Ccm Info Sheet

CCM Information sheet

1

Basic concepts and definitions Summation convention and index notation Vector:

v = vi ii, meaning v = vi i ii=1

3∑ .

Tensor:

T = Tij iiij, meaning T = Tiji ii jj=1

3∑

i=1

3∑ .

Matrix notation Vector:

v = [v] = [vx , vy , vz]

or in transposed form:

€

vT = [v]T =

vxvyvz

.

Tensor:

€

T = [T ] =

Txx Txy TxzTyx Tyy TyzTzx Tzy Tzz

.

Coordinates’ transformation

Consider two Cartesian coordinate systems Ox1x2x3 and Ox1' x2' x3' having the same pole O.

Cosines of angles between the coordinate axes of the two systems are denoted as aij meaning aij = cos(xi

' , x j ) according to:

€

A =

x1 x2 x3

x1' a11 a12 a13

x2' a21 a22 a23

x3' a31 a32 a33

or A =

a11 a12 a13a21 a22 a23a31 a32 a33

.


2

Position vector of an arbitrary point P can be written as

x = x1i1 + x2i2 + x3i3 = x1' i1' + x2

' i2' + x3

' i3'

,

and the relation between the coordinates of the point as

xi' = aij x j ⇒ xi = a ji x j

' or x' = Ax ⇒ x = AT x '.

Matrix A is called rotation matrix and it is orthogonal, i.e. A-1=AT. Also, det(A)=1. Formal definition of tensors

vi' = aij vj ⇒ vi = aji v j

' or v' = Av ⇒ v = AT v'.

This can be generalised giving an equation for the transformation of tensor T

€

Tij' = aipa jqTpq or T ' = ATAT .

Alternative definition of tensors v = T ⋅u or vi = Tijuj or v = Tu .

Algebraic operations Scalar multiplication

The product of a scalar s and a vector v is a vector

sv = sviii = (svx, svy, svz),

and the product of a scalar s and a tensor T is a tensor

sT = sTij iiij =

€

sTxx sTxy sTxzsTyx sTyy sTyzsTzx sTzy sTzz

.

Addition and subtraction

The sum of two vectors a and b is a vector

a + b = (ai + bi)ii = (ax + bx, ay + by, az + bz),

and the sum of two tensors S and T is a tensor

S + T = (Sij + Tij)iiij =

€

Sxx + Txx Sxy + Txy Sxz + TxzSyx + Tyx Syy + Tyy Syz + TyzSzx + Tzx Szy + Tzy Szz + Tzz

.


3

Multiplication of two vectors 1) Scalar (inner, dot) product of two vectors, whose result is a scalar

a ⋅b = ai bi = ax bx + ay by + az bz.

2) Vector (cross) product of two vectors, whose result is a vector

a x b = eijk aj bk ii =

€

i j kax ay azbx by bz

,

where eijk is a permutation (Levy-Civita) symbol defined as

eijk =(i1 × i2) ⋅i3= 1 if ijk is an even permutation of indices (123, 231, 312);−1 if ijk is an odd permutation of indices (132, 321, 213); 0 if any two indices are equal to each other.

3) Tensor (dyadic) product of two vectors, whose result is a tensor (dyad)

ab = aibj iiij =

€

axbxi i+ axbyij+ axbzik+aybx ji+a yby jj+a ybz jk+azbxki+azbykj+ azbzkk

=

axbx axby axbzaybx ayby aybzazbx azby azbz

.

Multiplication of a vector and a tensor 1) Scalar (inner, dot) product of a vector v and a tensor T, whose result is a vector

v ⋅T = vi Tij ij =

€

(vxTxx + v yTyx + vzTzx)i+(vxTxy + v yTyy + vzTzy)j+(vxTxz + vyTyz + vzTzz)k

=

vxTxx + vyTyx + vzTzxvxTxy + vyTyy + vzTzyvxTxz + v yTyz + vzTzz

.

2) Scalar product of a vector v and a dyad ab, whose result is a vector

v ⋅(ab) = viaibji j ,

and the following is valid

v ⋅(ab) = (v ⋅ a)b .

3) Vector product of a vector v and a tensor T, whose result is a tensor

€

v × T = eijk viTjl i k i l .


