1. 7/18/05Princeton Elasticity Lectures Nonlinear Elasticity

2. 7/18/05Princeton Elasticity Lectures Outline Some basics of nonlinear elasticity Nonlinear elasticity of biopolymer networks Nematic elastomers

3. 7/18/05Princeton Elasticity Lectures What is Elasticity Description of distortions of rigid bodies and the energy, forces, and fluctuations arising from these distortions. Describes mechanics of extended bodies from the macroscopic to the microscopic, from bridges to the cytoskeleton.

4. 7/18/05Princeton Elasticity Lectures Classical Lagrangian Description Reference material in D dimensions described by a continuum of mass points x. Neighbors of points do not change under distortion Material distorted to new positions R(x) Cauchy deformation tensor

5. 7/18/05Princeton Elasticity Lectures Linear and Nonlinear Elasticity Linear : Small deformations – L near 1 Nonlinear: Large deformations – L >>1 Why nonlinear? Systems can undergo large deformations – rubbers, polymer networks, … Non-linear theory needed to understand properties of statically strained materials Non-linearities can renormalize nature of elasticity Elegant an complex theory of interest in its own right Why now: New interest in biological materials under large strain Liquid crystal elastomers – exotic nonlinear behavior Old subject but difficult to penetrate – worth a fresh look

6. 7/18/05Princeton Elasticity Lectures Deformations and Strain Complete information about shape of body in R(x)= x +u(x); u= const. – translation no energy. No energy cost unless u(x) varies in space. For slow variations, use the Cauchy deformation tensor Volume preserving stretch along z-axis

7. 7/18/05Princeton Elasticity Lectures Simple shear strain Constant Volume, but note stretching of sides originally along x or y. Rotate Not equivalent to Note: L is not symmetric

8. 7/18/05Princeton Elasticity Lectures Pure Shear Pure shear: symmetric deformation tensor with unit determinant – equivalent to stretch along 45 deg.

9. 7/18/05Princeton Elasticity Lectures Pure shear as stretch

10. 7/18/05Princeton Elasticity Lectures Pure to simple shear

11. 7/18/05Princeton Elasticity Lectures Cauchy Saint-Venant Strain u ab is invariant under rotations in the target space but transforms as a tensor under rotations in the reference space. It contains no information about orientation of object. R=Reference space T=Target space Symmetric!

12. 7/18/05Princeton Elasticity Lectures Elastic energy The elastic energy should be invariant under rigid rotations in the target space: if is a function of u ab. This energy is automatically invariant under rotations in target space. It must also be invariant under the point- group operations of the reference space. These place constraints on the form of the elastic constants. Note there can be a linear “stress”-like term. This can be removed (except for transverse random components) by redefinition of the reference space

13. 7/18/05Princeton Elasticity Lectures Elastic modulus tensor K abcd is the elastic constant or elastic modulus tensor. It has inherent symmetry and symmetries of the reference space. Isotropic system Uniaxial (n = unit vector along uniaxial direction)

14. 7/18/05Princeton Elasticity Lectures Isotropic and Uniaxial Solid Isotropic: free energy density f has two harmonic elastic constants Uniaxial: five harmonic elastic constants m = shear modulus; B = bulk modulus

15. 7/18/05Princeton Elasticity Lectures Force and stress I external force density – vector in target space. The stress tensor s ia is mixed. This is the engineering or 1 st Piola-Kirchhoff stress tensor = force per area of reference space. It is not necessarily symmetric! s ab II is the second Piola-Kirchhoff stress tensor - symmetric Note: In a linearized theory, s ia = s ia II

16. 7/18/05Princeton Elasticity Lectures Cauchy stress The Cauchy stress is the familiar force per unit area in the target space. It is a symmetric tensor in the target space. Symmetric as required

17. 7/18/05Princeton Elasticity Lectures Coupling to other fields We are often interested in the coupling of target-space vectors like an electric field or the nematic director to elastic strain. How is this done? The strain tensor u ab is a scalar in the target space, and it can only couple to target-space scalars, not vectors. Answer lies in the polar decomposition theorem

18. 7/18/05Princeton Elasticity Lectures Target-reference conversion To linear order in u, O ia has a term proportional to the antisymmetric part of the strain matrix.

19. 7/18/05Princeton Elasticity Lectures Strain and Rotation Simple Shear Symmetric shear Rotation

20. 7/18/05Princeton Elasticity Lectures Sample couplings Coupling of electric field to strain Free energy no longer depends on the strain u ab only. The electric field defines a direction in the target space as it should Energy depends on both symmetric and anti-symmetric parts of h’

21. 7/18/05Princeton Elasticity Lectures Biopolymer Networks cortical actin gelneurofilament network

22. 7/18/05Princeton Elasticity Lectures Characteristics of Networks Off Lattice Complex links, semi-flexible rather than random-walk polymers Locally randomly inhomogeneous and anisotropic but globally homogeneous and isotropic Complex frequency-dependent rheology Striking non-linear elasticity

23. 7/18/05Princeton Elasticity Lectures Goals Strain Hardening (more resistance to deformation with increasing strain) – physiological importance Formalisms for treating nonlinear elasticity of random lattices –Affine approximation –Non-affine

