^{1,a)}and M. Paul Lettinga

^{1}

### Abstract

The nonlinear yielding responses of three theoretical models, including the Bingham, a modified Bingham, and Giesekus models, to large-amplitude oscillatory shear are investigated under the framework proposed recently by Rogers *et al.* (2011). Under this framework, basis states are allowed to wax and wane throughout an oscillation, an approach that conflicts directly with the assumptions of all Fourier-like linear algebraic approaches. More physical yielding descriptions of the nonlinear waveforms are attained by viewing the responses as representing purely elastic to purely viscous sequences of physical processes. These interpretations are compared with, and contrasted with, results obtained from linear algebraic analysis methods: Fourier-transform rheology; and the Chebyshev description of the so-called elastic and viscous stress components σ′ and σ″. Further, we show that the discrepancies between the built-in model responses and parameters, and the interpretations of the Chebyshev and Fourier coefficients are directly related to misinterpretations of σ′ and σ″ as being the elastic and viscous stress contributions. We extend these ideas and discuss how every linear algebraic analysis is likely to conflate information from predominantly elastic and viscous processes when a material yields.

I. INTRODUCTION

A. LAOS in the literature

B. Motivation and manuscript layout

II. RESULTS AND DISCUSSION

A. The Bingham model

B. The modified Bingham model

C. The Giesekus model

D. Discussion on the SD method and Chebyshev extension as applied to the examples given

E. A comment on the presentation methods of LAOS data

F. Comments on Fourier transformation as applied to LAOS responses

III. CONCLUDING REMARKS

### Key Topics

- Elasticity
- 79.0
- Viscosity
- 60.0
- Shear rate dependent viscosity
- 47.0
- Elastic moduli
- 24.0
- Materials analysis
- 16.0

## Figures

The geometrical method of determining σ′. The stress orbit intersects the vertical line at the square points, the centre of which, marked by a circle, represents the value of σ′ at that strain.

The geometrical method of determining σ′. The stress orbit intersects the vertical line at the square points, the centre of which, marked by a circle, represents the value of σ′ at that strain.

A representative stress response of the Bingham model with relevant quantities from the SD method shown.

A representative stress response of the Bingham model with relevant quantities from the SD method shown.

The stress response of our modified Bingham model with relevant quantities from the SD method shown.

The stress response of our modified Bingham model with relevant quantities from the SD method shown.

Left—Linear viscoelastic relaxation spectrum of the Giesekus model showing plateau modulus of 10 Pa and a relaxation time of 1 s. Right—Steady-shear flow curve showing predicted responses based on η_{ 0 }, η_{ s } (angled dotted lines) and the inverses of the two relaxation times λ_{1} and λ_{2} (vertical dashed lines).

Left—Linear viscoelastic relaxation spectrum of the Giesekus model showing plateau modulus of 10 Pa and a relaxation time of 1 s. Right—Steady-shear flow curve showing predicted responses based on η_{ 0 }, η_{ s } (angled dotted lines) and the inverses of the two relaxation times λ_{1} and λ_{2} (vertical dashed lines).

Representative waveforms of the Giesekus model calculated under oscillatory shear conditions of γ_{ 0 } = 178 and ω = 0.1 (left), 1 (centre), and 10 (right) rad · s^{−1} displayed in elastic Lissajous-Bowditch curves. Colored lines show the calculated stress response from startup, which quickly becomes the “steady-state” response. Solid black lines indicate σ′ and dashed and dotted black lines are described in the text.

Representative waveforms of the Giesekus model calculated under oscillatory shear conditions of γ_{ 0 } = 178 and ω = 0.1 (left), 1 (centre), and 10 (right) rad · s^{−1} displayed in elastic Lissajous-Bowditch curves. Colored lines show the calculated stress response from startup, which quickly becomes the “steady-state” response. Solid black lines indicate σ′ and dashed and dotted black lines are described in the text.

The differential modulus and differential viscosity (see text) measured from calculated responses of the Giesekus model to a strain amplitude of 178, some of which are shown in Fig. 5. The polymer shear modulus of 10 Pa, indicated by the horizontal dotted line, is never fully recovered in the frequency regime investigated, while the zero-rate viscosity of 10 Pa · s is achieved at low frequencies, as expected.

