Non-Linear Elastic Behaviour

This approach is clearly applicable to rubbers with low mechanical hysteresis, which exhibit non-linear elastic behaviour. However, because the energy release rate Gj is defined specifically for the case of linear elastic fracture, we deflne a new parameter J for the non-linear case ... 

It is likely that most biomaterials possess non-linear elastic properties. However, in the absence of detailed measurements of the relevant properties it is not necessary to resort to complicated non-linear theories of viscoelasticity. A simple dashpot-and-spring Maxwell model of viscoelasticity will provide a good basis to consider the main features of the behaviour of the soft-solid walls of most biomaterials in the flow field of a typical bioprocess equipment. 

The theory outlined above is rigorous only for infinitesimal elastic deformation. Creep of polymeric materials is explicitly concerned with time dependence and implicitly with finite strains and therefore nonlinear behaviour. The nature of the non-linear behaviour is complex and varies not only from material to material but also with direction within a given sample of material, i.e. the non-linearity of behaviour is anisotropic. It is found, therefore, that on a particular definition of strain the behaviour of a sample may appear to be linear in one direction and significantly non-linear in another. Such a phenomenon is demonstrated in results presented below. 

A brief outline of the extension of the formalism of the classical theory of elasticity for the description of non-linear viscoelastic behaviour was... 

In Section 10.2 the effect of materials symmetry on the number of independent compliance constants Sij for linear elastic behaviour was presented. For the case of fibre symmetry, eqn. (3), we have in particular, Si3 = Sai = S23 = S32. For the linear viscoelastic case Rogers and Pipkin were able to show theoretically that without recourse to the arguments of irreversible thermodynamics it was not possible to show that Si3 = S31 and S23 = S32. Further the validity of all these equalities must be in doubt in non-linear behaviour at finite strains. 

The rigorous design method is based on generally accepted closed-form models. The adhesive behaviour in the models is assumed to be linearly elastic. Only the formulae used in the calculation of the temporary maximum joint resistance require the complete shear stress—shear strain curve or the elastic—plastic model of the adhesive to be known. As adhesives typically have a non-linear shear behaviour, using only the linear part of the stress—strain curve brings added conservatism to the models with respect to the actual joint resistance. 

Comas-Cardona S, Le Grognec P, Binetruy C, Krawczak P. Unidirectional compression of fibre reinforcements. Part 1 A non-linear elastic-plastic behaviour. Composites Science andTechnology,20017,67(3-4) 507—514. EX)I http //dx.doi. org/10.1016/j.compsci tech.2006.08.017. 

Abstract This chapter deals with the non-linear viscoelastic behaviour of rubber-rubber blend composites and nanocomposites with fillers of different particle size. The dynamic viscoelastic behaviour of the composites has been discussed with reference to the filler geometry, distribution, size and loading. The filler characteristics such as particle size, geometry, specific surface area and the surface structural features are found to be the key parameters influencing the Payne effect. Non-Unear decrease of storage modulus with increasing strain has been observed for the unfilled vulcanizates. The addition of spherical or near-spherical filler particles always increase the level of both the linear and the non-linear viscoelastic properties. However, the addition of high-aspect-ratio, fiber-like fillers increase the elasticity as well as the viscosity. 

It appears, then, that the mechanical degradation process is intimately connected with the molecular structure of the macromolecule and the resulting fluid rheology that arises from this structure. For a flexible coil macromolecule, such as HPAM or polyethylene oxide, the polymer solutions are known to display viscoelastic behaviour (see Chapter 3) and thus a liquid relaxation time, may be defined as the time for the fluid to respond to the changing flow field in the porous medium. It may be computed from several possible models (Rouse, 1953 Warner, 1972 Durst et al, 1982 Haas and Durst, 1982 Bird et al. 1987). The finite extendible non-linear elastic (FENE) (Warner, 1972 Bird et al, 1987a Haas and Durst, 1982 Durst et al, 1982) dumbbell model of the polymer molecule may be used to find the relaxation time, tg, as it is known that this model provides a good description of HPAM flow in porous media (Durst et al, 1982 Haas and Durst, 1982) the expression for fe is ... 

It will be convenient to discuss these various aspects separately as follows (1) behaviour at large strains in Chapters 3 and 4 (finite elasticity and rubber-like behaviour, respectively) (2) time-dependent behaviour in Chapters 5-7 and 10 (viscoelastic behaviour) (3) the behaviour of oriented polymers in Chapters 8 and 9 (mechanical anisotropy) (4) non-linearity in Chapter 11 (non-linear viscoelastic behaviour) (5) the non-recoverable behaviour in Chapter 12 (plasticity and yield) and (6) fracture in Chapter 13 (breaking phenomena). However, it should be recognised that we cannot hold to an exact separation and that there are many places where these aspects overlap and can be brought together by the physical mechanisms, which underlie the phenomenological description. 

Fig. 5 [88]. In this case, the fracture parameter is expressed as J, the energy release rate, rather than K. Note that a relationship similar to that of Eq. 18, relates the energy release rate, 7, to the stress intensity, K. In ref. [88], J was used rather than G to emphasize that some of the specimens studied behaved in a non-linear elastic manner (and thus the J integral was used to calculate the energy release rate). For linear elastic behaviour, G is identical to 7. The mode I fracture toughness (i.e. = 0°) is significantly lower than the mode 11 fracture toughness...

