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In search of physical meaning: defining transient parameters for nonlinear viscoelasticity


A complete set of model-independent viscoelastic functions for understanding responses to transient nonlinear rheological tests is presented, using large-amplitude oscillatory shear strain as a model nonlinear protocol. The derivation makes no assumptions about symmetries, and is therefore applicable to the responses to any input, allowing researchers to unambiguously define time-dependent moduli, viscosities, compliances, fluidities, and normal stress coefficients. A legend for interpreting the dynamic trajectories in modulus space is provided, along with explicit definitions of the rates at which the moduli change. These provide a quantitative mechanism to identify when, and by how much, a material response stiffens, softens, thickens, or thins while being deformed. In addition to providing analytical expressions for the moduli, the derivation requires the definition of a conceptually new term. This means there exist three, not two, time-dependent nonlinear viscoelastic functions by which any response can be fully described. The third function accounts for nonlinear properties such as yield stresses and the shifting of the strain equilibrium. This complete analysis scheme is unique in making a distinction between the strains in the lab and material frames. The quantitative sequence of physical process analysis, which is fully developed in this work, allows for comprehensive physical interpretations of responses to transient deformations of any kind to be made, including the steady alternance responses to large-amplitude oscillatory shear (LAOS), time-dependent oscillatory shear startup responses, and thixotropic and anti-thixotropic responses.

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This work was supported by start-up funds from the Department of Chemical and Biomolecular Engineering at the University of Illinois at Urbana-Champaign. I am grateful for helpful and insightful comments from Florian Nettesheim, Roney Thompson, and Paulo de Souza Mendes.

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Correspondence to Simon A. Rogers.


Appendix 1

Dimensions, units, stress-controlled experiments, and time-dependent viscosities, compliances, and fluidities

The derivations presented in the main body of the text have been carried out with strain-controlled experiments in mind, where one measures the stress response to oscillating strains and strain-rates and seeks to define moduli. Nonlinear viscoelastic functions in the form of moduli or viscosities from strain-controlled experiments, or compliances or fluidities from responses to stress-controlled tests, in addition to normal stresses from any tests can be derived via a generic route presented in the “A generic definition of time-dependent nonlinear viscoelastic functions” section. In this appendix, the different native spaces in which responses reside are discussed. The generic \( \left[\begin{array}{ccc}\hfill x\hfill & \hfill y\hfill & \hfill z\hfill \end{array}\right] \) notation will be employed to refer to the particular spaces. The ability to provide a single derivation for all nonlinear viscoelastic parameters is brought about by making the analogy between the linear-regime description of a rheological response and Eq. 7, the equation of the general form of a plane, which is repeated here:

$$ ax+ by+ cz+ d=0. $$

Equation 7 is the general form of the equation of a plane in \( \left[\begin{array}{ccc}\hfill x\hfill & \hfill y\hfill & \hfill z\hfill \end{array}\right] \)-space that has a normal \( \boldsymbol{n}=\left[\begin{array}{ccc}\hfill a\hfill & \hfill b\hfill & \hfill c\hfill \end{array}\right] \) and a vertical displacement of d. A trajectory in this plane therefore has a binormal vector given by \( \boldsymbol{B}=\left[\begin{array}{ccc}\hfill a\hfill & \hfill b\hfill & \hfill c\hfill \end{array}\right] \).

The displacement term, d, reduces to zero in the linear regime and can therefore be neglected in our discussion of spaces without loss of generality. In a generalization of the procedure that leads to Eqs. 5 and 6 from Eqs. 3 and 4, Eq. 72 is interpreted as describing a single response dimension, z(t), as a sum of two terms each consisting of a single viscoelastic shear material function multiplied by one of the excitation dimensions, x(t) or y(t). The formalism need not be restricted to shear conditions, though, and corresponding tension, compression, and torsional parameters can be derived without hindrance.

Application of a sinusoidal strain (Eq. 1) can equally be thought of as application of a cosinusoidal strain rate (Eq. 2). The material functions that describe the stress response to a strain rate are viscosities, meaning that the stress response can be generically described by

$$ \sigma \left(\omega, t\right)={\eta}^{\mathit{\prime}}\left(\omega \right)\dot{\gamma}(t)+{\eta}^{\mathit{{\prime\prime}}}\left(\omega \right)\omega \gamma (t) $$

Comparison with Eq. 72 suggests that \( \left[\begin{array}{ccc}\hfill \dot{\gamma}(t)\hfill & \hfill \omega \gamma (t)\hfill & \hfill \sigma (t)\hfill \end{array}\right] \) is the natural space in which to define time-dependent dynamic viscosities. Further, comparing the coefficients of the strain and rate directions in Eqs. 3 and 73 leads to the familiar scaling of dynamic moduli being equal to the product of the angular frequency and the dynamic viscosities.

