Instantaneous modulus

The resilient modulus (instantaneous or total) is determined by the average values obtained by two loading positions and for three specimens, per type of bituminous mixture. 

The shear relaxation modulus instantaneous value G(0) The equilibrium shear modulus (G = 0 for a fluid)... 

Example 10 Response to Instantaneous Valve Closing Compute the wave speed and maximum pressure rise for instantaneous valve closing, with an initial velocity of 2,0 m/s, in a 4-in Schedule 40 steel pipe with elastic modulus 207 X 10 Pa, Repeat for a plastic pipe of the same dimensions, with E = 1.4 X 10 Pa. The liquid is water with P = 2.2 X 10 Pa and p = 1,000 kg/m. For the steel pipe, D = 102,3 mm, b = 6,02 mm, and the wave speed is... 

Different viscoelastic materials may have considerably different creep behavior at the same temperature. A given viscoelastic material may have considerably different creep behavior at different temperatures. Viscoelastic creep data are necessary and extremely important in designing products that must bear long-term loads. It is inappropriate to use an instantaneous (short load) modulus of elasticity to design such structures because they do not reflect the effects of creep. Viscoelastic creep modulus, on the other hand, allows one to estimate the total material strain that will result from a given applied stress acting for a given time at the anticipated use temperature of the structure. 

Second, the creep modulus, also known as the apparent modulus or viscous modulus when graphed on log-log paper, is normally a straight line and lends itself to extrapolation for longer periods of time. The apparent modulus should be differentiated from the modulus given in the data sheets, which is an instantaneous or static value derived from the testing machine, per ASTM D 638. 

Immediately the load is applied, the specimen elongates corresponding to an instantaneous elastic modulus. This is followed by a relatively fast rate of creep, which gradually decreases to a smaller constant creep rate. Typically this region of constant creep in thermoplastics essentially corresponds to... 

Copolymerization of ethylene and styrene by the INSITE technology from Dow generates a new family of ethylene-styrene interpolymers. Polymers with up to 50-wt% styrene are semicrystalline. The stress-strain behavior of the low-crystallinity polymers at ambient temperature exhibits elastomeric characteristics with low initial modulus, a gradual increase in the slope of the stress-strain curve at the higher strain and the fast instantaneous recovery [67], Similarly, ethylene-butylene copolymers may also be prepared. 

The viscoelastic behavior of concentrated (20% w/w)aqueous polystryene latex dispersions (particle radius 92nm), in the presence of physically adsorbed poly(vinyl alcohol), has been investigated as a function of surface coverage by the polymer using creep measurements. From the creep curves both the instantaneous shear modulus, G0, and residual viscosity, nQ, were calculated. 

In order to obtain a general model of the creep and recovery functions we need to use a Kelvin model or a Kelvin kernel and retardation spectrum L. However, there are some additional subtleties that need to be accounted for. One of the features of a Maxwell model is that it possesses a high frequency limit to the shear modulus. This means there is an instantaneous response at all strains. The response of a simple Kelvin model is shown in Equation 4.80 ... 

The experiment here is a small rapid shear-strain at time zero - after this the shear stress in a viscoelastic liquid will not vanish instantaneously, but decay as a characteristic function with time. When normalised by the strain to yield the dimensions of modulus, this is G(f). 

In this introduction, the viscoelastic properties of polymers are represented as the summation of mechanical analog responses to applied stress. This discussion is thus only intended to be very introductory. Any in-depth discussion of polymer viscoelasticity involves the use of tensors, and this high-level mathematics topic is beyond the scope of what will be presented in this book. Earlier in the chapter the concept of elastic and viscous properties of polymers was briefly introduced. A purely viscous response can be represented by a mechanical dash pot, as shown in Fig. 3.10(a). This purely viscous response is normally the response of interest in routine extruder calculations. For those familiar with the suspension of an automobile, this would represent the shock absorber in the front suspension. If a stress is applied to this element it will continue to elongate as long as the stress is applied. When the stress is removed there will be no recovery in the strain that has occurred. The next mechanical element is the spring (Fig. 3.10[b]), and it represents a purely elastic response of the polymer. If a stress is applied to this element, the element will elongate until the strain and the force are in equilibrium with the stress, and then the element will remain at that strain until the stress is removed. The strain is inversely proportional to the spring modulus. The initial strain and the total strain recovery upon removal of the stress are considered to be instantaneous. 

Moreover, real polymers are thought to have five regions that relate the stress relaxation modulus of fluid and solid models to temperature as shown in Fig. 3.13. In a stress relaxation test the polymer is strained instantaneously to a strain e, and the resulting stress is measured as it relaxes with time. Below the a solid model should be used. Above the Tg but near the 7/, a rubbery viscoelastic model should be used, and at high temperatures well above the rubbery plateau a fluid model may be used. These regions of stress relaxation modulus relate to the specific volume as a function of temperature and can be related to the Williams-Landel-Ferry (WLF) equation [10]. 

The first term again represents drag in steady motion at the instantaneous velocity, with Cd an empirical function of Re as in Chapter 5. The other terms represent contributions from added mass and history, with empirical coefficients, Aa and Ah, to account for differences from creeping flow. From measurements of the drag on a sphere executing simple harmonic motion in a liquid, Aa and Ah appeared to depend only on the acceleration modulus according to ... 

The second question concerns the amplitude ratios between the reflected and incident waves. For both homogeneous and inhomogeneous incident waves these ratios must have a modulus equal to unity, because no energy loss through the instantaneous reflection process at x = 0 is expected. 

The value G(0) is called the instantaneous modulus, and will be denoted by G°. 

During aging, there are changes in most textural and physical properties of the gel. Inorganic gels are viscoelastic materials responding to a load with an instantaneous elastic strain and a continuous viscous deformation. Because the condensation reaction creates additional bridging bonds, the stiffness of the gel network increases, as does the elastic modulus, the viscosity, and the modulus of rupture. 

Above its melting point of 327° C, polytetrafluoroethylene has some properties more like a rubber than a liquid. The instantaneous Young s modulus is 2—3 X 107 dynes/cm2, and the melt viscosity is about 10u poises at 380° C (Nishioka and Watanabe). Because of this very high melt viscosity, it is not feasible to process the polymer by conventional extrusion or injection molding. Instead, techniques similar to those of powder metallurgy are employed. These involve three basic steps. 

The stability of the shape of an article is partly determined by the value of its modulus of elasticity, but the required value of this parameter is not reached instantaneously and homogeneously throughout the volume of an article. Therefore, estimates of material stability must be made from the distribution of the elastic modulus throughout the volume of an article and its evolution in time. In this approach, the modulus must reach the necessary minimum value at any point in the material before the mold can be opened. 

The parameter R is applicable for the case of instantaneous change in surface temperature (infinite h) for conditions of rapid heat transfer R is for a relatively low Biot modulus ( jl< 2) for conditions of slow heat transfer R" is for a constant heating or cooling rate.88 defines the minimum temperature difference to produce fracture under conditions of infinite heat-transfer coefficient, i.e. A = 1. The parameter Ris inversely proportional to a. Alow value of a is therefore essential for good thermal stress resistance. The coefficient of thermal expansion normally increases with increasing temperature however, thermal conductivity decreases. 

---