Armin Toffler, The whole is more than the sum of the parts. Aristotle, BC. Prigogine I. and Stengers I. Order out of Chaos (1984)

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2 OUTLINE 1. Introduction 2. Complexity 3. Waves in microstructured materials internal variables examples 4. Biophysics / biomechanics 5. Complexity around snow 6. Final remarks 2

3 One of the most highly developed skills in contemporary western civilization is dissection: the split-up of problems into their smallest possible components. We are good at it. So good, we often forget to put the pieces back together again The whole is more than the sum of the parts Prigogine I. and Stengers I. Order out of Chaos (1984) Armin Toffler, 1984 Nicolis G. and Nicolis C. Foundations of Complex Systems (2007) Aristotle, BC 3

4 COMPLEXITY complex systems are comprised of many different parts which are connected in multiple ways; complex systems produce global emergent structures generated by local interactions; emergence occurs far from equilibrium; complex systems are typically nonlinear; emergence usually occurs at the edge of chaos. 4

5 MECHANICS CLASSICAL EXAMPLES three body system double pendulum Lorenz attractor turbulence..... and many more 5

6 MECHANICS OF MICROSTRUCTURED SOLIDS Problems: discrete vrs continuum? nonlinearit-y/ies? description of microstructure (s)? interaction between constituents? 6

7 SCALES Reproduced from: T.S. Gates, G.M. Odegard, S.J.V. Frankland, T.C. Clancy, Computational materials: Multiscale modeling and simulation of nanostructured materials. Composites Science and Technology, 65,

8 METHODS: discretization homogenization advanced continuum theories Eringen, Mindlin,... Maugin pseudomomentum internal variables... 8

9 BALANCE LAWS Canonical (material) momentum balance P t x Div R b f int f ext f inh, P material momentum, b material Eshelby stress, material inhomogeneity force f inh, material external (body) force f ext, material internal force f int 9

10 Energy conservation Sθ Q x R t h int, h int : T : F W t x the second law int Sθ S θ h θk R S the entropy flux, S the entropy density per unit reference volume, θ absolute temperature, K extra entropy flux, T the first Piola Kirchhoff tensor, F deformation gradient. 10

11 INTERNAL VARIABLES (1) observable internal strains, displacements, etc. describe the internal structure of the material see Maugin, Muschik, 1994 damage parameter orientation of liquid crystals dislocations etc. In this formalism : internal variables might not be inertial. 11

12 INTERNAL VARIABLES (2) Microstructured solids Single / dual variables Berezovski, Engelbrecht, Maugin, 2008 Dual variables α, β second order tensors Free energy W W W F, θ, α, α, β, R R β 12

13 INTERNAL VARIABLES (3) From dissipation inequality ~ 11 α A L L L β ~ 21 B L L L depend on state variables ~ ~ A, B related to W A simple non-dissipative process α L 12 ~ B, β L ~ A ~ A ~ B A special case α W independent of L L A ~ R inertia taken into account! 13

14 MICROMORPHIC ELASTICITY General theory Mindlin (1964) 1D models Engelbrecht and Pastrone (2003) internal variables approach Berezovski et al. (2008) Free energy function W quadratic linear model cubic nonlinear model 14

15 MINDLIN MODEL Mindlin model two balance laws Internal variable one balance law dissipation inequality ρ u tt α u xx N u x u xx A ψ x I ψ tt Cψ xx M ψ x ψ xx A u x Bψ Ψ microdeformation Ψ internal variable 15

16 16 MINDLIN MODEL exact, nonlinear, nondimensional XX 2 XX M XX XX TT 2 B 2 A X 2 X N XX A TT U k 2 1 U c c U c c U k 2 1 U c c 1 U

17 MULTIPLE SCALES macrostructure microstructure 1 microstructure 2 17

18 MULTIPLE SCALES linear u tt c c u p c u c c 0 A xx 1 A1 u tt 1 A2 xx xx p 2 1 c 2 A1 p 2 2 c 2 A2 2 u c u tt 2 xx xxxx 18

19 COMPLEXITY OF MICROSTRUCTURED SOLIDS hierarchy soliton emergence solitons asymmetric solitonic structures patterns of trajectories cf. fluids interaction of solitons 19

20 BIOPHYSICS / BIOMECHANICS biological systems need energy exchange with surrounding environment; systems are far from the thermodynamic equilibrium; processes operate over different time and space scales, include many hierarchies; in physical terms: nonlinearities, dissipation, activity/excitability. 20

21 HIERARCHIES IN BIOPHYSICS Structural hierarchies: atom molecule cell tissue organ human sarcomeres myofibrils fibres myocardium heart Process hierarchies: oxygen consumption energy transfer Ca 2+ signals cross bridge motion contraction 21

22 EXAMPLES 22

23 EXAMPLES 23

24 FUNCTIONAL HIERARCHIES concept of hierarchical internal variables... γ β β α α observable variables (stress, strain) 24

25 CARDIAC CONTRACTION Ca 2+ signals all activated crossbridges force producing crossbridges active stress 25

26 NERVE PULSE TRANSMISSION 26

27 NERVE PULSE TRANSMISSION Hodgkin Huxley model three phenomenological variables FitzHugh Nagumo model one recovery variable In terms of continuum theory these are internal variables Maugin Engelbrecht (1994) 27

28 COMPLEXITY AROUND SNOW 28

29 PROBLEMS athlete physiology biomechanics of muscles kinematics equipment technology mechanics chemistry snow granular (?) material phase transformation 29

30 ATHLETE: HEART 30

31 CARDIAC CONTRACTION actin connection myosin cross-bridges Z-line 31

32 CARDIAC CONTRACTION A B D C A to B attaching C to D detaching D to A resetting B to C swivelling 32

33 KINEMATICS 33

34 FORCES 34

35 TECHNOLOGY: skis, poles bindings, boots clothing waxes 35

36 TECHNOLOGY: EXAMPLES 36

37 SNOW 37

38 SNOW 38

39 SNOW 39

40 SNOW: REALITY 40

41 SNOW: CALCULATIONS 41

42 Andrus Veerpalu COMPLEXITY AROUND SNOW 42

43 CENS: ANNUAL REPORTS See also 43

44 CENS: BOOKS 44

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