L’Approccio Frattale alla Meccanica non Locale

(The fractal approach to non-local mechanics)

Università degli Studi di Palermo, ItalyDipartimento di Ingegneria Strutturale, Aerospaziale e Geotecnica (DISAG), Viale delle Scienze Ed.8, 90128 Palermo. email:mario.dipaola@unipa.it

Mario Di Paola

OUTLINESOUTLINES

•• LATTICE THEORYLATTICE THEORY

•• GRADIENT AND STRONG NONGRADIENT AND STRONG NON--LOCAL THEORYLOCAL THEORY

•• PHYSICALLYPHYSICALLY--BASED APPROACH TO NONBASED APPROACH TO NON--LOCAL LOCAL MECHANICSMECHANICS

•• SELECTION OF THE DECAYING FUNCTIONSELECTION OF THE DECAYING FUNCTION

•• FRACTALS FRACTALS vsvs FRACTIONAL FRACTIONAL

•• CONCLUSIONSCONCLUSIONS

The Classical Continuum Mechanics (1D) Case

( )f x V∆

jN1jN +

x∆ ( )1j jN N f x V+ − = ∆V A x∆ = ∆

( )jNf x x

A

∆= − ∆

( ) ( )d xf x

dx

σ= −

0x∆ →

Equilibrium of solid element:

duE E

dxσ ε= =

Constitutive Equation (LOCAL)

( )2

2

f xd u

dx E= −

Governing equation of the 1D solid

Boundary conditions: ( )0 0 0 ; 0EA F u uε = − =

( ) ; L L LEA F u L uε = =

x∆

FF− ( )f x V∆

RENORMALIZATION (WILSON 1972)

0ℓ 0bℓ

0H 1H 2H nH

20b ℓ

1b >

The presence of microstructure in real-materials

Continuum Mechanics Approach

Homogeneous and often isotropic elastic material

0HnH

The Molecular Dynamics ApproachEach particle has three degree offreedom and its motion is ruled by the Newtons’ law:

( )1 10 O nm− ( ) ( ) ( )t t t+ =Mu Ku Fɺɺ

( ) 3nt ∈ℜu Displacement vector3 3n n×∈ℜM Mass matrix

3 3n n×∈ℜK Elastic bonds matrix

NANOSCALE

MESOSCALE ( )0.001 1 O mµ−

TERANUMBERS DEGREE OF FREEDOM INVOLVED !!!

( )1210O

The need for an enriched continuum mechanics

•In the late fifties and mid-sixties the basis of a generalized continuum mechanics had been proposed considering the inner microstructure

The theory of micromorphic continuum

The Lattice Model of Materials (NN)

( )1 12lj j j jk u u u F+ −− + = −

- Mass of Lattice Atoms

- Lattice Elastic Constantk

m

• Material properties often described at molecular level (Born-Von Karman):

kk k

j1j − 1j +

a

m

juru

jF( )1 j jk u u−− ( )1 j jk u u+ −

a - Lattice Distance

aEAkk l ==

juru

lk

j1j − 1j +

a

m

• A point-spring equivalent model:

lk lkjF

External loadjF =

( )2

2

f xd u

dx E= −

The Continuum Equivalence of the Lattice Models

• As it is implicitely assumed that the same kind of interaction exists at each smaller scale (EUCLIDEAN OBJECT) .

