Appendix I

Elements of Tensor Calculus

Key Word Appendix 1. Elements of Tensor Calculus 695 In Brief

Given a veetor spaee E and its dual spaee E*, t4e not ion of tensor arises when we study the multilinear forms on the spaee F defined as the order n Cartesian produet of E and E* (Seet. 1). One obvious way of produeing sueh forms eonsists in eonsidering those forms whose values are given by the produet of the values taken by n linear forms on E or E*. An n-linear form on F obtained in this way is ealled a product tensor of rank n, and it is written by inserting the symbol @ between eaeh of the forms on E or E* that make it up, these being themselves elements of E* or E, respeetively (Seet. 2). This eonstruetion does not generate the whole spaee of n-linear forms on F. However, it ean be used to produee a basis for the veetor spaee of n-linear forms on F, for example, given a basis of E and the dual basis for E*. All the forms we seek, known as tensors (of rank n) on F, are generated by linear eombination from sueh a basis (Seet. 3). The eorresponding veetor spaee is identified with a tensor produet of order n of E and E* . Two fundamental operations are defined on the tensors. The first is the tensor product, denoted by @. For two tensors of rank p and q, defined on spaees Fp and Fq , their tensor produet is a tensor of rank (p + q) on the product spaee F of Fp and Fq (or the other way round). This generalises the method for eonstrueting produet tensors from elements of E* and E (Seet. 2). The seeond is the contraction. This allows us, in eertain eireumstanees, to obtain tensors of rank (n - 2), (n - 4), and so forth, on spaees Fn - 2 , ... , from a tensor of rank n on F (Seet. 3). These operations ean be eombined. We obtain then the contracted product of two tensors, of great importanee in meehanics (Seet. 4). We know that for a spaee E with Euelidean strueture, there is a eanon• ieal (natural) isomorphism whieh allows us to identify E with its dual E*, defined by substituting the sealar produet in E for the duality produet. This isomorphism ean also be used to show that the Cartesian produet spaees of E and E* of arbitrary order n are themselves isomorphie. It follows that the tensor spaees defined on these produets are also isomorphie. Henee, we identify a linear form on E, element of E*, with its assoeiated veetor, ele• ment of E, using the eanonieal isomorphism, thereby introdueing the not ion of Euelidean veetor. In the same way, we shall identify the 2n tensors of rank 696 Appendix I. Elements of Tensor Calculus n which correspond to one another by the isomorphism with that tensor amongst them which is an element of the tensor product of rank n of E with itself: this gives us the corresponding Euclidean tensor. The two fundamental operations introduced earlier can be carried over in a consistent way to the Euclidean tensors. Contraction is always possible in this case; the associated rules are simplified and can all be expressed in terms of the scalar product on E (Sect. 5). One important application of tensor calculus is associated with the fact that we can generalise the ideas of gradient and divergence to higher orders. For a field of rank n tensors, defined on an affine space for which E is the associated vector space, the gradient at a point is the tensor of rank (n + 1) whose contracted product with an infinitesimal vector (differential element) in E gives the corresponding differential increment in the tensor field at this point. The divergence is obtained by contracting the gradient tensor. It is therefore a tensor of rank (n - 1). The divergence theorem transforming a fiux type surface integral into a volume integral is then extended to tensors of any rank (Sect. 6). Appendix I. Elements of Tensor Calculus 697

Main Notation

Notation Meaning First cited (, ) Duality product (1.2) 0 Tensor product (2.1) sj, Kronecker symbol (2.6) T == Ti/§.i0e*j 0§.k Tensor (3.3) det Determinant (3.16) tr Trace (3.17) Transposition symbol (3.18) 8 Contracted product on the last index of (4.3) the tensor before 8 and the first index of the tensor coming after it 0 Doubly contracted product on the (4.14) two indices next to 0, and on the two indices next to those G Metrie tensor (5.1) Scalar product (5.2) Contraeted produet for Euelidean (5.32) tensors: the same rule as for 8 coneerning indices Doubly contracted product for (5.36) Euclidean tensors: same rule as for o eoneerning indices T.. Euelidean tensor Seet. 5.8 t. Matrix for T.. in an orlhonormal basis Seet. 5.9

DllC Derivative along veetor Y1. (6.3) \7 Gradient (6.4) div Divergenee (6.8) 698 Appendix 1. Elements of Tensor Calculus

1 Tensors on a Veetor Spaee ...... 699 1.1 Definition ...... 699 1.2 First Rank Tensors ...... 700 1.3 Seeond Rank Tensors ...... 700 2 Tensor Produet of Tensors ...... 701 2.1 Definition ...... 701 2.2 Examples ...... 701 2.3 Produet Tensors ...... 702 3 Tensor Components ...... 703 3.1 Definition ...... 703 3.2 Change of Basis ...... 704 3.3 Mixed Seeond Rank Tensors ...... 705 3.4 Twiee Contravariant or Twiee Covariant Seeond Rank Tensors 707 3.5 Components of a Tensor Product ...... 708 4 Contraction ...... 708 4.1 Definition of the Contraction of a Tensor ...... 708 4.2 Contraeted Multiplieation ...... 709 4.3 Doubly Contraeted Produet of Two Tensors ...... 711 4.4 Total Contraction of a Tensor Produet ...... 713 4.5 Defining Tensors by Duality ...... 713 4.6 Invariants of a Mixed Seeond Rank Tensor ...... 714 5 Tensors on a Euclidean Vector Spaee ...... 714 5.1 Definition of a Euelidean Spaee ...... 714 5.2 Applieation: Deformation in a Linear Mapping ...... 715 5.3 Isomorphism Between E and E* ...... 715 5.4 Covariant Form of Veetors in E ...... 717 5.5 First Rank Euelidean Tensors and the Contracted Product .. . 718 5.6 Seeond Rank Euclidean Tensors of Simple Produet Form and their Contraeted Produets ...... 719 5.7 Second Rank Euelidean Tensors ...... 721 5.8 Rank n Euelidean Tensors ...... 726 5.9 Choice of Orthonormal Basis in E ...... 726 5.10 Prineipal Axes and Principal Values of a Real Symmetrie Seeond Rank Euclidean Tensor ...... 727 6 Tensor Fields ...... 729 6.1 Definition ...... 729 6.2 Derivative and Gradient of a Tensor Field ...... 729 6.3 Divergenee of a Tensor Field ...... 731 6.4 Curvilinear Coordinates ...... 732 Summary of Main Formulas ...... 737 1. Tensors on a Vector Space 699 Elements of Tensor Calculus

Without being too concerned with the mathematical formalism, which is available in many other works, the aim here is to provide the reader with sufficient basic knowledge to be able to use the tensor calculus in the context of 3-dimensional continuum mechanics as presented earlier in the book. This introduction to tensor calculus is organised into three parts. The first is de• voted to the definition of tensors on a vector space, presentation of their basic properties and discussion of the main operations of tensor calculus (Sects. 1 to 4). The second part deals with Euclidean tensors (Sect. 5), and the third tackles the quest ion of tensor fields and their derivatives. Concerning immediate applications to the subject of the present book, the second and third parts (for Euclidean tensors) will be the most relevant. For this reason, the summary of the main formulas given at the end of this appendix only presents results referring to Euclidean tensors. However, it seems preferable to provide an initial discussion that brings out the role of duality, before introducing a Euclidean structure.

1. Tensors on a Vector Space

1.1 Definition

Let E denote a vector space of finite dimension n (over ~ or C) and let E* be the dual of E. Then a p times contravariant and q times covariant tensor is any multilinear form T defined on (E*)P x (E)q. Denote by u*(i) p arbitrary vectors in E* i = 1, ... ,p , 12.(j) q arbitrary vectors in E j = 1, ... ,q . Such a multilinear form associates the following scalar with the vector argu• ments u*(i) and 1I.(j)' taken in this order:

T(u*(l), ... ,u*(p) ,12.(1)' ... ,12.(q)) . (1.1) The sum (p + q) is called the rank of the tensor. The pair of numbers (p, q) is called its type. The order in which the vector arguments occur in T must be specified in the definition of the form. In this presentation, we have chosen to order the arguments by first taking the vectors in E* and then taking those in E. It is clear that the set of all tensors of given type (p, q), and corresponding to the same order of vector arguments, can be supplied with a vector space structure. 700 Appendix I. Elements of Tensor Calculus

As an example, let us examine the first and second rank tensors, these being widely used in continuum mechanics.

Notation: To begin with, let (u*, '1)) = ÜL, u*) (1.2) denote the duality product between a vector u* in E* and a vector :l!. in E.

1.2 First Rank Tensors

First Rank Contravariant Tensor From the definition, this is a linear form T on E*, identified classically with a vector 'L of E by means of the duality product. We write

Vu* E E*, T(u*) = ('L, u*). (1.3) First rank contravariant tensors are just vectors in E

First Rank Covariant Tensor From the definition, this is a linear form on E, which is therefore identified with a vector of E*. First rank covariant tensors are just vectors in E*

1.3 Second Rank Tensors

Second Rank Covariant Tensor This is a bilinear form on E xE. In continuum mechanics, strain and strain rate tensors are second rank covariant tensors (see Chaps. 11 and 111).

Second Rank Contravariant Tensor This is a bilinear form on E* x E*. In continuum mechanics, stress tensors are second rank contravariant ten• sors (see Chap. V).

Mixed Covariant-Contravariant Second Rank Tensor This is abilinear form T on E* x E associating the scalar T( u* , :l!.) with two arbitrary vectors u* in E* and :l!. in E. It can be used to define by duality a linear mapping 'P from E into E:

Vu* E E* , V:l!. E E, T(u*,:l!.) = (u*, rp(:l!.)). (1.4) Conversely, given a linear mapping rp from E into E, we can define a mixed second rank covariant-contravariant tensor T by (1.4). Such definitions are commonplace in continuum mechanics. 2. Tensor Product of Tensors 701

In particular, we define the inverse tensor to the tensor T, denoted T-I, as the tensor associated with the linear mapping cp-I inverse to cp, assuming it exists. This is another mixed covariant-contravariant tensor:

(1.5)

An analogous process is possible with regard to linear mappings from E* into E*.

2. Tensor Product of Tensors

2.1 Definition

Consider as an example the two tensors T' , trilinear form on E x E* xE, T" , bilinear form on E* xE, and define the tensor T, tensor product of T' and T", denoted T = T ' 0T" , by

The tensor product operation is distributive with respect to addition (on the left and the right), and associative. Note that it is not commutative. (T is also called the outer product of T' and T".)

2.2 Examples

Tensor Product of Two Vectors in E If g, !2 E Ethen T = g 0!2 is defined by a rule similar to (2.1):

'V u*(I) , u*(2) E E* , (g 0 !2) (u*(I) , u*(2)) = (g, u*(1)) (!2 , u*(2)). (2.2)

Thus T = g 0!2 is abilinear form on (E*)2, also called a dyad.

Tensor Product of a Vector in E with a Vector in E* If gE E and b* E E* , T = g 0 b* is defined by a rule analogous to (2.1), yielding

'Vu* E E* , 'VQ E E, (g 0 !2*)(u*, Q) = (g, u*)(Q, b*) . (2.3)

We see that T = g 0 b* is a mixed covariant-contravariant tensor. 702 Appendix I. Elements of Tensor Calculus

The corresponding linear mapping cp from E into E, defined by (1.4), can be found explicitly in this case. According to (2.3), T(u*, Q) can also be written

T(u*, Q) = (u*, g(Q, b*)) .

Hence,

(2.4)

2.3 Product Tensors

Let T be a p-contravariant and q-covariant tensor of rank n = p + q. We say that T is a product tensor if it can be put into the form of a tensor product, in the desired order, of p vectors from E and q vectors from E*. Now let ü:.d be a basis for E, and define the dual basis {e*k} for E* by *j)_s:J (f.i , e - U i , (2.5) where 5f is the Kronecker symbol taking values

5{ = 1 if i = j , 5{ = 0 if i =f. j . (2.6)

We can then decompose the vectors of E and E* in terms of these bases:

Q=v§.e,e (2.7)

(2.8) adopting the repeated or dummy index convention, according to which we sum over pairs 01 repeated indices when one appears as an upper index and the other as a lower index. 1 As an example, let us consider the product tensor

(2.9)

We have

I::/U*(l) , U*(2) E E*, I::/Q E E, (2.10) e. ® e*j ® e )(u*(l) v u*(2)) u(1)vj u(2) . (-, -k , -, = ,k

1 The eonvention eoneerning the position of the indices is also eommon usage. As we shall see later on, its systematie nature helps with reading and writing formulas (see for example Seet. 3.2). 3. Tensor Components 703

3. Tensor Components

3.1 Definition

As an example, let T be a rank 3 tensor, 1-contravariant, 1-covariant and 1-contravariant. Using the bases {~k} and {e*k} introduced above, define

T( e *i ,f:.j' e *k) = Ti j k ,Z,],.. k = 1 , ... , n . (3.1)

Since T is linear, it follows that

(3.2)

using the summation convention for repeated indices wh ich will be taken as understood from this point on, unless otherwise stated. Comparing (3.2) with (2.10), we obtain the key formulas

T == T i / f:.i (>9 e*j (>9 f:.k (3.3) Ti .k T(e*i e. e*k) J = , -J ' (3.4)

Equation (3.3) gives the decomposition of an arbitrary 1-contravariant, 1-covariant, 1-contravariant tensor T in terms of the n 3 product tensors f:.i (>9 e* j (>9 f:.k (i , j , k = 1, ... ,n). Moreover, (2.10) guarantees thc linear independence of these n 3 product tensors. Indeed, we observe, using (2.5), that

Hence the product tensors f:.i (>9 e*j (>9 f:.k (i, j, k = 1, ... ,n) constitute a basis fOT" the vectOT" space of tensors T of type indicated above. The Ti/are the components of T in this basis. If u*(1) , u*(2) , 1!. denote arbitrary vector arguments of T, decomposed as in (2.7) and (2.8), we immediately obtain the value of T( u*(1) , 1!., u*(2)) using (2.10) and (3.3):

(3.6)

Needless to say, the results given here for a particular type of tensor T are of completely general scope. 704 Appendix I. Elements of Tensor Calculus

Note The idea of the tensor product of tensors was introduced in Sect. 2.1. In a more general mathematical presentation, we define the tensor product of spaces. It can then be shown that the vector space of tensors T, chosen as an example above to illustrate how tensors can be decomposed, is isomorphie to the tensor product of vector spaces E and E* written E 0 E* 0 E. It is convenient to identify these two spaces, so that we have, for the above tensors T,

TE E0E*0E,

a notation that is consistent with the decomposition formula (3.3). We shall adopt this notation throughout the following.

3.2 Change of Basis

Once again, as an example, let us consider the same tensors T in the space E0E* 0E. Apart from the dual bases {~d and {e*k} of E and E* used in the last section, let us suppose that we have another pair of dual bases on these same spaces, denoted {~~} and {e*/k} . Then the vectors in the new basis {~D for the vectors of E can be expressed in terms of the old basis {~k} by a formula oftype

(3.7)

The oof are the components of the vectors in the new basis for E in terms of the vectors in the old basis. Inverting this formula, we can express the old basis in terms of the new one:

(3.8)

where

(3.9) since the matrices with coefficients 001 and ßt are clearly inverses of one another. The relations between the new and old dual bases for E* are obtained by identification using the above formulas in the expression for the duality product:

*/j /) - d (e , ~ R - UR .

This gives us the new dual basis in terms of the old:

(3.10) 3. Tensor Components 705

and the inverse formula

(3.11)

Having established all these change of basis formulas, we would now like to investigate how T is expressed in the basis {~~ Q9 e*'j Q9 ~U:

T = T,i .k e' iO. e*'j iO. e' . J -" VY VY_k The new components are easily obtained by starting with (3.3), then substi• tuting in (3.8) and (3.11) and using the distributivity of the tensor product of tensors. It follows that

(3.12) with inverse formulas

(3.13)

Applying these formulas to the case of a first rank tensor T = Ti e*i belonging to E*, the linear forms on E, and therefore a 1-covariant tensor (Sect. 1.2), we find

T'i = af Te· This explains the terminology: the tensor is said to be covariant because its components, which carry lower indices, transform under change of basis in the same way as the original basis itself, in which the vectors also carry lower indices. For a first rank tensor T = Tj ~j in E, we obtain

T'j = ß~ T k , which explains, in the same way, why the tensor is said to be contravariant. The general formula for an arbitrary tensor T can be remembered in the following way. For each lower index, i.e., each degree of covariance, introduce an a factor, and for each upper index, i.e., each degree of contravariance, introduce aß factor. 2

3.3 Mixed Second Rank Tensors

Let T be an element of E Q9 E* and cp the linear mapping of E into E specified by (1.4). Then let cpi j denote the coefficients of the matrix3 of cp in the basis {~k} of E, so that

2 In practice, when we change basis in this way, it is often more convenient to reproduce the above type of argument, obtaining the formula by identification, rather than to apply the general formula (3.12). 3 The upper index numbers the rows, and the lower numbers the columns. 706 Appendix I. Elements of Tensor Calculus

V j j '11 - = v -J'e . rp(v)- = rpi J-<. v e· . (3.14)

Furthermore, let Ti j be the components of T in the basis {~m ® e*n} of E ® E*. Then, using (3.4) and (1.4),

T ij = T(e*i, f.j) = (e*i , rp (f.j)) , and with (3.14),

'IIi,j=l, ... ,n. (3.15)

Hence, the components 0/ T in the basis {f.m ® e*n} are identical to the matrix coefficients of the linear mapping from E into E associated with T in the basis {f.d. With regard to arbitrary changes 0/ basis {f.d for E, and provided we always use the dual basis {e*k} for E*, it follows that the components Ti j of T exhibit all the known properties of the matrix coefficients of a linear mapping. In particular, we know that in these basis changes, certain polynomial expressions are invariant. Recall that det [rpi j 1 is an invariant, and therefore

det [Ti j 1= det T is an invariant . (3.16)

Likewise for all the coefficients of the characteristic polynomial in A, ob• tained as the determinant of the matrix of the linear mapping from E into E defined by

'IIY.. E E ~ rp(y") - Ay" E E (A an arbitrary scalar) .

These coefficients form a basis for the n independent polynomial invariants of degrees 1 to n in rpi j. Among these, apart from det [ rpi j 1 of degree n, we find tr [rpi j 1w hich is of degree 1:

T 'i i = tr T is an invariant . (3.17)

(This can be checked directly, since by (3.12) we know that T'i i = ßj af Tj k = 8% Tj k = T kk.) This trace operation is a particular case of the contraction, to be studied shortly. Likewise, in mechanics we shall often use a basis of n independent polynomial invariants of degrees 1 to n, differing from the one mentioned above, obtained by contraction (see Sect. 4.6). We also introduce the transposed tensor tT of T. This is the element 0/ E* ® E defined by

'II u* E E*, 'IIY.. E E , { (3.18) tT(y", u*) = T(u*, Y..) .

tT can be written in the form 3. Tensor Components 707

tT = (tT)·j e*i (9 e· ~ -J ' so that, by (3.4) and (3.18),

(3.19)

If T is a produet tensor T = a* (9 Q, it is clear that tT = f2. (9 a* .