Correspondingly, the generic linear-regime descriptions of the measured strain and strain-rate responses to an applied oscillatory stress define the dynamic compliances and fluidities:

$$ \gamma \left(\omega, t\right)={J}^{\mathit{\prime}}\left(\omega \right)\sigma (t)-{J}^{\mathit{{\prime\prime}}}\left(\omega \right)\frac{1}{\omega}\dot{\sigma}\left(\boldsymbol{t}\right) $$
$$ \dot{\gamma}\left(\omega, t\right)={\phi}^{\prime}\left(\omega \right)\sigma (t)+{\phi}^{{\prime\prime}}\left(\omega \right)\frac{1}{\omega}\dot{\sigma}(t) $$

Comparisons of Eqs. 74 and 75 with Eq. 72 suggest that the spaces in which dynamic compliances and fluidities are naturally defined are therefore \( \left[\begin{array}{ccc}\hfill \sigma (t)\hfill & \hfill -\dot{\sigma}(t)/\omega \hfill & \hfill \gamma (t)\hfill \end{array}\right] \) and \( \left[\begin{array}{ccc}\hfill \sigma (t)\hfill & \hfill \dot{\sigma}(t)/\omega \hfill & \hfill \dot{\gamma}(t)\hfill \end{array}\right] \), respectively. The x dimensions of these spaces are the same, while the y dimensions are different by a factor of −1.

Under the general construction of Eq. 13, the dimensions of the space must all have the same units, which are those of the perceived applied perturbation (in the formalism suggested here, the perceived applied perturbation is always set as being the x-dimension). In the case of moduli (Eq. 3), the perceived applied perturbation is in the strain, and therefore, the dimensions of the native space are unit-less. If the strain-rate is thought of as being the perturbing quantity (Eq. 72), then the dimensions of the native space take units of inverse time. Interestingly, in the cases of compliances and fluidities (Eqs. 73 and 74), both native spaces take units of stress, reflecting the identical perturbing parameter.

In the soft matter community, a perturbation of stress-rate is rarely considered and so is hardly used by the vast majority of soft matter scientists. The official Society of Rheology list of symbols and nomenclature (Dealy et al. 2013), for instance, does not contain an entry for shear stress-rate. Stress-rate, however, is not a completely unused parameter. Evans (1974) linked the rate at which stress was applied to the growth of cracks in brittle materials, and Bacabac et al. (2004) linked the stress rate of interstitial fluid through the lacuno-canalicular network to the adaptive response of bone cells. Viscoelastic material functions infrequently used by rheologists that are defined in terms of the stress-rate could be incorporated into the current framework in an identical manner by following the procedure outlined above.

Viewing responses to strain- and stress-controlled stimuli as existing within different spaces provides an intuitive explanation as to why nonlinear compliances and moduli are not simply inverses of each other, or clearly related. In the linear regime, however, the response of a given material or model is independent of whether the excitation is strain- or stress-controlled. The well-known relations between linear-regime shear moduli and compliances can therefore be deduced in the familiar way (Eqs. 27, 28, 29 and 30 of Chapter 1 of Ferry 1980).

Appendix 2

The Frenet–Serret apparatus

The language used throughout this work to describe the space curves that result from oscillatory excitations is that of the Frenet–Serret apparatus. The Frenet–Serret frame consists of the mutually orthogonal tangent, principal normal, and binormal vectors. Here, definitions of those vectors are presented, along with the scalars, referred to as the curvature and torsion of a response, that define how the frame reorients along a curve. The collection of the three vectors defining the frame and the two scalars defining how the frame reorients comprises the Frenet–Serret apparatus. Only issues salient to this work will be presented in this appendix. A full treatment of the geometrical concepts applied to space curves can be found elsewhere (Chapter 2 of Pressley 2010).

The space curve of a response within the three-dimensional Euclidean space ℝ3 is defined as

$$ \boldsymbol{A}=\left[\begin{array}{ccc}\hfill {A}_x\hfill & \hfill {A}_y\hfill & \hfill {A}_z\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill x\hfill & \hfill y\hfill & \hfill z\hfill \end{array}\right], $$

where x, y, and z refer to the specific dimensions of the space as discussed in Appendix 1. The time dependence of all terms is implied.

