L Approccio Frattale alla Meccanica non Locale (The fractal approach to non-local mechanics) Mario Di Paola

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1 L Approccio Frattae aa Meccanica non Locae (The fracta approach to non-oca mechanics) Mario Di Paoa Università degi Studi di Paermo, Itay Dipartimento di Ingegneria Strutturae, Aerospaziae e Geotecnica (DISAG), Viae dee Scienze Ed.8, 908 Paermo. emai:mario.dipaoa@unipa.it

2 OUTLINES LATTICE THEORY GRADIENT AND STRONG NON-LOCAL THEORY PHYSICALLY-BASED APPROACH TO NON-LOCAL MECHANICS SELECTION OF THE DECAYING FUNCTION FRACTALS vs FRACTIONAL CONCLUSIONS

3 The Cassica Continuum Mechanics (D) Case x F f ( x) V F Equiibrium of soid eement: x V = A x N N f x V = + N N + f ( x) V N A = f x x x 0 dσ dx x = f x Constitutive Equation (LOCAL) σ = Eε = du E dx Boundary conditions: ; ( 0) ; Governing equation of the D soid d u dx EAε = F u = u EAε = F u L = u L L L f = ( x) E

4 The presence of microstructure in rea-materias RENORMALIZATION (WILSON 97) H0 H H Hn b 0 0 b > b 0 Continuum Mechanics Approach Homogeneous and often isotropic eastic materia H 0 H n

5 The Moecuar Dynamics Approach NANOSCALE O ( 0 nm) Each partice has three degree of freedom and its motion is rued by the Newtons aw: u 3n ( t) R M R R 3n 3n 3n 3n Mu ɺɺ t + u t = F t Dispacement vector Mass matrix Eastic bonds matrix MESOSCALE O ( 0.00 µ m) ( 0 ) TERANUMBERS O DEGREE OF FREEDOM INVOLVED!!!

6 The need for an enriched continuum mechanics In the ate fifties and mid-sixties the basis of a generaized continuum mechanics had been proposed considering the inner microstructure The theory of micromorphic continuum

7 The Lattice Mode of Materias (NN) Materia properties often described at moecuar eve (Born-Von arman): a k m u k k u r A point-spring equivaent mode: m a - Mass of Lattice Atoms k F m u k + k + u r u k u F k u + u + k u u + u = F k - Lattice Eastic Constant k = k = EA a a - Lattice Distance F = Externa oad

8 The Continuum Equivaence of the Lattice Modes Derivation of the Waves Equation for an -D mode ( ) u k ( u ) + u k u a = x F = f A x m + EA ( u ) u u f A x + + = x x ( n ) d u dx = f ( x) As x 0 it is impicitey assumed that the same kind of interaction exists at each smaer scae (EUCLIDEAN OBJECT). Lattice eements may exchange interactions sti at distance a (NN) 0 body force fied f x = EA k = ; m = ρ A x x E

9 The Non-Loca Easticity Theories GRADIENT (weak non-ocaity) σ d d ( x ) = E ε ( x ) + E ( x ) E ( x ).. dx ε + dx ε + σ ( x) ε NON-LOCAL Non-oca mode INTEGRAL (strong non-ocaity) Axia stress E, E,E Eastic modui g( x, ξ ) Axia strain σ x = Eε x + g( x, ξ ) ε( ξ )dξ Attenuation function 9

