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    Introduction to Continuum Mechanics12

    1. Preliminaries

    The following notation is adopted in which Vis the translation (vector) spaceof a real three-dimensional Euclidean point spaceE:Lin: the linear space of linear transformations (tensors) fromV toV.InvLin: the group of invertible tensors.Sy m ={A Lin : A = AT}, where superscript T denotes the transpose:linear space of symmetric tensors; also, the linear operation of symmetriza-tion on Lin.Sy m+ =

    {A

    Sym : u

    Au > 0

    } for u

    = 0, u

    V: the positive-definite

    tensors.Skw ={A Lin : AT =A}: the linear space of skew tensors; also, thelinear operation of skew-symmetrization on Lin.Orth ={A InvLin : AT = A1}, where A1 is the inverse of A: thegroup of orthogonal tensors.Orth+ ={AOrth: JA = 1}: the group of rotations.Here and in the following chapter on balance laws, both indicial notationas well as bold notation are used to represent vector and tensor fields. Thecomponents in the indicial notation are written with respect to the Cartesiancoordinate system. Indices denoted with roman alphabets vary from one

    to three but those denoted with Greek alphabets vary from one to two.Einsteins summation convention is assumed unless an exception is explicitlystated. Leteijk be the three dimensional permutation symbol, i.e. eijk = 1 oreijk =1 when (i,j,k) is an even or odd permutation of (1, 2, 3), respectively,and eijk = 0 otherwise.The determinant and cofactor ofAare denoted by JA and A

    , respectively,where A = JAA

    T if A InvLin. It follows easily that (AB) = AB.Further, Lin is equipped with the Euclidean inner product and norm definedby A B= tr(ABT) and|A|2 =A A, respectively, where tr() is the traceoperator. We make frequent use of relations like ABC = ACT B =C

    T

    AT

    B and AB CD= ABDT

    C, etc., which follow easily from tr A=tr AT and tr(AB) = tr(BA). It is well known that Lin =Sym Skw, thedirect sum ofS ym and Skw.The tensor product a bof vectors{a, b} Vis defined by (a b)v= (b v)a for all v inV, where b v is the standardinner product of vectors.

    1A part of these notes have been taken from Anurag Gupta and David Steigmann,Chapters on Kinematics and Balance Laws, in Continuum Mechanics: Encyclopediaof Life Support Systems (EOLSS), Developed under the auspices of the UNESCO, EolssPublishers, Oxford ,UK. (Forthcoming)

    2For any comments on these notes contact Anurag Gupta at [email protected]

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    A fourth order tensor is a linear transformation

    A : Lin

    Lin. Its

    operation on a second order tensor is represented by A =A[B] for{A, B} Lin. In terms of indices, this is written as Aij =AijklBkl. A fourth orderunit tensor I is defined as I[A] = A for every A Lin. In components,Iijkl =ikjl . A tensor productA Bof two second order tensors is a fourthorder tensor defined by (A B)[C] = (C B)A, for C Lin. The majortranspose of a fourth order tensorA is a fourth order tensorAT defined byAT[B]A = B A[A]. Moreover,A has a major symmetry ifAT =A,or in indicesAijkl =Aklij. It has a minor symmetry of the first kind, if itis symmetric with respect to first two indices, i.e. if (A[B])T =A[B], forB Lin. The tensorA has a minor symmetry of a second kind, if it issymmetric with respect to last two indices, i.e. ifA[BT] =A[B]. A fourthorder tensorAis invertible if there exists another fourth order tensorBsuchthatAB = I, or in componentsAijklBklmn = Iijmn. The tensorB is thencalled the inverse ofAand is denoted byA1.2. Body, configurations, and motion

    The geometrical structure of a physical body is independent of a frame ofreference, and therefore the body(in continuum mechanics) is usually takento be a three dimensional differential manifold. We denote such a manifoldby B and call its elements material points. At every material point X

    B

    we have an associated tangent spaceTXwhich is a three dimensional vectorspace representing a neighborhood of X. On the other hand, the body isobserved and tested in a (three dimensional) Euclidean frame of referenceE,which requires us to endow the body B with a class C of bijective mappings,: B E (the subscript is used to indicate the mapping employed). Wecall these mappings the configurationsof the body B. The spatial position(X) E denotes the placewhich a material point X B occupies inE.The translation space ofE is a three dimensional inner product space, andis denoted byV.We introduce a fixed reference configuration, relative to which the notions of

    displacement and strain can be defined. Let C be a reference configura-tion. The configuration need not be a configuration occupied by B at anytime and therefore can be arbitrary as long as it belongs to C.The motionof a body B is defined as a one-parameter family of configura-tions, t : BR E. Such a motion assigns a placex Eto the materialpoint X B at time t. We write this as

    x= t(X)(X, t). (1)The reference configuration assigns a placeX EtoX, so we can express

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    x as a function ofX,

    x= (1(X), t)(X, t), (2)where:E R E denotes a mapping from the reference configurationto the configuration of the body at time t.The displacementu: B R V (Vcan be identified with eitherV orV)of a material point Xwith respect to the reference configuration at timet is defined as

    u(X, t) =(X, t) (X). (3)The particle velocity v : B R V and the particle acceleration a :B

    R

    V are defined as

    v(X, t) =

    t(X, t) (4)

    and

    a(X, t) = 2

    t2(X, t), (5)

    respectively. Displacement, particle velocity and particle acceleration can allbe alternatively expressed as functions on (B) by using the inverse 1 :E B. Such functions exist in a one-to-one relation with the functionsexpressed in the equations above. We write

    u(X, t)u(1(X), t)v(X, t)v(1(X), t) (6)a(X, t)a(1(X), t).

    We can similarly write these functions as

    u(x, t)u(1t (x), t)v(x, t)v(1t (x), t) (7)a(x, t)a(1t (x), t).

    We define the material time derivativeas the derivative of a function withrespect to time for a fixed material point. For an arbitrary scalar functionf : B R R, we denote its material time derivative as f. Thus,

    f=

    tf(X, t)|X, (8)

    where the notation|Xdenotes the evaluation of the derivative at a fixed X.Iff is instead given in terms ofx, i.e. iff= f((X, t), t), we write

    f=

    tf(x, t)|x +(gradf) v, (9)

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    where t

    f(x, t)

    |x is the spatial time derivative(at a fixed x) and grad f is

    the spatial gradient (gradient is defined below). Therefore, if the particlevelocity is a function of spatial position x, then the particle acceleration isa=

    tv(x, t)|x +Lv, where L= gradv is the spatial velocity gradient.

