Small-strain elasticity

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-... 

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). 

Since these material constants are defined for a perfectly isotropic and homogeneous material within an ideally elastic, small strain regime, application of such mechanical concepts to single molecules should only be done with reservations. 

It is consistent with the approximations of small strain theory made in Section A.7 to neglect the higher-order terms, and to consider the elastic moduli to be constant. Stated in another way, it would be inconsistent with the use of the small deformation strain tensor to consider the stress relation to be nonlinear. The previous theory has included such nonlinearity because the theory will later be generalized to large deformations, where variable moduli are the rule. 

This linear relationship between stress and strain is a very handy one when calculating the response of a solid to stress, but it must be remembered that most solids are elastic only to very small strains up to about 0.001. Beyond that some break and some become plastic - and this we will discuss in later chapters. A few solids like rubber are elastic up to very much larger strains of order 4 or 5, but they cease to be linearly elastic (that is the stress is no longer proportional to the strain) after a strain of about 0.01. 

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... 

When a foam is compressed, the stress-strain curve shows three regions (Fig. 25.9). At small strains the foam deforms in a linear-elastic way there is then a plateau of deformation at almost constant stress and finally there is a region of densification as the cell walls crush together. 

Linear-elasticity, of course, is limited to small strains (5% or less). Elastomeric foams can be compressed far more than this. The deformation is still recoverable (and thus elastic) but is non-linear, giving the plateau on Fig. 25.9. It is caused by the elastic... 

Once values for R , Rp, and AEg are calculated at a given strain, the np product is extracted and individual values for n and p are determined from Eq. (4.19). The conductivity can then be calculated from eq. (4.18) after the mobilities are calculated. The hole mobility is the principal uncertainty since it has only been measured at small strains. In order to fit data obtained from elastic shock-loading experiments, a hole-mobility cutoff ratio is used as a parameter along with an unknown shear deformation potential. A best fit is then determined from the data for the cutoff ratio and the deformation potential. 

The laminate strains have been reduced to e, Sy, and by virtue of the Kirchhoff hypothesis. That is, ez = Yw -r 0- small strains (linear elasticity), the remaining strains are ned in terms of displacements as... 

Bronze (7%) (Cu/Sn) typically 5-10% Sn often with added P or Zn to aid casting and impart superior elasticity and strain resistance. Gun metal is 85% Cu, 5% Sn, 5% Zn and 5% Pb. Coinage metal and brass also often contain small amounts of Sn. World production of bronzes approaches SOO 000 tonnes pa. 

In comparing the correlation sought between MH and E one should emphasize the following while the plastic deformation of lamellae at larger strains when measuring MH depends primarily on crystal thickness and perfection in case of the elastic modulus the major role is played by the amorphous layer reinforced by tie molecules, which is elastically deformed at small strains. Figure 17 illustrates de... 

In reality the ideal elastic rubber does not exist. Real rubbery materials do have a small element of viscosity about their mechanical behaviour, even though their behaviour is dominated by the elastic element. Even so, real rubbers only demonstrate essentially elastic behaviour, i.e. instantaneous strain proportional to the applied stress, at small strains. 

The mechanical concepts of stress are outlined in Fig. 1, with the axes reversed from that employed by mechanical engineers. The three salient features of a stress-strain response curve are shown in Fig. la. Initial increases in stress cause small strains but beyond a threshold, the yield stress, increasing stress causes ever increasing strains until the ultimate stress, at which point fracture occurs. The concept of the yield stress is more clearly realised when material is subjected to a stress and then relaxed to zero stress (Fig. Ih). In this case a strain is developed but is reversed perfectly - elastically - to zero strain at zero stress. In contrast, when the applied stress exceeds the yield stress (Fig. Ic) and the stress relaxes to zero, the strain does not return to zero. The material has irreversibly -plastically - extended. The extent of this plastic strain defines the residual strain. 

The modulus of elasticity of a material it is the ratio of the stress to the strain produced by the stress in the material. Hooke s law is obeyed by metals but mbber obeys Hooke s law only at small strains in shear. At low strains up to about 15% the stress-strain curve is almost linear, but above 15% the stress and strain are no longer proportional. See Modulus. 

The experimental data to be considered are shown in Figure 1. They refer to previously published data on hexamethylene diisocyanate(HDI) reacting with polyoxypropylene(POP) triols and tetrols in bulk and in nitrobenzene(5-7,12) that is, to RA2 + RBj polymerisations. is the molar mass of chains between elastically effective junction points. A/Mj. has been determined directly from small-strain compression measurements on swollen and dry networks using the equations... 

The lines Pr e Pr c indicated in the figures and it is apparent that the inequality is satisfied only for affine behaviour (Figure 4), as expected for the small-strain measurements used. The condition Pr e Pr c need not be met if a significant number of complex ring structures are formed pre-gel which then become elastic-ally active when incorporated in the gel. The present interpretation of the plots in Figures 4 and 5 makes the simplifying assumption that all pre-gel ring structures remain elastically inactive in the final network. 

To this point, we have limited the discussion to small strains—that is, small deviations from the equilibrium bond distance, such that all imposed deformations are completely recoverable. This is the elastic response region, one that virtually all materials possess (see Figure 5.8). What happens at larger deformations, however, is dependent to some... 

In crystalline solutions, the developing interfaces are initially coherent—strains are continuous across interfaces. Unless defects such as anticoherency dislocations intervene, the interfaces will remain coherent until a critical stress is attained and the dislocations are nucleated. For small-strain fluctuations, the system can be assumed to remain coherent and the resulting elastic coherency energy can be derived.9... 

A viscoelastic material is represented by a combination of elastic and viscous bodies. If a controlled sinusoidal small strain (y) is applied to a viscoelastic material as follows ... 

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). 

The purpose of this chapter is to remind the reader of the basis of the theory of elasticity, to outline some of its principal results and to discuss to what extent the classical theory can be applied to polymeric systems. We shall begin by reviewing the definitions of stress and strain and the compliance and stiffness matrices for linear elastic bodies at small strains. We shall then state several important exact solutions of these equations under idealised loading conditions and briefly discuss the changes introduced if realistic loading conditions are considered. We shall then move on to a discussion of viscoelasticity and its application to real materials. 

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. 

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