3.4 Twice Contravariant or Twice Covariant Second Rank Tensors

As an example, we eonsider twiee eovariant seeond rank tensors. • Symmetrie Tensors By definition, T E E* (9 E* is symmetrie if

't:I '11.' , '11." E E , T('Q' , '11.") = T('Q" , '11.') , so that far any basis {~J of E and dual basis {e*j} of E*, (3.3) and (3.4) imply

(3.20)

• Antisymmetrie Tensors Likewise, T E E* (9 E* is antisymmetrie if

W'V'Q ,'11. " E E , T(''11., '11. ") = - T("')'11., '11. , so that as above (3.21 )

• Any tensor T E E* (9 E* ean be uniqucly expressed as the sum of a symmetrie tensor Ts and an antisymmetrie tensor Ta, both in E* (9 E*: (3.22) Indeed, these tensors Ts and Ta are uniquely defined by

't:I'Q', '11." E E, Ts('Q', '11.") = ~[T('Q', '11.") + T('Q", '11.') 1 , { (3.23) 't:I '11.' , '11." E E , Ta ('11.' , '11.") = ~ [T ('11.' , '11.") - T ('11." , '11.') 1, whieh implies for any basis {~j} of E and dual basis {e*k} of E*,

(3.24)

• The same results hold for twiee contravariant tensors, provided that we raise the two indices. 708 Appendix 1. Elements of Tensor Calculus

3.5 Components of a Tensor Product

As an example, consider T' E E* ® E ® E* and T" E E ® E*:

j e. T' = T' • k e*i!G'U -J !G'U e*k , T" = T"R.m~ ® e*m .

Then it is clear that for T = T' ® T" ,

where

(3.25)

Hence, in the particular case of a product tensor T such as

we have simply

(3.26)

4. Contraction

4.1 Definition of the Contraction of a Tensor

As an example, consider a tensor T in E ® E* ® E* ® E. Let Ü~k} be a basis for E and {e*j} the dual basis for E*. Then the object Tc defined by4

VY..E E,VU*E E*, Tc(u*,Y..) =T(U*,fi'Y..' e*i) (4.1) is independent of the choice of basis {fk} and is in fact a tensor in E ® E*. Indeed, if {f~} denotes another basis of E, then using the notation in Sect. 3.2,

T( U * , f I i , Y.., e *'i) - O!ij ßik T( u,* fj , Y.., e *k) , implying by (3.9),

T( U * , f I. • , Y.., e *'i) - T( u,* fj , Y.., e *j) . ---- 4 Note the summation over repeated indices. 4. Contraction 709

The object Tc specified by (4.1) is thus intrinsically defined: it is a bilinear form on E* x E, i.e., a 1-contravariant 1-covariant tensor. The tensor Tc is said to be a contraction of T over the vector arguments 2 and 4, or over indices 2 and 4. Note that contraction can only be carried out for indices corresponding to one vector argument in E and the other in E*. In terms of components, it is a simple matter to check that

i i k (Tc) j = T kj (4.2)

(summation over indices 2 and 4, one upper and the other lower). The definition given for this example is actually completely general. The contraction of a rank n tensor of type (p, q) produces a tensor of rank (n - 2) and type (p - 1, q - 1).

4.2 Contracted Multiplication

The contracted product of two tensors TI! and T' is found by first carrying out the tensor product T = TI! (>9 T' and then contracting over one index of TI! and one index of T'. In the most common case, the tensor product T = TI! (>9 T' is contracted over the last index of TI! and the first index of T', provided of course that this operation is allowed, i.e., that one of these indices is raised and the other lowered. The result of this contracted multiplication will be denoted

Tc = TI! 8T'. (4.3)

The operation (4.3) is often encountered in mechanics and we shall ex• amine several particular cases below. Note that the contracted product is distributive over addition on the right and on the left. (It is called an inner product.)

Contracted Product of Q E E and b* E E* We have

i T = -a (>9 b* = a bJ -,e· (>9 e*j , whence,

which is just the duality product (Q, b* ):

Q 8 b* = (Q, b* ) . (4.4) 710 Appendix I. Elements of Tensor Calculus

Contracted Product of TEE Q9 E* and Q E E Now

whence,

which is just the image vector in E of 12. under the linear mapping

T 812. =

Note also that if we consider the tensor product of 12. and tT, i.e., 12. ® tT, its contraction yields

12. 8 tT = T 8 12. =

Contracted Product of Two Mixed Second Rank Tensors Consider T' and T" E E ® E* , and the corresponding linear mappings

Tc = T" 8T'

is itself a mixed second rank tensor in E ® E* and that, if

i ('T' )i T"i T'k "i ,k (

It also follows that

I;j 12. E E, (T" 8 T') 812. = T" 8 (T' 812.) , (4.8)

which expresses associativity of the contracted multiplication in this particu• lar case. As usual, associativity allows us to write express ions of the following type without bracketing:

T'" 8 T" 8 T' 8 12. , etc.

In particular, consider TEE ® E* and the inverse tensor T~l defined in Sect. 1.3. As an immediate application of (4.8), we then have 4. Contraction 711

T-1 8 T = T 8 T- 1 = I, (4.9)

where I is the tensor in E (9 E* associated with the identity map from E to E. In terms of components, it is clear that

(4.10)

Finally, it is an immediate consequence of (4.6) and (4.8) that, if T' and T" are two elements of E (9 E*, then

t(T' 8 T") = tT" 8 tT' . (4.11)

Doubly Contracted Product of T E E* 0 E* , 'Q' E E and 'Q" E E Referring back to the definition (4.3) of the symbol 8 given above, we see that the notation

can be unambiguously interpreted and corresponds to the double contraction of the tensor 1I' = Y..' 129 T 129 y.." over indices 1 and 2 and over indices 3 and 4. Then Tc is just the scalar

rr _ T 'i "j 1 c - ij V V , (4.12)

which means that, by (3.6),

(4.13)

4.3 Doubly Contracted Product of Two Tensors

Consider two tensors T' and T" of rank greater than or equal to 2, and take the tensor product T = T" Q9T'. The doubly contracted product, denoted 0, corresponds to the double contraction of T over the last index 0] T" and the first index 0] T', and then over the penultimate index 0] T" and the second index o]T', provided that these contractions are possible, i.e., provided that the pairs of indices mentioned always include one upper index and one lower index.5

5 The convention we have adopted here with regard to the indices involved in the double contraction symbolised by CD will be maintained in Sects. 5.6 and 5.7 for Euclidean tensors. Their double contraction will be symbolised by " : ". Other conventions are used in the literature. There are some cases, particularly concerning Euclidean tensors, where the double contraction, symbolised in the sam8; way, operates first over the penultimate index of T" and the first index of T', then over the last index of T" and the second index of T'. It is therefore wise to check the exact meaning of the notation in different cases. 712 Appendix I. Elements of Tensor Calculus

We then write Tc = T" OT' . (4.14) The doubly eontraeted product is clearly distributive over addition on the right and on the left.

Doubly Contracted Product of a Twice Covariant Tensor and a Twice Contravariant Tensor Consider two seeond rank tensors

A = aij e*i (>9 e*j E E* (>9 E* and B = bij f:.i (>9 f:.j E E (>9 E . For these two tensors, the doubly eontracted product defined by (4.14) is written

Tc = AOB = aij lJi . (4.15) This sealar ean also be identified with

A 0 B = tr (A 8 B) . (4.16) Note that, in this ease, the doubly eontraeted produet is eommutative: AOB=BOA. Applying the results of Sect. 3.4, we ean express eaeh of the tensors A and B in the form of a sum of its symmetrie and antisymmetrie parts, obtained from (3.23):

A =As+Aa, { (4.17) B = Bs +Ba .

It ean then be eheeked, for example by writing As 0 Ba in the form (4.15) and using the eharaeteristie properties of A s and Ba, that

AOBa = (as)ij(ba)ji = (as)ij(-ba)ij = -(as)ji(ba)ij = -AOBa , whieh implies that

As 0 Ba = 0 and likewise Aa 0 Bs = 0 . (4.18) We have thus shown that A 0 B = A 0 Bs + Aa 0 Ba . (4.19) The doubly eontracted product 0 ean therefore be viewed as a duality product between the spaees E* (>9 E* and E (>9 E. Equation (4.19) gives the expression when elements of E* (>9 E* are deeomposed into symmetrie and antisymmetric tensors aeeording to (4.17). 4. Contraction 713

4.4 Total Contraction of a Tensor Product

Quite generally, given two rank n tensors A and ß, the first being p times con• travariant and q times covariant and the second being q times contravariant and p times covariant, we may totally contract the tensor product T = A®ß. All pairs of opposite type indiees over which contraction is carried out must be stated. We then obtain a scalar Tc. The doubly contracted product of a twice covariant tensor and a twice contravariant tensor discussed in the last section is clearly a case in point, as is the contracted product of a vector in E and a vector in E*.

4.5 Defining Tensors by Duality

As an example, consider some tensor A in E®E* ®E. Let X be an arbitrary tensor in E* ® E ® E*. We take the tensor product

( 4.20)

The total contraction of this tensor product over the pairs of indices 1 and 4, 2 and 5, 3 and 6 gives a scalar Tc which is a linear function of X. Therefore, starting with the given tensor A in E ® E* ® E, we can define a linear form a on E* ® E ® E*, that is, an element of (E* ® E ® E*)*. E ® E* ® E and (E* ® E ® E*) * are isomorphie. In continuum mechanics (see for example Chap. V, Sect. 3 for the repre• sentation of internal forces), we shall use the converse of this result: given a linear form a on a vector space of p times contravariant and q times covariant tensors X, we use it to define a tensor A of opposite type. (We will specify the pairs of indices to be contracted together in the total contraction of the tensor prod uct (4.20).) In particular, a linear form a on E ® E defines a tensor A of E* ® E* through the duality product 0, using the formula

\lXE E®E AOX = a(X). (4.21)

If the linear form a is only defined on the subspace of symmetrie tensors in E ® E, that is, for X E (E ® E)s, then the tensor A associated with a by the relation

\IX E (E®E)s, AOX = a(X) (4.22) is only determined up to addition of an arbitrary antisymmetrie tensor. This is because, by (4.19), equation (4.22) only determines the symmetrie part of A. 714 Appendix 1. Elements of Tensor Calculus

4.6 Invariants of a Mixed Second Rank Tensor

If T denotes a mixed contravariant-covariant tensor, we saw in Seet. 3.3 that the n coefficients of the characteristic polynomial A, written det [Ti j - A 6J l, are invariant under any change of basis {~d for E, with {e*k} being the dual basis for E*. It is common practice in mechanics to substitute another set of n inde• pendent polynomial invariants, of degrees 1 to n in the components Ti j, for the classic set of n polynomial invariants (tr T, ... , det T) which are also of degrees 1 to n in the components Ti j. These are obtained by the following total contractions:

(4.23)

11. . In = -tr (T 8 T 8 T 8 ... 8 T) = - T"j Tl k ... TP i . n n

5. Tensors on a Euclidean Vector Space

5.1 Definition of a Euclidean Space

The vector space E has a Euclidean strueture if E x E is equipped with a fundamental positive definite symmetrie bilinear form, denoted G and called the sealar produet. Using the notation of (4.13), we can write

(5.1)

and adopting the usual notation for the scalar produet,

\-IvQ,Q I " E E , G(Q,Q I ") =Q.QI" , (5.2)

an expression to which we shall return in Seet. 5.5. G is called the metrie tensor. Let gij denote its components relative to a basis {~d of E and the dual basis {e*k} of E*. Then as a consequence of (3.3) and (3.4):

(5.3) 5. Tensors on a Euelidean Veetor Spaee 715

5.2 Application: Deformation in a Linear Mapping

Let

mixed tensor in E®E*. We would like to evaluate, for two arbitrary vectors '!2-' , 'Q", the sealar produet of their images in E under

(5.4)

Defining the tensor C = tF 0 G 0 F in E* ® E*, it ean be eheeked that (5.4) simplifies by assoeiativity to (5.5)

The tensor C = tF 0 G 0 F is the twiee covariant tensor on E ® E giving the sealar produet of the images under

5.3 Isomorphism Between E and E*

The Euclidean structure of E leads to a canonical isomorphism between E and its dual E*. This isomorphism "I is defined by

V'Q' E E , V'Q E E , 'Q' . 'Q = ('Q' , 'Y('Q) ) , (5.6) where 'Y('Q) is the image under "I of'Q in E*. 6 Introducing the contracted product G 0) 'Q, it is clear from (5.6) that

(5.7)

With the basis {~J for E and the dual basis {~*k} for E*, let 'Yij be the matrix coefficients of "I, defined by

'Y(~j) = 'Yij e*i . (5.8)

Comparing with (5.7), it follows that

gij = 'Yij . (5.9)

Hence, the matrix coefficients of the isomorphism "I relative to the dual bases {~i} and {e*k} of E and E* are equal to the components of the metric tensor in the basis {e*i ® e* j}. The isomorphism "I naturally induces a Euclidean structure on E*. The fundamental bilinear form G* on E* x E* is defined as the image of the

6 In other words, the isomorphism "y assoeiates with 'Q the linear form "Y('Q) whose duality produet with any veetor 'Q' in E is just the sealar produet of'Q with this same veetor 'Q'. 716 Appendix 1. Elements of Tensor Calculus bilinear form G on E x E: for two elements U*(l) , u*(2) of E* , its value is the scalar product of their inverse images in E under T

(5.10)

If gij are the components of G* for bases {~k} and {e*k}, then by (3.3) and (3.4),

= gij e· ® e· -< -J' (5.11) = G*(e*i, e*j) .

It is useful to introduce the vectors in E, denoted f.k , which are images under 1'-1, inverse isomorphism to l' , of the vectors e*k in the basis for E* dual to {f.d:

(5.12)

It is then an obvious consequence of (5.11) and (5.10) that

(5.13) whilst (5.6) and (5.12) imply that f.i .f.j = (f.i' e*j), and hence,

(5.14)

The vectors f.i form a basis {f.k} for E called the dual in E of the basis {f.d: each vector f:.i in the basis {f.k} is therefore 'orthogonal' to (n - 1) vectors in the original basis {f.k} and such that its scalar product with the nth vector is equal to 1 (see Figs. 1 and 2).

E*

( e e o$1 >= {j J _t' I

f!..' = y'. ( * J)

Fig. 1. A basis for E and its duals in E* and E 5. Tensors on a Euclidean Vector Space 717

The eomponents ryi j of the inverse isomorphism "1-1 are given by

ry-l(e*j) = ryi j fi = fj ,

and we deduee from (5.13) and (5.14) that

ij ",kj ei . e ",ij g = I - -k = I , (5.15) so that

j j -e = gi -te· . (5.16) Then, eomparing (5.9) and (5.15), k' . gik 9 J = öf (5.17) and

(5.18)

(reealling that G and G* are symmetrie). The main formulas are:

j i j gtJ .. - -ie . -j e , gi - -e '-e gik gk j = ö{ (5.19) ei=gij . e.=g·ej - -Je ' -t tJ_

Let us just note that if the original basis {fd is onhonormal, it is iden• tical to its dual basis {fk} in E.

5.4 Covariant Form of Vectors in E

The eonstruetion of the dual basis in E deseribed above ean be used to define an assoeiated eovariant form for any veetor in E. We express

(5.20) where the Up are ealled the covariant components of the veetor 11c. This terminology is justified by the fact that the eomponents Up defined by (5.20) are also the eomponents of the linear form ry(11c) relative to the dual basis {e*k} for E*:

ry(11c) = up ry(fR) = Up e*P .

These eomponents therefore transform as the eomponents of a eovariant vee• tor, by the rule established in Seet. 3.2. Figure 2 illustrates (5.19) and (5.20) in the speeifie ease E = ]R2. 718 Appendix 1. Elements of Tensor Calculus

I! "-"-,,- I~i 1= 1 / "- / "- / 2 '" ~1·~2=-sine e // "- / "- / / " i / "" k 1 = 1/ cos "- e " "- " " " "-

~1

Fig. 2. E = ]R2, with a normed basis Ü:~J. The contravariant components are the oblique coordinates along fl and f2 ; the covariant components are the lengths of the orthogonal projections of 1! in the directions of fl and f2. For example, Ul = 1!·fl

5.5 First Rank Euclidean Tensors and the Contracted Product

The isomorphism between E and E* means that any element 1& of E can be given two interpretations: • its original interpretation as a vector in E, • its dual interpretation as a linear form on E, via the scalar product. The covariant representation of the elements in E illustrates this dual interpretation, as we saw in the last section:

1! = uC 0 expresses the vector aspect , (5.21 ) 1& = Uc lexpresses the dual aspect . (5.22)

Definition Bearing this in mind, we now introduce the not ion of a first rank Euclidean tensor. Any vector 1& in E (first rank contravariant tensor) and the covariant tensor u* = '")'(1&) associated with it by the canonical isomorphism will from now on be treated as a single tensor called a first rank Euclidean tensor. It is identified with the vector 1& in E, and we shall see that it will be possible to carry out all operations previously defined for first rank tensors. The Euclidean tensor 1& can be decomposed either by (5.21) or by (5.22), which are called the contravariant and co variant representations of 1&.

Contracted Product Considering two first rank Euclidean tensors 1& and :!!., we define their con• tracted product by observing that

(5.23) 5. Tensors on a Euclidean Vector Space 719

This value, obtained by contracting one of the vectors with the linear form associated with the other, is the obvious thing to define as the contracted product of Euclidean tensors 'lJ and 1!.. Moreover, a glance at (5.23) shows that 'lJ 8 v* = u* 81!. is just the scalar product 'lJ.1!. of vectors 'lJ and 1!. in E. Consequently, we can now use the symbol " . " to denote the contraetion of two first rank Euclidean tensors: _ i j_ i_i 'lJ .1!. - u 9ij V - U Vi - Ui V . (5.24) Use of the symbol ". " emphasises the fact that the contracted product of two first rank Euclidean tensors is simply obtained by calculating the scalar product of the vectors 'lJ and 1!. in E, given (5.19) and (5.20). Hence, for example:

'lJ'1!. = (ui ~i)' (vj ~j) = ui vj(~i '~j) = 9ij ui vj , { or alternatively , (5.25) i j i j i --u. V = (u -,e.). (v'J -e ) = u v J·(e -, .. -e ) = u V·", = u· Vi which shows why it was useful to introduce the covariant representation.

