Chapter III Elasticity: soft modes, bending and nonaffinity I) A crash course in elasticity theory 1. Stress 2. Mechanical equilibrium 3. Strain 4. Linear elasticity II) Affine deformation of a hyperstatic network III) Bending-dominated elasticity of a hypostatic network IV) Conclusions References: Landau & Lifschitz, Theory of elasticity, Butterworth Heinemann (1986) R. Aris, Vectors, tensors, and the basic equations of fluid mechanics, Dover (1989) Broedersz et al. Nat. Phys. 7, 983 (2011)
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Chapter IIIElasticity: soft modes, bending and nonaffinity
I) A crash course in elasticity theory1. Stress2. Mechanical equilibrium3. Strain4. Linear elasticity
II) Affine deformation of a hyperstatic network
III) Bending-dominated elasticity of a hypostatic network
IV) Conclusions
References:Landau & Lifschitz, Theory of elasticity, Butterworth Heinemann (1986)R. Aris, Vectors, tensors, and the basic equations of fluid mechanics, Dover (1989)Broedersz et al. Nat. Phys. 7, 983 (2011)
Stretching-dominated network
Bending-dominated network
Criticality and isostaticity in fibre networks
network at restnonaffine deformationunder simple shear(linear response)
three-dimensional version
Mechanics and non-affine strain fluctuations
shea
r m
odul
us (
2D)
shea
r m
odul
us (
3D)
nona
ffini
typa
ram
eter
Γ (
2D
)
nona
ffini
typa
ram
eter
Γ (
3D
)
p p
p p
Scaling analysis of the mechanics and anomalous elasticity
Finite-size scaling
Phase diagram
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