A Every student of engineeringfrequently comes across the word function and as well as the words mapping and operator. Most likely he/she has seen the mathematical definition of a function in an introductory mathematics course and perhaps also been told that mapping and operator are basically syn- onymous terms. Nevertheless, since in applied mathematics, mechanics and engineering, the term function is mostly (or at least frequently) used in a slightly vague sense, connected to a traditional style of notation that may be confusing (see below), it seems as if the more precise concept of a mathemat- ical function becomes diffuse to many students. While I do not object to this common use of the notion of a function – it may be economical – I do believe that it is very useful to constantly have the exact mathematical definition in the background. Therefore, we will give such a definition together with the prerequisite notion of a set. A set is a collection of ob jects. In mathematics, an ob ject is frequently a natural or real number, a vector or point (see Appendix B), or a function (yet to be defined). When used in a physical model these objects will usually have referents in the real world. The objects are said to be members or elements of the set. In general, a set is denoted by a letter, say A, and if an object x is a member of A we write x ∈ A and the symbol ∈ should be read as “is an element of ” or “belongs to”. We can define a particular set by simply listing all of its elements. Usually curly brackets are used for this. Say that A is the set of all naturalnumbers from 1 to 4, then A = {1, 2, 3, 4}. (A.1) A.1 The Notion of a Set Sets and Functions 175
Transcript
A
Every student of engineering frequently comes across the word function andas well as the words mapping and operator. Most likely he/she has seen themathematical definition of a function in an introductory mathematics courseand perhaps also been told that mapping and operator are basically syn-onymous terms. Nevertheless, since in applied mathematics, mechanics andengineering, the term function is mostly (or at least frequently) used in aslightly vague sense, connected to a traditional style of notation that may beconfusing (see below), it seems as if the more precise concept of a mathemat-ical function becomes diffuse to many students. While I do not object to thiscommon use of the notion of a function – it may be economical – I do believethat it is very useful to constantly have the exact mathematical definition inthe background. Therefore, we will give such a definition together with theprerequisite notion of a set.
A set is a collection of objects. In mathematics, an object is frequently anatural or real number, a vector or point (see Appendix B), or a function (yetto be defined). When used in a physical model these objects will usually havereferents in the real world. The objects are said to be members or elementsof the set. In general, a set is denoted by a letter, say A, and if an object x isa member of A we write
x ∈ A
and the symbol ∈ should be read as “is an element of” or “belongs to”.We can define a particular set by simply listing all of its elements. Usually
curly brackets are used for this. Say that A is the set of all natural numbersfrom 1 to 4, then
A = 1, 2, 3, 4. (A.1)
A.1 The Notion of a Set
Sets and Functions
175
176
Another way of defining a set is by giving rules that identify its elements. Forinstance, the set (A.1) could be written
A = x : x is an integer, 1 ≤ x ≤ 4.
in this text we never use this type of definition for a set. Furthermore, thereare some symbols that can be taken as predefined definitions of sets. Forinstance, the symbol R is the set of real numbers, and R
2 is the set of pairsof real numbers. Similarly, R
n is the set of n-tuples of real numbers.If A and B are two sets, we say that A is a subset of B if each element
of A is also in B. This is denoted by the symbol ⊂ and we write
A ⊂ B.
The explicitly defined sets that we are using in this text are
(i) the physical Euclidean space E and subsets of this space, and(ii) bodies, denoted B, whose elements are material points; depending on what
body model is considered, the material points are natural numbers, realnumbers, pairs of real numbers or points in a subset of E .
functions, are alluded to in this text, but I have not felt a necessity to introduceparticular notations for these sets.
A function is a rule that associates each element in a set A to an element ina set B. We write
f : A → B
and think of f as the rule. If y ∈ B is the element that is associated to x ∈ Awe also write
y = f(x)
and y is called the image of x under f or the value of f at x. The set A is thedomain and B the co-domain of the function. The subset of B consistingof all the images under f is called the range of the function, denoted f(A).Figure A.1 gives an illustration of the concept of a function.
Note that a function consists of three objects: the rule f , the domain Aand the co-domain B. However, a common practice is to talk about f asthe function and let A and B be understood from the context. It is also fairlycommon to call f(x) the function, which following the scheme introduced hereis far from correct since f(x) is not even a rule but rather an element of theco-domain. A reason for this usage is that it gives a shorthand for indicatingwhich symbol is used for elements of the domain. A further common usage,
A Sets and Functions
The colon should be read as “such that” and the comma “and”. However,
Clearly, other sets, for instance, sets (sometimes called classes) of continuous
A.2 The Notion of a Function
177
Fig. A.1. The concept of a function
which may be confusing, but is sometimes useful, is to use the same symbolfor both the rule and the image, i.e., one writes
y = y(x).
The Notion of a Function
B
Many facts of mechanics are formulated by means of three-dimensional vec-tors. Examples of such vectors are position vectors, velocity vectors, accelera-tion vectors and force vectors. We call these vectors geometric vectors andnotationally they are distinguished from other mathematical objects, such asscalars and points, by use of bold face letters. In the overwhelming majorityof textbooks in mechanics and algebra, geometric vectors are conceived (oreven defined) as directed line segments. This is a very useful mental image ofa vector that gives considerable insight and intuitive feeling to problems ofmechanics and other scientific fields. However, it should be realized that suchan image is just but one possible interpretation of the mathematical structurerepresented by vectors and the operations related to vectors. In terms of theview of physical theories presented in Chapter 1, the directed line segment (inthe physical sense of the word) is a real world referent of the mathematicallystructure defined within the conceptual world. In the following we define anaxiomatic structure that can stand by itself, irrespective of the interpretationor which physical referent is chosen. The presentation can be seen as a com-plement to the treatment of most beginners’ textbooks. In fact, the commonintuitive definitions of geometric vectors and the space in which mechanicstakes place, should be sufficient for the understanding most of this text. Onthe other hand, the insight on how models of mechanics (and in this casegeometry) interplay with the real world should be strengthened by the follow-ing sections. Besides, it ties up some loose ends usually identified by seriousstudent of mechanics.
