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Table of contents
Abstract
1 Introduction
2 Theory
2.1 Hyperelastic composites and e®ective behavior
2.1.1 Hyperelastic materials
2.1.2 E®ective behavior
2.1.3 Constitutive hypotheses
2.1.4 Macroscopic and microscopic instabilities
2.2 Bounds and estimates
2.3 Second-order homogenization method
2.3.1 On the speci¯c choice of the variables F(r) and L(r) for isotropic phases
2.4 E®ective behavior of two-phase hyperelastic composites with \particulate » microstructures
2.4.1 Classical bounds
2.4.2 The linear comparison composite
2.4.3 Second-order homogenization estimates: compliant particles
2.4.4 Second-order homogenization estimates: porous elastomers
2.4.5 Second-order homogenization estimates: rigid particles
2.5 Microstructure evolution
2.5.1 Range of validity of the HS-type second-order estimates
2.6 Macroscopic stability
2.7 Concluding remarks
2.8 Appendix I. Overall objectivity of fW
2.9 Appendix II. On the relation S = @fW=@F
2.10 Appendix III. Microscopic and macroscopic instabilities in periodic elastomers
2.11 Appendix IV. On the limit of eL as F ! I
2.12 Appendix V. Earlier versions of the second-order homogenization method
2.12.1 Tangent second-order estimates
2.12.2 Second-order estimates with °uctuations: F(r) = F(r)
2.13 Appendix VI. The tensor P for cylindrical ¯bers and laminates
3 Porous elastomers: cylindrical voids, random microstructure
3.1 Plane-strain loading of transversely isotropic, random porous elastomers
3.1.1 Second-order homogenization estimates
3.1.2 Tangent second-order homogenization estimates
3.1.3 Comparisons with exact results
3.1.4 Loss of strong ellipticity
3.2 Results for plane-strain loading: random porous elastomers
3.2.1 Hydrostatic loading
3.2.2 Uniaxial loading
3.2.3 Pure shear loading
3.2.4 Failure surfaces
3.3 Concluding remarks
3.4 Appendix I. In-plane components of the tensor P for cylindrical inclusions with circular cross-section
3.5 Appendix II. Second-order estimates for transversely isotropic porous elastomers with incompressible Neo-Hookean matrix phase
3.6 Appendix III. Coe±cients associated with the incompressible limit for the second-order estimate of Neo-Hookean porous elastomers
3.7 Appendix IV. Tangent second-order estimates for transversely isotropic porous elastomers with incompressible Neo-Hookean matrix phase
4 Porous elastomers: cylindrical voids, periodic microstructure
4.1 Plane-strain loading of periodic porous elastomers
4.1.1 Second-order homogenization estimates
4.1.2 Loss of strong ellipticity
4.2 Results for plane-strain loading: periodic porous elastomers
4.2.1 Hydrostatic loading
4.2.2 Aligned uniaxial loading
4.2.3 Failure surfaces
4.3 Concluding remarks
4.4 Appendix I. Expressions for the microstructural tensor P
4.5 Appendix II. Onset of percolation
5 Porous elastomers: spherical voids
5.1 Overall behavior of isotropic porous elastomers
5.1.1 Earlier estimates
5.1.2 Second-order homogenization estimates
5.1.3 Small-strain elastic moduli
5.1.4 Exact evolution of porosity
5.1.5 Loss of strong ellipticity
5.2 Results and discussion
5.2.1 Axisymmetric loadings
5.2.2 Plane-strain loadings
5.3 Concluding remarks
5.4 Appendix I. Second-order estimates for isotropic porous elastomers with compressible matrix phases
5.5 Appendix II. Second-order estimates for isotropic porous elastomers with incompressible matrix phases
6 Hyperelastic laminates
6.1 E®ective behavior of hyperelastic laminates
6.1.1 Tangent second-order homogenization estimates
6.1.2 Microstructure evolution
6.2 Plane-strain loading of Neo-Hookean laminates
6.3 Results and discussion
6.3.1 Aligned pure shear
6.3.2 Pure shear at an angle
6.4 Concluding remarks
7 Reinforced elastomers: cylindrical ¯bers, random microstructure
7.1 Plane-strain loading of ¯ber-reinforced, random elastomers
7.1.1 Second-order homogenization estimates: compliant ¯bers
7.1.2 Second-order homogenization estimates: rigid ¯bers
7.1.3 Loss of strong ellipticity
7.2 Results for plane-strain loading: random reinforced elastomers
7.2.1 Pure shear: circular rigid ¯bers and incompressible matrix
7.2.2 Aligned pure shear: rigid ¯bers and incompressible matrix
7.2.3 Pure shear at an angle: rigid ¯bers and incompressible matrix
7.2.4 Aligned pure shear: compliant ¯bers and compressible matrix
7.2.5 Simple shear: rigid ¯bers and incompressible matrix
7.3 Concluding remarks
7.4 Appendix I. Incompressibility limit for rigidly reinforced elastomers: cylindrical ¯bers
8 Closure
Appendices
A Second-order homogenization estimates incorporating ¯eld °uctuations in ¯nite elasticity1
A.1 Hyperelastic composites and e®ective behavior
A.2 The second-order variational procedure
A.3 Application to particle-reinforced elastomers
A.3.1 Lower bounds
A.3.2 Second-order estimates
A.4 Plane strain loading of transversely isotropic, ¯ber-reinforced Neo-Hookean composites
A.4.1 Formulation
A.4.2 Results
A.5 Concluding remarks
Bibliography
