiaj = xiaj = viaj = vixk xkaj .The time derivative of J thus gives rise to

J =

d

dt

x1a1

d

dt

x1a2

d

dt

x1a3

x2a1

x2a2

x2a3

x3

a1

x3

a2

x3

a3

+ . . . =v1xk

xka1

xka2

xka3

x2a1

x2a2

x2a3

x3

a1

x3

a2

x3

a3

+ . . .

=v1x1

J +v2x2

J +v3x3

J = Jivi = J v,

where the dots refer to similar determinants with the second or the third row differentiatedinstead of the first one. The transition from the second to the third expression for J in theprevious derivation comes about because only the value k = 1 gives a nontrivial result forthe first determinant, and similarly for the other two determinants. Once J is known, weobtain

d

dt V (x, t) dV = V0 + v JdV0 = V + v dV,or(t) =

V

( + v)dV

This expression will be used several times later on. It is instructive, however, to expand thedefinition of the material time derivative and get

=

V

(t + v + v)dV

= V

t dV + V ( v)dV.

The second volume integral can be changed into a surface integral by using the Gauss-Ostrogradski theorem

=

V

t dV +

V

n ( v)dS,

where V is the boundary of V and n is an outer unit normal to V. The expression for thus consists of two parts: one related to the time change of inside V, the other one givingthe total flux of v through the boundary V.


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2.1 Mass density and equation of continuity

Let us start by choosing a reference frame (coordinate system) Ox1x2x3 for the physicalspace. Next, we can think of a lump of matter, or a portion of a liquid or gas, which inthe continuum concept fills a domain D0 in space having a volume V0 and with boundaryD0. In an interior point P the description of the continuum and its behaviour requires thedefinition and knowledge of certain physical properties. How do we proceed?

First, we introduce the notion ofmass density. This is done in a way which might also serveto get other required properties in P. The continuum enclosed in D0 has a total mass M0.The notions of mass and volume should be familiar from theoretical mechanics and give nodifficulties at this level of the description. The average mass density in each interior pointof D0 is simply

average = M0V0

.

The mass density in P would then be somehow the limit of this expression, if the volumearound P were tending to zero. To see what this implies, consider a sequence of domains Dienclosed in each other, as shown in Figure 2.1. The matter enclosed in a domain Di withvolume Vi has a mass Mi. The domains Di can be fictitious and need not at all have physicalboundaries, they are just a theoretical construction used to define the concept of mass density.This is a stratagem we will often resort to in the development of continuum mechanics:mentally isolating part of the continuum under consideration. Each of the domains Dishould contain P, of course. With the sequence of domains we get a corresponding sequence

of average mass densities. The mass density in P is then defined as the limit of thissequence,

i =MiVi

i (P) (Mi 0, Vi 0).This limit should exist and be unique, regardless of the sequence of domains Di used toconstruct the limit. Nevertheless, if we go back for a moment to the physical side of the

P

DiD0

Figure 2.1. Sequence of domains enclosing P.


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argument, it is clear that even an elementary volume around P should still contain a largeenough number of molecules in order not to violate the whole idea of a continuum. Indeed,

if volumes of molecular or intra-molecular scale are considered, chopping off part of sucha volume will result in a discontinuous change in the mass density whenever a moleculeis discarded. Here comes a natural limitation of the kind of phenomena one can hope todescribe adequately with a continuum theory: phenomena that are on a scale which is largecompared to the atomic interparticle distance and a fortiori to the size of an atom or amolecule.

This process of defining a mass density can be repeated for each interior point P. Theresult is a mass density (x, t) defined everywhere in D0 and by virtue of the ever presentcontinuum hypothesis, this mass density is a continuous function of x in D0 at all times.In a similar vein other densities can be defined, such as an energy density or a momentum

density. Instead of the preceding, more intuitive way of defining the mass density, one canalso use measure theory to define a mass density in a unique way, from the moment onedeals with a continuum such that one can associate with every volume V in D0 a numbergiving the mass in V. What are needed are properties of a measure defined in D0 suchthat in particular

(U U) = (U) + (U) (U U = ; U, U D0),(U) 0,() = 0.

Clearly, both volume and mass are two such measures defined in D0, mass only in the con-tinuum limit which precludes masses concentrated in one point. Then the Radon-Nikodymtheorem states the existence of a unique (mass) density such that mass and volume arerelated according to

M(D) =

D

dM =

D

dV.

This enables us at once to get the total mass contained in a domain D D0 with volumeV. Henceforth, such integrals will be written as volume integrals over V rather than over D:

M =

V

dV.

If the volume V of a continuum is followed in its motion, it is assumed that V will alwaysconsist of the same material particles, since it is precisely their motion which is constitutingthe motion ofV. Hence the mass in V is invariant when V is followed in its motion and thematerial time derivative of M vanishes,

M = 0.

This expresses the first conservation law, the conservation of mass , in extensive form for aspecific volume V,

M = V(t + (v))dV = V( + v)dV = 0.


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If the conservation of mass is to hold throughout the continuum for arbitrary volumes, evenimagined ones, then the integrands in the previous integrals have to vanish everywhere,

which yields the point or differential form of mass conservation,

t + (v) = 0 = + v

This equation is known as the equation of continuity.

2.2 Forces and conservation of linear momentum

From point mechanics one knows Newtons (second) law in the form: the time derivative

of the total linear momentum of all particles equals the sum of all forces acting upon theseparticles. Going then to the notion of rigid bodies with a finite extent, one had Eulers lawsas the appropriate generalizations of Newtons law of motion.

Continuum mechanics can be viewed as a limit of point mechanics with an ever increasingnumber of particles of ever decreasing mass, hence it would seem natural to postulate arephrasing of Eulers first law for rigid bodies: the material (i.e. total) time derivative of thetotal linear momentum of the mass contained in a volume V is equal to the sum of all forcesacting upon V. The translation of this principle into mathematical form requires a suitabledefinition of the total linear momentum of the mass in V, which we take as

P = V v dV,as well as a suitable definition of all forces acting upon V. Instead of Eulers first law for arigid body, we get for a continuum something formally very similar,

P =d

dt

V

v dV = F.