    Derivatives By fieldswe mean scalar, vector and tensor valued functionsdefined on position (x or X) and time (t). In the following we are mainlyconcerned with the derivatives with respect to the position and thereforedependence of fields on time is suppressed.A scalar-valued field (X) :E R is differentiable at X0 U(X0), where

    U(X0)

    E is an open neighborhood ofX0, if there exists a unique c

    V

    such that

    (X) =(X0) + c(X0) (X X0) + o(|X X0|), (10)

    where o() 0 as 0. We call c(X0) =|X0 (or(X0)) the gradient

    of at X0. Consider a curve X(u) inE parameterized by u R. Let(u) =(X(u)) and X1 = X(u1), X0 =X(u0) for{u1, u0} R. Then from(10),

    (u1) (u0) =(X0) (X1 X0) + o(|X1 X0|). (11)Moreover X1

    X0 = X

    (u0)(u1

    u0) +o(

    |u1

    u0

    |), where X(u0) is the

    derivative ofX with respect to u at u = u0. Therefore,|X1 X0|= O(|u1 u0|) and consequently we can rewrite (11)

    (u1) (u0)u1 u0 =(X0) X

    (u0) +o(|u1 u0|)

    u1 u0 . (12)

    For u1 u0 we obtain the chain rule, (u0) =(X(u0)) X(u0), whichcan also be expressed as d

    du=(X) dX

    du or

    d(X) =(X) dX. (13)

    A vector-valued field v(X) :E Vis differentiable at X0 U(X0) if thereexists a unique tensor l:V Vsuch that

    v(X) =v(X0) + l(X0)(X X0) + r, (14)

    where|r|= o(|XX0|). We calll(X0) =v|X0 (orv(X0)) thegradientofvat X0. The chain rule in this case can be obtained following the procedurepreceding equation (13):

    dv(X) = (v)dX. (15)

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    The divergenceof a vector field is a scalar defined by

    Div v= tr(v). (16)

    The curlof a vector field is a vector defined by

    (Curl v) c= Div(v c) (17)for any fixed c V.Differentiability of a tensor-valued function is defined in a similar manner.In particular, for a tensor A(X) :ELin, we write

    dA(X) = (A)dX. (18)The divergenceofA is the vector defined by

    (Div A) c= Div(ATc) (19)

    for any fixed c V. The superscriptTdenotes the transpose. The curlofAis the tensor defined by

    (Curl A)c= Curl(ATc) (20)

    for any fixed c V.Finally, if the fields are expressed as functions ofxrather thanX, the variousdefinitions above remain valid. We instead denote the gradient, divergenceand curl operators by grad, div, and curl, respectively.

    Derivatives of functions on tensor spaces A functionf(A) :Lin Ris said to be differentiable at ALin if there exists a linear mapping Af(which maps tensors in Lin to scalars) such that for all BLin (B shouldbelong to some open neighborhood ofA)

    f(A + B)

    f(A) =Af(A)

    B + o(B). (21)

    This definition is equivalent to

    Af(A) B= dds

    f(A + sB)|s=0. (22)

    The map Afis called the Frechet derivative, or simply the derivative. ThetermAf(A) B, however, is called the Gateaux derivative or the directionalderivative in the direction of B. A similar definition holds for vector andtensor valued functions (and also for vector arguments).

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    3. Deformation Gradient

    If the mapping (X, t) is differentiable with respect to X, then we definethe deformation gradient by

    F=. (23)Since (X, t) is invertible for each X E, the deformation gradient Fbelongs to a family of invertible linear maps from the translation space ofEto the translation space ofE, i.e. FInvLin. This follows from the inversefunction theorem (Rudin, W. Principles of Mathematical Analysis, 3rd Ed.,McGraw-Hill (1976), page 221). For{X, Y} E equation (14) becomes

    (Y, t) =(X, t) + F(X, t)(Y X) + r (24)and the chain rule (15) takes the form (for fixed t)

    dx= FdX, (25)

    where the differentials dX and dx belong to the translation spacesV at XandV at x, respectively.We now obtain relationships for transforming infinitesimal area and volumeelements. LetdX1 V and dX2 V be two linearly independent infinites-imal line elements at X. An infinitesimal area element can be constructed

    using these line elements, with area given by da =|dX1 dX2| and the as-sociated direction given by the unit normalnsuch thatnda=dX1dX2.In the configuration t the line elements dX1 and dX2 are transformed intoline elements dx1 V and dx2 V, respectively at x = (X, t). Weobtain, using relation (25), dx1 = FdX1 and dx2 = FdX2. The area elementconstructed using these line elements has area da =|dx1dx2| with unitnormal ngiven by nda= dx1 dx2. Therefore,

    nda = FdX1 FdX2= F(dX1

    dX2)

    = Fnda. (26)

    As FInvLin, we haveF =JFF

    T. (27)

    Consider a third line element dX3 Vat X such that the set {dX1, dX2, dX3}is linearly independent and positively oriented. The infinitesimal volume el-ement associated with the reference configuration is then given by dv =

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    dX1

    dX2

    dX3. In configuration t the volume element at x= (X, t) is

    dv=dx1 dx2 dx3 with dx3 = FdX3. Therefore,dv = FdX1 FdX2 FdX3

    = FdX1 F(dX2 dX3)= JFdv (28)

    and accordingly, if is a configuration that could be attained in the course ofthe motion ofB, we requireJF >0 to ensure that a volume in correspondsto a volume in .

    Material curves Consider a curve C E and parameterize it with areal number s R such that C : R E. We call C a material curve. Itsplacement in the configuration is denoted bycand we usesto parameterizeit such that c : R E. Using the definition of the deformation gradientand assuming the mappings C and c to be differentiable, we write

    x(s) =FX(s). (29)

    Ifsis the arc-length onC, then the vector X(s) defines a unit tangent vector(denoted M) to the curve C at arc-length station s. Let x(s) = m with

    |m

    |= 1 and =

    |x(s)

    |. Substituting these in (29), we obtain

    m= FM. (30)

    Since F InvLin, FM= 0 and therefore > 0. We call (s, t) the localstretch ofC. It follows from (30) that

    2 =|m|2 =FM FM= M CM, (31)whereC = FTF:V V is the Right Cauchy Green tensor. The tensor Cis symmetric and positive definite, i.e. CS ym+. Indeed, CT = (FTF)T =FTF = C and for arbitrary a V, aCa = FaFa =|Fa|2 > 0, asJF= 0. Similarly, if we rewrite (30) as