5.6 Second Rank Euclidean Tensors of Simple Product Form and their Contracted Products

The isomorphism '/ set up between E and E* naturally implies that E x E , E x E* , E* x E and E* x E* are all isomorphie. It therefore follows that the spaces of second rank tensors E* Q9 E* , E* Q9 E , E Q9 E* and E Q9 E are also isomorphie to one another. In order to examine the consequences of these isomorphisms, we begin by considering simple product tensors. Letting Q and Qdenote two vectors in E , the above isomorphisms between the tensor spaces relate the product tensors

a Q9 b a Q9 b* a* Q9 band a* Q9 b* { (5.26) :he~~ :* = ,/~Q) a:d b* = ,/(Q) , ' in such a way that, considering for example T = Q Q9 Q and T' = a* Q9 b* , we have

V'lJ, 1!. E E T'('lJ, 1!.) = T(u*, v*) , (5.27)

where u* = ,/('lJ) and v* = '/(1!.) . Writing out T(u*, v*) and T'('lJ, 1!.) using (2.2), (4.4) and (4.13) yields T(u* , v*) = u* 8 T 8 v* = (Q 8 u*)(Q 8 v*) (5.28) and T' ('lJ, 1!.) = 'lJ 8 T' 8 1!. = (a * 8 'lJ) (b* 8 1!.) . (5.29) 720 Appendix 1. Elements of Tensor Calculus

Definition Comparing these relations with (5.23) and (5.24) which define the contracted product of two first rank Euclidean tensors, we can consistently justify the following. • We define the second rank simple product Euclidean tensor 'L, identified with the tensor g (29 Qin E (29 E , for which (5.26) gives the four associated expressions:

• We use the symbol" . " to denote the contraction of this tensor with a first rank Euclidean tensor (maintaining our previous conventions concerning the relevant indices, see (4.3) and (4.13) in Sect. 4.2):

(5.30)

• Explicitly, this expression becomes

(5.31)

• This involves the contracted products of the first rank Euclidean tensors g and 1&, Q and 1l., which can be calculated as discussed above with regard to (5.25).

Contracted Product of a Second Rank Euclidean Tensor in Simple Product Form with a First Rank Euclidean Tensor Likewise we may check that the notation

(5.32) is consistent for the contracted product of a second rank Euclidean tensor in simple product form and a first rank Euclidean tensor, defined as the contracted product according to (4.5) of the mixed form associated with 'L in E (29 E* and the vector 1l. in E. Explicitly, -

(5.33) which can be calculated using (5.24) and (5.25).

Contracted Product of Two Second Rank Euclidean Tensors in Simple Product Form We can also define the contracted product of two second rank Euclidean tensors in simple product form, 'L" = g" (29 I{ and 'L' = g' (29 Q', as the contraction of the product of the ~ixed tensors assoc~ted with 'L" and 'L' in E (29 E*. We then check, using (5.32) and (5.33), that the notation 5. Tensors on a Euclidean Vector Space 721

'r = t' .'r' = (rl' 181 fl') . ([i 181 fl) (5.34) for this product is unambiguous. Explicitly,

'r = (Tl' . g')g" 181 fl , (5.35)

which is easy to calculate in terms of Euclidean tensors.

Doubly Contracted Product of Two Second Rank Euclidean Tensors in Simple Product Form The doubly contracted product of the above Euclidean tensors 'I'.." and 'I'..' is defined as the doubly contracted product given by (4.4) of twoexpresSions associated with 'I'.." and 'I'..', respectively, which have appropriately opposite types. For this product, we use the notation

'r" : 'r' = (g" I8Ifl') : (g' 181 fl.') , (5.36) which is indeed unambiguous (in the sense that it is independent of any particular choice of associated expression for 'r" and 'r'). We then have

(g" 181 fl.") : (g' 181 fl.') = (g" . fl.') (fl." . g') , (5.37) allowing straightforward calculation.

5.7 Second Rank Euclidean Tensors

The above results concerning second rank tensors in simple product form have shown that: • we may introduce a notion of Euclidean tensor, identified as an element of E 181 E to which four different expressions (5.26) are associated by the canonical isomorphism "( ; • all contraction operations defined in Sect. 4 over opposite type indices, are still well defined in the case of Euclidean tensors ; • and all these operations can be simply expressed in terms of the scalar product on E. These basic results can be extended to general second rank tensors if we decompose into simple product tensors (Sect. 3) and use distributivity of the tensor product over tensor addition.

Definition The isomorphism between E 181 E, ... ,E* 181 E* induced by the isomorphism 'Y is established by equations such as (5.27). Considering the tensor

Tij e· e . T = -t 181 -J in E 181 E , (5.38) 722 Appendix I. Elements of Tensor Calculus the following tensors are associated with it in E ® E* , in E* ® E, and in E* ® E* :

T' = T i k fi ® e*k with Tik = Tij gjk , { Tkj e*k ® TI! = -Je" with Tkj = gki Tij , (5.39) T III = T kR e*k ® e*R with T kR = gki Tij gjR .

The Euclidean tensor corresponding to these fOUf associated tensors is T.., identified with the element of E ® E, -

T = Tij e" ® e" . (5.40) = -'-J The distributivity ofthe tensor product over addition (Sect. 2.1) is clearly conserved for Euclidean tensors through this definition, so that we may de• compose the vectors fi and fj in (5.40) in terms of the dual basis {fk} ac• cording to (5.19). We then obtain new expressions for r, such as

so that by (5.39),

T i k = Tij gjk , Tkj = gki Tij , (5.41) TkR = gki Tij gjR .

We shall say that (5.40) and (5.41) constitute the four representations of the Euclidean tensor T... This yields twice contravariant, 1-contravariant 1-covariant, 1-covariant l=Contravariant, and twice covariant representations. The terminology is justified, as in Sect. 5.4, by comparing these expressions with those of the various tensors associated with T. We must emphasise, however, that (5.40) and (5.41) represent fOUf expressions for the same tensor T.. in E ® E.

Contracted Products The notion of second rank Euclidean tensor has thus been introduced in the most general case. All definitions and results referring to simple product tensors can now be generalised, by noting in particular the distributivity of the contracted product of Euclidean tensors over addition, which allows straightforward calculation of contracted products.

As an example: ij k Ic..Y.= (T fi®f~).(V fk) = Tij k (e " . ) e" = Tij gJ" k k e = Tij " e" = Ti k k e" v -J -ek -t v -1. vJ -7. v -1. , and also Ic.".J;.' = (T"ij fi ® fj). (T'kt fk ® ft) - T"ij T'kt( fj ·fk ) fi K/,'GIft - T"ij gjk T'kt(fi K/,'GIft ) - T"ijT' j t fi 'GIft,K/, 5. Tensors on a Euclidean Vector Space 723

and so on, or alternatively, r"·r' = (T"ij ~i 0 ~j). (T'ki ~k 0 ~i) = T'\ T'j i ~i 0l . For the doubly contracted product,

=T" '=. T' - (Tij ~i'

The above examples show that contraetions are always possible for Euclid• ean tensors. They are calculated by adopting suitable representations for the tensors involved, in which the relevant indices form pairs with one lower and the other upper. It can also be seen that a sure and systematic method for calculating contracted products, when not familiar with this type of exercise, simply consists in writing out the scalar products in E corresponding to the various contractions.

Transposition From the definition given in Seet. 3.3 and from the characteristic property (4.6), we define the transpose of the Euclidean tensor r, denoted tr, by the relation

1::/ 'J!.. , 'J!.. 'J!.. • (5.42) T..- = tT.- . This is equivalent to saying that the mixed tensor associated with tT. in E* ® E is the transpose of the mixed tensor associated with T. in E ® E* . Since r has representations r = Tij f.i ® f.j = T ij f.i ® f.j = Tijf.i ® f.j = Tij f.i ® f.j' the representations of tr are

(5.43)

This shows that, for the twice contravariant and twice covariant represen• tations, the coefficients of T. and tT. correspond by permutation of the two indices. Furthermore, for the mixed representations, the order of the two indices is permuted whilst they conserve their upper or lower positions. 724 Appendix I. Elements of Tensor Calculus

It is interesting at this stage to return to the calculation of the dilatation tensor for a linear mapping of E into E, first considered in Sect. 5.2. It follows that 'PClO . 'Pe!!.") = (!;;. .1!.') . (!;;. .1!.") = 1!.' . ~ .1!." with ~ = t!;;.. ~ .

Q can be written out explicitly, for example, in theform Q = (CF)i k f;i ® f;k)'

(Ff j f;f ® f;j), whence Q = (tF)/ gkf F f j f;i ® f;j and we retrleve the expression in (5.5). By (5.43), this b~omes: ~ = F k i gkf F f j f;i ® f;j.

Symmetrie and Antisymmetrie Seeond Rank Euelidean Tensors A Euclidean tensor T. is said to be symmetrie if its associated tensors in E®E or E* ® E* are symmetrie (see Sect. 3.4). (Note that the symmetry of one of these implies the symmetry of the other.) For such a tensor, we thus have the following symmetry relations for the twice contravariant, twice covariant and mixed representations:

(5.44)

The symmetry of'r can also be expressed through the eharaeteristie prop• erty

(5.45)

In the same way, we define the antisymmetrie Euclidean tensors for which the above formulas (5.44 and 5.45 ) are modified by adjoining a minus sign:

y,.'r.7l. = -7l..'r.y" (5.46) Tij = -Tji , Tji = -Tij .

Comparing these formulas with the results given above for the transposition, the characteristic properties can be expressed in the form

T. symmetrie T=tT { --- -, (5.47) 'r antisymmetrie T. = _tT..

Using the approach adopted in Sect. 3.4, a general second rank Euclidean tensor can be decomposed into its symmetrie and antisymmetrie parts, with the corresponding formulas:

T =T +T == ==8 ==a'

j=T = ~2(= T+ tT=), (5.48) T = ~(T - tT) . =a 2 = = 5. Tensors on a Euclidean Vector Space 725

Now taking any two tensors T.." and T..' decomposed according to (5.48), their doubly contracted product can be written in the form

T" : T' = T" : T' + T" : T' (5.49) ======s ==s ==a ==a' since

T" : T' = 0 and T": T' = 0 . ==s ==a ==a ==s

N otational Convention The bilinear form G on E xE, an element of E* 0 E* , corresponds to the Euclidean tensor identified with the element gij ~i 0 ~j of E 0 E. We shall use the notation 1!, for this Euclidean tensor:

11. - ij ,0, _ s;i ,0, j _ d i,o, _ i,o, j = - 9 ~i '61 ~j - Vj ~i '61 ~ - U i ~ '61 ~j - gij ~ '61 ~ • (5.50)

This notation is justified by the fact that

VT.., - T...1!,=T...- --

Invariants of a Second Rank Euclidean Tensor The invariants, defined by (4.23), of the mixed tensors associated with a Euclidean tensor T.. in E 0 E* and E* 0 E are equal. They can be written in terms of Euclidea~ tensors in the following way: 7

T.. : 11. tr T.. , h = - --= 1 1 2 h = -2 --T.. : T.. = -2 tr (T..- ) , 1 1 h = "3 G~;:·;r;) : ;r; = "3 tr (;r;3) , (5.51)

This provides a set of principal invariants of T... We mayaIso define detT.. in an analogous-way, by applying (3.16) to the mixed tensors associated with T.. in E 0 E* or E* 0 E. It is invariant under change of basis, and we also have

det(;r;" .;r;') = det;r;" x det;r;'.

7 With the convention adopted here for the indices involved in the double con• traction, denoted by " . " and specified in Sect. 4.3, note that cd B tr (4·[4) , (4 .[4) : g = tr (4·[4· g) , etc. 726 Appendix I. Elements of Tensor Calculus

The importance of principal invariants in mechanics (see Chaps. VI and VII, for example) is a consequence of our next result.

Isotropie Function of aSymmetrie Seeond Rank Euelidean Tensor Consider a scalar-valued funetion of the tensor ';[. Such a funetion is invariant under change of basis. This means that the vaiue of this funetion for a given tensor ,;[, although it may result from algebraic calculations involving the components of';[ relative to some basis {~d in E, is actually independent of the choice of bäSis. In order to emphasise this fact, we shall say that it is an isotropie funetion of 'I/ We now have the following representation theorem far symmetrie tensors ';[: Any (isotropie) funCtion of the symmetrie tensor';[ ean be expressed as a function of the prineipal invariants ofb (or an equivalent set). This clearly applies to det b'

5.8 Rank n Euelidean Tensors

The construetion for second rank Euclidean tensors in the previous seetion can be repeated for Euclidean tensors of arbitrary rank n. A Euclidean tensor of rank n will be denoted by a letter (usually upper case) underlined n times. This notation would so on become awkward if n were large. However, for the applications we shall make in mechanics, it turns out to be quite convenient and allows us to distinguish at a glance the scalar, vector or tensor nature of the mathematical objeets arising in our formulas. It thus provides a quick way of checking that formulas are homogeneous from the tensorial viewpoint.

5.9 Choice of Orthonormal Basis in E

As remarked in Seet. 5.3, if we choose an orthonormal basis Ü~k} in E, this implies that the corresponding dual basis in E, {~k} will coincide with it. Then by (5.19),

_ i ~i = ~ , _ d _ gi j 9 ij - Ui - ,

and it follows that far any Euclidean tensor all the various representations also coincide:

'T') _ 'T'. . _ TijkR. _ .L t kR. - .LtJkR. -- ...

In other words, if the basis is orthonormal, the position of the indices beeomes irrelevant.

8 See Chap. VI (Sect. 2.7 and 4.2) for an explanation of this terminology. 5. Tensors on a Euclidean Vector Space 727

When only orthonormal bases are considered, Euclidean tensors are called Cartesian tensors. The formulas for change of basis and transformation of components given in Sect. 3.2 are then considerably simplified. We will thus write (all indices lowered)

where by the orthonormality of the basis ü~d ,

Since the basis Ü{ k} is itself orthonormal, the inverse formula is then,

For an arbitrary Cartesian tensor (in this case we take b as an example), the formula for the transformation of components becomes

(5.52)

For Euclidean tensors of first and second rank, contraction operations can be written in matrix form. When the basis is orthonormal, we write a tilde over the tensor symbol to denote the matrix of the tensor, e.g., t.. denotes the matrix of the tensor 'b. (first index numbering the row, second inde;Z-the column). Similarly, f!. denotes the single column matrix of the vector Q. (These matrices are the cor• responding tables of coefficients relative to the orthonormal basis.) Using a dot to denote matrix products, we thus obtain

(~.Q) = t·f!., ;;---;- -" -, - ('b. . 'b. . Q) = 'b. . 'b. .Q ,

(T=.Q ') . (T=.Q ") = t-,Q . tmb·=·Q T- -" .

5.10 Principal Axes and Principal Values of a Real Symmetrie Second Rank Euclidean Tensor

Consider a real symmetrie second rank Euclidean tensor T... We aim to find the eigenvectors and eigenvalues of the linear mapping associated with T.., i.e., nonzero vectors Y.i and scalars Ai(i = 1,2, ... ,n) such that -

no summation . (5.53)

The Ai are found by solving the polynomial equation in A (Sect. 4.6) 728 Appendix I. Elements of Tensor Calculus

det(r - All) = 0 , whose roots are real or complex conjugate pairs. The symmetry of r implies the following classic results. • The eigenveetors corresponding to two distinct eigenvalues are orthogonal.

Indeed we have

(5.54) and also, by symmetry of r (see (5.45)),

(5.55)

• All the eigenvalues are real.

If Ai is an eigenvalue and Xi its complex conjugate, the corresponding eigenvectors are also complex conjugates: r·1!.i = Ai1!.i => r· 'fZi = .6.i 'fZi since r is real, so that (5.54) and (5.55) give

whose only solutions are Ai =1= Ai => 1!.i = 0 and 1!.i =1= 0 => Ai = Ai .

The analysis of the case where some eigenvalues are repeated (degener• acy) is standard and it can be shown that we can always construct a ba• sis of orthogonal eigenvectors ~l , ~2 , ... '~n corresponding to n eigenvalues Al, A2 , ... ,An (possibly repeated). In this basis and its dual basis {~k}, 'L can be written in a simple way using (5.53): -

r = Al ~l Q9 ~l + A2 ~2 Q9 ~2 + ... + An ~n Q9 ~n , { (5.56) r = Al ~l Q9 ~l + A2 ~2 Q9 ~2 + ... + An ~n Q9 ~n .

This implies for the bilinear form ll· 'L .12.

(5.57)

It is convenient to choose ~l '~l ... '~n orihonormal (see Seet. 5.9). In this case,

r = Al ~l Q9 ~l + ... + An ~n Q9 ~n , ll· 'L.12. = Al Ul VI + ... + An Un V n , and the quadratic form ll· r.ll becomes 6. Tensor Fields 729

The directions defined by the eigenvectors and the corresponding eigen• values of the linear mapping associated with T.., are called the principal axes and principal values of the Euclidean tensor 'f. The invariants of'I:. are simply expressed in terms of the principal values :

(5.59)

and also

(5.60)

6. Tensor Fields

6.1 Definition

We now assume, as is the case in the context of continuum mechanics, that the vector space E used to define the tensors in Sect. 1.1 is the vector space associated with an affine space. Hence, each point M of the affine space is characterised by its position vector M or ±, relative to some fixed origin in the space. We define a tensor field on this affine space by associating a tensor of specified type with each point M. The value of the tensor field at the field point M, that is, the corresponding tensor, will be denoted T(±). We will usually use H to symbolise the vector space of tensors T(±) and we shall refer to this as an H-tensor field. Let {'~d be a basis for E, and {e*k} the dual basis in E*. The position vector ± has coordinates xk in the basis {.~k}.

6.2 Derivative and Gradient of a Tensor Field

We consider a tensor field T(±) and assume its components to be differen• tiable functions of the coordinates x k of the position vector ±. Then let W be an arbitrary vector in E. At the field point M, we can define the derivative of the field T along the vector w. It is the H-tensor 730 Appendix I. Elements of Tensor Calculus denoted D:!Q. T(X.), defined by taking the limit

- 1· T(~ + ).1Q) - T(~) H D wX-1mT( ) E. (6.1) -- A--+O ). The existence of the limit is guaranteed by our assumption about the differentiability of the components of T(~). We can then show that D:!Q. T(~) is a linear function of 1Q . In particular, putting the vectors in the basis {~d for 1Q, we obtain

aT(~) (6.2) D-ke T(~) = -aX k ' and by linearity

D T(x) = aT(~) w k = (D T(x) Q9 e*k) 8 w . (6.3) :!Q. - a~ ~ --

This formula brings out the tensor D~k T(~) Q9 e*k associated with the linear mapping of E into H

1Q --+ D:!Q. T(~) .