Geometrically, it is more natural to define derivatives of space curves with respect to arc length. However, because the rheological data collected from oscillatory experiments are recorded discretely at evenly spaced time intervals, the natural derivatives of interest in the current case are those with respect to time. The definitions contained in this appendix will use the usual dot notation to represent differentiation with respect to time, as opposed to the dash notation which would indicate differentiation with respect to arc length.

From the perspective of an observer traveling along the space curve (or one who imagines they are), the tangent vector points in the direction of instantaneous travel and is defined as

$$ \boldsymbol{T}=\frac{\dot{\boldsymbol{A}}}{\left\Vert \dot{\boldsymbol{A}}\right\Vert } $$

The normalized temporal derivative of the tangent vector is the principal normal vector:

$$ \boldsymbol{N}=\frac{\dot{\boldsymbol{T}}}{\left\Vert \dot{\boldsymbol{T}}\right\Vert }=\frac{\dot{\boldsymbol{A}}\times \left(\ddot{\boldsymbol{A}}\times \dot{\boldsymbol{A}}\right)}{\left\Vert \dot{\boldsymbol{A}}\right\Vert \left\Vert \ddot{\boldsymbol{A}}\times \dot{\boldsymbol{A}}\right\Vert } $$

The tangent and principal normal vectors span the osculating plane, the plane that has a second-order contact with the space curve at the point t. An n-th order contact is defined by the two objects having the same (n − 1)th derivative. Second-order contact between the osculating plane and the trajectory is therefore the case where the first derivatives of each are the same.

The orientation of the osculating plane is defined by the binormal vector, which is therefore orthogonal to both the tangent and principal normal vectors:

$$ \boldsymbol{B}=\boldsymbol{T}\times \boldsymbol{N}=\frac{\dot{\boldsymbol{A}}\times \ddot{\boldsymbol{A}}}{\left\Vert \dot{\boldsymbol{A}}\times \ddot{\boldsymbol{A}}\right\Vert } $$

The Frenet–Serret frame, made up of the set of T, N, and Bvectors, is therefore a right-handed orthonormal basis that spans ℝ3:

$$ \boldsymbol{B}=\boldsymbol{T}\times \boldsymbol{N};\boldsymbol{T}=\boldsymbol{N}\times \boldsymbol{B};\boldsymbol{N}=\boldsymbol{B}\times \boldsymbol{T} $$

A visual representation of a generic linear viscoelastic response is shown in Fig. 9 along with the Frenet–Serret frame at two instances separated by a quarter of a period.

Fig. 9
figure 9

A generic linear viscoelastic response is shown as a black solid line. The Frenet–Serret frame, which is defined by the tangent (red), principal normal (green), and binormal (blue) vectors, is displayed at two instances a quarter of a period apart. While the tangent and principal normal vectors change direction, the binormal vector remains oriented in the same direction throughout the oscillatory period (color figure online)

There are two scalars that define how the Frenet–Serret frame reorients along an arbitrary space curve. The two scalars are the curvature, κ, and the torsion, τ. Generally speaking, the curvature measures the extent to which a curve is not a straight line, and the torsion measures the extent to which the curve does not sit within a plane.

The curvature is defined as the projection of the derivative of the tangent vector onto the principal normal vector, and can therefore be thought of as the rate at which the tangent vector rotates about the binormal vector:

$$ \dot{\boldsymbol{T}}=\left\Vert \dot{\boldsymbol{A}}\right\Vert \kappa N $$

The term \( \left|\right|\dot{\boldsymbol{A}}\left|\right| \)results from the desire to work with the temporal derivatives, rather than the arc-length derivatives which naturally account for the changes in the magnitude of A. In the case of oscillatory rheology, the mutually orthogonal lab-frame oscillatory inputs, x and y, ensure that the response sits on the surface of a cylinder, which in turn means the curvature is always non-zero.

The torsion is defined as the negative projection of the derivative of the binormal vector onto the principal normal vector, and can therefore be thought of as the rate at which the binormal vector rotates about the tangent vector:

$$ \tau =-\left\Vert \dot{\boldsymbol{A}}\right\Vert \boldsymbol{N}\cdot \dot{\boldsymbol{B}}=\frac{\left(\dot{\boldsymbol{A}}\times \ddot{\boldsymbol{A}}\right)\cdot \overset{\dddot{}}{\boldsymbol{A}}}{\left\Vert \dot{\boldsymbol{A}}\times \ddot{\boldsymbol{A}}\right\Vert {}^2} $$

The geometric interpretations of the curvature and torsion are emphasized in Fig. 1. The tangent vector points in the direction of travel around the curve, and the circular arrows indicating curvature and torsion point in the directions of positive values.