    1

    M = F1

    m, we can arrive atthe (symmetric and positive definite) Left Cauchy Green tensorB= FFT :V Vsuch that2 =m B1m. We can useCto calculate the deformedlength of a material curve and the deformed angle between two materialcurves. Given an infinitesimal element of the material curve dX= Mds, itsdeformed length is|dx|= FM FMds= ds and therefore the deformedlength of a material curve with reference arc-length s1 s0 is

    lc(t) =

    s1s0

    (s, t)ds. (32)

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    Consider two material curves intersecting at X with associated unit tan-

    gent vectors M1 and M2, respectively. Let 1 and 2 be the local stretchescorresponding to the two curves and let be the angle between the tangentvectors of the deformed curve at x. We then write,12cos = FM1FM2 =M1 CM2 and, on using (31), obtain

    cos = M1 CM2

    (M1 CM1)(M2 CM2). (33)

    Finally, we introduce two definitions of extensional strain: The first, denotedeCand defined byeC=

    12

    (C1), yields 12

    (21) =M eCM, where1Linis the identity transformation. Therefore, eC :V V characterizes therelative local stretch with respect to the reference configuration. It is knownas the relative Lagrange strainor the Green-St.Venant strain. Alternatively,to characterize local stretch relative to the current configuration, we defineeB =

    12

    (1 B1), and obtain 12

    (1 2) = 12

    m (1 B1)m. The tensoreB :V V is called the relative Eulerian strainor the Almansi-Hamelstrain tensor. The two strain tensors are related by eC=F

    TeBF.Using equations (3) and (23), we can obtain the deformation gradient fromthe displacement field, F= 1 + u. For small deformations|u| 1 andconsequently1 and eC 12(u+uT) ( denotes the small deformationapproximation). The two strain measures are asymptotically coincident in

    this approximation.

    Principal stretches We would now like to identify the material curvesalong which the local stretch assumes extreme values and obtain these ex-tremals from C. Define f(M) = MCM at fixed C. We therefore havef(M) > 0 (from (31)), for M S ={v V :|v| = 1}. Since f(M) is acontinuous function, defined on a compact set, a theorem in analysis (Rudin,W. ibid., page 89) yields the existence of M1 S and M2 S such thatf(M1) = min

    MSf(M) 21 and f(M2) = max

    MSf(M) 22, respectively. Our

    aim is to compute 21 and 22 for a given C. These are extremal values of

    f(M) and thus render f(M) stationary, i.e. df(M) = 0, or CM dM= 0 forM {M1, M2}. Furthermore, the identityM M= 1 implies M dM= 0and therefore dMM at each M S. SinceS is a two dimensional man-ifold with dM belonging to its tangent space, the vector M represents theunit vector normal toSat M S. As a result of these arguments, for some1, 2 R we can write, CM1 = 1M1 and CM2 = 2M2. Evidently, 1and 2 are equal to

    21 and

    22, respectively (1=M1 CM1=f(M1) =21,

    etc.), the largest and smallest eigenvalues ofC, respectively, and thus

    CM1 = 21M1, and CM2=

    22M2. (34)

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    In general, for

    R and M

    S, we can solve the eigenvalue problem

    CM = 2M to obtain three real values for 2. If{EA} is an orthonormalbasis forV and if we set CAB = EA CEB, then we can conclude that theeigenvalues bound the diagonal entries of the matrix{CAB}; i.e.

    21min{C11, C22, C33} max{C11, C22, C33} 22. (35)

    Two theorems for symmetric tensors According to the spectral theo-rem, for every A Sym, there exists an orthonormal basis{ui} V(i =1, 2, 3) and numbers i Rsuch that

    A=3

    i=1

    iui ui. (36)

    The numbers i are the principal values associated with the tensor A andcan be obtained as the roots of the characteristic equation det(A 1) = 0with R. We now prove this assertion. Let and u be a principal value(eigenvalue) and the corresponding principal vector (eigenvector) associatedwith A. Allow them to be complex, i.e. = a+ ib and u = a+ ib forsome{a, b} R and{a, b} R3 with i = 1. Therefore Au= u. Wealso have Au = u, where an over-bar represents the complex conjugate.

    Since A is symmetric, we can write u Au = u Au or 0 = ( )u u.This implies = , as u u > 0. We now have to prove the existence

    of orthonormal{ui} such that (36) holds. For eigenvalues1, 2 and theircorresponding eigenvectors u1, u2, we have Au1=1u1and Au2 = 2u2. AsAis symmetric,u1 Au2=u2 Au1and thus 0 = (12)u1 u2. If1=2,thenu1 and u2 are mutually orthogonal. Therefore if{i}are distinct,{ui}necessarily forms an orthonormal set. If 1= 2 = 3, then u1u2 = 0.Define u3=u1 u2, so that{ui}is orthonormal. The vector u3 is the thirdprincipal vector of A. Indeed Au3 =

    3

    i=1(uiAu3)ui =

    3

    i=1(u3Aui)ui =

    (u3

    Au3)u3 where in the first equality, the vector Au3 is expressed in termsof the basis vectors{ui}. In the second equality, the symmetry ofAis usedand in the third equality, the relations Au = u ( = 1, 2) and theorthonormality of{ui} are employed. Finally, if1 = 2 = 3 = , we canpick any orthonormal basis inVand in this case A= 1.According to the square root theorem, for every A Sym+, there exists aunique tensor G Sym+ such that A = G2. By the spectral theorem wehave a representation (36) for A withi >0 (due to the positive definiteness

    of A). Define G=3

    i=1

    iui ui. Then, G2 = GG=

    3i=1

    i(Gui) ui =

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    3i=1iui ui= A and it is obvious that G is symmetric and positive definite.To prove uniqueness we assume that there exists a symmetric and positivedefinite tensor G such that G2 = A = G2 and show that G = G. Let ube an eigenvector of A with eigenvalue > 0. Then (G2 1)u = 0 or(G +

    1)v= 0, where v = (G 1)u. This requires v = 0 as otherwise

    becomes an eigenvalue ofG, contradicting the positive definiteness ofG. Therefore Gu=

    u and similarly Gu=

    u. Thus Gui = Gui and

    since an arbitrary vector fcan be expressed as a linear combination of{ui},we obtain Gf= Gf. This implies G= G.

