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## Statistical theory

One of the simplest models to describe rubber elasticity of a polymer network is the affine model originally proposed by Kuhn and developed by Wall , Flory, James and Guth and accurately described by Treloar4. The two main assumptions are: the affine model and the Gaussian statistics of the polymer chains. The first implies that the deformation of each network strand is identical to the deformation of the macroscopic solid. The Gaussian assumption considers the entropy of the network as the sum of the entropies od individual chains and can be described by Gaussian statistics. Furthermore, the network is considered ideal so that enthalpic force are neglected (f H=0) , the volume change is equal to zero, the chains are flexible (T>Tg) and there is no chain slip or strain induced crystallization. The entropy of a single chain with n links of equal length l with completely random orientation within the chain, can be expressed as Equation 2‐1 Where, c is an arbitrary constant with no physical meaning, (function of chain length) and r is the end-to-end distance of the chain. Within the limit of the model assumption, given Li0 the polymer network dimension in the -i direction in the 2-11 Elastomers undeformed state. If the network is deformed by the factor i the dimension on the deformed network is described as: Li= i*Li0 Equation 2‐2.

If each network strand is composed by N monomers and both end of the strands are deformed affinely to the macroscopic network, we can define the initial end-to-end vector R0 and the projections along the -i direction of the deformed R as : ri = iri0 Equation 2‐3.

For an ideal network, in absence of any enthalpic contributions (which means that the presence of crosslinking is ignored), the total Helmholtz free energy of the system can be written as W=-T S. Where S is given by the sum of the N single chain entropy difference between the undeformed and deformed state: S s.

Combining Equation 2-3 and Equation 2-1 the total work of deformation or, elastically stored energy per unit volume W takes the form: W= Equation 2‐4.

And the shear modulus can be expressed for as: Equation 2‐5.

where is the density of the rubber, R the gas constant , M0 the molecular weight of the monomer and the number of monomers per elastic strands, i.e. between crosslink points. Equation 2-4 is the fundamental expression defining the elastic properties of rubbers in the Gaussian approximation regime and has the huge advantage to enable the derivation of the stress-strain relationship for any type of applied strain. Additionally, it involves only a single physical parameter or elastic constant: G which contains the dependence of the materials structure. Equation 2-4 indicates that, as far as the Gaussian approach is valid, the elastic response of the rubber is independent of the chemistry of the polymer chains.

### The case of uniaxial tension and limit of Gaussian theory

We can use the Gaussian theory to predict the stress strain behaviour in the specific case of uniaxial tension. According to incompressibility condition ( ∗ ∗ =1), the following holds: ; ; The strain energy density calculated with Equation takes the form: Equation 2-4 2‐6.

And the force per unit cross-sectional area becomes: = Equation 2‐7.

Experimental observations of stress-strain relationship in conventional elastomers anyway, reveal substantial deviations from the theoretical behaviour of Equation 2-7 . Figure 2 -2 shows the comparison between stress-elongation behaviour predicted by statistical Gaussian theory (full line) with experimental behaviour observed for a vulcanized NR sample4. The model works quite well over a limited range of deformations below =1.5. The deviation at higher strain is due to the effect of limited chain extensibility that is not accounted in the Gaussian theory. Additionally, in NR and other stereo- regular elastomers strain induced crystallization effects will also increase the material stiffness and dominate at high strain. Unfortunately, the stiffening of crosslinked polymer chains cannot be predicted by simple considerations on the crosslinking density of the network. Several non-gaussian theories have been provided to describe the large strain behaviours of elastomers that can be mainly divided in: phenomenological models, invariant-based and physical models. Nevertheless, non-gaussian approaches are less general than the gaussian approach that, despite failing at large strain, maintains the advantage to provide an easy understanding of the relationship between the stress-strain curve for different type of loading (uniaxial, biaxial, pure shear, etc. ) 2-13 Elastomers.

#### Viscoelastic behaviour of elastomers

In the previous sub-chapters, we mainly explored the ideal behaviour of an elastomer that is assumed to have a perfect reversible character. In practice, real elastomers are viscoelastic materials and exhibit time dependent and hysteretic behaviour. The viscoelastic behaviour of elastomers mainly originates from their own architecture made of several long macromolecules that can slide on each -other generating complex phenomena of friction and resistance to motion. As far as the energy of the system is above such energetic barrier the material is soft and the polymer chains have a high mobility. When the energy of the system is reduced (decreasing temperature for example) the polymer chains do not have enough thermal energy to overcome the molecular friction and the material behaves as a hard solid and becomes difficult to deform. A typical manifestation of the viscoelastic behaviour of elastomers is the variation of the complex modulus G* at low strain with applied temperature (or frequency). G* can be divided in an elastic component G’ and in a viscous component G’’.

