Shear nonlinear behavior

The nonlinear shear stress-shear strain behavior typical of fiber-reinforced composite materials is ignored, i.e., the behavior is regarded as linear. 

On the other hand, for aircraft and spacecraft structures, real laminate behavior is pretty typically linear. Laminate behavior is reasonably linear even with some 45° layers which you would expect to contribute their nonlinear shear deformation characteristic to the overall laminate and degrade its relative performance. If you go beyond the behavior of a laminate and look at a large structure, typically the load-response characteristics are linear. Even around a cutout, linear behavior exists. Beyond that apparent linear performance of many laminates, you might not like to operate in some kind of a nonlinear response regime. Certainly not when in a fatigue environment and probably not in a creep environment either would you like to operate in a nonlinear behavior range. 

PEs, as other polymers, exhibit nonlinear behavior in their viscous and elastic properties under practical processing conditions, i.e., at high-shear stresses. The MFI value is, therefore, of little importance in polymer processing as it is determined at a fixed low-shear rate and does not provide information on melt elasticity [38,39]. In order to understand the processing behavior of polymers, studies on melt viscosity are done in the high-shear rate range viz. 100-1000 s . Additionally, it is important to measure the elastic property of a polymer under similar conditions to achieve consistent product quality in terms of residual stress and/or dimensional accuracy of the processed product. 

Let us return to the reduction of shear stress at the crack tip due to the emission of dislocations. Figure 14-9 illustrates a possible stress reduction mechanism. It can be seen that the tip of a crack is no longer atomically sharp after a dislocation has been emitted. It is the interaction of the external stress field with that of the newly formed dislocations which creates the local stress responsible for further crack growth. Thus, the plastic deformation normally impedes embrittlement because the dislocations screen the crack from the external stress. Theoretical calculations are difficult because the lattice distortions of both tension and shear near the crack tip are large so that nonlinear behavior is expected. In addition, surface effects have to be included. 

Acoustic cavitation (AC), formation of pulsating cavities in a fluid, occurs when a powerful ultrasound is applied to a non-viscous fluid. The cavities are formed when the variable acoustic pressure in the rarefaction phase exceeds the cohesive strength of the fluid. Under acoustic treatment (AT), cavities grow to resonance dimensions conditioned by frequency, amplitude of oscillations, stiffness properties and external conditions, and start to pulsate synchronously (self-consistently) with acoustic pressure in the medium. The cavities undergo significant strains (compared to their dimensions) and their size decreases under compression up to collapsing. This nonlinear behavior determines the active, destructional character of the cavities near which significant shear velocities, local pressure and temperature bursts occur in the fluid. Cavitation determines the specific character of acoustic treatment of the fluid and effects upon objects resident in the fluid, as well as all consequences of these effects. 

Although attempts to measure and interpret nonlinear behavior are potentially useful, there are few reports in the literature on the measurement of the nonlinear viscoelastic properties of foods. This has been due to a lack of both suitable instrumentation and suitably developed theory nonlinear behavior, the predominant form of which is the exhibition of normal stresses, and a dependence of viscosity on shear rate, is much more complex than linear behavior (Gunasekaran and Ak, 2002). 

Considering a mass of ceramic powder about to be molded or pressed into shape, the forces necessary and the speeds possible are determined by mechanical properties of the diy powder, paste, or suspension. For any material, the elastic moduli for tension (Young s modulus), shear, and bulk compression are the mechanical properties of interest. These mechanical properties are schematically shown in Figure 12.1 with their defining equations. These moduli are mechanical characteristics of elastic materials in general and are applicable at relatively low applied forces for ceramic powders. At higher applied forces, nonlinear behavior results, comprising the flow of the ceramic powder particles over one another, plastic deformation of the particles, and rupture of... 

The nonlinear damping function, hiy), measured in step shearing on star polymers follows the Doi-Edwards prediction, just as h(y) does for linear polymers. Although one end of each arm of the star is anchored, the other end is free to retract, leading to the same strain softening as in linear polymers (Pearson 1987). Molecules with strands that are anchored at two ends, such as molecules with the topology of an H, a comb, or a gel fractal, are expected to show nonlinear behavior that is very different from that of stars (McLeish 1988a Bick and McLeish 1996). 

Linearity in relaxation experiments holds only at small shear strains. A schematic diagram illustrating the linear and nonlinear behavior in relaxation experiments is shown in Figure 5.7. 

Figure 5,7 Sketch illustrating the transition from linear to nonlinear behavior in shear relaxation experiments. Note that the data must be obtained by a series of stress relaxation experiments.

A mathematical expression relating forces and deformation motions in a material is known as a constitutive equation. However, the establishment of constitutive equations can be a rather difficult task in most cases. For example, the dependence of both the viscosity and the memory effects of polymer melts and concentrated solutions on the shear rate renders it difficult to establish constitute equations, even in the cases of simple geometries. A rigorous treatment of the flow of these materials requires the use of fluid mechanics theories related to the nonlinear behavior of complex materials. However, in this chapter we aim only to emphasize important qualitative aspects of the flow of polymer melts and solutions that, conventionally interpreted, may explain the nonlinear behavior of polymers for some types of flows. Numerous books are available in which the reader will find rigorous approaches, and the corresponding references, to the subject matter discussed here (1-16). 

