Strain phase angle shift

From the amplitudes of stress and strain and the phase angle shift, one can obtain the following viscoelastic parameters. 

Dynamic (oscillatory) measurements A sinusoidal stress or strain with amphtudes (Tjj and is appHed at a frequency a> (rads ), and the stress and strain are measured simultaneously. For a viscoelastic system, as is the case with most formulations, the stress and strain amplitudes oscillate with the same frequency, but out of phase. The phase angle shift S is measured from the time shift of the strain and stress sine waves. From a, y and S, it is possible to obtain the complex modulus j G, the storage modulus G (the elastic component), and the loss modulus G" (the viscous component). The results are obtained as a function of strain ampHtude and frequency. 

Elastic response This occurs when the maximum of the stress amplitude is at the same position as the maximum of the strain amplitude (no energy dissipation). In this case, there is no time shift between the stress and strain sine waves. Viscous response This occurs when the maximum of the stress is at the point of maximum shear rate (i.e., the inflection point), where there is maximum energy dissipation. In this case, the strain and stress sine waves are shifted by (referred to as the phase angle shift, 5, which in this case is 90°). 

Consider the case of a viscoelastic system, for which the sine waves of strain and stress are shown in Figure 20.11. The frequency co is in rads , and the time shift between the strain and stress sine waves is At. The phase angle shift S is given by (in dimensionless units of radians). 

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... 

In a viscoelastic system (such as the case with a flocculated suspension), the stress oscillates with the same frequency, but out-of-phase from the strain. From measurement of the time shift between strain and stress amplitudes (At) one can obtain the phase angle shift 3,... 

From the amplitudes of stress and strain and the phase-angle shift one can obtain the various viscoelastic parameters The complex modulus G, the storage modulus (the elastic component of the complex modulus) G, the loss modulus (the viscous component of the complex modulus) G", tan 3 and the dynamic viscosity t]. ... 

From the time shift between the sine waves of the stress and strain. At, the phase angle shift <5 is calculated as... 

In dynamic (oscillatory) measurements, one applies a sinusoidal strain or stress (with amplitudes yo or < o and frequency co in rad s ) and the stress or strain is measured simultaneously. For a viscoelastic system, the stress oscillates with the same frequency as the strain, but out of phase. From the time shift of stress and strain, one can calculate the phase angle shift <5. This allows one to obtain the various viscoelastic parameters G (the complex modulus), G (the storage modulus, i.e. the elastic component of the complex modulus) and G" (the loss modulus or the viscous component of the complex modulus). These viscoelastic parameters are measured as a function of strain amplitude (at constant frequency) to obtain the linear viscoelastic region, whereby G, G and G" are independent of the applied strain until a critical strain above which G and G begin to decrease with further increase of strain, whereas G" shows an increase. Below y the structure of the system is not broken down, whereas above y the structure begins to break. From G and one can obtain the cohesive energy density of the structure... 

In dynamic (oscillator) measurements, a sinusoidal strain, with frequency v in Hz or 0) in rad s (o) = 2nv) is applied to the cup (of a concentric cylinder) or plate (of a cone and plate) and the stress is measured simultaneously on the bob or the cone which are connected to a torque bar. The angular displacement of the cup or the plate is measured using a transducer. For a viscoelastic system, such as is the case with a cosmetic emulsion, the stress oscillates with the same frequency as the strain, but out of phase [23]. This is illustrated in Fig. 1.12 which shows the stress and strain sine waves for a viscoelastic system. From the time shift between the sine waves of the stress and strain. At, the phase angle shift 8 is calculated. 

At = time shift for sine waves of stress and strain At 0) = 6 phase angle shift 0) = frequency in radian s ... 

For viscoelastic materials, the phase angle shift (3) between stress and strain occurs somewhere between the elastic and viscous extremes. The stress signal generated by a viscoelastic material can be separated into two components an elastic stress (cr ) that is in phase with strain, and a viscous stress (cr" ) that is in phase with the strain rate (d7/dt) but 90f out of phase with strain. The elastic and viscous stresses are sometimes referred to as the in-phase and out-of-phase stresses, respectively. The elastic stress is a measure of the degree to whidi the material behaves as an elastic solid. The viscous stress is a measure of the degree to which the material behaves as an ideal fluid. By separating the stress into these components, hoth strain amplitude and strain rate dependence of a material can he simultaneously measured. We can resume this paragraph by a set of equation ... 

The ratio of the loss modulus to the storage modulus is the tangent of the phase angle shift d between the stress and strain vectors, thus ... 

A technique for performing dynamic mechanical measurements in which the sample is oscillated mechanically at a fixed frequency. Storage modulus and damping are calculated from the applied strain and the resultant stress and shift in phase angle. 

When using small deformation rheology there are several useful parameters that may be obtained to describe a material the complex modulus (G ), storage modulus (G ), loss modulus (G") and the tangent of the phase shift or phase angle (tan 5). These values must be taken from within the LVR, and are obtained using a dynamic oscillatory rheometer (Rao 1999). Outside the LVR, important information may be obtained such as the yield stress and yield strain. 

This mode is used for accurate determination of the frequency dependence of materials and prediction of end-use product performance. In the fixed frequency mode applied stress (i.e., a force per unit area that tends to deform the body, usually expressed in Pa (N/m)) forces the sample to undergo sinusoidal oscillation at a frequency and amplitude (strain), i.e., the deformation from a specified reference state, measured as the ratio of the deformation to the total value of the dimension in which the strain occurs. Strain is non-dimensional, but is frequently expressed in reference values (such as %strain) selected by the operator. Energy dissipation in the sample causes the sample strain to be out of phase with the applied stress (Figure 15.2(a)). In other words, since the sample is viscoelastic, the maximum strain does not occur at the same instant as maximum stress. This phase shift or lag, defined as phase angle (6), is measured and used with known sample geometry and driver energy to calculate the viscoelastic properties of the sample. 

The influence of the vinyl content on the viscoelastic behaviour of polybutadienes is shown in Figure 4. Measurements of tan S, the phase angle between stress and strain under sinusoidal deformation, have been performed by Dynamic Mechanical Spectrometry (Rheometrics). Looking at the shift along the temperature axis due to the different vinyl content, a maximum vinyl content of 72% has been chosen, since beyond this limit the polymer can hardly be regarded as a rubber. 

Phase angle, phase shift between stress and strain vectors Tensile strain [—]... 

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