0x∆ →

( )n → ∞

• Lattice elements may exchange interactions still at distance (NN)a

0x∆ →

( )1 12j j j j

EAu u u f A x

x + −− + = − ∆∆

j jF f A x= ∆

( )1 j jk u u−− ( )1 j jk u u+ −

• Derivation of the Waves Equation for an 1-D model

; EA

k m A xx

ρ= = ∆∆

j1j − 1j +

a x= ∆

m( ) body force fieldf x =

The Non-Local Elasticity Theories

NON-LOCALNON-LOCAL

Non-local model

GRADIENT (weak non-locality) INTEGRAL (strong non-locality)2

1 2 2ld d

( x ) E ( x ) E ( x ) E ( x ) ..dx dx

σ ε ε ε= + + + ( ) ( )x E x g( x, ) ( )dσ ε ξ ε ξ ξ= + ∫

εElastic moduli

Axial strain

Attenuation function

( )xσ Axial stress

1 2l ,E E ,E g( x, )ξ9

Essentialreferences

• Benvenuti E., Borino G., Tralli A., 2002, A thermodynamically consistent non local formulation of damaging

materials, European Journal of Mechanics /A Solids, Vol.21, 535-553.

• Kröner E., 1967, Elasticity theory of material with long range cohesive forces, International Journal of Solids

and Structures, Vol. 3, 731-742.

• Eringen A.C., Edelen D.G.B., 1972,On nonlocal elasticity,International Journal Engineering Science, Vol. 10,

233-248.

• Aifantis E.C., 1984,On the microstructural origin of certain inelastic models, Trans. ASME, J. Eng. Mat. Trchn.,

Vol. 106, 326-330.

• Polizzotto C., Borino G., 1998, A thermodynamics-based formulation of gradient-dependent plasticity,

European Journal of Mechanics A/Solids, Vol.17, 741-761.

Integral non-local models

Gradient non-local models

Fractal Mechanics

• Carpinteri A., 1994, Fractal nature of material microstructure and size effects on apparent mechanical

properties, Mechanics of Materials, Vol.18, 89-101.

• Carpinteri A., Chiaia B., Cornetti P., 2001, Static-Kinematic duality and the principle of virtu al work in the

mechanics of fractal media, Computer Methods in Applied Mechanics and Engineering, Vol. 191, 3-19.

• Carpinteri A., Cornetti P., 2002, A fractional calculus approach to the description of stress and strain

localization in fractal media, Chaos Solitons and Fractals, Vol.13, 85-94.

• Epstein M., Sniatycki J., 2006, Fractal Mechanics, Physica D, Vol. 220, 54-68.

A Different approach

THE TEAM

Prof. Mario Di Paola, DISAG, Università di Palermo

Prof. Antonina Pirrotta, DISAG, Università di Palermo

Dr. Massimiliano Zingales, DISAG, Università di Palermo

Dr. Giuseppe Failla, MecMat, Reggio Calabria

Dr. Alba Sofi, Dip. Arte, Scienza e Tecnica del costruire, Reggio Calabria

Dr. Giulio Cottone, DISAG, Università di Palermo

Dr. Francesco Marino , MecMat, Università di Reggio Calabria

Dott. Gianvito Inzerillo, DISAG, Università di Palermo

The proposed model (Bounded domain)

x

x∆

A

( )jf x

( )jf x A x∆

jN 1jN +

jQ

jV A

hV jV

( )h, jQ ( )h,jQ

hV

( )h, jQ ( )h, jQ

( ) ( ) ( ), , ,h j h j j hh j h jQ q V V q V V= =

( ) ( )( ) ( ), ,h jh j h j h jq sign x x u u g x x= − −

( ) ( )1

, ,

1

j

h j h jj

h j h

Q Q Q

−∞

= + =−∞

= −∑ ∑

Di Paola M., Zingales M., 2008, Long-Range Cohesive Interactions of Non-Local Continuum Faced by Fractional Calculus,

International Journal of Solids and Structures, Vol.45, 5642-5659.

Di Paola M., Failla G., Zingales M., 2009, Physically-Based Approach to the Mechanics of Strong Non-Local Linear

Elasticity Theory, Journal of Elasticity, Vol. 97, 103-130.