    Polar decomposition theorem Every F InvLin can be uniquely de-composed in terms of tensors{U, V} Sym+ and a rotation R Orth+such that

    F= RU= VR. (37)

    The first of these equalities can be proved by using the right Cauchy Greentensor C= FTF. By the square root theorem there exists a unique symmetricpositive definite tensor U such that U2 =C. Define R= FU1. It follows,thatRTR= 1 and det R= 1 (since det F= det U=

    det C), and therefore

    R is a proper orthogonal tensor. The relation F = VR can be provedsimilarly via the left Cauchy Green tensor B = FFT. Relation (37) allows

    us to decompose a deformation into a stretch and a pure rotation. Let unitvectors M V and m V and the scalar be such that m= FM (fromequation (30)). Using the decomposition F = RUand defining M = UMwe find that =| M| and m = RM

    |M|. Thus, F stretches (and rotates) M

    to M, and then rotates M to the direction of m. If is a principal valueof U with M as the corresponding principal vector, then UM = M. Asa result, M = M, and therefore in such a case, F stretches M to M,and then rotates it to m, with m= RM. Consider three material curvesintersecting at X E with mutually orthogonal tangent vectors{Mi}. If

    {Mi

    } coincide with the principal vectors of U at X, then the curves will

    undergo (locally) a pure stretch, followed by a rigid rotation, with tangentvectors to the deformed curves remaining mutually orthogonal at x = (X).If on the other hand {Mi} are not the principal vectors ofU, thenUMiis nolonger parallel to Mi and consequently Uchanges the angle between{Mi}.The tangent vectors to the deformed curves, therefore, are not orthogonalat x. The decomposition can be understood in the opposite order (rotationfollowed by a stretch) if we consider the VR decomposition instead of theRUdecomposition ofF.

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    Principal invariants The characteristic equation forA

    Lin is

    0 = det(A 1) =3 + 2I1(A) I2(A) + I3(A), (38)

    where

    I1(A) = tr A

    I2(A) = tr A =

    1

    2[(tr A)2 tr A2] (39)

    I3(A) = det A

    are the principal invariants of A. A physically meaningful interpretation

    of these invariants follows by identifying A with U Sym+ which appearsin the polar decomposition (37) of the deformation gradient. In terms ofthe eigenvalues of U (denoted by i > 0), we obtain from (39), I1(U) =1+ 2+ 3, I2(U) = 12+ 13+ 23 and I3(U) = 123. Therefore,if the edges of a unit cube are aligned with the eigenvectors of U, thenI1(U) is the sum of the lengths of three mutually orthogonal edges afterdeformation, I2(U) is the sum of the areas of three mutually orthogonalsides after deformation, and I3(U) is the deformed volume.According to the Cayley-Hamilton theorem, A satisfies its own characteristicequation, i.e.

    A3 + I1(A)A2 I2(A)A + I3(A)1= 0. (40)

    We now prove this theorem. Let D = ((A 1))T, where R is suchthat det(A 1)= 0 but otherwise arbitrary. Since A 1 is invertible,we have D= det(A 1)(A 1)1 or D(A 1) = det(A 1)1. Theright hand side of this relation is cubic in and the term A 1 is linearin . Therefore D has to be quadratic in (by a theorem on factorizationof polynomials). Let D = D0+ D1 + D2

    2 for some D0, D1 and D2. Then(D0+ D1+D2

    2)(A 1) = det(A 1)1 = (3 +2I1 I2+ I3)1.Matching coefficients of various powers of between the first and the lastterm and eliminatingD0, D1and D2from these, we get the required relation(40). The coefficients of all the powers of have to vanish since otherwisewe would obtain a polynomial (of order 3) in , which could then be solvedto obtain roots for , contradicting the premise that Ris arbitrary.

    Velocity gradient We can use the chain rule for differentiation to writethe gradient of the velocity field with respect to Xas

    v(X, t) =LF, (41)

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    whereL= gradv:

    V

    V is the spatial velocity gradient. Under sufficient

    continuity of the motion we havev = F and therefore L = FF1. Wecan decompose L into DS ym (rate of deformation tensor) and WS kw(vorticity tensor). The material time derivative of the right and the leftCauchy-Green tensor can be obtained as

    C= 2FTDF, B= LB + BLT. (42)

    Indeed, C = FTF+ FTF = FTLTF+ FTLF and B = FFT + FFT =LFFT + FFTLT.For a fixed material curve with unit tangent vector M recall relation (30),i.e. m= FM. As a result

    =m Dm, (43)

    where we have used M= 0, m m= 0 (which follows from m m= 1) andm Wm= 0 (since m Wm= WTm m =Wm m). We also obtain m = Lm m, which on using (43) and the decomposition of L intosymmetric and skew parts, reduces to

    m= Dm (m Dm)m + Wm. (44)If m should coincide with a principal vector of D with principal value ,thenDm = m. The relations (43) and (44) in this case give =

    = (ln )

    and m = Wm, respectively. Therefore, when the unit tangent m to thedeformed material curve instantaneously aligns with a principal vector ofD,the corresponding principal value of D is the rate of the natural logarithmof the stretch associated with the material curve. Moreover, the vorticitytensor W then characterizes the spin of the material element instantaneouslyaligned with a principal vector.Associated with WS kw there exists a vector w V (the axial vector ofW) such that, Wa= w a for all a V. This fact can be proved by firstobtaining the canonical form for a skew tensor. The characteristic equation

    for W has three roots and therefore at least one of them is real (complexroots occur in a pair). Let this real eigenvalue be and let f V be thecorresponding eigenvector. Then Wf=f. But this implies= Wf f= 0and soWf= 0. Choose{g, h} Vsuch that{f, g, h}forms a right handedorthonormal basis forV. The canonical form for Wis then given by

    W= (h g g h), (45)where =h Wg. The canonical form (45) can been proved by rememberingthatWT =W, Wf=0 and a Wa= 0 for all a V. Then W = W1 =

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    W(f

    f+g

    g + h

    h) = Wg

    g + Wh

    h. Note that Wg = h,

    since Wg f =g Wf= 0 and Wg g = 0. Similarly Wh=g. Thiscompletes the proof.Letw = f. Then on using (45) for arbitrarya, we obtainWa= ((ga)h(ha)g) =((ga)(fg)(ha)(hf)) =f((fa)f+(ga)g+(ha)h) =w a.IfW is the skew part of the spatial velocity gradient, then the axial vectorw is given in terms of the velocity field v as

    w=1

    2curlv. (46)

    The vector w is also called the vorticity vector. This relation can be provedby considering two constant but otherwise arbitrary vectors g and h. Then2Wg h= ((gradv) (gradv)T)g h= div((v h)g (v g)h) = div(v (g h)) = curlv g h= (curlv g) h. Using the arbitrariness ofh andthe relation Wg= w g we obtain equation (46).Finally, we interpret the off-diagonal terms of D on an orthogonal basis.Consider two intersecting material curves with tangent vectors M1 and M2at the point of intersection. In the current configuration, they map to m1and m2 with local stretches 1 and 2, respectively. Let cos = m1m2.Then, (sin )= (m1 Dm1+m2 Dm2)(m1 m2)2m1 Dm2, where relation(44) has been used. If sin = 1 (i.e. m

    1 m

    2= 0), we have =

    2m

    1 Dm

    2.

