^{1,a)}

### Abstract

An entirely new way of analyzing linear and nonlinear oscillatory material responses is presented. The new quantitative sequence of physical processes (SPP) method views generic oscillatory responses within the Frenet-Serret frame as sequences of planar 2D curves embedded in the 3D space defined by the strain, strain rate, and stress axes. Associated with the curve within each so-called “osculating” plane is a binormal vector that wholly determines the orientation of the plane. Physically meaningful information is obtained by calculating the angles between a modified form of the binormal vector and two reference vectors from local information at any point in a cycle. Information from a full period of oscillation is not a requirement of this technique, so that moduli can be calculated from partial or incomplete oscillations. Time-dependent phase and complex modulus information, or dynamic moduli information are obtainable throughout a period for arbitrary oscillatory responses. The SPP method is applied to the Bingham plastic model, power-law fluids, the Herschel–Bulkley model, and the nonlinear Giesekus model to accustom readers to its function. Application of the SPP method to these nonlinear models necessitates the refinement of some common language, as well as changes to the way results from strain amplitude sweeps are displayed and discussed.

The author acknowledges useful mathematical discussions with Gerhard Naegele, Marco Heinen, and Claudio Contreras-Aburto. Insightful and informative discussions with Pavlik Lettinga and Pierre Ballesta are also acknowledged. The author thanks Dimitris Vlassopoulos and Jan Dhont for comments on the manuscript. Financial support from the EU through FP7, project Nanodirect (Grant No. NMP4-SL-2008-213948) is gratefully acknowledged. This paper is dedicated to the memory of Professor Sir Paul Callaghan, a wonderful friend and mentor. Rārangi maunga, tū tonu, tū tonu. Rārangi tangata, ngaro noa, ngaro noa. (You have gone, but your mountain is everlasting.)

I. INTRODUCTION

II. FUNDAMENTAL CONCEPT

III. EXTRACTING PHYSICAL INFORMATION

A. Phase information

B. Complex modulus information

IV. EXTENSION TO NONLINEAR RESPONSES

V. NAMING AND APPLICATION

A. The Bingham model

B. Power-law fluids

C. The Herschel–Bulkley model

D. The Giesekus model

VI. DISCUSSION AND CONCLUSIONS

### Key Topics

- Elastic moduli
- 27.0
- Elasticity
- 27.0
- Shear rate dependent viscosity
- 18.0
- Materials analysis
- 16.0
- Stress strain relations
- 13.0

## Figures

A linear viscoelastic response plotted in the 3D space defined by the strain, rate, and stress axes. 2D projections of the periodic orbit are displayed as a plan and elevations on the left, and in perspective 3D on the right. Curved arrows indicate the direction of orbit. The negative binormal vector, −B, and reference vectors s_{1} and s_{2} (see text) are indicated.

A linear viscoelastic response plotted in the 3D space defined by the strain, rate, and stress axes. 2D projections of the periodic orbit are displayed as a plan and elevations on the left, and in perspective 3D on the right. Curved arrows indicate the direction of orbit. The negative binormal vector, −B, and reference vectors s_{1} and s_{2} (see text) are indicated.

A purely elastic response (a) and a purely viscous response (b). In each case, the gradient is equal to the tangent of the angle between the negative binormal vector and the reference vector s_{2}.

A purely elastic response (a) and a purely viscous response (b). In each case, the gradient is equal to the tangent of the angle between the negative binormal vector and the reference vector s_{2}.

The response of the Bingham model with an elastic modulus , plastic viscosity , and yield strain showing (a) the time trace of the stress for a complete period, (b) the phase angle as a function of time, (c) the magnitude of the complex modulus as a function of time, (d) the elastic Lissajous projection, (e) the viscous Lissajous projection, and (f) the full 3D orbit.

The response of the Bingham model with an elastic modulus , plastic viscosity , and yield strain showing (a) the time trace of the stress for a complete period, (b) the phase angle as a function of time, (c) the magnitude of the complex modulus as a function of time, (d) the elastic Lissajous projection, (e) the viscous Lissajous projection, and (f) the full 3D orbit.

Full 3D orbits of power-law fluid responses for various values of the power-law index *n*. The perfect plastic response corresponds to a power-law index of zero, so that the stress is a signum function of the shear rate.

Full 3D orbits of power-law fluid responses for various values of the power-law index *n*. The perfect plastic response corresponds to a power-law index of zero, so that the stress is a signum function of the shear rate.