The presentation starts by defining Euclidean vector space and then usingthis structure to define Euclidean point space. Thus, vectors comes beforepoints, which is in contrast to the elementary treatment where points andspace are assumed at the outset.
The presentation is inspired by Chadwick [6] and the appendix in Trusdell[17]. Other useful references are Bowen and Wang [3, 4] and Uhlhorn [20].
Euclidean Point and Vector Spaces
179
180 B
From a mathematical point of view geometric vectors are elements of an ori-ented three-dimensional Euclidean vector space. We will introduce thismathematical structure in a step-wise fashion starting with axioms for addi-tion, and scalar multiples, then adding the scalar product axioms and finally,introducing the vector or cross product which leads to the idea of orientation.
A vector space V over the real numbers R (real numbers are typicallydenoted α and β) is a set of vectors (vectors are typically denoted u, v andw) and the following rules or axioms:
Addition axioms To every pair of vectors u and v there corresponds avector u + v, called the sum, with the properties:1. u + v = v + u (commutative property).2. u + (v + w) = (u + v) + w (associative property).3. There exists a vector 0 such that u + 0 = u for all u ∈ V .4. For every u ∈ V there exists −u ∈ V such that u + (−u) = 0.
Scalar multiple axioms To every vector u and real number α there corre-sponds a vector αu, called the product of u and α, with the properties:5. 1u = u.6. α(βu) = (αβ)u (associative property).7. (α + β)u = αu + βu (distributive property).8. α(u + v) = αu + αv (distributive property).
for u + (−v) and it holds that −u = (−1)u.A set of n vectors v1, v2, . . . ,vn is linearly independent if there is no
linear combination which equals to zero except the trivial one, i.e.,
α1v1 + α2v2 + . . . + αnvn = 0
is true only for α1 = α2 = . . . = αn = 0. A set of vectors which is not linearlyindependent is called linearly dependent. If there exists at least one set ofn linearly independent vectors in V , but no such set with n + 1 vectors, thenV is said to be n-dimensional.
In an n-dimensional vector space V , a set of n linearly independent vectorsis called a basis. Let b1, b2, . . . , bn be such a basis. It can then be shownthat any vector u ∈ V can be uniquely represented in the basis, i.e., there areunique real numbers α1, α2, . . . , αn such that
u = α1b1 + α2b2 + . . . + αnbn =n∑
i=1
αibi.
There are many referents of the mathematical structure of a vector spacein the real world. The reader may verify that straight directed line segments1
1 To be more precise, equivalence classes of directed line segments form a vectorspace.
Euclidean Point and Vector Spaces
B.1 Euclidean Vector Space
Note that the +’s in 7 represent different operations. Usually we write u−v
B.1 181
drawn on a paper are physical referents of two-dimensional vectors, given thatthe parallelogram law represents addition and the product operation is takenas a scaling (with direction) of the directed line segments. This interpretationis carried over to the three-dimensional case in an obvious way, perhaps bythinking of vectors as rigid thin rods.
A vector space becomes an Euclidean vector space if, in addition tothe operations of addition and scalar multiple introduced above, there existsa scalar product as follows:
Scalar product axioms To every pair of vectors u and v there correspondsa real number u · v, called the scalar product of u and v, with the prop-erties:9. u · v = v · u (commutative property).10. (αu + βv) · w = α(u · w) + β(v · w) (linearity).11. u ·u ≥ 0, and u ·u = 0 if and only if u = 0 (positive-definiteness).
(or absolute value) of a vector u, denoted |u|, is defined by
|u| =√
u · u.
A consequence of the scalar product axioms is Cauchy’s inequality
|u|2|v|2 ≥ (u · v)2,
where equality holds if and only if u and v are linearly dependent. Thisinequality allows us to introduce the notion of angle between vectors, denotedby φ, which is defined by the equation
u · v = |u||v| cosφ, 0 ≤ φ ≤ π. (B.1)
In a Euclidean vector space we distinguish a particular class of bases,namely the orthonormal bases. A basis e1, e2, . . . ,en is orthonormal if
ei · ej = δij , i, j = 1, . . . , n, (B.2)
where δij is Kronecker’s delta, which takes the value 1 when i = j, and 0when i = j. In the following we will deal with three-dimensional spaces only;a vector u can be written as
u = u1e1 + u2e2 + u3e3 =3∑
i=1
uiei, (B.3)
where (u1, u2, u3) are the components of u relative to the orthonormal basise1, e2, e3. These components can be expressed as
ui = u · ei, i = 1, 2, 3,
Euclidean Vector Space
Two vectors u and v are said to be orthogonal if u ·v = 0; the magnitude
182 B
which is found from (B.3) by using (B.2). Thus, we can write equation (B.3)as
u =
3∑i=1
(u · ei)ei. (B.4)
The representation (B.3) can be used to calculate the scalar productbetween two vectors. Let u and v have the components (u1, u2, u3) and(v1, v2, v3), respectively. Then, by using (B.2) and Axiom 10, one finds that
u · v =
(3∑
i=1
uiei
)·⎛⎝⎛⎛
3∑j=1
vjej
⎞⎠⎞⎞
=
3∑i=1
3∑j=1
uivjei · ej =
3∑i=1
3∑j=1
uivjδij =
3∑i=1
uivi. (B.5)
introduced thus far.We will now add the vector or cross product operation to our Euclidean
vector space.
vector u × v, called the vector product of u and v, with the properties:12. u × v = −v × u (anti-commutative property).13. (αu + βv) × w = α(u × w) + β(v × w) (linearity).14. u · (u × v) = 0.15. (u × v) · (u × v) = (u · u)(v · v) − (u · v)2.
