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CH.5. BALANCE PRINCIPLESContinuum Mechanics Course (MMC) - ETSECCPB - UPC

Overview

Balance Principles

Convective Flux or Flux by Mass Transport

Local and Material Derivative of a Volume Integral

Conservation of Mass Spatial Form

Material Form

Reynolds Transport Theorem Reynolds Lemma

General Balance Equation

Linear Momentum Balance Global Form

Local Form

2

Overview (cont’d)

Angular Momentum Balance Global Spatial

Local Form

Mechanical Energy Balance External Mechanical Power

Mechanical Energy Balance

External Thermal Power

Energy Balance Thermodynamic Concepts

First Law of Thermodynamics Internal Energy Balance in Local and Global Forms

Reversible and Irreversible Processes

Second Law of Thermodynamics Clausius-Planck Inequality

3

Overview (cont’d)

Governing Equations Governing Equations

Constitutive Equations

The Uncoupled Thermo-mechanical Problem

4

5

Ch.5. Balance Principles

5.1. Balance Principles

The following principles govern the way stress and deformation vary in the neighborhood of a point with time.

The conservation/balance principles: Conservation of mass Linear momentum balance principle Angular momentum balance principle Energy balance principle or first thermodynamic balance principle

The restriction principle: Second thermodynamic law

The mathematical expressions of these principles will be given in, Global (or integral) form Local (or strong) form

Balance Principles

REMARKThese principles are always valid, regardless of the type of material and the range of displacements or deformations.

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7


5.2. Convective Flux

The term convection is associated to mass transport, i.e., particle movement. Properties associated to mass will be transported with the mass when

there is mass transport (particles motion)

Convective flux of an arbitrary property through a control surface :

Convection

SS

amountof crossing

unitof timeA

convective transport

A

S

8

Consider: An arbitrary property of a continuum medium (of any tensor order)

The description of the amount of the property per unit of mass, (specific content of the property ) .

The volume of particles crossing a differential surface during theinterval is

Then, The amount of the property per unit of mass crossing the differential

surface per unit of time is:


,t xA

dV dS dh dt dSdm dV dSdt

v nv n

S

dmd dSdt

v n

d Vd S

,t t d t

A

9

inflowoutflow0 v n

0 v n

Consider: An arbitrary property of a continuum medium (of any tensor order) The specific content of (the amount

per unit of mass) .

Then, The convective flux of through a spatial surface, , with unit

normal is:

If the surface is a closed surface, , the net convective flux is:


A

,t x

A

Sn

S st dS v n

V Vt dS

v nS V

= outflow - inflow

Where: is velocityis density

v

A

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Convective Flux

REMARK 1The convective flux through a material surface is always null.

REMARK 2Non-convective flux (advection, diffusion, conduction). Some properties can be transported without being associated to a certain mass of particles. Examples of non-convective transport are: heat transfer by conduction, electric current flow, etc.

Non-convective transport of a certain property is characterized by the non-convective flux vector (or tensor) : , tq x

;s s

dS dS q n v nconvectiveflnon- convectiveflu ux xconvective flux vector

11

non-convective flux vector

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Example

Compute the magnitude and the convective flux which correspond to the following properties:

a) volumeb) massc) linear momentumd) kinetic energy

S

12

Example - Solution

a) If the arbitrary property is the volume of the particles:

The magnitude “property content per unit of mass” is volume per unit of mass, i.e., the inverse of density:

The convective flux of the volume of the particles through the surface is:

VA

1VM

1S s s

dS dS

v n v n

SV

VOLUME FLUX

S st dS v n

13

Example - Solution

b) If the arbitrary property is the mass of the particles:

The magnitude “property per unit of mass” is mass per unit of mass, i.e., the unit value:

The convective flux of the mass of the particles through the surface is:

MA

1MM

1S s sdS dS v n v n

SM

MASS FLUX

S st dS v n

14

Example - Solution

c) If the arbitrary property is the linear momentum of the particles:

The magnitude “property per unit of mass” is mass times velocity per unit of mass, i.e., velocity:

The convective flux of the linear momentum of the particles through the surface is:

vMA

MM

v v

S sdS v v n

SM v

MOMENTUM FLUX

S st dS v n

15

Example - Solution

d) If the arbitrary property is the kinetic energy of the particles:

The magnitude “property per unit of mass” is kinetic energy per unit of mass, i.e.:

The convective flux of the kinetic energy of the particles through the surface is:

212

vMA

2

2

1122

M

M

vv

212S s

dS v v n

S21

2M v

KINETIC ENERGY FLUX

S st dS v n

16

17


5.3. Local and Material Derivative of a Volume Integral


The description of the amount of the property per unit of volume

(density of the property ),

The total amount of the property in an arbitrary volume is:

The time derivative of this volume integral is:

Derivative of a Volume Integral

A

,t xREMARK

and are related through .