Two things remain to be done, the computation of P and a proper definition of F. It isstraightforward to find that

P =

d

dt V v dV = V (v + v + v v) dV = V v dV.The simplification in the integrand comes from invoking the continuity equation and is aresult which is valid for more general integrands besides v. As we will need this propertyseveral times afterwards, we check that for a tensor T of any order

d

dt

V

TdV

=

V

T+ T + T v

dV =

V

TdV,

as soon as the equation of continuity is valid. Hence Newtons law now stands in the form

V v dV = F.


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As far as the forces are concerned, one usually distinguishes two kinds of forces: the body orvolume forces and the surface forces.

Body forces arise as a consequence of action at a distance and are intuitively thought ofas the generalization of external forces in point mechanics. Typical examples are gravity,electromagnetic forces and the like. Such forces act in each point of the continuum andwe will characterize them through a force density per unit volume f(x, t), or else through anormalized force density b(x, t) with the dimensions of an acceleration. The relation betweenboth densities is

f(x, t) = (x, t) b(x, t).

The sum of the body forces acting upon a volume V of the continuum is thus

Fbody = V fdV = V b dV.In the case of gravity one finds that

Fgrav = Mg =

V

dV

g =

V

g dV

and hence

f(x, t) = (x, t) g,

b(x, t) = g.

Surface forces, on the other hand, act on the surface of a continuum, and could in somerare cases belong to the external forces, such as when the atmospheric pressure has to beconsidered at the surface of a body. Usually, however, surface forces have to be introducedto translate the action of one part of the continuum upon another, if part of the continuumis considered separately but yet in the same state of motion and deformation. In this sensethe surface forces could be thought of as generalizing the idea of internal forces in a setof particles, and they represent something as the cohesion of the material. These forcescancel each other in each interior point as the result of the law of action and reaction, andonly survive with a non-null resultant at the boundary of the continuum, whether the real

physical boundary of a lump of matter or a fictitious boundary which is thought of isolatingpart of the continuum from the rest.

Think of a point P, interior in the volume V0, and imagine an internal volume V in such away that P lies on the boundary V of V, which we assume to be continuous. Define n asa unit vector in the external direction, normal to V, and passing through P. As part ofV0this volume V is in a certain state of motion and deformation. If we now want to describeonly V and forget about the remainder of V0 without changing the motion or deformationofV, the action which the mass in V0\V exerts on V has to be taken into account explicitly.Consider now an elementary surface element S ( V) at P, as shown in Figure 2.2, andcall F(n) the total force exerted by V0

\V upon V, acting through S, and N(n)O the


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P

SVV0n

Figure 2.2. Surface elements around P.

resulting moment (with respect to O). Let S shrink to zero and we get, according to thestress postulate of Euler-Cauchy, a unique and well defined limit:

limS0F(n)

S= t(n)

limS0N(n)O

S= 0

(S 0 n S, P S V).The vector t(n) is called stress vector, associated with the choice of the normal n in P, and it

has the dimension of force per area. Note that this postulate tells us that there is no stressmoment density.

Several remarks are in order here. First of all, the Euler-Cauchy postulate is a postulate (i.e.can not be proven), even though introduced on fairly obvious physical grounds. Furthermore,the stress vector t(n) is uniquely defined under this hypothesis regardless of the choice of Vor its boundary, as long as n is the external unit vector in P normal to V. The argumentto have no net stress moment is based intuitively on the shrinking of the dimensions of thesurface around P to zero. We will follow this argument for the sake of simplicity, althoughit is by no means required to construct a consistent theory. Suffice it here to mention thatfor so-called Cosserat continua the stress moment density is taken into account in a fully

coherent development of the continuum theory. The principle of action and reaction in aninterior point P means that

t(n) = t(n).To know the stress field fully at this stage is equivalent to knowing a triple infinity of stressvectors at each interior point P. Obviously, something simpler will be necessary for a moreworkable description.

The sum of the surface forces acting upon a continuum with volume V and boundary V is

Fsurface = Vt(n) dS.


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If we return to the law of motion, which was left in the form

V v dV = F,

we find V

v dV =

V

fdV +

V

t(n) dS.

For the present, no point or differential form of this conservation law is possible. This willhave to wait until after the introduction of the stress tensor.

2.3 Stress tensor

In an interior point P the body force density f or b is a given quantity. On the other hand,with each choice of the normal n in P there corresponds a stress vector t(n). It is generallyadmitted that there exists a linear relation between both, expressed in terms of a tensor Tof order two, such that

t(n) = n TThis tensor T is called the stress tensor and the knowledge of the stress tensor determinesthe stress field in P completely. We are now in a position to further transcribe the equationsexpressing the conservation of linear momentum.

The surface integral appearing in the momentum equation can be transformed into a volumeintegral with the help of the Gauss-Ostrogradski theorem,

V

t(n)dS =

V

n T dS =

V

T dV

or componentwise, V

t(n)j dS =

V

niTij dS =

V

iTij dV.

The momentum equation itself thus becomes

V v dV = V fdV + V T dV,and it is now possible to go to the differential or pointwise form of the equation expressingthe conservation of momentum:

v = f + T

or equivalently in index notation,

vj = fj + iTij.

These are known under the name of equations of motion.


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2.4 Center of mass

We define, with respect to the given reference frame and again in analogy to what is knownfrom point mechanics and rigid body mechanics, the center of mass of a continuum containedin V as

xC =

V

x dVV

dV=

1

M

V

x dV ,

with M the total mass of V. When the material time derivative of this expression is taken,we use the fact that M remains constant and get

xC =1

M

d

dt

V

x dV =1

M

V

v dV,

with the help of the equation of continuity. HenceP = MxC,

just as for a rigid body. Taking once again the material derivative, we find with the help ofV

v dV =

V

fdV +

V

t(n) dS

that

xC =1

M

d

dt V v dV =

1

MV v dV

= 1M

V

fdV + 1M

V

t(n)dS = 1M

F.

