Tensile strain Simple extension

The continuous chain model includes a description of the yielding phenomenon that occurs in the tensile curve of polymer fibres between a strain of 0.005 and 0.025 [ 1 ]. Up to the yield point the fibre extension is practically elastic. For larger strains, the extension is composed of an elastic, viscoelastic and plastic contribution. The yield of the tensile curve is explained by a simple yield mechanism based on Schmid s law for shear deformation of the domains. This law states that, for an anisotropic material, plastic deformation starts at a critical value of the resolved shear stress, ry =/g, along a slip plane. It has been... 

One of the simplest criteria specific to the internal port cracking failure mode is based on the uniaxial strain capability in simple tension. Since the material properties are known to be strain rate- and temperature-dependent, tests are conducted under various conditions, and a failure strain boundary is generated. Strain at rupture is plotted against a variable such as reduced time, and any strain requirement which falls outside of the boundary will lead to rupture, and any condition inside will be considered safe. Ad hoc criteria have been proposed, such as that of Landel (55) in which the failure strain eL is defined as the ratio of the maximum true stress to the initial modulus, where the true stress is defined as the product of the extension ratio and the engineering stress —i.e., breaks down at low strain rates and higher temperatures. Milloway and Wiegand (68) suggested that motor strain should be less than half of the uniaxial tensile strain at failure at 0.74 min.-1. This criterion was based on 41 small motor tests. 

DMA experiments are performed under conditions of very small strain so that the material response is in the linear viscoelastic range. This means that the magnitude of stress and strain are linearly related and the deformation behavior is completely described by the complex modulus function, which is a function of time only. The theory applies both for the case of a tensile deformation or simple extension and for shear. In the latter case the comparable modulus is with components G ico) and G" co). As a first-order approximation, E = 3G. The theory is developed assuming deformation under isothermal conditions, and temperature does not appear (nor is implicit) as a variable. 

Derive the stress-strain relation for a simple extension Ai = A produced by a force applied in the 1 direction, and hence show that the low strain tensile modulus for this rubber is given by is = 6(Ci -1- C2). 

This analysis turns out to be a simple extension of that for the deflection of a linear elastic cantilever. Young s modulus is replaced by the appropriate viscoelastic counterpart—in this case the tensile stress relaxation modulus. That this is the appropriate viscoelastic property to be employed can be thought of as arising from the fact that, when the cantilever is subjected to constant deflection, every element of it is subjected to a constant tensile strain (i.e. to tensile stress relaxation conditions). 

The left-hand equation states that for the set of failure phenomena under consideration, the initial stress tensor is eqnivalent to a stress of simple uniaxial extension that causes the maximum tensile strain caused by the single component of the stress tensor to be eqnivalent to the maximum tensile strain resulting from all the other components of the initial stress tensor. The right-hand eqnation establishes that failure occurs when the nniaxial extension reaches the value of o,-... 

The state of strain in large deformations is commonly described either by the principal extension ratios, Xi, X2, X3, deflned in the notation of Chapter 1 as X,- = 1 + Uilxi, with the coordinate axes suitably oriented, or by three strain invariants whose values are independent of the coordinate system. In simple extension, Xi = 1 + e, where e is the (practical) tensile strain U jx cf. equation 8 above), not to be confused with the e in equations 3,4 and 6. Most of this section is concerned with simple extension. 

In a very extensive study of both stress relaxation and dynamic mechanical properties in simple extension, on single crystal mats of fractions of linear polyethylene, Takayanagi and collaborators were able to combine data at different temperatures by reduced variables over most of the range from 16°C up to the temperature of crystallization and also to show that the dynamic and transient data corresponded fairly closely, provided the latter were corrected for nonlinear behavior by an extrapolation procedure to zero strain. It is characteristic of crystalline polymers that departures from linear viscoelastic behavior appear at very small strains, and are sometimes significant in stress relaxation even at a tensile strain of = 0.001. In dynamic measurements, the strains are usually small enough to fall within the linear range. 

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

PP bead foams were subjected to oblique impacts (167), in which the material was compressed and sheared. This strain combination could occur when a cycle helmet hit a road surface. The results were compared with simple shear tests at low strain rates and to uniaxial compressive tests at impact strain rates. The observed shear hardening was greatest when there was no imposed density increase and practically zero when the angle of impact was less than 15 degrees. The shear hardening appeared to be a unique function of the main tensile extension ratio and was a polymer contribution, whereas the volumetric hardening was due to the isothermal compression of the cell gas. Eoam material models for FEA needed to be reformulated to consider the physics of the hardening mechanisms, so their... 

For gum rubbers and lightly filled compounds, the Mooney-Rivlin equation often models the tensile stress-strain curve well up to extensions of 150% or more. However, for more highly filled compounds (and almost always for commercially important compounds) this simple function only works well up to about 50% strain. A much better fit over an extended strain range can be obtained by taking the next logical term in the infinite series of the general expression. Using ... 

In our introductory material, we talked about stresses and strains in simple stretching or tensile experiments. We neglected the fact that if you stretch a material lengthways, it will also contract in the direction peipen-. dicular to the applied stress. The amount of this contraction is proportional to the extension (Figure 13-9) and we can write Equations 13-3 ... 

In conclusion, when such an adhesive is debonded from a high energy surface such as steel, the high-strain properties of the adhesive control the formation and extension of the fibrillar structure which provides the bulk of the work necessary to detach the adhesive from the surface, and hence the major part of the peel force. We have seen that the level of the plateau stress can be predicted quantitatively by a simple tensile test. From the studies on cavitation, we know that the nominal stress at the plateau corresponds also to the cavity growth stress for large initial defects. 

Material functions must however be considered with respect to the mode of deformation and whether the applied strain is constant or not in time. Two simple modes of deformation can be considered simple shear and uniaxial extension. When the applied strain (or strain rate) is constant, then one considers steady material functions, e.g. q(y,T) or ri (e,T), respectively the shear and extensional viscosity functions. When the strain (purposely) varies with time, the only material functions that can realistically be considered from an experimental point of view are the so-called dynamic functions, e.g. G ((D,y,T) and ri (a), y,T) or E (o),y,T) and qg(o),y, T) where the complex modulus G (and its associated complex viscosity T] ) specifically refers to shear deformation, whilst E and stand for tensile deformation. It is worth noting here that shear and tensile dynamic deformations can be applied to solid systems with currently available instruments, whUst in the case of molten or fluid systems, only shear dynamic deformation can practically be experimented. There are indeed experimental and instrumental contingencies that severely limit the study of polymer materials in the conditions of nonlinear viscoelasticity, relevant to processing. 

In most cases simple determination of tensile strength is not sufficient. Elongation at break and often full stress-strain behaviour analysis is required. Elongation of the sample must be measured. Many problems are associated with the accurate measurement of extension of polymers [11]. There is a wide range of extensions to be measured, from less than 1% for stiff materials like GP polystyrene to over 1000% for certain elastomer systems. There is a significant... 

For a single slip the slip direction in the crystal rotates always towards the direction of maximum extension in uniaxial tension it rotates towards the tensile axis, while in uniaxial compression it rotates away from the compression axis. The angle through which the crystal rotates is a simple function of the applied strain [107]. It must be mentioned that the slip takes place when the resolved shear stress on the slip plane reaches a critical value known as critical resolved shear stress. Currently the critical resolved shear stresses for slips are well known for only few polymers. 

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