The proposed model (continue…) ( )jf x A x∆

jN 1jN +

jQ

jV A

( ) ( ) ( ) ( ) ( )2

2

d u xE A u x u g x d f x

dxξ ξ ξ

∞

−∞

− − − = − ∫UNBOUNDED

DOMAIN

( ) ( ) ( ) ( ) ( ) ( )1

2 2, ,

1 1

0

jm

h j h jj j j j j

h j h

N Q f x A x N q A x q A x f x A x

−

= + =

∆ + + ∆ = ∆ + ∆ − ∆ + ∆ =∑ ∑

( ) ( )l x N x Aσ =

( ) ( ) ( ) ( ) ( )2

20

Ld u xE A u x u g x d f x

dxξ ξ ξ − − − = − ∫

BOUNDED DOMAIN

Mechanical interpretation of non-locality

LOCAL MODEL F F

L

( ) ( ) ( ) ( );x N x A x N x E Aσ ε= =

xAEk l ∆=m

Lx =∆

fuK =l

( )2 2

20

j j

x

u( x ) f A x f xd ulim A

E Ex dx∆ →

∆ ∆ = − ⇒ = −

∆

( )f x lklk lk lkF

xm-1

1 2 F

xm1 2 j 1−m

... ... 0

2 ... 0

... ... ... ... ...

... ... 2

0 ... ...

l l

l l l

l

l l l

l l

K K

K K K

K K K

K K

− − − =

− − −

K

14

[ ][ ]

1 2

1

Tm

Tm

u u ... u

f ... ... f x

=

= ∆

u

f

Equilibrium of jth node

x∆

F(0) F(L) F(L)F(0) ( )jf x A x∆

fuK =

nll KKK +=

Mechanical interpretation of non-locality II

NON-LOCAL MODEL

15

24K

13K

14K

l K l K l K

3 4 1 2

4 F 1 F

nl

nl

K 12

nl K 23

nl K 34

nl + + +

nl

( )2 2nljh j hK A x g x x= ∆ −

11 12 13 1

22 23 2

1 1 1

... ...

... ...

... ... ... ...

SYM

nl nl nl nlm

nl nl nlm

nl

nl nlm m m m

nlmm

K K K K

K K K

K K

K

− − −

− − − − − =

−

K

( )2 2nljh j hK A x g x x= ∆ −

1

m

nl nljj jh

hh j

K K=≠

=∑

j h≠

The stress-strain Relations and the overall Cauchy stress

x∆

x∆ x∆

( ) ( ) ( )1 1 2 12

1 mnlj j l

j

x K u u K u uA

σ=

= − + −

∑

( ) ( ) ( ) ( )1 1 2 2 3 23 2

1 m mnl nlj j j j l

j j

x K u u K u u K u uA

σ= =

= − + − + −

∑ ∑

2x x x∆ < < ∆

0 x x< < ∆

( ) ( ) ( )

( ) ( ) ( ) ( )

11 1

21

1 1

1 m rnlhj j h l r r

j r h

m rr r

h j j hj r h

x K u u K u uA

u uE A u u g x x x

x

σ += + =

−

= + =

= − + −

−= − − − ∆ ∆

∑ ∑

∑∑

GENERALIZING ( )1r x x r x∆ < < + ∆

1nlmK

23nlK

23+ nllK K

2nlmK

1 1nlmK −

12nl

lK K+

0x∆ →

1nlmK

( )1r x x r x∆ < < + ∆

0x∆ →AT THE LIMIT

( ) ( ) ( )( ) ( )2 1

1 2 1 2 1 20

L x

:x :

dux E A u u g d d

dx ξ ξσ ξ ξ ξ ξ ξ ξ= − − −∫ ∫

The stress-strain relations and the overall Cauchy stress (II)

( ) ( ) ( ) ( ) ( )( )1l nlx x x N x Q x

Aσ σ σ= + = +

( ) ( )l

N x dux E

A dxσ = = ( ) ( ) ( ) ( )( ) ( )

2 1

1 2 1 2 1 2

: :0

L x

nl

x

Q xx A u u g d d

A ξ ξ

σ ξ ξ ξ ξ ξ ξ= = − − −∫ ∫

( ) ( ) ( )

