Atthemacroscale,thematerialisusuallyconsideredasahomogeneouscon4nuum(orlocallyhomogeneous).Inotherwords,eachsmallneighborhoodofamaterialpointcanbeconsideredashomogeneous,withphysicalquan44esthatareconstantwithintheneighborhood(constantvaluesmaybedifferentpointbypointifthematerialislocallyhomogeneous).Ontheotherhand,atthemicrolevelmaterialsaregenerallyheterogeneous(themorphologyconsistsofdis4nguishablecomponentsorphases,namelyassociatedtoinclusions,grains,interfaces,cavi4es,fibers,etc.).Moreover,eachmicro-phaseexhibitsafurtherheterogeneityatthenanoscale(molecularscale).
• Computational homogenization of structured thin sheets and shells: application of second-order homogenization principles to through-thickness representative volume elements, en-abling its application to shell-type continua.
• Computational homogenization of interface problems, which is now emerging.
These lecture notes focus on the basics underlying each of these categories, with an outreachto some of the extensions listed above. Note that there is a vast amount of recent literature onother multi-scale (and multi-physics) methods [80–85], often partially connected to the subjectsaddressed in these lecture notes.Cartesian tensors and associated tensor products will be used throughout these lecture notes, mak-ing use of a Cartesian vector basis {e⃗1, e⃗2, e⃗3}. Using the Einstein summation rule for repeatedindices, the following conventions are used in the notations of well-known tensor products
C = a⃗ b⃗ = aibj e⃗i e⃗j
C = A·B = AijBjk e⃗i e⃗k
C = 4A : B = AijklBlk e⃗i e⃗j
C = 4A... 4B = AiklmBmlkj e⃗i e⃗j
1.2 Underlying principles and assumptions
1.2.1 Scale separationAt the macro-scale, the material is considered as a homogeneous continuum, whereas at the mi-cro level it is generally heterogeneous (the morphology consists of distinguishable componentsor phases, i.e. inclusions, grains, interfaces, cavities, etc.). This is schematically illustrated infigure 1.1. The microscopic length scale is much larger than the molecular dimensions ℓdiscrete,
Figure 1.1: Macroscopic continuum point representation (M) in relation to its underlyingheterogeneous microstructure.
so that a continuum approach is justified for every constituent. At the same time, in the contextof the principle of separation of scales, the microscopic length scale ℓmicro is assumed to be muchsmaller than the characteristic length ℓmacro over which the size of the macroscopic loading variesin space, i.e.
ℓdiscrete ≪ ℓmicro ≪ ℓmacro (1.1)
Note that it is not the size of the macroscopic domain which is important, but rather the spatialvariation of the kinematic fields and stress fields within that domain.
1.6
`macro
`micro
`discrete
Introduc.ontoHomogeniza.onTechniques
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 1
Scalesepara.on
• Computational homogenization of structured thin sheets and shells: application of second-order homogenization principles to through-thickness representative volume elements, en-abling its application to shell-type continua.
• Computational homogenization of interface problems, which is now emerging.
These lecture notes focus on the basics underlying each of these categories, with an outreachto some of the extensions listed above. Note that there is a vast amount of recent literature onother multi-scale (and multi-physics) methods [80–85], often partially connected to the subjectsaddressed in these lecture notes.Cartesian tensors and associated tensor products will be used throughout these lecture notes, mak-ing use of a Cartesian vector basis {e⃗1, e⃗2, e⃗3}. Using the Einstein summation rule for repeatedindices, the following conventions are used in the notations of well-known tensor products
C = a⃗ b⃗ = aibj e⃗i e⃗j
C = A·B = AijBjk e⃗i e⃗k
C = 4A : B = AijklBlk e⃗i e⃗j
C = 4A... 4B = AiklmBmlkj e⃗i e⃗j
1.2 Underlying principles and assumptions
1.2.1 Scale separationAt the macro-scale, the material is considered as a homogeneous continuum, whereas at the mi-cro level it is generally heterogeneous (the morphology consists of distinguishable componentsor phases, i.e. inclusions, grains, interfaces, cavities, etc.). This is schematically illustrated infigure 1.1. The microscopic length scale is much larger than the molecular dimensions ℓdiscrete,
Figure 1.1: Macroscopic continuum point representation (M) in relation to its underlyingheterogeneous microstructure.
so that a continuum approach is justified for every constituent. At the same time, in the contextof the principle of separation of scales, the microscopic length scale ℓmicro is assumed to be muchsmaller than the characteristic length ℓmacro over which the size of the macroscopic loading variesin space, i.e.
