Sandra Carillo. SAPIENZA Università diroma. Singular Kernel Problems in Materials with Memory. . p.1

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1 . p.1 Sandra Carillo SAPIENZA Università diroma Singular Kernel Problems in Materials with Memory

2 Sandra Carillo SAPIENZA Università diroma Singular Kernel Problems in Materials with Memory talk s outline... Results... bibliography : Work in progress... Perspectives & Open Problems... S.C. V.Valente & G.Vergara Caffarelli, 211, 212a, 212b; G.Amendola, S.C. & A.Manes, 21; G.Amendola, S.C., J.M. Golden & A.Manes, preprint 213; G.Amendola, S.C., 24; S.C., 25, 21, 211a & 211b.. p.1

3 Sandra Carillo SAPIENZA Università diroma Singular Kernel Problems in Materials with Memory talk s outline The Viscoelastic Material Model... Results... bibliography : Work in progress... Perspectives & Open Problems... S.C. V.Valente & G.Vergara Caffarelli, 211, 212a, 212b; G.Amendola, S.C. & A.Manes, 21; G.Amendola, S.C., J.M. Golden & A.Manes, preprint 213; G.Amendola, S.C., 24; S.C., 25, 21, 211a & 211b.. p.1

4 Sandra Carillo SAPIENZA Università diroma Singular Kernel Problems in Materials with Memory talk s outline The Viscoelastic Material Model Singular Memory Kernel... Results... bibliography : Work in progress... Perspectives & Open Problems... S.C. V.Valente & G.Vergara Caffarelli, 211, 212a, 212b; G.Amendola, S.C. & A.Manes, 21; G.Amendola, S.C., J.M. Golden & A.Manes, preprint 213; G.Amendola, S.C., 24; S.C., 25, 21, 211a & 211b.. p.1

5 Sandra Carillo SAPIENZA Università diroma Singular Kernel Problems in Materials with Memory talk s outline The Viscoelastic Material Model Singular Memory Kernel Magneto-Viscoelasticity... Results... Existence & Unicity of Solutions bibliography : Work in progress... Perspectives & Open Problems... S.C. V.Valente & G.Vergara Caffarelli, 211, 212a, 212b; G.Amendola, S.C. & A.Manes, 21; G.Amendola, S.C., J.M. Golden & A.Manes, preprint 213; G.Amendola, S.C., 24; S.C., 25, 21, 211a & 211b.. p.1

6 bibliography : Sandra Carillo SAPIENZA Università diroma Singular Kernel Problems in Materials with Memory talk s outline The Viscoelastic Material Model Singular Memory Kernel Magneto-Viscoelasticity... Results... Existence & Unicity of Solutions Asymptotic behaviour of Solutions Work in progress... Perspectives & Open Problems... S.C. V.Valente & G.Vergara Caffarelli, 211, 212a, 212b; G.Amendola, S.C. & A.Manes, 21; G.Amendola, S.C., J.M. Golden & A.Manes, preprint 213; G.Amendola, S.C., 24; S.C., 25, 21, 211a & 211b.. p.1

7 General Framework of the Problem the viscoelastic material model FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.2

8 General Framework of the Problem the viscoelastic material model Ω R 3 connected set: the body configuration; FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.2

9 General Framework of the Problem the viscoelastic material model Ω R 3 connected set: the body configuration; viscoelastic body = shape change of Ω according to linear viscoelasticity; FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.2

10 General Framework of the Problem the viscoelastic material model Ω R 3 connected set: the body configuration; viscoelastic body = shape change of Ω according to linear viscoelasticity; memory effects = the deformation depends on time via present and past times; FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.2

11 General Framework of the Problem the viscoelastic material model Ω R 3 connected set: the body configuration; viscoelastic body = shape change of Ω according to linear viscoelasticity; memory effects = the deformation depends on time via present and past times; passive environment = environment s status is not affected by the body s one. FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.2

12 General Framework of the Problem the viscoelastic material model Ω R 3 connected set: the body configuration; viscoelastic body = shape change of Ω according to linear viscoelasticity; memory effects = the deformation depends on time via present and past times; passive environment = environment s status is not affected by the body s one. no space dependence... in addition... key references i.e. x-independence M. Fabrizio, A. Morro, 1992; G. Gentili, 21 FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.2