( ) ( ) ( )( )

11 1

21

1 1

1 m rnlhj j h l r r

j r h

m rr r

h j j hj r h

x K u u K u uA

u uE A u u g x x x

x

σ += + =

−

= + =

= − + −

−= − − − ∆ ∆

∑∑

∑ ∑

Comparisons between the Eringen model and the proposed model of long-range interactions

( ) ( ) ( ) ( ) b

a

x E x C g x dσ ε ε ξ ξ ξ= + −∫( ) ( ) ( ) ( ) ( )

1 2

2 1 1 2 1 2

b x

x a

x E x A u u g d dξ ξ

σ ε ξ ξ ξ ξ ξ ξ= =

= − − − ∫ ∫

( ) ( )expK j m j mg x x C x x λ− = − −(POWER-LAW, HELMOLTZ)

(M. Di Paola, G. Failla, M. Zingales, 2009, Phisically-Based Approach to the Mechanics of Strong Non-LocalElasticity, Journal of Elasticity, Vol. 97,103-130. )

ERINGEN (1972)

Mechanically-based(2008)

UNBOUNDED DOMAINS

( ) ( ) ( )2 expx E x C A x dσ ε λ ε ξ ξ λ ξ∞

−∞

= + − − ∫

Comparisons between the Gradient and the proposed model of long-range interactions

( ) ( ) ( ) ( ) ( )2

2

d u xE A u x u g x d f x

dxξ ξ ξ

∞

−∞

− − − = − ∫

( ) ( ) ( )2 2

22 21

j

j jj

d u x d u xE r f x

dx dx

∞

=

− = −∑

( ) ( )2

2 2j

j

Ar x g x d

j !ξ ξ ξ

∞

−∞

= − −∫

Taylor series expansion of about location x

( )u x C∞∈

( )u x

Di Paola M., Marino F., Zingales M., 2009, A Generalized Model of Elastic Foundation based on Long-Range Interactions: Integral and Fractional Model, International Journal of Solids and Structures, 46, 3124-3137.

The elastic problem of the 1D Continuum with long-range forces

( ) ( ) ( )( ) ( )

( )

l nl

d x dx x f x

dx dxdu

xdx

σσ σ

ε

= + = −

= Compatibility

Equilibrium

Constitutive

BOUNDARY CONDITIONS

( ) ( )00 Lu u u L u= =

( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )0 0 00

l nlL L LL

l nl

A x A x x N x F

A x A x x N x F

σ σ σ

σ σ σ

= + = =

= + = = −

Kinematic

Static

Di Paola M., Pirrotta A., Zingales M., 2010, Mechanically-based approach to non-local elasticity: Variational principles ,

International Journal of Solids and Structures, Vol.47, 539-548.

( ) ( ) ( )( ) ( )2 1

1 2 1 2 1 20

L x

:x :

dux E A u u g d d

dx ξ ξσ ξ ξ ξ ξ ξ ξ= − − −∫ ∫

( ) ( )( ) ( ) ( )2

20

Ld u

E A u x u g x d f xdx

ξ ξ ξ− − − =∫

( ) ( ) ( ) ( ) ( ) ( )00 0 0 ; l l LEA A F EA L L A L Fε σ ε σ= = − = =

• The decaying function must be symmetric and must belong to the class of monotonically decreasing function of the arguments as from lattice theory.

