(A brief) Introduction to Continuum Mechanics

(A brief) Introduction to Continuum Mechanics

Inigo Arregui, Jose M. Rodrıguez Seijo([email protected], [email protected])

Department of MathematicsUniversity of A Coruna, Spain

June, 2018

ROMSOC Project

Bibliography

M. E. GURTIN, An Introduction to Continuum Mechanics. AcademicPress, 1981.

O. LOPEZ POUSO, “An Introduction to Continuum Mechanics”, deM.E. Gurtin. Ejercicios resueltos. Publicaciones docentes delDepartamento de Matematica Aplicada. Universidad de Santiago deCompostela, 2002.

A. BERMUDEZ DE CASTRO, Continuum Thermomechanics.Birkhauser, 2005.

1 Introduction

2 Kinematics

3 Conservation laws

1 IntroductionPoints, vectors and tensorsSpectral, Cayley–Hamilton and polar decomposition theoremsTensor differentiationGradient. Divergence. CurlDivergence and Stokes theorems

2 Kinematics

3 Conservation laws


Introduction

Points, vectors and tensors


We will work in the three–dimensional Euclidean point space, E

Associated vector space: (IR3,+, ·)≡ V

Cartesian system of coordinates: orthonormal basisei= e1,e2,e3 and a point o called origin.

Other (non cartesian) systems of coordinates are also possible: polar,cylindrical and spherical coordinates, for example

Inner product in V : (·, ·) : V ×V −→ IR

(u,v) = u ·v = (u ·v) = |u| |v| cos(u,v) , |u|= (u ·u)1/2

Components of a vector: ui = u · ei

Thus, u ·v =3∑

i=1ui vi = ui vi (Euler notation)

Coordinates of a point x: xi = (x−o) · ei


Introduction


Recall that every linear application f : IRn −→ IRm has an associated matrixA ∈Mm×n, such that f (u) = Au

DefinitionWe call span of vectors u,v, . . . ,w the subspace of V consisting of allpossible linear combinations of those vectors:

spu,v, . . . ,w= αu+βv+ · · ·+ γw/α,β , . . . ,γ ∈ IR

DefinitionFor a given vector v, we call v⊥ = u ∈ V /u ·v = 0 the subspace in Vof all vectors perpendicular to v

Example

Given v = (1 1 0)T and w = (1 2 1)T ,

v⊥ = u ∈ V /u1 =−u2w⊥ = u ∈ V /u2 =−(u1 +u3)/2


Introduction


DefinitionA tensor is any linear transformation from V into V .

For example,

S : V −→ V v1v2v3

−→

w1w2w3

=

v1 + v2− v3v1− v2− v3v1 + v2 + v3

If we consider the sum of tensors and the product of a scalar by a tensor:

(S+T)v = Sv+Tv (αS)v = α(Sv)

then the set of tensors is a vector space

Null tensor: 0v = 0, ∀v ∈ V

Identity tensor: Iv = v, ∀v ∈ V


Introduction


Components of a tensor: Sij = ei ·Sej

Thus, v = Su ⇐⇒ vi =3∑

j=1Sij uj

We call [S] the matrix:

[S] =

S11 S12 S13S21 S22 S23S31 S32 S33

;

and it is the matrix associated to the linear application SIn particular,

[I] =

1 0 00 1 00 0 1

, [0] =

0 0 00 0 00 0 0


Introduction


DefinitionWe define the product of two tensors S and T as ST = ST, i.e.,

(ST)v = S(Tv) , ∀v ∈ V

Standard notation: S2 = SS, S3 = SSS, etc.[ST] = [S][T]In general, ST 6= TS.

If ST = TS, we say that S and T conmute. For example,

[S] =

1 0 20 1 02 0 1

[T] =

1 0 30 1 03 0 1

[ST] = [TS]


Introduction


DefinitionWe call transpose of S, ST , the unique tensor such that:

(Su) ·v = u · (STv) , ∀u,v ∈ V

It stands:

(S+T)T = ST +TT , (ST)T = TT ST , (α S)T = α ST ,

(ST)T = S , [ST ] = [S]T , (ST)ij = Sji

A tensor is symmetric if S = ST

A tensor is skew if S =−ST

If S is symmetric, Su ·v = u ·Sv = Sv ·uFor every tensor, S = E+W where E is symmetric and W is skew

E is the symmetric part of SW is the skew part of S


Introduction


DefinitionThe tensor product of two vectors a and b is the tensor a⊗b such that:

(a⊗b)v = (b ·v)a

In terms of components,

(a⊗b)v=

(b1v1 +b2v2 +b3v3)a1(b1v1 +b2v2 +b3v3)a2(b1v1 +b2v2 +b3v3)a3

=

b1a1 b2a1 b3a1b1a2 b2a2 b3a2b1a3 b2a3 b3a3

v1v2v3

so that the components of the tensor product are: [a⊗b]ij = ai bj

(a⊗b)T = b⊗a(a⊗b)(c⊗d) = (b · c)a⊗d

3∑

i=1ei⊗ ei = I

S =3∑

i,j=1Sij ei⊗ ej


Introduction


Let e be a unit vector. Then,the projection of v in the direction of e is:

(e⊗ e)v = (v · e)e

The projection of v on the orthogonal plane to e is:

(I− e⊗ e)v = v− (v · e)e


Introduction



Introduction



Introduction



Introduction


DefinitionWe call trace the linear application such that the image of a tensor S is thescalar TrS and, in particular,

Tr(u⊗v) = u ·v , ∀u,v ∈ V

Tr : Lin −→ IR u1v1 u1v2 u1v3u2v1 u2v2 u2v3u3v1 u3v2 u3v3

−→ u1v1 +u2v2 +u3v3

From the linearity of the trace, we deduce:

TrS = Tr

[3

∑i,j=1

Sijei⊗ ej

]=

3

∑i,j=1

SijTr(ei⊗ ej) =3

∑i,j=1

Sij ei · ej =3

∑i=1

Sii

TrST = TrSTr(ST) = Tr(TS)


Introduction


DefinitionWe define the inner product of two tensors as:

S ·T = Tr(STT)

In terms of the components,

S ·T = Tr

S11 S21 S31S12 S22 S32S13 S23 S33

T11 T12 T13T21 T22 T23T31 T32 T33

=3

∑i,j=1

Sij Tij

S ·T = T ·SI ·S = TrSR · (ST) = (STR) ·T = (RTT) ·Su ·Sv = S · (u⊗v)(a⊗b) · (u⊗v) = (a ·u)(b ·v)


Introduction


DefinitionWe define the determinant of a tensor S as the determinant of matrix [S]:

detS = det [S]

DefinitionA tensor S is invertible if it exists another tensor S−1, called the inverse ofS, such that:

SS−1 = S−1S = I

S is invertible if and only if detS 6= 0Moreover,

det(ST) = (detS)(detT) (ST)−1 = T−1S−1

detST = detS det(S−1) = (detS)−1 (S−1)T = (ST)−1

Notation: S−T = (S−1)T


Introduction


DefinitionA tensor Q is orthogonal if it preserves inner products:

(Qu) · (Qv) = u ·v , ∀u,v ∈ V

A necessary and sufficient condition for Q to be orthogonal is:

QQT = QTQ = I ⇐⇒ QT = Q−1

An orthogonal tensor with positive determinant is called a rotationAn orthogonal tensor is either a rotation, either a rotation multiplied by−IIf R 6= I is a rotation, then the set of all vectors v such that Rv = v is aone-dimensional vector subspace, called the axis of R

DefinitionA tensor S is positive definite if

v · (Sv)> 0 , ∀v 6= 0


Introduction


Notations:Lin: set of all tensorsLin+ = S/det(S)> 0Sym: set of symmetric tensorsSkw: set of skew tensorsPsym: set of symmetric and definite positive tensorsOrth: set of orthogonal tensorsOrth+: set of rotations


Introduction


There are two cross products in IR3. We consider the one with the positiveorientation

If the basis ei is positive oriented, we have e3 = e1× e2 and:

u×v=

∣∣∣∣∣∣e1 e2 e3u1 u2 u3v1 v2 v3

∣∣∣∣∣∣=(u2v3−u3v2)e1+(u3v1−u1v3)e2+(u1v2−u2v1)e3

Thus, u×v =−v×u , u×u = 0

u · (v×w) = w · (u×v) = v · (w×u)

If u, v and w are linearly independent, the magnitude of the scalar u · (v×w)(i.e., |u · (v×w)|) represents the volume of the parallelepiped P determinedby u, v and w.Moreover,

detS =Su · (Sv×Sw)

u · (v×w), |detS|= vol(S(P))

vol(P)


Introduction



Introduction


Given a skew tensor W, there exists a unique vector w such that

Wv = w×v , ∀v ∈ V (1)

and viceversa. w is called the axial vector of W.

If tensor W is given by:

[W] =

0 −γ β

γ 0 −α

−β α 0

then

w1 = α , w2 = β , w3 = γ

The ker of W (the vectors v ∈ V such that Wv = 0) is a onedimensional subspace generated by w, and it is called the axis of W


Introduction


Let us introduce:the Kronecker delta,

δij =

1 , if i = j0 , if i 6= j

the skew tensor

εijk =

1 , if (i, j,k) is an even permutation of 1,2,3−1 , if (i, j,k) is an odd permutation of 1,2,30 , otherwise

Thus,

detA = εijkai1aj2ak3 =

= ε123a11a22a33 + ε132a11a32a23 + ε213a21a12a33+

+ ε231a21a32a13 + ε312a31a12a23 + ε321a31a22a13

(u×v)i = εijkujvk


Introduction


Summary on vectors and tensors

Vector space, V

inner productcross product

Tensor: linear transformation from V onto V

associated matrix, components, sum, product by a scalartensor product (ST), transpose (ST ), tensor product of two vectors (a⊗b),inner product (S ·T)determinantprojection on a vector, traceorthogonal tensors, positive definite tensors


Introduction

Spectral, Cayley–Hamilton and polar decomposition theorems

Spectral, Cayley–Hamilton and polar decompositiontheorems

DefinitionA scalar ω is an eigenvalue of a tensor S if it exists a (unit) vector e suchthat Se = ωe. In this case, e is an eigenvector of S.

The characteristic space of S relative to ω is the subspace of Vconsisting of all vectors in v such that Sv = ωv. If this subspace isn-dimensional, we say that ω has multiplicity n

The spectrum of S is the sequence ω1,ω2, . . ., where ω1 ≤ ω2 ≤ ·· ·are the sorted eigenvalues of S and repeated as many times as indicatedby their respective multiplicity.The eigenvalues of a positive definite tensor are strictly positive.The characteristic spaces of a symmetric tensor are orthogonal to eachother.