13 Quantities of Interest E = E (t) T = T (t) G = G (t) G = G () strain tensor stress tensor relaxation modulus initial relaxation modulus FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.3

14 Quantities of Interest E = E (t) T = T (t) G = G (t) G = G () strain tensor stress tensor relaxation modulus initial relaxation modulus furthermore...constitutive assumptions FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.3

15 Quantities of Interest E = E (t) T = T (t) G = G (t) G = G () strain tensor stress tensor relaxation modulus initial relaxation modulus furthermore...constitutive assumptions T (t) = G(τ) Ė (t τ) dτ, G(t) =G + t Ġ(s) ds FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.3

16 Quantities of Interest E = E (t) T = T (t) G = G (t) G = G () strain tensor stress tensor relaxation modulus initial relaxation modulus T (t) = furthermore...constitutive assumptions G(τ) Ė (t τ) dτ, G(t) =G + t Ġ(s) ds or equivalently, when E t (τ) denotes the strain past history T (t) =G E(t)+ Ġ(τ) E t (τ) dτ, E t (τ) :=E(t τ) FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.3

17 Regular Memory Kernel 1-dimensional Viscoelastic Problem Classical Problem S.C, V.Valente, G.Vergara Caffarelli 211, 212 (3-D) u tt = G()u xx + t Ġ(t τ)u xx (τ)dτ + f u(, ) = u, u t (, ) = u 1 in Ω ; u =on Σ= Ω (,T) u displacement and f external force (history of the material included) Ġ L 1 (IR + ), G(t) =G + Ġ(s) ds, G( ) = lim G(t) t Hence, G enjoys the fading memory property ε > ã = a (ε, E t ) IR + s.t. a >ã, Ġ(s + a)e t (s) ds <ε t FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.4

18 Regular Memory Kernel 1-dimensional Viscoelastic Problem Classical Problem S.C, V.Valente, G.Vergara Caffarelli 211, 212 (3-D) u tt = G()u xx + t Ġ(t τ)u xx (τ)dτ + f u(, ) = u, u t (, ) = u 1 in Ω ; u =on Σ= Ω (,T) u displacement and f external force (history of the material included) Ġ L 1 (IR + ), G(t) =G + Ġ(s) ds, G( ) = lim G(t) t Hence, G enjoys the fading memory property ε > ã = a (ε, E t ) IR + s.t. a >ã, Ġ(s + a)e t (s) ds <ε t FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.5

19 Singular Memory Kernel 1-dimensional Viscoelastic Problem Idea S.C, V.Valente, G.Vergara Caffarelli 212b relax the assumptions on G: require G L 1 (,T), T IR... hence it may be lim G(t) =+ t + construct regular approximated problems:... but how? find approximated solutions u ε ; show u := lim u ε ε recognize uniqueness of u solution to the singular problem. FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.6

20 Approximation Strategy: Idea Approximated Problems approximated problems <ε<1,g ε ( ) =G(ε + ) u ε tt = G ε () u ε xx + t Ġ ε (t τ) u ε xx(τ) dτ + f, i.c. & b.c. integral formulation of the problems u ε (t) = t K ε (t τ)u ε xx(τ)dτ + u 1 t + u + t dτ τ f(ξ)dξ, K(ξ) := ξ G(τ)dτ well defined since G L 1 (,T), T IR. FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.7

21 Existence of the limit Solution Theorem Given u ε solution to the integral problem P ε t t τ P ε : u ε (t) = then K ε (t τ)u ε xx(τ)dτ + u 1 t + u + dτ f(ξ)dξ, u(t) = lim ε u ε (t) in L 2 (Q), Q =Ω (,T). Proof (Idea): weak formulation, on introduction of test functions ϕ H 1 (Ω (,T)) s.t. ϕ =on Ω; consider separately the terms without ε; the terms with u ε & K ε prove convergence via Lebesque s Theorem. FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.8

22 Solution uniqueness Theorem u(t) = The integral problem t K(t τ)u xx (τ)dτ + u 1 t + u + t dτ τ f(ξ)dξ, Proof admits a unique weak solution. (by contradiction) Let w := v ṽ, v ṽ, wherev, ṽ are weak solutions = w satisfies t w(t)ψ(x, t)dxdt = ψ xx (x, t) K(t τ)w(τ)dτdxdt R recall the b.c.s w(,t)=w(π, t) = t (,T), choose the test function ψ(x, t) and w m (x, t) via ψ(x, t) =ϕ(t)sin(mx), w m (x, t) = α m (t)sin(mx); R m= orthogonality combined with Gronwall s Lemma implies m N, T [ t ] ϕ(t) α m (t) K(t τ)m 2 α m (t)dτ dt = = α m (t) =. FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.9

