Stress, time dependence

Rheological properties of mayonnaise have been studied using different rheological techniques steady shear rate-shear stress, time dependent shear rate-shear stress, stress growth and decay at a constant shear rate, dynamic viscoelastic behavior, and creep-compliance viscoelastic behavior. More studies have been devoted to the study of rheological properties of mayonnaise than of salad dressings, probably because the former is a more stable emulsion and exhibits complex viscous and viscoelastic rheological behavior. 

In all these stresses, time dependence can be separated from the strain. For instance, we have... 

Habas and co-workers [43] carried out an investigation into the viscoelastic properties of aqueous solutions of a four-branched polyethylene oxide - polypropylene oxide - polyethylene oxide triblock copolymer in the unimer range using TMA, DMTA and small-angle neutron scattering. The aqueous solutions were characterised in the crystalline phase and a modified version of Eyring s theory was used to describe the stress-time dependence of ordered solutions. 

The transport properties of polymeric materials which distinguish them most from other materials are their flow properties or rheological behavior. There are many differences between the flow properties of a polymeric fluid and typical low molecular weight fluids such as water, benzene, sulfuric acid, and other fluids, which we classify as Newtonian. Newtonian fluids can be characterized by a single flow property called viscosity (/r) and their density (p). Polymeric fluids, on the other hand, exhibit a viscosity function that depends on shear rate or shear stress, time-dependent rheological properties, viscoelastic behavior such as elastic recoil (memory), additional normal stresses in shear flow, and an extensional viscosity that is not simply related to the shear viscosity, to name a few differences. 

Thermal power plant components operated at high temperatures (>500°C) and pressures, such as superheater headers, steamline sections and Y-junctions, deserve great attention for both operation safety and plant availability concerns. In particular, during plant operation transients -startups, shutdowns or load transients - the above components may undergo high rates of temperature / pressure variations and, consequently, non-negligible time-dependent stresses which, in turn, may locally destabilize existing cracks and cause the release of acoustic emission. 

Continuum theory has also been applied to analyse tire dynamics of flow of nematics [77, 80, 81 and 82]. The equations provide tire time-dependent velocity, director and pressure fields. These can be detennined from equations for tire fluid acceleration (in tenns of tire total stress tensor split into reversible and viscous parts), tire rate of change of director in tenns of tire velocity gradients and tire molecular field and tire incompressibility condition [20]. 

In Chapter VI, Ohm and Deumens present their electron nuclear dynamics (END) time-dependent, nonadiabatic, theoretical, and computational approach to the study of molecular processes. This approach stresses the analysis of such processes in terms of dynamical, time-evolving states rather than stationary molecular states. Thus, rovibrational and scattering states are reduced to less prominent roles as is the case in most modem wavepacket treatments of molecular reaction dynamics. Unlike most theoretical methods, END also relegates electronic stationary states, potential energy surfaces, adiabatic and diabatic descriptions, and nonadiabatic coupling terms to the background in favor of a dynamic, time-evolving description of all electrons. 

Adding time-dependent terms to the equations the simulation is treated as an initial value problem in which at a given reference time all stresses are zero the steady-state solution can be found iteratively. 

The elastic and viscoelastic properties of materials are less familiar in chemistry than many other physical properties hence it is necessary to spend a fair amount of time describing the experiments and the observed response of the polymer. There are a large number of possible modes of deformation that might be considered We shall consider only elongation and shear. For each of these we consider the stress associated with a unit strain and the strain associated with a unit stress the former is called the modulus, the latter the compliance. Experiments can be time independent (equilibrium), time dependent (transient), or periodic (dynamic). Just to define and describe these basic combinations takes us into a fair amount of detail and affords some possibilities for confusion. Pay close attention to the definitions of terms and symbols. 

The situation is not so simple when these various parameters are time dependent. In the latter case, the moduli, designated by E(t)and G(t), are evaluated by examining the (time dependent) value of o needed to maintain a constant strain 7o- By constrast, the time-dependent compliances D(t) and J(t)are determined by measuring the time-dependent strain associated with a constant stress Oq. Thus whether the deformation mode is tension or shear, the modulus is a measure of the stress required to produce a unit strain. Likewise, the compliance is a measure of the strain associated with a unit stress. As required by these definitions, the units of compliance are the reciprocals of the units of the moduli m in the SI system. 

Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

Flexural stress SiC mpture curves are shown in Figure 3 (27). AU. the forms tend to be fairly resistant to time-dependent failure by elevated temperature creep. In addition, SiC shows outstanding resistance to oxidation even at 1200°C as a result of formation of a protective high purity siUca surface layer (28). 

Another aspect of plasticity is the time dependent progressive deformation under constant load, known as creep. This process occurs when a fiber is loaded above the yield value and continues over several logarithmic decades of time. The extension under fixed load, or creep, is analogous to the relaxation of stress under fixed extension. Stress relaxation is the process whereby the stress that is generated as a result of a deformation is dissipated as a function of time. Both of these time dependent processes are reflections of plastic flow resulting from various molecular motions in the fiber. As a direct consequence of creep and stress relaxation, the shape of a stress—strain curve is in many cases strongly dependent on the rate of deformation, as is illustrated in Figure 6. 