Axiom 14 expresses that the vector u × v is orthogonal to both u and v.By using (B.1) one finds that Axiom 15 can be rewritten as
|u × v| = |u||v| sinφ.
absolute value of the vector product u×v represents the area of the parallel-ogram spanned by u and v.
The triple scalar product of three vectors u, v and w is defined asu · (v ×w). A consequence of the vector product axioms is that the followingholds:
u · (v × w) = v · (w × u) = w · (u × v)
= −u · (w × v) = −w · (v × u) = −v · (u × w), (B.6)
which will be used to show the following important results concerning therelations between vectors in an orthonormal base e1, e2, e3:
This formula shows that the scalar product is uniquely defined by the axioms
Thus, when thinking of vectors as oriented line segments, we see that the
ector product axioms To every pair of vectors u and v there corresponds aV
B.1 183
As will be commented on below, the ± sign indicates an indeterminancy inthe vector product as defined by the axioms. To prove (B.7) we first concludefrom Axiom 15 that
Thus, comparing (B.9) and (B.10) with (B.8) one finds
(e1 × e2) · e3 = ±1
and equations (B.7), which we attempted to prove, follow from (B.9) and(B.10).
Using Axiom 12 and equations (B.7) one finds that the following holdstrue:
ei × ej = ±3∑
k=1
εijkek, i, j = 1, 2, 3, (B.11)
where εijk is the permutation symbol, which takes the value 1 when i, jand k is a cyclic permutation of 1, 2 and 3; the value -1 when i, j and k is anon-cyclic permutation of 1, 2 and 3; and zero otherwise.
The representation (B.3) was used above in (B.5) to calculate the scalarproduct. It will be used also to calculate the vector product: from (B.3), (B.11)and Axioms 13 and 14 one finds
u × v =
(3∑
i=1
uiei
)×
⎛⎝⎛⎛
3∑j=1
vjej
⎞⎠⎞⎞
=
3∑i=1
3∑j=1
uivjei × ej = ±3∑
i=1
3∑j=1
3∑k=1
εijkuivjek. (B.12)
It also follows from this result and (B.5) that the triple scalar product can bewritten as
u · (v × w) = ±3∑
i=1
3∑j=1
3∑k=1
εijkuivjwk = ± det
∣∣∣∣∣∣∣∣∣∣∣∣∣∣u1 u2 u3
v1 v2 v3
w1 w2 w3
∣∣∣∣∣∣∣∣∣∣∣∣∣∣ . (B.13)
Euclidean Vector Space
The ± sign in the last two equations indicates that the axioms do not uniq-determine the vector product, contrary to the case with the scalar product.uely
184 B
However, the indeterminacy is not dramatic: there are, in a sense, two vectorproducts obeying the axioms, one being the negative of the other. We resolvethe indeterminacy by dividing the set of orthonormal bases into two classes,the right-handed and the left-handed bases. We take a particular base,say e1, e2, e3, and require that for this base it holds that
ei × ej =
3∑k=1
εijkek, i, j = 1, 2, 3, (B.14)
i.e., we take the plus sign in (B.11). All other bases can now be compared tothis base by calculating its triple product. From (B.4) we have for any basee1, e2, e3:
ei =
3∑i=1
(ei · ej)ej , i = 1, 2, 3.
i.e., Axioms 10 and 13, one finds that
(e1 × e2) · e3 =
3∑i=1
3∑j=1
3∑k=1
εijk(e1 · ei)(e2 · ej)(e3 · ek).
take e1, e2, e3 to belong to the same class of bases as e1, e2, e3, the right-handed bases, and we use the + sign in (B.12) when calculating the vectorproduct. If the product is 1 2 3
bases and we use the − sign in (B.12). The definitions of right-handedness andleft-handedness are extended to all linearly independent sets of three vectorsby looking at the sign of its triple scalar product.
The referents of left-handed and right-handed bases in the real world, whenthe directed line segment interpretation is chosen, should be well known fromelementary courses: for a right-handed base e1, e2 and e3 correspond to thethumb, the index finger and the middle finger of the right hand, respectively.
A three-dimensional Euclidean vector space with the operation of vectorproduct and definitions of right-handedness and left-handedness is an ori-ented three-dimensional Euclidean vector space.
A further useful connection between the operations of scalar and vectorproducts can be established. We may arrive at this by recalling the so-calledε-δ identity
3∑i=1
εijkεist = δjsδ δkt − δjtδ δks,
which can be verified by direct trial. By using (B.5) and (B.12) we find throughthis identity that any vectors u, v and w satisfy
u × (v × w) = (u · w)v − (u · v)w. (B.15)
Euclidean Point and Vector Spaces
ducts,
As has been found above, this triple scalar product equals 1 or -1. If it is 1, we
−1, e ,e ,e belongs to the class of left-handed
By using this equation, (B.14) and linearity in both scalar and vector pro-
B.2 185
Note that this identity does not need the definition of right-handedness sinceminus signs cancel out on the left hand side.
Geometric vectors viewed as directed line segments are conceived as connect-ing points in space. Thus, we seem to think that it is obvious that there is aspace of points in which mechanics goes on and this space is frequently calledphysical space. In the following we will define this space mathematically. Wewill then use the structure of Euclidean vector space defined in the previoussubsection and the result will be a Euclidean point space.