Introduction


Theorem (spectral)

Let S be a symmetric tensor. Then, there exists an orthonormal basis of Vconsisting of eigenvectors of S. Moreover, for each orthonormal basise1,e2,e3 of eigenvectors of S, the corresponding eigenvalues (ω1,ω2,ω3)form, when ordered, the spectrum of S and

S =3

∑i=1

ωi ei⊗ ei (2.1)

Reciprocally, if S has the form (2.1), e1,e2,e3 being an orthonormal basis,then ω1,ω2,ω3 are the eigenvalues of S and e1,e2,e3 are the correspondingeigenvectors.

(2.1) is the spectral decomposition of S


Introduction


Moreover:(a) S has exactly three different eigenvalues if and only if its characteristic

spaces are three lines orthogonal to each other(b) S has exactly two different eigenvalues if it admits the following

representation:

S = ω1e⊗ e+ω2(I− e⊗ e), |e|= 1, ω1 6= ω2 (2.2)

Then, ω2 and ω2 are two different eigenvalues and the characteristicspaces are spe and e⊥, respectively. Reciprocally, if spe ande⊥ (|e|= 1) are the characteristic spaces, then S has the form (2.2)

(c) S has an only eigenvector if and only if: S = ωI.In this case, ω is the aigenvector and V is the correspondingcharacteristic space.


Introduction


The matrix of S relative to the basis of autovectors ei is:

[S] =

ω1 0 00 ω2 00 0 ω3

Theorem (of the square root)

Let C ∈ Psym. There exists an only tensor U ∈ Psym such that U2 = C, and we write√C = U.

Theorem (polar decomposition)

Let F ∈ Lin+. There exist U,V ∈ Psym and R ∈ Orth+ such that:

F = RU = VR . (2.5)

Moreover, each of those decompositions is unique; in fact,

U =√

FTF , V =√

FFT (2.6)

We call the representation F = RU (resp. F = VR) the right polardecomposition (resp. left polar decomposition) of F.


Introduction


Proposition

Given a tensor S,

det(S−ωI) =−ω3 + ı1(S)ω2− ı2(S)ω + ı3(S) , ∀ω ∈ IR (2.7)

where:

ı1(S) = TrS

ı2(S) =12

[(TrS)2−Tr(S2)

]ı3(S) = detS

DefinitionThe sequence

IS = ı1(S), ı2(S), ı3(S)

is called the relation of principal invariants of S


Introduction


Proposition

If S ∈ Sym, then IS is completely determined by the spectrum ω1,ω2,ω3of S. Actually,

ı1(S) = ω1 +ω2 +ω3

ı2(S) = ω1ω2 +ω2ω3 +ω1ω3

ı3(S) = ω1ω2ω3

Corollary

Let S,T ∈ Sym and assume IS = IT. Then S and T have the samespectrum.

Theorem (Cayley–Hamilton)

Every tensor S satisfies its own characteristic equation:

S3− ı1(S)S2 + ı2(S)S− ı3(S)I = 0 (2.12)


Introduction


Summary

Eigenvalues and eigenvectors, characteristic spacesSpectral theorem (for symmetric tensors)

Square root theorem: if C ∈ Psym,

∃U ∈ Psym s.t. U2 = C

Polar decomposition theorem: if F ∈ Lin+,

∃U,V ∈ Psym,R ∈ Orth+ s.t. F = RU = VR

Principal invariants of a tensor, Cayley–Hamilton theorem


Introduction

Tensor differentiation


Let V be a vector space.

DefinitionA norm in V is an application ‖·‖ : V −→ IR+ verifying:

1 ‖v‖= 0 ⇐⇒ v = 02 ‖λv‖= |λ | ‖v‖, ∀v ∈ V , ∀λ ∈ IR

3 ‖v+w‖ ≤ ‖v‖+‖w‖, ∀v,w ∈ V

Thus, V is a normed vector space

The most usual norms:‖v‖1 =

n∑

i=1|vi|

‖v‖2 =

[n∑

i=1v2

i

]1/2

‖v‖p =

[n∑

i=1|vi|p

]1/p

‖v‖∞= max

i=1,...,n|vi|


Introduction


Let U and W be normed vector spaces, and let f be defined in aneighborhood of zero in U and have values in V

DefinitionWe say that f(u) approaches zero faster than u (and we write f(u) = o(u))if

limu→0u6=0

‖f(u)‖W‖u‖U

= 0

Similarly, f(u) = g(u)+o(u) means f(u)−g(u) = o(u)For example, let f : IR3→ IR3 be given by:

f(u) =

u1u2u1u3−u2

2

Then, by a simple change to spherical coordinates,

limu→0

√u2

1u22 +u2

1u23 +u4

2√u2

1 +u22 +u2

3

= limρ→0

ρ2√. . .ρ

= 0


Introduction


Let g be a scalar, vector, tensor or point valued function defined on an openinterval D ∈ IR

DefinitionWe call derivative of g in t the following limit, if it exists:

g(t) =ddt

g(t)≡ limα→0

1α

[g(t+α)−g(t)] (1)

in which case we say that g is differentiable in t.

If g is point valued, then g(t+α)−g(t) is a vector and, thus, g(t) is avectorThe derivative of a vector function is a vectorThe derivative of a tensor function is a tensor

DefinitionWe say that g is smooth in D if g(t) exists for all t ∈D and function g iscontinuous in D


Introduction


If g is differentiable in t, then (1) implies:

limα→0

1α[g(t+α)−g(t)−α g(t)] = 0

or, in an equivalent way,

g(t+α) = g(t)+α g(t)+o(α) (3.2)

Clearly, α g(t) is linear on α; thus, g(t+α)−g(t) equals a linear term on α

plus a term that approaches zero faster than α


Introduction


Let U and W finite dimension, normed vector spaces. Let D ⊂U be openand let g : D −→W .

DefinitionWe say that g is differentiable in x ∈D if it exists a linear applicationDg(x) : U −→W such that:

g(x+u) = g(x)+Dg(x)[u]+o(u)

when u→ 0.

Proposition

If Dg(x) exists, then it is unique and for every u

Dg(x)[u] = limα→0α∈IR

g(x+αu)−g(x)α

=d

dαg(x+αu)

∣∣∣∣α=0


Introduction


DefinitionThe linear application (tensor) Dg(x) is called differential of g at x

DefinitionWe say that g is of class C 1 (or smooth) if g is differentiable at each point ofD and Dg is continuous. Similarly, we say that g is of class C 2 if g and Dgare of class C 1, . . .


Introduction


Examples:

Let ϕ : V → IR be defined by ϕ(v) = v ·v. Then,

ϕ(v+u) = (v+u) · (v+u) = v ·v+2v ·u+u ·u = ϕ(v)+2v ·u+o(u)

and: Dϕ(v)[u] = 2v ·uWe could also have computed the jacobian matrix and deduce:

Dϕ(v)[u] = Jϕ (v)u = (2v1 2v2 2v3)

u1u2u3

= 2v ·u

Let G : Lin→ Lin be defined by G(A) = A2. Then,

G(A+U) = (A+U)2 = A2 +AU+UA+U2 =

= G(A)+AU+UA+o(U)

thus: DG(A)[U] = AU+UA


Introduction


Theorem (Smooth-inverse)

Let D be an open subset of a finite-dimensional normed vector space U . Letg : D −→U be a C n (n≥ 1) bijection, and let us assume that the lineartransformation Dg(x) : U −→U is invertible for all x ∈D . Then, g−1 is ofclass C n.

If g : D ⊂ E −→ E , the theorem remains with Dg(x) : V −→ V .

TheoremLet ϕ the scalar function defined on the set of all invertible tensors A by:

ϕ(A) = detA .

Then, ϕ is regular and

Dϕ(A)[U] = (detA)Tr(UA−1) , ∀U ∈ Lin (3.8)


Introduction


TheoremThe function H : Psym→ Psym defined by:

H(C) =√

C

is smooth.


Introduction


Proposition (Product rule)

Let f and g be differentiable in x ∈D . Then, their product h = π(f,g) isdifferentiable in x and

Dh(x)[u] = π(f(x),Dg(x)[u])+π(Df(x)[u],g(x)) (3.9)

for every u ∈U .

Proposition

Let ϕ , v, w, S and T smooth functions defined in an open subset of IR. Let usassume that ϕ is scalar valued, v and w are vector valued, and S and T aretensor valued. Then:

(ϕv) = ϕ v+ ϕv(v ·w) = v · w+ v ·w(TS) = TS+ TS

(T ·S) = T · S+ T ·S(Sv) = Sv+ Sv


Introduction


Let U , G and F be finte-dimensional normed vector spaces (or euclideanpoint spaces). Let C and D be open subsets of G and U , respectively, and

g : D ⊂U −→ G , f : C ⊂ G −→F

such that Im(g)⊂ C .

Proposition (Chain rule)

Let g be differentiable in x ∈D and f differentiable in y = g(x). Then,h = fg is differentiable at x, and

Dh(x) = Df(y)Dg(x)

i.e.,

Dh(x)[u] = Df(g(x)) [Dg(x)[u]] , ∀u ∈U


Introduction


In the particular case U = IR, g and h are functions of a real variableand, replacing x by t, we get:

Dh(t)[α] = αh(t), Dg(t)[α] = α g(t)

for each α ∈ IR.

As h(t) = f(g(t)), we deduce:

ddt

f(g(t)) = Df(g(t))[g(t)] (3.12)

Example:

g(t) =

t2

2t5t−1

f(x,y,z) =

y2 + z2

x2 + z2

x2 + y2

Df(x,y,z)[v] =

0 2y 2z2x 0 2z2x 2y 0

v1v2v3

Df(g(t))[g(t)] =

0 4t 10t−22t2 0 10t−22t2 4t 0

2t25

=

58t−104t3 +50t−10

4t3 +8t


Introduction


Proposition

Let S be a smooth tensor function defined in an open set D ⊂ IR. Then,

(ST ) = (S)T ≡ ST

If, moreover, S(t) is invertible for every t ∈D , we have:

(detS) = (detS)Tr(SS−1) (3.14)


Introduction

Gradient. Divergence. Curl


We will consider functions defined on an open set, R ⊂ E .

DefinitionA function on R is called a scalar, vector, tensor o point field if its valuesare, respectively, scalars, vectors, tensors or points.

We will usually denote by:ϕ , a scalar fieldu, v, vector fieldsS, T, tensor fields


Introduction


Let ϕ be a smooth scalar field defined on R.For each x ∈R ⊂ E , Dϕ(x) is a linear application from V into IR and,thanks to representation theorem for linear forms, there exists an onlyvector a(x) such that Dϕ(x)[u] = a(x) ·u.We call gradient of ϕ in x the vector ∇ϕ(x) such that:

Dϕ(x)[u] = ∇ϕ(x) ·u

As ϕ(x+u) = ϕ(x)+Dϕ(x)[u]+o(u), we deduce:

ϕ(x+u) = ϕ(x)+∇ϕ(x) ·u+o(u)


Introduction


Let v be a smooth vector field defined on R.Dv(x) is a linear transformation from V into V and, hence, a tensor

We denote Dv(x) by ∇v(x) and we write

∇v(x)u = Dv(x)[u]

The tensor ∇v(x) is called gradient of v in x.

