# Nonlinear Elasticity

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Nonlinear Elasticity. Outline. Some basics of nonlinear elasticity Nonlinear elasticity of biopolymer networks Nematic elastomers. What is Elasticity. Description of distortions of rigid bodies and the energy, forces, and fluctuations arising from these distortions. - PowerPoint PPT Presentation

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• Nonlinear Elasticity

• OutlineSome basics of nonlinear elasticityNonlinear elasticity of biopolymer networksNematic elastomers

• What is ElasticityDescription 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.

• Classical Lagrangian DescriptionReference material in D dimensions described by a continuum of mass points x. Neighbors of points do not change under distortionMaterial distorted to new positions R(x)Cauchy deformation tensor

• Linear and Nonlinear ElasticityLinear: Small deformations L near 1Nonlinear: Large deformations L >>1Why 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 rightWhy 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

• Deformations and StrainComplete 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 tensorVolume preserving stretch along z-axis

• Simple shear strainConstant Volume, but note stretching of sides originally along x or y.RotateNot equivalent toNote: L is not symmetric

• Pure ShearPure shear: symmetric deformation tensor with unit determinant equivalent to stretch along 45 deg.

• Pure shear as stretch

• Pure to simple shear

• Cauchy Saint-Venant Strainuab 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 spaceT=Target spaceSymmetric!

• Elastic energyThe elastic energy should be invariant under rigid rotations in the target space: if is a function of uab.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

• Elastic modulus tensorKabcd is the elastic constant or elastic modulus tensor. It has inherent symmetry and symmetries of the reference space. Isotropic systemUniaxial (n = unit vector along uniaxial direction)

• Isotropic and Uniaxial SolidIsotropic: free energy density f has two harmonic elastic constantsUniaxial: five harmonic elastic constantsm = shear modulus;B = bulk modulus

• Force and stress Iexternal force density vector in target space. The stress tensor sia is mixed. This is the engineering or 1st Piola-Kirchhoff stress tensor = force per area of reference space. It is not necessarily symmetric!sabII is the second Piola-Kirchhoff stress tensor - symmetric Note: In a linearized theory, sia = siaII

• Cauchy stressThe 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

• Coupling to other fieldsWe 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 uab 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

• Target-reference conversionTo linear order in u, Oia has a term proportional to the antisymmetric part of the strain matrix.

• Strain and RotationSimple ShearSymmetric shearRotation

• Sample couplingsCoupling of electric field to strainFree energy no longer depends on the strain uab only. The electric field defines a direction in the target space as it shouldEnergy depends on both symmetric and anti-symmetric parts of h

• Biopolymer Networkscortical actin gelneurofilament network

• Characteristics of NetworksOff LatticeComplex links, semi-flexible rather than random-walk polymersLocally randomly inhomogeneous and anisotropic but globally homogeneous and isotropicComplex frequency-dependent rheology Striking non-linear elasticity

• GoalsStrain Hardening (more resistance to deformation with increasing strain) physiological importanceFormalisms for treating nonlinear elasticity of random latticesAffine approximationNon-affine

• Different NetworksMax strain ~.25 except for vimentin and NFMax stretch:L(L)/L~1.13 at 45 deg to normal

• Semi-microscopic modelsRandom or periodic crosslinked network: Elastic energy resides in bonds (links or strands) connecting nodesRb = separation of nodes in bond bVb(| Rb |) = free energy of bond bnb = Number of bonds per unit volume of reference lattice

• Affine TransformationsReference network:Positions R0Strained target network:Ri=LijR0jDepends only on uij

• Example: RubberPurely entropic forceAverage is over the end-to-end separation in a random walk: random direction, Gaussian magnitude

• Rubber : Incompressible StretchUnstable: nonentropic forces between atoms needed to stabilize; Simply impose incompressibility constraint.

• Rubber: stress -strainEngineering stressPhysical StressAR= area in reference spaceA = AR/L = Area in target space Y=Youngs modulus

• General CaseEngineering stress: not symmetricPhysical Cauchy Stress: SymmetricCentral force

• Semi-flexible Stretchable Linkt = unit tangentv = stretch

• Length-force expressionsL(t,K) = equilibrium length at given t and K

• Force-extension Curves

• Scaling at Small StrainG'/G' (0)Strain/strain8zero parameter fit to everythingTheoretical curve: calculated from K-1=0

• 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

• Examples of LC Gels1. Liquid Crystal Elastomers - Weakly crosslinked liquid crystal polymers4. Glasses with orientational order 2. Tanaka gels with hard-rod dispersion3. Anisotropic membranesNematicSmectic-C

• Properties ILarge thermoelastic effects - Large thermally induced strains - artificial musclesCourtesy of Eugene Terentjev300% strain

• Properties IILarge strain in smalltemperature rangeTerentjev

• Properties IIISoft or Semi-soft elasticityVanishing xz shear modulusSoft stress-strain for stressperpendicular to orderWarner Finkelmann

• Model for Isotropic-Nematic trans.m approaches zero signals a transition to a nematic state with a nonvanishing

• Spontaneous Symmetry BreakingPhase transition to anisotropicstate as m goes to zeroDirection of n0 isarbitrarySymmetric-Tracelesspart

• Strain of New Phasedu is the deviation ofthe strain relative to the original reference frame R from u0u is the strain relative to the new state at points x du is linearly proportionalto u

• Elasticity of New PhaseRotation of anisotropy direction costs no energyC5=0 because ofrotational invarianceThis 2nd order expansionis invariant under all U but only infinitesimal V

• Soft Extensional ElasticityStrain uxx can be converted to a zero energy rotation by developing strains uzz and uxz until uxx =(r-1)/2

• Frozen anisotropy: Semi-softSystem 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=0f (u) : isotropicRotation

• Semi-soft stress-strainWard IdentitySecond Piola-Kirchoff stress tensor.

• Semi-soft ExtensionsNot 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-TerentjevStripes form in real systems: semi-soft, BCBreak rotational symmetryFinkelmann, et al., J. Phys. II 7, 1059 (1997);Warner, J. Mech. Phys. Solids 47, 1355 (1999)Note: Semi-softness only visible in nonlinear properties

• Softness with DirectorDirector relaxes to zeroNa= unit vector along uniaxial direction in reference space; layer normal in a locked SmA phaseRed: SmA-SmC transition