Figure 2-3 shows the typical variation of G’ and loss modulus or tan for a viscoelastic network where we can distinguish three regimes: T<Tg the material behaves as a hard solid and the elastic modulus is high ≈ GPa T≈Tg the material is in the transition state between a hard and soft solid where chain can move but are highly constrained. This generates a maximum in the dissipation indicated by the peak in tan (=G’’/G’).

T>Tg the material is in a rubbery and soft state. Polymer chains can ideally flow unless they are kept together by some reticulation point as for vulcanised elastomers.

One typical difference between pristine and crosslinked elastomers is the Young’s modulus dependence on temperature. Figure 2 -3 shows that the drop in the modulus for the un-crosslinked elastomer is replaced by a plateau.

**Reinforcing effect on linear modulus**

The most simple model, aiming to understand the increase on the modulus by the filler addition, is the Einstein-Smallwood 8 equation. Einstein equation was not originally meant for rubber but to model the effect of filler on the viscosity of dilute solution and Smallwood adapted his equation to explain the increase of the linear modulus of filled rubber using the concept of hydrodynamic effect. This derives from the fact that the filler is a rigid phase and unable to be deformed under the applied load. The Einstein-Smallwood equation consider only one filler-parameter that is its volume fraction .

Where, E is the modulus of suspension, E0 is the modulus of the incompressible rubber. Equation 3-1 is valid for an infinitely dilute system and accounts for the individual contribution of the particles to the reinforcement but is not followed by rubber compounds where interaction between filler and matrix cannot be neglected. A modified version by Guth and Gold 9 who introduced a second-order term accounting for weak interactions.= 0(1+2.5 +14.1 2) Equation 3-2.

Equation 3-2 is still excessively simplistic and must often be corrected taking in to account other additionally effects including the filler-polymer interactions (in-rubber structure) and elastic properties of the polymer after the vulcanization (polymer network contribution). Generally, the contribution to the modulus is divided in deformation-dependent and deformation-independent as will be discussed in 3.4.1.

**Dissipative mechanisms in filled rubbers**

Unfilled rubbers at low strain and far from Tg are mainly elastic and the most significant viscoelastic contribution is chain friction. Real rubbers anyway, are never used alone and always contain several additives and a certain amount of filler as previously explained. The addition of filler is irremediably related to an increase in viscous dissipation within the rubber system both at small and large strain.

**Payne effect**

Filled rubbers present a significant drop of the dynamic shear modulus under strain that is generally accompanied by a maximum in dissipation indicated by the peak in tan . The maximum in tan , generally occurs at some tenths of strain, a value that is likely to be close to the values of macroscopic deformation used in cyclic fatigue in filled elastomers. The reduction of modulus with strain was first studied by Payne 10,11 who interpreted it as the result of the breakage of physical bonds between filler particles, for example van der Waals or London forces, neglecting the role of the matrix itself. Actually, there are several parameters contributing to the Payne effect as: filler content, surface modification of the fillers and morphological characteristic ( volume fraction, filler shape and aspect ratio) 12. A more recent insight of Payne effect includes the presence of a glassy layer around the filler particles where polymer has reduced mobility. Such glassy layer forms a percolating structure that is responsible for the strain induced softening 13. Generally, all the effects related to the structure of the material, that contribute to the strain amplitude dependency, are referred as “filler network” while polymer network, hydrodynamic effect and in-rubber structure are generally indicated as non-strain dependent as schematically shown in Figure 3-6.