The simplest model is the statistical theory of rubber-like elasticity, also called the affine model or neo-Hookean in the solids mechanics community. It predicts the nonlinear behavior at high strains of a rubber in uniaxial extension with Fq. (1), where ctn is the nominal stress defined as F/Aq, with F the tensile force and Aq the initial cross-section of the adhesive layer, A is the extension ratio, and G is the shear modulus. 

Some modifications of the melt flow behavior of thermoplastics that can be observed depending on filler concentration are a yield-like behavior (i.e., in these cases, there is no flow until a finite value of the stress is reached), a reduction in die swell, a decrease of the shear rate value where nonlinear flow takes place, and wall slip or nearwall slip flow behavior [14, 27, 46]. Other reported effects of flllers on the rheology of molten polymers are an increase of both the shear thinning behavior and the zero-shear-rate viscosity with the filler loading and a decrease in the dependence of the filler on viscosity near the glass transition temperature [18, 47-49]. 

Letwimolnun et al. [2007] used two models to explain the transient and steady-state shear behavior of PP nanocomposites. The first model was a simplified version of the stmcture network model proposed by Yziquel et al. [1999] describing the nonlinear behavior of concentrated suspensions composed of interactive particles. The flow properties were assumed to be controlled by the simultaneous breakdown and buildup of suspension microstructure. In this approach, the stress was described by a modified upper-convected Jeffery s model with a modulus and viscosity that are functions of the suspension structure. The Yziquel et al. model might be written ... 

Murayama, N., Network theory of the nonlinear behavior of polymer melts.. Simple shear flow,... 

The nonlinear behavior of soil, reduction in stifihess and increase in the damping with increase in the shear strain, is accounted by selecting the standard modulus reduction and damping curves. In the present study, the standard modulus reduction and damping curves proposed by Sun et al (1988) are selected based on the soil characteristics and confining pressure which is depicted in Figure 10. 

A nonlinear path-dependent constitutive model for the soil mainly depends on the shear stress-shear strain relationship, which is extended to three-dimensional generic conditions and assumed to follow Masing s rule for the soil hysteresis. The soil is idealized as an assembly of a finite number of elasto-perfectly plastic elements connected in parallel as shown in Fig. 25.3 (Okhovat et al. 2009, Mohammed and Maekawa 2012 Mohammed et al. 2012a). The nonlinear behavior of the soil system in liquefaction is assumed as in undrained state, since its drainage takes much longer than the duration of an earthquake (Towhata 2008). The soil undrained behavior is shown in Fig. 25.4. 

Melt Viscosity. The study of the viscosity of poljmier melts (47-62) is important for the manufacturer who must supply suitable materials and for the fabrication engineer who must select polymers and fabrication methods. Thus, melt viscosity as a fimction of temperature, pressure, rate of flow, and poljmier molecular weight and structure is of considerable practical importance. Polymer melts exhibit elastic as well as viscous properties. This is evident in the swell of the polymer melt upon emergence from an extrusion die, a behavior that results from the recovery of stored elastic energy plus normal stress effects. Theoretical developments include a constitutive equation that correctly captures nonlinear behavior in both elongation and shear (63,64). 

The utility of the K-BKZ theory arises from several aspects of the model. First, it does capture many of the features, described below, of the behavior of polymeric melts and fluids subjected to large deformations or high shear rates. That is, it captures many of the nonlinear behaviors described above for steady flows as well as behaviors in transient conditions. In addition, imlike the more general multiple integral constitutive models (108,109), the experimental data required to determine the material properties are not overly burdensome. In fact, the information required is the single-step stress relaxation response in the mode of deformation of interest (72). If one is only interested in, eg, simple shear, then experiments need only be performed in simple shear and the exact form for U I, /2, ) need not be obtained. Furthermore, because the structure of the K-BKZ model is similar to that of finite elasticity theory, if a full three-dimensional characterization of the material is needed, some of the simplilying aspects of finite elasticity theories that have been developed over the years can be applied to the behavior of the viscoelastic fluid description provided by the K-BKZ model. One such example is the use of the VL form (98) of the strain energy function discussed above (110). The next section shows some comparisons of the material response predicted by the K-BKZ theory with actual experimental data. 

M. Saphiannikova, F. R. Costa, U. Wagenknecht, and G. Heinrich, Nonlinear behavior of polyethylene/layered double hydroxide nanocomposites under shear flow. Polymer Science Series A, 50 (2008), 573-82. 

In contrast to the two previous cases, plastic flow (plasticity) is characterized by the absence of proportionality between the stress and the strain, that is, plasticity represents a case of nonlinear behavior. For plastic bodies subjected to stresses below the critical value, t < t (the so-called shear yield point), the rate of strain is zero (dy/dt = 0). Plastic flow starts at the yield stress, t = t, and does not require further increase in stress (Figure 3.6). Similar to viscous flow, plastic flow is thermodynamically and mechanically irreversible. However, in contrast to the prior case, the rate of energy dissipation in plastic flow is proportional to the rate of strain ... 

The first manifestation of nonlinear behavior with increasing strain or strain rate is the appearance of normal stress differences in shearing deformation. For steady-state shear flow at small shear rates, several nonlinear models ° predict the relation for the primary normal stress difference given as equation 62 of Chapter 1, which, combined with equation 54, gives... 

In addition to non-Newtonian shear viscosity, which has been mentioned in Sections C2 and C5. there are a wide variety of nonlinear phenomena observed in various types of deformation, especially shear and simple (uniaxial) extension for deformations with large strains and large strain rates. Some of these have been mentioned in Chapters I and 3. Nonlinear behavior in shear has been studied mostly... 

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