    Therefore the off-diagonal terms ofDon an orthogonal basis are proportionalto the rate of change of the angle between tangents to the deformed materialcurves instantaneously aligned with the orthogonal elements of the basis.

    Homogeneous deformation We call a deformation homogeneous if theassociated deformation gradient F(X, t) is independent of X. The motionthen takes the simple form

    x= (X, t) =F(t)X + c(t) (47)

    with c V. Thus, for Y E,y= (Y, t) =F(t)Y+ c(t) (48)

    and therefore,(Y, t) =(X, t) + F(t)(Y X). (49)

    Comparing this to equation (24) we note that every deformation is approxi-mated by a homogeneous deformation in any sufficiently small neighborhoodof a material point. Homogeneous deformation is characterized by severalproperties:

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    (i) Material planes deform into planes and parallel planes map to parallel

    planes. A material plane is represented by XN = D, where X Vrepresents a vector extending from the origin to points on the plane, N isthe constant normal of the plane, and D is the (constant) perpendiculardistance from the origin to the plane. As a result we obtain, using (47),D = F1(xc)N or xFTN = d, where d = D + cFTN is theperpendicular distance from the origin (inE) to the deformed plane. Thevector FTN is parallel to the deformed normal (see equation (26)).(ii) Straight material lines deform into straight lines and parallel lines mapto parallel lines. The equation of a straight material line is given by X =SM + X0 for some S R+, X0 V (arc-length), where M is the constant(unit) tangent to the line. Using (47) we can obtain x = FX + c= sm + x0,where m= FM

    |FM|, s = S|FM| and x0 = FX0+ c. This too, is the equation

    of a straight line.(iii) A spherical material surface is mapped to an ellipsoidal surface. Letp0=Y Xand p = y x. Thus for a homogeneous deformation, we have,according to (49), p= Fp0. A spherical material surface can be representedby p0 p0= 1 (with Xas the center and Y Xas the radius vector). Thiscan be rewritten as, F1p F1p= 1 or p B1p= 1. In spectral form, letB=

    3

    i=12i vivi. Then the relationp B1p= 1 represents an ellipsoid with

    axes vi V

    and semi-axis lengths i

    R.

    4. Rotation tensors and rigid body motion

    Rotations A tensor Q :V V is orthogonal if for arbitrary{a, b} Vwe have Qa Qb= a b. As a result, QTQ = 1 = QQT and| det Q| = 1.If det Q = 1 then Q is called proper orthogonal or a rotation. Note thatQT(Q 1) =(Q 1)T and therefore ifQ is a rotation, det(Q 1) = 0.Thus, there exists a nonzero f Vsuch that Qf=f. The vector f is calledthe axis ofQ and is unaffected by the action ofQ. We can take fto be a unit

    vector without any loss of generality. Let{f, g, h} V be a right handedorthonormal basis forV. Then Qg f = Qg Qf = g QTQf = g f = 0and similarly Qh f= 0. In addition, Qg Qh = g h = 0. Furthermore,fQgQh = QfQgQh = f gh (since QfQgQh

    fgh det Q = 1).

    Therefore{f, Qg, Qh} forms a right handed orthonormal basis inV. Asshown above, Qg is orthogonal to f, and consequently we can write Qg =ag+bhfor some {a, b} R. But |Qg|=Qg Qg=g g= 1. Therefore,there exists R such that a = cos and b = sin . We then obtainQh= fQg= sin g+cos h. We write Q= Q1 = Q(ff+gg+hh),

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    or

    Q = f f+ Qg g+ Qh h= f f+ cos (g g + h h) + sin (h g g h). (50)

    This expression is known as the Rodrigues representation formula. Severalinteresting facts regarding rotations are now stated:(i) Every rotation Q (Q= 1) has a unique axis. Let v V be such thatQv = v. Using (50) we obtain (v g cos v h sin )g+ (v g sin + v h cos )h = (vg)g+ (vh)h, which in turn results into a system of twosimultaneous equations,

    cos 1 sin

    sin cos 1

    v gv h

    =

    00

    . (51)

    Assume{v g, v h} ={0, 0}. This requires 0 = (cos 1)2 +sin2 = 2(1 cos ) or cos = 1. In such a case (50) reduces to Q= ff+gg+hh= 1.Thus for Q= 1 we require{v g, v h} ={0, 0} and thus v = (v f)f, i.e.v is parallel to f.(ii) There exists WS kw such that its axial vector coincides with the axisofQ and moreover

    Q= 1 + sin W+ (1 cos )W2. (52)To see this, let W = hggh. Then according to (45), W is skewwith = Wg h = 1 and axial vector w = f = f, which is also the axisof Q. Consider W2 = WhgWgh =(gg + hh) and thusf f=1 + W2. On substituting these relations in (50) we get the requiredformula (52).(iii) Every rotation Q is expressible as Q = exp(W), with W Skw asdefined in (ii) above and R. The exponential of a tensor is defined bythe power series exp W =

    n=0

    1n!

    Wn. Using this definition we can expand

    exp(W),

    exp(W) =n=0

    n

    n!Wn =1 +

    m=1

    2m

    2m!W2m +

    m=0

    2m+1

    (2m + 1)!W2m+1. (53)

    Note that W2m = (1)m+1W2 and W2m+1 = (1)mW (m = 1, 2, 3, ..).Both of these claims can be proved using induction. The relation (53) thentakes the form

    exp(W) =1 +

    m=0

    (1)m2m+1(2m + 1)!

    W+

    m=1

    (1)m+12m2m!