We call divergence of the vector field v the scalar field

divv = Tr(∇v)


Introduction


DefinitionWe define the divergence of a smooth tensor field S, divS, as the only vectorfield verifying:

(divS) ·a = div(STa) (1)

for every constant vector a.

In general,

ST a =

S11a1 +S21a2 +S31a3S12a1 +S22a2 +S32a3S13a1 +S23a2 +S33a3

;

the previous definition is valid for every constant vector a, hence:

div(ST a) = S11,1a1 +S21,1a2 +S31,1a3 +S12,2a1 +S22,2a2 +S32,2a3+

+S13,3a1 +S23,3a2 +S33,3a3 =

= (S11,1 +S12,2 +S13,3)a1 +(S21,1 +S22,2 +S23,3)a2+

+(S31,1 +S32,2 +S33,3)a3 = divS ·a

thus:

divS =

S11,1 +S12,2 +S13,3S21,1 +S22,2 +S23,3S31,1 +S32,2 +S33,3


Introduction


Proposition

Let ϕ , v, w and S be smooth fields, ϕ scalar, v and w vectors, and S tensor.Then

(a) ∇(ϕv) = ϕ ∇v+v⊗∇ϕ (4.2a)(b) div(ϕv) = ϕ divv+v ·∇ϕ (4.2b)

(c) ∇(v ·w) = (∇w)Tv+(∇v)Tw (4.2c)(d) div(v⊗w) = (divw)v+(∇v)w (4.2d)

(e) div(STv) = S ·∇v+v ·divS (4.2e)(f ) div(ϕS) = ϕ divS+S∇ϕ (4.2f )

Proposition

If v is a vector field of class C 2,

div(∇vT) = ∇(div v) (4.5)


Introduction


DefinitionThe curl of v, which we denote curlv, is the only vector field verifying:

(∇v−∇vT)a = (curlv)×a , ∀a ∈ V

curlv(x) is the axial vector of the skew tensor ∇v(x)−∇v(x)T

it is also denoted by rot v, or by ∇×v.

The components of vector curlv are (α,β ,γ), with

α =∂v3

∂x2− ∂v2

∂x3, β =

∂v1

∂x3− ∂v3

∂x1, γ =

∂v2

∂x1− ∂v1

∂x2

as the skew part of ∇v, W =12(∇v−∇vT

), has components:

[W] =12

0 −γ β

γ 0 −α

−β α 0

Thus, (curlv)i =

12

εijk(vk,j− vj,k) = εijkvk,j


Introduction


Let Φ a scalar field of class C 2.

DefinitionWe define the laplacian ∆Φ of Φ as:

∆Φ = div∇Φ

If ∆Φ = 0, we say that Φ is harmonic


Introduction


Proposition

Let v a vector field of class C 2 such that divv = 0 and curlv = 0. Then v isharmonic.

Let Φ be a scalar, vector or tensor field. Then,

DΦ(x)[ei] = limα→0

1α[Φ(x+αei)−Φ(x)] =

∂Φ

∂xi(x)

If ϕ is a scalar field, v is a vector field and S is a tensor field,

(∇ϕ)i =∂ϕ

∂xidiv v = ∑

i

∂vi

∂xi∆ϕ = ∑

i

∂ 2ϕ

∂x2i

(∇v)ij =∂vi

∂xj(div S)i = ∑

j

∂Sij

∂xj(∆v)j = ∆vj


Introduction


Let f and g be smooth point fields and let h = fg. In this case, the chainrule stands:

∇h(x) = ∇f(y)∇g(x)

where y = g(x).

Let ϕ a scalar field, v a vector field, and g a point field of real variable, all ofthem smooth. The chain rule stands:

ddt

ϕ(g(t)) = ∇ϕ(g(t)) · g(t)

ddt

v(g(t)) = ∇v(g(t)) g(t)


Introduction


DefinitionA curve in R ⊂ E is a smooth application

c : [0,1]−→R

such that c is never null.

c is closed if c(0) = c(1).The length of c is the number

long(c) =∫ 1

0|c(σ)| dσ

Let v a continuous vector field in R. We define the integral of v along c as∫cv(x) ·dx =

∫ 1

0v(c(σ)) · c(σ)dσ


Introduction


Similarly, for a continuous tensor field S on R we define the integral along cas ∫

cS(x)dx =

∫ 1

0S(c(σ))c(σ)dσ

If ϕ is a smooth scalar field on R,∫c∇ϕ(x) ·dx =

∫ 1

0∇ϕ(c(σ)) · c(σ)dσ =

∫ 1

0

ddσ

ϕ(c(σ))dσ

= ϕ(c(1))−ϕ(c(0)) ;

thus, if c is a closed curve we have:∫c∇ϕ(x) ·dx = 0


Introduction


Given a subset R ⊂ E , we denote ∂R its boundary andR its interior.

We say that R is:connected if any two points in R can be connected by a curve in R;simply connected if any closed curve in R can be continuouslydeformed to a point without leaving R; i.e., given any closed curvec : [0,1]→R, there exists a continuous function f : [0,1]×[0,1]→Rand a point y ∈R such that, for every σ ∈ [0,1], f(σ ,0) = c(σ),f(σ ,1) = y, f(0,σ) = f(1,σ); or, in a different way, R has not holes.

R ⊂ E is an open region if it is an open and connected set in E ; the closureof an open region is called a closed region.


Introduction


Let R ⊂ E be a closed region.

A field Φ is smooth in R if Φ is smooth inR and Φ and ∇Φ have

continuous extensions to all of R; in this case, we also write Φ and ∇Φ fortheir extensions.

Similarly, we define Φ ∈ C N(R): the partial derivatives of order less orequal than N have to be continuous (i.e., have continuous extensions) in R.


Introduction


Proposition

A vector field v = ∇ϕ (ϕ ∈ C 2) verifies: curlv = 0.

The reciprocal also stands:

Theorem (potential)

Let v a smooth vector field in a simply (open or closed) connected region R,and assume:

curlv = 0

Then, there exists a scalar field ϕ of class C 2 in R such that

v = ∇ϕ


Introduction


Proposition

Let f be a smooth point or vector field on a region R, and assume F = ∇f isconstant in R. Then,

f(x) = f(y)+F(x−y) , ∀x,y ∈R (4.9)

Thanks to (4.9), every point or vector field f with constant gradient F can bewritten as:

f(x) = a+F(x−x0)

with a ∈ V (or a ∈ E if f is a point field) and x0 ∈ E . Moreover, x0 can bechosen in an arbitrary way (and a depends on the choice of x0).


Introduction

Divergence and Stokes theorems


R is a smooth region if it is a closed region with piecewise smoothboundary ∂R.

R may be bounded or unboundedIf it is bounded, we denote vol(R) its volume.

Theorem (divergence)

Let R be a bounded smooth region and let ϕ : R→ IR, v : R→ V andS : R→ Lin be smooth fields. Then,∫

∂RϕndA =

∫R

∇ϕ dV∫∂R

v ·ndA =∫

RdivvdV∫

∂RSndA =

∫R

divSdV

where n is the outwards unit normal field on ∂R.


Introduction


Theorem (localization)

Let Φ be a continuous scalar or vector field defined on an open subsetR ⊂ E . Let x0 ∈R; then,

Φ(x0) = limδ→0

1vol(Ωδ )

∫Ωδ

ΦdV (1)

where Ωδ (δ > 0) is th closed ball of radius δ centered in x0. Hence, if∫Ω

ΦdV = 0

for every closed ball Ω⊂R, then

Φ = 0


Introduction


If we combine localization and divergence theorems, we get:

divv(x) = limδ→0

1vol(Ωδ )

∫∂Ωδ

v ·ndA

divS(x) = limδ→0

1vol(Ωδ )

∫∂Ωδ

SndA .

Thus, the divergence (of v or S) in a point is the field flux per unit volumethrough the boundary of balls centered in the point, when the radius of theballs approaches zero.


Introduction


Theorem (Stokes)

Let v a smooth vector field on an open set R ⊂ E . Let Ω be a disc in R, n aunit normal to Ω, and c : [0,1]→R the boundary of Ω, oriented such that

[c(0)×c(σ)] ·n > 0 for all 0 < σ < 1 .

Then, ∫Ω

(curlv) ·ndA =∫

cv ·dx


Introduction


The integral∫

cv ·dx represents the circulation of v around c.

If x0 ∈ E is the center of Ω and δ > 0 is its radius (we write Ω = Ωδ andc = cδ ), then the combination of Stokes theorem and the “plane” version ofthe localization theorem leads us to:

(curlv)(x0) ·n = limδ→0

1A(Ωδ )

∫cδ

v ·dx

where A(Ωδ ) is the area of Ωδ

Thus, the magnitude of curlv is the circulation per unit area in that plane.


Introduction


Summary on vector and tensor differentiation (I)

Derivative of a function of one variableg : V → V is differentiable in x ∈ V if


Smooth functions

Product rule:

Dh(x)[u] = π(f(x),Dg(x)[u])+π(Df(x)[u],g(x))

Chain rule:

Dh(x)[u] = Df(g(x)) [Dg(x)[u]] , ∀u ∈U


Introduction


Summary on vector and tensor differentiation (II)


(∇ϕ)i =∂ϕ

∂xidiv v = ∑

i

∂vi

∂xi∆ϕ = ∑

i

∂ 2ϕ

∂x2i

(∇v)ij =∂vi

∂xj(div S)i = ∑

j

∂Sij


(curlv)i = εijkvk,j

Divergence theoremLocalization theoremStokes theorem

1 Introduction

2 KinematicsBodies, deformations, strainsSmall deformationsMotionsTypes of motions. Spin. Stretch rateTransport theorems. Volume. Isochoric motionsSpin. Circulation. Vorticity

3 Conservation laws


Kinematics

Bodies, deformations, strains


DefinitionA body B is a regular region of E . We will refer to B as referenceconfiguration.Points p ∈B are called material points and bounded regular subregions ofB are called parts of B.

Continuum mechanics is the study of deformations of bodies. From themathematical point of view, a body is deformed via an application f thatcarries each material point p ∈B into a position x = f(p).


Kinematics


DefinitionA deformation of a body B is a one–to–one smooth mapping f which mapsB into a closed region in E , f(B), and which satisfies det∇f > 0.

The requirement that the body not penetrate itself is expressed by theassumption that f be one–to–one.det∇f represents, locally, the volume after the deformation per unitoriginal volume; hence, we assume that det∇f 6= 0.Moreover, a deformation with det∇f < 0 cannot be reached by acontinuous process starting in the reference configuration.

