Cavitation and fracture in nonlinear elasticity

Transcription

1 Cavitation and fracture in nonlinear elasticity Duvan Henao Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie - CNRS Work under the supervision of John M. Ball University of Oxford In collaboration with Carlos Mora-Corral (BCAM), Xianmin Xu (CAS)

2 Radial traction in incompressible nonlinear elasticity ν P det Du(x) = 1, x B(0, 1) \ {0} Div T R (x) = 0, x B(0, 1) \ {0} ( TR (x) ) ν(x) = Pν(x), x B(0, 1) A where T(ɛ, θ)ν(θ) ɛ 0 0, θ (0, 2π) T R (x) = DW ( Du) p(x) cof Du(x) T(x) = DW ( Du ) Du T p(x)1 with energy density W : M n n 1 R.

3 Radially symmetric cavitation J. Ball (1982) u(x) = u(r) x r, r = x u u n 1 = r n 1 u(r) = ( A n + r n) 1 n p(x) = ( r/u ) n 1 Φ 1(r) T (r) T (r) = v(1) v(r) 1 dˆφ dv v n 1 dv Gent & Lindley (1959) v n 1 (v n 1) 2 dˆφ(v) dv L 1 (1 + δ, ) For W (F) = µ p F p, this is p < n.

4 Variational formulation In the previous example, Du A/r and u W 1,p for p < n. { } min µ Du p dx : det Du = 1 u A Ω min (µ Du p + γ(det Du)) dx, u A Ω (e.g., γ(t) = (t 1) 2 µ log t) u must be one-to-one a.e., det Du > 0 a.e. A W 1,p? A SBV? Brittle fracture (Ambrosio & Braides 1995, Francfort and Marigo 1998) u 2 dx + H n 1 (J u) min u SBV Ω Fracture of titanium alloys under impact loading Petrinic et al., 2006

5 Simulation of cavitation (with Xianmin Xu) Stored-energy function: W (F) = F (det F 1) 2 c log(det F) Boundary condition: u(x) = λx at B(0, 1) Domain with holes of radius ɛ = 0.01 Model of Sivaloganathan and Spector (2000)

6 Weak continuity of the determinant (Ball & Murat 1984) λ u 1 B.C.: u(x) = λx on.. u 2 λ λ λ λ. Du j = = Du 1 u 1 ν dh n 1 λ1 = λ1 u h λ Hence u j u in W 1,p, but 1 = det Du j det Du = λ 2

7 Per u(ω) as a surface energy (Müller & Spector 1995) min W (Du) dx + Per u(ω) u W 1,p Ω λ λ u 2 λ λ u k opens k 2 cavities, each of perimeter 2πA/k. area of the created surface 2πAk k

8 det Du j det Du and invertibility (Müller & Spector 1995) Suppose u j u in W 1,p, each u j satisfies det Du j > 0 a.e. and is one-to-one a.e. Suppose, further, that det Du j θ for some θ L 1, Per u j (Ω) is uniformly bounded, and u is one-to-one a.e. Then, θ = det Du a.e.

9 Invertibility of limits of one-to-one a.e. maps u j a sequence of one-to-one a.e. maps, det u j > 0 a.e. u j u in W 1,p, p > n u is one-to-one a.e. (Ciarlet & Nečas, 1985). u j u in SBV, det u j det u in L 1 u is one-to-one a.e. (Giacomini & Ponsiglione, 2008). Classical results on the weak continuity of the determinants (Ball 1977, Müller-Tang-Yan 1994, Ambrosio 1994, Fusco-Leone-March- -Verde 2006) incompatible with cavitation.

10 Invertibility of limits of one-to-one a.e. maps Let E(u) := [ sup Dx f cof u + (div y f) det u ] dx. f(x,y) 1 Ω Theorem (H.-M.C. 2009): Suppose that u j, u are approximately differentiable a.e. in Ω u j is one-to-one a.e. for every j, and det u j > 0 a.e. cof u j L 1 is bounded, u j u a.e., det u j is equi-integrable sup E(u j ) <. Then, u is one-to-one a.e. and det u j det u in L 1. If, moreover, {u j } SBV, sup H n 1 (J uj ) <, sup u j L 2 <, and cof u j is equi-integrable, then det u j det u.