4

Multiplication of two tensors 1) Scalar (inner, double-dot) product of two tensors, whose result is a scalar

S:T = Sij Tji = SxxTxx + SxyTyx + SxzTzx +

SyxTxy + SyyTyy + SyzTzy +

SzxTxz + SzyTyz + SzzTzz.

2) Vector product of two tensors, whose result is a vector

€

S × T = eijk S jlTlk i i.

3) Scalar product of two tensors, whose result is a tensor S ⋅T = SikTkj iiij.

Identity tensor

Tensor I for which v = I ⋅v = v ⋅ I ,

is called identity tensor and is equal to:

Ι =δij iiij =

€

1 0 00 1 00 0 1

.

Symbol δij is known as the Kronecker delta, and is defined as

€

δij = ii ⋅ i j =1 if i = j0 if i ≠ j

.

Trace and determinant of a tensor

The sum of the diagonal elements of the tensor T, for which the symbol tr is used, is termed the trace of the tensor T tr T = Tii = Txx + Tyy + Tzz,

while the symbol det denotes the determinant of the tensor T, which is defined as

det (T) = det (Tij) =

€

Txx Txy TxzTyx Tyy TyzTzx Tzy Tzz

.


5

Symmetric and anti-symmetric tensors

A tensor whose elements are symmetric with respect to the main diagonal, i.e.

T = TT or Tij = Tji (i, j = 1,2,3)

is called a symmetric tensor, and a tensor where

T = – TT or Tij = − Tji (i, j = 1,2,3)

is called an anti-symmetric (or skew-symmetric) tensor. Tensor TT = Tjiiiij is called the transpose of T. Every tensor may be expressed as a sum of one symmetric and one anti-symmetric tensor in one and only one way

T = 12(T +TT )+ 1

2(T −TT )= [ 1

2(Tij + Tji )+ 1

2(Tij − Tji ) ] iiij.

Ts Ta Tijs Tij

a Spherical tensor and tensor deviator

It is sometimes convenient to split a tensor into two parts, one called the spherical tensor and the other the tensor deviator:

T = Ts + Td = TmΙ + (T – TmΙ) = [Tmδij + (Tij – Tmδij)] iiij =

€

Tm 0 0 0 Tm 0 0 0 Tm

+

€

Txx − Tm Txy Txz Tyx Tyy −Tm Tyz Tzx Tzy Tzz −Tm

,

Spherical tensor Tensor deviator where

Tm = 13

tr T = 13

Tii = 13(Txx + Tyy + Tzz ) .

Vector and tensor invariants

A vector has only one invariant, its length or intensity

|v| = v ⋅v = vivi = vx2 + vy

2 +vz2 .

A (symmetric) tensor, has three invariants. They are related to the so-called characteristic equation det (T – T I) = det (Tij – Τ δij) = 0

which can be written in the form

Τ3 – ITΤ2 – IITΤ – IIIT = 0,


6

where IT = tr T= Tii,

IIT = 12(T : T − tr2T) = 1

2[TijTji − (Tii )

2 ] ,

IIIT = det (T )=

€

1

6eijk e pqrTipTjqTkr

are the first, second and third invariant of the tensor T. Principal tensor components and principal directions

T(1), T(2), T(3) are termed the principal tensor components. For each T(k) (k = 1, 2,3) equation

(T − T(k)I) ⋅l(k) = 0 or (Tij − T(k)δij ) ⋅l j

(k ) = 0 (k = 1,2,3; no summation on k)

has non-trivial roots (l1(k) ,l2

(k ),l3(k) ) which represent components of the three principal

directions. If coordinate axes are taken along the principal directions, the tensor T becomes a diagonal tensor

T =

€

T(1) 0 0 0 T(2) 0 0 0 T(3)

.

Reduction of a cubic to ‘standard’ form: for the equation:

€

x 3 + ax 2 + bx + c = 0

let

€

x = xo −a3

then the equation becomes:

€

x03 = Ax0 + B

where

€

A =a2

3− b and B =

ab3−

2a3

27− c

Inverse and orthogonal tensors

Tensor S for which

S ⋅T = T ⋅S = I or SikTkj = TikSkj = δ ij .

is called inverse tensor of tensor T and is denoted as S = T-1. Tensor T is called direct tensor of tensor T-1. The sufficient condition for the existence of an inverse tensor is det(T)≠0. Tensor T for which T-1= TT , i.e. for which

T ⋅TT = TT ⋅T = I or TikTjk = TkiTkj =δ ij ,

is called orthogonal tensor.