The vector u(p) = f(p)−p represents the displacement of p.When u is constant, f is a translation: f(p) = p+u


Kinematics


Deformations

f(p) =

1−p21

1−p21−p2

2

2+12

p23

f(p) =

1+3p11.2+2p2−p1p3

1+p23


Kinematics


Translation

f(p) =

p1 +0.5p2 +1.0p3 +0.6


Kinematics


DefinitionThe tensor

F(p) = ∇f(p)

is called gradient of deformation

As det∇f > 0, we have: F(p) ∈ Lin+, ∀p ∈B.

DefinitionA deformation is homogeneous if F is constant.

Thanks to (4.9), every homogeneous deformation can be written as:

f(p) = f(q)+F(p−q) , ∀p,q ∈B (6.4)

Reciprocally, every field f, on points of B, in the form (6.4) withF ∈ Lin+ and constant is a homogeneous deformation.


Kinematics


Homogeneous deformations

f(p) =

1+3p1−p2 +2p32+p2−4p3

1+1.5p3


Kinematics


Proposition (about homogeneous deformations)

(HD1) Given a point q ∈ E and a tensor F ∈ Lin+, there exists a uniquehomogeneous deformation f with ∇f = F and q fixed (i.e., f(q) = q).

(HD2) If f and g are homogeneous deformations, fg is also homogeneousand:

∇(fg) = (∇f)(∇g) .

Moreover, if f and g have q ∈ E fixed, then so does fg.

Proposition

Let f be a homogeneous deformation. Given a point q ∈ E , we candecompose f as:

f = d1 g = gd2

where g is a homogeneous deformation with q fixed, while d1 and d2 aretranslations. Further, each of these decompositions is unique.


Kinematics


Some homogeneous deformations with a point fixed:(a) Rotation around q:

f(p) = q+R(p−q) , with R ∈ Orth+

(b) Stretch from q:

f(p) = q+U(p−q) , with U ∈ Psym


Kinematics


Rotation

f(p) =

cos(4π/5)p1 + sin(4π/5)p2−sin(4π/5)p1 + cos(4π/5)p2

p3


Kinematics


Stretching

q =

120

, U =

2 1 −11 3 0−1 0 2


Kinematics


In particular, a stretch such that:

U = I+(λ −1)e⊗ e

with λ > 0 and |e|= 1, then f is an extension (or elongation) of magnitudeλ in the direction of e.

In this case, in a system of coordinates with e1 = e the matrix of U is:

[U] =

λ 0 00 1 00 0 1

and the corresponding displacement has components (u1,0,0) with

u1(p) = (λ −1)(p1−q1)


Kinematics


Extension

q =

−220

, U =

2 0 00 1 00 0 1


Kinematics


Propositions (HD1) and (HD2) of homogeneous deformations, jointly withthe polar decomposition theorem, leads us to:

Proposition

Let f be a homogeneous deformation with q fixed. Then, f admits thedecompositions

f = g s1 = s2 g ,

where g is a rotation around q, while s1 and s2 are stretches from q.Moreover, each of these decompositions is unique. In fact, if F = RU = VRis the polar decomposition of F = ∇f, then:

∇g = R, ∇s1 = U, ∇s2 = V


Kinematics



Kinematics


Proposition

Every stretch f from q can be decomposed in a sequence of three extensionsfrom q in mutually orthogonal directions. The amounts and directions of theextensions are the eigenvalues and eigenvectors of U = ∇f, and theextensions may be performed in any order.


Kinematics


Each stretching can be decomposed as the sequence of three extensionsof intensity λ1, λ2 y λ3.λi are called the principal stretches

U and V having the same spectre, the stretches s1 and s2 inf = g s1 = s2 g have the same principal stretches

From now on, until the end of this section, f is a general deformation of B.

As f is a bijection, there exists f−1 : f(B)→B. Further, det∇f > 0 so that∇f(p) is invertible for each p ∈B. Thanks to the smooth inverse theorem,f−1 is also smooth.

Moreover,

f(B) = f(B) (6.5a)

f(∂B) = ∂ f(B) (6.5b)


Kinematics


If we develop deformation f : B→ E around a point q ∈B, we have:

f(p) = f(q)+F(q)(p−q)+o(p−q)

where F(q) = ∇f(q) is the gradient of deformation at q.

Thus, in a neighborhood of q and to within an error o(p−q), thedeformation behaves as a homogeneous deformation.

If F = RU = VR is the pointwise polar decomposition of F, then R is therotation tensor, U is the right stretch tensor and V is the left stretchtensor of deformation f.

R(p) measures the the local rigid rotation of points near p, while U(p)and V(p) measure the local stretches from p.


Kinematics


As U =√

FTF and V =√

FFT , their computation is often difficult.

DefinitionWe introduce the right and left Cauchy–Green strain tensors, C and B,defined by:

C = U2 = FTF , B = V2 = FFT ,

which components are:

Cij =3

∑k=1

∂ fk∂pi

∂ fk∂pj

, Bij =3

∑k=1

∂ fi∂pk

∂ fj∂pk

AsV = RURT B = RCRT

and R ∈ Orth+, we have:

IV = IU , IB = IC


Kinematics


DefinitionA deformation is rigid if it preserves the distances. Actually, f is rigid if

|f(p)− f(q)|= |p−q| ∀p,q ∈B

f is rigid if and only if it is homogeneous and ∇f is a rotation

Theorem (Characterization of rigid deformations)

The following statements are equivalent:

(a) f is a rigid deformation.

(b) f has the formf(p) = f(q)+R(p−q)

for every p,q ∈B, with R ∈ Orth+ constant.

(c) F(p) is a rotation for every p ∈B.

(d) U(p) = I for every p ∈B.

(e) For every curve c in B, long(c) = long(f c).


Kinematics


Proposition

Let f be a deformation of B and let ϕ be a continuous scalar field on f(B).Given a part P of B, we have:∫

f(P)ϕ(x)dVx =

∫P

ϕ(f(p))detF(p)dVp∫∂ f(P)

ϕ(x)m(x)dAx =∫

∂Pϕ(f(p))G(p)n(p)dAp

where m and n are, respectively, the outwards unit normal vector to ∂ f(P)and ∂P , and

G = (detF)F−T


Kinematics


Given a part P ,

vol(f(P)) =∫

f(P)dV =

∫P

detFdV

and, thanks to the localization theorem,

(detF)(p) = limδ→0

vol(f(Ωδ ))

vol(Ωδ )

where Ωδ is a closed ball of radius δ and centered in p.

Thus, detF is the volume after deformation per unit of original volume.


Kinematics


DefinitionA deformation f is isochoric (or it preserves the volume) if for any part Pof B we have:

vol(f(P)) = vol(P)

Proposition

A deformation is isochoric if and only if

detF = 1


Kinematics


Summary on deformations

Material points (p) and spatial positions (x), x = f(p)Gradient of deformation: F = ∇fSome kinds of deformations:

translation (constant displacement)homogeneous (constant F)rotation (F ∈ Orth+)stretching (F ∈ Psym)extensionrigidisochoric (detF = 1)

Decomposition: F = RU = VRCauchy–Green strain tensors:

C = U2 = FTF, B = V2 = FFT


Kinematics

Small deformations

Small deformations

As f(p) = p+u(p), we have: F = I+∇uHence, Cauchy–Green tensors are

C = I+∇u+∇uT +∇uT ∇uB = I+∇u+∇uT +∇u∇uT

When the deformation is rigid, C = B = I, and:

∇u+∇uT +∇uT∇u = ∇u+∇uT +∇u∇uT = 0 . (7.3)

Moreover, in this case ∇u is constant as F is constant.

We will study deformations with small ∇u.


Kinematics

Small deformations

The tensor fieldE =

12(∇u+∇uT)

is called infinitesimal deformation

The infinitesimal deformation verifies:

C = I+2E+∇uT∇u

B = I+2E+∇u∇uT


Kinematics

Small deformations

An infinitesimal rigid displacement of B is a vector field u on B with ∇uconstant and skew, i.e.

u(p) = u(q)+W(p−q)

for all p,q ∈B, with W skew

As a consequence, we have u(p) = u(q)+ω×(p−q), where ω is theaxial vector of W.If f is a rigid deformation, then

C = I =⇒ 2E =−∇uT∇u ;

if u is an infinitesimal rigid displacement, E = 0An infinitesimal rigid displacement is not the displacement of a rigiddeformation; it is only a good approximation when |∇u| is small.


Kinematics

Small deformations

Theorem (Characterization of infinitesimal rigid displacements)

Let u be a smooth vector field on B. The following statements areequivalent:

(a) u is an infinitesimal rigid displacement

(b) u verifies the projection property: for all p,q ∈B,

(p−q) · [u(p)−u(q)] = 0

(c) ∇u(p) is skew for all p ∈B

(d) the infinitesimal deformation is E(p) = 0 for all p ∈B.


Kinematics

Small deformations

Summary on small deformations

|∇u| ≈ 0 =⇒ B = C = I+∇u+∇uT

Infinitesimal deformation: E = 12

(∇u+∇uT

)Infinitesimal rigid displacement:u on B, with ∇u constant and skew


Kinematics

Motions

Motions

Let B be a body.

DefinitionA motion of B is a function of class C 3

x : B×IR −→ E

where x(·, t), for each fixed t, is a deformation of B.

Thus, a motion is a smooth parametric family of deformations, time tbeing the parameter.We will denote

x = x(p, t)the position ocuppied by the material point p at time instant t.We call

Bt = x(B, t)

the region ocuppied by the body B at time t.


Kinematics

Motions

x(p, t) =

p1−p2t+2tp2 +3t2

p3 +2sin t


Kinematics

Motions

Trajectory: T = (x, t)/x ∈Bt, t ∈ IRFor each t, x(·, t) is a bijection from B onto Bt; thus, it has an inverse

p(·, t) : Bt −→B

such thatx(p(x, t), t) = x, p(x(p, t), t) = p

Given (x, t) ∈T ,p = p(x, t)

is the material point in the position x at time t.The application p : T −→B is also called reference mapping of themotionVelocity and acceleration:

x(p, t) =∂

∂ tx(p, t) ; x(p, t) =

∂ 2

∂ t2 x(p, t)

Using the reference mapping p, we can describe the velocity x(p, t) as afunction v(x, t) of the position x and time tThe mapping v : T → V defined by: v(x, t) = x(p(x, t), t) is the spatialdescription of the velocity.v(x, t) is the velocity of the material point ocuppying position x at time t


Kinematics

Motions

Any field related to the motion can be described as a function of thematerial point and time, with domain B× IR, or as a function of theposition and time, with domain T

Notation: a material field is a mapping on B× IR, while a spatial fieldis a mapping on T

For example, the field x is material and the field v is spatial

Spatial description Φs of a material field (p, t) 7→Φ(p, t):

Φs(x, t) = Φ(p(x, t), t)

Material description Ωm of a spatial field (x, t) 7→Ω(x, t):

Ωm(p, t) = Ω(x(p, t), t)

Clearly, (Φs)m = Φ and (Ωm)s = Ω


Kinematics

Motions

Given a material field Φ, we write:Φ, material derivative with respect to time of Φ: Φ(p, t) = ∂

∂ t Φ(p, t)(derivative with respect to time t for p fixed)

∇Φ, material gradient of Φ: ∇Φ(p, t) = ∇pΦ(p, t)(gradient with respect to p for t fixed)

The material field F = ∇x is called the deformation gradient in themotion x

Given a spatial field Ω, we denote:

Ω′, spatial time derivative of Ω: Ω′(x, t) =∂

∂ tΩ(x, t)

(derivative with respect to time t with x fixed)

gradΩ, spatial gradient of Ω: gradΩ(x, t) = ∇xΩ(x, t)(gradient with respect to x for fixed t)


Kinematics

Motions

We define the spatial divergence and the spatial curl, div and curl , asthe divergence and curl defined for spatial fields, so that the underlyinggradient is the spatial gradient, grad

Similarly, Div and Curl denote the material divergence and thematerial curl, defined for material fields and computed by means ofthe material gradient, ∇.