A Euclidean point space E is a set of points (points are typically denotedx, y and z) which is related to a Euclidean vector space V in the followingway:
(i)from x to y, which is denoted y − x, i.e. v = y − x.
(ii) y − x = (y − z) + (z − x).(iii) Given an arbitrary point o ∈ E and vector r ∈ V there exists a unique
point x ∈ E such that r = x − o.
the position vector, respectively. The space V is known as the translationspace of E .
Axiom (ii) is the parallelogram law for addition of vectors. Axiom (iii) saysthat given an origin o there is a one-to-one correspondence between positionvectors and points. Therefore, we may say that the dimension of the pointspace equals that of the corresponding vector space. The distance betweenpoints x and y is the magnitude of the corresponding vector, i.e. |x − y|.Furthermore, Axiom (iii) indicates that points and vectors may be added togive a point, i.e. we may write x = u + y. Note, however, that the addition oftwo points is not a meaningful concept.
The following results are proved from the axioms:
x − x = o, x − y = −(y − x).
B.2 Euclidean Point Space
Every pair of points (x, y) in E defines a vector v ∈ V, called the vector
andThe point o and the vector r in Axiom (iii) may be called the origin
Euclidean Point Space
C
In the following we will introduce the concept of a tensor, and present thoseresults from algebra and analysis that are used in this book. The aim is to makethe presentation reasonably self-contained, but readers are recommended toconsult a more complete mathematical textbook if they are encountering thesenotions for the first time. Note, however, that the mathematics presentedhere is mostly needed for the derivation and analysis of the three-dimensionalmodel. The presentations of the discrete and the one-dimensional models canbe followed by using elementary mathematics of geometric vectors.
C.1 Algebra
In this text a tensor will be a linear mapping from a three-dimensional Euclid-ean vector space to the same space, i.e., a mapping that linearly assigns avector v to any vector u.
That a mapping f is linear means that
f(α1u1 + α2u2) = α1f(u1) + α2f(u2) (C.1)
for all vectors u1 and u2, and all real numbers α1 and α2.We frequently write the action of a tensor T on a vector u without the
bracket notation that is used for general functions, i.e., T assigns to eachvector u a vector
v = Tu. (C.2)
Note that since vectors are geometric objects that are independent of anyparticular coordinate system or base, it is clear from (C.2) that tensors alsohave this property. This is the reason why it is logical to represent tensors bybold face notation. However, given an orthonormal base e1, e2, e3, tensorcomponents can be derived and used to represent tensors. To show this, westart by representing the two vectors u and v in (C.2) in components:
representations depend on the basis while the left hand sides of these relationsdo not.
Since an orthonormal base is related to a cartesian coordinate system, thecomponents TijTT and ui are frequently called cartesian components.
There are many definitions of tensors in the literature. The definition usedhere actually defines what is sometimes called a second-order tensor. In thisterminology a vector would be a first-order tensor. A very common definitionis based on satisfaction of rules for how tensor components transform under
Tensors and Some Mathematical Background
Equation (C.3) defines the components of T , and given this definition,and (C.4) are equivalent equations. We may say that the matrix in (C.4)
Obviously, it is not strictly correct to use = instead of ∼ here, since matrix
C.1 Algebra 189
change of coordinate system. If these rules are satisfied, equations based ontensor components become form invariant to coordinate change. The presentdefinition is automatically insensitive to coordinate change since vectors arecoordinate independent.
A member of a special class of tensors is formed from two vectors: thetensor (open or vector) product, u ⊗ v, of the two vectors u and v, is atensor defined by the property that
(u ⊗ v)w = (u · w)v
for any vector w. From (C.3) it follows that the components of the tensorproduct u⊗v are uivj (i, j = 1, 2, 3). It may be shown that any tensor T canbe written as
T =3∑
i=1
3∑j=1
TijTT ei ⊗ ej . (C.5)
i j
aI, defined by the property Iv = v for all vectors v, can be represented as
I =
3∑i=1
ei ⊗ ei. (C.6)
The product ST of two tensors S and T is defined by
(ST )v = S(T v)
for all vectors v. In particular, for tensors a ⊗ b, c ⊗ d and T it holds that
(a ⊗ b)(c ⊗ d) = (b · c)a ⊗ d, (C.7)
T (a ⊗ b) = (Ta) ⊗ b. (C.8)
The trace of a tensor is denoted tr T and is a linear operator on tensorswith the property
tr (u ⊗ v) = u · vfor all vectors u and v. By (C.5) and the linearity of tr we find
tr T =
3∑i=1
TiiTT .
orthonormal base.The transpose of a tensor T is denoted T T and is defined by
Tu · v = u · T T v (C.9)
space of tensors. As a particular case of (C.5) we have that the identity tensor(i, j = 1, 2, 3) may be seen as a basis for
That is, the trace of a tensor is the sum of its diagonal components in an
This equation indicates that e ⊗e
190 C
for all vectors u and v. In terms of components we may write
T T =3∑
i=1
3∑j=1
TjiTT ei ⊗ ej ,
and for the tensor product and arbitrary tensors S and T it holds that
(u ⊗ v)T = v ⊗ u, (C.10)
(ST )T = T T ST . (C.11)
By using (C.8), (C.10) and (C.11) we find
(u ⊗ c)T = u ⊗ (T T v). (C.12)
A tensor Q is orthogonal if
QQT = QT Q = I. (C.13)
Thus, for an orthogonal tensor we get the inverse by forming the transpose.The determinant of a tensor is the determinant of any of its matrix
representations, see Section 3.4. From (C.13) and standard properties of the1.
The orthogonal tensors that have a determinant valued 1 are called properorthogonal tensors. In Section 12.3 it was shown that proper orthogonaltensors represent rotations in the physical space.