This tensor in H Q9 E* is called the gradient of the tensor field at the point M. It is denoted VT(~):

VT(~) = D~k T(~) Q9 e*k = a~;~) Q9 e*k (6.4) and (6.5)

Gradient of a Euclidean Tensor Field We now assume that E is equipped with a Euclidean structure and consider the field of Euclidean tensors with associated tensor space H. For concrete• ness, we shall consider the field T.. The above definition of the gradient can be transposed unambiguously and the main formulas are (without specific reference to the ~ dependence):

(6.6)

or in differential form , (6.7) 6. Tensor Fields 731

6.3 Divergence of a Tensor Field

Let us suppose that the field of H-tensors involves tensors whose last index is contmvariant. We then define at each point M the divergence of the field T by the contraction of the tensor VT(;r) E H ® E* over its two last indices. Hence, using (6.4),

(6.8)

which corresponds to the double contraction of VT(;r) with the tensor I in E®E*:

This yields

(6.9)

In this way we obtain a new field of tensors which no longer possesses the last contravariant index of the original tensor field. In the particular case where T(;r) E H = E ® E, for example, this gives a vector

div T(;r) E E ,

and we can check that

(6.10)

Divergence of a Euclidean Tensor Field The not ion of divergence is primarily used in the case where E is equipped with a Euclidean structure. Therefore, making this hypothesis from now on, we shaIl simplify the presentation by only considering Euclidean tensors. For a Euclidean tensor field ';L, the divergence operation is always possible. It is defined by

div I = VT : 1l , (6.11) - - an obvious extension of (6.9), which amounts to applying (6.9) to any of the tensor fields associated with r for which such an application is possible. This definition leads to a generalisation of the divergence theorem from the weIl known case ofvector fields (first rank Euclidean tensors) to Euclidean tensors of arbitrary rank. 732 Appendix I. Elements of Tensor Calculus

As an example, consider

In this case, ijk · T - öT - d· (Tijk ) dlV = - fci Q9 fcj öxk - fci Q9 fcj lV fck·

We thus see that, for a volume D, with the standard conditions for ap• plying the divergence theorem to each of the vectors Tijk fck with fixed i and j, i.e., provided that the Tijk are class Cl, it follows that

div r dD = fci Q9 fcj r (Tijk fck) . '!1. da , (6.12) 1n - Jan where da denotes the surface element of öD and '!1. its outward normal. Gathering the terms in (6.12), we recognise the tensor r on the right hand side, so that

(6.13)

This is actually valid for any rank of Euclidean tensor.

6.4 Curvilinear Coordinates

The above formulas for the gradient and divergence of a tensor field, involving derivatives with respect to the variables Xi , apply to the case where points in the Euclidean space are labelled by the coordinates Xi of their position vec• tors in a coordinate system with fixed origin and fixed basis {fcJ (Cartesian coordinates) . We sometimes need to consider tensor fields in which the points Mare labelled by parameters rJi defining a system of curvilinear coordinates in the Euclidean space. This is the case, for example, for the widely used cylindrical and spherical coordinates. Tensors T(;.f.) are then defined by their components in a local basis made up of a basis of vectors in E that are tangent to the co ordinate curves at M, and its dual basis. These components are given as junctions oj the curvilinear coordinates. Clearly, when we determine the components of VT(;r:.) in the local basis at point M, we shall encounter two types of terms arising from different origins :

• terms due to differentiation of the components of T(;r:.) with respect to rJi , • terms due to the fact that the local basis itself changes with varying rJi . 6. Tensor Fields 733

This is not the place to go into the standard formulas (Christoffel symbols) for this operation, known as covariant differentiation. The applications we shall meet would not justify such an analysis. The reader will find here all formulas relevant to the main types of curvilinear coordinates used, limited to the expressions for gradients and divergences of Euclidean tensor fields that are strictly necessary for the discussions in the text. These expressions are established by using (6.7) and identifying components. We shall examine two simple 2-dimensional examples of this type of cal• culation.

Gradient of a Vector Function in Polar Coordinates The position of a point M in ]R2 is specified by parameters

TJl = r, TJ2 = B . (6.14) Coordinate curves are circles cent red on 0 and straight lines radiating out from 0 (Fig. 3).

Fig. 3. Polar coordinates

At each point of M, we consider the local orthonormal basis consisting of vectors tangent to the co ordinate curves at this point:

A vector function U is usually defined in this polar coordinate system by the formula

(6.15) where the vectors f:.r and f:.o are themselves functions of rand B. A convenient way of calculating '\lU consists in using the differential form of (6.7): - äU äU '\lU . dM = dU = är dr + äB dB , { (6.16) \i dM = f:.r dr + f:.o r dB .

The change in the local basis is characterised by 734 Appendix 1. Elements of Tensor Calculus

(6.17)

We therefore have au aUr aUe -VU . f.r = -ar = -a r f.r + -a r f.e, laU 1 (aUr ) 1 (aUe ) VU . f.e = ;: ae =;: ae - Ue f.r + ;: ae + Ur f.e·

Identifying components, this implies that VU is given by

(6.18)

Gradient of a Vector Function in Arbitrary Orthogonal Curvilinear Coordinates Generalising the previous example, Fig. 4 represents the co ordinate curves of an arbitrary orthogonal curvilinear coordinate system: along each 1]1 curve, 1]1 varies whilst 1]2 remains constant, and along each 1]2 curve, 1]2 varies whilst 1]1 remains constant.

Fig. 4. Arbitrary orthogonal curvilinear 111 coordinates

At each point M labelled by coordinates 1]1 and 1]2 ,

(6.19) we define the naturallocal basis consisting of vectors EI, E 2 tangent to the coordinate curves, using the differential formula

(6.20) 6. Tensor Fields 735

Referring to (6.19), we also write

(6.21)

We now assume that the coordinate curves are numbered 1 or 2 in such a way that (EI' E 2 ) = +n /2 as shown in the figure. The loeal orthonormal basis collinear with the natural basis is spec,ified by unit vectors f2 , f2: (6.22)

The components of dM in this basis are dS 1 and dS 2 where SI and S2 denote the path length parametrisation along the coordinate curves passing through the points M: (6.23)

A vector function U is usually defined in this system of curvilinear coor• dinates by the formula

(6.24) where the vectors fl and f2 in the local ort ho normal basis are themselves functions of 1]1 and 1]2· In order to calculate VU, we write

(6.25)

Let R 1 (resp. R 2 ) denote the radius of curvature at M of the co ordinate curve 1]1 (resp. 1]2), measured positively9 along E 2 (resp. -Ed· The change in the local orthonormal basis is characterised by the standard formulas

(6.26)

In more suggestive notation, 10

9 This me ans that if !.pI is the angle between ~1 and a fixed direction, we have R1 = dsdd!.p1 . Likewise with !.p2 for ~2 : R2 = ds2/d!.p2. 10 J!..- = _1_J!..- and J!..- = _1_J!..-. OSl I E.1 I aT/I OS2 I E.2 I OT/2 The function !l.. cannot be defined as a function of SI and S2 (for genuine curvi• linear coordinates) because SI and S2 do not constitute a coordinate system. The meaning of the partial derivatives with respect to SI and S2 is given by (6.27). 736 Appendix 1. Elements of Tensor Calculus o and D e = -;:;;- . (6.27) -2 US2

We then have oU VU. EI = DEI U = !l• - - U1]1

= ~h (~~: - U2 LI EIl) + f2 (~~: + UI ~Il EI I) , oU VU. E 2 = Dll22 U = 01]2

= ~I (~~~ - U2 ~21 E 2 1) + ~2 (~~: + UI ~21 E 2 1) Hence the expression for VU:

(6.28)

From this result, the following expression for div U is immediate:

div U = oUl _ U2 + oU2 + UI . - OSI R I OS2 R 2

The curl of the field U can be calculated from the fact that, in ffi.3,

VJl. E ffi.3 , (curlU) 1\Jl. = (VU - tVU).Jl..

Then in the present case, Summary of Main Formulas 737

Summary of Main Formulas

Primar basi in E: {~} Dual basis in E: LI<} _, . fi = 6~ , !';i . $:.} = g,} • fl' .!l.} = g'J

Euclid 'an ten or (repr ntations) - T,)k ,,0. ,0. - T' I< .." -, ,0. - - T. ' =T - !... 'öl f} 'öl ~k - } ~i 'öl E" 'öl fl< - ... - ,}I<_ = g'J 181 e = 6} (,' 181 e = 6' e 181 e1 = g, c' 181 e1 n= -, -) , - -) J -, - J - -

Tensor product (example)

~®!J. = (Ai) k ~,®~®~I<)®(Bmn fm®fn) = A') I< E m.. ~t®

!! 181 ~k ® ~'" 0 -n

Contracte.d product (examples) 4·~ = (A'/~, ®~ ®fk) ' (Emn!lm 181 -n) - = A'/ Ern" C~I< ' _m)f, 0~ ®f" = A'/ gkm Bm"f, ®~ ®f..

~'k = (A') I< !l, ®~ ®f,,)· ( ","_'" ®_") = Ai/C",nk" . ft'") ft, ®f:" ®f:.. = A'/C,," _, ®~ 181ft ..

Doubly contracted product (cxampl )

~: ~ = (A'/ _, ®~ 0ft,,): (Bm .. f m ®!;n)

= A'/ Bffi"(tl< .fm)(~ ·fn)f, = A'/ gkm B m ) f, 738 Appendix I. Elements of Tensor Calculus

co nd rank tensor (properti.'i and rnain results)

• r·~ = ."Ü:) .p : lin ar m8pping of B into E who'e mat.rix in , Il!' I a..-;is {f:k} h. ' cOE'fficient· T' J '

. 'b,T I=b-I'b=~ r 1 : invl;'rse t 'nwr, Orr 'ponding to - I (ir it xists),

• r, t' = I. tT. tr: tran,'pos cl ten 'or CT).) = T)I . ('T)') = T)' . eT),1 = Tl, ....

t (Q . IJ) = I IJ. ~

• 'Ylnnwtric- t{'I1SOr : AntisYlll1ll , ric l l1 'or: b = _Ir;

'b = I; + 4:, •. 1:.. = ~(r; + 'r)

• lnvarianls

I 1 tr b=b:~

2 12 ~ ~ Ir (r ) = ~ r : r l 'I~~ tl' (ra)=~(b'b):b

f n !tr = n

Gradient of a tensor fteld (exampl )

Divcrgence of a ten or fteld (exalllple)

divZ;d!2 = { Z;.!l.da (div rg ne th r m) in{ ian Appendix II

Differential Operators: Basic Formulas 742 Appendix 11. Differential Operators: Basie Formulas

1 Orthonormal Cartesian Coordinates ...... 743 1.1 Coordinates ...... 743 1.2 Veetor Field ...... 743 1.3 Sealar Function ...... 744 1.4 Seeond Rank Tensor Field ...... 744 2 General Cartesian Coordinates...... 744 2.1 Coordinates ...... 744 2.2 Veetor Field ...... 745 2.3 Sealar Function ...... 745 2.4 Seeond Rank Tensor Field ...... 745 3 Cylindrieal Coordinates ...... 746 3.1 Parametrisation ...... 746 3.2 Vector Field ...... 746 3.3 Sealar Function ...... 747 3.4 Symmetrie Seeond Rank Tensor Field ...... 747 4 Spherieal Coordinates ...... 747 4.1 Parametrisation ...... 747 4.2 Veetor Field ...... 748 4.3 Sealar Function ...... 749 4.4 Symmetrie Seeond Rank Tensor Field ...... 749 1. Orthonormal Cartesian Coordinates 743 Differential Operators: Basic Formulas

1. Orthonormal Cartesian Coordinates

1.1 Coordinates

The coordinates of a point Mare xi(i = 1, 2, 3) also denoted x, y, z. These are the components of GM with respect to the basis f:i(i = 1, 2, 3) also denoted (f:x , f:y , f:z):

GM = Xif:i'

z

ez o ex '~,,--:_----,---/--- y " I ,// " I / ------~~/ Fig. 1. Orthonormal Cartesian x coordinates

1.2 "ector Field

In this case contravariant and covariant components are equal, and the bases {f:J and {f:i} are identical:

1!.(~) = vx(~) f:x + Vy(~) f:y + vz(~) f:z, OVi V'v(~) = ox. (~) f:i Q9 f:j , J 744 Appendix 11. Differential Operators: Basic Formulas

OVx oVx oVx ---ox oy oz oV y oV y oV y \7v = ox oy oz oVz oVz oV z ---ox oy oz

1.3 Scalar Function f(M) = f(;r.) , of \7f = ~~i' - UXi

1.4 Second Rank Tensor Field

2. General Cartesian Coordinates

2.1 Coordinates

The coordinates of a point M are xi (i = 1, 2,3), components of GM with respect to the basis ~i (i = 1, 2, 3):

--GM = xie·.-, 20 General Cartesian Coordinates 745

3

------". 1 Fig. 2. General Cartesian coordinates

2.2 "ector Field

2 i A _ d O (n ) _ kj 0 V ~ - IV v V - 9 oxkox j fi 0

2.3 Scalar Function

f(M) = f (~),

of i V' f = -;:;-:- f , - ux'

2 - d O ij L.1Af - IV (nf)v -_ 9 ;::) 0.;::)f . - ux'uxJ

2.4 Second Rank Tensor Field

oTij div 'I:. = ox j fi' 746 Appendix 11. Differential Operators: Basic Formulas

3. Cylindrical Coordinates

3.1 Parametrisation

The position of a point M described by parameters r, e, z (Fig. 3). The local orthonormal basis is .121' , fe , fz:

dM =f1'dr+ferdß+fzdz.

Its variation is given by

0.121' =0 Ofe =0 Ofz = 0 or ' or ' or '

0.121' = 0 Ofz = 0 oz ' oz .

z

8M ~r = 8r IBM ~o = ;: 8B o.~~ ____+- ____~y 8M ~z = 8z

x e

Fig. 3. Cylindrical coordinates

3.2 "ector Field

A vector 1!. at the point M is expanded in terms of the loeal orthonormal basis .121' , fe, f z . Its components are V1' , Ve, Vz: 4. Spherical Coordinates 747

In this basis,

8vr ! (8vr _ vo) 8r r 8e \jv = 8vo 1 ( 8Vo + ) Fr -:; ae V r 8vz 18vz --- 8r r 8e

3.3 Scalar Function 1 (M) = 1 (r, e, z),

81 181 81 \7 1 = 8r f'.r + -:; 8e f'.o + 8z f'.z ,

3.4 Symmetrie Second Rank Tensor Field 'lJM) ='[Jr, e, z) =Tij(r, e, Z)f'.iQ9f'.j We quote only the result for div 'I:: (r, e, z):

d· T ( e ) _ (8Trr ! 8Tro 8Trz Trr - TM) IV -_ r, ,z - 8 r + r 8e + 8 z + r f'.r 8Tor 1 8Too 8Toz 2Tro) + ( --+---+--+-- eo 8r r 8e 8z r- 8Tzr 1 8Tzo 8Tzz Tzr ) + ( 8r + -:; 7iB + Tz + -:;:- f'.z·

4. Spherical Coordinates

4.1 Parametrisation The position of a point M is deseribed by parameters r, e, ep (Fig. 4). The loeal orthonormal basis is f'.r , f'.o , ~

z er

oM e f r = or / / r 10M / fe = ;: oe o / ------.... y 1 oM e = ----= -

Fig. 4. Spherical coordinates

Its variation is given by

Ofr =0 o~ =0 Ofep = 0 or ' or ' or

Ofe Of

4.2 Vector Field

A vector 1!. at the point M is expanded in terms of the loeal orthonormal basis (fr' fe, f

In this basis,

OVr 1 1 oVr ~ (ov r _ ve) -(---v) or r oB r sin e o

. oVr 1 aVe 1 oV

. ( ) ( 2 ( 1 Ö. 1 öV

A 2 (övr Ve cos 0 öV

2 (ÖVr öve V

4.3 Sealar Funetion

1 (M) = 1 (r, 0, rp) ,

öl 1 öl 1 öl -'V' 1 = -;:)ur f r + -r u!:l0 fe + -'-0 r sm ~urp f

4.4 Symmetrie Seeond Rank Tensor Field

div'l::.(r,O,rp)=

öTrr 1 öTre 1 öTr

öTer 1 öTee 1 öTe

Elements of Plane Elasticity

Key Word lan ' rain. Plan 2-dimcnsional probl ß1 ', r ' fundi n, Airy function, 2-dim nsion I gorn tri al rnp tibilit , Transverse geometrical compatibilily. Thin sli approximation. ------~ Appendix III. Elements of Plane Elasticity 753 In Brief

Plane strain equilibrium problems are characterised by the fact that their solution involves a displacement field parallel to a plane (Oxy) and indepen• dent of the orthogonal transverse coordinate (z). When the system concerned is made from a homogeneous material with linear, isotropie thermoelastic be• haviour, the thermal, mechanical and geometrical properties defining such a problem can be specified. The 3-dimensional linearised thermoelastic equi• librium problem reduces to a 2-dimensional problem within a cross-section of the system. It thus becomes a problem of regular linearised thermoelastic equilibrium in which the 2-dimensional linearised stress and strain tensors are related by the governing equations of 2-dimensional elastic behaviour for plane strain, deduced from the 3-dimensional thermoelastic behaviour of the relevant material. Solution of the original 3-dimensional thermoelastic equi• librium problem is achieved by completing the stress field with a principal component along Oz, calculated from the 2-dimensional field (Sect. 2). Plane stress equilibrium problems, on the other hand, are characterised by a property of the solution stress field: it must be planar, parallel to Oxy and independent of the z coordinate. When the system is made from a homoge• neous and isotropie linear thermoelastic material, we can specify the general form of the problem. We are then led to solve a 2-dimensional problem re• stricted to a cross-section of the system under study, and this turns out to be identical to the one brought out in the plane strain problem, except with regard to the relation between the 2-dimensional linearised stress and strain tensors. The relevant law of 2-dimensional elastic behaviour is indeed the plane stress law, also deduced from the governing equations of 3-dimensional thermoelastic behaviour for the material in quest ion. The original thermoelas• tic equilibrium problem is solved by completing the displacement field with a transverse component in the Oz direction. The conditions under which this is possible turn out to be extremely restrictive owing to the geometri• cal compatibility of the transverse strain. Plane stress linearised thermoelastic equilibrium problems are generally only valid in the 'thin slice' approximation (Sect. 3). Appendix 111. Elements of Plane Elasticity 755

Main Notation

Notation Meaning First cited

E 2 ({f2) Body force per unit mass parallel to Oxy (2.4) ~2 ({f2) 2-dimensional displacement (2.10) grad2 2-dimensional gradient (2.15) div2 2-dimensional divergence (2.15) ~){f2) 2-dimensional strain (2.16) 2-dimensional part of ~ (3.13) g2 ({f2) 2-dimensional part of Q. (2.20) 2-dimensional stress (3.8) ,12 2-dimensional Laplacian (2.40) V({f2) Potential for body forces per unit mass (2.42) cjJ({f2 ) Airy function (2.43) ,12,12 2-dimensional bilaplacian (2.45) 756 Appendix 111. Elements of Plane Elasticity

1 Plane Problems...... 757 2 Plane Strain Thermoelastie Equilibrium ...... 757 2.1 Plane Linearised Strain Tensor ...... 757 2.2 Plane Strain Displacement Field ...... 758 2.3 Plane Strain Thermoelastie Equilibrium in a Homogeneous and Isotropie Material ...... 758 2.4 Solution by the Displacement Method ...... 760 2.5 Solution by the Stress Method ...... 764 2.6 Remarks on the Plane Strain Two-Dimensional Problem ..... 767 2.7 Two-Dimensional Beltrami~Michell Equation ...... 767 2.8 Body Forces Deriving from a Potential. Airy Function ...... 768 2.9 Cylindrical Tube Under Press ure ...... 770 3 Plane Stress Thermoelastie Equilibrium...... 771 3.1 Plane Stress Tensor ...... 771 3.2 Plane Stress Field ...... 771 3.3 Plane Stress Thermoelastic Equilibrium in a Homogeneous and Isotropie Material ...... 771 3.4 Solution ...... 773 3.5 Cylindrieal Tube Under Pressure ...... 778 Summary of Main Formulas ...... 780 2. Plane Strain Thermoelastic Equilibrium 757 Elements of Plane Elasticity

1. Plane Problems

The aim here is to give an elementary description of linear thermoelastic equilibrium problems for the case of homogeneous and isotropie materials under plane elasticity conditions. In this brief presentation, our main tasks will be: • to define 3-dimensional plane stmin and plane stress problems in ther• moelasticity, • to show how solution of such problems reduces to or leads us to solution of a 2-dimensional problem in thermoelastic equilibrium, and hence a significant simplification through reduction in the number of unknown functions and variables, in particular by introducing the stress function or Airy function. The reader wishing to go into furt her detail, in particular as regards the use of functions of a complex variable, will find the subject treated in many classic textbooks. It should also be said that the concepts of plane strain and plane stress problems introduced in this way have wider application, going beyond the hypotheses of elastic behaviour, homogeneity and isotropy for the material in quest ion.