To calculate the derivative of the principal normal vector, combine Eqs. 80, 81, and 82:

$$ \dot{\boldsymbol{N}}=\dot{\boldsymbol{B}}\times \boldsymbol{T}+\boldsymbol{B}\times \dot{\boldsymbol{T}}=-\left\Vert \dot{\boldsymbol{A}}\right\Vert \tau N\times \boldsymbol{T}+\boldsymbol{B}\times \left\Vert \dot{\boldsymbol{A}}\right\Vert \kappa N=\left\Vert \dot{\boldsymbol{A}}\right\Vert \left(-\kappa T+\tau B\right) $$

Equations 81, 82, and 83 are compactly summarized by Eq. 21:

$$ \left[\begin{array}{c}\hfill \dot{\boldsymbol{T}}\hfill \\ {}\hfill \dot{\boldsymbol{N}}\hfill \\ {}\hfill \dot{\boldsymbol{B}}\hfill \end{array}\right]=\left\Vert \dot{\boldsymbol{A}}\right\Vert \left[\begin{array}{ccc}\hfill 0\hfill & \hfill \boldsymbol{\kappa} \hfill & \hfill 0\hfill \\ {}\hfill -\boldsymbol{\kappa} \hfill & \hfill 0\hfill & \hfill \boldsymbol{\tau} \hfill \\ {}\hfill 0\hfill & \hfill -\boldsymbol{\tau} \hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{c}\hfill \boldsymbol{T}\hfill \\ {}\hfill \boldsymbol{N}\hfill \\ {}\hfill \boldsymbol{B}\hfill \end{array}\right] $$

Appendix 3

An alternative illustration of the need for the displacement term

The definitions presented in Eqs. 18, 19, 22 and 23 made no assumptions regarding the form of the excitation. If this condition is relaxed and a calculation is made where sinusoidal excitations are assumed, an interesting result is obtained that confirms the need for the displacement term. An alternative way to proceed from Eq. 16 that makes use of the symmetries of sinusoidal excitations and their derivatives is to note that the binormal vector is defined by the vector cross-product of the first and second temporal derivatives of the measured trajectory, \( \dot{\boldsymbol{A}} \) and \( \ddot{\boldsymbol{A}} \) (see Appendix 2), and can therefore be written in terms of the temporal derivatives of A γ , \( {A}_{\dot{\gamma}/\omega} \), and A σ :

$$ \boldsymbol{B}\propto \dot{\boldsymbol{A}}\times \ddot{\boldsymbol{A}}=\left[\begin{array}{ccc}\hfill {\dot{\boldsymbol{A}}}_{\dot{\gamma}/\omega}\overset{\cdot \cdot }{A_{\sigma}}-{\ddot{A}}_{\dot{\gamma}/\omega}\dot{A_{\sigma}}\hfill & \hfill -\left(\dot{A_{\gamma}}\overset{\cdot \cdot }{A_{\sigma}}-\overset{\cdot \cdot }{A_{\gamma}}\dot{A_{\sigma}}\right)\hfill & \hfill \dot{A_{\gamma}}{\ddot{A}}_{\dot{\gamma}/\boldsymbol{\omega}}-\overset{\cdot \cdot }{A_{\gamma}}{\dot{A}}_{\dot{\gamma}/\boldsymbol{\omega}}\hfill \end{array}\right] $$

In the derivatives of the sinusoidal and cosinusoidal inputs, Eqs. 1 and 2 can be simplified as follows:

$$ \dot{A_{\gamma}}={\omega A}_{\dot{\gamma}/\omega};\overset{\cdot \cdot }{A_{\gamma}}=-{\omega}^2{\boldsymbol{A}}_{\gamma};{\ \dot{A}}_{\dot{\gamma}/\omega}=-{\omega A}_{\gamma};{\ \ddot{A}}_{\dot{\gamma}/\omega}=-{\omega}^2{A}_{\dot{\gamma}/\boldsymbol{\omega}} $$