23 Free Energy solution s existence relies on Free Energy; the same in magneto-viscoelasticity (1 D);... and 3 D too; see M. Fabrizio, C. Giorgi, V. Pata... and many other authors... next steps 2D(in progress), then, 3 D singular problem; 1 D singular magneto-viscoelasticity problem; 2-3 D singular magneto-viscoelasticity problem. FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.1

24 General Framework of the Problem the magnetic model FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.11

25 General Framework of the Problem the magnetic model Ω R 3 connected set: the body configuration; FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.11

26 General Framework of the Problem the magnetic model Ω R 3 connected set: the body configuration; magnetic body = magnetization of Ω changes according to the Landau Lifshitz equation: FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.11

27 General Framework of the Problem the magnetic model Ω R 3 connected set: the body configuration; magnetic body = magnetization of Ω changes according to the Landau Lifshitz equation: in Gilbert form, where m represents the magnetization vector γ 1 m t m (a m m t )=, m =1, γ,a IR + FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.11

28 General Framework of the Problem the magnetic model Ω R 3 connected set: the body configuration; magnetic body = magnetization of Ω changes according to the Landau Lifshitz equation: in Gilbert form, where m represents the magnetization vector γ 1 m t m (a m m t )=, m =1, γ,a IR + references M. Bertsch, P. Podio Guidugli and V. Valente, 21 T. L. Gilbert, 1955 FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.11

29 General Framework of the Problem the magneto-viscoelasticity model FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.12

30 General Framework of the Problem the magneto-viscoelasticity model interaction between the two different effects; FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.12

31 General Framework of the Problem the magneto-viscoelasticity model interaction between the two different effects; model equations 3dimensional FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.12

32 General Framework of the Problem the magneto-viscoelasticity model interaction between the two different effects; model equations ρü div 3 dimensional γ 1 ṁ m (a m ṁ Lm u) = ( t G() u + Ġ(t τ) u(τ)dτ + 1 )=f 2 Lm m FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.12

33 General Framework of the Problem the magneto-viscoelasticity model interaction between the two different effects; model equations ρü div model equations 3 dimensional γ 1 ṁ m (a m ṁ Lm u) = ( t G() u + Ġ(t τ) u(τ)dτ + 1 )=f 2 Lm m u tt G()u xx m t + m m 2 1 ε 1 dimensional t Ġ(t τ)u xx (τ)dτ λ 2 + λλ(m)u x m xx =, ( ) Λ(m) m = f, x FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.12

34 General Framework of the Problem the magneto-viscoelasticity model interaction between the two different effects; model equations ρü div model equations 3 dimensional γ 1 ṁ m (a m ṁ Lm u) = ( t G() u + Ġ(t τ) u(τ)dτ + 1 )=f 2 Lm m u tt G()u xx m t + m m 2 1 ε 1 dimensional t Ġ(t τ)u xx (τ)dτ λ 2 + λλ(m)u x m xx =, ( ) Λ(m) m = f, x references S. C., V. Valente, G. Vergara Caffarelli, D.C.D.S. Series S 212 S. C., V. Valente, G. Vergara Caffarelli, Appl. Anal. 211 FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.12

35 The evolution problem 3-dim ρü div γ 1 ṁ m (a m ṁ Lm u) = ( G() u + t (1) Ġ(t τ) u(τ)dτ Lm m )=f FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.13

36 The evolution problem 3-dim ρü div γ 1 ṁ m (a m ṁ Lm u) = ( G() u + t (1) Ġ(t τ) u(τ)dτ Lm m )=f initial m() = m, m =1, u() = u, u() = u 1 (2) and boundary conditions m ν Ω =, u Ω =. (3) FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.13

37 The evolution problem 1-dim u tt G()u xx m t + m m 2 1 ε t Ġ(t τ)u xx (τ)dτ λ 2 (Λ(m) m) x = f, + λλ(m)u x m xx =, FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.14