2. Plane Strain Thermoelastic Equilibrium

2.1 Plane Linearised Strain Tensor Let Oxyz be an orthonormal coordinate system. We say that the tensor of linearised strains ~ is plane, parallel to Oxy, if it takes the form

(2.1a) which can also be written1

a,ß = x,y, (2.1b) introducing Greek indices a, ß for basis vectors in the plane Oxy.

1 Summation over repeated indices. 758 Appendix 111. Elements of Plane Elasticity

2.2 Plane Strain Displaeement Field

Relative to some referenee configuration, the displacement field ~ of a 3- dimensional eontinuous system is said to be plane strain, parallel to Oxy, if it has the following properties: denoting its eomponents by ~x ,~y ,~z,

~z = 0, { (2.2) ~x and ~y independent of z , or alternatively:

~z = 0 (2.3) 8~x = 8~y == 0 8z 8z

These hypotheses clearly imply that at eaeh point the eorresponding lin• earised strain tensor is plane, parallel to Oxy, with the form given in (2.1). Moreover the linearised strain field is independent of z.

2.3 Plane Strain Thermoelastie Equilibrium in a Homogeneous and Isotropie Material

We shall be eoneerned here with thermoelastie equilibrium problems in homo• geneous and isotropie solid materials, whose solution results in a displacement field { of form (2.2). Needless to say, a problem is not generally posed in the form of a plane strain problem as such. However, given the isotropy of the material, the type of information available (body forees and temperature ehanges within the objeet, and surfaee forees and boundary displacements) lead us to seek a solution of form (2.2). This assumption is justified aposteriori when the solution has been eonstrueted (see Chap. VIII, Seet. 4.4). Given the results established in Chap. VIII (Seet. 4.3), we shall rest riet ourselves to isothermal elastie equilibrium problems, treating either an orig• inally posed isothermal problem or, if the initial problem eontains thermal variations, the assoeiated isothermal problem. In the following, assumptions eoneerning the data of the problem therefore refer to the assoeiated isother• mal problem. Reeall that the displacement fields solving a thermoelastie equi• librium problem and its associated isothermal problem are identical: henee, the notion of plane strain problem is maintained in the two eases.

Statement of the Plane Strain Elastie Equilibrium Problem The system treated, of volume fl, is a eylinder with axis parallel to the Oz axis, height C and eross-seetion S. So and Sp denote the end eross-seetions, loeated at z = 0 and z = C, respeetively (Fig. 1). The elastie equilibrium problem is eharaeterised in the following way. 2. Plane Strain Thermoelastic Equilibrium 759

y O @. ------+

!!

x

Fig. 1. Plane strain elastic equilibrium problem

• Body Forces Body forces are parallel to Oxy and independent of z. They may therefore be defined on any cross-section S of n, in particular on So: where .Je2 denotes the projection onto Oxy of the position vector of the field point in the cross-section S, and F 2 is a 2-component vector parallel to the plane Oxy of So.

• Boundary Conditions on So and Se So and Se are assumed to be surfaces Sf,z and the component ~z of { must be zero: (2.5)

So and Se are assumed to be surfaces STx and STy ' and the components Tx and Ty of the surface force 'L must be zero:

Tx (.Je) = T1 (.Je) = 0 , { (2.6) Ty (.Je) = T: (.Je) = 0, on So and Se .

• Boundary Conditions on Lateral Sur/ace Data on the lateral surface of n are independent of z. They are defined as functions of the 2-dimensional position vector .Je2 on the boundary 8So of So. Regarding the condition in the Oz direction, 8So is decomposed into complementary sets (8Sok and (8Sohz' The corresponding data are zero: (2.7)

(2.8)

The two other conditions concern components of vectors ~ (.Je) and 'L (.Je) in two orthogonal directions spanning the plane Oxy and independent of z , at each point of (8n - So - Se). Corresponding boundary conditions are 760 Appendix 111. Elements of Plane Elasticity

r.iÜr) = r.? (:.Jd on (ÖSOhi , {J!;) = ~f (,I2) on (ÖSO)!;i , (2.9) (öSok U (öSok = öSo, (ÖSOhi n (öSok = 0, i = 1,2,

where directions i = 1 and 2 usually correspond to the normal and the tangent to öSo at the field point M (Fig. 1).

Note When a thermoelastic equilibrium problem involves temperature variations, the above conditions must be satisfied in the associated isothermal problem. It is worth noting that this happens in the particular case when the 'mechanical' characteristics of the initial problem themselves take the form given by (2.4 to 2.9) and that the temperature change field T is independent of z.

2.4 Solution by the Displacement Method

Form of the Displacement Field The form of the above conditions (2.4 to 2.9) lead us to seek a solution by the displacement method, assuming at the outset that the field ~ takes the form (2.2): plane strain displacement field parallel to Oxy and independent of z. This displacement field fl can be specified in terms of a 2-dimensional vector field {2 (;r2) on the cross-section So:

(2.10)

We then follow through the solution procedure shown in Fig. 7 of Chap. VIII, using the Navier equation.

Boundary Conditions on Displacements Given the form of {, it is clear that

~z (;r) = 0 on So and Sc , { (2.11) ~z(;r)=O on (öfl-So-Sc) , thus satisfying boundary conditions (2.5) and (2.7). The two other boundary conditions on the displacements, arising from (2.9), apply to the field {2 on (ÖSO)!;i:

(2.12) 2. Plane Strain Thermoelastic Equilibrium 761

Navier Equation The 3-dimensional Navier equation (Chap. VIII, Sect. 5.2)

(A + JL) grad (div {) + JL div (grad ~) + pF = 0 (2.13)

yields a first scalar equation along Oz

(2.14)

which is automatically satisfied, and two other scalar equations along Ox and Oy:

(A + JL) ! (div {) + JL Ll~x + p Fx = 0 , { (A+JL): uy (div~)+JLLl~y+pFy=O,-

which constitutes a 2-dimensional vector equation for the field {2(;r2) in the plane Oxy:

(2.15)

Associated Stress Field The linearised strain field corresponding to this displacement field ~ is, by construction, a plane strain field parallel to Oxy and independent oCz:

(2.16) where ~2 , defined by the 2-dimensional formula

(2.17) is the 2-dimensional linearised strain field of the field {2 in the plane Oxy. The stress field g associated with ~ by the 3-dimensional constitutive equation

(2.18) is also independent of z. Furthermore, given that [satisfies (2.1), substitution into (2.18) shows that at every point of [2, g takes the form

(2.19)

2 The lower index 2 indicates that differential operators refer to the Oxy plane. 762 Appendix 111. Elements of Plane Elasticity where aaß and a zz are independent of z. The field f!.. is therefore specified on [l in terms of the scalar field a zz and the 2-dimensional tensor field f!..2 defined on So:

g (:r.) = g2 (:r.2) + a zz (.:r.2) ~z Q9 ~z , { (2.20) .:r.2 E So, .:r. = .:r.2 + z ~z' 0 ~ z ~ C .

Rewriting the constitutive equation (2.18) in this way, we find that the fields ~2 and g2 are related at each point of So by the 2-dimensional relation

(2.21) where we have defined ll.2 = ~a Q9 ~a (a = x, y). It follows immediately that, since ~2 is 2-dimensionäI, the field a zz can be inferred from the field f!..2 by the relation -

(2.22) which states that E zz (.:r.) = o.

Boundary Conditions on Stresses According to (2.20), the field f!.. automatically satisfies the boundary condi- tions on So and Sc: -

azx (.:r.) == 0 = T~ (.:r.) on So and Sc , { (2.23) a zy (.:r.) == 0 = T: (.:r.) on So and Sc . On the lateral surface, denoting the normal to (a[l - So - Sc) by Tl (.:r.) = Tl2 (.:r.2), we have relative to the Oz direction,

~z . g . Tl (.:r.) = ~z . g2 (.:r.2) . Tl2 (:r.2) = 0 , (2.24) which satisfies (2.8). Boundary conditions (2.9) become

(2.25) and therefore involve the components of the 2-dimensional stress vector T..2 (.:r.2) = g2 (.:r.2) . Tl2 (.:r.2) at the field point On aSo.

Summary We thus find that solving the 3-dimensional elastic equilibrium problem under conditions (2.4-2.9) reduces to solving a 2-dimensional elastic equilibrium problem posed on the cross-section So of the system. 2. Plane Strain Thermoelastic Equilibrium 763

The main unknown is a 2-dimensional plane strain displacement field {2. It is a function of .J:.2' the 2-dimensional spatial coordinate on So, and is constructed from the 2-dimensional strain and stress fields, f:2 and ~2, respectively. 3 -- The equations for the 2-dimensional problem are gathered below (simpli• fying the notation): 2-dimensional Navier equation

(2.26)

2-dimensional geometrical equation

~2 = (grad2 6 + t grad2 6)/2, (2.27)

2-dimensional elastic constitutive equation

(2.28)

2-dimensional boundary conditions

on (ÖSO)~i , (2.29)

on (ÖSO)ri . (2.30)

The solution to the 3-dimensional problem, with 3-dimensional displace• ment field { and stress field g on n, is inferred from fields {2 and g2 by

{(.J:.) = {2 (.J:.2) , (2.31 ) { g (.J:.) = g2 (.J:.2) + v (tr g2 (.J:.2)) f z ® f z , (2.32)

.J:.2 E So , .J:. = .J:.2 + Z f z , 0 ::; Z ::; C .

The solution procedure is shown in Fig. 2.

3 The tensorial nature of these fields on the cross-section S is clearly intrinsic, that is, independent of the chosen orthonormal coordinate system Oxy. 764 Appendix III. Elements of Plane Elasticity

SOlllfioll procedure Field b.c. equat;oTl$

Priflcipallll1kllOWII ~ 2 ...... ~~:.~.~~ ...... (2.29)

(2.27)

(2.2 )

g2 ...... (2. 0) (2. I), (2.32) Fig. 2. Plane strain elastic g,i equilibrium problem: solution by displacement method

2.5 Solution by the Stress Method

Form of the Stress Field Considering the results obtained by the previous method, we seek to construct the field Q. solving the 3-dimensional problem from a 2-dimensional field Q.2 , which is a function of ;[2 alone on the cross-section So of the system, via (2.32) :

g (;[) = g2 (;[2) + v (tr g2 (;[2) hz ® ~z , { ;[2 E So, ;[ = ;[2 + Z ~z' 0:::; z :::; f .

We use the stress method whose solution procedure is illustrated in Fig. 8, Chap. VIII for the 3-dimensional problem.

Boundary Conditions on Stresses By the same arguments as in the previous section, we see that the boundary conditions for the stresses on So and SR, and also on the lateral surface for the component in the Oz direction, are automatically satisfied. The two remaining conditions for the stresses on the lateral surface can be written in the form (2.25).

Equilibrium Equations The 3-dimensional equilibrium equation

div g + P F = 0 on [2 (2.33) 2. Plane Strain Thermoelastic Equilibrium 765

yields an equation in the Oz direction that is automatically satisfied, and two scalar equations in the plane Oxy representing a 2-dimensional vector equation for the field g2:

div2 g2 + P F 2 = 0 on So . (2.34) Associated Strain Field The strain field f:. associated with Q. can be written at each point of [l in - - the form

(2.35)

Substituting in the expression (2.32) for g (;r.) , we obtain

l+v ~ (;r) = --y (g2 (;r2) - v (tr g2 (;r2)) Jk2) , (2.36) ;r2 E So , ;r = ;r2 + z f z ,OS; z S; e , which shows that this field is plane, parallel to Oxy, and independent of z. (This is indeed the opposite calculation to the one carried out in the last section!) Once again, we shall write

(2.37)

Geometrical Compatibility Conditions The geometrical compatibility conditions (Chap. II, Sect. 6) for a field f:. of form (2.37) reduce to a single scalar equation with second order partial derivatives, the others being automatically satisfied:

2 2 cPsxx + 8 syy _ 2 8 sxy = 0 (Oxy orthonormal) . (2.38) 8y2 8x2 8x8y

This equation expresses geometrical compatibility of the components saß of f:. in the plane Oxy, that is, geometrical compatibility of the 2-dimensional Aeld f:.2 within its plane.

Displacement Field If (2.38) is satisfied, then integration over [l of the field ~ given by (2.36) leads to fields ~ of form

~ (;r) = ~2 (;r2) + arbitrary rigid body motion S.P.H. , { (2.39) ;r2 E So , ;r = ;r2 + z f z ,OS; z S; e ,

where ~2 (;r2) is a 2-dimensional field in the plane Oxy, and a function of ;r2' obtained by 2-dimensional integration of the field ~2. 766 Appendix III. Elements of Plane Elasticity

Boundary Conditions on Displacements Boundary conditions (2.5) on So and SR reduce the arbitrary rigid body motion on the right hand side of (2.39) to an arbitrary rigid body motion in the plane Oxy, and this can be included in {2. The component ~z of { is identically zero, whilst the remaining components of ~ are independent of z. The condition on the lateral surface in the Oz direction is then automat• ically satisfied. It remains for {2 to satisfy the boundary conditions (2.12) on (äSok· This generally suffices to determine the arbitrary rigid body motion in the plane Oxy.

Summary As expected, solving the 3-dimensional problem stated in Seet. 2.3 by the stress method amounts to solving the same 2-dimensional problem that was developed in Seet. 2.4 on the basis of the displaeement method. The main unknown is then the 2-dimensional field Q:2, a funetion of ;[2 on the eross• seetion So. The 2-dimensional fields ~2 andthen {2 are thus construeted. The system of equations required for this solution is: 2-dimensional equilibrium equation (2.34)

div2g2+pF2=0,

2-dimensional elastic eonstitutive equation (2.36)

equation for 2-dimensional geometrieal eompatibility (2.38)

2 2 ä2 ä c;xx + ä c;yy _ 2 c;xy = 0 (Oxy ort ho normal) , ä y2 Bx2 äxäy

2-dimensional geometrie al equation (2.27)

2-dimensional boundary eonditions (2.29) and (2.30)

on (äSok ,

on (äSO)Ti •

The solution (Q:, ~) of the 3-dimensional problem follows from the solution (g2, {2) of the-2-dimensional problem by the relations (2.31) and (2.32). The solution proeedure is represented in Fig. 3. 2. Plane Strain Thermoelastic Equilibrium 767

SoLution procedure Field b.c. equalion

(2.34) (2.30) Principal unkllown g-2 ...... • • ...... H ......

(2.36)

g, 2 ...... (2.38) J (2.27)

~ 2 ...... (2.29) (2.31) , (2.32) Fig. 3. Plane strain elastic g { equilibrium problem: solution by the stress method

2.6 Remarks on the Plane Strain Two-Dimensional Problem

It is important to note that the mutually inverse relations (2.28) and (2.36) are the 2-dimensional elastie eonstitutive equations to be used for plane strain problems. They express the relation between the 2-dimensional tensors ~2 and Q.2 for the problem assoeiated with the 3-dimensional problem stated in Seet:-2.3. This plane problem, speeified by (2.26) to (2.30), or by (2.34, 2.36, 2.27, 2.29 and 2.30), has preeisely the same strueture as the 3-dimensional lin• earised elastie equilibrium problems studied in Chap. VIII. The form of the boundary eonditions guarantees that the problem is regular and well posed. 4

2.7 Two-Dimensional Beltrami-Michell Equation

As in Chap. VIII (Seet. 6.2), we ean express the eondition (2.38) in terms of g2 by means of (2.36). Henee,

82a xx 82a yy () 82a xy 2 2 - v.d2 tr Q.2 2- (2.40) -8y + -8x - = 8 x 8 y .

Using (2.34), this equation rearranges 5 to give the sealar equation

.d2 (tr g2) + 1 ~ v P div2 F 2 = 0 , (2.41 )

4 Uniqueness of the solution follows in the same way as it does for 3-dimensional problems. This is because the positive definite character of the quadratic form from which (2.28) is derived is guaranteed by isothermal stability in the natural state of the elastic material: (3 A + 2 p,) > 0 and p, > 0 =} (A + p,) > 0 and p, > o. 5 Adding (2.38), the derivative with respect to x of the x component of equation (2.34) and the derivative with respect to y of the y component of (2.34). 768 Appendix III. Elements of Plane Elastieity known as the Beltrami-Michell equation for the plane strain elastic equilib• rium problem. We thus arrive at the solution procedure shown in Fig. 4.

oltllion procedure Field b.c. equatioll

Principalllllknown g2 (2.34) ,(2.41) (2.30)

(2.36)

f (2.27)

~2 (2.29) (2.3 1) , (2.32) Fig. 4. Plane strain elasticity problem: solution by the g i stress method ; Beltrami• Michell equation

Referring to Figs. 2 or 3, we may observe that there is no redundaney in the field equations between the equilibrium relations and the geometrical eompatibility eondition or the Beltrami-Miehell equation for the 2-dimensional problem (3 sealar field equations for the symmetrie 2-dimensional tensor field Q:2). It is interesting to examine the eoherenee of this result as far as the Beltrami=-Miehell equations are eoneerned, by referring to the solution proeedure for the 3-dimensional problem il• lustrated in Fig. 10, Chap. VIII. Coneerning the six 3-dimensional Beltrami-Miehell equations, we find that, given the form of (2.32) for g, • the (x, z) and (y, z) equations are identieally satisfied, • the (z, z) equation is just (2.41) , • the (x, x) , (x, y) and (y, y) equations are eaeh equivalent to (2.41) when (2.34) is satisfied. In other words, using the terminology in Chap. VIII (Seet. 6.2), equation (2.41) is indeed the single Beltrami-Michell equation associated with this problem, pro• vided that the equilibrium equations (2.34) are taken as field equations, and not merely as boundary eonditions in the way shown in Fig. 10, Chap. VIII, for the 3-dimensional ease.

