Quasistatic Nonlinear Viscoelasticity and Gradient Flows

Transcription

1 Quasistatic Nonlinear Viscoelasticity and Gradient Flows Yasemin Şengül University of Coimbra PIRE - OxMOS Workshop on Pattern Formation and Multiscale Phenomena in Materials University of Oxford September 2011

2 Introduction We will study the quasistatic equation for one-dimensional nonlinear viscoelasticity of strain-rate type ( σ(yx ) + S(y x, y xt ) ) = 0, x (0, 1), t [0, T ]. x

3 Introduction We will study the quasistatic equation for one-dimensional nonlinear viscoelasticity of strain-rate type ( σ(yx ) + S(y x, y xt ) ) = 0, x (0, 1), t [0, T ]. x Nonlinear viscoelasticity is a prototype for the study of the dynamics of microstructure observed during solid phase transformations and the main postulate is that σ is not an increasing function so that is not convex. W (y x ) = yx 0 σ(z)dz

4 Outline Microstructure in Solids Energetic interpretation The Model Nonlinear viscoelasticity One-dimensional case The problem Assumptions λ-convexity Finitely many initial data General initial data Relation with the theory of gradient flows Equivalence of the theories

5 Microstructure in Solids Microtwins in Ni65 Al35 (Boullay & Schryvers) Microstructure in CuZnAl (M.Morin)!c

6 Energetic Interpretation single crystals subject to deformation y W depends on the local change of the lattice measured by the deformation gradient Dy The total energy of the body is given by I(y) = W (Dy) dx. Ω A double-well energy density

7 The Model The equation of nonlinear viscoelasticity is y tt Div DW (Dy) Div S(Dy, Dy t ) = 0 where F = Dy(x, t), F iα = y i x α deformation gradient W : M 3 3 [0, ] stored-energy function T R (Dy, Dy t ) = DW (Dy) + S(Dy, Dy t ) Piola-Kirchhoff stress tensor.

8 One-dimensional case In one space dimension we have y tt = ( σ(y x ) + S(y x, y xt ) ), x (0, 1), t [0, T ]. x We will consider the quasistatic case when S(y x, y xt ) = y xt.

9 One-dimensional case In one space dimension we have y tt = ( σ(y x ) + S(y x, y xt ) ), x (0, 1), t [0, T ]. x We will consider the quasistatic case when Boundary conditions: S(y x, y xt ) = y xt. y(0, t) = 0 and (σ + S)(1, t) = 0 y(0, t) = 0 and y(1, t) = µ > 0.

10 One-dimensional case In one space dimension we have y tt = ( σ(y x ) + S(y x, y xt ) ), x (0, 1), t [0, T ]. x We will consider the quasistatic case when Boundary conditions: S(y x, y xt ) = y xt. y(0, t) = 0 and (σ + S)(1, t) = 0 y(0, t) = 0 and y(1, t) = µ > 0.

11 The problem We have ( ) σ(yx ) + y xt x = 0. Using the boundary conditions and putting p = y x we get p t (x, t) = σ(p(x, t)) Therefore, the problem we study becomes p t (x, t) = σ(p(x, t)) + (P ) p(x, 0) = p 0 (x) 1 p(x, t) dx = µ. 0 σ(p(y, t)) dy. 1 0 σ(p(y, t)) dy

12 Assumptions We assume the following for W : σ(p) σ(q) L(C) p q whenever 1 C p, q C (LC) Behaviour at infinity: - W (p) is convex for sufficiently large p (Conv ) - σ(p) > 0 for sufficiently large p (Pos) Behaviour near zero: - W (p) is strictly convex for sufficiently small p (Conv 0 ) - σ(p) as p 0 +. (Neg)

13 Assumptions We assume the following for W : σ(p) σ(q) L(C) p q whenever 1 C p, q C (LC) Behaviour at infinity: - W (p) is convex for sufficiently large p (Conv ) - σ(p) > 0 for sufficiently large p (Pos) Behaviour near zero: - W (p) is strictly convex for sufficiently small p (Conv 0 ) - σ(p) as p 0 +. (Neg) Remark : Their relation with the natural assumptions on W for extreme deformations : W (p) as p 0 +,.

14 λ-convexity We say that W is λ-convex if v W (v) + λ 2 v 2 is convex for some λ R. Remark : λ > 0 if W is not convex.

15 λ-convexity We say that W is λ-convex if v W (v) + λ 2 v 2 is convex for some λ R. Remark : λ > 0 if W is not convex. Lemma Assume that W C(0, ) and satisfies (LC), (Conv ) and (Conv 0 ). Then, W is λ-convex for some λ > 0.

16 Initial Data with Finitely Many Values Assume that initial data p 0 (x) is given by p 0 (x) = N p 0i χ Ei (x) i=1 where meas(e i ) = µ i, i µ i = 1, and E i are mutually disjoint subsets of (0, 1).

17 Initial Data with Finitely Many Values Proposition Assume that the initial data p 0 (x) is given as above and (LC), (Conv ), (Pos), (Conv 0 ) and (Neg) hold. Then, there exists a unique global solution p N (x, t) C([0, T ]; L 1 (0, 1)) to problem (P ). Furthermore, there exist functions ε(t) and E(t), independent of N, such that satisfying ε(0) = 0 and ε(t) > 0 for t > 0, E(t) as t 0 and E(t) 0 for t 0 ε(t) < p N (x, t) < E(t) for all t > 0.

18 The General Case Proposition (Approximation) Assume that p 0N (x) p 0 (x) in L 2 (0, 1). Then, there exists a p(x, t) such that p N (x, t) p(x, t) in C([0, T ]; L 2 (0, 1)) as N.

