Interface and contact problems

Transcription

1 Interface and contact problems Heiko Gimperlein Heriot Watt University and Maxwell Institute, Edinburgh and Universität Paderborn TU Wien May 30, 2016 H. Gimperlein (Edinburgh) Interface and contact problems TU Wien 1 / 7

2 Which equations describe materials in contact? Basic principle: Energy is additive For a system composed of two subsystems Total Energy = Energy of 1 + Energy of 2 + Interaction betw. 1/2. In elastic problems, the interaction might correspond to friction between 1/2, surface energy etc. localized on Γ c =contact area (Interaction betw. 1/2) = dx Γ C Nature minimizes Total Energy over physically allowed configurations. In this course this will lead to unconstrained, smooth variational problems PDE variational problems not C 1 or constrained variational inequalities H. Gimperlein (Edinburgh) Interface and contact problems TU Wien 2 / 7

3 Energy H. Gimperlein (Edinburgh) Numerical Analysis of Real Materials AIMS 5 / 23

4 Total energy Energy(u) 1 2 ε(u) : Cε(u) dx = W(ε(u(x))) dx Here ε(u) = 1 2 ( u+ ut ), u = x1 u 1 x2 u 1 x3 u 1 x1 u 2 x2 u 2 x3 u 2 x1 u 3 x2 u 3 x3 u 3. The integrand is 3 i,j,k,l=1 ε ij C ijkl ε kl. Large u quadratic approximation bad. Properties of energy density physics W(ε(u)) convex function larger deformations require larger force W(ε(u)) nonconvex function: phase transitions! H. Gimperlein (Edinburgh) Interface and contact problems TU Wien 3 / 7

5 Nature minimizes energy For simplicity, we will mostly consider a toy problem: u : R scalar function. u instead of ε(u) = 1 2 ( u+ ut ). simplest energy (with force f ) is given by Energy(u) = 1 (D(x) u(x)) u(x) dx 2 Nature finds (local) minimizers of the energy: f(x)u(x) dx denergy du What does denergy du mean? For all physically reasonable h : R d E(u+th) dt t=0 H. Gimperlein (Edinburgh) Interface and contact problems TU Wien 4 / 7

6 Nature minimizes energy differential equations Energy(u) = 1 2 (D u) u dx fu dx R n bounded, force f, both nice. D = 1, n unit normal vector to. Theorem u minimizes Energy(v) = 1 2 ( v)2 fv over H1 () u H 1 v () solves v = f, n H. Gimperlein (Edinburgh) Interface and contact problems TU Wien 5 / 7

7 Nature minimizes energy differential equations Energy(u) = 1 2 (D u) u dx fu dx R n bounded, force f, both nice. D = 1, n unit normal vector to. Theorem u minimizes Energy(v) = 1 2 ( v)2 fv over H1 () u H 1 v () solves v = f, n Proof: h H 1 () E(u+h) E(u) = 1 ( u+ h) 2 f(u+h) = u h fh+ 1 ( h) 2 2 = ( u+f)h+ (n u)h+ 1 2 ( u) 2 + ( h) 2 H. Gimperlein (Edinburgh) Interface and contact problems TU Wien 5 / 7 fu

8 Nature minimizes energy differential equations Theorem u minimizes Energy(v) = 1 2 ( v)2 fv over H1 () u H 1 v () solves v = f, n Proof: h H 1 () E(u+h) E(u) = 1 ( u+ h) 2 f(u+h) 1 ( u) 2 + fu 2 2 = u h fh+ 1 ( h) 2 2 = ( u+f)h+ (n u)h+ 1 ( h) 2 2 = If u = f, v n E(u+h) E(u) ( h)2 0 H. Gimperlein (Edinburgh) Interface and contact problems TU Wien 5 / 7

9 Nature minimizes energy differential equations Theorem u minimizes Energy(v) = 1 2 ( v)2 fv over H1 () u H 1 v () solves v = f, n Proof: h H 1 () E(u+h) E(u) = 1 ( u+ h) 2 f(u+h) 1 ( u) 2 + fu 2 2 = u h fh+ 1 ( h) 2 2 = ( u+f)h+ (n u)h+ 1 ( h) 2 2 = If u = f, v n E(u+h) E(u) ( h)2 0 = E(u+λh) E(u) = λ ( ( u+f)h+ (n u)h) + λ2 2 ( h)2 0. For all λ R ( u+f)h+ (n u)h. H. Gimperlein (Edinburgh) Interface and contact problems TU Wien 5 / 7

10 Nature minimizes energy differential equations Theorem u minimizes Energy(v) = 1 2 ( v)2 fv over H1 () u H 1 v () solves v = f, n = E(u+λh) E(u) = λ ( ( u+f)h+ (n u)h) + λ2 2 For all λ R ( u+f)h+ (n u)h. For all h H0 1 ( u+f)h, therefore u+f a.e. Thus, for all h H 1 (n u)h, therefore n u a.e. ( h)2 0. H. Gimperlein (Edinburgh) Interface and contact problems TU Wien 5 / 7

11 Nature minimizes energy differential equations Theorem u minimizes Energy(v) = 1 2 ( v)2 fv over H1 () u H 1 v () solves v = f, n inhomogeneous/anisotropic energy: E(u) = 1 2 ( u(x))t D(x) u(x) fu, D = Dt > 0 div (D u) = f H. Gimperlein (Edinburgh) Interface and contact problems TU Wien 5 / 7

12 Examples of nonlinear operators E(u) = 1 2 ( u(x))t D(x) u(x) fu div (D u) = f p-laplace: E(u) = 1 p u p fu div ( u p 2 u) = f, p (1, ) p < 2 hair gel, glaciers, p > 2 thick emulsion of sand and water phase transitions / bistable materials: double well potential E(u) = (from S. Müller) u (1, 0) 2 u (0, 1) 2 fu H. Gimperlein (Edinburgh) Interface and contact problems TU Wien 6 / 7

13 Boundary conditions Neumann and Dirichlet: Contact: Signorini (= nonpenetration, wall) and friction ( wall) H. Gimperlein (Edinburgh) Interface and contact problems TU Wien 7 / 7