2.8 Body Forces Deriving from a Potential. Airy Function

We assurne that body forces P, of form (2.4), derive from a 2-dimensional potential:

(2.42) 2. Plane Strain Thermoelastic Equilibrium 769

We can then solve the equilibrium equation (2.34) by introducing a scalar function cjJ(Ji.2) called the Airy junction or stress junction 6, and setting

82cjJ (J xx = 8y 2 + P V , 82cjJ (J yy = 8x2 + P V , (2.43) 82cjJ (J xy = - 8x 8y .

We have

tr g2 = LJ.2cjJ + 2pV, (2.44)

and the Beltrami-Michell equation becomes, in terms of cjJ, 1- 2v LJ.2LJ.2cjJ + --p LJ.2 V = 0 . (2.45) I-v The problem thus reduces to determining the function cjJ satisfying (2.45) and boundary conditions (2.30) and (2.29). In particular, if body forces derive from a harmonie potential LJ.2 V = 0, as happens when they are constant, equation (2.45) becomes

(2.46) implying that the Airy function is biharmonic. 7 Clearly, it will only be easy to formulate the problem that determines the stress function if all boundary conditions of the 2-dimensional problem actually refer to the stresses, that is, assume the form (2.30). Note that the definition (2.43) of Q.2 from the stress function cjJ can be put into intrinsic tensor form -

(2.47)

We can then obtain the components of g2 in polar coordinates. The result is

6 G.B. Airy (1801-1892). 7 LhLl2 4> = (84 4)/8x4 ) + 2(84 4)/8x2 8y 2) + (84 4)/8y 4) = 0 (zero bilaplacian). 770 Appendix III. Elements of Plane Elasticity whence, 02 lo (Jrr = r 2 O()2 + ~ or + p V , 02 (2.48) (J(J(J = or 2 + p V ,

(J r(J = - :r (~ ~:) .

2.9 Cylindrical Tube Under Pressure The isothermal elastic equilibrium of a cylindrical tube subject to internal and external pressure discussed in Chap. IX (Sect. 7), in the case where boundary conditions require zero displacement of the end cross-sections along the cylinder axis, is an example of a plane strain elastic equilibrium problem. Using the notation in Chap. IX (Sect. 7.1, Fig. 13), it can indeed be checked that the problem is posed in precisely the same way as described in Sect. 2.3 above. Body forces F=O in agreement with (2.4) .

Boundary conditions on So and Sg

~~ = 0 in agreement with (2.5) , (2.49) r;!=o, rt=o in agreement with (2.6) . (2.50) Boundary conditions on the lateral surface

d T = po f r for r = ro , (2.51) d 'L = -P1 f r for r = r1 , in agreement with (2.8) and (2.9) . (2.52) The solution is, in cylindrical coordinates (without specifying the values of constants A and B determined by boundary conditions for r = ro and r = rI)

c= (A(1+V)(1-2V) ~) (2.53) -<, E +2 /-Lr f r , which does indeed take the form of (2.31), and B B I7rr = A - 2 ' 17ee = A+ 2' I7zz = 2 vA, other 17ij = 0 , (2.54) r r which agrees with (2.32) if we set B g2 (~2) = AJk2 - r 2 (fr Q9 f r - fe Q9 fe) (2.55) and (2.56) The corresponding biharmonic Airy function is

r 2 r/J( r, 8) = A 2 - B In r . (2.57) 3. Plane Stress Thermoelastic Equilibrium 771

3. Plane Stress Thermoelastic Equilibrium

3.1 Plane Stress Tensor

The stress tensor at a point is said to be plane, parallel to Oxy, if it takes the form

(3.1a)

or alternatively,

a,ß = x,y. (3.1b)

3.2 Plane Stress Field gis a plane stress field, parallel to Oxy, if it fulfills the following eonditions:

Q. is independent of z , { (3.2) g is plane, parallel to Oxy at every point , so that:

axz=ayz=azz=O

oaxx oaxy oayy _ (3.3) --=--=---0 oz oz oz-

3.3 Plane Stress Thermoelastie Equilibrium Problem in a Homogeneous and Isotropie Material

We shall eonsider elastie equilibrium problems for a homogeneous and iso• tropie solid material, in whieh the solution results in a stress field Q. of form (3.2). - As in Seet. 2.3 eoneerning plane strain problems, it is clear that a problem is not generally posed apriori as a 'plane stress' problem. It is rather the form of the available information that leads us to seek a solution in whieh the stress field satisfies eonditions (3.2). The hypothesis is then justified aposteriori by the existenee of such a solution. We shaH see that problems for whieh such a hypothesis is strictly justifiable are uneommon (Seet. 3.4). In the following, we shaH restrict diseussion to isothermal elastie equilib• rium problems in order to simplify the presentation. We should nevertheless note that the arguments put forward in Seet. 2.3 are no longer valid. The no• tion of plane stress is not systematieaHy maintained between a thermoelastie equilibrium problem with temperature variations and its assoeiated isother• mal problem. 772 Appendix III. Elements of Plane Elasticity

Statement of the Plane Stress Elastic Equilibrium Problem We eonsider a eylindrieal system, of volume [l, with axis parallel to Oz (Fig. 5). All notation is as in Seet. 2.3.

y O ®· ------

x

Fig. 5. Plane stress elastic equilibrium problem

The elastie equilibrium problem is then determined by the following eon• ditions.

• Body Forces These are parallel to Oxy and independent of z:

F (22) = F 2 (zd, 222 E So, 22 = 222 + Z f z ' 0 ~ z ~ C . (3.4)

• Boundary Conditions on So and Sg So and Sg are stress free surfaees: r.d = 0 on So and Sg. (3.5)

• Boundary Conditions on the Lateral Surface Boundary eonditions on the lateral surfaee of [l are independent of z and expressed on the boundary 8So of So. In the Oz direetion, the eondition eoneerns, at eaeh point of 8So, the eomponent Tz of the stress vector, whieh is required to be zero: T: (222) = 0 on 8So . (3.6) The two remaining boundary eonditions are identieal to (2.9):

Ti (22) = Tid (222) on (8Sok , ~d22) = ~f (222) on (8Sok , (3.7) (8Sok U (8S0 )Ei = 8So , (8Sohi n (8S0 )Ei = 0 , i = 1,2, 3. Plane Stress Thermoelastic Equilibrium 773

In practice, boundary conditions (3.7) only usually affect the stress vector: (aSok is empty, i = 1, 2.

Note In thermoelastic equilibrium problems with temperature variations, plane stress conditions are satisfied if 'mechanical' conditions for the problem have the form indicated in (3.4 to 3.7) and if the specified temperature change field T(~) is independent of z.

3.4 Solution

We follow through the stress method shown step by step in the diagram of Fig. 8, Chap. VIII, as in Sect. 2.5 above.

Form of the Stress Field The stress field is assumed apriori to take the form (3.2), i.e., we assume a plane stress field parallel to Oxy and independent of z. This 3-dimensional stress field on [2 can be specified in terms of a 2-dimensional tensor field Q:2 on So: -

Boundary Conditions on Stresses The boundary conditions for the stresses given by (3.5) on So and SjI, and (3.6) on aso are automatically satisfied thanks to the form (3.2) chosen for Q:. The two remaining boundary conditions in (3.7) for the stresses on the lateral surface can be expressed, as in Sects. 2.4 and 2.5, in the same way as (2.25):

(3.9) thus restricting the components of the 2-dimensional stress vector 7:..2 (~2) = g2 (~2) . Ib (~2) .

Equilibrium Equations The equilibrium equation in the Oz direction is automatically satisfied. The other two equations, in the plane Oxy, are just the 2-dimensional vector equation for the field g2 already stated in Sect. 2.5:

(3.10) 774 Appendix 111. Elements of Plane Elasticity

Associated Strain Field The strain field [ associated with !l. by the elastic constitutive equation is specified at each point of f2 by -

l+v v ~ ür) = ~ g (!f) - E ( tr g (!f) ) 11 . (3.11)

With the expression (3.8) for g (!f), we find that ~ (!f) can be written

(3.12) where Eaß and Ezz are independent of z. The field [ can thus be specified on f2 in terms of the scalar field E zz and the 2-dimens2onal tensor field [2, both defined on So:

~ (!f) = ~2 (!f2) + Ezz (!f2) f z 0 f z , { (3.13) !f2 E So, !f = !f2 + Z f z ' 0:<:; z :<:; C .

Ey writing out the elastic equations in this way, we see that [2 and !l.2 are associated at each point of So by the 2-dimensional relation --

(3.14) whilst the field E zz is calculated from g2 by:

(3.15)

Geometrical Compatibility Condition. Given the form (3.13) of the strain field [, the geometrical compatibility con• ditions reduce to four scalar equations with second order partial derivatives. We must satisfy (3.16), already stated in Sect. 2.5:

2 2 8 Exx + 8 Eyy _ 282EXY = 0 . (3.16) 8 y 2 8x2 8x8y

This expresses geometrical compatibility of the components Eaß of [ in the plane Oxy, i.e., the integrability of the 2-dimensional field [2 in its plane. We must also satisfy the three equations -

2 2 2 8 Ezz = 0 8 Ezz = 0 8 tz = 0 (3.17) 8x2 ' 8x8y '8y 2 ' which require Ezz to be a linear (affine) function of coordinates x and y:

(3.18) 3. Plane Stress Thermoelastic Equilibrium 775

Equations (3.17) express the geometrical compatibility of the transverse strain Ezz (X2) in (3.13).

Displacement Field If equation (3.16) holds, integration of the compatible field ~2 over So deter• mines, up to a rigid body motion (S.P.H.) parallel to Oxy,-a 2-dimensional displacement field {2 which is a function of the position variable ;[2. If the three equations (3.17) are satisfied, it becomes possible to integrate the field ~ over fl, thereby determining the 3-dimensional displacement field {in the furm

z2 { (;[) = {2 (;[2) - (a f- x + b f- y ) 2 + (a x + b y + c) Z f-z 1 + arbitrary rigid body motion (S.P.H.) , (3.19)

;[2 E So, ;[ = ;[2 + Z f-z ' 0:::; Z :::; P .

Boundary Conditions on Displacements The displacement field ~ must satisfy the boundary conditions on (aSO)!;i stated in (3.7), which only refer to the components of the displacement field { (;[) in the plane Oxy, that is,

Z2 {2 (;[2) - (af-x + bf-y ) 2 + rigid body displacement S.P.H. in Oxy .

Note that if (aso)t;, is nonempty, the independence of conditions (3.7) from Z concerns the term (a f-x + b f- y ). In particular, if the two components of ~ in the plane Oxy are given at one point of aso then according to (3.18), Ezz-(;[2) is constant.

Summary Gathering the above results, we note that solving the elastic equilibrium prob• lem by the stress method under conditions dictated by (3.4 to 3.7) amounts to solving a 2-dimensional elastic equilibrium problem on a cross-section So of the system, together with supplementary conditions imposed by transverse geometrie al compatibility of strains and compatibility of displacements with boundary conditions on the lateral surface if the case arises.

• Two-Dimensional Plane Stress Problem The equations of the 2-dimensional problem are as follows: 2-dimensional equilibrium equation (3.10)

div2 g2 + pF2 = 0 ,

2-dimensional elastic constitutive equation (3.14) 776 Appendix III. Elements of Plane Elasticity

1+v V ~2 = ~ g2 - E (tr g2) 1k2 ,

2-dimensional eompatibility equation (3.16)

(Oxy orthonormal) ,

2-dimensional geometrieal equation

(3.20)

2-dimensional boundary eonditions on stresses (3.9)

• Geometrical Compatibility of the Transverse Strain and Compatibility of Displacements with Boundary Conditions In addition to the above equations, we also have the transverse strain equation (3.15)

the eompatibility equations (3.17) for the transverse strain

and the boundary eonditions on the displaeements (3.7)

• Solution Procedure Figure 6 shows the overall solution proeedure. As far as the 2-dimensional stress field fZ:2 and its assoeiated 2-dimensional strain field §:2 are eoneerned, the problem is formally identieal to the one stated in Seet. 2.5, whieh aimed to solve the plane strain problem by the stress method (Fig. 3). The 2-dimensional elastie eonstitutive equation for plane strain in Fig. 3 is replaeed in Fig. 6 by the 2-dimensional elastie eonstitutive equation for plane stress (3.14). This amounts to saying that, for a given system made from a homoge• neous and isotropie elastie material with elastie eonstants E and v, if no boundary eondition on the lateral surfaee affeets displaeements, solution of the 2-dimensional plane strain problem yields the solution to the eorrespond• ing 2-dimensional plane stress problem via a simple formal transformation. 3. Plane Stress Thermoelastic Equilibrium 777

Ollllioll procedure Field h.c. equorion

Prmcipal unknoH'1I ~2 ...... t.:.!.9.L .... ,,<3.9)

(3.14) (3. 15)

g; 2 E<2 .....(?:!.~t!.Q:. ~? ) J <3.20) Q; ...... (3.7) Fig. 6. Plane stress elastic equilibrium problem: solution by the stress method

The solution to the second is obtained by making the following replacement for the terms occurring in the first:

E (1 + 2 v) / (1 + v) 2 for E , { (3.21) v / (1 + v) for v . Using the correspondence (3.21), we can also obtain in a straightforward manner the Beltrami-Michell equation, for the 2-dimensional plane stress problem, counterpart to (2.41), (3.22) describing the 2-dimensional geometrical compatibility of ~2' When body forces derive from a potential V, we may onee again introduce the Airy function by the same formulas (2.43). Q:2 then satisfies (3.10). The Airy function

Note that incompatibilities involving strains, and displacements should the need arise, are of a transverse nature, that is, they are related to the z coordinate. Hence, in practice, it is assumed that these equations may not be satisfied when the solid is thin, like a plate: this is the thin slice approximation (R small compared with the minimal diameter of So). The fact that (3.17) no longer holds means that, stacking thin slices one on top of the other, their deformations are not compatible. The lower face of one detaches from the upper face of the next. A thick solid is not simply a stack of thin slices. 8

3.5 Cylindrical Tube Under Press ure

The isothermal elastic equilibrium problem far a cylindrical tube subject to internal and external pressures, presented in Seet. 2.9, provides an example of plane stress elastic equilibrium when surface forces are required to be zero on the end cross• seetions So and Se. Using the same notation, it can be checked that conditions correspond to those in the statement of the problem specified in Sect. 3.3. Indeed (2.49) is replaced by

T~ =0, (3.24) which, together with (2.50), agrees with (3.5). Note also that (2.51) and (2.52) agree with (3.6), introducing no conditions on displacements in (3.7).

The solution can be written in cylindrical coordinates, with the same constants A and B as in Seet. 2.9, determined from boundary conditions (2.51) and (2.52):

B B CTrr = A - 2 ' CTOO = A+ 2' CTzz = 0, other CTij = 0 , (3.25) r r which does indeed take the form (3.8) with

(3.26) and

(3.27) which takes the form (3.19) with

c _ (A(l-V) ~) b - E r + 2/1 r ~r' (3.28)

8 This shows how impartant it is in the plane stress elastic equilibrium problem to follow the solution procedure for the stress method in the form displayed in Fig. 8 of Chap. VIII, where compatibility equations are given far strains. It is then possible to assess, from a mechanical point of view, the approximate nature of this type of solution, a task rendered more difficult by direct use of the Beltrami-Michell equations. We can also understand why solution by the displacement method is not considered in this case. 3. Plane Stress Thermoelastic Equilibrium 779

We can also check the relation (3.15) between Czz and tr fZ2 • Note that fZ2 is constant, indicating that for this probl;m conditions (3.17) hold rigorously: transverse geometrical compatibility of strains is guaranteed without any kind of thin slice approximation. Finally, we can check that expression (3.28) for ~2 follows from (2.53) via the transformation (3.21). 780 Appendix III. Elements of Plane Elasticity

Summary of Main Formulas

• Plane troin

Equilibrium equation

2-dim Ilsional elastic constitutive equation

g2 = A (tr ~2):!!:2 + 2 J.' ~ l+v ~2 = --e (~ - v(tr ~):!!:2)

avi r equation

Geometrical compatibility equation

ß2czz + ß2clIlI _ 2ß2czlI = 0 (Oxy orthonormal) {}y2 8x2 8x {}y

Beltrami- Michell equation

Airy function, if E.2 = - grad2 V Summary of Main Formulas 781

• Plam trcss

l~qllilihrium cquation

1-uiml'llsional cl!l8tie eon titutivc equation

G ~m lrical ornpatibility equation 2 2 2 a erz + () e"" _ 2{) er" = 0 (Oxy orthollormal) [)y2 [);r2 {)x Dy

Beltrami lieh 11 <'quation

iry fUllction, if E2 = -grad2 V

[)2dJ (J"zr 2 + pV = ull" [)2tfJ (J"~~ = D.r2 + pV [)2tfJ (J"ru = - {)Jo{)y .J2..:l2tfJ+(1-v)p..l2V=O

• l'ron ' llion /mn! Plane Strain Lo Plant~ trc s

E replaecd by E(I + 2//)/(1 + //)2

11 replac d by 11/(1 + //) Bibliography

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Absolute temperature 299, 311 - change of 704-705 Acceleration 5, 17, 85, 105, 130 - dual 702, 715, 717, 737 - Coriolis 143 - local orthonormal 735 - quantity of 139 - natural local 734 Actual motion 150, 157, 158, 177, - orthonormal 717, 726, 728 234, 593, 653 - primary 737 Adiabatie evolution 319 Beam 416 Airy function 757, 768, 770, 777, 780, - continuous 640, 689 781 - curvature 673 Angle of twist 365, 408, 482 - curved 635 Angular distortion 46 - slender 661 - rate 85,93 - statics 579, 592-613, 628 Angular velocity, virtual 149 - straight 626, 635, 640, 656-663, Anisotropie material 303, 343 672, 681, 691, 692 - composite 391 Beltrami equations 412 - in compression 471 Beltrami-Miehell equations 398-401, Antisymmetric 420, 435, 778 - Euclidean tensor 724 - two-dimensional 767, 777, 780, 781 - tensor 707, 738 Bending 590 Apparent - circular 448 - compliance 515 - combined 555 - compression modulus 557 - moment 433, 443, 450, 574, 605, - elastic compliance tensor 558 626, 630, 637, 643, 657 - modulus 471, 515, 522 - normal 442-450, 465 Arch 579, 635 - of a rod 537 - funicular 590 - of cantilever beam 417 - low curvature 665 - off-axis 451-454, 660 - thrust 636 - with axial loading 454 Archimedes' principle 241 Bernoulli's theorem 244 Assembly Bernoulli, Daniel 346 - boundary condition 623 Bernoulli, Jakob 346 - joint 618 Biharmonic function 769 Axial loading, of a rod 454 Bilaplacian 769 Bilateral constraint 375 Balance equation 113, 309 Binding force 530, 545, 548 Ball-and-socket Body force 187, 191, 192 - connection 590 Boundary conditions 196, 206, 371, - joint 617, 619, 632 374, 378, 380, 490, 618-621, 759 - support 616 - assembly 623 Barotropic fluid 245 - endpoint 614-618 Basis - mixed 374, 563, 616 790 Subjeet Index