The differential modulus and differential viscosity (see text) measured from calculated responses of the Giesekus model to a strain amplitude of 178, some of which are shown in Fig. 5. The polymer shear modulus of 10 Pa, indicated by the horizontal dotted line, is never fully recovered in the frequency regime investigated, while the zero-rate viscosity of 10 Pa · s is achieved at low frequencies, as expected.

Positive oscillatory stress (filled color symbols) superimposed on the steady-shear flow curve (unfilled squares).

Positive oscillatory stress (filled color symbols) superimposed on the steady-shear flow curve (unfilled squares).

Three representative waveforms [in black with σ′ in blue (a–c) and σ″ in red (d–f)] of responses that are best viewed as resulting from a sequence of physical processes: (a and d) The Bingham model, (b and e) the modified Bingham model (see text) and (c and f) the Giesekus model (response to γ_{ 0 } = 178, ωλ = 1). The portions in the dotted boxes show the parts of σ′ and σ″ that conflate stresses from elastic and viscous mechanisms.

Three representative waveforms [in black with σ′ in blue (a–c) and σ″ in red (d–f)] of responses that are best viewed as resulting from a sequence of physical processes: (a and d) The Bingham model, (b and e) the modified Bingham model (see text) and (c and f) the Giesekus model (response to γ_{ 0 } = 178, ωλ = 1). The portions in the dotted boxes show the parts of σ′ and σ″ that conflate stresses from elastic and viscous mechanisms.

LAOS data can be displayed as one-dimensional closed traces in a three-dimensional space defined by the strain (red), strain-rate (blue), and stress (black) axes. Rather than viewing these traces as fixed in time, the SPP framework suggests rotating the space from elastic (E) to viscous (V) presentations (left to right) as a function of time.

LAOS data can be displayed as one-dimensional closed traces in a three-dimensional space defined by the strain (red), strain-rate (blue), and stress (black) axes. Rather than viewing these traces as fixed in time, the SPP framework suggests rotating the space from elastic (E) to viscous (V) presentations (left to right) as a function of time.

The three models used in this work (a—Bingham/b—modified Bingham/c—Giesekus response to γ_{ 0 } = 178, ωλ = 1) presented in the 3D space defined by the stress, strain, and shear rate axes. Projections onto the elastic (stress–strain) and viscous (stress–rate) planes previously presented in Figs. 2, 3, and 8 are also displayed. Under the SPP framework, the best viewing of this data includes a rotation as the responses change from elastic (red lines) to viscous (blue lines).

The three models used in this work (a—Bingham/b—modified Bingham/c—Giesekus response to γ_{ 0 } = 178, ωλ = 1) presented in the 3D space defined by the stress, strain, and shear rate axes. Projections onto the elastic (stress–strain) and viscous (stress–rate) planes previously presented in Figs. 2, 3, and 8 are also displayed. Under the SPP framework, the best viewing of this data includes a rotation as the responses change from elastic (red lines) to viscous (blue lines).

Three linear viscoelastic stress responses, characterized by δ = π/8 (top row), π/4 (middle row), and 3π/8 (bottom row) are rotated by angles −θ as displayed at the top. No area is enclosed by the curves when the angle of rotation is equal to the phase angle of the response, i.e. when θ = δ.

Three linear viscoelastic stress responses, characterized by δ = π/8 (top row), π/4 (middle row), and 3π/8 (bottom row) are rotated by angles −θ as displayed at the top. No area is enclosed by the curves when the angle of rotation is equal to the phase angle of the response, i.e. when θ = δ.

The unsigned integral as a function of rotation angle for four normalized responses. (a) Linear response defined by sin(ω*t* + 3π/8), (b) the Bingham model, (c) the modified Bingham model, and (d) the Giesekus model. Vertical dashed lines indicate the angular value of δ_{1}, the phase of the first harmonic. The insets display the projections where minimum area is enclosed.

The unsigned integral as a function of rotation angle for four normalized responses. (a) Linear response defined by sin(ω*t* + 3π/8), (b) the Bingham model, (c) the modified Bingham model, and (d) the Giesekus model. Vertical dashed lines indicate the angular value of δ_{1}, the phase of the first harmonic. The insets display the projections where minimum area is enclosed.

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