Material field Φ Spatial field Ω

Domain: B×IR TArguments: (p, t) (x, t)Gradient: ∇Φ gradΩ

Time derivative: Φ Ω′

Divergence: DivΦ divΩ

Curl: CurlΦ curlΩ


Kinematics

Motions

It is also convenient to define the time derivative with respect to time Ω ofa spatial field Ω, such that Ω represents the derivative of Ω leaving fixed thematerial point p.

Thus, to compute Ω:we transform Ω to its material description Ωm

we calculate the material derivative with respect to time (Ωm), andwe come back to the spatial description ((Ωm))s,

i.e.,Ω = ((Ωm))s (1)

or:

Ω(x, t) =∂

∂ tΩ(x(p, t), t)

∣∣∣∣p=p(x,t)


Kinematics

Motions

Let Φ be a smooth material field and Ω a smooth spatial field. Then,

(Φ)s = (Φs)≡ Φs (Ω)m = (Ωm)≡ Ωm

In particular, for Ω = v:

(v)m = (vm) = x

thus v is the spatial description of the acceleration

Let ϕ and u be smooth spatial fields, ϕ being scalar and u vector. Then:ϕ = ϕ ′+v ·gradϕ

u = u′+(gradu)v(8.4)

In particular,v = v′+(gradv)v (8.5)

We define the position vector r : E → V as r(x) = x−o.Let us consider the position vector as a spatial field: r(x, t) = r(x) foreach (x, t) ∈T . Then

r = v


Kinematics

Motions

Let u be a smooth spatial vector field. Then, ∇(um) = (gradu)mFWe call velocity gradient the spatial field L = gradvThe velocity gradient verifies:

F = LmF , F = (grad v)mF

The function s : IR−→ E defined by s(t) = x(p, t) is called trajectory(or pathline) of the material point p

Clearly, s is a solution of the differential equation:

s(t) = v(s(t), t)

If we fixed the time in t = τ and look at the integral curves of the vectorfield v(·,τ), we get the streamlines of the motion at instant τ

Thus, each streamline is a solution s of the differential equation

s(λ ) = v(s(λ ),τ)


Kinematics

Motions

Example

Let us consider the motion x : (p, t) ∈B×IR −→ x(p, t) = x ∈ E , defined by:

x1 = p1et2 , x2 = p2et, x3 = p3.

Then the deformation gradient F is

[F] = [∇x] =

et2 0 00 et 00 0 1

and the velocity is

x=

2p1tet2

p2et

0


Kinematics

Motions

The reference application p is:

p1 = x1e−t2 , p2 = x2e−t , p3 = x3 ,

the spatial description of the velocity is:

v1(x, t) = 2x1t , v2(x, t) = x2 , v3(x, t) = 0 ,

and the velocity gradient is

[L(x, t)] =

2t 0 00 1 00 0 0

The streamlines of the motion at instant t are the solutions of the followingsystem:

s1(λ ) = 2ts1(λ ) , s2(λ ) = s2(λ ) , s3(λ ) = 0,

thus,s1(λ ) = y1e2tλ , s2(λ ) = y2eλ , s3(λ ) = y3,

is the streamline that passes through (y1,y2,y3) for λ = 0.


Kinematics

Motions

Summary on motions

Motion: x : B×IR→ E , x(p, t) = xTrajectory: T = (x, t)/x ∈Bt, t ∈ IRVelocity and acceleration

x(p, t) =∂

∂ tx(p, t) , x(p, t) =

∂ 2

∂ t2 x(p, t)

Spatial and material descriptionVelocity gradient, L = gradvPathline of a material particleStreamline


Kinematics

Types of motions. Spin. Stretch rate


DefinitionA motion x is steady (or stationnary) if:

Bt = B0, ∀tv′ = 0 on the trajectory T

The body will ocuppy the region B0 at every moment, henceT = B0×IR

The velocity only depends on the position, thus the material particlesthat cross a given position x all cross x with the same velocity v(x)The velocity of a material point varies with time, depending on theposition occupied at each instant: x(p, t) = v(x(p, t))In a steady motion, the velocity field is tangent to the boundary:

v(x) is tangent to ∂B0 , ∀x ∈ ∂B0


Kinematics


In a steady motion, every streamline is a pathline, and viceversa.

Let Φ be a smooth field on the pathline of a steady motion. Φ is steadyif Φ′ = 0

Let ϕ be a smooth, scalar, steady field on the pathline of a steadymotion. The following statements are equivalent:

(a) ϕ is constant on the streamlines:ddt

ϕ(s(t)) = 0, ∀t(b) ϕ = 0(c) v ·gradϕ = 0


Kinematics


DefinitionA motion x is rigid if:

∂

∂ t|x(p, t)−x(q, t)|= 0 , ∀p,q , ∀t (9.4)

The magnitude δ (t) = |x(p, t)−x(q, t)| represents the time distancebetween the material points p and qDuring a rigid motion, the distance between material points is constantalong timeSimilarly, the angle θ(t) between the material points a, p and q is theangle between vectors x(a, t)−x(p, t) and x(q, t)−x(p, t)


Kinematics


Characterization of rigid motionsLet x be a motion, and let v be its velocity field. The followingstatements are equivalent:

(a) x is a rigid motion(b) v(t) has the form of an infinitesimal rigid motion: ∀t,

v(x, t) = v(y, t)+W(t)(x−y) , ∀x,y ∈Bt,W(t) ∈ Skw

(c) the velocity gradient L(x, t) is skew for each (x, t) ∈T

Let ω(t) the axial vector of W(t); then,

v(x, t) = v(y, t)+ω(t)×(x−y) ,

which is the classical expression for the velocity field of a rigid motionThe vector function ω is called the angular velocity


Kinematics


Let us assume ω 6= 0. For y fixed, the velocity field

x 7→ ω×(x−y)

vanishes at positions x on the line y+αω /α ∈ IR, and represents arigid rotation around that line.

Thus, for a fixed y, the velocity v is the sum of:a uniform velocity field, with constant value v(y)a rigid rotation around the line passing through y with the direction ofvector ω .

We call `= sp ω the spin axisLet us remark that `= `(t) can also be defined as the set of vectors e suchthat: We = 0


Kinematics


Let us now consider a general velocity field v.As L = gradv,

v(x) = v(y)+L(y)(x−y)+o(x−y)

when x→ y, where y is a given positionLet D and W be, respectively, the symmetric and skew parts of L:

D =12(L+LT) =

12(gradv+gradvT)

W =12(L−LT) =

12(gradv−gradvT) .

Then, L = D+W and:

v(x) = v(y)+W(y)(x−y)+D(y)(x−y)+o(x−y) .

Thus, in points close to y, and within an error o(x−y), a velocity fieldv is the sum of a rigid velocity field

x 7→ v(y)+W(y)(x−y)

and a velocity field of the form

x 7→ D(y)(x−y) .


Kinematics


DefinitionWe call W(y, t) and D(y, t) the spin and the stretching, respectively, and wecall the spin axis in (y, t) the subspace `⊂ V of all vectors e such thatW(y, t)e = 0.

` is one dimensional if W(y, t) 6= 0.The acceleration v verifies:

v = v′+12

grad(v2)+2Wv , (9.10a)

v = v′+12

grad(v2)+(curlv)×v . (9.10b)

DefinitionA motion is plane if the velocity field has the form:v(x, t) = v1(x1,x2, t)e1 + v2(x1,x2, t)e2 in some cartesian reference system.

Proposition

In a plane motion, WD+DW = (divv)W


Kinematics


Summary on motions

Steady motion: Bt = B0, v′ = 0Rigid motion: the distance between material points is constant alongtimeStretching and spin:

D =12(L+LT) , W =

12(L−LT)

Plane motion


Kinematics

Transport theorems. Volume. Isochoric motions


Let x be a motion of body B. Let P ⊂B be a part of the body.

DefinitionWe call volume of P at time t:

vol(Pt) =∫

Pt

dV =∫

Pt

dVx =∫

P(detF)dVp

Theorem (of volume transport)

For every part P ⊂B and every instant t,

ddt

vol(Pt) =∫

P(detF)· dV =

∫Pt

(divv)dV =∫

∂Pt

v ·ndA


Kinematics


DefinitionA motion is isochoric if:

ddt

vol(Pt) = 0 , ∀P ⊂B, ∀t

Theorem (of characterization of isochoric motions)

The following statements are equivalent:

the motion x is isochoric

(detF)· = 0divv = 0

∀P ⊂B, ∀t,∫

∂Pt

v ·ndA = 0.

Rigid motions are isochoric, as their velocity gradient is skew.For a motion to be isochoric, the volume of every part should beconstant along the motion; but it is not necessary that the volume ofeach part be equal to its volume in the reference configuration.


Kinematics


Theorem (Reynolds transport theorem)

Let Φ be a smooth field, either scalar or vector. Then, for every part P andevery time t,

(a)ddt

∫Pt

ΦdV =∫

Pt

(Φ+Φdivv

)dV (10.5a)

(b)ddt

∫Pt

ΦdV =∫

Pt

Φ′ dV +

∫∂Pt

Φv ·ndA (10.5b)

Let us remark that:∫Pt

Φ′ dV =

∫Pt

∂

∂ tΦ(x, t)dVx =

[d

dτ

∫Pt

Φ(x,τ)dVnx

]τ=t

.

