CH.5. BALANCE PRINCIPLES
Continuum 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 5.1. Balance Principles
Ch.5. Balance Principles
5 Balance Principles
The following principles govern the way stress and deformation vary in the neighborhood of a point with time. REMARK The conservation/balance principles: These principles are always Conservation of mass valid, regardless of the type of material and the range of Linear momentum balance principle displacements or deformations. 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
6 5.2. Convective Flux
Ch.5. Balance Principles
7 Convection
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 transport
Convective flux of an arbitrary property A through a control surface S :
amountofA crossing S S unitof time
8 Convective Flux or Flux by Mass Transport
Consider: An arbitrary property A of a continuum medium (of any tensor order) The description of the amount of the property per unit of mass, x , t (specific content of the property A ) . The volume of particles dV crossing a differential surface dS during the interval tt , dt is dV dS dhvn dt dS dm dVvn dSdt
Then, The amount of the property per unit of mass crossing the differential dm surface per unit of time is: ddS vn S dt
9 Convective Flux or Flux by Mass Transport
Consider: An arbitrary property A of a continuum medium (of any tensor order) inflow vn 0 outflow The specific content of A (the amount vn0 per unit of mass) x , t .
Then, The convective flux of A through a spatial surface, S , with unit normal n is: v is velocity tdS vn Where: S s is density
If the surface is a closed surface, SV , the net convective flux is: tdS vn = outflow - inflow V V
10 MMC - ETSECCPB - UPC 11/11/2015 Convective Flux
REMARK 1 The convective flux through a material surface is always null.
REMARK 2 Non-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) qx , t : non- convectiveflux qndS; convectiveflux vn dS ss non-convective flux convective vector flux vector
11 MMC - ETSECCPB - UPC 11/11/2015 Example
Compute the magnitude and the convective flux S which correspond to the following properties: a) volume b) mass c) linear momentum d) kinetic energy
12 tdS vn Example - Solution S s
a) If the arbitrary property is the volume of the particles: A V
The magnitude “property content per unit of mass” is volume per unit of mass, i.e., the inverse of density:
V 1 M
The convective flux of the volume of the particles V through the surface S is:
1 vn dS vn dS VOLUME FLUX S ss
13 tdS vn Example - Solution S s
b) If the arbitrary property is the mass of the particles: A M
The magnitude “property per unit of mass” is mass per unit of mass, i.e., the unit value:
M 1 M
The convective flux of the mass of the particles M through the surface S is:
1 vn dS vn dS MASS FLUX S ss
14 tdS vn Example - Solution S s
c) If the arbitrary property is the linear momentum of the particles: A M v
The magnitude “property per unit of mass” is mass times velocity per unit of mass, i.e., velocity:
M v v M
The convective flux of the linear momentum of the particles M v through the surface S is:
vvndS MOMENTUM FLUX S s
15 tdS vn Example - Solution S s
d) If the arbitrary property is the kinetic energy of the particles: 1 A M v2 2 The magnitude “property per unit of mass” is kinetic energy per unit of mass, i.e.: 1 M v2 1 2 v2 M 2 1 The convective flux of the kinetic energy of the particles M v 2 through the surface S is: 2 1 vvn2 dS KINETIC ENERGY FLUX S s 2
16 5.3. Local and Material Derivative of a Volume Integral Ch.5. Balance Principles
17 Derivative of a Volume Integral
Consider: An arbitrary property A of a continuum medium (of any tensor order) The description of the amount of the property per unit of volume (density of the property A ), x , t REMARK and are related through . The total amount of the property in an arbitrary volume V is: Qt Qt x, t dV V Qtt The time derivative of this volume integral is: Qt t Qt Qt lim t 0 t
18 MMC - ETSECCPB - UPC 11/11/2015 Local Derivative of a Volume Integral
Consider: Qt The volume integral Qt x, t dV V Qtt Control Volume, V The local derivative of Qt is: xx,,ttdV tdV local not REMARK x,limtdV VV t 0 derivative t V t The volume is fixed in space (control volume). It can be computed as: xx,ttdV , tdV Qt t Qt VV x,tdV lim lim tt00 ttV t [,xxtt ,] tdV xx,,tt t x , t limV limdV dV tt00ttt VV x,t t 19 Material Derivative of a Volume Integral