C.2 Analysis
In this section we consider fields over the Euclidean point space E . By a fieldwe mean a mapping from (a subset of) E to a set of either scalars, points,vectors or tensors. Mechanics and thermomechanics contains many examplesof such fields:
(i) The temperature inside a solid or fluid body (which represents a subsetof E) varies from point to point and is a scalar field.
(ii) t
a region B ⊂ E into E and is thus a point field.(iii) The velocity of a fluid or solid contained in a region of E is a typical
vector field.(iv) Prime examples of tensor fields are stresses and strains inside a three-
dimensional body.
We are generally interested in how fields vary over E . To measure this, theconcept of a gradient is introduced. We are interested in gradients of scalar,point and vector fields. Let ϕ be a scalar field. The gradient of ϕ at x ∈ E isa vector field ∇ϕ(x) which satisfies
Tensors and Some Mathematical Background
−determinant it follows that an orthogonal tensor has the determinant 1 or
The placement map φ for a three-dimensional body (see Chapter 3) maps
C.2 Analysis 191
ϕ(x + u) = ϕ(x) + ∇ϕ(x) · u + o(u),
where o(u) is “small o of u” and is any function which approaches zero fasterthan its argument. The cartesian component representation of the gradient ofa scalar field is
∇ϕ(x) ∼
⎡⎢⎡⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎢⎢
∂ϕ
∂x1
∂ϕ
∂x2
∂ϕ
∂x3
⎤⎥⎤⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎥⎥.
Next, let v be a vector field. The gradient of v at x ∈ E is a tensor field∇v(x) which satisfies
v(x + u) = v(x) + ∇v(x)u + o(u).
∇u(x) ∼
⎡⎢⎡⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎢⎢
∂u1
∂x1
∂u1
∂x2
∂u1
∂x3
∂u2
∂x1
∂u2
∂x2
∂u2
∂x3
∂u3
∂x1
∂u3
∂x2
∂u3
∂x3
⎤⎥⎤⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎥⎥.
example is the deformation gradient F , see Chapter 3.Having defined the gradient of a vector field v, we can define the diver-
gence of the same field. It is denoted div v, and is a scalar field definedby
div v = tr ∇v.
In cartesian components div v has a simple representation:
div v =∂v1
∂x1+
∂v2
∂x2+
∂v3
∂x3.
We can also define the divergence of a tensor field T : divT is a vector fielddefined by
(div T ) · v = div(T T v)
for every constant vector v. In cartesian components it holds that
div T ∼
⎡⎢⎡⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎢⎢
∂T11TT
∂x1+
∂T12TT
∂x2+
∂T13TT
∂x3
∂T21TT
∂x1+
∂T22TT
∂x2+
∂T23TT
∂x3
∂T31TT
∂x1+
∂T32TT
∂x2+
∂T33TT
∂x3
⎤⎥⎤⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎥⎥.
The cartesian component representation of the gradient of a vector field is
The gradient of a point field is defined as that of a vector field. A prime
192 C
The curl of a vector field v is denoted by curlv, and is a vector field withthe property (∇v − (∇v)T
)a = (curl v) × a (C.14)
for every vector a. In cartesian components it holds that
curl v ∼
⎡⎢⎡⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
∂a3
∂x2− ∂a2
∂x3
∂a1
∂x3− ∂a3
∂x1
∂a2
∂x1− ∂a1
∂x2
⎤⎥⎤⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.
We also have the following useful expression for curl v in the orthonormalbasis e1, e2, e3:
curl v =3∑
i=1
3∑j=1
3∑k=1
εijkei
∂vk
∂xj
. (C.15)
In the development of three-dimensional continuum models we sometimesneed to take gradients, divergences and curls of products of scalar and vectorfields. Therefore, we state some of the rules for such operations. Let ϕ be ascalar valued field and let u and v be two vector valued fields. Then
div (ϕv) = ϕ div v + v · ∇ϕ, (C.16)
∇(v · u) = (∇v)T u + (∇u)T v, (C.17)
curl (ϕv) = ϕ curl v + ∇ϕ × v. (C.18)
We will also need thatdiv (∇v)T = ∇div v, (C.19)
and that for any constant tensor A,
∇(Av) = A∇v. (C.20)
The formulas (C.17) through (C.20) are easily established by writing themin equivalent cartesian component forms. For instance, in such components(C.17) reads
3∑i=1
∂(viui)
∂xj
=
3∑i=1
∂vi
∂xj
ui +
3∑i=1
∂ui
∂xj
vi, j = 1, 2, 3,
which is obviously true from the elementary product rule for differentiation.
Tensors and Some Mathematical Background
C.2 Analysis 193
C.2.1 Divergence theorems
The physical meaning of the divergence operation is best understood fromintegral theorems (divergence theorems). Such theorems are given with refer-ence to a region R ⊂ E with boundary ∂R. This region has to satisfy someregularity properties which are too technical to be stated here. There are dif-ferent divergence theorems depending on whether we are considering scalar,vector or tensor fields and whether we are, e.g., considering scalar or vectorproducts. We give four different versions in the following theorem.
Theorem 1 Let ϕ, v and T be smooth scalar, vector and tensor fields on R,
respectively. Then ∫∂
∫∫R
ϕn dA =
∫R
∫∫∇ϕ dV, (C.21)
∫∂
∫∫R
v · n dA =
∫R
∫∫div v dV, (C.22)
∫∂
∫∫R
n × v dA =
∫R
∫∫curl v dV, (C.23)
∫∂
∫∫R
Tn dA =
∫R
∫∫div T dV, (C.24)
where n is the outward unit normal vector of ∂R.