    W2. (54)

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    Recalling that sin =

    m=0

    (1)m

    (2m+1)!2m+1 and (1

    cos ) =

    m=1

    (1)m+1

    2m! 2m,

    and substituting these in (54) we obtain an expression for exp(W) whichcoincides with the right hand side of (52).(iv) For a fixed W Skw, the rotation Q() = exp(W) uniquely solvesthe following initial value problem:

    Q() =WQ(), Q(0) =1. (55)

    Using definition (53) for exp(W), we can obtain Q() =n=1

    n1

    (n1)!Wn =

    W

    n=0

    n

    (n)!W

    n

    and therefore Q

    () = WQ. It is easy to see that Q(0) = 1.Thus, the rotation Q satisfies (55). The uniqueness of the solution for (55)results from the theory of ordinary differential equations (Coddington, E. A.& Levinson, N. Theory of Ordinary Differential Equations, Krieger (1984)).

    Rigid body motion The motion of a body B is rigidif the distance be-tween any pair of material points remains invariant. For arbitrary{X, Y} E we then have

    |(Y, t) (X, t)|=|Y X|. (56)We now show that the rigid body motion is homogeneous and the associateddeformation gradient is a proper orthogonal tensor. Fix X and then differ-entiate both sides of (56) with respect to Y. We obtain FT(Y, t)((Y, t) (X, t)) = Y X. Now fix Y and differentiate this relation with re-spect to X. We obtain FT(Y, t)F(X, t) = 1 for all{X, Y} E. SetX = Y to obtain FT(X, t)F(X, t) = 1. Since det F > 0, it follows thatdet F= 1. We have thus proved that F is proper orthogonal. Furthermore,F(X, t) = FT(Y, t) = F(Y, t), where the second equality is a consequenceof the orthogonality ofF. Since Xand Y are arbitrary, we conclude that Fis homogeneous. DenoteF(X, t) = Q(t), where QOrth+. Equation (47)then takes the form

    x= Q(t)X + c(t), (57)

    where x = (X, t). The spatial velocity gradient in this case is given byL(t) = QQT, which is skew. Therefore the rate of deformation tensor, whichis the symmetric part of L, vanishes i.e. D = 0 and the vorticity tensor isW(t) = QQT. The relation gradv= W(t) can then be integrated to get

    v(x, t) =W(t)x + d(t) (58)

    for some d V. This follows directly from (57).

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    We have shown that the vanishing of the rate of deformation tensor, i.e.

    D = 0, is a necessary condition for rigid motion. The proof for sufficiencyis now given. An equivalent condition for rigid body motion is obtained bytaking the material time derivative of (56), yielding

    (v(Y, t) v(X, t)) ((Y, t) (X, t)) = 0. (59)We will obtain (59) by assuming L to be skew and thus prove our assertion.Lety = (Y, t) andx = (X, t) be two points in some open neighborhoodofE. The equation of a straight line L E connecting x and y is given byz(u) =x + u(y x), where 0u1. Then,

    v(Y, t) v(X, t) = 1(L)

    (v)dZ= 1(L)

    LFdZ=L

    L(z)dz (60)

    or

    v(Y, t) v(X, t) = 10

    L(z(u))z(u)du=

    10

    L(z(u))(y x)du (61)

    and therefore

    (v(Y, t) v(X, t)) ((Y, t) (X, t)) = 10

    (y x) L(y x)du, (62)

    which vanishes becauseL is skew. This complete the proof.

    5. Singular surfaces

    By a singular surface, we refer to a surface in the body across which jumpdiscontinuities are allowed for various fields (and their derivatives) whichotherwise are continuous in the body. The jump of a field (say ) across asingular surface is denoted by

    = + , (63)

    where

    +

    and

    are the limit values of as it approaches the singularsurface from either side. The + side is taken to be the one into which thenormal to the surface points. Let be another piecewise continuous field.The following relation, which can be verified by direct substitution using(63), will find much use in our later developments

    = + , (64)where

    = + +

    2 . (65)

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    A two dimensional surface which evolves in time is given by

    St ={X(B) :(X, t) = 0}, (66)

    where : (B) R R is a continuously differentiable function. The ref-erential normal to the surface and the referential normal velocity are definedby

    N(X, t) =|| and

    U(X, t) =

    || , (67)

    respectively. The second of these definitions is motivated towards the end ofthis section. An immediate consequence of these definitions is

    N=(1 N N)U U(N)N. (68)

    Indeed, we have from (67)1

    N = ||

    ||2

    ||

    =

    ||(1 N N) and (69)

    N = 2||

    ||2

    2||

    =2|| N (2)N|| . (70)

    On the other hand, (67)2 yields

    U=|| +

    ||2

    2||

    =|| U(2)N

    || , (71)

    where2 =() Sym. Combining these relations we obtain (68).The tensor 1 N N is the orthogonal projection ontoV and is denotedby P. It is easy to check that PT = Pand PP= P.

    Derivatives We now define surface derivatives for scalar, vector and tensorvalued functions which are defined on the surface St. Let f denote a scalar,vector or tensor valued function onSt. The function f is differentiable atXSt iffhas an extension fto a neighborhood N ofX, which is differentiableat X in the classical sense and is equal to f for XSt. The surface gradientof f at XSt is then defined by

    Sf(X) =f(X)P(X). (72)

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    Let v : St

    Vand A : St

    Lin be respectively, vector and tensor valued

    functions on the surface St. We define thesurface divergenceas a scalar fieldDivSv and a vector field DivSA by

    DivSv = tr(Sv)c DivSA = DivS(ATc) (73)

    for any fixedc V. Moreover, we call vtangentialifPv = v and Asuperficialif AP = A. We define the curvature tensor L by (the normal N and itsextension to a neighborhood ofSt are both denoted by the same symbol)

    L =S

    N, (74)

    or L =N(1 N N). Therefore

    tr L = Div N + (N)N N= Div N, (75)

    where we have used (N)TN = 0, which follows from NN = 1. SinceSNP =NPP =NP =SN, the curvature tensor is superficial. Fur-thermore, using (70) we have

    L = N(1 N N)= 2|| N (

    2

    )N||

    (1 N N)

    = 1||{

    2 N (2)N (2)N N + (2)N NN N}and consequently we infer that L = LT and LN = 0. Therefore, N is aprincipal direction of L with the corresponding principal value being zero.Since L is symmetric, the spectral theorem implies that it has three realeigenvalues with mutually orthogonal eigenvectors. We have already obtainedzero as an eigenvalue (with Nas the eigenvector). Let the other eigenvalues

    be

    1and

    2, whose corresponding eigenvectors lie in the plane normal toN

    .The mean and the Gaussian curvature associated with the surface are thendefined as

    H=1

    2(1+ 2), and K=12, (76)

    respectively.A function : (t, t + ) E, >0, is said to be a normal curvethroughXSt at time t if for each (t , t + ),

    () =U((), )N((), ). (77)

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    The function () is therefore the position parameterized by . We define

    the normal time derivativeof a function on St by

    v(X, t) = dv((), )

    d

    =t

    . (78)

    The relation (68) can therefore be written as N=SU.Remark: We will assume that an extension of a surface field to a neighbor-hood of the surface exists, and will abuse the notation to use the same symbolfor the field and its extension.