Consider: The volume integral Qt x, t dV V
Q tt The material derivative of Qt is: Qt material not d derivative x,tdV dt VV t xx,,ttdV tdV REMARK Vt() t Vt () lim The volume is mobile in space t 0 t and can move, rotate and It can be proven that: deform (material volume). dd xvvv,tdV dV dV dV dV dtt t dt VVt V V V V material local convective derivative of derivative of derivative of the integral the integral the integral
20 5.4. Conservation of Mass
Ch.5. Balance Principles
21 Principle of Mass Conservation
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 M t of the system satisfies:
MMttt 0
Where: M ttdVVV x, V tt t M tt x, ttdVV V V tt tt tt 22 Conservation of Mass in Spatial Form
Conservation of mass requires that the material time derivative of the massM t be zero for any region of a material volume, M MMtt t d tdVVVtlim 0 , t 0 tdtVVVtt The global or integral spatial form of mass conservation principle: dd xv,tdV ( ) dV dt dt VVt V dd (,)xvtdV dV 0 V V , t VVV VV dttt dt By a localization process we obtain the local or differential spatial form of mass conservation principle: for V dV(,)x t (localization process) CONTINUITY dt(,)xx (,) t EQUATION ()(,)vxttVt ()(,)0 vx x , dt t 23 Conservation of Mass in Material Form
d F 1 d F Fv () v Consider the relations: dt F dt dV F dV0 The global or integral material form of mass conservation principle can be rewritten as: dd1(,)dtFFX X t (,) vFXdV () dV ((,)) t dV0 V dt dtF dt t t VV0 (,)X t F dV0 ||(,)FXt t t FX,tdV 0 V V , t t 000 VV00 The local material form of mass conservation principle reads : FX,0t XF XFX 0 tt0 tt0 t X Vt0 , t F 1 t 24 5.5. Reynolds Transport Theorem
Ch.5. Balance Principles
25 d v 0 Reynolds Lemma dt
Consider: An arbitrary property A of a continuum medium (of any tensor order) The spatial description of the amount of the property per unit of mass, x,t The amount of the property A in the continuum body at time t for an arbitrary material volume is: Qt dV VVt
Using the material time derivative leads to, dd dd Q t dV()()vv dV dV dtVV V dt V dt dt t dd Thus, dt dt =0 (continuity equation) dd dV dV REYNOLDS dt dt LEMMA VVt V 26 MMC - ETSECCPB - UPC 11/11/2015 d xv,tdV dV dV dt t Reynolds Transport TheoremVVt V V
The amount of the property A in the continuum body at time t for an arbitrary fixed control volume is: Qt dV V Using the material time derivative leads to, d dV dV v dV dt t VVt V V d d dV nv dV dt V V dt
And, introducing the Reynolds Lemma V and Divergence Theorem: dV d dV dV vn dS VVdt t V eˆ dV 3 ˆ REMARK e2 V The Divergence Theorem: eˆ1 vnvvndV dS dS VVV 27 d dV dV vn dS dt t Reynolds Transport TheoremVV V
The eq. can be rewritten as: d dV dV vn dS REYNOLDS TRANSPORT tdtVV V THEOREM
Net outward flux of A Rate of change of the total through the surface V that amount of A . within the surrounds the control volume V. d control volume V at time t. dt
Rate of change of the amount of A in a V material volume which instantaneously dV coincides with the control volume V.
eˆ dV 3 ˆ e2 V
eˆ1
28 Reynolds Transport Theorem
d dV dV vn dS REYNOLDS TRANSPORT tdtVV V THEOREM (integral form)
d d dV dV vn dS dt tdt VV V V ( v) dV ( ) dV dV V V t d ( ) dV [ ( v)] dV V V t tdt VV VV eˆ dV 3 ˆ e2 V d eˆ ( ) ( vx) Vt 1 tdt REYNOLDS TRANSPORT THEOREM (local form)
29 5.6. General Balance Equation
Ch.5. Balance Principles
30 General Balance Equation
Consider: An arbitrary property A of a continuum medium (of any tensor order) The amount of the property per unit of mass, x,t The rate of change per unit of time of the amount of A in the control volume V is due to:
a) Generation of the property per unit mas and time time due to a source: kt A (,)x 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: jx(,)t A non-convective So, the global form of the general balance equation is: flux vector dV k dV vn dS j n dS t AA VV V V acb
31 dV k dV vn dS j n dS t AA General BalanceVV Equation V V
The global form is rewritten using the Divergence Theorem and the definition of local derivative: dV vn dS t VV vjdV k dV AA VVt d (Reynolds Theorem) dt d AA dV k j dV V V t dt VV VV The local spatial form of the general balance equation is: REMARK d d k j ()j 0 k dt AA For only convective transport A then dt A and the variation of the contents of in a given particle
is only due to the internal generation k A .