For the physical interpretation of (C.22) we may think of v as a velocityfield. The scalar v ·ndA is then the rate of volume of material entering R at aparticular surface point, and the right hand side of (C.22) is the total rate ofchange of volume. Thus, the equality in (C.22) means that div v is a measureof this volume change, and since the region R can be made arbitrarily small,it is a local such measure. This fact can be appreciated by writing (C.22) (ina somewhat lose mathematical style) as∫
v · n dA = div v dV,
for an infinitesimal volume element dV . A more precise indication of thesefacts are given in relation to the issue of isochoric (volume preserving) motionsin Chapter 7.
The divergence theorem (C.24) can be given a similar physical interpre-tation. As is clear in Chapter 7, if T is taken as the stress tensor, then Tn
represents the force per area on the surface element with unit outward normaln. By writing (C.24) for an infinitesimal element dV as∫
Tn dA = div T dV,
one concludes that divT is a local measure of the surface force on the element.
194 C
C.2.2 Localization theorems
The universal principles of mechanics are global principles relating to a con-glomerate of material points. However, complete models that we solve, ana-lytically or numerically, are mostly of local character relating to each point.In the discrete model of mechanics, this transfer from global to local is triv-ial, but for continuum models more subtle mathematics is needed. Here wepresent two mathematical theorems that have the required function. The firstis useful for the one-dimensional model while the second relates to the three-dimensional one.
Theorem 2 Let f(x) be a continuous function on an interval [c, d] of the real
line. If ∫ b
a
∫∫f(x) dx = 0
for all a and b such that c ≤ a < b ≤ d, then, for all x in the interval [c, d],then
f(x) = 0.
Theorem 3 Let f(x) be a continuous function on a subset R of E. If∫Ω
∫∫f(x) dV = 0
for every Ω ⊂ R, then
f(x) = 0.
The term continuous function should here be understood as a scalar, vectoror tensor field.
C.2.3 Fundamental theorems of variational calculus
This text does not in general concern variational formulations of the problemsencountered. Such formulations are seen as steps in the numerical modelingthat naturally comes after the initial formulation of a complete problem andwhich we do not treat. However, an exception is Chapter 9 on small dis-placement theories, where work principles, of variational nature, are used anddiscussed. To derive consequences of work principles one generally needs thenotion of separating duality which is classically related to the so-called funda-mental theorem of variational calculus. Below we give two special versions ofthis theorem: the first one is useful for one-dimensional models and the secondfor three-dimensional ones.
Theorem 4 Let f(x) be a continuous function on an interval [a, b] of the real
line. If ∫ b
a
∫∫f(x)g(x) dx = 0
Tensors and Some Mathematical Background
C.2 Analysis 195
for all continuous functions g(x), then,
f(x) = 0.
Theorem 5 Let S(x) be a continuous field of symmetric tensors on a subset
R of E. If ∫R
∫∫S(x) : T (x) dV = 0
for all continuous fields of symmetric tensors T (x), then
S(x) = 0.
Exercises
that
S : T = S :1
2(T + T T ).
definition of curl, i.e., (C.14), and (C.17). Identity (11.11) follows directly from(C.19).
C.1. Let S and T be tensors, where S is symmetric and T is arbitrary. Show
C.2. Prove the identities (C.16) through (C.20).
C.3. Prove the identities (11.9) and (11.11). Identity (11.9) follows from the
D
In Appendix B we defined the vector space V as a vector space over the real
numbers. This means that distances in E are real numbers, which may seemsomewhat in contrast to the convention that distance, as an experimentallymeasured quantity, has a physical dimension ([length]) and is not purely anumerical value (real number). However, if one regards the unit of lengthas fixed, it is clearly sufficient to give the numerical value of a distance tocharacterize it, and this is a motivation for the choice of structure of AppendixB.
Similar arguments hold for other concepts such as force, velocity, acceler-ation and mass: it is assumed that units of length, time, mass and force arefixed, allowing velocity, acceleration and force to be considered as belongingto the same oriented three-dimensional Euclidean vector space, denoted by Vin this text. This is also the reason that vector and scalar products betweenvelocity vectors and force vectors are valid operations.
Note also in this context that units of length, time, mass and force mustbe connected so as to make Euler’s laws dimensionally correct, i.e.,
where the square brackets indicate physical dimension.
Note on Physical Dimensions
197
E
Notation
General conventions are that
• vectors and tensors are shown in bold face italic letters – examples: v, T ;• points and scalars are shown in light face letters – examples: x, y;• vector spaces, point spaces, bodies and parts of bodies are shown in script
letters – examples: E , B, P , V .
A material accelerationa spatial accelerationα pipe cross section areaB bodyBt subset of E occupied by body at time tb force per unit volumeC subset of E representing a pipec torquec speed of soundcurl curlD rate of deformationdet determinantdiv divergencedVXVV material integration elementdVxVV spatial integration elementE physical space or space of generalized strainsE strain tensor or unit vectorE Young’s moduluse unit vectorε strain componentsF deformation gradient
Meaning of Main Symbols
199
200 E Notation
f forceφt placementϕ rotation of beam cross-sectionh angular momentumJ inertia tensorκ curvature and compression modulusL velocity gradientM massmX mass of discrete material point Xm couplen normal vectoro originω angular velocityΩ body angular velocityP part of bodyp linear momentump pressureψ naturally parametrized curveQ orthogonal tensorq force per unit lengthR the real lineR control domainr position vectorρ mass per unit length or volume (density)s traction vectors natural coordinateσ stress componentT stress tensort timetr traceu displacement vectorU force potentialV material velocityv spatial velocityV material speedv spatial speedVol volumeX material point belonging to Bx spatial point belonging to Exc center of mass∇ gradient⊗ tensor product× vector product· scalar product
φ material time derivative of φφ′ spatial time derivative of φ
References
1. Antman, S.S. (1995) Nonlinear Problems of Elasticity, Springer, New York.2. M.P. and Sigmund, O. (2003) Topology Optimization: Theory,
Methods and Applications, Springer, Berlin.3. Bowen, R.M. and Wang, C.-C. (1976) Introduction to Vectors and Tensors,
Volume 1: Linear and Multilinear Algebra, Plenum Press, New York.4. Bowen, R.M. and Wang, C.-C. (1976) Introduction to Vectors and Tensors,
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George Allen & Unwin Ltd, London.7. Curnier, A. (2004) Mecanique des solides d´M´M eformables´ , Presses polytechniques
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http://www.claymath.org/millennium/.9. Fox, E.A. (1967) Mechanics, Harper & Row, New York.