Thus, (10.5b) asserts that the rate at which the integral of Φ over Pt ischanging is equal to the rate computed as if Pt were fixed in it currentposition plus the rate at which Φ is carried out of this region across itsboundary


Kinematics


Summary

Volume of a part P:

vol(Pt) =∫

Pt

dV =∫

Pt

dVx =∫

P(detF)dVp

Theorem of volume transport:

ddt

vol(Pt) =∫

P(detF)· dV =

∫Pt

(divv)dV =∫

∂Pt

v ·ndA

Isochoric motion:ddt

vol(Pt) = 0 , ∀P ⊂B, ∀t

Reynolds transport theorem:

ddt

∫Pt

ΦdV =∫

Pt

(Φ+Φdivv

)dV

=∫

Pt

Φ′ dV +

∫∂Pt

Φv ·ndA


Kinematics

Spin. Circulation. Vorticity


We have seen that a rigid motion is characterized by a skew velocitygradient: L = gradv = D+W = W.For a general motion, the spin W(x, t) describes the rigid rotation of thematerial points occupying positions close to x.

Let J be the skew part of the acceleration gradient:

J =12(grad v−grad vT)

Given a spatial tensor field G, let: GF = FTGmF.

Theorem (of spin transport)

The spin W verifies the differential equations:

(WF)· = JF (11.2a)

W+DW+WD = J (11.2b)


Kinematics


DefinitionA motion is irrotational if W = 0, which implies curlv = 0.

DefinitionA spatial vector field g is gradient of a potential (or it derives from apotential) if it exists a spatial scalar field α such that:

g(x, t) = gradα(x, t) , ∀(x, t) ∈T .

Proposition

If the velocity v derives from a potential, then J = 0.

The hypothesis of the acceleration deriving from a potential is valid, forexample, for non viscous fluids under conservative volume forces.


Kinematics


Theorem (Lagrange–Cauchy)

A motion which acceleration derives from a potential is irrotational if it isirrotational at a given instant.

Proposition

In a plane and isochoric motion which acceleration derives from a potential,we have W = 0.


Kinematics


Let x be a motion of B.

We call material curve a curve c on B.A material point on the curve is given by c(σ), σ ∈ [0,1]. In an instant t, thematerial curve c(σ) will occupy positions x(c(σ), t) and, hence, the materialpoints of the curve form another curve:

ct(σ) = x(c(σ), t) , σ ∈ [0,1]

in Bt.

DefinitionIf c (and, hence, ct) is closed, we call circulation on c in instant t theintegral: ∫

ct

v(x, t) ·dx .

The previous expression is the line integral of a vector: it is the sum oftangential components of vector v.


Kinematics


Theorem (of circulation transport)

If c is a closed material curve,

ddt

∫ct

v(x, t) ·dx =∫

ct

v(x, t) ·dx .

DefinitionA motion preserves the circulation if:

ddt

∫ct

v(x, t) ·dx = 0

for every closed material curve c and for every time instant.

Theorem (Kelvin)

If the acceleration derives from a potential, the motion preserves thecirculation.


Kinematics


DefinitionA curve h in Bt is a line of vortex in instant t if the tangent to the curve atevery point x ∈ h has the direction of the spin of the motion in (x, t).

As the spin axis in (x, t) is the set of vectors e such that W(x, t)e = 0,the curve h is a line of vortex if and only if:

W(h(σ), t)dh(σ)

dσ= 0 , ∀σ ∈ [0,1] .

Theorem (of vorticity transport)

If the acceleration derives from a potential, the lines of vortex aretransported with the motion; i.e., if ct is a line of vortex at an instant t = τ ,then it is a line of vortex at every instant t.

TheoremThe velocity field of an isochoric and irrotational motion is harmonic.


Kinematics


Summary


Irrotational motion: W = 0 ⇐⇒ curlv = 0Gradient of a potential vector field: g = gradα

Cauchy–Lagrange theorem

Circulation of a vector field on a curveTheorem of circulation transport, Kelvin theoremLine of vortex, Theorem of vorticity transport

1 Introduction

2 Kinematics

3 Conservation lawsConservation of massLinear and angular moments. Center of massForces. Stress. Momentum balanceConsequences of balance of momentumConservation of energy


Conservation laws

Conservation of mass


Let B be a body.

DefinitionA distribution of mass of B is a family of smooth density fieldsρf : f(B)−→ IR+ each of them relative to a deformation f, such that:∫

f(P)ρf dV =

∫g(P)

ρg dV = m(P) , ∀P ⊂B , ∀f,g ∈ Lin+ (1)

ρf(x) represents the density at position x = f(p) in the deformation f;(1) is an expression of the conservation of masswe call ρ0 the density field relative to the deformation such thatf(p) = p, ∀p ∈B (reference configuration); thanks to the localizationtheorem,

ρ0(p) = limδ→0

1vol(Ωδ )

∫Ωδ

ρ dV = limδ→0

m(Ωδ )

vol(Ωδ ),

δ centered in p


Conservation laws



Let f be a deformation of B and let F = ∇f. Then, for x = f(p),

ρf(x)detF(p) = ρ0(p) (2)

If we consider a pathline T , we define the density of the motion x as:

ρ : T −→ IR+

(x, t) −→ ρ(x, t) = ρx(·,t)(x)

Thanks to (1), we have:

m(P) =∫

Pt

ρ(x, t)dVx ≡∫

Pt

ρ dV

For every part P ⊂B and every instant t > 0,

ddt

∫Pt

ρ dV = 0


Conservation laws



If F = ∇x is the gradient of deformation of the motion and x = x(p, t),then:

ρ(x, t)detF(p, t) = ρ0(p) (3)

ρ = (ρ0/detF)s (spatial description)Thanks to the regularity lemma, ρ is smooth in T

Theorem (Local form of conservation of mass)ρ +ρ divv = 0ρ ′+div(ρv) = 0

Recall that a motion is isochoric if and only if divv = 0. Thus, amotion is isochoric if and only if ρ = 0


Conservation laws


Control volume

DefinitionWe call control volume in instant t a bounded region R ⊂Bτ , for everyinstant τ in an neigborhood of t.


Conservation laws


Control volume

DefinitionWe call control volume in instant t a bounded region R ⊂Bτ , for everyinstant τ in an neigborhood of t.


Conservation laws


Control volume

Due to the theorem of divergence,∫

Rdiv(ρv)dV =

∫∂R

ρv ·ndA

Thus, ∫R

ρ′(x, t)dVx =

∫R

∂

∂ tρ(x, t)dVx =

ddt

∫R

ρ(x, t)dVx

because R is independent of time.

Theorem (Conservation of mass in a control volume)

Let R be a control volume in instant t. Then,

ddt

∫R

ρ(x, t)dVx =−∫

∂Rρ(x, t)v(x, t) ·n(x)dAx

ρv ·n represents the flux of mass per unit of area; the velocity of massincrease in the volume R is equal to the mass flux entering R throughits boundary


Conservation laws


LemmaLet Φ be a continuous spatial field. Then, for any part P ,∫

Pt

Φ(x, t)ρ(x, t)dVx =∫

PΦm(p, t)ρ0(p)dVp

TheoremLet Φ be a smooth spatial field. Then, for any part P ⊂B,

ddt

∫Pt

Φρ dV =∫

Pt

Φρ dV

this result is equivalent to considering the measure of mass ρ dV as aconstanta particular case is Φ = 1, for which we find again:

ddt

∫Pt

ρ dV = 0


Conservation laws


Summary on mass conservation

Distribution of massConservation of mass:

ρ +ρ divv = 0ρ ′+div(ρv) = 0

Conservation of mass in a control volume:

ddt

∫R

ρ(x, t)dVx =−∫



Conservation laws

Linear and angular moments. Center of mass


Let x be a motion of a body B, and let P ⊂B.

DefinitionWe call linear moment l(P, t) and angular moment a(P, t) (with respectto the origin o) of P at instant t to:

l(P, t) =∫

Pt

vρ dV

a(P, t) =∫

Pt

(r×v)ρ dV

where r : E −→ V is the position vector given by r(x) = x−o

Proposition

For every part P and every time instant t > 0, we have:

l(P, t) =∫

Pt

vρ dV a(P, t) =∫

Pt

(r×v)ρ dV (4)


Conservation laws


Let B be a bounded body (m(B)<+∞).

DefinitionWe call center of mass α(t) of B at instant t the point defined by:

α(t)−o =1

m(B)

∫Bt

rρ dV (5)

the previous definition is independent of the origin o

α(t) =1

m(B)

∫Bt

vρ dV

such that α(t) represents the mean velocity of B

with the previous result,

l(B, t) = m(B) α(t) ; (6)

the linear moment of a body B is equal to the linear moment of aparticle of mass m(B) occupying the position of the center of mass ofB


Conservation laws


Summary on moments

Linear moment of P at instant t:

l(P, t) =∫

Pt

vρ dV

Angular moment of P at instant t:

a(P, t) =∫

Pt

(r×v)ρ dV

For every part P and every time instant t > 0:

l(P, t) =∫

Pt

vρ dV , a(P, t) =∫

Pt

(r×v)ρ dV

Center of mass:α(t)−o =

1m(B)

∫Bt

rρ dV


Conservation laws

Forces. Stress. Momentum balance


In a motion, the mechanical interactions between parts of a body, or betweenthe body and its environment, are described by means of forces. We willconsider three kinds of forces:

contact forces, between parts of a bodycontact forces, exerted on the boundary by its environmentvolume forces, exterted on the interior points of a body by theenvironment.

Hypothesis (Cauchy)

There exists density of surface forces s(n,x, t) defined for each unit vector nat every point (x, t) ∈T (motion trajectory). This field of forces has thefollowing property: if S is an orientable surface in Bt, with (positive) unitnormal n in x, then s(n,x, t) is the force, per unit of area, exerted through Son the material on the negative side of S by the material lying at the positiveside. If, moreover, C is an oriented surface tangent to S in x, with the sameunit normal, then the force per unit of area in x is the same in C and S .


Conservation laws



Conservation laws



Conservation laws


Let P,D ⊂B be two different parts such that St = Pt ∩Dt. The contactforce exerted by D on P at instant t is given by:∫

St

s(nx,x, t)dAx ≡∫

St

s(n)dA ,

where we have denoted nx the unit normal exterior to ∂Pt at x.The force, per unit of area, exerted by then environment on the points of theboundary of B is called surface traction:∫

∂Bt

s(n)dA

The volumen forces, exerted by the environment on the interior points of B,are determined by a vector field:

b : T −→ V

such that b(x, t) is the force, per unit of volume, exerted by the environmenton the position x at instant t. For every part P ⊂B,∫

Pt

b(x, t)dVx ≡∫

Pt

bdV

is the fraction, not due to contact, of the total force exerted by theenvironment on part P .


Conservation laws


Let N be the set of unit vectors.

DefinitionBy a system of forces on B during a motion with trajectory T we mean apair (s,b) of functions:

s : N ×T −→ V b : T −→ V

such that:

s(n,x, t) is smooth with respect to x ∈Bt at every t > 0, ∀n ∈N

b(x, t) is continuous in x ∈Bt for every t > 0.

We call s the surface force and b the volume force.

DefinitionWe define the force f and the moment m in P ⊂B at instant t as:

f(P, t) =∫

∂Pt

s(n)dA+∫

Pt

bdV (7)

m(P, t) =∫

∂Pt

r×s(n)dA+∫

Pt

r×bdV .