32 MMC - ETSECCPB - UPC 11/11/2015 Example
If the property A is associated to mass AM , then: The amount of the property per unit of mass is 1 .
The mass generation source term is k M 0 . The mass conservation principle states mass cannot be generated.
The non-convective flux vector is j M 0 . Mass cannot be transported in a non-convective form. d k j 0 dt AA 0 0 Then, the local spatial form of the general balance equation is: d ( ) ( v)0 ()0v dt t t 11 d Two equivalent forms of ()vvx 0 Vt the continuity equation. tdt
33 5.7. Linear Momentum Balance
Ch.5. Balance Principles
34 Linear Momentum in Classical Mechanics
Applying Newton’s 2nd Law to the discrete system formed by n particles, the resulting force acting on the system is:
nn n dvi Rfatmmiiii ii11 i 1dt Resulting force mass conservation on the system dm principle: i 0 nn dt d dmi dtP miivv i dtii11 dt dt
P t linear momentum For a system in equilibrium, R 0, t :
dtP 0 P tcnt CONSERVATION OF THE dt LINEAR MOMENTUM
35 n Linear Momentum P tm iiv in Continuum Mechanics i1
The linear momentum of a material volume V t of a continuum mediumP with mass M is:
ttdttdVvx ,,, M x vx M V
ddVM
36 Linear Momentum Balance Principle
The time-variation of the linear momentum of a material volume is equal to the resultant force acting on the material volume.
dtP d vR dV t dt dt Vt
Where: body forces RbttdVdS VV surface forces
If the body is in equilibrium, the linear momentum is conserved: dtP R t 0 0 tcnt dt
37 P Global Form of the Linear Momentum Balance Principle
The global form of the linear momentum balance principle: d dtP RbttdVdSdVVVt v , VV VVdt V VV dt tt Pt
Introducing tn and using the Divergence Theorem, tn dS dS dV VV V So, the global form is rewritten: bt dV dS VV VV d bv +dV dV V V , t dt 38 VV Vtt VV MMC - ETSECCPB - UPC 11/11/2015 Local Form of the Linear Momentum Balance Principle
Applying Reynolds Lemma to the global form of the principle: ddv bv dV dV dV V V , t dt dt VV Vtt VV VV
Localizing, the local spatial form of the linear momentum balance principle reads: VdVt(,)x dtvx(,) (,)xbxtt (,) axx (,) tVt , dt LOCAL FORM OF THE LINEAR MOMENTUM BALANCE (CAUCHY’S EQUATION OF MOTION)
39 5.8. Angular Momentum Balance
Ch.5. Balance Principles
40 Angular Momentum in Classical Mechanics
Applying Newton’s 2nd Law to the discrete system formed by n particles, the resulting torque acting on the system is:
nn dvi MrfrOiiiitm ii11dt =0 nn n ddddri L rviiimm ii v rv iii m dtii11 dt dt i 1 dt v Lt i angular momentum dtL M t O dt
For a system in equilibrium, M O 0, t : dtL CONSERVATION OF THE 0 t Ltcnt dt ANGULAR MOMENTUM
41 Angular Momentum in Continuum Mechanics
L The angular momentum of a material volume V t of a continuum medium with mass M is: tttdtttdVrx ,, vx M rx ,,, x vx M V
ddVM
Where r is the position vector with respect to a fixed point.