This states that the acceleration of the center of mass times the total mass is equal to thesum of the forces acting upon V. The details of the distribution of matter in V seem tobe forgotten, and the actual shape of the volume only plays a role when computing thesum of the forces. This allows a coarser description of deformable media in terms of thetotal mass, center of mass and total forces. Precisely this property has ensured the successof the classical point mechanics and the mechanics of rigid bodies, where the deformation isnegligible compared to other characteristics, or when all the mass of a body can be thoughtof as being concentrated in one point, its center of mass.

2.5 Conservation of angular momentum

In point or rigid body mechanics the conservation of angular momentum is formulated asfollows: the time derivative of the total angular momentum equals the resulting momentumof all forces acting upon the system, all momenta being computed with respect to a givenfixed point. A similar principle will be invoked in continuum mechanics. The total angularmomentum (with respect to chosen the origin) of a continuum in a volume V is

LO = V x v dV.


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The total momentum results only from the momenta of the forces acting upon V and is

NO = V x fdV + V x t(n) dS.The conservation of angular momentum for continua is hence

LO = NO,

formally reminiscent of classical mechanics, or in more explicit form,V

x v dV =

V

x fdV +

V

x t(n) dS,

sinced

dt V x v dV = V d

dt(x

v) dV = V x v dV.One has to change the surface integral into a volume integral according to the theorem of

Gauss-Ostrogradski,V

x t(n)

idS =

V

(x (n T))i dS

=

V

ijk xj nTk dS =

V

(ijk xj Tk) dV

=

V

ijk (j Tk + xjTk) dV =

V

( T + x ( T))i dV.

The computation was done in index notation because in vector notation it is well-nigh

impossible and would be at any rate confusing. There remains in the angular momentumequation that

V

x v dV =

V

x fdV +

V

(x T + T) dV

or with the help of the equation of motion,V

T dV = 0.

If stress momenta would have been included in the description, this result would no longerbe valid. Again one finds the differential or point form of the equation expressing the

conservation of angular momentum,

T = 0 or ijk Tjk = 0.There follows that

ipqijk Tjk = pjqkTjk pkqj Tjk = Tpq Tqp = 0,and conversely, so that

T = 0 Tt = T.What remains of the conservation of angular momentum is that the stress tensor has to bea symmetric tensor. Such a result would not exist for Cosserat continua where a stress

momentum distribution is present.


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2.6 Conservation of energy

Many things could be said about energy, but most of them straightaway involve thermody-namical considerations. As we restrict ourselves in the next chapters to problems which canbe studied from an almost purely mechanical point of view (and there are enough interestingproblems falling in this category) we will give the balance of energy in this paragraph invery simple terms. We recall that the forces acting upon the part of the continuum underconsideration were the body forces f and the stress forces t(n). The total mechanical powerthen is

P=

V

f v dV +

V

t(n) v dS,

which can be computed to be

P= V

f v dV + V

n T v dS

=

V

f v dV +

V

(T v) dV

=

V

f v dV +

V

( T) v + Tt :v dV

=

V

(f + T) v dV +

V

T :v dV.

We define the rate of deformation tensor D, which will be treated in more detail in the next

chapter, as

D =1

2

v + (v)t

,

i.e. it is the symmetric part ofv. In view of the symmetry of T it is easy to prove that

T :v = T :1

2

v + (v)t

+ T :

1

2

v (v)t = T : D.

This will be used, together with the equation of motion, to rewrite

P= V(f + T) v dV + V T :v dV

as

P=

V

1

2

dv2

dtdV +

V

T : D dV =d

dt

V

1

2v2 dV +

V

T : D dV.

With the definition of the kinetic energy as

T =

V

1

2v2 dV,

we find that

P=T+ V T : D dV,


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stating that the total mechanical power equals the change in kinetic energy per unit time plussomething which will be related to the behaviour of the medium as a deformable continuum

(and is sometimes called the stress power).The kinetic energy is not the only form of energy for a continuum. There is the internalenergy, E, characterized by a density of internal energy:

E=

V

dV.

The remainder in the energy balance, the non-mechanical power, will be lumped togetherand called the heat Q, in such a way that the total mechanical power plus the heat is givenby the time variation of the kinetic plus the internal energy:

P+ Q = T+ E

This is the first law of thermodynamics. Comparing both expressions for P, one finds that

Q +

V

T : D dV =d

dt

V

dV

.

To get a point form of this balance of energy requires some additional hypotheses about howthe heat changes. We will split the heat contained in a volume V in two parts, one connectedwith the internal heat sources or sinks, with density (possibly from a radiation field), andone connected with the heat flux through the boundary, with heat flux vector q, such that

Q =

V

dV

V

n q dS .

As a result, we finally obtain that

= q + T : D

The time rate of change in the internal energy is given by the heat source density, the

divergence of the heat flux and the stress power T : D. For adiabatic processes, wherethere are no heat sources/sinks nor heat fluxes, the stress power changes the internal energydensity in a unique way,

= T : D.

This will be as far as we need to push the reasoning about energy for the purpose of thisand subsequent chapters.


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2.7 Basic equations

We are now in a position to collect the basic equations obtained sofar: the equation ofcontinuity, the equation of motion and the symmetry of the stress tensor:

+ v = t + (v) = 0v = f + T = b + TTt = T

To this one could add the energy balance, but we leave it out of the discussion, as it simplydetermines the internal energy once T and v (and hence v and D) are known. If in thepreceding set of equations one counts in scalar components, the equation of continuity is one

equation in four unknowns, the mass density and the three components of the velocity. Theequation of motion corresponds to three scalar equations in nine additional unknowns, thecomponents of the stress tensor, because the volume force density f has to be considered agiven quantity, as the example of gravity shows. Finally, the symmetry of the stress tensoradds three further equations.