10 Gradient non-oca modes Essentia references Aifantis E.C., 984, On the microstructura origin of certain ineastic modes, Trans. ASME, J. Eng. Mat. Trchn., Vo. 06, Poizzotto C., Borino G., 998, A thermodynamics-based formuation of gradient-dependent pasticity, European Journa of Mechanics A/Soids, Vo.7, Integra non-oca modes Benvenuti E., Borino G., Trai A., 00, A thermodynamicay consistent non oca formuation of damaging materias, European Journa of Mechanics /A Soids, Vo., röner E., 967, Easticity theory of materia with ong range cohesive forces, Internationa Journa of Soids and Structures, Vo. 3, Eringen A.C., Edeen D.G.B., 97, On nooca easticity, Internationa Journa Engineering Science, Vo. 0, Fracta Mechanics Carpinteri A., 994, Fracta nature of materia microstructure and size effects on apparent mechanica properties, Mechanics of Materias, Vo.8, Carpinteri A., Chiaia B., Cornetti P., 00, Static-inematic duaity and the principe of virtua work in the mechanics of fracta media, Computer Methods in Appied Mechanics and Engineering, Vo. 9, 3-9. Carpinteri A., Cornetti P., 00, A fractiona cacuus approach to the description of stress and strain ocaization in fracta media, Chaos Soitons and Fractas, Vo.3, Epstein M., Sniatycki J., 006, Fracta Mechanics, Physica D, Vo. 0,

11 A Different approach THE TEAM Prof. Mario Di Paoa, DISAG, Università di Paermo Prof. Antonina Pirrotta, DISAG, Università di Paermo Dr. Massimiiano Zingaes, DISAG, Università di Paermo Dr. Giuseppe Faia, MecMat, Reggio Caabria Dr. Aba Sofi, Dip. Arte, Scienza e Tecnica de costruire, Reggio Caabria Dr. Giuio Cottone, DISAG, Università di Paermo Dr. Francesco Marino, MecMat, Università di Reggio Caabria Dott. Gianvito Inzerio, DISAG, Università di Paermo

12 The proposed mode (Bounded domain) f ( x ) A x x V h V V h f ( x ) A x ( h,) Q ( h,) Q ( h,) Q ( h,) Q N V A N + ( h, ) ( h, ) (, h ) Q = q V V = q V V h h ( h, ) = ( h )( h ) ( h, ) q sign x x u u g x x Q h= + h= ( h, ) ( h, ) Q = Q Q Di Paoa M., ZingaesM., 008, Long-Range Cohesive Interactions of Non-Loca Continuum Faced by Fractiona Cacuus, Internationa Journa of Soids and Structures, Vo.45, Di Paoa M., FaiaG., ZingaesM., 009, Physicay-Based Approach to the Mechanics of Strong Non-Loca Linear Easticity Theory, Journa of Easticity, Vo. 97,

13 The proposed mode (continue ) ( ) f x A x N V A N + σ ( x) = N ( x) A Q m ( h, ) ( h, ) N + Q + f x A x = N + q A x q ( A x) + f ( x ) A x = h= + h= 0 d u x L 0 ( ξ ) ( ξ ) ξ E A u x u g x d = f x dx BOUNDED DOMAIN d u x ( ξ ) ( ξ ) ξ E A u x u g x d = f x dx UNBOUNDED DOMAIN

14 Mechanica interpretation of non-ocaity LOCAL MODEL F(0) F f x F(L) F f x A x F k k k k F(L) F k = E A x F(0) x = L m L mm- x x x m σ = ; ε = x N x A x N x E A u = f u f T T = [ u u... u m ] [ ] = f f x m Equiibrium of th = node im x 0 A u( x ) f A x d u f x = = x E dx E 4

15 Mechanica interpretation of non-ocaity II NON-LOCAL MODEL = A x g x x h h h = A x g x x h h F F 4 m = h= h = + h m m = u = f SYM m m m m mm 5

16 The stress-strain Reations and the overa Cauchy stress m 3 + σ m ( x) = u u + ( u u ) A = 0 < x < x x m m m + 3 σ m m ( x) = ( u u ) + ( u u ) + ( u3 u ) A = 3 = x < x < x x x m r σ + A = r+ h= ( x) = h ( u uh ) + ( ur ur ) ( u u ) GENERALIZING ( ) = x = r+ h= m r ( h ) ( h ) r r E A u u g x x x r x < x < r + x x 0