42 Angular Momentum Balance Principle
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. dtL d rv dV M t dt dt O VVt
Where: torque due to body forces MrbrttdVdS O VV torque due to surface forces
43 Global Form of the Angular Momentum Balance Principle
The global form of the angular momentum balance principle: d rb dV rt dS rv dV dt VVVVt
Introducing tn and using the Divergence Theorem, rt dS rn dS rTT n dS r n dS VV V V r T dV V It can be proven that, REMARK T e is the Levi-Civita rrm ijk permutation symbol. ˆ memmi i; ie ijk jk
44 MMC - ETSECCPB - UPC 11/11/2015 Global Form of the Angular Momentum Balance Principle
Applying Reynolds Lemma to the right-hand term of the global form equation: Reynold's Lemma dd d rv dV rv dV rv dV dt dt dt VVtt VV V ddrv=0 d v vrdV r dV dt dt dt VV v
Then, the global form is rewritten: dv rb e eˆ dV r dV ijk jk i VVdt
45 Local Form of the Angular Momentum Balance Principle
Rearranging the equation: =0 (Cauchy’s Eq.) dv rbm dV 0(,), mx0 t dV V V t VV VVdt
Localizing
mx(,)tm 0ie ijk jk 0 ; i ,,j kVt 1,2,3; x t , i 10ee 123 23 132 32 23 32 11 11 12 13 i 20ee 12 22 23 231 31 213 13 31 13 11 13 23 33 i 30ee312 12 321 21 12 21 11 T (,)xxxtt (,) Vtt , SYMMETRY OF THE CAUCHY’S STRESS TENSOR 46 MMC - ETSECCPB - UPC 11/11/2015 5.9. Mechanical Energy Balance
Ch.5. Balance Principles
47 Power
Power, Wt , 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 E : dtE Wt dt 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.
48 External Mechanical Power
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: Pt bv dV tv dS e VV
dr b dV dt v dr t dS dt v
49 Mechanical Energy Balance
Using tn and the Divergence Theorem, the traction contribution reads, Divergence Theorem tvvndS dS v dV vv: dV VV V V n l Taking into account the identity ldw : spatial velocity =0 gradient tensor :l :d :w
So, tvdS v dV : d dV VV V
50 dv b Mechanical Energy Balance dt
Substituting and collecting terms, the external mechanical power in spatial form is, tv dS V Pte bv dV vddV : dV V VV dv bvdV :: d dV v dV d dV VVVVdv dt dd112 dt vv (v) dt22 dt v v
Reynold's Lemma dd11 P t(v)22 dVdV:d (v) dVdV :d e VVdt22 dt VV
51 Mechanical Energy Balance. Theorem of the expended power. Stress power
d 1 P tdVdSdVdVbv tvv2 :d e VV dt 2 VVt V
external mechanical power K P kinetic energy entering the medium stress power d Pt K t P Theorem of the expended e dt mechanical power REMARK The 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 ( d0 ) .
52 External Thermal 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. incoming heat qx(,)tdS n unit of time V heat conduction flux vector
Internal heat sources heat generated by an internal source rtdV(,)x unit of time V specific internal heat production
53 External Thermal Power
The external thermal power is incoming heat in the continuum medium per unit of time. In spatial form it is defined as: Qt rdV qn dS( r q) dV e VVV nq dS V q) dV V where: qx ,t is the heat flux per unit of spatial surface area. rtx, is an internal heat source rate per unit of mass.
54 Total Power
The total power entering the continuous medium is:
d 1 P Qv2 dV :d dV r dV q n dS eedt 2 VVt V V V
55 5.10. Energy Balance
Ch.5. Balance Principles
56 Thermodynamic Concepts
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: ISOLATED SYSTEM OPEN SYSTEM Thermodynamic space
MATTER
HEAT
A thermodynamic system is characterized and defined by a set of
thermodynamic variables 1, 2, ....n which define the thermodynamic space
The set of thermodynamic variables necessary to uniquely define a system is called the thermodynamic state of a system.
57 Thermodynamic Concepts
A thermodynamic process is the energetic development of a thermodynamic 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.