To sum up, we have at our disposal seven scalar equations containing thirteen unknown scalarquantities. The theoretical description is thus not closed at all. This should not be surprising,because we only expressed very general principles which are supposed to hold for all kindsof continua, be they elastic material or fluids, all of which can be encountered in variousguises. The closure of the theoretical description will only be obtained by the introduction ofconstitutive equations, where something more specific is said about given classes of continua.Constitutive equations will have to wait, however, until chapter 4, because first deformationhas to be treated and this will occupy chapter 3. The remainder of the present chapter willbe devoted to some properties of the stress tensor as a symmetric tensor.

2.8 Eigenvalue problem for symmetric tensors

We recall the interpretation of a tensor of order two as a linear operator mapping a vectoru into another one. One can then ask the question if and when u and T u can be parallel,i.e. if and when there exists a scalar such that

T u = u.If this is possible, is called an eigenvalue and u is the corresponding eigenvector of T. Theterminology has been carried over from what is used in matrix algebra. In index notationwe have that

Tijuj = ui,

which form in three-dimensional space a set of three homogeneous equations in the threecomponents ofu. If an eigenvector u is to be found at all, the determinant of the coefficientsmust vanish,

det(Tij ij) = det(T 1) = 0.


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This is the characteristic equation of T which yields in general three, possibly coinciding,values for . There are three eigenvalues thus, and with each one corresponds an eigenvector.

We know that a symmetric tensor T has three real eigenvalues. If they are different fromeach other, the corresponding eigenvectors form an orthogonal system. If the eigenvaluesare not all different, then it is still possible to construct three eigenvectors which form anorthogonal system. In this orthogonal system the representation of T is diagonal, with thethree eigenvalues as diagonal elements,

T =

1 0 00 2 00 0 3

.Expansion of the characteristic equation leads to

3 IT2 + IIT IIIT = 0,

where the three (principal) invariants of T are given by

IT = tr T = Tii,

IIT =1

2(TiiTjj TijTji) = 1

2tr (T tr T T2),

IIIT = det T.

From

T u = uthere follows after repeated multiplication with T that

Tm u = mu.

Since each eigenvalue satisfies the characteristic equation and in view of the diagonal repre-sentation, T itself satisfies its own characteristic equation,

T3 ITT2 + IITT IIIT1 = 0

This is known as the Hamilton-Cayley equation and it can be used to express higher powersof T as combinations of T2, T and 1.

2.9 Normal and shear stresses

The symmetry of the stress tensor has several interesting consequences. First comes theconcept of normal stress, associated with a direction n at a point P, which is defined asthe component of the stress vector t(n) along n itself. This component can be written inequivalent ways as

(n)N = n t(n) = nn : T = T : nn = n T n.


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The normal stress is the same for both the inner and the outer normal to a given plane,

(n)

N =

(n)

N ,

and the stress is simply called the traction or pressure according to whether (n)N is positive

or negative. The shear stress is the component of the stress vector t(n) perpendicular to thenormal, in the plane itself,

t(n)S = t(n) (n)N n.

Note thatm t(n) = m T n = n t(m)

i.e.: the projection of the stress vector t(n) associated with a direction n upon a direction mequals the projection of the stress vector t(m) associated with m upon n. For two given or-

thogonal directions m and n with their associated stress vectors one speaks of the reciprocityof the shear stresses, given by

m t(n)S = n t(m)S = mn : T = nm : T (m n).All these properties simply follow from the symmetry of T.

In general one has for a given direction n at a point both normal and shear stresses. Arelevant question in this context is whether there exist planes in the continuum for whichthere is a normal but no shear stress. The corresponding stress vector t(n) should be parallelto the normal n for such planes, i.e.

t(n)

= n,which amounts to

T n = n T = n.Thus is required to be a solution of

det(T 1) = det(Tij ij) = 0.

We are then dealing, of course, with the eigenvalue problem for the stress tensor, for whichcan use the results for symmetric tensors recalled in the previous section. The (real) eigen-values of T are called the principal stresses, and it is possible to find a coordinate system in

which the stress tensor has a diagonal matrix representation. In such a coordinate system(based on a system of principal axes for stress) the coordinate planes at least satisfy therequirement that there be no shear stress along these planes. According to whether someeigenvalues are equal or not, or vanish or not, different states of stress exist. They aresummarized in Table 2.1.

The coordinate planes in the system of principal axes carry no shear stresses. In the converseproblem one could ask for which directions the shear stresses are maximum, or more general,extremum. For simplicity, the computations will be done in the system of principal axes.We start from the magnitude squared of the shear stress vector,

2S = t(n) Nn t(n) Nn = t(n) t(n) 2N.


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state of stress T in principal axes t(n) in principal axes invariants

1. general stress 1 0 00 2 00 0 3

n11n22n33

IIIT = 0 or det T = 0with possibly1 = 2 but 2 = 3

2. plane stress

1 0 00 2 00 0 0

n11n220

IIIT = 0t(n) plane (1,2) IIT = 0

3. linear stress 1 0 00 0 00 0 0

n1100

IIT = 0 = IIITt(n) e(1) IT = tr T = 0

4. spherical stress

0 00 00 0

n1n2n3

= n1n2

n3

IIT = 13(IT)21 = 2 = 3 = t(n) = n (n) IIIT = 127(IT)3T = 1

Table 2.1. Different stress states

With respect to principal axes we have

N = n2i i, t

(n) = n (i summed, not),

and hence2S = n

2i

2i (n2i i)2.

The extrema of this function have to be sought depending on the direction of n, with theconstraint that n be a unit vector,

nini = n2 = 1.

With the use of a Lagrangian multiplier we have to find the extrema of the function

= 2S (nini 1)

as a function of the components of n. The condition that be extremum is

n= 0, ( = 1, 2, 3)


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which gives the set of equations

n = 2n2 2(n2j j)2n 2n

= 2n

2 2(n211 + n222 + n233)

= 0 ( not summed).