ℓdiscrete ≪ ℓmicro ≪ ℓmacro (1.1)
Note that it is not the size of the macroscopic domain which is important, but rather the spatialvariation of the kinematic fields and stress fields within that domain.
1.6
`macro
`micro
`discrete
`discrete
⌧ `micro
⌧ `macro
Acon4nuumapproachisjus4fiedforeverycons4tuent
Acon4nuumapproachisjus4fiedatthemacroscale
Notethatthecharacteris4clengthisnotthesizeofthemacroscopicdomainbutthespa4alvaria4onofthekinema4cfieldsandstressfieldswithinthatdomain.
`macro
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 2
Undertheassump4onofthescalesepara4on,homogeniza4ontechniquesaimtodefineequivalenthomogeneous(orlocallyhomogeneous)modelsofanheterogenousmedium.
RVE=Representa4veVolumeElement
Homogeniza.onaim
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 3
Mul.scalehomogeniza.on
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 4
1.2.2 Local periodicityMost of the homogenization approaches rely on the assumption of global periodicity of the mi-crostructure, implying that the whole macroscopic domain consists of spatially repeated unit cells.In a computational homogenization approach, a more realistic assumption is made, which is com-monly denoted by local periodicity. According to this assumption, the microstructure can havedifferent morphologies corresponding to different macroscopic points, whereas it repeats itselfonly in a small vicinity of each individual macroscopic point. The concepts of local and globalperiodicity are schematically illustrated in figure 1.2. The assumption of local periodicity adoptedin the computational homogenization allows to incorporate a non-uniform distribution of the mi-crostructure at the macroscopic level (e.g. in functionally graded materials). Note that the local
(a) local periodicity (b) global periodicityFigure 1.2: Local periodicity (a) versus global periodicity (b).
periodicity assumption is directly linked to the principle of separation of scales.
1.2.3 Homogenization principlesThe basic principles of computational homogenization have gradually evolved from the conceptsemployed in other homogenization methods and well fit into the four-step homogenization schemeestablished by Suquet [57]:
1. definition of a microstructural representative volume element (RVE), of which the constitutivebehaviour of individual constituents is assumed to be known;
2. formulation of the microscopic boundary conditions from the macroscopic input variables andtheir application on the RVE (macro-to-micro transition);
3. calculation of the macroscopic output variables from the analysis of the deformed microstruc-tural RVE (micro-to-macro transition);
4. obtaining the (numerical) relation between the macroscopic input and output variables.
The main ideas of the first-order computational homogenization have been established in [41, 57,58, 69, 86] and further developed and improved in more recent works [1, 59, 60, 62, 64–66, 70, 73].
1.2.4 Computational homogenization schemeA computational homogenization generally departs from the computation of a macroscopic defor-mation (gradient) tensor FM, which is calculated for every material point of the macrostructure(e.g. the integration points within a macroscopic finite element domain). Here and in the following
1.7
Homogeniza.onprinciples
RVEexamples
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 5
1.2.2 Local periodicityMost of the homogenization approaches rely on the assumption of global periodicity of the mi-crostructure, implying that the whole macroscopic domain consists of spatially repeated unit cells.In a computational homogenization approach, a more realistic assumption is made, which is com-monly denoted by local periodicity. According to this assumption, the microstructure can havedifferent morphologies corresponding to different macroscopic points, whereas it repeats itselfonly in a small vicinity of each individual macroscopic point. The concepts of local and globalperiodicity are schematically illustrated in figure 1.2. The assumption of local periodicity adoptedin the computational homogenization allows to incorporate a non-uniform distribution of the mi-crostructure at the macroscopic level (e.g. in functionally graded materials). Note that the local
(a) local periodicity (b) global periodicityFigure 1.2: Local periodicity (a) versus global periodicity (b).
periodicity assumption is directly linked to the principle of separation of scales.
