Small-strain elastic behavior, polymer

The equilibrium small-strain elastic behavior of an "incompressible" rubbery network polymer can be specified by a single number—either the shear modulus or the Young s modulus (which for an incompressible elastomer is equal to 3. This modulus being known, the stress-strain behavior in uniaxial tension, biaxial tension, shear, or compression can be calculated in a simple manner. (If compressibility is taken into account, two moduli are required and the bulk modulus. ) The relation between elastic properties and molecular architecture becomes a simple relation between two numbers the shear modulus and the cross-link density (or the... 

Polymers of all types, glassy or semi-crystalline, have a much more protracted transition from small-strain elastic behavior to fully developed plastic flow than do metals, which can stretch the transition over a quite large strain of the order of 0.05. This is a consequence of the much lower level of crystallinity in polymers than in metals and because the thermally assisted unit inelastic transformation events, occurring primarily in the amorphous component, are in the form of isolated sessile shear transformations in relatively equi-axed small-volume... 

Since the stiffness of the bonds transfers to the stiffness of the whole filler network, the small strain elastic modulus of highly filled composites is expected to reflect the specific properties of the filler-filler bonds. In particular, the small strain modulus increases with decreasing gap size during heat treatment as observed in Fig. 32a. Furthermore, it exhibits the same temperature dependence as that of the bonds, i.e., the characteristic Arrhenius behavior typical for glassy polymers. Note however that the stiffness of the filler network is also strongly affected by its global structure on mesoscopic length scales. This will be considered in more detail in the next section. 

While we do not want to give a sophisticated model including all the effects found in the mechanical behavior of polymers, we restrict ourselves to the simplest case, namely to an elastic small-strain model at constant temperature. Therefore, the governing variables are the linear strain tensor [Eq. (13)] derived from the spatial gradient of the displacement field u, and the microstructural parameter k and its gradient. The free energy density is assumed to be a function of the form of Eq. (14). 

At small strains, polymers (both amorphous and crystalline) show essentially linear elastic behavior. The strain observed in this phase arises from bond angle deformation and bond stretching it is recoverable on removing the applied stress. The slope of this initial portion of the stress-strain curve is the elastic modulus. With further increase in strain, strain-induced softening occurs, resulting in a reduction of the instantaneous modulus (i.e., slope decreases). Strain-softening phenomenon is attributed to uncoiling... 

Hard elastic behavior is a manifestation of a bulk-microfibril superstructure. A substantial surface energy component of the stress exists in these materials, independent of strain at high tension. As a result, significant changes in the equilibrium stress occurs when the polymers, under load, are subjected to changes in environmental surface tension. An apparent requirement for this surface tension component is load bearing microfibrils with sufficiently small radii. Evidently, a maximum average fibril diameter exists whereby hard elastic behavior may occur in polymers with these structures. 

Elastomers are a class of polymers that can be repeatedly strained and then return to the approximate original length on release of the load. Traditional elastomers such as rubber are able to achieve this elastic behavior by having a low glass transition temperature and a small number of chemical crosslinks that form a permanent network... 

For many applications, however, the small-strain behavior is at least as important as the large deformations. As the prediction of elastic behavior of heterogeneous materials is the most known and well-established, it is worth to know if the cmrent micromechaiiical models are able to predict the behavior of semicrystalline polymers. Firstly, the elastic properties of the composite components should be determined. 

Let s start by looking at a simple polymer, polyethylene, that has a lot going on in its stress/strain plots (Figure 13-38). Flexible, semi-crystalline polymers such as this (where the T of the amorphous domains is below room temperature) usually display a considerable amount of yielding or cold-drawing, as long as they are not stretched too quickly. For small deformations, Hookean elastic-type behavior (more or less) is observed, but beyond what is called the yield point irreversible deformation occurs. 

Polymers which yield extensively under stress exhibit nonlinear stress-strain behavior. This invalidates the application of linear elastic fracture mechanics. It is usually assumed that the LEFM approach can be used if the size of the plastic zone is small compared to the dimensions of the object. Alternative concepts have been proposed for rating the fracture resistance of tougher polymers, like polyolelins, but empirical pendulum impact or dart drop tests are deeply entrenched forjudging such behavior. 

Uny et also reported the chemical synthesis of protein polymers based on the (Val-Pro- Ala-Val-Gly) repeat sequence in which glycine is replaced by the D-alanine residue. The hetero-chiral Pro- Ala diad would be erqrected on the basis of stereochemical considerations to adopt a type-II p-tum conformation. Stmctural analyses of small-molecule "Pro- Ala turn models support the formation of the type-II p-mm conformation in solution and the solid state. Polymers based on the (Val-Pro- Ala-Val-Gly) repeat sequence display a thermo-reversible phase transition similar to the corresponding polypeptides derived from the parent (Val-Pro-Gly-Val-Gly) sequence, albeit with a shift of the Tt to approximately 5-10 ° G below the latter due to a slight inaease in hydrophobic character due to the presence of the alanine residue. NMR spectroscopic analyses of the (Val-Pro- Ala-Val-Gly) polymer suggest that the repeat unit retains the p-tum stmcture on the basis of comparison to the corresponding behavior of the (Val-Pro-Gly-Val-Gly) polymer. Stress-strain measurements on cross-linked matrices of the (Val-Pro- Ala-Val-Gly) polymer indicate an elastomeric mechanical response in which the elastic modulus does value in comparison to the (Val-Pro-Gly-Val-Gly) polymer. These smdies of glycine suhstitution support the hypothesis that type-II p-tum formation can he associated with the development of elastomeric behavior with native elastins and elastin-derived polypeptide sequences. Several investigators have proposed that the (Val-Pro-Gly-Val-Gly) pentapeptide represents the minimal viscoelastic unit... 

The stress-strain curves of hydrated IPMCs (Fig. 2.13 (b)) show soft plastic behavior, low range of tensile stress, and elastic deformation with just one yield point. This could be due to the water molecules in the Nafion. The tensile strength reduced in each case due to the disconnecting of the polymer chains and the volume increase from swelling. Also, water acts as a good plasticizer, even in small quantities, due to its low Tg. 

---