• One-Dimensional Geometry

( ) ( ) ( ), ,g x g x g xξ ξ ξ= = −

Helmoltz

Bi-Helmoltz

Kröner-Eringen model for unbounded solid

Local Cauchy stress equilibrates the external tractions

(Euclidean)

The Distance-Decaying function

The decaying function: The Fractional Problem

• Fractional Power-Law: ( ) ( ) 1

1

1

c Eg x

xα

αξα ξ +− =

Γ − −

• The Fractional Differential Problem:

( ) ( )2

02 L

d u( x ) f ( x )ˆ ˆc u ( x ) u ( x )Edx

α αα + − − + = −

D D

( )( ) ( )( ) ( )

( )( )

( )( ) ( )( ) ( )

( )( )

1

1

ˆ1

ˆ1

x

aa

b

bx

f x ff x d

x

f x ff x d

x

αα

αα

ξα ξα ξ

ξα ξα ξ

+

−

+

+

−=

Γ − −

−=

Γ − −

∫

∫

D

D

( )( ) ( ) ( ) ( ) ( ) ( )( )ˆ ˆ ; f x f x f x f xα α α α+ + − −= =D D D D ,a b→ −∞ → ∞

• Integral Parts of Marchaud Fractional Derivative

Unbounded Domains

0 1α≤ ≤

Numerical applicationFree-Free bar

21 .40 c m mαα

−=

200 L m m=

-100 -60 -20 20 60 100

x

-0.004

0

0.004

0.008

0.012

ε(x)

proposed modelpoint-spring modellocal model

F

L

F

2E=7200 daN mm

2

10

0.5

100

F KN

A mm

α===

23( ) ( )2

02 L

d u( x ) f ( x )ˆ ˆc u ( x ) u ( x )Edx

α αα + − − + = −

D D

• In real materials the strains ARE NOT UNIFORM in tensile specimen under uniform stress

OK

The 3D Non-Local Elasticity: The long-range forces

The relative displacement

( ) ( ) ( ),k k ku uη = −x xξ ξξ ξξ ξξ ξ

The director vector

( )( ) ( )

, k kk

k k k k

xr

x x

ξξ ξ

−=− −

x ξξξξ

The directional Jacobi tensor

( ),jk k jG r r g= x ξξξξ

The specific long-range force applied in a point x

( ) ( ) ( ), , ,=q x G x xξ ξ η ξξ ξ η ξξ ξ η ξξ ξ η ξDi Paola M., Failla G., Zingales M., 2010, The Mechanically-Based Approach to 3D Non-Local Elasticity Theory: Long-Range

Central Interactions , International Journal of Solids and Structures, doi: 10.1016/j.ijsolstr.2010.02.022.

The 3D Non-Local Elastic Problem

( ) ( ) ( ) ( )

( ) ( ) ( )( )( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

,

, ,

* *

1

2

, ,

2

, , , ,

lkj j k k

kj k j j k

k k k

lkj kj kj hh

k kj j

b f V

u u V

u u V

V

q g V

σ

ε

η

σ µ ε δ λ ε

η

= − − ∀ ∈ = + ∀ ∈ = − ∀ ∈ = + ∀ ∈

= ∀ ∈

x x x x

x x x x

x ξ ξ x x ξ

x x x x

x ξ x ξ x ξ x ξ

( ) ( )( ) ( ) ( )

k k c

lkj j nk f

u u S

n p Sσ

= ∀ ∈

= ∀ ∈

x x x

x x x

Field Equations & boundary Conditions

( ) ( ) ( )

( ) ( ) ( ) ( )

* 2 * *,

, ,

k i ik

kj j kV

u u

g dV b V

µ λ µ

η

∇ + +

+ = − ∈∫

x x

x ξ x ξ ξ x x

Euler-Lagrange Equations

( ) ( ) ( ) ( ) ( ) ( ), , ,k k kj jV V

f q dV g dVη= =∫ ∫x x ξ ξ x ξ x ξ ξ

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

x1 [m]

-1.5E-006

-1.2E-006

-9E-007

-6E-007

-3E-007

0

3E-007

6E-007

9E-007

1.2E-006

1.5E-006

Dis

pla

cem

ent u 1(x

1,-a/

2) a

nd u 1

(x1,0)