**Table of contents :**

**1 INTRODUCTION AND CHAPTERS ORGANIZATION **

1.1 INDUSTRIAL CONTEXT AND SCIENTIFIC QUESTION OF THE STUDY

1.2 ORGANIZATION OF THE STUDY

1.3 NOTE ON THE ADOPTED NOMENCLATURE

1.4 REFERENCES

**2 ELASTOMERS **

2.1 MAIN PROPERTIES OF ELASTOMER

2.2 DEFINITION OF ENTROPIC RUBBER ELASTICITY

2.3 STATISTICAL THEORY

2.4 VISCOELASTIC BEHAVIOUR OF ELASTOMERS

2.5 REFERENCES

**3 FILLED RUBBERS **

3.1 POLYMER MATRIX: SBR

3.2 VULCANIZATION

3.3 FILLER SYSTEM

3.4 DISSIPATIVE MECHANISMS IN FILLED RUBBERS

3.5 STRAIN INDUCED CRYSTALLIZATION

3.6 NANO-CAVITATION

3.7 CONCLUDING REMARKS

3.8 REFERENCES

**4 THERMOPLASTIC ELASTOMERS **

4.1 HISTORICAL SURVEY AND GENERIC CLASSIFICATION

4.2 MORPHOLOGY OF TPU

4.3 THERMODYNAMIC OF PHASE SEPARATION

4.4 STRENGTH OF TPU AND DEFORMATION MECHANISMS

4.5 INELASTICITY OF TPU

4.6 CONCLUDING REMARKS

4.7 REFERENCES

**5 FRACTURE AND FATIGUE IN SOFT MATERIALS **

5.1 BRIEF INTRODUCTION TO LINEAR ELASTIC FRACTURE MECHANIC

5.2 FROM LEFM TO FRACTURE MECHANIC OF SOFT MATERIALS

5.3 CYCLIC FATIGUE

5.4 STRATEGY TO IMPROVE FATIGUE RESISTANCE IN SOFT MATERIALS

5.5 CONCLUDING REMARKS

5.6 REFERENCES

**6 CYCLIC FATIGUE IN SBR: THE ROLE OF CRACK TIP **

6.1 ABSTRACT

6.2 INTRODUCTION

6.3 MATERIALS AND METHODS

6.4 RESULTS

6.5 CYCLIC FATIGUE TESTS

6.6 MULTI-SCALE CRACK TIP OBSERVATION

6.7 EXTENDED DISCUSSION

6.8 CONCLUSIONS

6.9 SUPPLEMENTARY INFORMATION

6.10 ACKNOWLEDGEMENTS

6.11 REFERENCES

**7 CYCLIC FATIGUE FAILURE OF TPU **

7.1 ABSTRACT

7.2 INTRODUCTION

7.3 MATERIALS AND METHODS

7.4 MATERIALS CHARACTERIZATION

7.5 DISCUSSION

7.6 CONCLUSION

7.7 ACKNOWLEDGEMENTS

7.8 REFERENCES

**8 MECHANICAL PROPERTIES OF SOFT TPU AND STRAIN INDUCED STRENGTHENING **

8.1 ABSTRACT

8.2 INTRODUCTION

8.3 MATERIALS AND METHODS

8.4 MECHANICAL TESTING AND STRUCTURAL INVESTIGATIONS

8.5 DAMAGE ANALYSIS IN CYCLIC LOADING

8.6 DISCUSSION ON THE DIFFERENCES BETWEEN TPU AND SBR

8.7 CONCLUSIONS

8.8 ACKNOWLEDGEMENTS

8.9 REFERENCES

**9 SELF-ORGANIZATION AT THE CRACK TIP AND CYCLIC FATIGUE IN TPU **

9.1 ABSTRACT

9.2 INTRODUCTION

9.3 MATERIALS AND METHODS:

9.4 RESULTS

9.5 DIFFERENCES BETWEEN MICROSTRUCTURE AT BULK AND CRACK TIP

9.6 DISCUSSION

9.7 CONCLUSION

9.8 ACKNOWLEDGMENTS

9.9 REFERENCES

9 EXTENDED SUPPLEMENTARY INFORMATION

9.1 X-RAY ANALYSIS

9.2 STRAIN-INDUCED STRUCTURAL CHANGES

9.3 RESIDUAL CRYSTALLINITY IN UNIAXIAL STRAINED TPU_XTAL

9.4 CYCLIC FATIGUE METHOD B

**10 GENERAL CONCLUSION AND PROSPECTS **

10.1 FINAL REMARKS AND FUTURE PERSPECTIVES

10.2 REFERENCES

**ANNEXES **

1. FTIR ANALYSIS

2. TOUGHNESS: EFFECT OF TEMPERATURE AND STRAIN RATE

1.1 EXPERIMENTAL CONDITIONS

3. CRACK EXTENSION AND BLUNTING AT HIGH STRETCH RATE

4. CREEP AND STRESS RELAXATION

5. REFERENCES