11 Existence of minimizers Theorem (H.-M.C. 2009): Suppose W is polyconvex and such that W (x, y, F) c F p + h 1( cof F ) + h 2(det F) with p 2, and h 1, h 2 of superlinear growth at infinity. Let I (u) = W (Du) dx + H 2 (J u) + E(u) + b.c. Ω A = {u SBV : u L p, u(x) K for a.e. x, Then I attains its infimum on A. det u > 0 a.e., u is one-to-one a.e.} Related to results in the theory of Cartesian Currents (Giaquinta-Modica-Souček 1998, Conti-De Lellis 2003, Mucci 2005)

12 Further aspects of the problem Created surface and jumps of the inverse Regularity of the inverse Lusin s condition Distributional determinant The topological degree

13 Invisible created surface Γ V (u) := u(ω) \ u( Ω), Γ I (u) := J u 1, Γ(u) := Γ V (u) + Γ I (u) Per u(ω) = sup g(y) 1 { E(u) = sup f(x,y) + { g ν dh n 1 + u( Ω) Γ g ν dh n 1 } f(u 1 (y), y) ν(y) dh n 1 Γ V [ f((u 1 ) + (y), y) f((u 1 ) (y), y) ] } ν(y) dh n 1 (y) Γ I Theorem: E(u) < u 1 SBV and E(u) = H n 1 (Γ V ) + 2H n 1 (Γ I ).

14 Regularity of the inverse Csörnyei, Hencl & Malý, 2009: Suppose u is a homeomorphism, u W 1,n 1 loc (Ω, R n ), and det Du > 0 a.e.. Then u 1 W 1,1 loc (u(ω), Rn ). Related results by Onninen (2006), Hencl & Koskela (2006), Hencl, Koskela & Malý (2006), Hencl, Koskela & Onninen (2007). If u is a homeomorphism, does it follow that E(u) = 0?

15 E(u) and Lusin s condition Theorem (H.-M.C.): Let u W 1,n 1 loc (Ω, R n ) be a homeomorphism, det Du > 0 a.e.. Then u satisfies Lusin s condition, E(u) = 0, and u 1 W 1,1 loc (u(ω), Rn ), or u does not satisfy Lusin s condition, E(u) =, and H n 1 ( u(ω d ) u(ω \ Ω d )) =. Example of a homeomorphism violating Lusin s condition given by Ponomarev (1981).

16 The Distributional determinant ν P For smooth maps, det Du = 1 Div ( (adj Du)Du). n Define Det Du(φ) := 1 [(adj Du(x))u(x)] Dφ(x) dx. n For the radial cavitation, Ω A Det Du = (det Du)L n + πa 2 δ 0. For a homeomorphism satisfying Lusin s condition, Det Du = det Du. Given φ C c(ω), g C c(r n, R n ), let f(x, y) := φ(x)g(y), then [ Dxf cof u + (div y f) det u ] dx = Ω Ω [ g(u(x)) ( cof u(x) ) φ(x) + φ(x)(div g)(u(x)) det u(x) ] dx = (div g) u det u Div ( (adj u)g u ), φ

17 The topological degree A D C B E Γ := u([0, 2π]), u:[0, 2π] R 2 u 1 (θ) = sin θ + cos 3 θ u 2 (θ) = cos θ sin θ cos2 θ u(r, θ) := u(θ) + rg(θ)ν Γ (u(θ)) Det Du = (det Du)L n + c δ 0 c = A B + C D + E = 0.

18 Non-interpenetration of matter Condition INV (Müller and Spector, 1995): deg(u, B, y) = 1 for a.e. y u(b). deg(u, B, y) = 0 for a.e. y u(ω \ B).

19 INV and Det = det u W 1,p for all p < n, u is one-to-one a.e., det Du > 0 a.e., and Det Du = det Du; still, E(u) > 0. Theorem: Det Du = det Du + INV E(u) = 0 ( H n 1 (J u 1) = 0).

20 Summary Large deformations low regularity. Slow growth at infinity lack of s.w.l.s.c. Invertibility and surface energies. BV and SBV regularity of the inverses, Cartesian currents. Rigorous notion of created surface; jumps of the inverses. Related to the distributional determinant, Lusin s condition, the topological degree, condition INV.