Strain-stiffening

Another problem which arises in a cyclic deformation under a large strain amplitude is that the modulus of material varies significantly during a cycle because of strain-stiffening or strain-loosening effects. Therefore, analysis of the viscoelastic properties in this case must include the variation of modulus during a cycle. 

Figure 7 Strain stiffening of cross-iinked cytoskeietai and other semi-fiexibie biopoiymer geis. Dynamic shear storage moduii S of F-actin, fibrin, coiiagen, vimentin, NF, and poiyacryiamide geis, piotted as a function of shear strain ampiitude y. Adapted from Storm, C. ...

Kang H, Wen Q, Janmey PAetal (2009) Nonlinear elasticity of stiff filament networks strain stiffening, negative normal stress, and filament alignment in fibrin gels. J Phys Chem B... 

In the stress-strain curve of a VGCF, there is a toe-in corresponding to an apparent modulus increase of 28%, termed strain stiffening, which occurs due to the improvement of the orientation of the fiber s graphitic planes as the load is apphed. A mean value for E [51] of fibers under 10 pm diameter is 237 49 GPa. Ruland [53] pictured graphite fibers... 

Strictly, the stress-strain curve is not linear and the modulus does tend to increase as the strain becomes larger, an effect termed strain stiffening. This effect dictates that the strain area selected for the modulus determination must be clearly stated otherwise the determined modulus of an identical fiber can be different (Figure 20.10). 

Figure 20.10 Strain stiffening. Source Reprinted from Toray technical literature.

The stress-strain curves in Fig. 4.39 to 4.41 have the characteristic shape for elastomers containing reinforcing particles. There is a rate-dependent flow stress associated with stress-activated segmental diffusion at the surface or interior of reinforcing particles, superimposed on the strain-stiffening hyperelastic stress response of the elastomeric matrix. Such a system is amenable to a quantitative description in terms... 

Adams, A. M., C. P. Buekley and D. P. Jones. (2000). Biaxial hot drawing of poly(ethylene terephthalate) measurements and modeling of strain-stiffening. Polymer, 41 771-786. 

Polymers with DBDl has a higher flow stress than polymers derived from conventional rigid isocyanates, but shows less pronounced strain stiffening. PU phase separation tendency is significantly more pronounced in the case of polyurethanic materials based on dibenzyl structures then in the case of similar hard segments based on classical isocyanates. 

In this case, the shear stress is linear in the shear strain. While more physically reasonable, this is not likely to provide a satisfactory representation for the large deformation shear response of many materials either, since most materials may be expected to stiffen with deformation. Note that the hypoelastic equation of grade zero (5.117) is not invariant to the choice of indifferent stress rate, the predicted response for simple shear depending on the choice which is made. 

The contribution to the stress from electromechanical coupling is readily estimated from the constitutive relation [Eq. (4.2)]. Under conditions of uniaxial strain and field, and for an open circuit, we find that the elastic stiffness is increased by the multiplying factor (1 -i- K ) where the square of the electromechanical coupling factor for uniaxial strain, is a measure of the stiffening effect of the electric field. Values of for various materials are for x-cut quartz, 0.0008, for z-cut lithium niobate, 0.055 for y-cut lithium niobate, 0.074 for barium titanate ceramic, 0.5 and for PZT-5H ceramic, 0.75. These examples show that electromechanical coupling effects can be expected to vary from barely detectable to quite substantial. 

Use the bounding techniques of elasticity to determine upper and lower bounds on the shear modulus, G, of a dispersion-stiffened composite materietl. Express the results In terms of the shear moduli of the constituents (G for the matrix and G for the dispersed particles) and their respective volume fractions (V and V,j). The representative volume element of the composite material should be subjected to a macroscopically uniform shear stress t which results in a macroscopically uniform shear strain y. 

Very active catalysts for the preparation of strictly alternating butadiene-propylene copolymers (BPR) consist of V0(0R)2C1/i-Bu Al (R = neopentyl). The CH3 side groups in BPR stiffen the polymer chain and were expected to promote the formation of strain-induced structures. The fact that we could not detect strain-induced crystallization is probably due to an atactic configuration of the propylene units. 

The ratio of shear stress to shear strain. A property which determines the rate at which elastomers stiffen as the temperature is lowered. The force required to twist the test piece through 90° is measured at each temperature and the modulus calculated from a formula. 

Stress-strain characteristics. Linear chain polymers are quite flexible and subject to creep or stretch. Branching or rings in the backbone have a stiffening effect. For example ... 

The frequency exaltation of the Kekule mode is mirrored by the structural manifestations in the twin states, discussed with reference to Figures 16 and 17. Thus, the repulsive jr-curve in the ground state softens the potential and thereby enables the ground-state molecule to distort along the Kekule mode when angular strain is exerted. In contrast, the attractive jr-curve in the twin excited state stiffens the potential and restores the local Deh symmetry of the benzene nucleus. The two physical effects are in perfect harmony and find a natural reflection in the VB model. 

A close connection exists between the presence of a flexible polymer skeleton and the flexibility of the bulk material. Macromolecular flexibility is often defined in terms of the glass-transition temperature, Tg. Below this temperature, the polymer is a glass, and the backbone bonds have insufficient thermal energy to undergo significant torsional motions. As the temperature is raised above the Y g, an onset of torsional motion occurs, such that individual molecules can now twist and yield to stress and strain. In this state the polymer is a quasi-liquid (an elastomer) unless the bulk material is stiffened by microcrystalfite formation. Thus, a polymer with a high Tt is believed to have a backbone that offers more resistance to bond torsion than a polymer with a low 7 g. 

The simple theory presented accounts for the fall-off of the modulus of polymers with increased extension. The stiffening of the polymer at very large distortion results from interactions between the extended polymer chains, which is called strain-induced crystallization. 

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