Deformation ratio

References 7, 64, and 72 through 75 discuss expected consequences at various levels of stmctural response deformation ratios or plastic hinge... 

Damage Level Deformation Ratio Plastic Hinge Rotation... 

On the contrary, are(j u/0red clearly shows some dependence on the structure of the crosslinks, changing from around 0.27 to 0.10 as the branching density z increases from 0.01 to 0.5. The different time scale of the experiments can not have effected the results, because is was proved that G is independent of frequency. The deformation ratio X is 1.00005 in case of torsional vibrations and 1.02-1.04 in case of uniaxial extension. Hence it ap-... 

The observed deviations from Gaussian stress-strain behaviour in compression were in the same sense as those predicted by the Mooney-Rivlin equation, with modulus increasing as deformation ratio(A) decreases. The Mooney-Rivlin equation is usually applied to tensile data but can also be applied compression data(33). According to the Mooney-Rivlin equation... 

O is the stress per unit unstrained area, G the shear modulus, A the deformation ratio, p the density of the dry network. iJ>2 volume fraction of polymer present in the network, V the volume at formation. A=1 for affine behaviour (expected) and 1-2/f for phantom behaviour(1,3). is the molar mass for the perfect network, essentially the molar mass of a chain of v bonds, the number which can form the smallest loop (5-7) see Figure 2. is equal to the... 

Note 2 The Xi are effectively deformation gradients, or, for finite deformations, the deformation ratios characterizing the deformation. 

Note 3 For elastomers, which are assumed incompressible, the modulus is often evaluated in uniaxial tensile or compressive deformation using X - as the strain function (where X is the uniaxial deformation ratio). In the limit of zero deformation the shear modulus is evaluated as... 

Figure 1.15. Orientation function f vs. total deformation ratio for four XL bands (above) and three...

In order to connect expressions for FJn, and Fei, it is necessary to define more precisely the reference state with respect to which deformation ratios a, reference state is defined by conditions of network preparation and in many cases it is close to the network state at the preparation conditions. The connection between G>N and otj (i = x, y, z) is given by ... 

One of the main consequence of Eq. (28) must be the independence of intrachain energy contribution on the deformation ratio. The majority of thermoelastic investigations that have been carefully analysed 7,20,29) confirm this theoretical conclusion. However, in some studies dealing with the thermoelasticity of NR, EPR and some other rubbers, a dependence of fu/f is found at small deformation ratios... 

The constant-temperature network partition function Q under a given deformation, specified by the deformation ratios %x, Xy and A, is obtained by assuming that there are no interactions between the chains other than those imposed by the presence of volume-less crosslinks. 

In Eq. (III-9) the deformation ratios are defined with respect to a reference state in which the chain dimensions are such that they do not exert any elastic forces on the crosslinks (state of normal coiling). In general, the chains in a network may not actually be in this state at the beginning of a deformation experiment, because the ciosslinking process may quite well exert a, largely unknown, influence on the chain dimensions. 

In a unidirectionally strained network the introduction of the experimental deformation ratios A Ay and Az with respect to the undeformed sample, is now performed as follows (cf. Eq. III-9)... 

To the authors knowledge there are no dry rubberlike networks which obey the stress-strain relation derived in the preceding section (Eq. III-l 1) over an appreciable range of deformation ratios (e.g. from Ax — 1 to 2), whereas this relation should hold for an ideal network at least over such a range before the finite extensibility of the chains impairs the derivation. 

Fig. 22 shows the same data of Fig. 21 plotted as force versus deformation ratio. If the data in compression are taken to follow Gaussian behaviour it is seen that a negative strain-dependent correction is needed... 

Fig. 22, The same data as in Fig. 21 plotted as force versus deformation ratio. If one identifies the compression modulus with the Gaussian constant 3 (A vk T/Lt) (( >o) the experimental curve in extension lies below the Gaussian curve. The C part of the Mooney-Rivlin curve in extension lies again below the experimental curve...

In the elastic free energy expression the deformation ratios are defined with respect to a reference state. This is defined as a state in which... 

A deformation ratio with respect to the reference state... 

The variation of the elastic free energy between the elastically deformed state and the initial undeformed state is the work accomplished by the force F applied to the sample. For a uniaxial deformation along the x axis yielding a macroscopic deformation ratio Ax ... 

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