    Compatibility conditions Central to the discussion on the kinematics ofsingular surfaces are the compatibility conditions which relate the deforma-tion gradient and the velocity field across the singular surface. Consider aclosed material curve C E such that it intersects the singular surface Stat two points, say p1 and p2. Let ACbe the area bounded by C and let =AC St be the line of intersection of this area with the singular surface.We parameterize by arc-length u such that the curve extends from p2 top1.In general we can write

    b= C

    FdX, (79)

    where a non-zero b E arises when F is incompatible (we assume for nowthatF is not expressible as a gradient). In dislocation theory b is referred toas the Burgers vector associated with the closed curve C. The integration inthe above relation is well defined since we assume F to be singular only overa set of zero Lebesgue measure (a finite number of points on a continuous lineconstitute such a set). According to Stokes theorem with a singular surface,

    b=

    C

    FdX=

    AC

    (Curl F)TNAdA

    FdX, (80)

    where NA is the unit normal associated with the area AC. A proof of thistheorem is given in the following chapter on balance laws. We discuss twoconsequences of the above relation:(i) Letb = 0. Then, Curl F= 0 in (B) \ St. We can show this by choosinga C such that =. The arbitrariness of AC (and thus of NA) and thelocalization theorem for surface integrals (see the following chapter) thenimply Curl F= 0 for all X(B) \ St. Equation (80) now reduces to

    0=

    FdX. (81)

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    Use the parametrization of to write dX= sdu, where s

    TSt(X) is a unit

    vector in the tangent plane TSt(X) toSt atX. The curveCcan be arbitrarilychosen and therefore is arbitrary. Use the arbitrariness of to localize(81), and obtain

    Fs= 0 (82)

    for all sTSt(X). Thus, there exists a vector k V such thatF= k N (83)

    on St, which is Hadamards compatibility condition for the deformation gra-dient.

    (ii) Let Curl F = 0 in (B)\St. Therefore there exists a vector fieldsuch that F = away from St. Note that might still suffer a jumpacross St. Let C

    + C =C, where C+ and C are two disjoint parts ofCwhich lie on the + and side ofSt, respectively. The + side is the oneinto which the normal Npoints. Therefore,

    C

    FdX =

    C+

    FdX +

    C

    FdX

    = +2 +1 + 12= 2

    1 =

    (u)du, (84)

    where+2 =+ (p2) etc. The negative sign in the last term above arises due

    to the orientation of , which extends from p2 to p1. On the other hand wehave in this case, from (80),

    C

    FdX=

    FdX. (85)

    Since (u) =s =Ss (as Ps = s), we obtain, on comparing

    equations (84) and (85) and using the arbitrariness of

    Fs=Ss (86)for all sTSt(X). Thus, there exists a vector k E such that

    F= k N + S (87)on St, which is the modified compatibility condition for the deformationgradient in the case when suffers a jump on the singular surface. If=const. then equation (87) reduces to Hadamards compatibility condition(83).

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    To obtain the compatibility condition for the velocity field, we apply the

    definition of the normal time derivative (cf. (77), (78)) on fields + and .We obtain

    (+ ) =d((), )

    d

    =t+

    =UF+N + v+ (88)

    and

    ( ) =d((), )

    d

    =t

    =UFN + v, (89)

    where (t, t+ ) in (88) and (t, t) in (89). Subtracting theserelations we get the compatibility condition for the velocity field,

    v+ UFN=

    . (90)

    For = const. (including the case when is continuous, i.e. = 0)this condition reduces to

    v+ UFN= 0 (91)

    or equivalentlyUF=v N. (92)

    Surface deformation gradient For a continuous motion across the sur-

    faceS

    t, we have

    (X, t

    ) =0

    for X

    S

    t, and in this case we can definethe surface deformation gradientF and the surface normal velocityv by

    F =S, v =. (93)It is then easy to check that

    F =FP, v =v + UFN, (94)

    where on the right hand side above, indicates that either + or can beused, a fact which can be verified using the compatibility conditions (83) and(91).

    The tensor F as defined above satisfies det F = 0 and F =0. That det F = 0can be verified using (94)1 and det P= 0. The cofactor F

    of F is defined byF(a b) = Fa Fbfor arbitrary vectors{a, b} V. Let{t1, t2} TSt(X)be two unit vectors in the tangent plane toStat XSt, such that {t1, t2, N}form an orthonormal basis at X. Then,

    FN= F(t1 t2) = Ft1 Ft2= Ft1 Ft2= (F)N, (95)

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    where in the second equality, relation (94)1 has been used. Furthermore, it is

    straightforward to check that Ft =0 (= 1, 2), since FN= 0. ThereforeF remains non-zero as long as (F)N does not vanish. Note that|FN|is equal to the ratio of the infinitesimal areas (on the singular surface) inthe current and the reference configuration. This follows immediately fromequations (26) and (95).

    Surface parametrization Consider X E in a neighborhood ofSt. Wecan then find a point XSt such that

    X= X + N, (96)

    where(t) R is a scalar. We parameterize the surface St by using a localcoordinate system (1, 2), where{1, 2} R. In terms of the new variables,X= X(1, 2, ), X= X(1, 2, t) andN = N(1, 2, t). Let X,=for =1, 2. We assume that the parametrization is such that the triad{1, 2, N}forms an orthonormal basis at X. In a sufficiently small neighborhood ofX= X(1, 2, ) it is possible to invert this to obtain (1, 2, ) =(X).Use (96) to obtain the differential ofX,

    dX= (+ N,)d+ Nd. (97)

    If we identify with 1 and 2, the principal directions of L (recall that N is

    the third principal direction, cf. (74) and the paragraph preceding (76)), wehave N, =(no summation implied over ), whereare the principalcurvatures associated with the surface. Therefore, ifA is the gradient of themap taking to Xthen dX= Ad and it follows from (97) that

    A= 11(1 1)1 1+ 22(1 2)2 2+ N N, (98)where = (no summation). Let =1122. Thus

    jAdet A= (1 2H+ 2K), (99)whereH and Kare defined in (76).Taking the differential of the function ,d =

    dX+dt, and substituting

    in it the expression for dXfrom (97) for a point near the surface, we obtain

    d= ((+ N,)d+ Nd) + dt. (100)On the surface, we have = 0 and d= 0. Consequently we obtain

    0 = d+ Nd+ dt= d+ Ndt + dt (101)

    on St, and noting the independence ofd and dt, we recover relations (67)along with the identification ofwith U.