Conservation laws


Axiom (momentum balance)

For every part P ⊂B and at every instant t > 0,f(P, t) = l(P, t)m(P, t) = a(P, t)

This axiom assumes the reference to whom magnitudes are expressed,and is called the inertial reference. The change of the reference impliesa different expression of the axiom.As a consequence of (7) and (6), a bounded body verifies:

f(B, t) = m(B) α(t)

Thus, the total force exerted on a finite body equals its mass times theacceleration of the center of mass.


Conservation laws


Thanks to (4), the momentum balance can be written as:∫∂Pt

s(n)dA+∫

Pt

bdV =∫

Pt

vρ dV∫∂Pt

r×s(n)dA+∫

Pt

r×bdV =∫

Pt

r×vρ dV (8)

which is usually known as forces equilibrium and momentumequilibrium.

If we introduce the total volume force:

b∗ = b−ρ v (9)

where −ρ v is the inertial force, and define:

f∗(P, t) =∫

∂Pt

s(n)dA+∫

Pt

b∗ dV

m∗(P, t) =∫

∂Pt

r×s(n)dA+∫

Pt

r×b∗ dV

then the momentum balance reads:

f∗(P, t) = 0 m∗(P, t) = 0


Conservation laws


Let us recall that an infinitesimal rigid displacement is a field w : E −→ Vof the form:

w(x) = w0 +W(x−o) , W ∈ Skw.

Theorem (Principle of Virtual Works)

Let (s,b) be a system of forces on B during a motion. A necessary andsufficient condition for the momentum equilibrium to verify is:∫

∂Pt

s(n) ·wdA+∫

Pt

b∗ ·wdV = 0 ∀P, ∀t

for all infinitesimal rigid displacement w.


Conservation laws


Theorem (Cauchy, existence of the stress tensor)

Let (s,b) be a system of forces for a body B during a motion. A necessaryand sufficient condition for the momentum equilibrium to be verified is theexistence of a spatial tensor field T (called Cauchy stress) such that:

for all vector n ∈N ,s(n) = Tn (10)

T is symmetric

T satisfies the motion equation: divT+b = ρ v


Conservation laws


Let T = T(x, t) be the stress in a given position and time instant, andn ∈N a unit vector. If Tn = σn, n is called principal direction andthe scalar σ is called principal stress. As T is symmetric, there existthree principal directions, orthogonal to each other, and thecorresponding principal stresses.Let be a plane surface with positive normal unit vector n(x). Thesurface force Tn can be decomposed in:

normal force: (n⊗n)Tn = (n ·Tn)nshear force: (I−n⊗n)Tn

Thus, n is a principal direction if and only if the corresponding shearforce vanishes.A fluid does not exerce shear forces: Tn is parallel to n, for every unitvector n, and is an eigenvector of T. Then, T has an only characteristicspace and T =−πI, where π is the pressure of the fluid. The force perunit of area at any surface inside the fluid is −πn.Other important situations:

pure traction (or compression): T = σ(e⊗ e), |e|= 1pure shear: T = τ(k⊗n+n⊗k), |k|= |n|= 1


Conservation laws


Summary on forces

Cauchy hypothesisSystem of forces: volume forces, surface forcesForce and moment in a part P

Momentum balance

Principle of Virtual WorksCauchy theorem: momentum balance and stress tensor T


Conservation laws

Consequences of balance of momentum


According to Cauchy theorem, each system of forces compatible with themomentum balance can be associated a symmetric field of tensors, T,verifying:

s(n) = Tn , ∀n ∈N

divT+b = ρ vReciprocally, given a motion x and a tensor field T, the system of forces iscompletely determined by:

s(n) = Tnb = ρ v−divT

DefinitionWe call dynamical process the pair (x,T), where:

x is a motion

T is a symmetric tensor field defined on the trajectory T of x

T(x, t) is smooth (C 1) in x ∈Bt


Conservation laws


DefinitionIf v and ρ are the velocity and density fields relative to motion x, the triplet(v,ρ,T) is called flux.

to each system of forces compatible with the momentum balance isassociated an only dynamical process (or flux), and viceversa

Theorem (Momentum balance for a control volume)

Let R be a control volume in instant t. In that instant,∫∂R

s(n)dA+∫

RbdV =

ddt

∫R

vρ dV +∫

∂R(ρv)v ·ndA∫

∂Rr×s(n)dA+

∫R

r×bdV =ddt

∫R

r×vρ dV +∫

∂Rr×(ρv)v ·ndA

the total force in the control volume∫

∂Rs(n)dA+

∫R

bdV

is equal to the variation of linear momentum plus the flux ofmomentum through the boundary ∂R


Conservation laws


DefinitionA flux (v,ρ,T) is stationnary if:

Bt = B0 , ∀tv′ = 0 , ρ ′ = 0 , T′ = 0

In this case, we call B0 the flux region.


Conservation laws


Theorem (of the power expended)

At every instant t > 0, and for every part P ⊂Bt,∫∂Pt

s(n) ·vdA+∫

Pt

b ·vdV =∫

Pt

T ·DdV +ddt

∫Pt

v2

2ρ dV (11)

DefinitionThe terms: ∫

Pt

v2

2ρ dV and

∫Pt

T ·DdV

are called, respectively, kinetic energy and elastic power of P in instant t

the power expended on P by the surface and volume forces equals theelastic power plus the variation of kinetic energy


Conservation laws


DefinitionA flux is potential if its velocity v is the gradient of a potential:

v = gradϕ

the potential ϕ ∈ C 2 is a spatial fieldthe potential fluxes are irrotationals: curlv = curlgradϕ = 0reciprocally, if a flux is irrotational and Bt is simply connected at anyinstant t, then the flux is potential

DefinitionA field of volume forces b is conservative if it exists a potential function β

such that:bρ=−gradβ

if the flux is also stationnary,

divT+b = ρ v =⇒ b′ = 0

and we will required, for the force field to be conservative, that β ′ = 0


Conservation laws


Theorem (Bernoulli)

Let (v,ρ,T) be a flux with tension T =−πI and conservative volume forcesof potential β . Then,

if the flux is potential,

grad(

ϕ′+

v2

2+β

)+

1ρ

gradπ = 0

if the flux is stationnary,

v ·grad(

v2

2+β

)+

1ρ

v ·gradπ = 0

if the flux is stationnary and irrotational,

grad(

v2

2+β

)+

1ρ

gradπ = 0


Conservation laws


Summary on momentum balance

Dynamical process, fluxMomentum balance in a control volumeStationary fluxTheorem of power expended: kinetic energy, elastic powerPotential fluxConservative volume forcesBernouilli theorem


Conservation laws

Conservation of energy


A body (and each of its parts) can receive heat at every instant:through its boundary, by conductionat the interior points, by radiation

Analogously to Cauchy hypothesis about contact forces, there is aCauchy hypothesis on heat transmission: we assume a density ofsurface heat g(n,x, t) for every n ∈N and every (x, t) ∈T , with thefollowing property: if S is an oriented surface in Bt, with outwardsunit positive normal n in x, then g(n,x, t) is the heat, per unity of areaand time, flowing from the negative side of S towards the positive sideof S

Moreover, we will assume the existence of a scalar field f (density ofinner heat) defined in T , such that f (x, t) is the heat, per unity ofvolume and time, provided by the environment to point x at time t.


Conservation laws


DefinitionA thermal system of a body B during a motion x (with trajectory T ) is acouple (g, f ) of functions: g : N ×T → R, f : T → R, such that:

g(n,x, t) is, for each n and for each t, a smooth function of x in Bt,

f (x, t) if, for each t, a continuous function of x in Bt.

We call g the surface heat and f the inner heat.

DefinitionWe call supplied heat to part P at time t:

Q(P, t) =−∫

∂Pt

g(n,x, t) dA+∫

Pt

f (x, t) dV ,

DefinitionThe evolution of a body B is said adiabatic if

ddt

Q(B, t) = 0 , ∀t


Conservation laws


The first principle of Thermodynamics, or energy conservation law,stablishes the existence of a scalar field E, called specific total energy,such that the increment of that field is equal to the power of exteriorforces applied to the body plus the increment of the supplied heat.Thanks to the theorem of supplied power and the definition of suppliedheat, we can write the first principle of Thermodynamics in thefollowing way:

Axiom (first principle of Thermodynamics, or energy conservation law)

Let us consider a system of forces (s,b) and a thermal system (g, f ) duringthe motion x of a body B. Then, there exists a scalar field E, called specifictotal energy, such that for every part P and every time instant t, we have:

ddt

∫Pt

ρE dV =∫

∂Pt

s(n) ·v dA+∫

Pt

b ·v dV−∫

∂Pt

g(n) dA+∫

Pt

f dV .

(12)


Conservation laws


Theorem (Cauchy)

Let us assume that the momentum equilibrium stands. A necessary andsufficient condition for the first principle of Thermodynamics to verify is theexistence of a spatial vector field q, called thermal flux, such that:

For every unit vector n

g(n,x, t) = q(x, t) ·n (13)

ρE = div(Tv)+b ·v−divq+ f . (14)

DefinitionWe call specific inner energy the scalar field

e = E− v2

2.


Conservation laws


From previous definition we deduce that, for each part P at instant t,∫Pt

ρE dV =∫Pt

ρe dV +∫Pt

ρv2

2dV

i.e., the total energy is equal to the inner energy plus the kinetic energy.

Proposition

For every part P and time t,

ddt

∫Pt

ρe dV =∫

Pt

T ·D dV−∫

∂Pt

g(n) dA+∫

Pt

f dV (15)

ρ e = T ·D−divq+ f . (16)

(14) and (16) are local versions of the energy conservation law. Theconservative versions are obtained adding these equations the massconservation law multiplied by E and e, respectively:

(ρE)′+div(ρEv) = div(Tv)+b ·v−divq+ f

(ρe)′+div(ρev) = T ·D−divq+ f .


Conservation laws


Axiom (Second principle of Thermodynamics)

There exists a scalar field s, called specific entropy, and a strictly positivescalar field θ , called absolute temperature, such that

ddt

∫Pt

ρs dV ≥−∫

∂Pt

q ·nθ

dA+∫

Pt

fθ

dV (17)

for every part P and time t.

DefinitionWe call entropy of part P at time t

S (P, t) =∫

Pt

ρs dV

while the sum−∫

∂Pt

q ·nθ

dA+∫

Pt

fθ

dV

is called the entropy, per unit of time, supplied to part P at instant t.


Conservation laws


The second principle of Thermodynamics stablishes that entropy cannotconserve, as it can grow more than the supplied by the environment.