38 The evolution problem 1-dim u tt G()u xx m t + m m 2 1 ε t Ġ(t τ)u xx (τ)dτ λ 2 (Λ(m) m) x = f, + λλ(m)u x m xx =, initial u(, ) = u, u t (, ) = u 1, m(, ) = m, m =1 inω and boundary conditions u =, m ν = on Σ= Ω (,T) FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.14

39 Quantities of Interest u := u (x,t) displacement vector m := m (x,t) magnetization vector G(s) ={G klmn (s)}, s [,T] visco-elasticity tensor L = {λ klmn } magneto-elasticity tensor E = {ɛ lm } strain tensor G klmn ɛ kl (u)ɛ mn (v) =G u v, {λ klmn m k m l } = Lm m λ ijkl = λ 1 δ ijkl + λ 2 δ ij δ kl + λ 3 (δ ik δ jl + δ il δ jk ) {λ klmn m k ɛ lm (u)} = Lm u λ klmn m k m l ɛ mn (u) =Lm m u ɛ lm (u) = 1 ( ) 2 ul,m + u m,l ɛ(u) deformation tensor furthemore...constitutive assumptions FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.15

40 Constitutive Assumptions FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.16

41 Constitutive Assumptions E ex (m) = 1 2 Ω a ij m k,i m k,j dω a ij = a ji a ij = aδ ji, a IR + exchange magnetization energy symmetric positive definite matrix diagonal matrix (most materials) FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.16

42 Constitutive Assumptions E ex (m) = 1 2 Ω a ij m k,i m k,j dω a ij = a ji a ij = aδ ji, a IR + exchange magnetization energy symmetric positive definite matrix diagonal matrix (most materials) E em (m, u) = 1 2 Ω λ ijkl m i m j ɛ kl (u)dω magneto-elastic energy λ ijkl = λ 1 δ ijkl + λ 2 δ ij δ kl + λ 3 (δ ik δ jl + δ il δ jk ); δ ijkl =1if i = j = k = l and δ ijkl =otherwise; λ 1,λ 2,λ 3 IR constants. FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.16

43 Constitutive Assumptions FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.17

44 Constitutive Assumptions 1 2 t dτ ( Ω E ve (u) = 1 G klmn ()ɛ kl ɛ mn dω+ 2 Ω ) Ġ klmn (t τ)ɛ kl (τ)ɛ mn (τ)dω viscoelastic energy FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.17

45 Constitutive Assumptions 1 2 t dτ ( Ω E ve (u) = 1 G klmn ()ɛ kl ɛ mn dω+ 2 Ω ) Ġ klmn (t τ)ɛ kl (τ)ɛ mn (τ)dω viscoelastic energy G klmn = G mnkl = G lkmn G klmn e kl e mn βe kl e kl, β >, e kl = e lk Ġ klmn e kl e mn Gklmn e kl e mn FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.17

46 Magneto-Viscoelasticity Model Results FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.18

47 Magneto-Viscoelasticity Model 1-dimensional problem (1) : u tt G()u xx m t + m m 2 1 ε t Results Ġ(t τ)u xx (τ)dτ λ 2 ( ) Λ(m) m = f, x + λλ(m)u x m xx = + i.b.c FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.18

48 Magneto-Viscoelasticity Model 1-dimensional problem (1) : u tt G()u xx m t + m m 2 1 ε t Results Ġ(t τ)u xx (τ)dτ λ 2 ( ) Λ(m) m = f, x + λλ(m)u x m xx = + i.b.c existence & uniqueness of the weak solution (1) FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.18

49 Magneto-Viscoelasticity Model 1-dimensional problem (1) : u tt G()u xx m t + m m 2 1 ε t Results Ġ(t τ)u xx (τ)dτ λ 2 ( ) Λ(m) m = f, x + λλ(m)u x m xx = + i.b.c existence & uniqueness of the weak solution (1) 3-dimensional problem (2) : ρü div ( G() u + γ 1 ṁ m (a m ṁ Lm u) = t Ġ(t τ) u(τ)dτ Lm m )=f + i.b.c FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.18

50 Magneto-Viscoelasticity Model 1-dimensional problem (1) : u tt G()u xx m t + m m 2 1 ε t Results Ġ(t τ)u xx (τ)dτ λ 2 ( ) Λ(m) m = f, x + λλ(m)u x m xx = + i.b.c existence & uniqueness of the weak solution (1) 3-dimensional problem (2) : ρü div ( G() u + γ 1 ṁ m (a m ṁ Lm u) = t existence of weak solution (2) Ġ(t τ) u(τ)dτ Lm m )=f + i.b.c FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.18