Galerkin methodFinite difference method

Local theory, β1=1.0

Local theory, β1=0.9

x2=-a/2

x2=0

O 1x

2x

The 3D Non-Local Elastic Problem: Some Preliminar results

The 3D Non-Local Elastic Problem: Some Preliminar results

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

x1 [m]

0

3E-007

6E-007

9E-007

1.2E-006

1.5E-006

1.8E-006

2.1E-006

2.4E-006

2.7E-006

Str

ain

ε1(

x 1,-

a/2)

an

d ε 1

(x1,

0)

Galerkin methodFinite difference method

Local theory, β1=1.0

Local theory, β1=0.9

x2=-a/2

x2=0

O 1x

2x

Research Developments

Di Paola M., Failla G., Zingales M.,2009, Physically-Based approach to the mechanics of strong non-local linear elasticity, Journal of Elasticity, 97, 103-130.

(Variational Formulation 1D)

Di Paola M., Failla G., Zingales M.,2010, The Mechanically-Based Approach to 3D Non-Local Elasticity Theory: Long-Range Central Interactions , International Journal of Solids and Structures, doi: 10.1016/j.ijsolstr.2010.02.022.

Di Paola M., Pirrotta A., Zingales M.,2010, Physically-Based approach to the mechanics of non-local continuum: Variational Principles, International Journal of Solids and Structures, 47, 539-548.

(ThermodynamicConsistency 1D)

Di Paola M., Marino F., Zingales M.,2009, A Generalized Model of Elastic Foundation based on Long-Range Interactions: Integral and Fractional Model, International Journal of Solids and Structures, 46, 3124-3137.

(Non-Local elasticFoundations)

Failla G., Santini A., Zingales M., 2010, Solution Strategies for 1D Elastic Continuum with Long-Range Cohesive Interactions: Smooth and Fractional Decay, Mechanics Research Communications, 37, 13-21.

(ApproximateSolutions 1D)

(3D Linear Elasticity)

Zingales M.,2009, Waves Propagation in 1D Elastic Solids in presence of Long-Range Central Interactions, Journal of Sound and Vibrations, (accepted).

(1D propagation)

Zingales M., Di Paola M., Inzerillo G.,2009, The Finite Element Method for the Physically-Based Model of Non-Local Continuum, International Journal for Numerical Methods in Engineering, (accepted)

(FEM)

Cottone G., Di Paola M., Zingales M.,2009, Elastic Waves Propagation in 1D Continuum with Fractionally-decaying Long-Range Interactions, Physica E, 42, 95-103.

(1D propagation: Fractional calculus)

Carpinteri A., Cornetti P., Sapora A., Di Paola M. Zingales M.,2009, Fractional calculus in solid

mechanics: local versus non-local approach, Physica Scipta, T136, Article number 014003(1D Comparison)

( )

( )( )

2, 1 , 121

1

22j j j jP

P

b VK K

l

γγ

γ+ +−∆= =

( ) ( ) ( ) ( )( ), 1 , 11

j j j j P PP P j jQ K u u+ +

+= −

( )

( ) ( )2 2 2

, 1

11

j j P P PP

j j

b A l b VK

lx xγ γ

β+

+

∆= =Γ − −

( ) ( ), 1 , 12j j j jh PK h Kγ+ +−=

1 / 2V Al∆ =

The Fractal Mechanical Model: The NN Lattice

• It corresponds to a mechanical, point-spring model as :

( )

( )( )

2, 1 , 122

2

33j j j jP

P

b VK K

l

γγ

γ+ +−∆= =

2 3V Al∆ =

( )

( )2

, 1 1j j PP

b VK

lγ

+ ∆=

( ) ( )( )

( )2

, 1 , 12j j j jP hh P

b VK h K

l hγ

γ+ +−∆

= =

/hV l hA∆ =

( )22 4

, 11

2 2j j p

h h

b Al lK

h h

γφ

−+ = =

[ ] [ ] ( ) [ ] [ ]40 0

ss s s s

h hh h hγφ −Φ = = Φ = Φ

1

4Hdγ

=− 0 4γ< <

• Elastic potential energy invariance at anyobservation scale

The Fractal Mechanical Model: The HB Dimension

Di Paola M., Zingales M., 2009, MultiScale Fractals and Fractional operators, A relevant example: Non-Local Elastic

Continuum, International Journal of Solids and Structures, (submitted).