The relations Eqs. 84 and 85 can therefore be used to simplify the expression of the displacement term given in Eq. 11 that involves only the A σ component and its second derivative:

$$ \frac{B_{\gamma}}{B_{\sigma}}{A}_{\gamma}+\frac{B_{\dot{\gamma}/\omega}}{B_{\sigma}}{A}_{\dot{\gamma}/\omega}+{A}_{\sigma}=\frac{1}{\omega^2}\overset{\cdot \cdot }{A_{\sigma}}+{A}_{\sigma} $$

The remarkably simple form of Eq. 86, coupled with Eq. 13, allows the generic description of shear stress responses to oscillatory strains given in Eq. 16 to be rewritten as

$$ \sigma (t)={G_t}^{\boldsymbol{\hbox{'}}}(t)\gamma \left(\boldsymbol{t}\right)+{G_t}^{\mathit{\hbox{'}}\boldsymbol{\hbox{'}}}(t)\dot{\gamma}(t)/\omega +\frac{1}{\omega^2}\ddot{\sigma}(t)+\sigma (t) $$

Canceling the σ(t) terms on either side of the equality leads to

$$ {G_t}^{\mathit{\hbox{'}}}(t)\gamma (t)+{G_t}^{\mathit{\hbox{'}}\mathit{\hbox{'}}}(t)\dot{\gamma}(t)/\omega =-\frac{1}{\omega^2}\ddot{\sigma}(t) $$

The second derivative of the stress in Eq. 89 eliminates any offsets, making this a statement that the viscoelastic parameters can only account for changes in the measured response relative to the inputs. Yield stresses and strains can therefore not be described by viscoelastic parameters such as moduli, viscosities, compliances, or fluidities defined in this way. It is noted here that Eq. 87 is derived from the general form of Eq. 16, which can be used to describe any response. The only assumptions that go into the derivation of Eq. 87 are listed in Eq. 84, which is valid for any oscillatory deformation. Equation 87 is therefore also applicable in a general sense. It is also possible to rephrase Eq. 87 in the form of Eq. 7, that is, in the general form of the equation of a plane. The nonlinear response moduli are directly interpretable in terms of the components of the binormal vector of the response in the same manner as before, but with the restriction of being in \( \left[\begin{array}{ccc}\hfill \gamma (t)\hfill & \hfill \dot{\gamma}(t)/\omega \hfill & \hfill \ddot{\sigma}(t)/{\omega}^2\hfill \end{array}\right] \)-space, and that the binormal of the trajectory, or the normal of the osculating plane, is equal to \( \left[\begin{array}{ccc}\hfill {G_t}^{\hbox{'}}(t)\hfill & \hfill {G_t}^{\hbox{'}\hbox{'}}(t)\hfill & \hfill 1\hfill \end{array}\right] \). It is also clear from Eq. 89 why there is a need to include the information about the displacement of the osculating plane in the full description of the response trajectory as given by Eq. 16—the moduli, which define the orientation of the plane, can only be used to reconstruct the second derivative of the stress, and not the whole stress response.

The linear-regime representation of the shear stress response to a sinusoidally varying strain is given by

$$ \sigma (t)={G}^{\mathit{\prime}}\left(\omega \right)\gamma (t)+{G}^{\mathit{\hbox{'}\hbox{'}}}\left(\omega \right)\dot{\gamma}(t)/\omega $$

The second derivative of this generic expression is (applying the results of Eq. 85)

$$ \ddot{\sigma}(t)=-{\omega}^2\left[{G}^{\mathit{\prime}}\left(\omega \right)\gamma (t)+{G}^{\mathit{\hbox{'}\hbox{'}}}\left(\omega \right)\dot{\gamma}(t)/\omega \right] $$

By comparing Eqs. 88 and 90, an identical conclusion to that previously drawn regarding the equality of the time-dependent response parameters and linear-regime material parameters is reached:

$$ \begin{array}{ccc}\hfill {G_t}^{\mathit{\hbox{'}}}(t)={G}^{\mathit{\hbox{'}}}\left(\omega \right)\hfill & \hfill \mathrm{and}\hfill & \hfill {G_t}^{\mathit{\hbox{'}}\mathit{\hbox{'}}}(t)={G}^{\mathit{\hbox{'}\hbox{'}}}\left(\omega \right)\hfill \end{array} $$

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Rogers, S.A. In search of physical meaning: defining transient parameters for nonlinear viscoelasticity. Rheol Acta 56, 501–525 (2017).

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  • Large-amplitude oscillatory shear
  • Maxwell fluid
  • Normal stresses
  • Oscillating flow
  • Plastic fluids
  • Yielding