The discussion is first given for the case when all eigenvalues of the stress tensor are differ-ent. Then it can be checked that at least one component of n has to vanish in order to find asolution, while on the other hand not all three components of n may vanish simultaneously.

A first class of solutions is with two components of n zero, the third component then being1, which yields

=

21 if n = e

(1),

2S = 0.

This type of solution yields nothing but the coordinate planes in the system of principalaxes, where the shear stresses vanish. The other class of solutions is found by putting onecomponent of n, say n3, zero and solving

21 21

n211 + n222 = 0,

22 22

n211 + n222 = 0,

n21 + n22 = 1,

for n1 and n2. This gives

n21 = n22 =

1

2, = 12,

2S =

1 2

2

2.

The other cases are analogous, and the principal shear stresses occur along the bisectorplanes of the system of principal axes,

n21 = n22 =

1

2

and n3 = 0

S =

2 12

n21 = n23 =

1

2and n2 = 0 S = 3 1

2

n22 = n23 =

1

2and n1 = 0 S = 3 2

2

(1 < 2 < 3)

These are the maximal shear stresses, and the discussion of the solutions is herewith ex-hausted, if we suppose that all i are different from each other.

Suitable but straightforward corrections can be made to the above reasoning if some of theeigenvalues coincide. We will leave this to the reader, except to say that in the case of threeequal eigenvalues

1 = 2 = 3 =


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one has a state of isotropic or spherical stress,

T = 1,

where every direction corresponds to an eigenvector and there are no shear stresses at all,only (equal) normal stresses.


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Chapter 3

Deformation

Finite and infinitesimal deformation are obtained by comparing initial andfinal states, after subtracting translation and rigid body rotation. In its

simplest form the deformation and rotation tensors are

E =1

2(u + u) and W =

1

2(u u)

Rates of strain similarly address the evolutionary aspects of deformation,with rate of strain and vorticity tensors

D =1

2(v + v) and =

1

2(v v)

Eigenvalues for the strain and rate of strain tensors give principal strainsand special deformations

29


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3.1 Finite deformation

Recall that the motion of a rigid body can be decomposed, for the purpose of the theoreticaldescription, into a translation and a rotation. For deformable bodies, which cannot beregarded as strictly rigid, the motion will have a third aspect, that of deformation.

Two ways of looking at the motion are possible. First, without bothering about the evolutionthrough intermediate states, one can simply compare the initial and the final states. Thiscomparison will yield a certain translation, rotation and deformation. On the other hand,one might be interested in the motion itself, and then one gets at each time a translationvelocity, an angular velocity and a rate of deformation.

In the present section we start with the comparison of the initial and final states and leavethe velocities for a subsequent section. In Figure 3.1 both states to be compared are drawn

with respect to the same reference system. Fix a reference frame with a Cartesian coordi-nate system. The coordinates in the initial state, a, are called the material or Lagrangiancoordinates, whereas those in the final state, x, are called the space or Eulerian coordinates.The particle formerly at P0 is displaced to P, and has a displacement vector u given by

u = x a.In view of the continuum hypotheses, we will suppose that the correspondence betweenP0 and P is one-to-one, which simply means that the continuum under consideration is

1

O

3

2

a

x

u

P0

P

da

Q0

db

R0

dx

Q

dy

R

V0

V

Figure 3.1. Comparison of initial and final states.


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nowhere torn apart nor joined onto another material. This requirement stems from the usualdeterministic view that the initial conditions uniquely determine the subsequent evolution of

a given particle. The one-to-one correspondence between P0 and P, which is now built intothe theory, demands quite some caution and adaptation before the theory can be applied toproblems such as fractures of elastic materials. As a further requirement, the displacementvector is a continuous function of the position in either the initial or the final state.

Using the material coordinates or, equivalently, tagging all particles by their initial positions,we find that

x = x(a, t),and thus for the displacement vector

u = u(a, t) = x(a, t) a.Conversely, using the space coordinates, we can express the original positions of the particlesnow at x through

a = a(x, t).Similarly, the displacement vector becomes

u = u(x, t) = x a(x, t).Both descriptions refer to the same comparison between an initial and a final state, hence thetransformations x = x(a, t) and a = a(x, t) should be each others inverse. This in particularimplies that the Jacobian determinant of the transformation be different from zero, and even,for material continua, that it be positive,

J = det

aixj > 0, J1 = det

xiaj > 0.

This is based on the reasoning that if the initial and final states are smoothly connected, thenthe Jacobian tends to 1 when the final state approaches the initial state. The uniquenessrequirements prevent the Jacobian from going through zero, so that it has to keep a positivesign. The transformation between initial and final state is orientation preserving.

In order to separate the translation and rotation aspects of the motion from the deformationaspects, we set out with three neighbouring points, P0, Q0 and R0 in the initial state andconsider the corresponding points in the final state, P, Q and R. How these particles gotthere is irrelevant for the time being. From P0 to Q0 we have the infinitesimal vector da,corresponding in the final state to dx from P to Q. With the help of the transformationsrelating a and x one gets

dx = da ax dxi = xi

ajdaj

,

da = dx xa dai

=

ai

xjdxj .


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Repeating the argument for the pair of points (P0, R0) and (P, R), denoting the separationvector resp. db and dy, we find analogously

dy = db ax dyi = xi

ajdbj

,

db = dy xa dbi = ai

xjdyj

.

For the simplicity of subsequent calculations we abbreviate the position gradients as

F =

a

x, F1 = G =

x

a,

and rewrite the respective differentials as

dx = da F, dy = db Fda = dx G, db = dy G.

The scalar product dx dy contains information about the distances between P and Q andP and R, as well as about the angle QP R. Subtracting similar information from the initialstate, contained in da db, we calculate successively

dx dy da db = (da F) (db F) da db= da F Ft db da db= da (F Ft 1) db = 2da L db.