17 The stress-strain reations and the overa Cauchy stress (II) m r σ + A = r+ h= ( x) = h ( u uh ) + ( ur ur ) ( u u ) = x = r+ h= m r ( h ) ( h ) r r E A u u g x x x ( ) r x < x < r + x AT THE LIMIT x 0 L x du σ x = E A u ξ u ξ g ξ ξ dξ dξ dx ξ :x ξ : 0 ( x) = ( x) + ( x) = ( N ( x) + Q ( x) ) σ σ σ A L x σ N ( x) du ( x) = = E Q( x) σ x = = A ( u( ξ ) u ( ξ )) g ( ξ ξ ) dξ dξ A dx A ξ : x ξ :0

18 Comparisons between the Eringen mode and the proposed mode of ong-range interactions b = + a σ x Eε x C ε ξ g x ξ dξ ERINGEN (97) b x ξ x ξ = ( ) σ x Eε x A u ξ u ξ g ξ ξ dξdξ = = a Mechanicay-based (008) (POWER-LAW, HELMOLTZ) UNBOUNDED DOMAINS ( ) = exp( ) g x x C x x λ m m σ x = Eε x + Cλ A ε ξ exp x ξ λ dξ (M. Di Paoa, G. Faia, M. Zingaes, 009, Phisicay-Based Approach to the Mechanics of Strong Non-Loca Easticity, Journa of Easticity, Vo. 97, )

19 Comparisons between the Gradient and the proposed mode of ong-range interactions d u x ( ξ ) ( ξ ) ξ E A u x u g x d = f x dx Tayor series expansion of u ( x) about ocation x d u x d u x = dx E r = f x dx u ( x) C A r = ( ξ x) g ( x ξ ) dξ! Di Paoa M., Marino F., Zingaes M., 009, A Generaized Mode of Eastic Foundation based on Long-Range Interactions: Integra and Fractiona Mode, Internationa Journa of Soids and Structures, 46,

20 The eastic probem of the D Continuum with ong-range forces L x du σ x = E A u ξ u ξ g ξ ξ dξ dξ dσ x d = + = dx dx du ε ( x) dx ( σ ( x) σ ( x) ) f ( x) Constitutive Equiibrium = Compatibiity dx ξ :x ξ : 0 u u u L u 0 = 0 = L σ σ σ BOUNDARY CONDITIONS ( ) L A x = A x + x = N x = F L L L σ σ σ A x = A x + x = N x = F inematic Static Di Paoa M., PirrottaA., ZingaesM., 00, Mechanicay-based approach to non-oca easticity: Variationa principes, Internationa Journa of Soids and Structures, Vo.47,

21 The Distance-Decaying function d u L 0 ( ( ξ )) ( ξ ) ξ E A u x u g x d = f x dx ( 0) ε ( 0) σ ( 0 ) ; ε σ EA = A = F EA L L = A L = F 0 L Loca Cauchy stress equiibrates the externa tractions The decaying function must be symmetric and must beong to the cass of monotonicay decreasing function of the arguments as from attice theory. (, ξ ) = ( ξ, ) = ( ξ ) g x g x g x One-Dimensiona Geometry (Eucidean) Hemotz Bi-Hemotz röner-eringen mode for unbounded soid

22 The decaying function: The Fractiona Probem Fractiona Power-Law: g ( x ξ ) cα E = Γ ( α ) x ξ + α 0 α The Fractiona Differentia Probem: ( ˆ α ) ( ˆ α D D ) d u( x ) c u ( x ) u ( x ) f ( x ) α + + = 0 L dx E Integra Parts of Marchaud Fractiona Derivative x ( ˆ f ( x ) f α α ( ξ ) D + ) = a Γ ( α α + ) a x ξ b ( ˆ f x f α α ξ D ) = b Γ ( + α α ) x ξ x f x d ξ f x d ξ Unbounded Domains ( α f )( x) ( ˆ α f )( x) ; ( α f )( x) ( ˆ α + = + = f )( x) D D D D a, b