7

Differential operations Gradient of a scalar field

Each scalar field can be associated with a vector field of its gradient

grad s = ∂s∂x j

i j =∂s∂xi+ ∂s

∂yj + ∂s

∂zk .

Scalar multiplication of gradient by unit vector lo = l/|l| gives derivative of the scalar field s in

direction l defined by vector l

∂s∂l= grad s ⋅lo.

Divergence, curl and gradient of a vector field

Each vector field can be associated with a scalar field of its divergence

div v =∂vj∂x j

=∂vx∂x

+∂vy∂y

+∂vz∂z

,

with a vector field of its curl:

curl v = eijk ∂vk∂x j

ii =

€

i j k∂

∂x ∂∂y

∂∂z

vx v y vz

,

as well as a tensor field of its gradient

grad v = ∂v∂xi + ∂v

∂yj+ ∂v

∂zk =

∂v j∂xi

ii i j =

∂vx∂xii+

∂vy∂xij+ ∂vz

∂xik +

∂vx∂yji+

∂vy∂yjj+ ∂vz

∂yjk+

∂vx∂zki +

∂vy∂zkj+ ∂vz

∂zkk

, or

€

grad v =

∂v x∂x

∂v y∂x

∂vz∂x

∂v x∂y

∂v y∂y

∂vz∂y

∂v x∂z

∂v y∂z

∂vz∂z

.


8

Divergence of a tensor field

Each tensor field can be associated with vector field of its divergence,

div T = ∂Tji∂x j

ii =

€

∂Txx∂x

+∂Tyx∂y

+∂Tzx∂z

i +

∂Txy∂x

+∂Tyy∂y

+∂Tzy∂z

j +

∂Txz∂x

+∂Tyz∂y

+∂Tzz∂z

k

€

=

∂Txx∂x

+∂Tyx∂y

+∂Tzx∂z

∂Txy∂x

+∂Tyy∂y

+∂Tzy∂z

∂Txz∂x

+∂Tyz∂y

+∂Tzz∂z

.

Del (nabla) operator Hamilton (del or nabla) operator for the Cartesian coordinates:

∇(...) = ∂(...)∂x j

i j =∂(...)∂x

i+ ∂(...)∂y

j+ ∂(...)∂z

k .

This operator has the properties of both a vector and a linear differential operator. It follows that:

grad s = ∇s, grad v = ∇ v,

div v = ∇ ⋅v, div T = ∇ ⋅T,

curl v = ∇ x v.

Some useful relationships

€

grad f (s) = f ' (s) grad s = f ' (s) ∇sdiv (sv) = s div v + v ⋅ grad s = s∇⋅ v + v ⋅ ∇s

div (T ⋅ v) = div T ⋅ v + T: grad v = (∇⋅T) ⋅ v + T: (∇v)div (ab) = b div a + a ⋅ grad b = b(∇ ⋅ a) + a ⋅ (∇b)curl (sv) = s curl v + grad s× v = s∇ × v +∇s× v

curl (a × b) = b ⋅ grad a + a div b −a ⋅ grad b −b div a


9

Integral theorems

In the field theory flux of vector field through a surface S is defined as

€

˙ V = v ⋅n dSS∫ = v ⋅ dS

S∫ ,

where n is the unit vector of the outer normal to the surface S, and circulation of the vector field along a closed curve C as

€

Γ = v ⋅ dxC∫ , where d x = dxi + dyj+ dzk ,

In connection with the above quantities one can prove the following two very important theorems:

1) Gauss' theorem Gauss' (Divergence) theorem relates the source of a field in a region V with the flux of that field through the boundary surface of the region S

€

v ⋅n dS = div v dVV∫

S∫ , or more generally ηn dS = ∇η dV

V∫

S∫ ,

where η = s, v or T, and ∇η stands for ∇s, ∇ ⋅v or ∇ ⋅T, respectively. 2) Stokes' theorem

According to Stokes' theorem, the circulation of a vector field along a regular closed curve C is equal to the flux of the curl of that vector field through an arbitrary simply connected (one-sided) surface S bounded by the curve C

€

v ⋅ dxC∫ = curl v ⋅n dS

S∫ .


