Complementi di scienza delle costruzioni - DidatticaWEB

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


Undertheassump4onofthescalesepara4on,homogeniza4ontechniquesaimtodefineequivalenthomogeneous(orlocallyhomogeneous)modelsofanheterogenousmedium.

RVE=Representa4veVolumeElement

Homogeniza.onaim


Mul.scalehomogeniza.on


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


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


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


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


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


Proper.esofhomogeneousneighbourhoods

�⌦

�⌃

s(P )

r(P )

P

�⌦

�⌃

n(P ) tn(P )

P

tn

(P ) = �o


= const

s(P ) = "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)



Homogeneousboundarycondi.onsonanheterogeneousneighbourhood

�⌦

�⌃

�⌦

�⌃

s(P )

n(P ) tn(P )

r(P )

P

P

tn

(P ) = �o


= const

s(P ) = "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)



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


�⌦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⌃

(*)

(*)


)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


,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)


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



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


Hill-Mandelcondi.on:energyconsistency


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

) � · " = � · "


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 )�⌃


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