The symmetric tensor L is called the finite or nonlinear Lagrangian strain tensor of Greenand StVenant. That we do indeed find a tensor can be proved in different ways, eitherdirectly on the definition

L =1

2(F Ft 1)

or via the fact that indx

dy

da

db = 2da

L

db

L yields a scalar for an arbitrary choice of the (infinitesimal) vectors da and db.

In a similar vein we get, via the Eulerian description,

dx dy da db = dx dy (dx G) (dy G)= dx (1 G Gt) dy = 2dx E dy,

and the tensor E is the finite or nonlinear Eulerian strain tensor of Almansi and Cauchy.The definition

E =1

2

(1

G

Gt)


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shows that it is also a symmetric tensor.

To see indeed that L and E merit the name of strain tensors, or put differently, that these

tensors measure pure strain or deformation, we let for a while Q0 and R0, or Q and R, referto the same particle and find that

dx dx da da = dx2 da2 = 2L : da da = 2E : dx dx.

The left-hand side of this expression refers only to the distance between two particles in theinitial state and to the distance between the same two particles in the final state. Note thatfor rigid bodies we could pick any two (not even neighbouring) particles and, by definitionof a rigid body, the distance always remains invariant. This implies that for rigid bodies Land E have to vanish (i.e. F and G are represented by orthogonal matrices). Conversely, if L

and E vanish in the neighbourhood of a particle P, we find that the distance between P anda neighbouring particle Q remains invariant during the motion, hence the neighbourhoodof P behaves as if it were part of rigid body. This shows that L and E indeed measure thedeformation or the strain around P0 or P.

Instead of expressing L and E in components of the coordinates, we may just as well use thedisplacements. Combining

u = u(a, t) = x(a, t) awith the original definition of L gives us in index notation

Ljk

=1

2 ij + uiajik + uiak jk=

1

2

ijik + ij

uiak

+ ikuiaj

+uiaj

uiak

jk

=1

2

ujak

+ukaj

+uiaj

uiak

.

By similarly substitutingu = u(x, t) = x a(x, t)

into the expression for E we obtain

Ejk =12jk ij ui

xj

ik uixk

=

1

2

jk ijik + ij ui

xk+ ik

uixj

uixj

uixk

=

1

2

ujxk

+ukxj

uixj

uixk

.

These expressions will be used in the next section to determine the small strain tensors.


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3.2 Infinitesimal deformation or small strain

If the components of the gradient of the displacement are sufficiently small, in such a way thatthe products of those components can be neglected compared to the components themselves,uiaj uiak

ujak or uixj uixk

ujxk ,

one says that the continuum is subjected to a small or infinitesimal strain or deformation.Applying the above approximation in the last obtained expressions for L and E reduce theseto

Ljk =1

2

uj

ak+

uk

aj ,Ejk =

1

2

ujxk

+ukxj

.

Written in tensor notation this is

L =1

2

au + (au)t ,

E =1

2

x

u + (x

u)t

,

and it is obvious that the distinction between a Lagrangian and an Eulerian description canusually be omitted, the coordinates then being designated by x in both cases. Keeping thisin mind, henceforth we will use E to denote also the linear or small strain tensor,

E =1

2

u + (u)t

=

1

2(u + u)

The notation u will be used instead of (u)t, whenever no confusion is possible, and isinspired by the fact that for ordinary vectors (vu)t is just uv.

Now E is the symmetric part of the gradient of the displacement, and the antisymmetric

part ofu will be called the rotation tensor W, with

W =1

2

u (u)t = 1

2(u u)

This decomposition ofu is unique:

u = E + W.

We yet have to show that, for small strain, W merits the name of rotation tensor. To do


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P0

P

Q0

Q

u

dx

uV0

V

Figure 3.2. Decomposition of neighbouring displacements.

so, the displacement u = u(Q0) of a particle Q0 in the neighbourhood of P0, as indicated

in Figure 3.2, is expressed through a Taylor expansion around the displacement of P0,

u = u(xQ0) = u(xP0 + dx) = u(xP0) + dx u(xP0)= u + dx u = u + dx E + dx W.

The first term on the right-hand side of the last equation is the translation part of themotion, the second term refers to the pure deformation and the third part is nothing but therotation. To see this, we introduce the concept of the dual vector of an antisymmetric tensorof order two such as W. The notion of a dual vector is restricted to a three-dimensionalspace, because only there a vector and an antisymmetric tensor of order two both have three

components, so that a one-to-one correspondence between the information contained in theseis possible. For an antisymmetric tensor with components Wk, define in a unique way thevector w with components

wj =1

2jkWk.

Conversely, it follows that

pqj wj =1

2pqjjkWk =

1

2pkqWk 1

2pqkWk

=1

2Wpq 1

2Wqp = Wpq,


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or summarized together,

wj =1

2 jkWk, Wk = kj wj.

If W vanishes, so does w and vice versa. Furthermore, one can check that

(v W)i = vkWki = vkkij wj = ijk wj vk = (w v)iand hence the expression for u can be rewritten as

u = u + w dx + E dx.

This shows indeed that W is giving the rotation part of the motion, namely a rotation over

an angle w. The explicit expression of w is

wj =1

2jkWk =

1

4jk(ku uk)

=1

4jkku +

1

4jkuk =

1

2jkku =

1

2( u)j

or

W =1

2(u u) w = 1

2 u

3.3 Geometric interpretation of small strain

For the interpretation of the diagonal elements ofE, we return to the Lagrangian view point,

dx dx da da = 2E : dada

and take da parallel to the first axis,

dx2 da2 = 2E11da2,

so that

E11 =dx2 da2

2da2.

This yields several expressions which are equivalent within the small strain approximation,

E11 =(dx + da)(dx da)

2da2 2da(dx da)

2da2=

dx dada

.

This shows that the diagonal elements of the small strain tensor give the relative stretchingalong the coordinate axes. This is a quantity which can be measured and is often used inexperimental setups.


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tan =u1

x2

tan =u2x1

O1

2

Figure 3.3. Deformation of angles.