19 The General Case Proposition (Approximation) Assume that p 0N (x) p 0 (x) in L 2 (0, 1). Then, there exists a p(x, t) such that p N (x, t) p(x, t) in C([0, T ]; L 2 (0, 1)) as N. Proof : The main technical issue is to show the existence of a constant K > 0 such that for any p q, ( σ(p) σ(q) )( p q ) K (p q) 2, which is quite straightforward by λ-convexity.

20 The General Case Definition We say that p(x, t) is a solution of the initial boundary-value problem (P ) on (0, 1) (0, T ) if: p(x, t) C([0, T ]; L 2 (0, 1)) and σ(p(x, t)) L 1 (τ, T ; L 2 (0, 1)) where 0 < τ < T, p(x, t) > 0 for a.e. x (0, 1) and for all t [0, T ], p(, t) p 0 ( ) in L 2 (0, 1) as t 0, for a.e. x (0, 1) the identity p(x, t) p(x, s) = t holds for all 0 < s < t < T. s σ(p(x, τ)) dτ + t 1 s 0 σ(p(y, τ)) dy dτ

21 The General Case Theorem Assume that (LC), (Conv ), (Pos), (Conv 0 ) and (Neg) hold, and p 0 L 2 (0, 1), p 0 (x) > 0 for a.e. x (0, 1). Then, there exists a unique solution p(x, t) to problem (P ). Methods of proof: Using the global upper and lower bounds to pass to the limit, Following Brézis 1 for the analysis of the gradient flow equation. 1 H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973.

22 Relation with Theory of Gradient Flows Let H be an Hilbert space with inner product (, ) and norm. Given T > 0 and f : (0, T ) H, the gradient flow equation is (GF ) { ut (t) + φ(u(t)) f(t) for a.e. t (0, T ) u(0) = u 0 where φ : H (, ] is a proper and lower semicontinuous functional and φ: D( φ) H 2 H is its Fréchet subdifferential.

23 Relation with Theory of Gradient Flows Recall that the functional φ is said to be proper if D(φ) and the Fréchet subdifferential φ of φ at a point u D(φ) is defined as v φ(u) lim inf ω u φ(ω) φ(u) (v, ω u) ω u 0. When φ is assumed to be convex, φ is a maximal monotone operator and existence, uniqueness and regularity of solutions for (GF ) follow from the theory of nonlinear semigroups in Hilbert spaces developed by Brézis, Crandall & Pazy 2 and Komura 3. 2 M. G. Crandall, A. Pazy, Semi-groups of nonlinear contractions and dissipative sets, J. Funct. Anal., Y. Komura, Nonlinear semi-groups in Hilbert spaces, J. Math. Soc. Japan, 1967

24 Relation with Theory of Gradient Flows Definition Let f L 1 (0, T ; H). A function u C([0, T ]; H) is called a solution of (GF ) if u is differentiable a.e. on (0, T ), u(t) D(φ) for a.e. t (0, T ) and there exists g(t) φ(u(t)) for a.e. t (0, T ) such that u t (t) + g(t) = f(t) for a.e. t (0, T ).

25 Relation with Theory of Gradient Flows Definition Let f L 1 (0, T ; H). A function u C([0, T ]; H) is called a solution of (GF ) if u is differentiable a.e. on (0, T ), u(t) D(φ) for a.e. t (0, T ) and there exists g(t) φ(u(t)) for a.e. t (0, T ) such that u t (t) + g(t) = f(t) for a.e. t (0, T ). When φ is λ-convex, Fréchet subdifferential can be characterized by v φ(u) { u D(φ) and φ(ω) φ(u) (v, ω u) λ 2 ω u 2, ω H.

26 Equivalence of the theories We define the functional φ on L 2 (0, 1) as 1 1 W (p) dx, if p dx = µ, p > 0 a.e. φ(p) = 0 0 +, otherwise and its effective domain as { D(φ(p)) = p L 2 (0, 1) : p > 0 a.e., 1 0 W (p)dx <, 1 0 } pdx = µ.

27 Equivalence of the theories Proposition (The subdifferential characterization) Assume W C 1 (0, ) is λ-convex and W satisfies (LC). Then, for any p D(φ), we have v φ(p) W (p) L 2 (0, 1) and v = W (p) c for a constant c.

28 Equivalence of the theories Theorem (Equivalence) Assume that W C 1 (0, ) is λ-convex and (LC), (Conv ), (Pos), (Conv 0 ) and (Neg) are satisfied. Assume also that p 0 satisfies p 0 L 2 (0, 1), p 0 (x) > 0 for a.e. x (0, 1). Then, any solution of problem (P ) is a solution of (GF ) for almost every t (0, T ), and vice versa.

29 Equivalence of the theories Sketch of proof : 1. Take any solution p(t) of (GF ). Then, p(t) D(φ(p)) and there exists a g(t) φ(p) such that p t = g(t) a.e. in (0, T ). 2. By the characterization for the subdifferential we get g(t) = W (p(t)) c(t) and 1 0 p(t)dx = µ. 3. Taking the integral of both sides gives c(t) = 1 0 σ(p(t))dx.

30 Equivalence of the theories 4. Conversely, take any solution p(t) of (P ). It satisfies p t = σ(p(t)) This is equivalent to writing 0 σ(p(t)) dy, 1 0 p(t)dx = µ. 6. Set c(t) = p t = W (p(t)) c(t) for a.e. t (0, T ). 1 0 characterization. σ(p(t))dx and use the subdifferential

31 Summary well-posedness for a special case of nonlinear viscoelasticity of strain-rate type uniform upper and lower bounds equivalence with the existence theory of gradient flows explicit proof for well-posedness of the gradient flow equation in λ-convex case