Non-linear strain

Show that for a viscoelastic material in which the modulus is given by (t) = At ", there will be a non-linear strain response to a linear increase in stress with time. 

The function fj(y) represents the non-linear strain dependence of the deformed tube. The non-linear stress relaxation function in the reptation zone is thus... 

Fig. 16. Chain retraction in a tube under non-linear strain...

Other important parameters for the correlation between GJJ and GJ include the ductility or the failure strain, particularly the non-linear strain (Jordan and Bradley, 1988 Jordan et al., 1989) of the matrix resin, the bond strength of the fiber-matrix interface (Jordan and Bradley, 1987 Bradley 1989a, b), and the fiber V and their distributions in the composites (Hunston et al., 1987). A high failure strain promotes the intrinsic capacity of the resin to permit shear deformation, and is shown to increase the G and G. values almost linearly, the rate of increase being steeper for G j than for Gf. ... 

Jc Critical non-linear strain energy release rate (J m-2)... 

Besides the outstanding chemical characteristics of certain mesoscopic structures, they also possess a number of surprising physical characteristics. Typical examples are the initiation of premelting near dislocations, twin boundaries or grain boundaries (e g. Raterron et al. 1999, Jamnik and Maier 1997) and the movement of twin boundaries under external stress which leads to non-linear strain-stress relationships. It is the purpose of this review to focus on some of the characteristic features of mesocopic structures and to illustrate the generic results for the case of ferroeleastic twin patterns (Salje 1993). 

Thus, as well as with the abandonment of any product of two rotational angles by virtue of Remark 7.2, the remaining Green Lagrange strain components may be significantly simplified. Arranging the non-linear strain measures for finite displacements but small rotations of the beam in the vector egl( )i leads to... 

The non-linear strain measures for the most general comprehensible case, involving finite displacements but small rotations of the beam, are derived in Section 7.1 and given by Eq. (7.15). In accordance with the calculus of variations, see Funk [77], the virtual variant of these strain measures reads... 

As outlined above, the initial internal loads need to be known and, for this purpose, may be determined with the aid of the first-order theory developed so far. Due to the absence of non-linear strains related to both rotation and warping, only normal and transverse forces, as well as bending moments, have to be obtained. In accordance with the equilibrium equations and natural boundary conditions of Eq. (8.36), these can be expressed as... 

The points in the figures represent tj and the lines It] . Figure 6.10a shows that the linear form, equation (6.35), holds, and Figure 6.10b shows that the non-linear, strain-time correspondence, equation (6.37), does not hold. 

Fig. 8.2. Stress-strain behoviour for a non-linear elastic solid. The axes are calibrated for a material such as rubber.

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. 

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... 

Occasionally, materials are tested in tension by applying the loads in increments. If this method is used for plastics then special caution is needed because during the delay between applying the load and recording the strain, the material creeps. Therefore if the delay is not uniform there may appear to be excessive scatter or non-linearity in the material. In addition, the way in which the loads are applied constitutes a loading history which can affect the performance of the material. A test in which the increments are large would quite probably give results which are different from those obtained from a test in which the increments were small or variable. 

Quite often isochronous data is presented on log-log scales. One of the reasons for this is that on linear scales any slight, but possibly important, non-linearity between stress and strain may go unnoticed whereas the use of log-log scales will usually give a straight-line graph, the slope of which is an indication of the linearity of the material. If it is perfectly linear the slope will be 45°. If the material is non-linear the slope will be less than this. 

An alternative energy approach to the fracture of polymers has also been developed on the basis of non-linear elasticity. This assumes that a material without any cracks will have a uniform strain energy density (strain energy per unit volume). Let this be IIq. When there is a crack in the material this strain energy density will reduce to zero over an area as shown shaded in Fig. 2.65. This area will be given by ka where )k is a proportionality constant. Thus the loss of elastic energy due to the presence of the crack is given by... 

Because of the assumption that linear relations exist between shear stress and shear rate (equation 3.4) and between distortion and stress (equation 3.128), both of these models, namely the Maxwell and Voigt models, and all other such models involving combinations of springs and dashpots, are restricted to small strains and small strain rates. Accordingly, the equations describing these models are known as line viscoelastic equations. Several theoretical and semi-theoretical approaches are available to account for non-linear viscoelastic effects, and reference should be made to specialist works 14-16 for further details. 

FIGURE 1.2 Normal stress 22 in the central region of the bond line versus imposed shear strain y. (From Gent, A.N., Suh, J.B., and Kelly, III, S.G., Int. J. Non-Linear Mech., 42, 241, 2007. With permission.)... 

Figure 4. Fits of lattice strain model to experimental mineral-melt partition coefficients for (a) plagioclase (run 90-6 of Blundy and Wood 1994) and (b) elinopyroxene (ran DC23 of Blundy and Dalton 2000). Different valence cations, entering the large cation site of each mineral, are denoted by different symbols. The curves are non-linear least squares fits of Equation (1) to the data for each valence. Errors bars, when larger than symbol, are 1 s.d. Ionic radii in Vlll-fold coordination are taken from Shannon (1976).

Van Westrenen et al. (2001a) present a model of lanthanide and Sc partitioning between the garnet X-site and melt. The model is a variant of the lattice strain model of clinopyroxene-melt partitioning of Wood and Blundy (1997), and is based on 160 experimental garnet-melt pairs in the pressure-temperature range 2.5-7.5 GPa and 1450-1930°C. The model includes composition-sensitive expressions for and accounts for the non-linear variation in with composition, as follows ... 

For many materials, the application of a stress creates a strain rate in a linear fashion, i.e., the rate of strain is proportional to the applied stress. This linear relationship, which defines a Newtonian fluid, does not hold true for polymers. Most molten polymers respond to stresses in a non-linear fashion, such that the greater the applied stress the more effective the stress is at inducing a strain rate. This non-Newtonian behavior is referred to as shear thinning ... 

At high stresses and strains, non-linearity is observed. Strain hardening (an increasing modulus with increasing strain up to fracture) is normally observed with polymeric networks. Strain softening is observed with some metals and colloids until yield is observed. 

The term y(t,t ) is the shear strain at time t relative to the strain at time t. The use of a memory function has been adopted in polymer modelling. For example this approach is used by Doi and Edwards11 to describe linear responses of solution polymers which they extended to non-linear viscoelastic responses in both shear and extension. 

Firstly, it helps to provide a cross-check on whether the response of the material is linear or can be treated as such. Sometimes a material is so fragile that it is not possible to apply a low enough strain or stress to obtain a linear response. However, it is also possible to find non-linear responses with a stress/strain relationship that will allow satisfactory application of some of the basic features of linear viscoelasticity. Comparison between the transformed data and the experiment will indicate the validity of the application of linear models. 

For a concentrated system this represents the ratio of the diffusive timescale of the quiescent microstructure to the convection under an applied deforming field. Note again that we are defining this in terms of the stress which is, of course, the product of the shear rate and the apparent viscosity (i.e. this includes the multibody interactions in the concentrated system). As the Peclet number exceeds unity the convection is dominating. This is achieved by increasing our stress or strain. This is the region in which our systems behave as non-linear materials, where simple combinations of Newtonian or Hookean models will never satisfactorily describe the behaviour. Part of the reason for this is that the flow field appreciably alters the microstructure and results in many-body interactions. The coupling between all these interactions becomes both philosophically and computationally very difficult. 

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