Once the meaning of the diagonal elements is intuitively clear, we proceed to the off-diagonalelements. We use

dx dy da db = 2E : dadbwith da parallel to the first axis and db parallel to the second axis and find

dxdy cos dadb cos 2

= 2E12 da db.

Up to second order terms, we have that

2E12 =dx

da

dy

dbcos cos .

As shown in Figure 3.3, we can express the cosine of as follows

cos = cos

2

= sin( + ) +

tan + tan = u2x1

+u1x2

= 2E12 ,

where it has been taken into account that and are assumed to be very small. Hence wehave that the off-diagonal elements of the small strain tensor give half of the change in anglebetween two line elements which were perpendicular to each other before the deformation.This change in angle is called shear. The reason that there is a factor two is that notions likerelative stretching or shear evolved experimentally long before it was recognized that thesewere parts of what is now known to behave as a (small) strain tensor.

3.4 Rate of deformation

Up to now the discussion of the deformation was confined to a comparison between an initial

and a final state, without paying any attention to the intervening states or to the motion


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P

Q

dx

v

v

V

Figure 3.4. Velocities of neighbouring particles.

involved in getting from one state to the other. To get some idea of the velocities occurring,we Taylor decompose the velocity of a particle Q in the neighbourhood of a particle P, inmuch the same way as we did with the displacements before and as shown in Figure 3.4, sothat

v = v(xQ) = v(xP) + dx v(xP) = v + dx v= v +

1

2dx (v + v) + 1

2dx (v v) = v + dx D + dx

or alsov = v + dx + dx D.

The new quantities introduced in the above are respectively the rate of deformation tensorD,

D =1

2(v + v) or Dij =

1

2(ivj + jvi),

the vorticity tensor ,

=1

2(v v) or ij = 1

2(ivj j vi),

and its dual vector, the vorticity (vector)

=1

2 v or i = 1

2ijk jvk

The rate of deformation tensor D is a symmetric tensor, the vorticity tensor is anti-

symmetric, and they are respectively the symmetric and antisymmetric part of the velocity


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gradient tensorv. The decomposition ofv shows that the motion contains three differentelements, at least conceptually, since all aspects occur simultaneously in real motion. These

three different elements are a translation velocity v, a instantaneous rotation with angularvelocity and a pure rate of deformation given through D.

Another way of arriving at D is by taking the time derivative ofdxdxdada (= 2E : dxdx),which yields

d

dt(dx2 da2) = d

dt(dx dx) = 2dx d

dt(dx).

We need to compute separately

d

dt(dx) =

d

dt(da a

x) = da d

dt(a

x)

= da a ddtx = da av = dv = dx xv,

and finally get

d

dt(dx2 da2) = 2dx dv = 2dx dx :v

= 2dx dx : D + 2dx dx : = 2dx D dx.

Here again, the interpretation is that if the neighbourhood of a particle P instantaneouslybehaves as a rigid body, then the corresponding rate of deformation tensor vanishes.

In order to get some relation between the strain tensor E and the rate of deformation tensorD, we calculate that

2dx D dx = ddt

(dx2 da2) = ddt

(2dx E dx)= 2dv E dx + 2dx E dx + 2dx E dv= 2dx

E +v E + E (v)t

dx.

It follows that

D = E +v

E + E

(v)t

becausedx A dx = 0 (dx) A = 0,

provided A is a symmetric tensor. If the deformation is infinitesimal, the last two terms onthe right-hand side of the previous equation may be neglected and there only remains

D E.


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3.5 Principal strains and special deformations

Both tensors E and D are symmetric and we will confine the discussion to one of the tensors,say E. The eigenvalue problem for E,

E n = n,means that one is looking for those directions n which remain parallel to themselves underpure deformation, that is when neither translation nor rotation occurs. The eigenvalues arethe roots of

det(E 1) = 3 + IE 2 IIE + IIIE = 0

123

with invariants

IE = tr E, IIE =1

2tr

E tr E E2 , III E = det E.Since E is symmetric, the eigenvalues are real and there exists at least three mutually or-thogonal eigenvectors. After normalization, these eigenvectors can be used as the base of acoordinate system in which E is diagonal. There are thus at least three different directionswhich remain unchanged when the continuum is subjected to pure deformation.

The first invariant IE has a special meaning and is often called the coefficient of cubicdilatation in small strain. To see where this name comes from, one considers an elementaryparallelepiped around P0, with volume

dV0 = da (db dc),where da, db, dc are parallel to the three coordinate axes, respectively. Use

dx = da F = da (1 +au)= da + da E + w da = (1 + 1)da + w da

and analogous relations between dy and db, and between dz and dc, where the computationsare done in the principal coordinate system for E. We thus find that

dV = dx (dy dz)= [(1 +

1)da + w

da]

{[(1 +

2)db + w

db]

[(1 +

3)dc + w

dc]

} (1 + 1 + 2 + 3)da (db dc) + da [db (w dc)]+ da [(w db) dc] + (w da) (db dc)

= (1 + tr E) dV0.

The relative change in volume is thus

dV dV0dV0

= tr E.

According to whether some of the invariants of E vanish, we can perform a classificationreminiscent of the one given for the stress tensor T. We distinguish the following special

states of strain (and a similar survey could be given for the rate of deformation, of course):


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General deformation:III

E = 0,

1,

2 =

3(possibly

1=

2).

The stretching for a particular direction n is defined as

(n) = nn : E = ninjEji = n2i i,

where the last expression is only valid in a principal coordinate system based on thestrain tensor eigenvectors.

Plane deformation:IIIE = det E = 0, IIE = 0, 1,2 = 0, 3 = 0.

All displacements for pure deformation are parallel to one of the coordinate planes ofthe system of principal axes, here the (1,2)-plane.

Linear deformation:IIIE = IIE = 0, IE = 0, 1 = 0, 2,3 = 0.

All displacements for pure deformation are now parallel to one of the axes of theprincipal coordinate system, here the first one.