V

, V

Q t t dV x

0

limt

Q t t Q tQ t

t

A


Q t

Q t t

0 0

0 0

,

, , , lim lim

[ , , ], , ,

lim lim

V V

t tV

V

t tV V

tt

t t dV t dVQ t t Q t

t dVt t t

t t t dVt t t t

dV dVt t t

x

x xx

x xx x x

Q t

Q t t Control Volume, V

Consider: The volume integral

The local derivative of is:

It can be computed as:

Local Derivative of a Volume Integral

REMARKThe volume is fixed in space (control volume).

, V

Q t t dV x

0

, ,, lim

t

notV V

tV

t t dV t dVt dV

t

x xx

local

derivative

Q t

19

Q t Q t t

Consider: The volume integral

The material derivative of is:

It can be proven that:

Material Derivative of a Volume Integral

REMARKThe volume is mobile in space and can move, rotate and deform (material volume).

,V

Q t t dV x

( ) ( )

0

,

, ,lim

x

x xt

not

V V

V t t V t

t

d t dVdt

t t dV t dV

t

materialderivative

Q t

, x v v vV V V V Vt V

d dt dV dV dV dVdt t dt

dVt

derivative ofderivative of the integralthe integral

derivative ofthe integral

convectivelocal material

20

21


5.4. Conservation of Mass

It is postulated that during a motion there are neither mass sources nor mass sinks, so the mass of a continuum body is a conserved quantity (for any part of the body).

The total mass ofthe system satisfies:

Where:

Principle of Mass Conservation

0t t t M M

tM

,xt

t tVt t dV V V

M

,xt t

t t t tVt t t t dV V V

M

22

Conservation of mass requires that the material time derivative of the mass be zero for any region of a material volume,

The global or integral spatial form of mass conservation principle:

By a localization process we obtain the local or differential

spatial form of mass conservation principle:

Conservation of Mass in Spatial Form

tM

0

lim 0 ,t tV V Vt

t t t dt dV V V tt dt

M M

M

( , ) 0 ,t tV V V V V

d dt dV dV V V tdt dt

x v

( , )( , ) ( , )( )( , ) ( )( , ) 0 ,

xx xv x v x x

for V dV td t tt t V t

dt t

(localization process) CONTINUITY EQUATION

, ( ) V V Vt

d dt dV dVdt dt

x v

23

Consider the relations:

The global or integral material form of mass conservation principle can be rewritten as:

The local material form of mass conservation principle reads :

Conservation of Mass in Material Form

1( )F

vF

ddt

0 0

0

0

0 0 0

( , )1 ( , )( ) ( ( , ) )

, 0 ,

| |( , )

F F XXv F XF

F X

F X

VV V

V V

t

d td d tdV dV t dVdt dt dt t t

t dV V V tt

t

00 ,t

t

V t XF

0 0

1

, 0 t tt tt

t

F X X F X F X

24

( , )tt

X 0dVF

0

FF v

F

ddt

dV dV

25


5.5. Reynolds Transport Theorem


The spatial description of the amount of the property per unit of mass,

The amount of the property in the continuum body at time for an arbitrary material volume is:

Using the material time derivative leads to,

Thus,

Reynolds Lemma

A

,t xA

tV V

Q t dV

t

V V Vt

d ddV dVdt dt

REYNOLDS LEMMA

0ddt v

( ) ( ) V V V Vt

d d d dQ t dV dV dVdt dt dt dt

v v

=0(continuity equation)

d ddt dt


V

dV

VdV

ddt

1e2e3e

The amount of the property in the continuum body at time for an arbitrary fixed control volume is:

Using the material time derivative leads to,

And, introducing the Reynolds Lemmaand Divergence Theorem:

Reynolds Transport Theorem

A

V

Q t dV t

tV V V V

d dV dV dVdt t

v

v nV V V

d dV dV dSdt t

REMARKThe Divergence Theorem:

v n v v nV V V

dV dS dS

, x vV V V Vt

d t dV dV dVdt t

V

dV

n vV

d dVdt

27

V

dV

VdV

ddt

1e2e3e

The eq. can be rewritten as:


V V V

ddV dV dSt dt

v n REYNOLDS TRANSPORT THEOREM

Rate of change of the total amount of . within the

control volume V at time t.A

Rate of change of the amount of in a material volume which instantaneously

coincides with the control volume V.

A

Net outward flux of through the surface that

surrounds the control volume V.

A

v nV V V

d dV dV dSdt t

V

28

V

dV

VdV

ddt

1e2e3e


V V V

ddV dV dSt dt

v n REYNOLDS TRANSPORT THEOREM (integral form)

V V V

ddV dV dSt dt

v n

( ) ( )d V tt dt

v x

REYNOLDS TRANSPORT THEOREM (local form)

( ) [ ( )] V V V V

ddV dV V V tt dt

v

( ) V

dV v( ) V

dVt

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30


5.6. General Balance Equation

Consider: An arbitrary property of a

continuum medium (of any tensor order)

The amount of the property per unit of mass,

The rate of change per unit of time of the amount of in the control volume V is due to:a) Generation of the property per unit mas and time time due to a source: b) The convective (net incoming) flux across the surface of the volume.c) The non-convective (net incoming) flux across the surface of the volume:

So, the global form of the general balance equation is:


A

,t x

v n j nV V V V

dV k dV dS dSt

A A

a cb

A( , )xk tA

( , )j x t

non-convectiveflux vector

A

31

The global form is rewritten using the Divergence Theorem and the definition of local derivative:

The local spatial form of the general balance equation is:


jddt k A A

v n

v j

j

V V

V V

V V V V

dV dSt

dV k dVt

d dV k dV V V tdt

A A

A A

v n j nV V V V

dV k dV dS dSt

A A

REMARKFor only convective transport then and the variation of the contents of in a given particle is only due to the internal generation .

ddt k A( )j 0A

k A

ddt (Reynolds Theorem)


Example

If the property is associated to mass , then: The amount of the property per unit of mass is . The mass generation source term is .

The mass conservation principle states mass cannot be generated.

The non-convective flux vector is . Mass cannot be transported in a non-convective form.

Then, the local spatial form of the general balance equation is:

A A M

1

1 1

( ) ( ) 0ddt t

v

0k M

0j M

( ) 0t

v

00

0jddt k

A A

33

( ) 0d V tt dt

v v x

Two equivalent forms of the continuity equation.

34


5.7. Linear Momentum Balance

Applying Newton’s 2nd Law to the discrete system formed by nparticles, the resulting force acting on the system is:

For a system in equilibrium, :

Linear Momentum in Classical Mechanics

1 1 1

1 1

vR f a

v v

n n ni

i i i ii i i

n ni

i i ii i

dt m mdt

d tdmd mdt dt dt

Resulting forceon the system

P

mass conservation principle: 0idm

dt

0, t R

0d t

dt

P t cntP CONSERVATION OF THE

LINEAR MOMENTUM

35

tP linear momentum

The linear momentum of a material volume of a continuum medium with mass is:

Linear Momentumin Continuum Mechanics

MtV

, , ,v x x v xV

t t d t t dV M

MP

d dVM

1

vn

i ii

t m

P

36

The time-variation of the linear momentum of a material volume is equal to the resultant force acting on the material volume.

