Stress uniaxial deformations

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

Component stress tensor resulting from a tensile uniaxial deformation. 

Note 1 The stress tensor for a uniaxial deformation is given in Definition 3.1. 

Component stress tensor resulting from a compressive uniaxial deformation. Note See notes 1 and 2 of Definition 3.2. 

Neutron scattering experiments were also carried out on some model networks with labelled crosslinks, in the unswollen state13 26 To achieve such measurements the dry polystyrene networks are heated above their glass transition temperature, submitted to uniaxial deformation, quenched under stress and studied in the neutron scattering apparatus. 

If we exclude the pressure from the relation for the stresses (9.58) under the considered uniaxial deformation, we can obtain an expression for the extensional stress... 

For filler reinforced rubbers, both contributions of the free energy density Eq. (35) have to be considered and the strain amplification factor X, given by Eq. (39) differs from one. The nominal stress contributions of the cluster deformation are determined by oAtfJ=dWA/dzA, where the sum over all stretching directions, that differ for the up- and down cycle, have to be considered. For uniaxial deformations E =e, E2=Ej= +E) m- one obtains a positive contribution to the total nominal stress in stretching direction for the up-cycle if Eqs. (29)-(36) are used ... 

The usual design procedure is to couple a specific value of design stress with a conventional stress or strain analysis of the assumed structural idealisation. The uniaxial deformation behaviour is of special importance in thin-walled pipes, circular tanks and comparable systems under simple stress. 

In section 3.1.3. we proposed a simple model to calculate the anisotropic form factor of the chains in a uniaxially deformed polymer melt. The only parameters are the deformation ratio X of the entanglement network (which was assumed to be identical to the macroscopic recoverable strain) and the number n, of entanglements per chain. For a chain with dangling end submolecules the mean square dimension in a principal direction of orientation is then given by Eq. 19. As seen in section 3.1.3. for low stress levels n can be estimated from the plateau modulus and the molecular weight of the chain (n 5 por polymer SI). 

The affine predictions for both true and engineering stresses in uniaxial deformation at constant network volume can be rewritten using the shear modulus ... 

Demonstrate that the true stress in uniaxially deformed incompressible networks is the derivative of the free energy per unit volume FjV with respect of the logarithm of deformation A ... 

Demonstrate that for uniaxial deformation of an incompressible network, the stress is given by Eq. (7.64) where the crosslink and entanglement moduli Gx and Ge, respectively, are defined in Eqs (7.43) and (7.47). 

The effect was first observed after uniaxial deformation, but such deformation is not restricted to pure tension and compression. Plastic bending, for example, causes true macrostress (Fig. 16-2), but the deformation mode is predominantly a tension or compression of layers parallel to the neutral axis of the beam. The longitudinal residual stress indicated by x-rays is therefore the sum of true macrostress and pseudo-macrostress, and the x-ray result will be numerically larger at either surface than the result obtained by dissection. 

Here, we investigate the stress response to large uniaxial deformations of model single-protein films and protein plus surfactant mixed films. We show that the general structure of a compressed (expanded) protein film is very sensitive to the breakability of the protein-protein bonds. We then study the structural changes and mechanical response of a protein plus surfactant mixed film to large compression (expansion). We show that the nature of the protein-protein bond parameters also affects the overall displacement behavior of the coadsorbed surfactant during compression. 

Since we consider only uniaxial deformation in the x-direction, the normal component of the interfacial stress is the most affected. Therefore, we only report this component of S. It is worth noticing that a positive value of S means that the adsorbed film pulls in from the imaginary moving barrier, whereas a negative value implies that the film pushes out the barrier. 

A simple model of the permeation experiment with ethanol is now presented, where the distribution of effective stresses and contaminant concentration is uniform. Let us consider an axisymmetric specimen with the same dimensions as described above, subjected to uniaxial deformation, with zero vertical displacement at its base. The specimen is subjected initially to a vertical load applied on the upper boundary, and the pore pressure in the specimen is zero. The cases... 

The stress-strain relation for uniaxial deformations of an incompressible sample k ) has the form... 

Stress-strain measurements at uniaxial extension are the most frequently performed experiments on stress-strain behaviour, and the typical deviations from the phantom network behaviour, which can be observed in many experiments, provided the most important motivation for the development of theories of real networks. However, it has turned out that the stress-strain relations in uniaxial deformation are unable to distinguish between different models. This can be demonstrated by comparing Eqs. (49) and (54) with precise experimental data of Kawabata et al. on uniaxially stretched natural rubber crosslinked with sulphur. The corresponding stress-strain curves and the experimental points are shown in Fig. 4. The predictions of both... 

For uniaxial deformations of magnitude A one then writes Eq. (29.4) for the Mooney-Rivlin stress-strain response as ... 

Both the affine and the phantom network models predict that the reduced stress, [/ ], measured in uniaxial deformation is independent of the deformation ratio. However, it... 

An alternative model which also describes stress-strain data for larger deformation is presented by the Mooney-Rivlin equation [40, 41], The equation describes the rubber elasticity of a polymer network on the basis that the elastomeric sample is incompressible and isotropic in its unstrained state and that the sample behaves as Hookean solid in simple shear. In a Mooney-Rivlin plot of a uniaxial deformation, the experimental measured stress cr, divided by a factor derived from classical models, is plotted as function of the reciprocal deformation 1/A ... 

Fig. 29a and b. Simultaneous stress-strain and FTIR measurements during uniaxial deformation and recovery of a primarily amorphous PTMT film, a stress-strain curve of the mechanical treatment b FTIR spectra in the 1500-1400 cm 8(CH2) region alongside the wide angle X-ray diagrams of the original and a 600% drawn sample... 

The experimental set-up can also be used for the testing of stress relaxation in a plane state of stress. In this test a specified quantity of water is pumped rapidly into the device producing a certain arc height and associated multi-axial deformation. Care must be taken to prevent air cushion formation. The gradual decrease in pressure as a function of time can be obtained from the manometer reading. The test procedure and the evaluation of the relaxation test data can be performed in analogy to the stress relaxation test imder an imposed uniaxial deformation. Figure 3.13 shows such a stress relaxation curve at an imposed multi-axial deformation. 

Fig. 4.4. Isochronous stress-strain diagram of HDPE material for uniaxial deformation. Continuous line calculated from tensile test, dashed line data measured in creep tests. In an uniaxial deformation force is applied along one direction and only stress and strain components along this direction are considered, e.g. as in the tensile test (Sect 3.2.8). Source (Schmachtenberg 1985)...

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