- support 614-618, 623 Components - traetion 374 - oftensor 703-708 - veloeity 374 - of tensor produet 708 Boussinesq stress tensor 230 Composite material 237, 502 Bresse-Navier equations 634,668 - anisotropie 391 Buekling 692 - in eompression 469 - lateral 692 Compressible fiow 241, 242 - transverse 692 Compression 291 Bulk modulus 299, 337 - isothermal 522 - isotropie 273 Cable 651 - of anisotropie material 471 - staties 578, 591 - of eomposite solid 469 Canonieal isomorphism 715 - simple 271, 302 Cantilever beam 417, 674, 676 - test 342 Cartesian tensor 727 - with eonfinement 468 Castigliano theorem 531, 541-545, - with frietion 562 549, 553, 668, 674, 675, 690 Coneentrated forees 418, 581, 587 Cauehy reeiproeal theorem 258 Conduetion inequality 318, 322 Cauehy stress tensor 187, 208-211, Configuration 10 238, 278, 327 - eurrent 10, 18, 90, 158, 372, 651 - material frame indifferenee 226, 279 - generalised referenee 17, 65 Cauehy, A.L. 347 - initial 328, 372, 651 Cauehy-Green strain tensor 44 Cavity - referenee 10, 313, 325, 652 - eylindrieal 464 Conjugate shear stresses 258 - spherieal 459 Conneetion 621 Celerity 111 Conservation Centre of inertia 656 - law 223 Centre of mass theorem 170 - of mass 118-121, 123 Centroid, of eross seetion 439, 442 - of moment um 170, 175, 179, 222, Change of 226, 600 - basis 704-705 Constitutive law 280, 301, 367 - frame 148, 159, 180 - elastie 520, 579 Cireular bending of a rod 448 - for one-dimensional medium 578, Cireular eross seetion 409-411, 413, 594, 601, 613, 627, 651, 668 415, 416, 420 - for struetural member 639 Cireulation - geometrieal linearisation 334-340, - material derivative of 115 350 - of tensor quantity 116 - physieallinearisation 328-332, 349 - of veetor quantity 115 - thermoelastie 315-328, 349, 354, Clapeyron equation 520-522, 549, 369, 496-498, 503, 610, 614, 680 553-557, 562, 564, 567 Constitutive relations 541 Clausius-Duhem inequality 312,313, Constraint parameter 530 316,322 Contact foree 197, 206, 236, 586, 600, Closure eonditions 63, 397, 438, 444 604 Collapse meehanism 644 Continuity 8, 56, 59, 64 Colonnetti's theorem 548 - equation 118, 123, 367, 378 Compatibility eonditions 60-62, 386, - hypotheses 13-16 504, 513, 590, 600, 606 - of stress veetor 219 - two-dimensional 776 Continuous beam 640, 689 Complementary elastie potential 662 Continuum 7,23 Complementaryenergy 487, 504-511 Contracted produet - minimum principle 506, 513, 529, - of Euelidean tensors 718, 720, 722 552 - of tensors 709, 710, 737 Subject Index 791

Contraction - elliptical 425 - of a tensor 708, 737 - of a rod 402, 438, 447 - of tensor product 709, 737 polar moment of inertia 410, 411 - total 713 - rigid rotation 403 Contravariance 699, 705 - rotation 433 Contravariant representation 718 simply connected 412, 426, 438, 444 Convective term 105, 107, 110, 278 - small 590 Convective transport 37, 53, 69, 87, - torsional inertia 365, 407, 408, 410, 115, 116, 228, 280 411 - of a surface element 447 - transverse 592, 605 of a vector 40 - triangular 426 of a volume 40 - warping 404, 425, 427 - of an oriented surface 42, 438 Cubic symmetry 354, 424 - ofaxes 475 Curl 736 Convex Curvature - function 319, 320, 495 - low 665 - set 494 - of beam 673 Convexity 492, 494-496, 551 - of fibres 433 of potential 498 - of mean fibre 448 - strict 495, 498, 508, 510, 542, 543, - of rod 449 547, 551 - radius 79, 585, 735 -- of potential 498 -- of director curve 626, 654 Coordinates Curvilinear coordinates 579, 732-736 curvilinear 579, 732-736 Cylinder 468, 471, 472, 474, 562, 758, - cylindrical 70, 124, 214, 239, 421, 770, 772, 778 746-747 - heterogeneous 522 - general Cartesian 744-745 Cylindrical cavity 464 - generalised Lagrangian 29 Cylindrical coordinates 70, 124, 214, - material 13 239, 421, 746-747 - ort ho normal Cartesian 124, 214, Decoupled thermoelasticity 371, 372 239, 393, 743-744 Deformation 38, 54 spatial 13 - infinitesimal 328, 334, 376 - spherical 70, 125, 215, 239, 422, - reversible 342 747-749 Deviator 253, 265, 276, 282 Coriolis Deviatoric - acceleration 143 - plane 285 - force 144 - stress tensor 265 Corotational derivative 280 Differential rotation 365, 404, 407, - material frame indifference 281 408, 447, 482, 660 Cosserat continuum 237 Dirac distribution 110, 217 Cotler-Rivlin rate 281 Director curve 573, 577, 590, 593, Coulomb yield criterion 287 605, 607, 611, 651, 652, 665 Coulomb, C.A. 346 - of straight beam 657, 660 Couple per unit mass 234 - position of 656 Couple stress 237 - radius of curvature 626, 654 - tensor 237 - virtual stretch rate 583 Covariance 699, 705 Dirichlet problem 413 Covariant representation 718 Discontinuity Crack 15 - equation 587, 602, 612 Cross section - term 643 centre of inertia 409, 439, 442 Discontinuous - centroid 439, 442 - internal force field 586-587, - circular 409-411, 413, 415, 416, 420 601-602 792 Subject Index

- stress field 218 - defining tensors by 713 - virtual motion 603 Dummy index eonvention 702 - virtual velocity field 199-201, 215-217,588 Effeetive stress 278 Dislocation, serew 566 Eigenvalue 728 Displacement Eigenveetor 728 - distributor 649, 652 Elastic - function 395, 458, 464 - eomplianee 686 - method 373, 392-396, 403-407, - eomplianee tensor 487, 497 420, 424, 426, 456, 462, 467, 474, 492, -- apparent 558 513 - eonstitutive law 520, 579 - of one-dimensional medium 614 - domain 305 Displaeement field 35, 54, 660 -- of a strueture 684 - kinematically admissible 365, 386, - energy 519, 521, 553, 556, 568 392, 396, 491, 492, 499, 513, 531, 533, -- in combined bending 555 -- in torsion 555 669 - Navier-Bernoulli 680 - instability 692 - plane strain 758, 760 - limit 276, 278, 284, 304 - radial 456 - moduli 331, 333, 337 - potential, complementary 662 - rigid body 63, 503, 508 - strain energy 487, 502, 552 - virtual 487, 505 - stress energy 487, 507, 552, 662, Displaeement method 760, 778 669,690 Dissipation 299, 312 -- line density 649 - intrinsie 312, 318, 319, 322, 370 Elasticity tensor 299, 331, 487 - thermal 312 Elliptical cross section 425 Distortion, angular 46 Endpoint Distributed forees 580, 581 - boundary condition 614-618 Distribution 110, 201, 217, 220, 587, - free 615 589, 602, 603, 612, 626 - loaded 615 - Dirae 110, 217 Energy Distributor 137, 160, 179, 540, 608, - bounds 510, 552, 562, 563 609, 614 - equation 306-311, 349 - displaeement 649, 652 - theorem 667, 679 - field 163, 573, 593, 596 Entropy 299, 311, 315 -strain 649, 652 Equations of motion 195, 204, 214, - tensorial 164, 235 239, 245, 286, 290, 367 - veloeity 160 - weak formulation 225, 512 - virtual veloeity 628 Equilibrium Divergence - equations 235, 369, 377, 399, 583, - of Euelidean tensor field 731 587-588, 591, 598-599, 602, 607, 614, - of tensor field 731, 739 623, 625, 628, 629, 669, 671, 673 Divergence theorem 110, 176, 195, -- for joint 618 205, 207, 218, 220, 230, 235, 309, - isothermal elastic 389, 393, 398- 311, 406, 564, 739 402, 420, 435, 489, 519-526, 549, 553, - for Euclidean tensors 731 554, 758, 770, 771 Double shear 73 - non-isothermal 401 Doubly eontraeted produet - thermoelastic 381 - of Euclidean tensors 721 -- linearised 667-673 - of tensors 711, 737 Equivalent stress 253, 278, 282 Dry frietion 632 Euclidean Dual basis 702, 715, 717, 737 - space 714 Dualisation 144, 493 - tensor 39, 46, 94, 204, 319, 718 Duality 540 -- antisymmetrie 724 Subject Index 793

first rank 718 - equations 584 general rank 726 - triad 612 representations of 722, 737 Fringes 416 second rank 719, 721-726 Functional 501, 506 symmetrie 724, 727 Fundamental inequality 312, 313, 349 -- transpose 723, 738 Fundamental law of Euler critical force 692 - dynamics 139, 145, 149, 153, 166, Euler's theorem 171, 175, 179, 222, 179, 199, 207 243 statics 167, 586, 601 Eulerian Funicular arch 590 - description 18-22, 90, 103, 114 - kinematics 90-102 Galilean reference frame 137, 144, - strain rate 91, 122, 22~ 327, 359 145, 149, 170, 172, 173, 222, 223 Evolution Galileo 346 adiabatic 319 Geiringer equations 128 isothermal 319, 342 General Cartesian coordinates of a system 9 744-745 - quasi-static 369 Geometrie instability 528, 622, 636 Expansion tensor 35, 43, 54, 92, 321 Geometrical compatibility 59, 70, 99, Extension ratio 46 122, 154, 231, 397, 400, 468, 492, Extensometry 77 514, 623, 633, 634 External force 140, 141, 151, 629, 670 - for a structure 641, 669 - line density 585, 604 for a truss 682, 683 - on structural member 625 - for thermal deformations 78 External moment 670 - of deformations 688 - of virtual strain rates 638 - transverse 775, 779 Facet 208, 255, 266, 285, 291 - two-dimensional 765, 766, 768, 774, - octahedral 266 777, 780, 781 - principal 263, 269 - weak formulation 100, 225, 514 Fibre 404, 445 Geometrical linearisation 334-340, - curvature of 433 350 - mean 446,660 Gradient Finite element method 391, 516 - of Euclidean tensor field 730 Finite transformation - of tensor field 730, 739 - in torsion 480 - of transformation 35, 53, 69, 321 - of spherical shell 476 -- material frame indifference 56 - tension test 474 Green, D. 347 First law of thermodynamics 306, 349 Green-Lagrange strain tensor 35, 48, First rank tensor 700 54,57 Fluid source (or sink) 126 Flux Hadamard compatibility conditions - of tensor quantity 117 113, 396, 401 - of vector quantity 116 Heat Force method 531, 548 - equation 368, 371, 378 Fourier's law 370 - flux 299 Fracture -- vector 309, 311 - mechanics 15 - input 299, 307 - surface 15 - problem 381 Fragile materials 64, 455 Helmholtz Frame 644 - free energy 312 Free energy 299, 315 - theorem 130 - Helmholtz 312 Hencky equations 286 Frenet Hertz problem 379 794 Subjeet Index

Heterogeneous material 392, 395, 396, - struetural 591 401, 502, 507, 557 Intermediate principal stress 276, 288 Hexagonal symmetry 353 Internal Hinged - eonstraint 299, 317, 321-325, 327, - joint 619 351, 353, 355, 356, 369, 372, 476, 490, - rods 155 496, 498, 503, 510 Hollow sphere 456, 458, 466 - energy 299, 306, 315 - under pressure 539 - force 64, 140, 141, 153, 237, 578, - yield point 460 629,638 Homogeneous -- in one-dimensional model 607 - isotropie material 393, 398, 402, - heat souree 299, 307 415, 420, 423, 425, 426, 435, 456, 462, Intrinsie 468, 474, 562, 566, 656, 690, 758, 771 - derivative 279,359 - orthotropie material 424 - dissipation 318, 319, 322, 370 - tangent map 51 Invariants - transformation 38, 65 - of mixed seeond rank tensor 714, - transversely isotropie material 471 738 Homogenisation 8, 237, 561 - of seeond rank Euclidean tensor Hooke's law 346 725 Hooke, R. 305 - of the strain tensor 299, 326 Hydraulie analogy 24 - of the stress tensor 253, 263-265, Hydrostaties 199 276,282 Inverse tensor 701, 738 Irrotational motion 126, 127 Ideal internal eonstraint 324 Isoehorie motion 97 Ineompressibility 97, 317, 321, 328, Isometry, direet 38, 43, 49 351, 352, 355-357, 474, 476, 480 Isothermal Ineompressible flow 97, 120, 240, 241, - behaviour 656-663 243 - eompression 522 Ineffective tensor 324, 328, 351, 353 - elastie equilibrium 389, 393, Inextensibility 317, 321, 353 398-402, 420, 435, 489, 519-526, 549, Infinitesimal 553, 554, 758, 770, 771 - deformation 328, 334, 376 - evolution 319, 342 - rotation 63, 654 - loading 556 - transformation 57, 65, 70, 98, 101, - problem 391, 396, 401, 472, 760 334-340, 350, 376, 445 - stability 341, 498, 767 Infinitesimal deformation hypothesis - torsion 415, 555 376 Isotropie Infinitesimal transformation hypothesis - eompression 273 376, 480, 652, 660, 662, 692 - eompression test 342 Initial elastie domain 274 - eonstraint 321 Initial state - function 264, 326, 726 - natural 336, 378, 382, 392, 396, material 274, 275, 303, 315, 325, 415, 420, 423, 435, 467, 519-526, 332, 340, 350, 351, 354, 356, 357, 393, 553-556, 656-663 398-401, 480, 502, 508, 519, 523, 552, - preloaded 383, 415, 416, 511 561 - prestressed 338, 415, 416, 489, 511, - tension 273 663-665 Isotropy 349 - quasi-natural 337, 378 - group 324 - self-equilibrating 683 - spatial 12, 274, 280, 301 Initial stresses 331, 337, 378 - transverse 353, 354, 424, 471 Instability 479 - elastie 692 Jaeobian 5, 17 - geometrie 528, 622, 636 - determinant 14 Subject Index 795

Jaumann rate 280, 359 Linearity 303 Joint 613 Loading - assembly 618 - function 274 - ball-and-socket 617, 619, 632 - isotherm al 556 equilibrium equation 618 parameter 487, 531, 535, 537, 538, - hinged 619 540, 541, 548, 550, 553, 555, 556, 561, - pinned 619 563, 565, 668 rigid 619, 643 Local orthonormal basis 735 Love, A.E.H. 346 Kelvin theorem 130 Lower bound 515 Kinematic parameter 487, 535, 536, - for apparent modulus 524 538, 540, 541, 555, 556, 563 for loading parameter 550 Kinematically admissible displacement - for torsional inertia 565 field 365, 386, 392, 396, 491, 492, Reuss 558 499, 513, 531, 533, 669 Lower index 702 Kinematics Macroscopic scale 302 - Eulerian 90-102 Mariotte, E. 306 - Lagrangian 87-90 Marking of material elements 8, 23, Kinetic energy 137, 306 67, 68 - theorem 172, 176, 180, 223, 309 Mass conservation 118-121, 123 Kötter equations 289 Mass per unit volume 118 Material Lagrange multiplier 299, 323, 504, - coordinates 13 510 - element 14, 130 Lagrangian - gradient 88 - description 12-18,87, 102 - parcel 107 - kinematics 87-90 point 9, 23, 38, 139, 579, 580, 594 - rate of volume expansion 89 - surface 14 - strain rate 89, 122, 227 - vector 39, 96, 280 Lame constant 299, 333, 479, 502, 508 - volume 14 Laplacian 405 Material derivative 88, 313 Law of - in Eulerian description 103, 123 - action and reaction 140, 168 - in Lagrangian description 102 - mutual actions 142, 146, 148, 149, - of circulation 115 152, 166, 179, 195, 203, 235, 582, 597 - of flux 116 Legendre-Fenchel transform 299, 319, - of free energy 317 497, 507, 509, 510 - of mass integral 120 Line density - of point function 104 - of distributed forces 580, 581 of stress tensor 278 - of elastic stress energy 649 of vector 90 - of external force wrenches 595 of volume integral 105-115 - of external forces 585, 604 Material frame indifference 11-12, - of virtual rate of work 582 101,301 Linear thermoelasticity 328-346, 351 - of Cauchy stress tensor 226, 279 Linearisation - of corotational derivative 281 - geometrical 334-340, 350 - of transformation gradient 56 - physical 328-332, 349 - of virtual rate of work by internal Linearised forces 159 strain field 59 Material symmetry group 275, 324 - strain tensor 57, 98 Maxwell-Betti reciprocity theorem thermoelastic constitutive law 340, 525, 549, 553, 554, 557 496-498, 503 Mean fibre 446 - thermoelastic equilibrium 667-673 - curvature 448 796 Subject Index