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    Singular surface in the current configuration The image of the sin-

    gular surface St in the current configuration is give by

    st=(St, t) = {x(B) :(x, t) = 0}, where((X, t), t) =(X, t). (102)

    The scalar function : (B) R Ris assumed to be continuously differ-entiable with respect to its arguments. The normal to the surface st and thespatial normal velocity are defined by (cf. (67))

    n= grad

    |grad

    |, and

    u= 1| grad |

    t, (103)

    respectively, where t

    is the spatial time derivative of at a fixed x. Differ-entiate (102)2 to obtain

    = (F)T grad , and = grad v +

    t. (104)

    The following relations can then be obtained on combining (67), (103), and(104):

    n = (F)TN

    |(F)TN| , and

    u = n v + U|(F)TN| . (105)

    6. Integral theorems

    In this subsection we state and prove the localization theorem, the divergencetheorem, the Stokes theorem, and the transport theorem for volume and sur-face integrals. We have employed only elementary concepts from differentialgeometry in proving these theorems.

    Localization theorem for volume integrals Letbe a continuous func-tion defined on an open set R E. If for all closed sets R

    dV = 0, (106)

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    then(u) = 0 for all u

    R. To prove this, we start by defining

    I=

    (u0) 1Vs

    (u)dV

    = 1V

    s

    ((u0) (u))dV , (107)

    where s is a sphere of radius and volume V centered at u0 R. Atheorem in analysis (Rudin, W. Principles of Mathematical Analysis, 3rdEd., McGraw-Hill (1976), page 317) yields,

    I 1V

    s

    |(u0) (u)|dV

    1

    Vs

    supus

    |(u0)

    (u)

    |dV

    = maxus

    |(u0) (u)|, (108)

    where in (108)2, sup can be replaced by max due to continuity and compact-ness ofs. Since (u) is continuous, we get I0 as 0. It then followsfrom equation (107),

    (u0) = lim0

    1

    V

    s

    (u)dV = 0, (109)

    where the last equality is a consequence of (106). The point u0can be chosenarbitrarily, and thus we can conclude that (u) = 0 for allu

    R.

    Localization theorem for surface integrals Letbe a continuous func-tion defined on a surfaceF E. If for all surfaces F

    dA= 0, (110)

    then(u) = 0 for all u F. This can be proved using arguments similar tothose used above.

    Divergence theorem for smooth fields Let f, p and P be respectively,

    scalar, vector and tensor fields defined on (B)(t1, t2). Assume these fieldsto be continuously differentiable over (B). Then for any part (B)and at any time t(t1, t2)

    (f)dV =

    fNdA, (111)

    (Div p)dV =

    p NdA, (112)

    (Div P)dV =

    PNdA, (113)

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    where N

    V is the outward unit normal to the boundary of . We

    outline a brief proof for (112). Let {E1, E2, E3} Vbe an orthonormal basisforV. Therefore there exists{p1, p2, p3, X1, X2, X3} Rsuch that p = piEiand X = XiEi, with i {1, 2, 3}. Consider a cuboidR ={X E : A 0 is the absolute temperature. According to the second law

    si0, (211)with equality holding for a reversible process. This can be rewritten in theform of the Clausius-Duhem inequality, using equations (208)-(210) as

    d

    dt

    dV

    q

    dA +

    r

    dV. (212)

    Use the transport theorem and the divergence theorem to obtain the localform:

    Divq

    + r

    . (213)

    4By alocalregion, Prigogine (cf. Prigogine I., Introduction to irreversible thermodynam-ics) meant a macroscopic region containing enough molecules for microscopic fluctuationsto be negligible. In our context, such local regions are represented by material points.

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    The balance of mass has also been used in obtaining this local inequality.

    Use the balance of energy (203) to eliminate r from (213). Obtain

    e P F Divq

    + Divq. (214)

    By the chain rule of differentiation

    Divq

    =

    Div q

    q g

    2 , (215)

    with g=. Inequality (214) can then be rewritten as

    e P F + q g

    . (216)

    Define specific free energy density as

    f=e . (217)

    Thereforef= e . (218)

    It is then straightforward to rewrite equation (216) in terms of the free energydensity. Substitute e from (218) into equation (216) to obtain

    f P F + + q g

    0. (219)

    Bibliography and further reading

    Chadwick, P. Continuum Mechanics: Concise Theory and Problems, Doverpublications (1999). [This little but comprehensive book on continuum me-chanics is introductory in nature and provides an ideal starting point in thesubject].Gurtin, M. E. An Introduction to Continuum Mechanics, Academic Press(1981). [This book covers all the fundamental aspects of continuum me-chanics: kinematics, balance laws, constitutive laws. The treatment is fairlyrigorous and most of the results are supplemented with well written proofs].Liu, I S. Continuum Mechanics, Springer-Verlag (2002). [A recent book witha lucid style. It can be approached by a beginner as well as an expert on thesubject].Noll, W. A new mathematical theory of simple materials, Archive of Ratio-nal Mechanics and Analysis, 52, 62-92 (1972). [This paper introduces a new

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    framework for thermodynamics of continuous media which, in particular, is

    suited for irreversible phenomena].Silhavy, M. The Mechanics and Thermodynamics of Continuous Media,Springer-Verlag (1997). [This excellent monograph provides many directionsof study for the interested reader].Truesdell, C. A First Course in Rational Continuum Mechanics, Vol.1, Aca-demic Press (1977). [A mathematically rigorous attempt to construct a the-ory of continuum mechanics embedded with interesting historical accountson the subject].Truesdell, C. & Noll, W. The Non-Linear Field Theories of Mechanics,Springer-Verlag (2004). [An established classic in the subject. It remains

    unparalleled in the nature of its scope and depth].Truesdell, C. & Toupin, R. A. The classical field theories of mechanics,Handbuch der Physik (ed. S. Flugge) Vol. III/1, Springer (1960). [This isrecommended, in particular, for its treatment of singular surfaces and waves].