1.2.3 Homogenization principlesThe basic principles of computational homogenization have gradually evolved from the conceptsemployed in other homogenization methods and well fit into the four-step homogenization schemeestablished by Suquet [57]:
1. definition of a microstructural representative volume element (RVE), of which the constitutivebehaviour of individual constituents is assumed to be known;
2. formulation of the microscopic boundary conditions from the macroscopic input variables andtheir application on the RVE (macro-to-micro transition);
3. calculation of the macroscopic output variables from the analysis of the deformed microstruc-tural RVE (micro-to-macro transition);
4. obtaining the (numerical) relation between the macroscopic input and output variables.
The main ideas of the first-order computational homogenization have been established in [41, 57,58, 69, 86] and further developed and improved in more recent works [1, 59, 60, 62, 64–66, 70, 73].
1.2.4 Computational homogenization schemeA computational homogenization generally departs from the computation of a macroscopic defor-mation (gradient) tensor FM, which is calculated for every material point of the macrostructure(e.g. the integration points within a macroscopic finite element domain). Here and in the following
1.7
Mostofthehomogeniza4onapproachesrelyontheassump4onofglobalperiodicityofthemicrostructure,implyingthatthewholemacroscopicdomainconsistsofspa4allyrepeatedunitcells.Localperiodicitymaybeassumed,correspondingtoassumethatthemicrostructurehasdifferentmorphologiescorrespondingtodifferentmacroscopicpoints,whereasitrepeatsitselfonlyinasmallvicinityofeachindividualmacroscopicpoint.Theassump4onoflocalperiodicityallowstoincorporateanon-uniformdistribu4onofthemicrostructureatthemacroscopiclevel(e.g.infunc4onallygradedmaterials).
Periodicity ) RVE = unit cell
Periodicity
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 6
Therepresenta.vevolumeelement(RVE)preliminaryrequirements
Thephysicalandgeometricalproper4esofthemicrostructureareiden4fiedbyarepresenta4vevolumeelement(RVE).TheRVEneedstobe- largeenoughtocontainasufficientnumberofmicro-features;- representa4veofthemicrostructure;- suchthatitdoesnotintroducenon-exis4ngmacroscopicproper4es(suchasanisotropy
inamacroscopicallyisotropicmaterial).
RVE:asta.s.callyrepresenta.vesampleofthemicrostructure
Inthecaseofnon-regularandnon-uniformmicrostructuresuchadedini4onleadstoaconsiderablylargeRVE.Atthesame4metheRVEhastobesufficientlysmallinordertoberepresenta4veofaneighborhoodofthemacrosopicproblem.
RVE:thesmallestmicrostructuralvolumethatallowsonetorepresentina sufficientlyaccuratewaytheoverallmacroscopicproper.esofinterest
TheexistenceofaRVEassumesthatitispossibletoreplace
aheterogeneousmaterialwithanequivalenthomogeneousmaterial.UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 7
Bo
B
x
X
udxdX
Referenceconfigura4on
Currentconfigura4on
MACRO-MICROSCALETRANSITION
•Com
putationalhom
ogenizationofstructured
thinsheets
andshells:
applicationofsecond-
orderhom
ogenizationprinciples
tothrough-thickness
representativevolum
eelem
ents,en-
ablingitsapplication
toshell-type
continua.
•Com
putationalhomogenization
ofinterfaceproblem
s,which
isnowemerging.
Theselecture
notesfocus
onthe
basicsunderlying
eachofthese
categories,withanoutreach
tosom
eofthe
extensionslisted
above.Note
thatthereisavastam
ountofrecentliterature
on
othermulti-scale
(andmulti-physics)
methods
[80–85],oftenpartially
connectedtothe
subjects
addressedinthese
lecturenotes.
Cartesiantensorsand
associatedtensorproductsw
illbeused
throughouttheselecture
notes,mak-
inguse
ofaCartesian
vectorbasis{e⃗1,e⃗
2,e⃗3}.