51 Magneto-Viscoelasticity Model 1-dimensional problem (1) : u tt G()u xx m t + m m 2 1 ε t Results Ġ(t τ)u xx (τ)dτ λ 2 ( ) Λ(m) m = f, x + λλ(m)u x m xx = + i.b.c existence & uniqueness of the weak solution (1) 3-dimensional problem (2) : ρü div ( G() u + γ 1 ṁ m (a m ṁ Lm u) = t Ġ(t τ) u(τ)dτ Lm m )=f + i.b.c existence of weak solution (2) (1) S. C., V. Valente, G. Vergara Caffarelli, Appl. Anal. 211 (2) S. C., V. Valente, G. Vergara Caffarelli,, D.C.D.S. Series S 212 FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.18

52 Theorem (1-D. pb.) FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.19

53 Theorem (1-D. pb.) Assumptions: T> small enough (i.e. ε<λ 2 G(T )) FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.19

54 Theorem (1-D. pb.) Assumptions: T> small enough (i.e. ε<λ 2 G(T )) =! weak solution to the problem (1), (2), (3) such that: u C ([,T]; H 1 (Ω)) C1 ([,T]; L 2 (Ω)) m C ([,T]; H 1 (Ω)) L 2 (,T; H 2 (Ω)) m t L 2 (,T; L 2 (Ω)), FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.19

55 Theorem (1-D. pb.) Assumptions: T> small enough (i.e. ε<λ 2 G(T )) =! weak solution to the problem (1), (2), (3) such that: u C ([,T]; H 1 (Ω)) C1 ([,T]; L 2 (Ω)) m C ([,T]; H 1 (Ω)) L 2 (,T; H 2 (Ω)) m t L 2 (,T; L 2 (Ω)), key tools viscoelastic term viscoelastic free energy; Gronwall s lemma; FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.19

56 Theorem (1-D. pb.) Assumptions: T> small enough (i.e. ε<λ 2 G(T )) =! weak solution to the problem (1), (2), (3) such that: u C ([,T]; H 1 (Ω)) C1 ([,T]; L 2 (Ω)) m C ([,T]; H 1 (Ω)) L 2 (,T; H 2 (Ω)) m t L 2 (,T; L 2 (Ω)), key tools viscoelastic term viscoelastic free energy; Gronwall s lemma; provide an a priori estimate of the viscoelastic term 1 2 Ω ϕ x 2 dx Ω ϕ t 2 dx αe T C(f,ϕ,ϕ 1 ),α,c IR + FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.19

57 Theorem (1-D. pb.) Proof s Ingredients FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.2

58 Theorem (1-D. pb.) Proof s Ingredients a priori estimates (coupled system); FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.2

59 Theorem (1-D. pb.) Proof s Ingredients a priori estimates (coupled system); via two Lemmas FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.2

60 Theorem (1-D. pb.) Proof s Ingredients a priori estimates (coupled system); via two Lemmas ( the 2nd provides a uniform a priori estimate ) FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.2

61 Theorem (1-D. pb.) Proof s Ingredients a priori estimates (coupled system); via two Lemmas ( the 2nd provides a uniform a priori estimate ) fixed point theorem. FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.2

62 Theorem (3-D. pb.) Assumptions: u H 1 (Ω; R 3 ) u 1 L 2 (Ω; R 3 ) m H 1 (Ω; R 3 ), m =1a.e. in Ω f L 2 (Q; R 3 ), Q =Ω [,T] G(s) C 2 [,T] = (m, u) weak solution to the problem (1), (2), (3) such that: m H 1 (Q; R 3 ) with m =1 a.e. in Q u L 2 (,T; H(Ω; 1 R 3 )) and u L 2 (Q; R 3 ) (p, g) s.t. g H(Q; 1 R 3 ), p H 1, (Q; R 3 ), p() = p(t )=, [ γ 1ṁ p + a(m m) p + m (ṁ + Lm p u) ] dωdt = Q [ ρ u ġ +(G() u + 1 Lm m) g] dωdt + Q 2 ( t ) + Ġ(t τ) u(τ) g(τ)dτ dωdt f g dωdt = Q Q FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.21

