Elasticity, large strain

Figure 8.2 shows a non-linear elastic solid. Rubbers have a stress-strain curve like this, extending to very large strains (of order 5). The material is still elastic if unloaded, it follows the same path down as it did up, and all the energy stored, per unit volume, during loading is recovered on unloading - that is why catapults can be as lethal as they are. 

The piezoelectric constant studies are perhaps the most unique of the shock studies in the elastic range. The various investigations on quartz and lithium niobate represent perhaps the most detailed investigation ever conducted on shock-compressed matter. The direct measurement of the piezoelectric polarization at large strain has resulted in perhaps the most precise determinations of the linear constants for quartz and lithium niobate by any technique. The direct nature of the shock measurements is in sharp contrast to the ultrasonic studies in which the piezoelectric constants are determined indirectly as changes in wavespeed for various electrical boundary conditions. 

The concept of a welt defined elastic range to large strain is not realistic. The concept of well defined stress at which mechanical yielding occurs leading to well defined elastic-inelastic conditions is not realistic. Actually, such conclusions could well be anticipated from strength studies at atmospheric pressure, but there has been little explicit reason to consider the nonideal effects from the mechanical-response shock studies. 

Secondly, the Llo structure must be stable with respect to both small and large strains. The former is assured by the positive definiteness of elastic moduli. Clearly, an exhaustive test of the stability with respect to large strains cannot be made and the following two types of tests were performed. First, the energy of the structure was calculated as a function of strains corresponding to 20% changes in the lattice parameters a and... 

Before dealing with reinforcement of elastomers we have to introduce the basic molecular features of mbber elasticity. Then, we introduce—step-by-step—additional components into the model which consider the influence of reinforcing disordered solid fillers like carbon black or silica within a rabbery matrix. At this point, we will pay special attention to the incorporation of several additional kinds of complex interactions which then come into play polymer-filler and filler-filler interactions. We demonstrate how a model of reinforced elastomers in its present state allows a thorough description of the large-strain materials behavior of reinforced mbbers in several fields of technical applications. In this way we present a thoroughgoing line from molecular mechanisms to industrial applications of reinforced elastomers. 

In particular it can be shown that the dynamic flocculation model of stress softening and hysteresis fulfils a plausibility criterion, important, e.g., for finite element (FE) apphcations. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. From the simulations of stress-strain cycles at medium and large strain it can be concluded that the model of cluster breakdown and reaggregation for prestrained samples represents a fundamental micromechanical basis for the description of nonlinear viscoelasticity of filler-reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of tender but fragile filler clusters [24]. 

In non-polymeric materials the entropy change on deformation is minimal so that the intrinsic and stored elastic energies are the same at least for rapidly occurring events - but in polymers not only may the entropy contribution predominate but for large strains in rubbers the internal energy term is nearly negligible (but not at small strains where it may amount to 20% of the free energy). 

In classical elasticity (small strains) W is a quadratic function of the coefficients of infinitesimal strain ey, whereas in large strain elasticity the relationship is not quadratic and W is then expressed as a polynomial in the strain coefficients or, as is usual in continuum mechanics, as a polynomial in the nine components of the deformation gra-... 

Hooke s law relates stress (or strain) at a point to strain (or stress) at the same point and the structure of classical elasticity (see e.g. Love, Sokolnikoff) is built upon this linear relation. There are other relationships possible. One, as outlined above (see e.g. Green and Adkins) involves the large strain tensor Cjj which does not bear a simple relationship to the stress tensor, another involves the newer concepts of micropolar and micromorphic elasticity in which not only the stress but also the couple at a point must be related to the local variations of displacement and rotation. A third, which may prove to be very relevant to polymers, derives from non-local field theories in which not only the strain (or displacement) at a point but also that in the neighbourhood of the point needs to be taken into account. In polymers, where the chain is so much stiffer along its axis than any interchain stiffness (consequent upon the vastly different forces along and between chains) the displacement at any point is quite likely to be influenced by forces on chains some distance away. 

It is shown in works on large strain elasticity (Green and Adkins25) that when 0 is large there is a normal stress term proportional to 67. 

Our reason for stressing the concept of representative volume element is that it seems to provide a valuable dividing boundary between continuum theories and molecular or microscopic theories. For scales larger than the RVE we can use continuum mechanics (classical and large strain elasticity, linear and non-linear viscoelasticity) and derive from experiment useful and reproducible properties of the material as a whole and of the RVE in particular. Below the scale of the RVE we must consider the micromechanics if we can - which may still be analysable by continuum theories but which eventually must be studied by the consideration of the forces and displacements of polymer chains and their interactions. 

We should note that the above simple treatment totally ignores any departure from linear elastic behaviour at large strains.)... 

Equation (13-12) and (13-13) plus a small noise term—which introduces small imperfections into the macrolattice—complete the description of the the model. Because of the periodic nature of the elastic stress [Eq. (13-9)] in large-strain shearing, instabilities occur that allow layers to slip with respect to each other, and the strain y, then varies from layer to layer. Thus flow occurs by yielding a mechanism similar to that discussed in Section 1.5.3. 

Several cautions are, however, in order. Polymers are notorious for their time dependent behavior. Slow but persistent relaxation processes can result in glass transition type behavior (under stress) at temperatures well below the commonly quoted dilatometric or DTA glass transition temperature. Under such a condition the polymer is ductile, not brittle. Thus, the question of a brittle-ductile transition arises, a subject which this writer has discussed on occasion. It is then necessary to compare the propensity of a sample to fail by brittle crack propagation versus its tendency to fail (in service) by excessive creep. The use of linear elastic fracture mechanics addresses the first failure mode and not the second. If the brittle-ductile transition is kinetic in origin then at some stress a time always exists at which large strains will develop, provided that brittle failure does not intervene. 

---