Proposition (Clausius–Duhem inequality)

ρ s+div(q

θ

)− f

θ≥ 0 . (18)

Proposition

ρθ s−ρ e+T ·D− 1θ

q ·gradθ ≥ 0 . (19)

DefinitionWe call Helmholtz specific free energy the scalar field ψ defined by:

ψ = e− sθ . (20)

Clearly ψ = e− sθ − sθ . Entering in (19), we get ...


Conservation laws


Proposition

ρsθ +ρψ−T ·D+1θ

q ·gradθ ≤ 0 . (21)

DefinitionA thermodynamic process for a body B with reference density ρ0 is a setof eight functions

x : B×R→ E , T : T → Sym, b : T → V , e : T → R,θ : T → R+, q : T → V , f : T → R, s : T → R,

where x is a motion, T its trajectory, T ∈ C1(T ;Sym), b ∈ C0(T ;V ),e ∈ C1(T ;R), θ ∈ C1(T ;R), q ∈ C1(T ;V ), f ∈ C0(T ;R), s ∈ C1(T ;R),and verifying:

ρ v = divT+bρ e = T ·D−divq+ f

ρ(x, t)detF(p, t) = ρ0(p, t), with x = x(p, t) .


Conservation laws


A thermodynamic process is determined by the functions x, T, e, θ , q and s,as b and f can be deduced from the motion equations and the energyconservation, respectively:

ρ v = divT+bρ e = T ·D−divq+ f ,

where ρ(x, t)detF(p, t) = ρ0(p) for x = x(p, t).

DefinitionA material body is a triple (B,ρ0,C ) consisting of a body B, a massdistribution ρ0 and a family C of thermodynamic processes, called theconstitutive class of the body.


Conservation laws


DefinitionA Coleman–Noll material is a material body the constitutive class of whichconsists of all thermodynamic processes satisfying:

T(x, t) = T(F(p, t),s(x, t),p)+ ˆ(F(p, t),s(x, t),p)(L(x, t))e(x, t) = e(F(p, t),s(x, t),p)

θ(x, t) = θ(F(p, t),s(x, t),p)q(x, t) = q(F(p, t),s(x, t),gradθ(x, t),p)

with x = x(p, t) for some smooth enough functions T, ˆ, e, θ and q.


Conservation laws


Proposition

Let us consider a Coleman–Noll material with constitutive class C andassume there exists a smooth enough function s such that if s ∈ IR, F ∈ Lin+,θ ∈ IR+ and p ∈B, then

s = s(F,θ ,p) ⇐⇒ θ = θ(F,s,p) .

Then, all elements in C satisfy the second principle of thermodynamics ifand only if:

θ(F,s,p) =∂ e∂ s

(F,s,p)

T(F,s,p) =ρ0(p)det(F)

∂ e∂F

(F,s,p)FT

ˆ(F,s,p)(L) ·L≥ 0 (dissipation inequality)

q(F,s,w,p) ·w≤ 0


Conservation laws


Proposition

For a Coleman–Noll material, we have the following form of the energyequation:

ρθ s = `(L) ·D−divq+ f

Moreover, divq = gradθ · qθ+θ div

(qθ

)=⇒ ρ s =

1θ`(L) ·D− 1

θdivq+

fθ=

=1θ`(L) ·D− 1

θgradθ · q

θ−div

(qθ

)+

fθ

ρ s+div(q

θ

)− f

θ=

1θ

[`(L) ·D− 1

θ 2 gradθ ·q]=

1θ

Φ

Φ is the dissipation rate, and it is responsible for the production ofnon–reversible entropy.


Conservation laws


A thermodynamic process is:isentropic if s≡ 0; then:

ρ v−div T−div( ˆ(L)) = b

adiabatic if q≡ 0 and f ≡ 0Eulerian if `≡ 0

Let as assume the existence of a smooth enough function ψ such that thefree energy ψ = e− sθ can be computed as ψ = ψ(F,θ ,p). Then,

∂ψ

∂θ(F,θ ,p) =−s

∂ψ

∂F(F,θ ,p) =

∂ e∂F

(F,s,p) =detFρ0(p)

T(F,s,p)F−T

=⇒ T(F,θ ,p) =ρ0(p)detF

∂ψ

∂F(F,θ ,p)FT


Conservation laws


The mass, momentum and energy conservation equations of aColeman–Noll material are:

ρ +ρdivv = 0

ρ v = div[

ρ∂ψ

∂F(F,θ ,p)FT

]+div(`(D))+b

−ρθ

(∂ψ

∂θ(F,θ ,p)

)·= `(D) ·D−divq+ f

DefinitionThe specific heat at constant deformation is the scalar field defined by:

cF(x, t) = cF(F(p, t),θ(x, t),p)

cF =∂ e∂θ

(F,θ ,p)


Conservation laws


Taking into account that(∂ψ

∂θ(F,θ ,p)

)·=

∂ 2ψ

∂θ 2 θ +∂ 2ψ

∂θ∂F· F =

=− ∂ s∂θ

θ +detFρ0

∂ T∂θ

F−T · F =

=−cF

θθ +

detFρ0

∂ T∂θ·L

then, the energy conservation becomes:

ρcFθ −ρθdetFρ0

∂ T∂θ·L = `(D) ·D−divq+ f

ρcFθ = θ∂ T∂θ·L+ `(D) ·D−divq+ f

ρcFθ = θ∂ T∂θ·D+ `(D) ·D−divq+ f

ρcFθ =

[θ

∂ T∂θ

+ `(D)

]·D−divq+ f


Conservation laws


Summary on thermodynamics

Thermal system, supplied heatFirst Principle of Thermodynamics, energy conservation lawCauchy TheoremSecond Principle of Thermodynamics, entropy, Clausis–DuheminequalityThermodynamic process

Global summary


Conservation laws


Summary on vectors and tensors

Vector space, V

inner productcross product

Tensor: linear transformation from V onto V

associated matrix, components, sum, product by a scalartensor product (ST), transpose (ST ), tensor product of two vectors (a⊗b),inner product (S ·T)determinantprojection on a vector, traceorthogonal tensors, positive definite tensors


Conservation laws


Summary

Eigenvalues and eigenvectors, characteristic spacesSpectral theorem (for symmetric tensors)

Square root theorem: if C ∈ Psym,

∃U ∈ Psym s.t. U2 = C

Polar decomposition theorem: if F ∈ Lin+,

∃U,V ∈ Psym,R ∈ Orth+ s.t. F = RU = VR

Principal invariants of a tensor, Cayley–Hamilton theorem


Conservation laws


Summary on vector and tensor differentiation (I)

Derivative of a function of one variableg : V → V is differentiable in x ∈ V if


Smooth functions

Product rule:

Dh(x)[u] = π(f(x),Dg(x)[u])+π(Df(x)[u],g(x))

Chain rule:

Dh(x)[u] = Df(g(x)) [Dg(x)[u]] , ∀u ∈U


Conservation laws


Summary on vector and tensor differentiation (II)


(∇ϕ)i =∂ϕ

∂xidiv v = ∑

i

∂vi

∂xi∆ϕ = ∑

i

∂ 2ϕ

∂x2i

(∇v)ij =∂vi

∂xj(div S)i = ∑

j

∂Sij


(curlv)i = εijkvk,j

Divergence theoremLocalization theoremStokes theorem


Conservation laws


Summary on deformations

Material points (p) and spatial positions (x), x = f(p)Gradient of deformation: F = ∇fSome kinds of deformations:

translation (constant displacement)homogeneous (constant F)rotation (F ∈ Orth+)stretching (F ∈ Psym)extensionrigidisochoric (detF = 1)

Decomposition: F = RU = VRCauchy–Green strain tensors:

C = U2 = FTF, B = V2 = FFT


Conservation laws


Summary on small deformations

|∇u| ≈ 0 =⇒ B = C = I+∇u+∇uT

Infinitesimal deformation: E = 12

(∇u+∇uT

)Infinitesimal rigid displacement:u on B, with ∇u constant and skew


Conservation laws


Summary on motions

Motion: x : B×IR→ E , x(p, t) = xTrajectory: T = (x, t)/x ∈Bt, t ∈ IRVelocity and acceleration

x(p, t) =∂

∂ tx(p, t) , x(p, t) =

∂ 2

∂ t2 x(p, t)

Spatial and material descriptionVelocity gradient, L = gradvPathline of a material particleStreamline


Conservation laws


Summary on motions

Steady motion: Bt = B0, v′ = 0Rigid motion: the distance between material points is constant alongtimeStretching and spin:

D =12(L+LT) , W =

12(L−LT)

Plane motion


Conservation laws


Summary

Volume of a part P:

vol(Pt) =∫

Pt

dV =∫

Pt

dVx =∫

P(detF)dVp

Theorem of volume transport:

ddt

vol(Pt) =∫

P(detF)· dV =

∫Pt

(divv)dV =∫

∂Pt

v ·ndA

Isochoric motion:ddt

vol(Pt) = 0 , ∀P ⊂B, ∀t

Reynolds transport theorem:

ddt

∫Pt

ΦdV =∫

Pt

(Φ+Φdivv

)dV

=∫

Pt

Φ′ dV +

∫∂Pt

Φv ·ndA


Conservation laws


Summary


Irrotational motion: W = 0 ⇐⇒ curlv = 0Gradient of a potential vector field: g = gradα

Cauchy–Lagrange theorem

Circulation of a vector field on a curveTheorem of circulation transport, Kelvin theoremLine of vortex, Theorem of vorticity transport


Conservation laws


Summary on mass conservation

Distribution of massConservation of mass:

ρ +ρ divv = 0ρ ′+div(ρv) = 0

Conservation of mass in a control volume:

ddt

∫R

ρ(x, t)dVx =−∫



Conservation laws


Summary on moments

Linear moment of P at instant t:

l(P, t) =∫

Pt

vρ dV

Angular moment of P at instant t:

a(P, t) =∫

Pt

(r×v)ρ dV

For every part P and every time instant t > 0:

l(P, t) =∫

Pt

vρ dV , a(P, t) =∫

Pt

(r×v)ρ dV

Center of mass:α(t)−o =

1m(B)

∫Bt

rρ dV


Conservation laws


Summary on forces

Cauchy hypothesisSystem of forces: volume forces, surface forcesForce and moment in a part P

Momentum balance

Principle of Virtual WorksCauchy theorem: momentum balance and stress tensor T


Conservation laws


Summary on momentum balance

Dynamical process, fluxMomentum balance in a control volumeStationary fluxTheorem of power expended: kinetic energy, elastic powerPotential fluxConservative volume forcesBernouilli theorem


Conservation laws


Summary on thermodynamics

Thermal system, supplied heatFirst Principle of Thermodynamics, energy conservation lawCauchy TheoremSecond Principle of Thermodynamics, entropy, Clausis–DuheminequalityThermodynamic process


Inigo Arregui, Jose M. Rodrıguez Seijo([email protected], [email protected])

Department of MathematicsUniversity of A Coruna, Spain

June, 2018

ROMSOC Project

(A brief) Introduction to Continuum Mechanics

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