24. 7/18/05Princeton Elasticity Lectures Different Networks Max strain ~.25 except for vimentin and NF Max stretch: L( L )/L~1.13 at 45 deg to normal

25. 7/18/05Princeton Elasticity Lectures Semi-microscopic models Random or periodic crosslinked network: Elastic energy resides in bonds (links or strands) connecting nodes R b = separation of nodes in bond b V b (| R b |) = free energy of bond b n b = Number of bonds per unit volume of reference lattice

26. 7/18/05Princeton Elasticity Lectures Affine Transformations Reference network: Positions R 0 Strained target network: R i =L ij R 0j Depends only on u ij

27. 7/18/05Princeton Elasticity Lectures Example: Rubber Purely entropic force Average is over the end-to-end separation in a random walk: random direction, Gaussian magnitude

28. 7/18/05Princeton Elasticity Lectures Rubber : Incompressible Stretch Unstable: nonentropic forces between atoms needed to stabilize; Simply impose incompressibility constraint.

29. 7/18/05Princeton Elasticity Lectures Rubber: stress -strain Engineering stress Physical Stress A R = area in reference space A = A R /L = Area in target space Y=Young’s modulus

30. 7/18/05Princeton Elasticity Lectures General Case Engineering stress: not symmetric Physical Cauchy Stress: Symmetric Central force

31. 7/18/05Princeton Elasticity Lectures Semi-flexible Stretchable Link t = unit tangent v = stretch

32. 7/18/05Princeton Elasticity Lectures Length-force expressions L ( t, K ) = equilibrium length at given t and K

33. 7/18/05Princeton Elasticity Lectures Force-extension Curves

34. 7/18/05Princeton Elasticity Lectures Scaling at “Small” Strain G'/G' (0) Strain/strain8 zero parameter fit to everything Theoretical curve: calculated from K -1 =0

35. 7/18/05Princeton Elasticity Lectures What are Nematic Gels? Homogeneous Elastic media with broken rotational symmetry (uniaxial, biaxial) Most interesting - systems with broken symmetry that develops spontaneously from a homogeneous, isotropic elastic state

36. 7/18/05Princeton Elasticity Lectures Examples of LC Gels 1. Liquid Crystal Elastomers - Weakly crosslinked liquid crystal polymers 4. Glasses with orientational order 2. Tanaka gels with hard-rod dispersion 3. Anisotropic membranes Nematic Smectic-C

37. 7/18/05Princeton Elasticity Lectures Properties I Large thermoelastic effects - Large thermally induced strains - artificial muscles Courtesy of Eugene Terentjev 300% strain

38. 7/18/05Princeton Elasticity Lectures Properties II Large strain in small temperature range Terentjev

39. 7/18/05Princeton Elasticity Lectures Properties III Soft or “Semi-soft” elasticity Vanishing xz shear modulus Soft stress-strain for stress perpendicular to order Warner Finkelmann

40. 7/18/05Princeton Elasticity Lectures Model for Isotropic-Nematic trans. m approaches zero signals a transition to a nematic state with a nonvanishing

41. 7/18/05Princeton Elasticity Lectures Spontaneous Symmetry Breaking Phase transition to anisotropic state as m goes to zero Direction of n 0 is arbitrary Symmetric- Traceless part

42. 7/18/05Princeton Elasticity Lectures Strain of New Phase d u is the deviation of the strain relative to the original reference frame R from u 0 u’ is the strain relative to the new state at points x’ d u is linearly proportional to u’

43. 7/18/05Princeton Elasticity Lectures Elasticity of New Phase Rotation of anisotropy direction costs no energy C 5 =0 because of rotational invariance This 2nd order expansion is invariant under all U but only infinitesimal V

44. 7/18/05Princeton Elasticity Lectures Soft Extensional Elasticity Strain u xx can be converted to a zero energy rotation by developing strains u zz and u xz until u xx =(r-1)/2

45. 7/18/05Princeton Elasticity Lectures Frozen anisotropy: Semi-soft System is now uniaxial – why not simply use uniaxial elastic energy? This predicts linear stress-stain curve and misses lowering of energy by reorientation: Model Uniaxial system: Produces harmonic uniaxial energy for small strain but has nonlinear terms – reduces to isotropic when h=0 f (u) : isotropic Rotation

46. 7/18/05Princeton Elasticity Lectures Semi-soft stress-strain Ward Identity Second Piola-Kirchoff stress tensor.

47. 7/18/05Princeton Elasticity Lectures Semi-soft Extensions Not perfectly soft because of residual anisotropy arising from crosslinking in the the nematic phase - semi-soft. length of plateau depends on magnitude of spontaneous anisotropy r. Warner-Terentjev Stripes form in real systems: semi-soft, BC Break rotational symmetry Finkelmann, et al., J. Phys. II 7, 1059 (1997); Warner, J. Mech. Phys. Solids 47, 1355 (1999) Note: Semi-softness only visible in nonlinear properties

48. 7/18/05Princeton Elasticity Lectures Softness with Director Director relaxes to zero N a = unit vector along uniaxial direction in reference space; layer normal in a locked SmA phase Red: SmA-SmC transition