10

Stresses Body forces The concept of body force fb in point P of a continuum element understands the limit of the ratio of the resultant force ΔFb acting on all points of that element and the mass of that element Δm = ρΔV, when its volume ΔV tends to zero:

fb =

€

limΔm→0

ΔFbΔm

= limΔV→ 0

ΔFbρΔV

.

Consequently, the total body force acting on a given finite mass of continuum is

Fb =

€

fbdmm∫ =

€

ρfbdVV∫ .

Surface forces Analogously, the surface force in a point P is defined as

fs =

€

limΔS→0

ΔFsΔS

,

where ΔFs is the resultant of all forces acting on the surface ΔS. According to the Cauchy

principal, this limit tends to a finite value dFs/dS while at the same time the moment of dFs around the point P tends to zero. The total force acting on a finite surface is:

Fs =

€

fsdSS∫ .

Stress tensor

€

σ =

σ xx σxy σxz

σ yx σyy σyz

σ zx σ zy σzz

.

x

y

z

σxz

σxy σxx

σyy

σyx

σyz

σzz

σzx σzy

σyy σyx

σyzi

jk i

j ki

j

ki

j

k

--

-

Normal and tangential stresses


11

Deformation and flow Configurations

ξ

ξ

ξ

x

x

x

1

2

3

1

2

3

O O

PP

i 1

i 2i 3

i 1 i 2

i 3xξ

0

(0)

(0)(0)0

Initial (reference, undeformed) Current (deformed) configuration configuration

ξ = ξ1i1(0) +ξ2i2

(0) +ξ3i3(0) , x = x1i1 + x2i2 + x3i3 .

Material and spatial description

The motion of body can be described by: x = x(ξ,t) or xi = xi (ξk ,t) .

If the Jacobian of this transformation

J = det ∂xi∂ξ j

≠ 0 ,

then the inverse transformation exists such that: ξ = ξ(x,t) or ξ i = ξ i (xk ,t) .

Coordinates ξi can be regarded as fixed and are called material or Lagrangian coordinates.

Coordinates xi are called spatial or Eulerian coordinates. Displacement vector

u = P0

P = ui

(0)ii(0) = uiii ,

or in terms of x, ξ , and vector O0 O :

u = x −ξ +O0

O .


12

The coordinate systems O0ξ1ξ2ξ3 and Ox1x2x3 are normally chosen to coincide to each other, and then: u = x −ξ .

In the Lagrangian description, displacement u is represented as a function of ξ and t:

u(ξ, t) = x(ξ,t) − ξ ,

and in the Eulerian description as function of x and t: u(x, t) = x − ξ(x, t) .

Deformation Gradients

Deformation gradients

ξ

ξ

ξ

x

x

x

1

2

3

1

2

3

P Px

ξ

QQ

o

o

dxξξ+dξd

u

u+du

Deformed and undeformed configurations

dx = F ⋅dξ or dxi =∂xi∂ξ j

dξ j ,

and:

dξ =H ⋅ dx or dξi =∂ξi∂x j

dx j ,

where

F = (gradξx )T or Fij =

∂xi∂ξ j

,

is the material deformation gradient, while

H = (grad xξ)T or Hij =∂ξi∂x j

,

is the spatial deformation gradient.


13

F ⋅H =H ⋅F = I ,

and H = F-1 is inverse of the tensor F.

det(F) = 1det(H)

= J .

Displacement gradients du = dx − dξ = F⋅ dξ - I ⋅dξ = (F − I) ⋅dξ = J ⋅dξ ,

or du = dx − dξ = I ⋅dx -H ⋅dx = (I−H) ⋅dx = K ⋅dx ,

where:

J = (gradξu )T = [gradξ (x − ξ) ]T = F− I or Jij =

∂ui∂ξ j

=∂xi∂ξ j

−δ ij ,

is the material displacement gradient, and

K = (grad xu )T = [gradx (x − ξ) ]T = I−H or Kij =∂ui∂x j

= δ ij−∂ξi∂x j

,

is the spatial displacement gradient. Deformation tensors

Green and Cauchy deformation tensors

(dx)2 = dxT ⋅dx = (F⋅ dξ)T ⋅ (F ⋅dξ) = (dξT ⋅FT ) ⋅(F ⋅dξ) ,

or

(dx)2 = dξT ⋅G⋅ dξ or (dx)2 =∂xk∂ξi

∂xk∂ξ j

dξidξ j ,

where

G = FT ⋅F or Gij =∂xk∂ξi

∂xk∂ξ j

is the Green deformation tensor. Similarly

(dξ)2 = dξT ⋅ dξ = (H ⋅dx)T ⋅ (H ⋅dx) = (dxT ⋅HT ) ⋅(H⋅ dx) ,

or

(dξ)2 = dxT ⋅C⋅ dx or (dξ)2 =∂ξk∂xi

∂ξk∂x j

dxidx j ,


14

where

C = HT ⋅H or Cij =∂ξk∂xi

∂ξk∂x j

is the Cauchy deformation tensor. Similar to tensors F and H, tensors G and C are inverse of each other, i.e. C = G-1.

Lagrange and Euler tensors of finite strain

(dx)2 − (dξ)2 = dξT ⋅G⋅ dξ − dξT ⋅ I ⋅dξ = dξT ⋅(G − I) ⋅dξ = dξT ⋅ 2L ⋅ dξ ,

where:

L =12

(G− I) =12

(FT ⋅F− I) or Lij =12

∂xk∂ξi

∂xk∂ξ j

−δ ij

,

is Lagrange or Green finite strain tensor. Similarly

(dx)2 − (dξ)2 = dxT ⋅I ⋅ dx − dxT ⋅C ⋅dx = dxT ⋅(I −C) ⋅dx = dxT ⋅2E ⋅dx ,

where:

E =12

(I −C) =12

(I −HT ⋅H) or Eij =12δ ij −

∂ξk∂xi

∂ξk∂x j

,

is Euler or Almansi finite strain tensor.

F = J + I, and

€

L =12J + I( )T ⋅ J + I( ) − I( ) =

12J + JT + JT ⋅ J( ) , or

€

L =12gradξu( )

T+gradξu+ gradξu ⋅ gradξu( )

T[ ] , or

€

Lij =12∂u j

∂ξ i+∂ui∂ξ j

+∂uk∂ξ i

∂uk∂ξ j

.

Similarly,

€

E =12I− I−K( )T ⋅ I−K( )( ) =

12K +KT −KT ⋅K( ) , or

€

E =12gradxu( )T +gradxu− gradxu ⋅ gradxu( )T[ ] , or

€

Eij =12∂u j

∂xi+∂ui∂x j

−∂uk∂xi

∂uk∂x j

.


15

Small deformations For dx ≈ dξ , L and E reduce to the Cauchy small (or infinitesimal) strain tensor:

Ls = Es = 12gradu( )T + gradu[ ] , or

€

lij = εij =12

∂ui∂ξ j

+∂u j∂ξi

=12

∂ui∂x j

+∂u j∂xi

,

where the grad operator can be applied in either Lagrangian or Eulerian coordinates, and lower case letters l and ε are used to denote small strains.

If u = u(x,t) is expanded in the vicinity of P0 by using the Taylor series expansion and neglecting the higher order terms, one gets:

u = u0 + du = u0 + grad u( )0T ⋅dx or ui = ui0 +

∂ui∂x j

0dx j .

The tensor grad u( )0T can be expressed as the sum of its symmetric and anti-symmetric parts

giving:

u = u0 +

12

grad u( )T + grad u[ ]0 ⋅dx +12

grad u( )T − grad u[ ]0 ⋅dx= u0 + E0

s ⋅dx +Ω0 ⋅ dx,

where E0s is Cauchy small strain tensor and

Ω =12

grad u( )T − grad u[ ] or Ωij =12

∂ui∂x j

−∂uj∂xi

,

is linear rotation tensor, and:

€

K = grad u( )T

= Es +Ω or Kij = εij +Ωij .

If Ω is expressed in term of its rotation vector

€

ω =12

curl u or ω i =12eijk∂uk

∂x j

=12eijkΩkj ,

then:

u = u0 +ω× dx + E0s ⋅ dx .