Where:

If the body is in equilibrium, the linear momentum is conserved:

Linear Momentum Balance Principle

v RtV

d t d dV tdt dt

P

V V

t dV dS

R b tbody forces

surface forces

0t R 0

d tt cnt

dt

PP

37

The global form of the linear momentum balance principle:

Introducing and using the Divergence Theorem,

So, the global form is rewritten:

Global Form of the Linear Momentum Balance Principle

,R b t vt tV V V V V V V

t

d tdt dV dS dV V V tdt dt

P

P

t n

V V V

dS dS dV

t n

+ ,t t

V V V V

V V V V V

dV dS

ddV dV V V tdt

b t

b v 38 11/11/2015MMC - ETSECCPB - UPC

Applying Reynolds Lemma to the global form of the principle:

Localizing, the local spatial form of the linear momentum balance principle reads:

Local Form of the Linear Momentum Balance Principle

,t tV V V V V V V

d ddV dV dV V V tdt dt

vb v

( , )( , )( , ) ( , ) ( , ) ,

xv xx b x a x x

V dV td tt t t V t

dt

LOCAL FORM OF THE LINEAR MOMENTUM BALANCE

(CAUCHY’S EQUATION OF MOTION)

39

40


5.8. Angular Momentum Balance

Applying Newton’s 2nd Law to the discrete system formed by nparticles, the resulting torque acting on the system is:

For a system in equilibrium, :

Angular Momentum in Classical Mechanics

1 1

1 1 1

vM r f r

rr v v r v

n ni

O i i i ii i

n n ni

i i i i i i i ii i i

dt mdt

dd d dm m mdt dt dt dt

L

0,O t M 0

d tt

dt

L t cntL CONSERVATION OF THE

ANGULAR MOMENTUM

i v

=0

MO

d tt

dtL

41

tLangular momentum

The angular momentum of a material volume of a continuum medium with mass is:

Where is the position vector with respect to a fixed point.

Angular Momentumin Continuum Mechanics

MtV

, , , , ,r x v x r x x v xV

t t t d t t t dV M

ML

d dVM

r

42

The time-variation of the angular momentum of a material volume with respect to a fixed point is equal to the resultant moment with respect this fixed point.

Where:

Angular Momentum Balance Principle

r v Mt

OV V

d t d dV tdt dt

L

OV V

t dV dS

M r b r t

torque due to body forces

torque due to surface forces

43

The global form of the angular momentum balance principle:

Introducing and using the Divergence Theorem,

It can be proven that,

Global Form of the Angular Momentum Balance Principle

tV V V V

ddV dS dVdt

r b r t r v

t n

T T

V V V V

T

V

dS dS dS dS

dV

r t r n r n r n

r

ˆ ;

r r m

m e

T

i i i ijk jkm m

e

REMARK

is the Levi-Civitapermutation symbol.

ijke


Applying Reynolds Lemma to the right-hand term of the global form equation:

Then, the global form is rewritten:

Global Form of the Angular Momentum Balance Principle

ˆ vr b e rijk jk iV V

ddV dVdt

e

r v r v r v

r v vv r r

t tV V V V V

V V

d d ddV dV dVdt dt dt

d d ddV dVdt dt dt

Reynold'sLemma

v

=0

45

Rearranging the equation:

Localizing

Local Form of the Angular Momentum Balance Principle

0 ( , ) ,V V

V V

d dV t dV V V tdt

vr b m m x 0

=0 (Cauchy’s Eq.)

( , ) 0 ; , , 1, 2,3 ; ,m x 0 xi ijk jk tt m i j k V t e

123 23 132 32 23 32

231 31 213 13 31 13

312 12 321 21 12 21

1 1

1 1

1 1

1 0

2 0

3 0

i

i

i

e e

e e

e e

( , ) ( , ) ,Ttt t V t x x x

SYMMETRY OF THE CAUCHY’S STRESS TENSOR

11 12 13

12 22 23

13 23 33


47


5.9. Mechanical Energy Balance

Power, , is the work performed in the system per unit of time.

In some cases, the power is an exact time-differential of a function (then termed) energy :

It will be assumed that the continuous medium absorbs power from the exterior through: Mechanical Power: the work performed by the mechanical actions

(body and surface forces) acting on the medium. Thermal Power: the heat entering the medium.