58 State Function
Is a function 1 ,..., n uniquely valued in terms of the “thermodynamic state”
or, equivalently, in terms of the thermodynamic variables 12 ,,, n
Consider a function 12 , , that is not a state function, implicitly defined in the thermodynamic space by the differential form:
f112 ,,df 1 212 d 2 The thermodynamic processes 1 and 2 yield: BA f212(, ) 2 11 ' B B 12 BA'2122 f (, ) 22 For to be a state function, the differential form must an exact differential: d , i.e., must be integrable: The necessary and sufficient condition for this is the equality of cross-derivatives: f ,..., f ,..., in1 jn1 ij,1,... n d 59 ji First Law of Thermodynamics
POSTULATES: 1. There exists a state function E t named total energy of the system, such that its material time derivative is equal to the total power entering the system: dd1 EtPtQt:v 2 dV :d dVrdV q n dS dtee dt 2 VVt V V V Qt() Pte() e
2. There exists a function U t 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) ut x , : U tudV: REMARK V dE and d K are exact differentials, The variation of the total energy of the system is: therefore, so is ddd UEK . dd d EKUttt Then, the internal energy is a dt dt dt state function. 60 Global Form of the Internal Energy Balance
Introducing the expression for the total power into the first
postulate: K dd1 EtdVdVrdVdSv2 :d q n dt dt 2 VVt V V V
Comparing this to the expression in the second postulate: dd d EKUttt dt dt dt
The internal energy of the system must be: dd GLOBAL FORM U tudVdVrdVdS :d q n dt dt OF THE INTERNAL VVt V V V ENERGY BALANCE Qt Pt , e stress power external thermal power 61 Local Spatial Form of the Internal Energy Balance
Applying Reynolds Lemma to the global form of the balance equation, and using the Divergence Theorem: dd du U t u dV dV :d dV r dV q n dS dt dtV VV V VV dt VV VV VV tt tt U()t q dV V du dV :d dV r dV q dV V V t VVdt VV VV VV Then, the local spatial form of the linear momentum balance principle is obtained through localization VdVt (,) x as: du LOCAL FORM OF THE :d rVt q x , ENERGY BALANCE dt (Energy equation)
62 Second Law of Thermodynamics
The total energy is balanced in all thermodynamics processes following: ddEKU d Pt Q t eedt dt dt In an isolated system (no work can enter or exit the system) dE ddUK Pt Q t 0 0 eedt dt dt
However, it is not established if the energy exchange can happen in both senses or not: ddUK ddUK 00 00 dt dt dt dt
There is no restriction indicating if an imagined arbitrary process is physically possible or not.
63 Second Law of Thermodynamics
The concept of energy in the first law does not account for the observation that natural processes have a preferred direction of progress. For example:
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.
64 MMC - ETSECCPB - UPC 11/11/2015 Reversible and Irreversible Processes
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. REVERSIBLE PROCESS IRREVERSIBLE PROCESS
The second law of thermodynamics allows discriminating: IMPOSSIBLE thermodynamic processes REVERSIBLE POSSIBLE IRREVERSIBLE
65 Second Law of Thermodynamics
POSTULATES: 1. There exists a state function x,t denoted absolute temperature, which is always positive.
2. There exists a state function S 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 s : St() s(,)x tdV V The following inequality holds true: nd dd r q Global form of the 2 S() t s dV dVn dS Law of dt dt VV V Thermodynamics = reversible process > irreversible process
66 Second Law of Thermodynamics
SECOND LAW OF THERMODYNAMICS IN CONTINUUM MECHANICS The rate of the total entropy of the system is equal o greater than the rate of heat per unit of temperature
nd dd r q Global form of the 2 S() t s dV dVn dS Law of dt dt VV V Thermodynamics = reversible process e t > irreversible process rate of the total amount of the entity heat, per unit Qt rdV qn dS of time, (external thermal power) entering into the e VV system
r q rate of the total amount of the entity heat per unit tdVdS n e of absolute temperature, per unit of time (external VV heat/unit of temperature power) entering into the system
67 Second Law of Thermodynamics
Consider the decomposition of entropy into two (extensive) counterparts: Entropy generated inside the continuous medium:
ie St S t S t dS dSie dS dt dt dt