Viscous Hquids are classified based on their rheological behavior characterized by the relationship of shear stress with shear rate. Eor Newtonian Hquids, the viscosity represented by the ratio of shear stress to shear rate is independent of shear rate, whereas non-Newtonian Hquid viscosity changes with shear rate. Non-Newtonian Hquids are further divided into three categories time-independent, time-dependent, and viscoelastic. A detailed discussion of these rheologically complex Hquids is given elsewhere (see Rheological measurements). 

Time-dependent effects ate measured by determining the decay of shear stress as a function of time at one or more constant shear rates (Fig. 7)... 

A rotational viscometer connected to a recorder is used. After the sample is loaded and allowed to come to mechanical and thermal equiUbtium, the viscometer is turned on and the rotational speed is increased in steps, starting from the lowest speed. The resultant shear stress is recorded with time. On each speed change the shear stress reaches a maximum value and then decreases exponentially toward an equiUbrium level. The peak shear stress, which is obtained by extrapolating the curve to zero time, and the equiUbrium shear stress are indicative of the viscosity—shear behavior of unsheared and sheared material, respectively. The stress-decay curves are indicative of the time-dependent behavior. A rate constant for the relaxation process can be deterrnined at each shear rate. In addition, zero-time and equiUbrium shear stress values can be used to constmct a hysteresis loop that is similar to that shown in Figure 5, but unlike that plot, is independent of acceleration and time of shear. 

Figure 16 (145). For an elastic material (Fig. 16a), the resulting strain is instantaneous and constant until the stress is removed, at which time the material recovers and the strain immediately drops back to 2ero. In the case of the viscous fluid (Fig. 16b), the strain increases linearly with time. When the load is removed, the strain does not recover but remains constant. Deformation is permanent. The response of the viscoelastic material (Fig. 16c) draws from both kinds of behavior. An initial instantaneous (elastic) strain is followed by a time-dependent strain. When the stress is removed, the initial strain recovery is elastic, but full recovery is delayed to longer times by the viscous component.

It is important to differentiate between brittie and plastic deformations within materials. With brittie materials, the behavior is predominantiy elastic until the yield point is reached, at which breakage occurs. When fracture occurs as a result of a time-dependent strain, the material behaves in an inelastic manner. Most materials tend to be inelastic. Figure 1 shows a typical stress—strain diagram. The section A—B is the elastic region where the material obeys Hooke s law, and the slope of the line is Young s modulus. C is the yield point, where plastic deformation begins. The difference in strain between the yield point C and the ultimate yield point D gives a measure of the brittieness of the material, ie, the less difference in strain, the more brittie the material. 

Measurements of stress relaxation on tempering indicate that, in a plain carbon steel, residual stresses are significantly lowered by heating to temperatures as low as 150°C, but that temperatures of 480°C and above are required to reduce these stresses to adequately low values. The times and temperatures required for stress reUef depend on the high temperature yield strength of the steel, because stress reUef results from the localized plastic flow that occurs when the steel is heated to a temperature where its yield strength is less than the internal stress. This phenomenon may be affected markedly by composition, and particularly by alloy additions. 

Stress relaxation, while aUied to creep, is different (20—23). Stress relaxation is the time-dependent decrease in load (stress) at the contact displacement resulting from connector mating. Here, the initial elastic strain in the spring contact is replaced by time-dependent microplastic flow. Creep, on the other hand, relates to time-dependent geometry change (strain or displacement) under fixed load, which is a condition that does not apply to coimectors. 

Purely viscous fluids are further classified into time-independent and time-dependent fluids. For time-independent fluids, the shear stress depends only on the instantaneous shear rate. The shear stress for time-dependent fluids depends on the past history of the rate of deformation, as a result of structure or orientation buildup or breakdown during deformation. 

Time-dependent fluids are those for which structural rearrangements occur during deformation at a rate too slow to maintain equilibrium configurations. As a result, shear stress changes with duration of shear. Thixotropic fluids, such as mayonnaise, clay suspensions used as drilling muds, and some paints and inks, show decreasing shear stress with time at constant shear rate. A detailed description of thixotropic behavior and a list of thixotropic systems is found in Bauer and Colhns (ibid.). 

Hiestand Tableting Indices Likelihood of failure during decompression depends on the abihty of the material to relieve elastic-stress by plastic deformation without undergoing brittle fracture, and this is time dependent. Those which relieve stress rapidly are less... 

The jump conditions must be satisfied by a steady compression wave, but cannot be used by themselves to predict the behavior of a specific material under shock loading. For that, another equation is needed to independently relate pressure (more generally, the normal stress) to the density (or strain). This equation is a property of the material itself, and every material has its own unique description. When the material behind the shock wave is a uniform, equilibrium state, the equation that is used is the material s thermodynamic equation of state. A more general expression, which can include time-dependent and nonequilibrium behavior, is called the constitutive equation. 

These techniques have very important applications to some of the micro-structural effects discussed previously in this chapter. For example, time-resolved measurements of the actual lattice strain at the impact surface will give direct information on rate of departure from ideal elastic impact conditions. Recall that the stress tensor depends on the elastic (lattice) strains (7.4). Measurements of the type described above give stress relaxation directly, without all of the interpretational assumptions required of elastic-precursor-decay studies. 

Schatz, R., Shooman, M. and Shaw, L. 1974 Application of Time Dependent Stress-Strength Models of Non-Electrical and Electrical Systems. In Proceedings Reliability and Maintainability Symposium, 540-547. 

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