Spherical deformation:

1 = 2 = 3 = , E = 1.

For every direction there is no shear and the stretching is the same, equal to . Thisimplies that every linear dimension is multiplied by the same number and the trans-formation from the initial to the final state is a homothetic one, changing the volumeof the continuum but neither the form nor the shape. In every coordinate system thestrain tensor is a multiple of the unit tensor.

For a general state of strain it is possible to distinguish between changes in volume and inform. Every tensor, not just E or a symmetric one, can be split in a unique way into aspherical part and a remainder, with the restriction that the remainder be traceless. Such atraceless part is then called the deviatoric part of the tensor. Thus one finds for E that

E = 1 + E, =1

3tr E =

1

3 u, tr E = 0.

In the special case of E, the spherical part is a third of the coefficient of cubic dilatationand indicates a change in volume without a change in form (homothetic transformation).The deviatoric part E is a deformation where the form changes without a change in volume,since the coefficient of cubic dilatation for E vanishes. Every deformation thus consistsconceptually of two parts: a change in volume without change in form, and a change in form without change in volume. Furthermore, the principal axes for E and E coincide, as

the spherical part is diagonal in any representation.


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3.6 Compatibility conditions for small strain

Up to now, the definitions of E and D have been interpreted as a way of calculating E orD, given the displacements u or the velocities v. There are six scalar equations giving Eij(or Dij), starting from the three components ui (or vi). The inverse problem consists indetermining the displacement u when the strain tensor E is given. It is clear that withoutspecial precautions this will be impossible, as we will have twice as many equations asthere are unknowns. Consequently, not every tensor is admissible as a strain tensor. Somerelations between the components of E have to exist in order to allow the determination of u.This question is not only of mathematical relevance but has some experimental implications,as it is the deformations rather than the displacements which are usually measured in anexperiment. The mathematical compatibility conditions then amount to consistency relationsfor the experimental data.

The required compatibility conditions are found upon elimination of u from

1

2(u + u) = E.

Taking the curl of both sides gives

E = 12

(u + u) =1

2 u = w = (w)t,

with the help of the particular expression

w = 12 ufor the dual vector w of the rotation tensor W. From

( E)t =wwe find that

( E)t = 0

This is the required compatibility condition of StVenant, rewritten in index notation as

ikjmnkmEn = 0.

Due to the symmetry ofE, this is also symmetric and hence equivalent to six scalar equations,three of the form

212E12 = 11E22 + 22E11,

and three of the form12E33 = 3(1E23 + 2E13 3E12).

In order to completely close the discussion, we will show that the compatibility conditionof StVenant is not only necessary but also sufficient to deduce the structure of E, if E issupposed symmetric. From

( E)t = 0


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there follows at once that( E)t =d,

and hence also that E = d.

Taking the trace of this tensor yields

d = tr E = ijk jEki = 0,since the tensor of Levi-Civita is completely antisymmetric and we have assumed that E issymmetric. As d is divergence-free, it can be written as the curl of yet another vector e andthus

E = e.From this one finds that

E = e+f.

Because of the symmetry of E this implies that

e+f =e + f,

or also that(f e) = (f e).

The index notation,i(fj

ej) = j(fi

ei),

shows that the vector expression is equivalent to

(f e) = 0.Hence f e has to be a gradient of a scalar,

f e =,and consequently,

E = e+(e +) =e + 12+e + 12has the structure required from a strain tensor.

There is also a compatibility condition on W, which is much easier to express through itsdual vector:

w =1

2 u w = 0.

Henceijk iWjk = 0.

The foregoing discussion was carried out for E and W (or w), and a similar discussion can

be given, of course, for D and (or ).


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Chapter 4

Constitutive relations

Notions of frame-indifference are used to construct constitutive relationswhich connect strain or rate of strain with stress

General isotropic media are known as Reiner-Rivlin fluids, with constitutiverelation

T = C(, D) = 0(, ID, IID, IIID)1 + 1(, ID, IID, IIID)D

+ 2(, ID, IID, IIID)D2

Simplifications occur for linear and isotropic media, recovering well knownlaws as Navier-Stokes (fluids)

T =

p()1 + (tr D)1 + 2D

and Hooke (elastic solids)

T = (tr E)1 + 2E

More general constitutive laws are possible, as for viscoelastic media withmemory effects

T(t) =

+

C() : E(t ) d

where causality allows the existence of Kramers-Kronig relations

C() = 1P+

C() d

C() = 1

P+

C() d

44


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4.1 Youngs experiment

In the previous chapters the notions of stress and strain or deformation were introduced ingeneral terms and the discussion was given without any attempt to relate the stress in agiven continuum to the strain or deformation it was undergoing. Stress and strain occurin all continua, but the as yet incomplete description through the basic laws has to bevalid for many different classes of material. The stress response to a certain strain willvary from one material to another, not just in numerical values but also in the form of thefunctional dependency. Hence the name of constitutive relations for the various stress-straindependencies we will survey.

The simplest starting point is to suppose that the stress depends on the strain in a linear way,without other parameters such as the temperature or the mass density entering the relation.

To see where these ideas come from we will briefly, and for the present only superficially,sketch Youngs experiment, in which a prismatic bar with cross-section S is subjected toa force F in the principal direction of the bar, as shown in Figure 4.1. For several elasticmedia the result was that for moderate forces the elongation increased linearly with theoriginal length of the bar at rest and with the magnitude F of the force applied at one endof the bar, the other end remaining clamped in position. At the same time decreasedwith the cross-section S of the bar, all in direct proportionality. Accordingly the result is

FS

.

If we label the principal direction of the bar the first direction, we can call the relativeelongation in this direction the element E11 of the strain tensor for this problem, keeping inmind the intuitive geometric interpretation of the diagonal elements of the strain tensor forsmall strains. On the other hand, the ratio F/S is the force per unit surface area, with thedimensions of a pressure (or traction!), nothing else but T11. Hence with

E11 =

, T11 =

F

S

we hav

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