Using
theEinstein
summation
ruleforrepeated
indices,thefollow
ingconventions
areused
inthe
notationsofwell-know
ntensorproducts
C=
a⃗b⃗=
aibj
e⃗ie⃗
jC
=A·B
=AijB
jke⃗ie⃗
k
C=
4A:B
=Aijk
l Blke⃗ie⃗
jC
=4A...4B
=AiklmBmlkje⃗ie⃗
j
1.2Underlying
principlesandassum
ptions
1.2.1Scale
separationAtthe
macro-scale,the
materialis
consideredasahom
ogeneouscontinuum
,whereas
atthemi-
crolevelitis
generallyheterogeneous
(themorphology
consistsofdistinguishable
components
orphases,i.e.
inclusions,grains,interfaces,cavities,etc.).This
isschem
aticallyillustrated
in
figure1.1.
Themicroscopic
lengthscale
ismuch
largerthanthe
moleculardim
ensionsℓdiscrete,
Figure1.1:M
acroscopiccontinuum
pointrepresentation(M)in
relationtoitsunderlying
heterogeneousmicrostructure.
sothata
continuumapproach
isjustified
foreveryconstituent.
Atthe
sametime,in
thecontext
oftheprinciple
ofseparationofscales,the
microscopic
lengthscale
ℓmicro
isassumedtobemuch
smallerthan
thecharacteristic
lengthℓm
acrooverw
hichthesize
ofthemacroscopic
loadingvaries
inspace,i.e.
ℓdiscrete≪ℓm
icro≪ℓm
acro
(1.1)
Note
thatitisnotthe
sizeofthe
macroscopic
domain
which
isimportant,butratherthe
spatial
variationofthe
kinematic
fieldsandstressfieldsw
ithinthatdom
ain.1.6
Microscale
Macroscale
homogeneous deformation
of small neighbouroods
) FM = const over
macroscopic neighbouroods
�x = FM�X+O(|�X|2) ) x� xc = FM (X�Xc) +w
xc : current position of the reference neighbourood center Xc
w : microfluctuation field associated to a fine scale contribution
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 8
x(X, t) = X+ u(X, t)
) dx =
@x
@X= (I+ru) dX ⌘ FMdX
FM : deformation gradient tensor
MACRO-MICROSCALETRANSITION
x� x
c
= F
M
(X�X
c
) +w ) (dx)m
= (F
M
+rw)(dX)
m
= F
m
(dX)
m
) F
m
= F
M
+rw
volume average over the RVE of F
m
:
¯
F
m
=
1
�⌦
o
Z
�⌦o
F
m
d⌦ = F
M
+
1
�⌦
o
Z
�⌦o
rw d⌦
Z
�⌦o
rw d⌦ =
Z
�⌦o
wi,j d⌦ =
Z
�⌃o
winj d⌃ =
Z
�⌃o
w ⌦ n d⌃
�⌦
o
: reference RVE volume (whose reference boundary surface is �⌃
o
)
) F̄m
= FM
+1
�⌦o
Z
�⌃o
w ⌦ n d⌃
Remark:
Scaletransi.onrela.onship
F̄m = FM ,Z
�⌃o
w ⌦ n d⌃ = 0
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 9
Homogeneousboundarycondi.ons
�⌦
�⌃
�⌦
�⌃
s(P )
n(P ) tn(P )
r(P )
P
P
tn
(P ) = �o
n(P ) 8 P 2 �⌃ and with �o
= const
s(P ) = "o
r(P ) 8 P 2 �⌃ and with "o
= const
HomogeneousstrainBC
HomogeneousstressBC
MICRO-SCALEBOUNDARYCONDITIONS
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 10
Proper.esofhomogeneousneighbourhoods
�⌦
�⌃
s(P )
r(P )
P
�⌦
�⌃
n(P ) tn(P )
P
tn
(P ) = �o
n(P ) 8 P 2 �⌃ and with �o
= const
s(P ) = "o
r(P ) 8 P 2 �⌃ and with "o
= const
HomogeneousstrainBC
HomogeneousstressBC
If �⌦ is a homogeneous neighborhood ) C and S are constant 8 Q 2 �⌦
T = �o
8 Q 2 �⌦
D = S�o
= const
D = "o
8 Q 2 �⌦
T = C "o
= const
Notation
T : Cauchy stress on homogeneous neighborhoods (macroscale stress)
D : infinitesimal strain on homogeneous neighborhoods (macroscale strain)
MICRO-SCALEBOUNDARYCONDITIONS
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 11
Homogeneousboundarycondi.onsonanheterogeneousneighbourhood
�⌦
�⌃
�⌦
�⌃
s(P )
n(P ) tn(P )
r(P )
P
P
tn
(P ) = �o
n(P ) 8 P 2 �⌃ and with �o
= const
s(P ) = "o
r(P ) 8 P 2 �⌃ and with "o
= constHomogeneousstrainBC
HomogeneousstressBC
� = �(r) 6= const in �⌦
" = "(r) 6= const in �⌦
Average strain theorem
" =
1
�⌦
Z
�⌦"(r) d⌦ = "
o
Average stress theorem
� =
1
�⌦
Z
�⌦�(r) d⌦ = �
o
Notation
� : Cauchy stress on heterogeneous neighborhoods (microscale stress)
" : infinitesimal strain on heterogeneous neighborhoods (microscale strain)
MICRO-SCALEBOUNDARYCONDITIONS
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 12
Forthesakeofsemplicity,leta2-phasesheterogeneousRVEbeconsidered,byassumingthatequilibriumandperfectbondingbetweenphaseshold.