Power

W t

d tW t

dtE

E

48

The external mechanical power is the work done by the body forces and surface forces per unit of time. In spatial form it is defined as:

External Mechanical Power

e V VP t dV dS

b v t v

d dVdt

rb v

d dSdt

rt v

49

Using and the Divergence Theorem, the traction contribution reads,

Taking into account the identity :

So,


t n

:n

v n vt v v vV V V V

dS dS dV dV

lspatial velocity gradient tensor

l d w : l :d : w

=0

:V V V

dS dV dV

t v v d

50

DivergenceTheorem

Substituting and collecting terms, the external mechanical power in spatial form is,


:

: :

V Ve V

V V V V

V

dV dV

dS

P t dV

ddV dV dV dVdt

v d

t v

b v

vb v d v d

2

v

1 1( v )2 2

d ddt dt

v

v vddt

v

2 21 1( v ) ( v ) 2 2e

V V V V

d dP t dV dV dV dVdt dt

:d :d

Reynold'sLemma

ddt

vb

51

Mechanical Energy Balance. Theorem of the expended power. Stress power

21 v 2

t

e V VV V V

dP t dV dS dV dVdt

b v t v :d

external mechanical power entering the medium stress powerkinetic energy

edP t t Pdt K

REMARKThe stress power is the mechanical power entering the system which is not spent in changing the kinetic energy. It can be interpreted as the work by unit of time done by the stress in the deformation process of the medium.A rigid solid will produce zero stress power ( ) .

K

d 0

P

52

Theorem of the expended mechanical power

The external thermal power is incoming heat in the continuum medium per unit of time.

The incoming heat can be due to: Non-convective heat transfer across the

volume’s surface.

Internal heat sources


( , ) V

t dS

q x n

heat conduction flux vector

incoming heatunit of time

( , )V

r t dV xspecific

internal heat production

heat generated by an internal sourceunit of time

53

The external thermal power is incoming heat in the continuum medium per unit of time. In spatial form it is defined as:

where:is the heat flux per unit of spatial surface area.is an internal heat source rate per unit of mass.


)

( ) eV V V

V

V

dS

dV

Q t r dV dS r dV

n q

q

q n q

,r tx , tq x

54

The total power entering the continuous medium is:

Total Power

21 v 2e e

V V V V Vt

dP Q dV dV r dV dSdt

:d q n

55

56


5.10. Energy Balance

A thermodynamic system is a macroscopic region of the continuous medium, always formed by the same collection of continuous matter (material volume). It can be:

A thermodynamic system is characterized and defined by a set of thermodynamic variables which define the thermodynamic space

The set of thermodynamic variables necessary to uniquely define a system is called the thermodynamic state of a system.

Thermodynamic Concepts

HEAT

MATTER

ISOLATED SYSTEM OPEN SYSTEM

1,2,....n

57

Thermodynamic space

A thermodynamic process is the energetic development of athermodynamic system which undergoes successive thermodynamic states,changing from an initial state to a final state

Trajectory in the thermodynamic space. If the final state coincides with the initial state, it is a closed cycle process.

A state function is a scalar, vector or tensor entity defined univocally as a function of the thermodynamic variables for a given system. It is a property whose value does not depend on the path taken to reach that

specific value.

Thermodynamic Concepts

58

Is a function uniquely valued in terms of the “thermodynamic state” or, equivalently, in terms of the thermodynamic variables

Consider a function , that is not a state function, implicitly defined in the thermodynamic space by the differential form:

The thermodynamic processes and yield:

For to be a state function, the differential form must an exact differential: , i.e., must be integrable:

The necessary and sufficient condition for this is the equality of cross-derivatives:

State Function

1 2, 1 2, , , n

1 1 2 1 2 1 2 2, , f d f d

d

1 1'

1 2

2 2

2 1 2 2

' 2 1 2 2

( , )

( , )

B A

BB

B A

f

f

11 ,...,,...,, 1,...j ni n

j i

ffi j n

d

59

1,..., n

1 2

POSTULATES:1. There exists a state function named total energy of the system, such that its

material time derivative is equal to the total power entering the system:

2. There exists a function named the internal energy of the system, such that: It is an extensive property, so it can be defined in terms of a specific internal energy (or

internal energy per unit of mass) :

The variation of the total energy of the system is:

First Law of Thermodynamics

tE

2

( )( )

1: v 2

:d q ne eV V V V Vt

eQ teP t

d dt P t Q t dV dV r dV dSdt dt

E

tU

,u tx :

V

t u dV U

d d dt t tdt dt dt

E K U

REMARKand are exact differentials,

therefore, so is . Then, the internal energy is a state function.

dKdEd d d U E K

60

Introducing the expression for the total power into the first postulate:

Comparing this to the expression in the second postulate:

The internal energy of the system must be:

Global Form of the Internal Energy Balance

21 v 2

: d q nV V V V Vt

d dt dV dV r dV dSdt dt

E

K

: d q ntV V V V V

d dt u dV dV r dV dSdt dt

UGLOBAL FORM

OF THE INTERNAL ENERGY BALANCE

, external thermal power

eQ tstress power

P t

61

d d dt t tdt dt dt

E K U

Applying Reynolds Lemma to the global form of the balance equation, and using the Divergence Theorem:

Then, the local spatial form of the linear momentum balance principle is obtained through localization as:

Local Spatial Form of the Internal Energy Balance

,du r V tdt

:d q xLOCAL FORM OF THE

ENERGY BALANCE(Energy equation)

( ) q

: d q n

:d q

t t t tV V V V V V V V V V V V

V V V V V V V V

VdV

d d dut u dV dV dV r dV dSdt dt dt

du dV dV r dV dV V V tdt

t

U

U

62

( , )V dV t x

The total energy is balanced in all thermodynamics processes following:

In an isolated system (no work can enter or exit the system)

However, it is not established if the energy exchange can happen in both senses or not:

There is no restriction indicating if an imagined arbitrary process is physically possible or not.

Second Law of Thermodynamics

e ed d dP t Q tdt dt dt

E K U

0e edP t Q tdt

E 0d d

dt dt

U K

0 0d ddt dt

U K0 0d d

dt dt

U K

63

If a brake is applied on a spinning wheel, the speed is reduced due to the conversion of kinetic energy into heat (internal energy). This process never occurs the other way round.

Spontaneously, heat always flows to regions of lower temperature, never to regions of higher temperature.


The concept of energy in the first law does not account forthe observation that natural processes have a preferreddirection of progress. For example:


A reversible process can be “reversed” by means of infinitesimal changes in some property of the system. It is possible to return from the final state to the initial state along the same path.

A process that is not reversible is termed irreversible.

The second law of thermodynamics allows discriminating:

Reversible and Irreversible Processes

REVERSIBLE PROCESS IRREVERSIBLE PROCESS

65

REVERSIBLE

IRREVERSIBLE

IMPOSSIBLE

POSSIBLEthermodynamic processes

POSTULATES:1. There exists a state function denoted absolute temperature,

which is always positive.

2. There exists a state function named entropy, such that: It is an extensive property, so it can be defined in terms of a specific entropy

or entropy per unit of mass :

The following inequality holds true:


, t x

S

s( ) s ( , )

V

S t t dV x

( ) sV V V

d d rS t dV dV dSdt dt

q n

Global form of the 2nd

Law of Thermodynamics

= reversible process> irreversible process

66


( ) sV V V

d d rS t dV dV dSdt dt

q n

Global form of the 2nd

Law of Thermodynamics


eV V

Q t r dV dS

q nrate of the total amount of the entity heat, per unit of time, (external thermal power) entering into the system

eV V

rt dV dS

q n

rate of the total amount of the entity heat per unit of absolute temperature, per unit of time (external heat/unit of temperature power) entering into the system

e t

67

SECOND LAW OF THERMODYNAMICS IN CONTINUUM MECHANICSThe rate of the total entropy of the system is equal o greater than the rate of heat per unit of temperature

Consider the decomposition of entropy into two (extensive) counterparts: Entropy generated inside the continuous medium:

Entropy generated by interaction with the outside medium:


s ,i i

V

S t dV x

s ,e e

V

S t dV x

i e

i e

S t S t S t

dS dS dSdt dt dt

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If one establishes,

Then the following must hold true:

And thus,


0

i e

V V V V

dS dS dS dS r dV dS V V tdt dt dt dt

q n

e

eV V

dS r dV dSdt

q n

i e

V V

edSdt

dS dS dS r dV dSdt dt dt

q n

69

REPHRASED SECOND LAW OF THERMODYNAMICS :The internally generated entropy of the system , , never decreases along time ( )iS t

The previous eq. can be rewritten as:

Applying the Reynolds Lemma and the Divergence Theorem:

Then, the local spatial form of the second law of thermodynamics is:

Local Spatial Form of the Second Law of Thermodynamics

0t t

i

V V V V V V V V V Vt t

d d rs dV s dV dV dS V V tdt dt

q n

0

i

V V V V V V V V

ds ds rdV dV dV dV V V tdt dt

q

0 ,

ids ds r V tdt dt

q x


70

Local (spatial) form of the 2nd

Law of Thermodynamics(Clausius-Duhem inequality)

Considering that,

The Clausius-Duhem inequality can be written as

Local Spatial Form of the Second Law of Thermodynamics

2

1 1

q q q

2

1 1 0ids ds r

dt dt

q q

is s

ilocals i

conds

REMARK (Stronger postulate)Internally generated entropy can be generated locally, , or by thermal conduction, , and both must be non-negative.

iconds

ilocals

Because density and absolute temperature are always positive, it is deduced that , which is the mathematical expression for the fact that heat flows by conduction from the hot parts of the medium to the cold ones.

0 q

1 0rs

q

CLAUSIUS-PLANCK INEQUALITY 2

1 0

q

HEAT FLOW INEQUALITY

71

Substituting the internal energy balance equation given by

into the Clausius-Planck inequality,

yields,

Alternative Forms of the Clausius-Planck Inequality

:notdu u r

dt d q

: 0ilocals s r q

: 0u s d

:r u q d

: 0s u d

Clausius-Planck Inequality in terms of the

specific internal energy

72

The Helmholtz free energy per unit of mass or specific free energy, , is defined as:

Taking its material time derivative,

and introducing it into the Clausius-Planck inequality in terms of the specific internal energy:

Alternative Forms of the Clausius-Planck Inequality

Clausius-Planck Inequality in terms of the

specific free energy

: u s

: u s s u s s

: 0u s d : 0s d

REMARKFor infinitesimal deformation, , and the Clausius-Planck inequality becomes:

d

( ) 0s :

73

74


5.11. Governing Equations

Conservation of Mass. Continuity Equation.

1 eqn.

Governing Equations in Spatial Form

0 v

Linear Momentum Balance. First Cauchy’s Motion Equation.

3 eqns. b v

Angular Momentum Balance. Symmetry of Cauchy Stress Tensor.

3 eqns.T

Energy Balance. First Law of Thermodynamics.

1 eqn.:u r d q

Second Law of Thermodynamics. Clausius-Planck Inequality.

Heat flow inequality2 restrictions

0u s :d

2

1 0

q 8 PDE + 2 restrictions

75

The fundamental governing equations involve the following variables:

At least 11 equations more (assuming they do not involve new unknowns), are needed to solve the problem, plus a suitable set of boundary and initial conditions.

Cauchy’s stress tensor field

Governing Equations in Spatial Form

v

u

s

q

density 1 variable

velocity vector field 3 variables

9 variables

specific internal energy 1 variable

absolute temperature

heat flux per unit of surface vector field 3 variables

1 variable

specific entropy 1 variable19 scalar unknowns

76

Thermo-Mechanical Constitutive Equations. 6 eqns.

Constitutive Equations in Spatial Form

Thermal Constitutive Equation. Fourier’s Law of Conduction. 3 eqns.

State Equations. (1+p) eqns.

(19+p) PDE + (19+p) unknowns

, , v

, ,s s v 1 eqn.

, K q q v

, , 0 1,2,...,iF i p , , ,u f v

KineticHeat

Entropy Constitutive Equation.

set of new thermodynamic variables: . 1 2, ,..., p

REMARK 1The strain tensor is not considered an unknown as they can be obtained through the motion equations, i.e., . v

REMARK 2These equations are specific to each material.

77

Conservation of Mass. Continuity Mass Equation.

1 eqn.

The Coupled Thermo-Mechanical Problem

0 v


3 eqns.


1 eqn.


2 restrictions.

Mechanical constitutive equations. 6 eqns.( ( ), )v

16 scalar unknowns

10 equations

MMC - ETSECCPB - UPC78

The mechanical and thermal problem can be uncoupled if

The temperature distribution is known a priori or does not intervene in the thermo-mechanical constitutive equations.

The constitutive equations involved do not introduce new thermodynamic variables, .

Then, the mechanical problem can be solved independently.