Entropy generated by interaction with the outside medium:
ii StdV s, x V StdVee s, x V
68 Second Law of Thermodynamics
e If one establishes, dS r q dV n dS e dt VV
Then the following must hold true: dSie dS dS r q dV n dS dt dt dt VV e And thus, dS dt dSie dS dS dS r q dV n dS0 V V t dt dt dt dt VV VV
REPHRASED SECOND LAW OF THERMODYNAMICS : The internally generated entropy of the system , St i () , never decreases along time
69 Local Spatial Form of the Second Law of Thermodynamics
The previous eq. can be rewritten as:
ddi rq s dV s dV dVn dS0 V V t dt dt Vtt VVtt V VV VV VV Applying the Reynolds Lemma and the Divergence Theorem:
dsi ds r q dV dV dV dV0 V V t VVdt VV dt VV VV
Then, the local spatial form of the second law of thermodynamics is: nd dsi ds r q Local (spatial) form of the 2 0,x Vt Law of Thermodynamics dt dt (Clausius-Duhem inequality)
= reversible process > irreversible process
70 Local Spatial Form of the Second Law of Thermodynamics
Considering that, q 11 REMARK qq2 (Stronger postulate) Internally generated entropy can i The Clausius-Duhem inequality can be written as be generated locally, s , or by i local s i s thermal conduction, scond , and i ds ds r 11 both must be non-negative. qq2 0 dt dt si i local scond CLAUSIUS-PLANCK HEAT FLOW r 1 1 s q 0 q 0 INEQUALITY INEQUALITY 2
Because density and absolute temperature are always positive, it is deduced that q 0 , which is the mathematical expression for the fact that heat flows by conduction from the hot parts of the medium to the cold ones. 71 Alternative Forms of the Clausius-Planck Inequality
Substituting the internal energy balance equation given by du not ur :dq qd ru : dt
into the Clausius-Planck inequality,
i ssrlocal :0q
yields, su :0d us :0d Clausius-Planck Inequality in terms of the specific internal energy
72 Alternative Forms of the Clausius-Planck Inequality
The Helmholtz free energy per unit of mass or specific free energy, , is defined as: :us
Taking its material time derivative, : us s us s
and introducing it into the Clausius-Planck inequality in terms of the specific internal energy: us :0d s :0d REMARK Clausius-Planck Inequality For infinitesimal deformation, d , in terms of the and the Clausius-Planck inequality specific free energy becomes: () s : 0
73 5.11. Governing Equations
Ch.5. Balance Principles
74 Governing Equations in Spatial Form
Conservation of Mass. v 0 1 eqn. Continuity Equation.
Linear Momentum Balance. bv 3 eqns. First Cauchy’s Motion Equation.
T Angular Momentum Balance. 3 eqns. Symmetry of Cauchy Stress Tensor.
Energy Balance. ur :dq 1 eqn. First Law of Thermodynamics.
us :d 0 Second Law of Thermodynamics. 1 Clausius-Planck Inequality. 2 restrictions q 0 2 Heat flow inequality 8 PDE + 2 restrictions
75 Governing Equations in Spatial Form
The fundamental governing equations involve the following variables: density 1 variable
v velocity vector field 3 variables
Cauchy’s stress tensor field 9 variables
u specific internal energy 1 variable q heat flux per unit of surface vector field 3 variables
absolute temperature 1 variable 19 scalar s specific entropy 1 variable unknowns
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.
76 Constitutive Equations in Spatial Form
v,, Thermo-Mechanical Constitutive Equations. 6 eqns.
Entropy ss v,, Constitutive Equation. 1 eqn.
Thermal Constitutive Equation. qqv, K Fourier’s Law of Conduction. 3 eqns.
uf ,,,v Heat State Equations. (1+p) eqns. Fipi , , 0 1,2,..., Kinetic (19+p) PDE + set of new thermodynamic (19+p) unknowns variables: 12 , ,..., p . REMARK 1 REMARK 2 The strain tensor is not considered an unknown as they These equations are can be obtained through the motion equations, i.e., v . specific to each material.
77 The Coupled Thermo-Mechanical Problem
Conservation of Mass. v 0 1 eqn. Continuity Mass Equation.
16 scalar Linear Momentum Balance. 10 3 eqns. unknowns First Cauchy’s Motion Equation. equations
((),)v Mechanical constitutive equations. 6eqns.
Energy Balance. 1 eqn. First Law of Thermodynamics.
Second Law of Thermodynamics. 2 restrictions. Clausius-Planck Inequality.
78 MMC - ETSECCPB - UPC The Uncoupled Thermo-Mechanical Problem
The mechanical and thermal problem can be uncoupled if
The temperature distribution x , t 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.