�⌃1 = �⌃int
[�⌃ext
�⌃2 = �⌃int
on �⌃int
n1 = �n2
s1 = s2
�1 n1 = ��2 n2
�⌦1
�⌦2
�⌃int
�⌃ext
n1
n1
n2
interfacecompa4bilityInterfaceequilibrium
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 13
�⌦1
�⌦2
�⌃int
�⌃ext
n1
n1
n2
HomogeneousstrainBC
" =1
�⌦
Z
�⌦"(r) d⌦ = "
o
s(P ) = "o
r(P ) 8 P 2 �⌃ext
and with "o
= const
Averagestraintheorem
" =1
�⌦
Z
�⌦"(r) d⌦ =
1
�⌦
Z
�⌦sym(rs) d⌦ =
1
�⌦
Z
�⌦1
sym(rs1) d⌦+
Z
�⌦2
sym(rs2) d⌦
�
=1
�⌦
Z
�⌃1
sym(s1 ⌦ n1) d⌃+
Z
�⌃2
sym(s2 ⌦ n2) d⌃
�
=1
�⌦
Z
�⌃ext
sym(s1 ⌦ n1) d⌃+
Z
�⌃int
[sym(s1 ⌦ n1) + sym(s2 ⌦ n2)] d⌃
�
=1
�⌦
Z
�⌃ext
sym(s1 ⌦ n1) d⌃
�=
1
�⌦
Z
�⌃ext
sym("o
r⌦ n1) d⌃
�=
"o
�⌦
Z
�⌦sym(rr) d⌦
=�⌦
�⌦"o
I = "o
Proof
interfacecompa4bility
Z
⌦sym(rs) d⌦ =
1
2
Z
⌦(si,j + sj,i) d⌦ =
1
2
Z
⌃(sinj + sjni) d⌃
=1
2
Z
⌃(s⌦ n+ n⌦ s) d⌃ =
Z
⌃sym(s⌦ n) d⌃
(*)
(*)
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 14
)Z
�⌦� ·rv d⌦ =
Z
�⌦div(� v) d⌦ =
Z
�⌦1
div(�1 v) d⌦+
Z
�⌦2
div(�2 v) d⌦
=
Z
�⌃1
�1v · n1 d⌃+
Z
�⌃2
�2v · n2 d⌃ =
Z
�⌃1
�1n1 · v d⌃+
Z
�⌃2
�2n2 · v d⌃
=
Z
�⌃ext
�1n1 · v d⌃+
Z
�⌃int
(�1n1 · v + �2n2 · v) d⌃ =
Z
�⌃ext
�o
n1 · v d⌃
= �o
·Z
�⌃ext
v ⌦ n1 d⌃ = �o
·Z
�⌦rv d⌦
�⌦1
�⌦2
�⌃int
�⌃ext
n1
n1
n2
HomogeneousstressBC
Averagestresstheorem
Proof
Average stress theorem
� =
1
�⌦
Z
�⌦�(r) d⌦ = �
o
tn
(P ) = �o
n1(P ) 8 P 2 �⌃ext
and with �0 = const
8 vector v, v · div� = div(� v)� � ·rv ) since the equilibrium (div� = 0)
) 8 vector v, div(� v) = � ·rv
interfaceequilibrium
Since it has to hold 8 v, and also for any v such that rv = const, thereby
Z
�⌦� d⌦ ·rv = �
o
·rv�⌦ ) � = �o
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 15
,HeterogeneousneighborhoodofthemacroscopicmaterialpointM
Equivalenthomogeneous
neighborhoodofthemacroscopicmaterialpointM
Heterogeneousmedium
Equivalenthomogeneous
medium
Equivalentelas4cresponse
M M
RVE
C⇤(M) ?