10. Hestenes, D. (1992) Modeling games in the newtonian world,Am. J. Phys. 60(8), pp. 732–748.
The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim ofthis series is to provide lucid accounts written by authoritative researchers giving vision and insightin answering these questions on the subject of mechanics as it relates to solids. The scope of theseries covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics;variational formulations; computational mechanics; statics, kinematics and dynamics of rigid andelastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories ofelasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes;structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimentalmechanics; biomechanics and machine design.
1. R.T. Haftka, Z. Gurdal and M.P. Kamat:¨ Elements of Structural Optimization. 2nd rev.ed., 1990ISBN 0-7923-0608-2
2. J.J. Kalker: Three-Dimensional Elastic Bodies in Rolling Contact. 1990 ISBN 0-7923-0712-73. P. Karasudhi: Foundations of Solid Mechanics. 1991 ISBN 0-7923-0772-04. Not published5. Not published.6. J.F. Doyle: Static and Dynamic Analysis of Structures. With an Emphasis on Mechanics and
Computer Matrix Methods. 1991 ISBN 0-7923-1124-8; Pb 0-7923-1208-27. O.O. Ochoa and J.N. Reddy: Finite Element Analysis of Composite Laminates.
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ISBN 0-7923-1355-010. D.E. Grierson, A. Franchi and P. Riva (eds.): Progress in Structural Engineering. 1991
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ISBN 0-7923-1504-9; Pb 0-7923-1505-712. J.R. Barber: Elasticity. 1992 ISBN 0-7923-1609-6; Pb 0-7923-1610-X13. H.S. Tzou and G.L. Anderson (eds.): Intelligent Structural Systems. 1992
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24. G. Prathap: The Finite Element Method in Structural Mechanics. 1993 ISBN 0-7923-2492-725. J. Herskovits (ed.): Advances in Structural Optimization. 1995 ISBN 0-7923-2510-926. M.A. Gonzalez-Palacios and J. Angeles:´ Cam Synthesis. 1993 ISBN 0-7923-2536-227. W.S. Hall: The Boundary Element Method. 1993 ISBN 0-7923-2580-X28. J. Angeles, G. Hommel and P. Kovacs (eds.):´ Computational Kinematics. 1993
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1994 ISBN 0-7923-3041-235. A.P.S. Selvadurai (ed.): Mechanics of Poroelastic Media. 1996 ISBN 0-7923-3329-236. Z. Mroz, D. Weichert, S. Dorosz (eds.):´ Inelastic Behaviour of Structures under Variable
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ics and Inverse Problems. 1996 ISBN 0-7923-3849-942. J. Mencik:ˇ Mechanics of Components with Treated or Coated Surfaces. 1996
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74. J.-P. Merlet: Parallel Robots. 2000 ISBN 0-7923-6308-675. J.T. Pindera: Techniques of Tomographic Isodyne Stress Analysis. 2000 ISBN 0-7923-6388-476. G.A. Maugin, R. Drouot and F. Sidoroff (eds.): Continuum Thermomechanics. The Art and
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103. J.A.C. Martins and M.D.P. Monteiro Marques (eds.): Contact Mechanics. Proceedings of the3rd Contact Mechanics International Symposium, Praia da Consolacao, Peniche, Portugal,˜17-21 June 2001. 2002 ISBN 1-4020-0811-2
104. H.R. Drew and S. Pellegrino (eds.): New Approaches to Structural Mechanics, Shells andBiological Structures. 2002 ISBN 1-4020-0862-7
105. J.R. Vinson and R.L. Sierakowski: The Behavior of Structures Composed of Composite Ma-terials. Second Edition. 2002 ISBN 1-4020-0904-6
106. Not yet published.107. J.R. Barber: Elasticity. Second Edition. 2002 ISBN Hb 1-4020-0964-X; Pb 1-4020-0966-6108. C. Miehe (ed.): IUTAM Symposium on Computational Mechanics of Solid Materials at Large
Strains. Proceedings of the IUTAM Symposium held in Stuttgart, Germany, 20-24 August2001. 2003 ISBN 1-4020-1170-9
MechanicsSOLID MECHANICS AND ITS APPLICATIONS
Series Editor: G.M.L. Gladwell
109. P. Stahle and K.G. Sundin (eds.):˚ IUTAM Symposium on Field Analyses for Determinationof Material Parameters – Experimental and Numerical Aspects. Proceedings of the IUTAMSymposium held in Abisko National Park, Kiruna, Sweden, July 31 – August 4, 2000. 2003
ISBN 1-4020-1283-7110. N. Sri Namachchivaya and Y.K. Lin (eds.): IUTAM Symposium on Nonlinear Stochastic
Dynamics. Proceedings of the IUTAM Symposium held in Monticello, IL, USA, 26 – 30August, 2000. 2003 ISBN 1-4020-1471-6
111. H. Sobieckzky (ed.): IUTAM Symposium Transsonicum IV. Proceedings of the IUTAM Sym-VVposium held in Gottingen, Germany, 2–6 September 2002, 2003 ISBN 1-4020-1608-5¨
112. J.-C. Samin and P. Fisette: Symbolic Modeling of Multibody Systems. 2003ISBN 1-4020-1629-8
113. A.B. Movchan (ed.): IUTAM Symposium on Asymptotics, Singularities and Homogenisationin Problems of Mechanics. Proceedings of the IUTAM Symposium held in Liverpool, UnitedKingdom, 8-11 July 2002. 2003 ISBN 1-4020-1780-4
114. S. Ahzi, M. Cherkaoui, M.A. Khaleel, H.M. Zbib, M.A. Zikry and B. LaMatina (eds.): IUTAMSymposium on Multiscale Modeling and Characterization of Elastic-Inelastic Behavior ofEngineering Materials. Proceedings of the IUTAM Symposium held in Marrakech, Morocco,20-25 October 2002. 2004 ISBN 1-4020-1861-4
115. H. Kitagawa and Y. Shibutani (eds.): IUTAM Symposium on Mesoscopic Dynamics of FractureProcess and Materials Strength. Proceedings of the IUTAM Symposium held in Osaka, Japan,6-11 July 2003. Volume in celebration of Professor Kitagawa’s retirement. 2004
ISBN 1-4020-2037-6116. E.H. Dowell, R.L. Clark, D. Cox, H.C. Curtiss, Jr., K.C. Hall, D.A. Peters, R.H. Scanlan, E.