79 The Uncoupled Thermo-Mechanical Problem
Conservation of Mass. v 0 1 eqn. Continuity Mass Equation.
10 scalar Linear Momentum Balance. Mechanical 3 eqns. unknowns First Cauchy’s Motion Equation. problem
((),v ) Mechanical constitutive equations. 6eqns.
Energy Balance. 1 eqn. First Law of Thermodynamics. Thermal problem Second Law of Thermodynamics. 2 restrictions. Clausius-Planck Inequality.
80 The Uncoupled Thermo-Mechanical Problem
Then, the variables involved in the mechanical problem are:
density 1 variable
Mechanical v velocity vector field 3 variables variables Cauchy’s stress tensor field 6variables
u specific internal energy 1 variable q heat flux per unit of surface vector field 3 variables Thermal variables absolute temperature 1 variable
s specific entropy 1 variable
81 Summary
Ch.5. Balance Principles
82 Summary
The convective flux of A t through a spatial surface S with unit normal n is: A t tdS vn is an arbitrary property S s Where: x,t is the description of the amount of the property per unit of mass.
Time derivatives of a volume integral:
not local x,tdV inflow derivative vn 0 t V outflow vn0 not d material x,tdV derivative dt V
d xv,tdV dV dV dt t VVt V V
83 Summary (cont’d)
Conservation of mass: the mass of a continuum body is a conserved quantity. d dV v dV 0 Global spatial form dt VV Local spatial form v 0 (Continuity Equation)
Reynolds Lemma: dd dV dV dt dt VVt V
Reynolds Transport Theorem: dV dVvvn dS dV dS t VVV VV Divergence Theorem
84 Summary (cont’d)
Linear Momentum Balance: d bt dV dS v dV Global spatial form dt VVVVt dv Local spatial form bx + Vt , dt (Cauchy’s Equation of Motion)
Angular Momentum Balance: d rb dV rt dS rv dV Global spatial form dt VVVVt
T Local spatial form x Vt, (Symmetry of the Cauchy stress tensor)
85 Summary (cont’d)
Mechanical Energy Balance: d 1 P tbv dV tv dSv2 dV :d dV e VV dt 2 VVt V external mechanical power K P entering the medium kinetic energy stress power
External Thermal Power: qx ,t is the heat flux per unit of Qt rdV qn dS Where: e spatial surface area. VV rtx, is an internal heat source rate per unit of mass.
Total Power PQee
86 Summary (cont’d)
First Law of Thermodynamics. Internal Energy Balance.
Pt Qte dd tudVdVrdVdS :d q n Global spatial form dt dt VVt V V V
du Local spatial form :d rVt q x , dt (Energy Equation)
Second Law of Thermodynamics. = reversible process > irreversible process dd r q SdVdVdSs n Global spatial form dt dt VV V
i ds ds r q 0,x Vt Local spatial form dt dt (Clausius-Duhem inequality)
r 1 CLAUSIUS-PLANK s q 0 INEQUALITY
87 Summary (cont’d)
Governing equations of the thermo-mechanical problem: Conservation of Mass. v 0 1 eqn. Continuity Mass Equation.
Linear Momentum Balance. bv 3 eqns. First Cauchy’s Motion Equation. Angular Momentum Balance. Symmetry 8 PDE + T 3 eqns. of Cauchy Stress Tensor. 2 restrictions Energy Balance. ur :dq 1 eqn. First Law of Thermodynamics.
us :d 0 Second Law of Thermodynamics. 1 2 restrictions 2 q 0 Clausius-Planck Inequality.
19 scalar unknowns: , v , , u , q , , s .
88 Summary (cont’d)
Constitutive equations of the thermo-mechanical problem: Thermo-Mechanical 6 eqns. v,, Constitutive Equations.
Entropy ss v,, Constitutive Equation. 1 eqn. (19+p) PDE + (19+p) unknowns Thermal Constitutive Equation. Fourier’s qqv , K Law of Conduction. 3 eqns.
uf ,,,v Heat State Equations. (1+p) eqns. Kinetic Fi ,, 0ip 1,2,...,
set of new thermodynamic
variables: 12 , ,..., p . 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.
89