Rigid body Deformation motion


16

Motion and flow Rate of deformation and material derivative

The velocity of continuum in a given point of space:

dxdt

=DxDt

= v(x, t) .

In Lagrangian coordinates

DFDt

=DF1(ξ,t)Dt

=∂F1(ξ, t)∂t

,

while in Eulerian coordinates

DFDt

=DF2 (x, t)Dt

=∂F2∂t

+ gradF2 ⋅DxDt

,

or

DFDt

= ∂F∂t

+ v(x, t) ⋅ gradF .

Local Material change change

Velocity and acceleration

The velocity can be expressed via displacement vector:

v = DxDt

=D(ξ + u)Dt

=DuDt

.

In Lagrangian coordinates u = u(ξ ,t) so

v(ξ,t) = ∂x(ξ, t)∂t

=∂u(ξ,t)∂t

.

The acceleration in Lagrangian coordinates is given as:

a(ξ,t) = DvDt

=∂v∂t

=∂ 2x∂t 2

=∂2u∂t 2

,

while in Eulerian coordinates:

a(x,t) = DvDt

=∂v∂t+v ⋅grad v .

Material derivative of a volume integral. Reynolds transport theorem.

DDt

FdVV∫ =

∂F∂tdV + Fv ⋅ndS

S∫

V∫ .


17

Rate of deformation

x

x x1 2

3

P

P

xo

o

dx

x

vo

v

vd

Velocities of the continuum particles

v = v0 + dv = v0 + grad v( )0T ⋅ dx or vi = vi0 +

∂vi∂x j

0dx j ,

where

Y = grad v( )T or Yij =∂vi∂x j

,

is the spatial velocity gradient.

The tensor grad v( )0T can be decomposed into the sum of its symmetric and anti-symmetric

parts giving:

€

v = v0 +12

grad v( )T + grad v[ ]0⋅ dx +

12

grad v( )T −grad v[ ]0⋅ dx = v0 +D0 ⋅ dx +V0 ⋅ dx ,

where

D =12

grad v( )T + grad v[ ] or Dij =12

∂vi∂x j

+∂vj∂xi

,

is the deformation rate tensor, and

V =12

grad v( )T −grad v[ ] or Vij =12

∂vi∂x j

−∂v j∂xi

,

is the vorticity or spin tensor, and: Y = grad x v( )T =D +V or Yij = Dij + Vij .

If V is expressed in term of its vorticity vector

€

˙ ω =12

curl v or ˙ ω i =12eijk∂vk

∂x j

=12eijkVkj ,

then: v = v0 + ˙ ω × dx + D0 ⋅dx .

Rigid body Deformation motion


18

Fundamental laws of continuum mechanics Mass conservation

For a system (material form):

DDt

ρdV = 0V∫ .

The rate of change of the mass in the system

The differential form of the mass conservation law or the continuity equation :

ρJ = ρ0,

or:

DDt

ρJ( ) = 0 or DDt

ρdV( ) = 0 ,

since dV = J dV0 and dV0 = const.

For a control volume (spatial form):

∂ρ∂t

dV + ρv ⋅ndSS∫

V∫ = 0 ,

Change of mass Mass flux in the CV through CS

The differential form:

∂ρ∂t

+ div ρv( ) = 0 .

Local Convective change change

or:

∂ρ∂t

+ v ⋅ grad ρ + ρdiv v = DρDt

+ ρdiv v = 0 .

Conservation of linear momentum (1st Euler’s law, 2nd Newton’s law)


DDt

ρvdVV∫ = ρfbdV + σ ⋅ndS

S∫

V∫ .

Rate of change of Total body Total surface total linear momentum force force


19

The differential form of the momentum equation is:

ρDvDt

= ρfb + div σ .

Rate of change of Body Surface linear momentum force force


∂ ρv( )∂t

dV + v ρv ⋅n( )dSS∫

V∫ = ρfbdV + σ ⋅ndS

S∫

V∫ ,

Change of mom. Flux of mom. Total body Total surface in CV across CS force force

and:

∂ ρv( )∂t

+ div ρvv( ) = ρfb + div σ .