2

1

jp j

ub A f

xγ −

∆= −

∆

( )2

2p

d ub A f x

dx= − ( )

2

1p j

ub A f x

xγ −

∆ = −∆

3γ = 3???γ ≠Local Euclidean mechanics

Local Fractal mechanics

( ) ( ) ( ) ( )( ), 11 12 2j j h h h

h j j j j

lK u u u f A

h+

+ −− + = −

The Governing Operators• Horizontal Equilibrium of the Generic Element of the NN lattice:

jF

( ) ( ) ( )( ), 11

j j P PP j jK u u−

−− ( ) ( ) ( )( ), 11

j j P PP j jK u u+

+ − ( ) ( ) ( ) ( )( ), 1 , 11

j j j j P PP P j jQ K u u+ +

+= −

• The classical govering equation of the continuum mechanics have been obtainedwithout introducing contact, local, stress in the model. We argue that contact stress isobtained as the resultant ofshort-rangeforces between adjacent particles of solids.

x l h∆ =

The MultiScale Mechanical Fractal (MSF)

( )

( )2

, 1j j P PP

b VK

lγ

+ ∆=

( ) ( ), 1 , 121 2j j j j

PK Kγ+ +−=

( ) ( ), 2 , 121 2j j j j

PK K+ +−=

( ) ( ), 1 , 122 3j j j j

PK Kγ+ +−=

( ) ( )2

, 2 , 12

3

2j j j j

PK Kγ

γ

−+ +=

( ) ( ), 3 , 122 3j j j j

PK K+ +−=

• The interaction distance do not change as we refine the observation scale.

The MultiScale Mechanical Fractal: The Scaling Law • Physical interactions became negligible but cannot vanishbeyond distance l so that

they are mathematically zeroonly as l ∞The MSF is obtained as the union of self-similar elastic chains asn → ∞

( ) ( )2 22 2

, , 1j j i p j jh P

b A lh hK K

i l i

γ γ

γ γ γ

− −+ += =

1

4Hdγ

=− 0 4γ< <

( )

1

limn

jF j

nj

M p E→∞

=

= ∪

It maintains its self-similar natureat any resolution scaleand the scaling law of the springs between different levels:

The MSF Model: ENERGY INVARIANCE

• The Elastic potential energy at the r=1/n scale

( ) 4,

21 1

sn nj j is

h Phi i

nn n

i

γ

γφ−+−

= =

Φ = = Φ ∑ ∑

• The Elastic potential energy at the h=1 scale

( ),1 0 2

1 1

1sn ns

j j isP

i i i γφ +−

= =

Φ = = Φ ∑ ∑( ), 1 21

2j j

P PK l+Φ =

THE INVARIANCE CONDITION

( ) ( )4

, ,1 0 2 2

1 1 1 1

1ssn n n ns

j j i j j is sP P h h

i i i i

nn n

i i

γ

γ γφ φ−

+ +− −

= = = =

Φ = = Φ = Φ = = Φ ∑ ∑ ∑ ∑

1

4Hdγ

=−The HB dimension of the mechanical MSF

Q: What about operators ?• Equilibrium equations at the n observation level:

( ) ( ) ( )( ) ( ) ( ) ( )( )1

. .

