Measurements of Sinusoidally Varying Stress and Strain

Direct Measurements of Sinusoidally Varying Stress and Strain 

The simplest oscillatory experiment is based on driving one surface with a known periodic displacement and measuring the periodic force at the surface on the other side of the gap with a sensing device of negligible motion, such as a ceramic pi- 

Schematic diagram of simple shear vibrator of Miles. The displacement is specified at the moving plate and the force is measured at the sensor plate. 

With a similar principle, the apparatus of Chompff uses Segel-Pochettino geometry if of Fig. 5-1) with a thin foil to contain the viscoelastic liquid between the cylinders. The displacement of the inner cylinder is measured by a differential transformer and the force on the outer cylinder by a ferroelectric ceramic transducer. The frequency range is from 1 to 2500 Hz, and good temperature control is provided over an extremely wide range. The moduli G and G arp obtained as in equations 19 and 20, with the form factor b specified by equation 13. A related apparatus with continuous recording at constant frequency has been described by Date.  

To remain within the linear range of viscoelastic behavior, the periodic displacements must be kept small linearity can always be tested by varying the amplitude. The Rheometrics Spectrometer can also be operated at large amplitudes. 

---

B. DIRECT MEASUREMENTS OF SINUSOIDALLY VARYING STRESS AND STRAIN... 

Most of the various types of dynamic methods described in preceding chapters have been applied to fibers. The simplest principle, that of direct measurement of sinusoidally varying stress and strain, has been employed below 0.5 Hz by mechanical deformation and optical recording of both stress and strain with rotating mirrors, somewhat in the manner of the corresponding device of Roelig for soft rubberlike polymers (Chapter 6). The Rheovibron of Takayanagi is well adapted to measurements on fibers. At lower frequencies (down to 10 Hz) an instrument... 

The methods described in Sections D1 and D2 of Chapter 5 for tracing directly the sinusoidally varying stresses and strains, or for measuring the mechanical im-... 

The response of a material to an applied stress after very short times can be measured dynamically by applying a sinusoidally varying stress to the sample. A phase difference, which depends on the viscoelastic nature of the material, is set up between stress and strain. 

Finally, one of the most useful ways of measuring viscoelastic properties is dynamic mechanical analysis, or DMA. In this type of experiment, an oscillating stress is applied to the sample and the response is measured as a function of the frequency of the oscillation. By using different instruments this frequency can be varied over an enormous range. Actually, the sample is usually stretched a little bit and oscillated about this strain also, the stress necessary to produce an oscillatory strain of a given magnitude is the quantity usually measured. If the sample being oscillated happens to be perfectly elastic, so that its response is instantaneous, then the stress and strain would be completely in-phase. If a sinusoidal shear strain is imposed on the sample we have (Equation 13-72) ... 

In linear viscoelastic behavior the stress and strain both vary sinusoidally, although they may not be in phase with each other. Also, the stress amplitude is linearly proportional to the strain amplitude at given temperature and frequency. Then mechanical responses observed under different test conditions can be interrelated readily. The behavior of a material in one condition can be predicted from measurement made under different circumstances. 

Usually, the deformation of a sample undergoing oscillatory shear is monitored by measuring the sinusoidally-varying motion of a transducer-controlled driving smface in contact with the sample. However, in turning to the subsequent calculation of shear strain amplitude in dynamic measurements, it must be recognized that conversion of experimentally determined forces and displacements to the corresponding stresses and strains experienced by a sample can involve consideration of the role of sample inertia. 

Dynamic mechanical analysis (DMA) is a sensitive method for glass transition temperature measurement, for detection of side-chain and main-chain motions, and for local mode relaxation measurements. Loeal mode relaxation can not be measured by DSC. DMA can give information about the crosslinking process of modified phenolic prepolymer [218] and about the erosslinked material [132]. During DMA measurements, sinusoidally varying stress of frequency is applied to the sample. Frequency and the stress are connected by equation 57, where is the maximum stress amplitude and is the phase angle at which the stress proceeds the strain. 

A very good way to characterize and differentiate between elastomers and rigid plastics is by the measurement of dynamic mechanical properties. A most convenient method to study dynamic mechanical properties is to impose a small, sinusoidal shear or tensile strain and measure the resulting stress. Dynamic mechanical properties are most simply determined for a small sinusoidally varying strain, for which the response is a sinusoidally varying stress. An increase in frequency of the sinusoidal deformation is equivalent to an increase in strain rate. 

The viscoelastic properties of concentrated o/w and w/o emulsions were investigated using dynamic (oscillatory) measurements. For that purpose a Bohlin VOR (Bohlin Reologie, Lund, Sweden) instrument was used. Concentric cylinder platens were used and the measurements were carried out at 25 0.1 °C. In oscillatory measurements, the response in stress of a viscoelastic material subjected to a sinusoidally varying strain is monitored as a function of strain amplitude and frequency. The stress amplitude is also a sinusoidally varying function in time, but for a viscoelastic material it is shifted out of phase with the strain. The phase angle shift between stress and strain, 5, is given by... 

Dynamic Oscillatory Experiments The dynamic rheological properties of a polymeric solution can be determined by small-amplitude oscillation tests [2]. In small amplitude oscillatory measurements, a sinusoidally varying shear stress field is imposed on a fluid and the amplitude of the resulting shear strain and phase angle between the imposed stress and the strain is measured. The test is... 

A complete description of the viscoelastic properties of a material requires information over very long times. Creep and stress relaxation measurements are limited by inertial and experimental limitations at short times and by the patience of the investigator and structural changes in the lest material at very long times. To supplement these methods, the stress or the strain can be varied sinusoidally in a dynamic mechanical experiment. The frequency of this alternation is u cycles/s or m(= 27ri ) rad/s. An alternating experiment at frequency w is qualitatively equivalent to a creep or stress relaxation measurement at a time t = (I /w) sec. 

A stress-strain ellipse can be traced for sinusoidal deformations of a hard solid with the methods described previously after suitable modifications in the sample mounting. For example, the versatile apparatus of Philippoff described in Chapters 5 and 6 can be further modified for measurements of hard solids in flexure, using a configuration similar to that of (c) in Fig. 7-1. Configuration (a) is used by Koppelmann. The calculation of dynamic properties from equation 1 with both / and X2 sinusoidally varying is analogous to equations 19 and 20 of Chapter 5, except that now the two components of Young s modulus are determined ... 

In these cases the relative velocity of the shearing plates is not constant but varies in a sinusoidal manner so that the shear strain and the rate of shear strain are both cyclic, and the shear stress is also sinusoidal. For non-Newtonian fluids, the stress is out of phase with the rate of strain. In this situation a measured complex viscosity (rf) contains both the shear viscosity, or dynamic viscosity (t] ), related to the ordinary steady-state viscosity that measures the rate of energy dissipation, and an elastic component (the imaginary viscosity ij" that measures an elasticity or stored energy) ... 

The elastic nature of a fluid is characterized by dynamic mechanical or stress relaxation techniques. Dynamic mechanical (oscillatory) testing is a procedure in which a sample is sinusoidally strained and the resultant stress is measured. The shear stress T varies with the same frequency as the shear rate... 

Oscillatory shear can also be performed by varying the stress sinusoidally and measuring the resulting strain as a function of time. The results can then be interpreted in terms of the real and imaginary components of the complex compliance, ] =J -i J". These components, the storage and loss compliances, are simply related to the storage and loss moduli as shown below. 

---