Elastic behaviour, linear

Figure 8.1 shows the stress-strain curve of a material exhibiting perfectly linear elastic behaviour. This is the behaviour characterised by Hooke s Law (Chapter 3). All solids are linear elastic at small strains - by which we usually mean less than 0.001, or 0.1%. The slope of the stress-strain line, which is the same in compression as in tension, is of... 

In recent years impact testing of plastics has been rationalised to a certain extent by the use of fracture mechanics. The most successful results have been achieved by assuming that LEFM assumptions (bulk linear elastic behaviour and presence of sharp notch) apply during the Izod and Charpy testing of a plastic. 

Mechanical properties per se concerns with the qualities which determine the behaviour of a material towards applied forces. The ability to support weight without rupture or permanent deformation, to withstand impact without breaking, to be mechanically formed into different shapes - all these depend upon a combination of mechanical properties characteristic of metals. Four types of behaviour of a material under stress are very important linear or elastic behaviour, plastic behaviour, creep behaviour and fatigue behaviour. 

The reduced stress is defined as the force per cross-sectional area of the undeformed sample, divided by the term X-X- with X being the relative elongation L/L0. With undiluted rubber, this is not found experimentally. In most cases, however, the elastic behaviour in a moderate elongation range is satisfactorily described fcy the empirical Mooney-Rivlin equation, which predicts a linear dependence of on reciprocal elongation X- (32-34)... 

Let us first examine the behaviour shown in Fig. 2a. The initial part is linear and perfectly reversible till a true stress value denoted as crei. It corresponds to purely elastic behaviour. 

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]. 

The situation for amorphous linear polymers is sketched in Fig. 2.8a. If a polymeric glass is heated, it will begin to soften in the neighbourhood of the glass-rubber transition temperature (Tg) and become quite rubbery. On further heating the elastic behaviour diminishes, but it is only at temperatures more than 50° above the glass-rubber transition temperature that a shear stress will cause viscous flow to predominate over elastic deformation. 

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

Figure 9a shows a typical load-loadline displacement curve of a nylon compact tensiiin specimen. The slight deviation from linear elastic behaviour prior to fracture relates to the presence of a plastic deformation zone confined near the pre-crack tip (Fig. 9b). This stable crack propagation domain is adjoining a wide hackle zone (rapid crack growth domain) characteristic of a brittle fracture.

Substitution of Eqs. (7.15 a to c) and (7.16 a to c) into Eq. (7.17 a to c) yields the full expression for the relaxed stress. Assuming linear-elastic behaviour of the material these equations can then be substituted into the biaxial Hooke s law and solved for the relaxed normal strains er and ee at point P(l ,a). The measuring procedure involves the following steps (Vishay Precision Group, 2010) ... 

All teams assumed linear elastic behaviour for the different media in addition, SKI used a notension model for the buffer and the backfill. Because of the crucial role of the rock mass permeability, three values of its initial intrinsic permeability have been considered lO lO (base case) and lO m. Moreover, a variation of tbe permeability with respect either to the porosity tp or to the effective mean stress Om (SKI) has been assumed ... 

This model is based on the mean features of the Mohr-Coulomb model and is expressed with stress invariants [Maleki (1999)] instead of principal stresses. Until plasticity is reached, a linear elastic behaviour is assumed. It is fully described by the drained elastic bulk and shear moduli. The yield surface of the perfectly plastic model is given by equation 7. Function 7i(0) is chosen so that the shape of the criterion in the principal stress space is close to the Lade criterion. 

In section 6.2 the formalism of the elastic behaviour of an ideal linear elastic solid for small strains was considered. Rubbers may, however, be reversibly extended by hundreds of per cent, implying that a different approach is required. The previous ideas suggest a possible plausible generalisation, as follows. 

As discussed in section 6.2.2, the values of Young s modulus for isotropic glassy and semicrystalline polymers are typically two orders of magnitude lower than those of metals. These materials can be either brittle, leading to fracture at strains of a few per cent, or ductile, leading to large but non-recoverable deformation (see chapter 8). In contrast, for rubbers. Young s moduli are typically of order 1 MPa for small strains (fig. 6.6 shows that the load-extension curve is non-linear) and elastic, i.e. recoverable, extensions up to about 1000% are often possible. This shows that the fundamental mechanism for the elastic behaviour of rubbers must be quite different from that for metals and other types of solids. 

Just as linear viscoelastic behaviour with full recovery of strain is an idealisation of the behaviour of some real polymers under suitable conditions, so ideal yield behaviour may be imagined to conform to the following for stresses and strains below the yield point the material has time-indepen-dent linear elastic behaviour with a very low compliance and with full recovery of strain on removal of stress at a certain stress level, called the yield stress, the strain increases without further increase in the stress if the material has been strained beyond the yield stress there is no recovery of strain. This ideal behaviour is illustrated in fig. 8.1 and the differences between ideal viscoelastic creep and ideal yield behaviour are shown in table 8.1. 

The models have been developed mainly for semi-crystalline polymers, which in general show the largest mechanical anisotropy, but some of the discussion is equally relevant to oriented non-crystalline polymers. Although an oriented polymer is strictly a non-linear viscoelastic solid (see Chapters 10 and 11) the present discussion is restricted to theoretical models which represent linear elastic or linear viscoelastic behaviour. 

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. 

Three significant characteristic points of the stress-strain curve are distinguishable. The first kink describes the elastic limit, the second kink is the yield strength and the third point describes the breaking point of the foil for uniaxial short term exposure. Up to the elastic limit, ETFE-foU shows an almost linear elastic behaviour. Ftooke s law prevails (at least for shortterm loads). From the elastic limit point up to the yield strength, ETFE... 

Within the region of linear visco-elastic behaviour, an imposed stress of angular frequency, u>, results in a harmonic strain of amplitude proportional to the stress amplitude, and with phase lag 8 relative to the stress, which is independent of the applied stress amplitude [Ferry, 1980],... 

C/SiC composites show a quasi-ductile fracture behaviour, derived from mechanisms like crack deflection and fibre pullout. Figure 14 shows exemplarily these effects within a C/C-SiC composite. The linear-elastic behaviour of C/SiC is less pronounced than for example SiC/SiC composites due to the inherent microcracks in the matrix which occur during cooling-down from processing to room temperature because of the high thermal mismatch between C-fibres and SiC-matrix. 

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 ... 

The most popular method is the first one. In the multi-layered elastic analysis method, the system (pavement) is divided into a number of distinct layers where each layer has its own thickness and mechanical properties. Each of these distinct layers is considered as homogeneous with linear-elastic behaviour. 

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