The response of a power-law fluid of consistency *K* = 1 and power-law index *n* = 0.5 showing (a) the time trace of the stress for a complete period, (b) the phase angle as a function of time, (c) the magnitude of the complex modulus as a function of time, (d) the elastic Lissajous projection, (e) the viscous Lissajous projection, and (f) the full 3D orbit.

The response of a power-law fluid of consistency *K* = 1 and power-law index *n* = 0.5 showing (a) the time trace of the stress for a complete period, (b) the phase angle as a function of time, (c) the magnitude of the complex modulus as a function of time, (d) the elastic Lissajous projection, (e) the viscous Lissajous projection, and (f) the full 3D orbit.

The response of the Herschel–Bulkley model with modulus *G* = 2, consistency *K* = 0.5, and power-law index *n* = 0.5 showing (a) the time trace of the stress for a complete period, (b) the phase angle as a function of time, (c) the magnitude of the complex modulus as a function of time, (d) the elastic Lissajous projection, (e)the viscous Lissajous projection, and (f) the full 3D orbit.

The response of the Herschel–Bulkley model with modulus *G* = 2, consistency *K* = 0.5, and power-law index *n* = 0.5 showing (a) the time trace of the stress for a complete period, (b) the phase angle as a function of time, (c) the magnitude of the complex modulus as a function of time, (d) the elastic Lissajous projection, (e)the viscous Lissajous projection, and (f) the full 3D orbit.

The linear-regime relaxation spectrum of the Giesekus model with parameters as quoted in the text. The angular frequencies are normalized by the relaxation time λ_{1} = 1 s.

The linear-regime relaxation spectrum of the Giesekus model with parameters as quoted in the text. The angular frequencies are normalized by the relaxation time λ_{1} = 1 s.

Results of a strain sweep experiment are normally displayed in terms of G′ and G″. These reduce the information from an entire period to two points. Shown here are the calculated values from the response of the Giesekus model for three representative frequencies.

Results of a strain sweep experiment are normally displayed in terms of G′ and G″. These reduce the information from an entire period to two points. Shown here are the calculated values from the response of the Giesekus model for three representative frequencies.

The strain dependence [(a) and (b)] and strain-rate dependence [(c) and (d)] on strain amplitude and time throughout an oscillation for an applied angular frequency of 0.1 rad s^{−1}.

The strain dependence [(a) and (b)] and strain-rate dependence [(c) and (d)] on strain amplitude and time throughout an oscillation for an applied angular frequency of 0.1 rad s^{−1}.

The results of strain sweeps must be shown as surfaces using the new analysis method. Here, the grey/blue surface represents the loss modulus and the white/red surface represents the storage modulus to an input frequency of De = 0.1.

The results of strain sweeps must be shown as surfaces using the new analysis method. Here, the grey/blue surface represents the loss modulus and the white/red surface represents the storage modulus to an input frequency of De = 0.1.

The storage (white/red) and loss (grey/blue) moduli as functions of strain amplitude and time throughout an oscillation at an input frequency of De = 1 for the Giesekus model. The double peaks in R″ around times t/T ∼ 0.5 and 1 are artifacts of the interpolation scheme of the plotting software.

The storage (white/red) and loss (grey/blue) moduli as functions of strain amplitude and time throughout an oscillation at an input frequency of De = 1 for the Giesekus model. The double peaks in R″ around times t/T ∼ 0.5 and 1 are artifacts of the interpolation scheme of the plotting software.

The storage (white/red) and loss (grey/blue) moduli as functions of strain amplitude and time throughout an oscillation at an input frequency of De = 10 for the Giesekus model.

The storage (white/red) and loss (grey/blue) moduli as functions of strain amplitude and time throughout an oscillation at an input frequency of De = 10 for the Giesekus model.

The data of Figs. 10–12 are displayed as viewed from “above looking down.” Yielding can be determined from the points where R′ > R″ changes to R″ > R′. This is equivalent to moving vertically upward along the time axis and going from a light to a dark region. The minimum yielding amplitude is the smallest amplitude that displays this behavior and is marked with vertical dashed lines at each frequency.

The data of Figs. 10–12 are displayed as viewed from “above looking down.” Yielding can be determined from the points where R′ > R″ changes to R″ > R′. This is equivalent to moving vertically upward along the time axis and going from a light to a dark region. The minimum yielding amplitude is the smallest amplitude that displays this behavior and is marked with vertical dashed lines at each frequency.

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