The Uncoupled Thermo-Mechanical Problem

,t x

79

Conservation of Mass. Continuity Mass Equation.

1 eqn.


0 v


3 eqns.


1 eqn.


2 restrictions.

Mechanical problem

Thermal problem

Mechanical constitutive equations. 6 eqns.

10 scalar unknowns

( ( ), v )

80

Then, the variables involved in the mechanical problem are:


Cauchy’s stress tensor field

v

density 1 variable

velocity vector field 3 variables

6 variables

u

s

q

specific internal energy 1 variable

absolute temperature

heat flux per unit of surface vector field 3 variables

1 variable

specific entropy 1 variable

Mechanical variables

Thermal variables

81

82


Summary

The convective flux of through a spatial surface with unit normal is:

Time derivatives of a volume integral:

Summary

tA S n

S st dS v n Where:

is an arbitrary propertyis the description of the amount

of the property per unit of mass. ,t x tA

inflowoutflow0 v n

0 v n

, xnot

V

d t dVdt

materialderivative

, V V V Vt

d t dV dV dVdt t

x v

, t

xnot

V

t dV

localderivative

83

Conservation of mass: the mass of a continuum body is a conserved quantity.

Reynolds Lemma:

Reynolds Transport Theorem:

Summary (cont’d)

Global spatial form

Local spatial form (Continuity Equation)

0V V

d dV dVdt

v

0 v

V V Vt

d ddV dVdt dt

V V V V V

dV dV dS dV dSt

v v n

Divergence Theorem

84

Linear Momentum Balance:

Angular Momentum Balance:

Summary (cont’d)

Global spatial form

Local spatial form (Cauchy’s Equation of Motion)

Global spatial form

Local spatial form(Symmetry of the Cauchy stress tensor)

tV V V V

ddV dS dVdt

b t v

+ ,d V tdt

vb x

,T V t x

tV V V V

ddV dS dVdt

r b r t r v

85

Mechanical Energy Balance:

External Thermal Power:

Total Power

Summary (cont’d)

21 v 2

t

e V VV V V

dP t dV dS dV dVdt

b v t v :d

external mechanical power entering the medium stress powerkinetic energy

K P

eV V

Q t r dV dS

q n is the heat flux per unit of spatial surface area.

is an internal heat source rate per unit of mass. ,r tx

, tq xWhere:

e eP Q

86

First Law of Thermodynamics. Internal Energy Balance.

Second Law of Thermodynamics.

Summary (cont’d)

Global spatial form

Local spatial form (Energy Equation)

: d q ntV V V V V

d dt u dV dV r dV dSdt dt

eQ t P t

,du r V tdt

:d q x

Global spatial form

Local spatial form (Clausius-Duhem inequality)

sV V V

d d rS dV dV dSdt dt

q n

0 ,

ids ds r V tdt dt

q x


1 0qrs

CLAUSIUS-PLANK INEQUALITY

87

Governing equations of the thermo-mechanical problem:

19 scalar unknowns: , , , , , , .

Conservation of Mass. Continuity Mass Equation. 1 eqn.

Summary (cont’d)

0 v

Linear Momentum Balance. First Cauchy’s Motion Equation. 3 eqns.b v

Angular Momentum Balance. Symmetry of Cauchy Stress Tensor. 3 eqns.T

Energy Balance. First Law of Thermodynamics. 1 eqn.:d qu r

Second Law of Thermodynamics. Clausius-Planck Inequality. 2 restrictions

0:du s

2

1 0

q

8 PDE + 2 restrictions

v u q s

88

Constitutive equations of the thermo-mechanical problem:

The mechanical and thermal problem can be uncoupled if the temperature distribution is known a priori or does not intervene in the constitutive eqns. and if the constitutive eqns. involved do not introduce new thermodynamic variables.

Thermo-Mechanical Constitutive Equations. 6 eqns.

Summary (cont’d)

Thermal Constitutive Equation. Fourier’s Law of Conduction. 3 eqns.

State Equations. (1+p) eqns.

(19+p) PDE + (19+p) unknowns

, , v

, ,s s v 1 eqn.

, K q q v

, , 0 1,2,...,iF i p , , ,u f v

Kinetic

Heat

Entropy Constitutive Equation.

set of new thermodynamic variables: . 1 2, ,..., p

89

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