Simiu, F. Sisto and D. Tang: A Modern Course in Aeroelasticity. 4th Edition, 2004ISBN 1-4020-2039-2
117. T. Burczynski and A. Osyczka (eds.):´ IUTAM Symposium on Evolutionary Methods in Mechan-ics. Proceedings of the IUTAM Symposium held in Cracow, Poland, 24-27 September 2002.2004 ISBN 1-4020-2266-2
118. D. Iesan:¸ Thermoelastic Models of Continua. 2004 ISBN 1-4020-2309-X119. G.M.L. Gladwell: Inverse Problems in Vibration. Second Edition. 2004 ISBN 1-4020-2670-6120. J.R. Vinson: Plate and Panel Structures of Isotropic, Composite and Piezoelectric Materials,
Including Sandwich Construction. 2005 ISBN 1-4020-3110-6121. Forthcoming122. G. Rega and F. Vestroni (eds.): IUTAM Symposium on Chaotic Dynamics and Control of
Systems and Processes in Mechanics. Proceedings of the IUTAM Symposium held in Rome,Italy, 8–13 June 2003. 2005 ISBN 1-4020-3267-6
123. E.E. Gdoutos: Fracture Mechanics. An Introduction. 2nd edition. 2005 ISBN 1-4020-3267-6124. M.D. Gilchrist (ed.): IUTAM Symposium on Impact Biomechanics from Fundamental Insights
to Applications. 2005 ISBN 1-4020-3795-3125. J.M. Huyghe, P.A.C. Raats and S. C. Cowin (eds.): IUTAM Symposium on Physicochemical
and Electromechanical Interactions in Porous Media. 2005 ISBN 1-4020-3864-X126.
ISBN 1-4020-4033-4127. W. Yang (ed): IUTAM Symposium on Mechanics and Reliability of Actuating Materials.
Proceedings of the IUTAM Symposium held in Beijing, China, 1–3 September 2004. 2005ISBN 1-4020-4131-6
H. Ding, W. Chen and L. Zhang: Elasticity of Transversely Isotropic Materials. 2005
MechanicsSOLID MECHANICS AND ITS APPLICATIONS
Series Editor: G.M.L. Gladwell
128. J.-P. Merlet: Parallel Robots. 2006 ISBN 1-4020-4132-2129. G.E.A. Meier and K.R. Sreenivasan (eds.): IUTAM Symposium on One Hundred Years of
Boundary Layer Research. Proceedings of the IUTAM Symposium held at DLR-Gottingen,¨Germany, August 12–14, 2004. 2006 ISBN 1-4020-4149-7
130. H. Ulbrich and W. Gunthner (eds.):¨ IUTAM Symposium on Vibration Control of NonlinearMechanisms and Structures. 2006 ISBN 1-4020-4160-8
131. L. Librescu and O. Song: Thin-Walled Composite Beams. Theory and Application. 2006ISBN 1-4020-3457-1
132. G. Ben-Dor, A. Dubinsky and T. Elperin: Applied High-Speed Plate PenetrationDynamics. 2006 ISBN 1-4020-3452-0
133. X. Markenscoff and A. Gupta (eds.): Collected Works of J. D. Eshelby. Mechanics and Defectsand Heterogeneities. 2006 ISBN 1-4020-4416-X
134. R.W. Snidle and H.P. Evans (eds.): IUTAM Symposium on Elastohydrodynamics and Microelas-tohydrodynamics. Proceedings of the IUTAM Symposium held in Cardiff, UK, 1–3 September,2004. 2006 ISBN 1-4020-4532-8
135. T. Sadowski (ed.): IUTAM Symposium on Multiscale Modelling of Damage and FractureProcesses in Composite Materials. Proceedings of the IUTAM Symposium held in KazimierzDolny, Poland, 23–27 May 2005. 2006 ISBN 1-4020-4565-4
136. A. Preumont: Mechatronics. Dynamics of Electromechanical and Piezoelectric Systems. 2006ISBN 1-4020-4695-2
137. M.P. Bendsøe, N. Olhoff and O. Sigmund (eds.): IUTAM Symposium on Topological DesignOptimization of Structures, Machines and Materials. Status and Perspectives. 2006
ISBN 1-4020-4729-0138. A. Klarbring: Models of Mechanics. 2006 ISBN 1-4020-4834-3