23 Numerica appication Free-Free bar F L F E=700 dan mm In rea materias the strains ARE NOT UNIFORM in tensie specimen under uniform stress ε(x) F = 0 N α = 0.5 A = 00 mm O proposed mode point-spring mode oca mode c α =.4 0 m m α L = 00 m m x ( ˆ α ) ( ˆ α D D ) d u( x ) c u ( x ) u ( x ) f ( x ) α + + = 0 L dx E 3

24 The 3D Non-Loca Easticity: The ong-range forces The reative dispacement ( x, ξ ) u ( x) u ( ξ ) η = k k k The director vector r k ( x ξ) x ξ k k, = ( x ξ )( x ξ ) k k k k The directiona Jacobi tensor G = r r g x, ξ k k The specific ong-range force appied in a point x q x, ξ = G x, ξ η x, ξ Di Paoa M., FaiaG., ZingaesM., 00, The Mechanicay-Based Approach to 3D Non-Loca Easticity Theory: Long-Range Centra Interactions, Internationa Journa of Soids and Structures, doi: 0.06/.isostr

25 The 3D Non-Loca Eastic Probem Fied Equations & boundary Conditions Euer-Lagrange Equations ( x) u ( x) µ u + λ + µ * * * k i, ik + k ( xξ, ) η ( xξ, ) ( ξ) = k ( x) x V g dv b V ( ) σ k, x = bk x fk x x V f ( x k ) = qk (, ) dv = gk (, ) η (, ) dv xξ ξ V xξ xξ ξ V ε k ( x) = ( uk, ( x) u, k ) V + x x η k ( xξ, ) = uk ( ξ) uk ( x) xξ, V ( ) * * σ k ( x) = µ ε k ( x) + δ kλ ε hh ( x) x V qk ( xξ, ) = gk ( xξ, ) η ( xξ, ) xξ, V ( ) u x = u x x S σ x n = p x x S k k c k nk f

26 The 3D Non-Loca Eastic Probem: Some Preiminar resuts.5e-006.e-006 x 9E-007 Dispacement u (x,-a/) and u (x,0) 6E-007 3E E-007-6E-007-9E-007 O x x =-a/ x =0 Gaerkin method Finite difference method Loca theory, β =.0 -.E-006 Loca theory, β = E x [m]

27 The 3D Non-Loca Eastic Probem: Some Preiminar resuts x.7e-006 O x.4e-006 Strain ε (x,-a/) and ε (x,0).e-006.8e-006.5e-006.e-006 9E-007 6E-007 3E-007 Gaerkin method Finite difference method Loca theory, β =.0 Loca theory, β =0.9 x =-a/ x = x [m]

28 Research Deveopments Di Paoa M., Pirrotta A., Zingaes M., 00, Physicay-Based approach to the mechanics of non-oca continuum: Variationa Principes, Internationa Journa of Soids and Structures, 47, Di Paoa M., Faia G., Zingaes M., 009, Physicay-Based approach to the mechanics of strong non-oca inear easticity, Journa of Easticity, 97, Di Paoa M., Marino F., Zingaes M., 009, A Generaized Mode of Eastic Foundation based on Long-Range Interactions: Integra and Fractiona Mode, Internationa Journa of Soids and Structures, 46, Faia G., Santini A., Zingaes M., 00, Soution Strategies for D Eastic Continuum with Long- Range Cohesive Interactions: Smooth and Fractiona Decay, Mechanics Research Communications, 37, 3-. (Variationa Formuation D) (Thermodynamic Consistency D) (Non-Loca eastic Foundations) (Approximate Soutions D) Di Paoa M., Faia G., Zingaes M., 00, The Mechanicay-Based Approach to 3D Non-Loca Easticity Theory: Long-Range Centra Interactions, Internationa Journa of Soids and Structures, doi: 0.06/.isostr (3D Linear Easticity) Zingaes M., Di Paoa M., Inzerio G., 009, The Finite Eement Method for the Physicay-Based Mode of Non-Loca Continuum, Internationa Journa for Numerica Methods in Engineering, (accepted) Zingaes M., 009, Waves Propagation in D Eastic Soids in presence of Long-Range Centra Interactions, Journa of Sound and Vibrations, (accepted). (FEM) (D propagation) Cottone G., Di Paoa M., Zingaes M., 009, Eastic Waves Propagation in D Continuum with Fractionay-decaying Long-Range Interactions, Physica E, 4, Carpinteri A., Cornetti P., Sapora A., Di Paoa M. Zingaes M., 009, Fractiona cacuus in soid mechanics: oca versus non-oca approach, Physica Scipta, T36, Artice number (D propagation: Fractiona cacuus) (D Comparison)