1

j

j j i n n j j i n nn j j i n j i j j

i p r

K u u K u u F

− ∞+ +

+ +

=−∞ = +

− − + − = −∑ ∑

( ) ( )( )

( ) ( )( )

( )x

Px

u x u u x ub A d d f x

x xγ γ

ξ ξξ ξ

ξ ξ

∞

−∞

− −+ =

− − ∫ ∫

Di Paola M., Zingales M., 2010, The Multiscale Fractal Approach to Non-Local Elasticity: Direct Connection with

Fractional Operators, International Journal of Solids and Structures, (submitted).

( )

( )

22,

/

j j i pn

b A lK

nil nγ

+ =

j j

lF f A

n=

( ) ( )( )( )

( ) ( )( )( )

1 2 22 2

1

j n n n nP j j i P j i j

j

j j i j i ji i j

b A u u b A u ul l lf A

n n nx x x xγ γ

− ∞+ +

+ +=−∞ = +

− − − + = − − −∑ ∑

0x l n∆ = →

A: Marchaud Derivatives !! ( )( ) ( )( ) ( )Pb A u x u x f xα α+ −

+ = − D D

Q: What about operators ? A: Marchaud fractional derivative

( ) ( )( )

( ) ( )( )

( )x

Px

u x u u x ub A d d f x

x xγ γ

ξ ξξ ξ

ξ ξ

∞

−∞

− −+ =

− − ∫ ∫

Marchaud Fractional derivatives if:

( )1P

cb

Aαα

α=

Γ −

1 1

4 3Hdγ α

= =− −

( )( ) ( )( )

( ) ( )( ) ( )( ) ( )( ) ( )1 11

x

x

u x u u x ucd d c u x u x f x

x xα αα

α α α

ξ ξα ξ ξα ξ ξ

∞

+ + + −−∞

− − + = + = Γ − − −

∫ ∫ D D

1 3α− < <0 4γ< <

Di Paola M., Zingales M., 2009, MultiScale Fractals and Fractional operators, A relevant example: Non-Local Elastic

Continuum, International Journal of Solids and Structures, (submitted).

0 2α< ≤

Fractional Riesz-Weyl potential if: 1 0α− < ≤

Non-admissible for internal stress scaling: 2 3α< <

1γ α= +

[ ] 5c FLγα

−=

The role of the fractal dimension1

4Hdγ

=−

0 4γ< <Simple mechanical fractal: Euclidean solids only with classical differential operators

Multiscale mechanical fractals: Fractional-order operators.

The Euclidean case

( )( ) ( )( )

( ) ( )( ) ( )( ) ( )( ) ( )1 11

x

x

u x u u x ucd d c u x u x f x

x xα αα

α α α

ξ ξα ξ ξα ξ ξ

∞

+ + + −−∞

− − + = + = Γ − − −

∫ ∫ D D

1 1

4 3Hdγ α

= =− −

3γ =2α =

1Hd =

( )( ) ( )( ) ( )2

2 2

2 2 22

d uc u x u x c f x

dx+ − + = − = D D

22E c= ( )2

2

f xd u

dx E= −

The Equilibrium Equation of Cauchy solid

[ ] 4c FLαα

−=

Conclusions• Solid bodies with fractal mass distributions may be studied within the Mechanically-

Based model of Long-Range Interactions and some important conclusions may be withdrawn.

1. The introduction of fractal distributionof the mass density in the solid leads to a fractal mechanical modelrepresented by a point-spring model whose stiffness is power-law decreasing with the interdistance.

2. The assumption that only interactions with adjacent particleis included in the model leads toward a simple mechanical fractal that seems to be ruled by the local version of fractional operators. The order of the operators is connected with the fractal dimension of the mechanical fractal model (study in progress).

3. Assuming that long-range interactions are maintained at any resolution scale and that interactions extends to infinity a Multiscale Fractal mechanical model is obtained. The fractal dimension of the MSF coincides with that of the composing elastic chains.

4. The governing operators of the Multiscale Fractal mechanical modelare Marchaud-type fractional differential equations. Therefore we conclude that such operators rules the physics of multiscale fractal sets.

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