63 Theorem s Proof Outline: Approximated penalty problem: introduce a small positive parameter ε and consider in Q an approximated i.b.v. problem; consider the related Faedo-Galerkin approximation Convergence of the approximate solutions; γ 1 ṁ ε m ε + ṁ ε a m ε + Lm ε u ε + ε 1 ( m ε 2 1)m ε = ρü ε div ( G() u ε + t Approximated penalty problem Ġ(t τ) u ε (τ)dτ Lmε m ε ) = f u ε (, ) = u, u ε (, ) = u 1, m ε (, ) = m, m =1 inω u ε =, m ε ν = on Σ= Ω [,T] i.b.c FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.22

64 Comparison: 1-D & 3-D Results 1-D = existence & uniqueness 3-D = global existence... but no uniqueness FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.23

65 Comparison: 1-D & 3-D Results 1-D aprioriestimates; = existence & uniqueness 3-D = global existence... but no uniqueness FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.23

66 Comparison: 1-D & 3-D Results 1-D aprioriestimates; fixed point theorem = existence & uniqueness 3-D = global existence... but no uniqueness FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.23

67 Comparison: 1-D & 3-D Results 1-D aprioriestimates; fixed point theorem = existence & uniqueness 3-D Approximated penalty problem = global existence... but no uniqueness FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.23

68 Comparison: 1-D & 3-D Results 1-D aprioriestimates; fixed point theorem = existence & uniqueness 3-D Approximated penalty problem Faedo-Galerkin approximation = global existence... but no uniqueness FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.23

69 Comparison: 1-D & 3-D Results 1-D aprioriestimates; fixed point theorem = existence & uniqueness 3-D Approximated penalty problem Faedo-Galerkin approximation Local Existence = global existence... but no uniqueness FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.23

70 Comparison: 1-D & 3-D Results 1-D aprioriestimates; fixed point theorem = existence & uniqueness 3-D Approximated penalty problem Faedo-Galerkin approximation Local Existence Prolongation = global existence... but no uniqueness FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.23

71 References G.Amendola, S.C., J.M. Golden & A.Manes, preprint 213 G. Amendola, S.C., Quart.J.Mech.Appl. Math. 57(3), (24); K. Adolfsson, M. Enelund, P. Olsson, On the Fractional Order Model of Viscoelasticity, Mechanics of Time-Dependent Materials 9, (25), 15-34C. V. Berti, Existence and uniqueness for an integral-differential equation with singular kernel, Boll. Un. Mat. Italiana, Sez. B., 9-B, 26, Boltzmann, L., Zur theorie der elastichen nachwirkung, Annalen der physik und chemie 77, 1876, S. C., J. Nonlinear Math. Phys. 12, (25), suppl. 1; S.C, V.Valente, G.Vergara Caffarelli A result of existence and uniqueness for an integro-differential system in magneto-viscoelasticity, Applicable Analisys, , (9) n.ro 12, (211); S.C, V.Valente, G.Vergara Caffarelli An existence theorem for the magneto- -viscoelastic problem D.C.D.S. Series S., (5) 3, (212); S.C, V.Valente, G.Vergara Caffarelli, 212; FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.24

72 References (viscoelastic pb.) L. Deseri, M. Fabrizio, J.M. Golden, (24); Enelund, M.,Mähler, L., Runesson, K. and Josefson, B.L., Formulation and integration of the standard linear viscoelastic solid with fractional order rate laws, Int. J. Solids Struc., 36, 1999, Enelund, M. and Olsson, P., Damping described by fading memory Analysis and application to fractional derivative models, Int.J.Solids Str. 36, 1999, M.Fabrizio,inMathematical Models and Methods for Smart Materials, Ed.s: M. Fabrizio, B. Lazzari, A. Morro, World Sci. 23; M.Fabrizio,J.M.Golden,Quart. Appl. Math. 6 no.2(22); M. Fabrizio, J. M. Golden, Rend. Mat. Appl. 2 no. 7(2); G. Gentili,Quart. Appl. Math. 6 no. 1 (22); C. Giorgi, G. Gentili, Quart. Appl. Math. 51 (2) (1993); J.M.Golden,Quart. Appl. Math. 58 (2). C. Giorgi and A. Morro, Viscoelastic solids with unbounded relaxation function, Continuum Mechanics And Thermodynamics, 4 n.2, (1992), , FCPNLO Bilbao November the 7th, 213 Sandra Carillo p.25

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