Local Convective. Body Surface change change force force

In the index form in Cartesian coordinate system:

ρDviDt

= ρfb,i +∂σ ji∂x j

,

which can be written in the expanded form as:

ρDvxDt

= ρfb, x +∂σxx∂x

+∂σ yx∂y

+∂σzx∂z

,

ρDvyDt

= ρfb, y +∂σ xy∂x

+∂σ yy∂y

+∂σ zy∂z

,

ρDvzDt

= ρfb, z +∂σxz∂x

+∂σ yz∂y

+∂σ zz∂z

.

Energy conservation law (1st law of thermodynamics)


DDt

ρedVV∫ = − q ⋅ndS

S∫ + hdV +

V∫ ρfb ⋅ vdV + σ ⋅v( ) ⋅ndS

S∫

V∫ ,

Rate of change of Heat Heat Work done (power) total energy delivered source

e = u +12v ⋅ v = u + v

2

2.


20

The differential form of the energy balance equation takes the form:

ρDeDt

= − divq + h + ρfb ⋅v + divσ ⋅v( ) .

Rate of change Heat Heat Work by Work by of energy flux source body forces surface forces


∂ ρe( )∂t

dVV∫ + e ρv ⋅n( )dS

S∫ = − q ⋅ndS

S∫ + hdV +

V∫ ρfb ⋅vdV + σ⋅ v( ) ⋅ndS

S∫

V∫ ,

Change of Energy flux Heat Heat Work done (power) energy in CV through CS delivered source

and its differential form is:

€

∂ ρe( )∂t

+ div ρev( ) = −divq + h + ρfb ⋅ v + divσ ⋅ v( )

Local Convective Heat Heat Work by Work by change change flux source body forces surface forces

Equation of mechanical energy

ρDDt

v2

2

= ρfb ⋅v + divσ( ) ⋅ v.

Change of Work by Mechanical Mech. energy body forces work

Heat equation

ρDuDt

= −divq + h + σ : gradv .

Total change Heat Heat Deformation of internal energy flux source work

The index notation takes the form:

ρDuDt

= −∂qj∂x j

+ h +σij∂v j∂xi

,

and its expanded form is:

€

ρDuDt

= −∂qx∂x

+∂qy∂y

+∂qz∂z

+ h +σ xx

∂vx∂x

+ σ yx∂vx∂y

+ σzx∂vx∂z

+

σ xy∂vy∂x

+ σ yy∂vy∂y

+ σzy∂vy∂z

+

σ xz∂vz∂x

+ σ yz∂vz∂y

+σ zz∂vz∂z


21

Basic constitutive laws Linear elastic solids (Hooke’s law):

€

σ = 2µEs + λ trEs( ) − 3λ + 2µ( )α T −T0( )[ ]I

Newtonian viscous fluid (Stokes law):

€

σ = −pI− 23

µdivvI +2µD

Fourier’s law of heat conduction (valid for solids and fluids):

€

q = −k ⋅ gradT


22

Mathematical models Summary for thermo-elastic solids and Newtonian fluids Equations governing momentum and energy transport in a continuous media (for thermo-elastic solids and Newtonian fluids, respectively), can all be written in the form of a following generic transport equation:

∂∂t

ρBφdV + ρBφv ⋅ndSS∫

V∫ = Γφgradφ ⋅ndS

S∫ + qφS ⋅ndS

S∫ + qφVdV

V∫ ,

while the continuity equation is combined with momentum equation to obtain an equation for pressure (in fluids usually). Here, φ stands for the transported property, i.e. displacement u, velocity v or temperature T. The meaning of the properties Bφ and Γφ are given in the Table below. The term qφS contains parts of the mass or heat flux vector or the stress tensor, which are not included in Γφgradφ , while qφV contains the volumetric source terms.

The meaning of various terms in the generic transport equation φ Bφ Γφ qφV qφS

T cT k σ : gradv+ h 0

u ∂u∂t

µ ρfb µ gradu( )T + grad λdivu − 3λ + 2µ( )α T −T0( )[ ]I

v v µ ρfb − pI+ µ gradv( )T

In case of thermo-elastic solid, the continuity equation does not have to be considered (it serves for determining density which is constant). In most practical applications it can be assumed that not only the strains are small but also velocities, thus the (non-linear) convection term ρvv , as well as the mechanical energy dissipation term σ : gradv can be neglected.

Ccm Info Sheet

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