29 The Fracta Mechanica Mode: The NN Lattice It corresponds to a mechanica, point-spring mode as : + (, + ) (, + ) ( P) ( P) Q = u u P P b A b V (, + ) P P P P = = ( β ) ( ) Γ x x γ γ + V = A / γ, + P γ, + = = γ P ( ) b ( ) V ( ) γ, + P γ, + = = 3 γ P ( ) 3 b ( ) V ( ) V = A 3 (, + ) γ (, + ) = h h P

30 The Fracta Mechanica Mode: The HB Dimension, + bp V P = γ ( ) V = h / ha, + γ, + = = P ( ) bp ( Vh ) h γ ( h) h ( ) φ h 4 γ (, + ) bp A = h = h h Eastic potentia energy invariance at any observation scae d H 4 s [ ] h[ ] h( h γ Φ = φ = ) [ Φ ] = [ Φ ] = 4 γ 0 < γ < 4 h s s s s h 0 0 Di Paoa M., ZingaesM., 009, MutiScaeFractas and Fractiona operators, A reevant exampe: Non-Loca Eastic Continuum, Internationa Journa of Soids and Structures, (submitted).

31 The Governing Operators Horizonta Equiibrium of the Generic Eement of the NN attice: ( ) (, ) ( P) ( P) u u P F (, + ) ( P) ( P) (, + ) (, + ) ( P) ( P) P + u u ( ) + (, + ) ( h ) ( h ) ( h ) h u u + u = f A h + Q = u u P P Loca Eucidean mechanics p γ = 3 u γ 3??? bp A = f x γ d u dx = x = h u bp A = f ( x ) x γ b A f x Loca Fracta mechanics The cassica govering equation of the continuum mechanics have been obtained without introducing contact, oca, stress in the mode. We argue that contact stress is obtained as the resutant of short-range forces between adacent partices of soids.

32 The MutiScae Mechanica Fracta (MSF) The interaction distance do not change as we refine the observation scae. (, + ) bp VP P = γ (, + ) γ (, + ) = P (, + ) (, + ) = P (, + ) γ (, + ) = 3 P 3 γ = γ P (, + ) (, + ) (, + 3 ) (, + ) = 3 P

33 The MutiScae Mechanica Fracta: The Scaing Law Physica interactions became negigibe but cannot vanish beyond distance so that they are mathematicay zero oy as The MSF is obtained as the union of sefsimiar eastic chains as n M F n = im n = p E It maintains its sef-simiar nature at any resoution scae and the scaing aw of the springs between different eves: γ γ (, + i) p (, + ) h h b A h = = i i γ γ γ P d H 4 γ = 0 < γ < 4