When we consider non-linear material properties by a closed-form analysis such as Hart-Smith s, the limitation is how tractable is a realistic mathematical model of the stress-strain curve within an algebraic solution. With the finite-element techniques developed for adhesive joints by Adams and his co-workers, the limit becomes that of computing power. The high elastic stress and strain gradients at the ends of the adhesive layer need to be accommodated by three or four 8-node quadrilateral elements across the thickness. However, consideration of non-linear material behaviour requires a much larger computing effort on any given element. Thus, it becomes necessary to... 

If we now allow for non-linear adhesive behaviour, the high adhesive stress concentrations predicted by the linear elastic analysis will be relieved to some extent. Figure 54 shows the predicted spread of the yield zone of adhesive at the tension end of a double-lap joint as the load is increased. As would be expected, plastic flow begins near the adherend corner and the load corresponds to a joint efficiency of 21%. Each subsequent load increment represents an increase in joint efficiency of 4 4%. When elastic perfectly-plastic behaviour is assumed for the adhesive, a maximum strain criterion for failure seems appropriate. In Fig. 55 the joint efficiency is plotted against the maximum principal strain in the adhesive at each end of a double-lap joint. Assuming a failure strain for the adhesive of 5%, the analysis predicts a joint efficiency of 31% for a double-lap joint compared with 16% predicted by the linear elastic analysis. Similarly, the non-linear analysis predicts an efficiency of 39% for the double-scarf joint compared with 20% predicted by the linear elastic analysis. Although the predicted efficiencies are almost doubled by allowing for non-linear behaviour in the adhesive, failure in the adhesive is still predicted to be more probable than failure in the adherends (Table 5). 

The exact solutions of the linear elasticity theory only apply for small strains, and under idealised loading conditions, so that they should at best only be treated as approximations to the real behaviour of materials under test conditions. In order to describe a material fully we need to know all the elastic constants and, in the case of linear viscoelastic materials, relaxed and unrelaxed values of each, a distribution of relaxation times and an activation energy. While for non-linear viscoelastic materials we cannot obtain a full description of the mechanical properties. 

There is strong interest to analytically describe the fzme-dependence of polymer creep in order to extrapolate the deformation behaviour into otherwise inaccessible time-ranges. Several empirical and thermo-dynamical models have been proposed, such as the Andrade or Findley Potential equation [47,48] or the classical linear and non-linear visco-elastic theories ([36,37,49-51]). In the linear viscoelastic range Findley [48] and Schapery [49] successfully represent the (primary) creep compliance D(t) by a potential equation ... 

It was shown that the stress-induced orientational order is larger in a filled network than in an unfilled one [78]. Two effects explain this observation first, adsorption of network chains on filler particles leads to an increase of the effective crosslink density, and secondly, the microscopic deformation ratio differs from the macroscopic one, since part of the volume is occupied by solid filler particles. An important question for understanding the elastic properties of filled elastomeric systems, is to know to what extent the adsorption layer is affected by an external stress. Tong-time elastic relaxation and/or non-linearity in the elastic behaviour (Mullins effect, Payne effect) may be related to this question [79]. Just above the melting temperature Tm, it has been shown that local chain mobility in the adsorption layer decreases under stress, which may allow some elastic energy to be dissipated, (i.e., to relax). This may provide a mechanism for the reinforcement of filled PDMS networks [78]. 

In Figure 5 and 6, calculated displacements for the two nonlinear models are compared to the measured ones for three different multiple-point borehole extensometers. One general trend concerning the calculated results is an increasing relative displacement, from Anchor 1 to Anchor 4, with a delay in the development of the latter caused by the delay in the propagation of the thermal front. Non-linearities are more pronounced when using the ubiquitous joint model, whereas the brittle model leads to weakly nonlinear behaviour, close to the pure elastic one. 

Each block is modeled as linear, isotropic, homogeneous and elastic medium and subdivided with a mesh of constant-strain triangle finite-difference elements. Key factors affecting the hydraulic behaviour of fractures such as opening, closure, sliding and dilation of fractures are modeled by an elasto-perfectly plastic constitutive model of a fracture. A step-wise non-linear normal stress-normal closure relationship is adopted with a linear Mohr-Coulomb failure for shear (Figure 3). 

These are essentially independent effects a polymer may exhibit all or any of them and they will all be temperature-dependent. Section 6.2 is concerned with the small-strain elasticity of polymers on time-scales short enough for the viscoelastic behaviour to be neglected. Sections 6.3 and 6.4 are concerned with materials that exhibit large strains and nonlinearity but (to a good approximation) none of the other departures from the behaviour of the ideal elastic solid. These are rubber-like materials or elastomers. Chapter 7 deals with materials that exhibit time-dependent effects at small strains but none of the other departures from the behaviour of the ideal elastic sohd. These are linear viscoelastic materials. Chapter 8 deals with yield, i.e. non-recoverable deformation, but this book does not deal with materials that exhibit non-linear viscoelasticity. Chapters 10 and 11 consider anisotropic materials. 

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