S⇤(M) ?
T(M) = C⇤(M)D(M)
D(M) = S⇤(M)T(M)
Hp: constitutive response of each phase is known
�i(r) = Ci "i(r)
"i(r) = Si �i(r)
C(r) = C(Ci, i = 1, ...,#phases)
S(r) = S(Si, i = 1, ...,#phases)
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 16
Equivalenthomogeneousneighborhood
RVE
s(P )
HomogeneousstrainBC
s(P )
�(r) = C(r)"(r)r(Q) s.t. Q 2 RV E
s(P ) = "o
r(P )
� =1
�⌦
Z
�⌦�(r) d⌦ =
1
�⌦
Z
�⌦C(r)"(r) d⌦ = C⇤"
o
,
C⇤: equivalent macroscale sti↵ness tensor
C⇤
C(r)
Hill-Mandelcondi.ons
T = C⇤ D = C⇤ "o
= const
� = T
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 17
Equivalenthomogeneousneighborhood
RVE
HomogeneousstressBC
,
" =1
�⌦
Z
�⌦"(r) d⌦ =
1
�⌦
Z
�⌦S(r)�(r) d⌦ = S⇤�
o
tn(P ) tn(P )tn
(P ) = �o
n(P )
"(r) = S(r)�(r)r(Q) s.t. Q 2 RV E
S⇤
S(r)
Hill-Mandelcondi.ons
" = D
D = S⇤T = S⇤�o
= const
S⇤ : equivalent macroscale compliance tensor
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 18
Hill-Mandelcondi.on:energyconsistency
Equivalenthomogeneousneighborhood
RVE
,C(r), S(r) C⇤, S⇤Energe4cequivalence
Emicro
=1
2
Z
�⌦�(r) · "(r) d⌦ =
1
2�⌦� · "
Emacro
=1
2
Z
�⌦T ·D d⌦ =
1
2�⌦T ·D
If T = � and D = " ) Emicro
= Emacro
) � · " = � · "
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 19
Hill-Mandelcondi.on:energyconsistency
If T = � and D = " ) Emicro
= Emacro
Proof
HomogeneousstrainBC s(P ) = "o
r(P )
For equilibrium (and as in the proof of the average stress theorem)
� · " = � · sym(rs) = � · (rs) = div(�s)� s · div� = div(�s)
)Z
�⌦� · " d⌦ =
Z
�⌦div(�s) d⌦ =
Z
�⌃�s · n d⌃ =
Z
�⌃�n · s d⌃ = 2E
micro
Emicro
=1
2
Z
�⌃�s · n d⌃ =
1
2
Z
�⌃(�"
o
r) · n d⌃ =1
2
Z
�⌦div(�"
o
r) d⌦
=1
2
Z
�⌦[("
o
r) · div(�) + � ·r("o
r)] d⌦ =1
2
Z
�⌦� · "
o
rr d⌦ =1
2
Z
�⌦� · "
o
I d⌦
=1
2"o
·Z
�⌦� d⌦ =
1
2� · "�⌦ =
1
2T ·D�⌦ = E
macro
RVE
s(P )
�⌃
HomogeneousstressBC tn
(P ) = �o
n(P )
Emicro
=1
2
Z
�⌃�n · s d⌃ =
1
2
Z
�⌃� · (s⌦ n) d⌃ =
1
2�
o
·Z
�⌃sym(s⌦ n) d⌃
=1
2�
o
·Z
�⌦sym(rs) d⌃ =
1
2�
o
·Z
�⌦" d⌃ =
1
2� · "�⌦ =
1
2T ·D�⌦ = E
macro RVE
tn(P )�⌃
UniversitàdegliStudidiRoma“TorVergata”–GiuseppeVairo 20
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