34 The MSF Mode: ENERGY INVARIANCE The Eastic potentia energy at the h= scae Φ = P ( + ) P, n s n s (, + i) Φ = φ 0 = Φ i i i γ = = P s The Eastic potentia energy at the r=/n scae n γ 4 s (, + i) n Φ h = n φ h = n Φ P i γ i= i= THE INVARIANCE CONDITION 4 s n s s n n γ n s (, + i) n (, + i) s φ 0 P n P n φ γ γ h h i= i= i i= i i= Φ = = Φ = Φ = = Φ d H = 4 γ The HB dimension of the mechanica MSF n s

35 Q: What about operators? Equiibrium equations at the n observation eve: F = f A n (. + i) ( ( n) ( n) ) (. + i) ( ( n) ( n) ) = u u u u F n i n i i= p= r+ ( ) ( ) ( ξ ) γ ( x ξ ) (, i) ( ξ ) γ ( x ξ ) + p ( / ) x u x u u x u b A dξ dξ f x P + = x n ( ) ( ) b A = i n γ n n n n n bp A u u + i bp A u + i u f A γ γ x n n n i x + i x i + i x = = + + = x = n 0 A: Marchaud Derivatives!! b A α ( u )( x α P D ) ( u )( x ) f ( x + + D = ) Di Paoa M., Zingaes M., 00, The MutiscaeFracta Approach to Non-Loca Easticity: Direct Connection with Fractiona Operators, Internationa Journa of Soids and Structures, (submitted).

36 Q: What about operators? A: Marchaud fractiona derivative ( ξ ) γ ( x ξ ) ( ξ ) γ ( x ξ ) x u x u u x u b A dξ dξ f x P + = x d H = = 4 γ 3 α x u ( x) u ( ξ ) u ( x) u ( ξ ) α ξ α α α ( x ) ξ x ( x ) D D Marchaud Fractiona derivatives if: b αcα = Γ α P A ( ) γ = + α 5 [ c ] FL γ α = α αc α d + d c u x u x f x + + = + = + Γ( α ) ξ ξ 0 < γ < 4 < α < 3 0 < α Fractiona Riesz-Wey potentia if: < α 0 Non-admissibe for interna stress scaing: < α < 3 Di Paoa M., ZingaesM., 009, MutiScaeFractas and Fractiona operators, A reevant exampe: Non-Loca Eastic Continuum, Internationa Journa of Soids and Structures, (submitted).

37 d H 4 γ Simpe mechanica fracta: Eucidean soids oy with cassica differentia operators The roe of the fracta dimension = 0 < γ < 4 Mutiscae mechanica fractas: Fractionaorder operators.

38 The Eucidean case d H = = 4 γ 3 α x u ( x) u ( ξ ) u ( x) u ( ξ ) α ξ α α α ( x ) ξ x ( x ) D D αc α d + d c u x u x f x + + = + = + Γ( α ) ξ ξ α γ = 3 d H = α = 4 [ c ] FL α = α d u ( ) ( D + D ) = = c u x u x c f x + dx The Equiibrium Equation of Cauchy soid E d u = c dx = f E x

39 Concusions Soid bodies with fracta mass distributions may be studied within the Mechanicay- Based mode of Long-Range Interactions and some important concusions may be withdrawn.. The introduction of fracta distribution of the mass density in the soid eads to a fracta mechanica mode represented by a point-spring mode whose stiffness is power-aw decreasing with the interdistance.. The assumption that oy interactions with adacent partice is incuded in the mode eads toward a simpe mechanica fracta that seems to be rued by the oca version of fractiona operators. The order of the operators is connected with the fracta dimension of the mechanica fracta mode (study in progress). 3. Assuming that ong-range interactions are maintained at any resoution scae and that interactions extends to infinity a Mutiscae Fracta mechanica mode is obtained. The fracta dimension of the MSF coincides with that of the composing eastic chains. 4. The governing operators of the Mutiscae Fracta mechanica mode are Marchaud-type fractiona differentia equations. Therefore we concude that such operators rues the physics of mutiscae fracta sets.

40 THAN YOU FOR THE ATTENTION!