Sequence of Physical Processes in Rheology

• SPP is a geometric framework in oscillatory rheology that extracts time‐dependent elastic and viscous moduli by analyzing the process trajectory.
• The extended formulation maps measurements from the current to the reference configuration to separate geometric nonlinearities from intrinsic rheological responses at large strains.
• Incorporating techniques like Gram–Schmidt in an enlarged state space, SPP distinguishes finite-deformation artifacts from the material’s inherent behavior.

Sequence of Physical Processes (SPP) is a rheological framework for analyzing oscillatory deformation experiments by extracting transient viscoelastic moduli from the geometry of the measured process trajectory. In its usual form, SPP operates in the small-strain regime with the Cauchy stress tensor $\boldsymbol{\sigma}$ and the linear strain $\boldsymbol{\varepsilon}$. An extended formulation reformulates the analysis in the reference configuration with the second Piola–Kirchhoff stress tensor $\boldsymbol{S}$ and the Green–Lagrange strain tensor $\boldsymbol{e}$, with the explicit aim of separating geometric nonlinearities from rheological nonlinearities at large deformation (Bouthier et al., 2 Mar 2025).

1. Classical SPP in oscillatory rheology

In the usual SPP setting, an oscillatory shear input is imposed as

$\varepsilon(t)=\varepsilon_0 \sin(\omega t),$

and the measured stress is decomposed into in-phase and out-of-phase contributions. In the simplest linear-viscoelastic form,

$$\sigma = G_t' \, \varepsilon + G_t'' \, \frac{\dot\varepsilon}{\omega}.$$

The transient moduli $G_t'$ and $G_t''$ are intended to track the instantaneous elastic and viscous responses during a cycle (Bouthier et al., 2 Mar 2025).

The construction is geometric. The framework uses a Frenet/Serret-like local basis in a state space built from the relevant variables and their time derivatives, and from the geometry of the trajectory one extracts the time-dependent moduli. In this form, SPP is tied to the current configuration and to the assumption that geometric nonlinearities can be neglected. That restriction is central: the usual construction is effective for small deformations, but it does not distinguish intrinsic rheological nonlinearities from finite-deformation kinematic effects when the imposed amplitude becomes large (Bouthier et al., 2 Mar 2025).

2. Reference-configuration reformulation

The large-deformation extension switches from the current configuration to the reference configuration. The corresponding conjugate variables are the second Piola–Kirchhoff stress tensor $\boldsymbol{S}$ and the Green–Lagrange strain tensor $\boldsymbol{e}$, defined by

$\boldsymbol{\varepsilon}$0

and

$\boldsymbol{\varepsilon}$1

The power balance in the reference frame is

$\boldsymbol{\varepsilon}$2

so a constitutive law of the form $\boldsymbol{\varepsilon}$3 is objective and properly defined (Bouthier et al., 2 Mar 2025).

This reformulation is designed to remove geometric artifacts. For small deformations,

$\boldsymbol{\varepsilon}$4

so the extension reduces to the classical SPP limit. At finite strain, however, the reference-frame description can remain simple even when the observed stress in the current configuration exhibits extra components, higher harmonics, or apparent amplitude dependence. The central claim of the extension is therefore not that nonlinear effects disappear, but that some signatures ordinarily interpreted as rheological nonlinearity can instead arise from geometry alone (Bouthier et al., 2 Mar 2025).

3. Geometric corrections and enlarged state space

For simple oscillatory shear in the $\boldsymbol{\varepsilon}$5-plane, the Green–Lagrange strain contains a quadratic correction: $\boldsymbol{\varepsilon}$6 This $\boldsymbol{\varepsilon}$7 term is the essential geometric correction in the shear kinematics (Bouthier et al., 2 Mar 2025).

The extended SPP embeds the process in a five-dimensional state space,

$\boldsymbol{\varepsilon}$8

and constructs an instantaneous local orthonormal basis by Gram–Schmidt on the time derivatives $\boldsymbol{\varepsilon}$9. The resulting decomposition yields four transient moduli,

$\boldsymbol{S}$0

with the constitutive relation written schematically as

$\boldsymbol{S}$1

In this representation, $\boldsymbol{S}$2 and $\boldsymbol{S}$3 describe the linear elastic and viscous contributions, while $\boldsymbol{S}$4 and $\boldsymbol{S}$5 isolate the quadratic terms associated with finite-deformation geometry (Bouthier et al., 2 Mar 2025).

The significance of this extension is interpretive as much as computational. The additional quadratic terms are not automatically evidence of exotic constitutive physics. In the extended SPP framework, they can be traced to the geometry of the deformation itself.

4. Canonical viscoelastic models in the extended framework

The extension is illustrated on the linear Maxwell model and the linear Kelvin–Voigt model. In both cases, the reference-frame constitutive law remains linear, whereas the Cauchy-stress description acquires higher harmonics and normal-stress components (Bouthier et al., 2 Mar 2025).