- of a rod 660 Node 516 Mean normal stress 253, 265, 288 Non-isothermal equilibrium 401 Mean rigid body motion 96, 280 Normal bending of a rod 442-450, Menabrea theorem 548, 554 465 Mesh unit 516 Normal force 574, 605, 630, 668 Metric tensor 714 Normal stress 253, 256, 282, 287 Micropolar medium 234-237, 579, 600 - mean 253, 265, 288 Microstructure 23, 234, 317, 321, 353, - octahedral 266 354, 592 Numerical methods 516, 530 - transverse 611, 652, 656, 657, 660 Minimum potential theorem 530, Objectivity 11 545-549, 554, 669, 677, 682, 683, 688 Observer 9 Minimum principle 667 Octahedral - for complementary energy 506, 513, - facet 266 529, 552 - normal stress 266 - for displacements 499-502, 504, - shear stress 266 557, 560 Oedometric test 423, 561 - for displacements and stresses 509 Off-axis bending 660 - for potential energy 501, 512, 529, - of a rod 451-454 552 Ogden model 357,476 - for stresses 505-507, 510, 558, 560 Oldroyd rate 281 Mohr One-dimensional continuous medium - circles 267, 441 234, 557, 577 - plane 267 - constitutive law 578, 594, 601, 613, - representation 76, 266-274, 285, 627 287 - subsystem 580, 585, 587, 590, 595, Moire method 67 599, 602, 604 Momentum 169 Oriented surface 42 - conservation 170, 175, 179, 222, Orthogonal section 593, 605 226, 600 - cent re of inertia 656 Mooney-Rivlin model 357, 476 Orthonormal Multiple-scale model 8 - basis 717, 726, 728 Multiply connected domain 63 - Cartesian coordinates 124, 214, 239, 393, 743-744 Orthotropic material 344, 424 Natural initial state 336, 378, 382, 392, 396, 415, 420, 423, 435, 467, Parallel transport 230 519-526, 553-556, 656-663 Parametric problems 531-540, 545 Naturallocal basis 734 - examples of 536-540 Navier equation 393, 395, 420, 435, Particle 9, 23 457, 464, 467, 566, 761 Pathline 16, 19, 21, 22, 27-29 - two-dimensional 763, 780 Perfect fluid 199 Navier, C. 347 Perfectly bonded materials 395 N avier-Bernoulli Photoelasticity 416 - condition 610-614, 630, 634, 638, Physical linearisation 328-332, 349 653, 661, 664, 679 1[" plane 285 - displacement field 680 Piecewise - hypothesis 450, 661 - continuity 16, 17, 19, 110, 119, 190, Neo-Hookean 218, 387, 489, 551 - material 477, 478, 480 - continuous differentiability 16, 17, - model 357, 476 19, 110, 119, 190, 218, 387, 489, 551 Neumann problem 405,407,409 - geometrical compatibility 64 Neutral axis 445, 453 Pin-connected support 691 Newton's laws 140, 149 Pinned Subject Index 797

- joint 619 - values 727-729 - support 616 Principle of Piola-Kirchoff stress tensor 187, - local action 301 226-229, 239, 246, 279, 314, 315 material symmetries 324, 325 Piola-Lagrange stress tensor 187, spatial isotropy 315, 321 230-231 - superposition 382, 390, 440, 452, Plane 454 elasticity 757 - virtual power 494 - limit equilibrium 287, 289, 290 - virtual work 144, 149, 158, 179, - structures 625-627, 670 283, 292, 493, 588, 603, 624, 634, 639, Plane strain 641, 642, 678 - displacement field 758, 760 Product tensor 702, 708 problem 463, 757, 776, 780 Propagation velocity 85, 108, 111 - tensor 757 Pure shear 302 thermoelastic equilibrium 758 - stress state 272 Plane stress - field 771 Quantity of acceleration 139 - problem 417, 463, 757, 781 Quasi-natural initial state 337, 378 - tensor 771 Quasi-static Plate model 678 - evolution 369 Point, material 9, 23, 38, 139, 579, - thermoelastic process 419 580, 594 Radial Poisson ratio 299, 333, 344, 346, 398, - displacement field 456 438, 502, 508 - transformation 477 Poisson, S.D. 347 Radius Polar factorisation 50, 55, 58, 65, 316 - of curvature 79, 585, 735 Position vector 5, 10, 13 -- of director curve 626, 654 Potential 395, 498 - of torsion 585 complementary elastic 662 Rank of a tensor 699 - convexity of 498 Rate of - energy 487, 499-504, 510 - angular distortion 85, 93 -- minimum principle 501, 512, 529, - extension 93 552 - rotation vector 95 - strictly convex 498 - volume dilatation 96, 122, 126, 127 - thermodynamic 319, 330, 497 - volume expansion, Lagrangian 89 - two-dimensional 768, 777 Rate of change Poynting effect 481 - convective 105, 107 Preloaded initial state 383, 415, 416, - local 105, 107 511 Reciprocity theorem, Maxwell-Betti Pressure 189, 198, 238 525, 549, 553, 554, 557 Prestress tensor 497 Reduced elements 600, 605, 619 Prestressed initial state 338, 415, 416, Redundant unknowns 487, 528, 546, 489, 511, 663-665 548, 554, 623, 636, 639, 640, 668, 676 Primary basis 737 Reference frame 9 Principal - Galilean 137, 144, 145, 149, 170, axes 326, 727-729 172, 173, 222, 223 axes of inertia 442, 656 non-Galilean 143 - axes of stress 356, 414 Regularity 387 circles 269 Reissner principle 550 facets 263, 269 Repeated index convention 38, 140, - moments of inertia 442 702 stresses 253, 262, 265, 276 Representation theorem 263-265, - stretches 46, 66 275,726 798 Subject Index

Reuss lower bound 558 Second law of thermodynamics 311, Reversibility 303, 342 349 Right stretch tensor 51 Second order Rigid body - effect 481 - dis placement field 63, 503, 508 - thermoelasticity 351 - motion 10, 11, 99, 147, 158, Second rank 160-165, 280, 408, 608 - Euclidean tensor 719 - transformation 49, 56, 64, 65 - tensor 700, 738 - virtual motion 148 -- mixed 705 Rigid joint 643 -- twice contravariant 707 Rigid rotation 403 -- twice covariant 707 Ring 636, 685, 686 Self-equilibrating Ritz method 515 - distribution of forces 154 Rod 416, 590, 605, 656, 660, 663 - initial state 683 - bending of 537 - internal force field 622, 624, 634, - bending with axial loading 454 639, 684, 687 - circular bending of 448 - stress field 225, 231, 291, 379, 487, - combined bending and torsion of 489, 504, 511, 526-529, 535, 537, 539, 557 540, 554 - combined bending of 555 - stress state 383 - cross section 402, 447 Semi-inverse method 403, 443, 660 - curvature of 449 Semi-steady motion 22 - isothermal torsion of 555 Shear - normal bending of 442-450, 465 - lines 413, 425, 441 - off-axis bending of 451-454 - slender 690 - modulus 299, 333, 344, 346, 502, - tension-compression of 436-442, 508 465,536 - stress 253, 256, 282, 287, 408, 411, - tension-torsion of 440-442 427 - torsion of 402-416, 420, 425, 426, -- octahedral 266 480, 563 Shearing force 455,574,605,612,626, - yield point 630, 662, 679 -- in normal bending 450 Shell model 678 -- in simple compression 440 Shock wave 16, 119, 173-178, 219, -- in simple tension 440 220, 238 -- in torsion 413-416, 425, 427 Sign convention 258 Roller support 691 Simple Rotation 43, 50 - compression 271, 302 - differential 365, 404, 407, 408, 447, - shear 71 482, 660 - tension 271, 302, 303, 438 - infinitesimal 63, 654 Simply connected - of cross section 433 - cross section 412, 426, 438, 444 - rigid 403 - domain 62, 99 - vector 447 - solid 397 Slender Saint Venant - beam 661 - principle 416-418, 439, 449, 468, - object 416, 492, 577, 590, 607, 653 492, 566, 661 - rod 449, 690 - problem 455, 656, 679 - solid 655 Saint Venant, A. 403, 427 Slenderness ratio 446 Scalar product 714 Slip surface 15 Scale 7 Small displacement hypothesis 79, Screw dislocation 566 376, 404, 409, 446, 482, 655, 667 Subject Index 799

Small parallelipiped argument 213, - flow 120 219 - motion 21, 29 Small perturbation hypothesis 667 - regime 371, 372, 378 Small perturbation hypothesis (SPH) Stiff slender solid 590 377,384,409,419,423-425,445,449, Stiffness 592, 604 468, 489, 493, 503, 531, 651 Straight beam 626, 635, 640, 656-663, Small tetrahedron argument 212, 219 672, 681, 691, 692 Solid - complementary elastic potential - cylinder 468, 471, 472, 562 662 - sphere 459 - director curve 660 Spatial - thermoelastic behaviour 663-665 - coordinates 13 - thermoelastic constitutive law 664, - gradient 91 665 - isotropy 12, 274, 280, 301 Strain distributor 649, 652 -- principle of 315, 321 Strain ellipsoid 74 Sphere Strain field - hollow 456, 458, 466, 539 - solid 459 - geometrical compatibility 59, 492 Spherical - linearised 59 - cavity 459 Strain gauge 67 - shell 456, 458, 466 Strain parameter 535 -- finite transformation 476 Strain rate 85, 92, 122, 126, 127 -- thin 477,479 - Eulerian 91, 122, 227, 327, 359 Spherical coordinates 70, 125, 215, - Lagrangian 89, 122, 227 239, 422, 747-749 - virtual 203, 628 Spin Strain tensor 315, 321 - tensor 85, 95, 122, 126, 127, 280 - Cauchy-Green 44 -- virtual 203 - Green-Lagrange 35, 48, 54, 57 - vector 85, 95, 96, 122, 126, 127, - invariants of 299, 326 130,280 - linearised 35, 57, 98 Spring, helicoidal 686 - plane 757 Square symmetry 354 Streakline 16, 22 Stability Stream tube 120 - isotherm al 341, 498, 767 Streamline 20-22, 27-29, 120 - of material 319, 340-342, 353, 382 Strength condition 617, 641, 642 - of structures 668 Strength of materials 392, 545, 665, Static 673,678 - determinacy 528, 622, 627, 668, Stress 64 671, 674-676 - effective 278 - indeterminacy 528, 531, 545, 622, - equivalent 253, 278, 282 627, 668, 671, 676-678 -- degree of 528, 548, 622, 623, 636, - initial 331, 337, 378 639, 643, 668, 676, 678 - minimum principle for 505-507, Statically admissible stress field 365, 510, 558, 560 386, 392, 396, 490, 492, 505, 512, - normal 253, 256, 282, 287 531, 532 - principal 253, 262, 265, 276 Statics 579 - shear 253, 256, 282, 287, 408, 411, - fundamentallaw of 167, 586, 601 427 - of beams 579, 592-613, 628 - tangential 257 - of cables 578, 591 Stress concentration 418 - of wires 578-591, 597, 600, 602, Stress field 189 606, 628, 633 - discontinuous 218 Steady - plane 771 800 Subject Index

- self-equilibrating 225, 231, 291, - unilateral 617, 640 379, 487, 489, 504, 511, 526-529, 535, Surface couple 234 537, 539, 540, 554 Surface density - statically admissible 365, 386, 392, - of extern al forces 207, 221, 380 396, 490, 492, 505, 512, 531, 532 Surface force 187, 192, 193 Stress free surface 271, 374, 402, 405, Symmetrie 412, 414, 436, 442 - Euclidean tensor 724, 727 Stress function 365, 412, 757, 769 - tensor 707, 738 Stress method 396-401,412, 420, 423, System 139, 145, 154, 165, 191, 579, 424, 437, 443, 468, 492, 764-766, 773, 594 778 - of material points 144, 148, 150 Stress state - initial 378 Tangential stress 257 - pure shear 272 Temperature - self-equilibrating 383 - change 64, 78, 79, 299, 395, 649, - uniaxial 271 663, 683, 684, 687, 760, 773 Stress tensor 237 -- small 332, 376 - Boussinesq 230 - gradient 690, 691 - Cauchy 187, 208-211, 238, 278, 327 Tension 291 - in cable 689 - deviatorie 265 - in one-dimensional medium 573, - invariants of 253, 263-265, 276, 282 589 - material derivative of 278 - in wire 591 - Piola-Kirchoff 187, 226-229, 239, - isotropie 273 246, 279, 314, 315 - simple 271, 302, 303, 438 - Piola-Lagrange 187, 230-231 - test 342, 474 - plane 771 Tension-compression, of a rod Stress vector 187, 208-211, 238, 282 436-442, 465, 536, 657 - continuous 219 Tension-torsion, of a rod 440-442 Stretch 43, 45 Tensor Structural - antisymmetrie 707, 738 - analysis 392, 651-692 - Cartesian 727 - instability 591 - components 703-708 - stability 668 - contraction of 708 Structures 416, 531, 613-627 - defined by duality 713 - complex 637, 687 - Euclidean 718 - kinematic analysis of 623-624, 642 -- antisymmetrie 724 - plane 625-627, 670 -- first rank 718 - static analysis of 621-623, 642, 651 -- general rank 726 Substructure 545 -- representations of 722, 737 Substructuring 530 -- second rank 719, 721-726 Subsystem 141, 145, 148, 154, 165, -- symmetrie 724, 727 192, 196, 19~ 205, 206, 209, 309, 311 -- transpose 723, 738 - of one-dimensional medium 580, - first rank 700 585, 587, 590, 595, 599, 602, 604 - invariants 714, 725, 738 Superposition principle 382, 390, 440, - inverse to 701, 738 452, 454 - product 737 Support 621 - rank of 699 - ball-and-socket 616 - second rank 700, 738 - boundary condition 614-618, 623 -- mixed 705 - pin-connected 691 -- twiee contravariant 707 - pinned 616 -- twice covariant 707 - rigid 615 - symmetrie 707, 738 - roller 691 Tensor field 729-736 Subject Index 801

- divergence 731 Torque 403 - Euclidean Torsion 590 -- divergence 731 - elastic energy of 555 -- gradient of 730 - elementary theory of 563 - gradient of 730, 739 - isotherm al 415, 555 Tensor product - of a rod 402-416, 420, 425, 426, - components 708 480, 563 - contraction of 710, 722 - of a tube 425 - double contraction of 711, 721, 737 - problem 412, 440, 660 - oftensors 701 - radius of 585 total contraction of 713 - with finite transformation 480 Tensorial Torsional - distributor 164, 235 - inertia 365, 407, 408, 410, 411, 563 - wrench 164, 235 -- upper bound 411 Tetrahedron lemma 310 - rigidity 408 Thermal Traction coefficients 299, 331, 333 - boundary condition 374 - conductivity tensor 365, 370 - vector 209 deformation 79 Transformation 38 dissipation 312 - finite 474, 480 - effects 468, 472 - gradient 35, 53, 69, 321 expansion coefficient 299, 333, 508, - homogeneous 38, 65 690 - infinitesimal 57, 65, 70, 98, 101, - expansion tensor 487, 497 334-340, 350, 376, 445 - shock 395, 401 - radial 477 - strain 64, 78, 291 rigid body 49, 56, 64, 65 Thermodynamic potential 319, 330, Transpose of Euclidean tensor 723, 497 738 Thermodynamics 301, 306-314 Transverse - first law 306, 349 - buckling 692 - second law 311, 349 - cross section 592, 605 Thermoelastic - geometrical compatibility 775, 779 - constitutive law 610, 614 - microstructure 611, 652, 656, 657, - equilibrium 381 660 -- linearised 667-673 Transversely isotropie material 345, - process 367 353, 354, 424, 471 -- quasi-static 419 Tresca yield criterion 276, 282-286, Thermoelastic constitutive law 291, 415, 425, 427, 440, 441, 460 315-328, 349, 354, 369, 680 Triangular cross section 426 - for a system 543 Trochoidal sea swell 29 - for curved beam 665 Truesdell rate 279, 359 - for straight beam 664, 665 Truss 590, 632, 641 linearised 340, 496-498, 503 - pin-connected 682, 683 Thermoelasticity 49, 301 - planar 622, 633, 684 decoupled 371, 372 - statically determinate 675 linear 328-346, 351 Tube 302, 305, 411, 467, 566 - second order 351 - in torsion 425 - unconstrained 316,-320 - thin circular 464 - with internal constraints 321-324, - under press ure 461-464, 466, 770, 349 778 Thin slice approximation 778, 779 Twisting moment 365, 403, 407, 415, Thin spherical shell 477, 479 416, 440, 482, 555, 574, 605, 630, Thrust, arch 636 637,657 802 Subject Index

U nconstrained thermoelasticity - theorem 493, 499, 505, 531, 533, 316-320 551, 559, 655, 658, 680 Uniaxial stress state 271 Viscous fluid 199 Uniform extension 71 Voigt upper bound 558 Unilateral Volume expansion 14, 18, 41, 334 - constraint 375, 473, 617, 689 Von Mises yield criterion 277, 282, - support 617, 640 284, 285, 415, 425, 427, 440, 441, 460 Uniqueness theorem 79, 382, 391, Vortex 25 392, 397, 502, 504, 508, 510, 668 - lines 130 Unit extension 46 - point 75, 127 Upper bound 515 - surfaces 130 - for apparent modulus 524 Vorticity vector 96 - for loading parameter 550 - for torsional inertia 565 Warping 660 - Voigt 558 - function 365, 426, 482, 557 U pp er index 702 - of cross section 404, 425, 427 Water tunnel 24-26 Weak formulation 493 Vanishing load 674 - of equations of motion 225, 512 - method 544, 668 - of geometrical compatibility 100, Varga model 357 225, 514 Variational methods 391, 411, 489, Well-posed problem 372, 382, 408 492, 512-518 Wire 651 Vault 590 - arc 631, 632 Velo city 17, 18, 28, 88 - statics 578-591, 597, 600, 602, 606, - boundary condition 374 628, 633 - distributor 160 - tension in 591 - notation 22 Wrench 8,137,161,179,540,600,614 - virtual 137, 148, 493 - field 164, 594 Virtual - of body forces 234 - angular velocity 149 - of concentrated external forces 574 - displacement field 487, 505 - of external forces 137, 166, 193, - motion 137, 148, 151, 156, 190, 234, 573, 582, 596, 618 573, 643 -- line density 595 -- discontinuous 603 - of force system 162 -- rigid body 148 - of internal forces 137, 166, 574, 597, - motions, vector space of 580, 594 604, 609, 618, 625, 627, 652 - power 494 - of momenta 130, 137, 168 - spin tensor 203 - of quantities of acceleration 130, - strain rate 203, 628 137, 166, 168, 175 - stretch rate 573, 583 - of shear forces 567 - velocity 137, 148, 493 - of surface forces 235, 402, 439, 451, Virtual rate of work 454 - by external forces 137, 148, 158, - on cylinder bases 468, 481 191-194, 221, 234, 307, 580-582, - tensorial 164, 235 594-596 - by internal forces 137, 148, 158, Yield 194, 201, 203, 217, 228, 235, 283, - design 642, 644 582-583, 589, 596-597, 603, 606, 612 - function 274 - by momenta 173 - surface 441 - by quantities of acceleration 137, Yield criterion 148,158,174,191,579 - Coulomb 287 Virtual work - Tresca 276, 282-286, 291, 415, 425, - method 150, 156-160, 189, 578, 592 427, 440, 441, 460 Subject Index 803

- von Mises 277, 282, 284, 285, 415, - of rod in simple compression 440 425,427,440,441,460 - of rod in simple tension 440 Yield point 304, 379, 441 - of rod in torsion 413-416, 425, 427 - of hollow sphere under pressure 460 Young modulus 299, 333, 344, 346, - of rod in normal bending 450 398,438,471,502,508