Model	Reference-frame law	Characteristic SPP outcome
Maxwell	$\boldsymbol{S}$6	$\boldsymbol{S}$7 linear; $\boldsymbol{S}$8 quadratic and at $\boldsymbol{S}$9
Kelvin–Voigt	$\boldsymbol{e}$0	$\boldsymbol{e}$1 classical; $\boldsymbol{e}$2 quadratic with nonzero mean

For the Maxwell model, the shear and normal components satisfy

$\boldsymbol{e}$3

Under sinusoidal forcing, $\boldsymbol{e}$4 has the standard Maxwell response, whereas $\boldsymbol{e}$5 is an induced normal component quadratic in $\boldsymbol{e}$6 and oscillating at double frequency $\boldsymbol{e}$7. In the reference frame, the extracted moduli are constant: $\boldsymbol{e}$8 and

$\boldsymbol{e}$9

The same analysis shows that, after transformation back to the Cauchy stress, $\varepsilon(t)=\varepsilon_0 \sin(\omega t),$0 contains the first and third harmonics, $\varepsilon(t)=\varepsilon_0 \sin(\omega t),$1 contains zero, second, and fourth harmonics, and $\varepsilon(t)=\varepsilon_0 \sin(\omega t),$2 retains the double-frequency oscillation (Bouthier et al., 2 Mar 2025).

For the Kelvin–Voigt model,

$\varepsilon(t)=\varepsilon_0 \sin(\omega t),$3

so that

$\varepsilon(t)=\varepsilon_0 \sin(\omega t),$4

Here again, $\varepsilon(t)=\varepsilon_0 \sin(\omega t),$5 is the classical response and $\varepsilon(t)=\varepsilon_0 \sin(\omega t),$6 is a geometrically induced quadratic term. The extracted reference-frame moduli are constant: $\varepsilon(t)=\varepsilon_0 \sin(\omega t),$7 and

$\varepsilon(t)=\varepsilon_0 \sin(\omega t),$8

When mapped back to the current configuration, $\varepsilon(t)=\varepsilon_0 \sin(\omega t),$9 and $$\sigma = G_t' \, \varepsilon + G_t'' \, \frac{\dot\varepsilon}{\omega}.$$0 develop higher harmonics and amplitude-dependent offsets. The average normal component is reported as

$$\sigma = G_t' \, \varepsilon + G_t'' \, \frac{\dot\varepsilon}{\omega}.$$1

which is explicitly identified as a purely geometry-induced normal stress (Bouthier et al., 2 Mar 2025).

5. Interpretation of nonlinear signatures

The extended SPP framework rationalizes several behaviors that are often classified as nonlinear rheology. One class of effects is the generation of normal stresses under shear. Even when the constitutive law is linear in $$\sigma = G_t' \, \varepsilon + G_t'' \, \frac{\dot\varepsilon}{\omega}.$$2, finite deformation produces axial components such as $$\sigma = G_t' \, \varepsilon + G_t'' \, \frac{\dot\varepsilon}{\omega}.$$3 and $$\sigma = G_t' \, \varepsilon + G_t'' \, \frac{\dot\varepsilon}{\omega}.$$4. A second class is harmonic generation: after transformation to the Cauchy stress, the response can contain $$\sigma = G_t' \, \varepsilon + G_t'' \, \frac{\dot\varepsilon}{\omega}.$$5, $$\sigma = G_t' \, \varepsilon + G_t'' \, \frac{\dot\varepsilon}{\omega}.$$6, $$\sigma = G_t' \, \varepsilon + G_t'' \, \frac{\dot\varepsilon}{\omega}.$$7, and even $$\sigma = G_t' \, \varepsilon + G_t'' \, \frac{\dot\varepsilon}{\omega}.$$8-type content although the underlying constitutive law in the reference frame remains linear (Bouthier et al., 2 Mar 2025).

A further consequence concerns transient moduli extracted directly from the current configuration. The extended analysis highlights that Cole–Cole plots and time traces of $$\sigma = G_t' \, \varepsilon + G_t'' \, \frac{\dot\varepsilon}{\omega}.$$9 and $G_t'$0 computed from $G_t'$1 may show negative lobes or unusual loops. These are presented as possible geometric artifacts rather than necessary signs of thermodynamic inconsistency or true negative dissipation. In the same way, amplitude-dependent apparent stiffening or softening in the Cauchy frame can be produced by the configuration change itself, while the Piola–Lagrange description remains that of a linear Maxwell or Kelvin–Voigt medium (Bouthier et al., 2 Mar 2025).

The practical implication stated by the extension is methodological: before interpreting complex oscillatory-rheology signatures as material nonlinearity, one should first map the measured data back to the reference configuration and analyze $G_t'$2 versus $G_t'$3. The framework therefore functions as a separation principle between finite-strain geometry and intrinsic constitutive response.

6. Scope of the term and cross-disciplinary ambiguity

Within rheology, “Sequence of Physical Processes” denotes the trajectory-based framework summarized above. The same acronym, however, is used for unrelated concepts in other literatures, and those uses should be distinguished from the rheological framework.

Field	Meaning of “SPP”	Representative paper
Rheology	Sequence of Physical Processes	(Bouthier et al., 2 Mar 2025)
Astrophysics	sequence of physical separation processes in white dwarf cores	(García-Berro et al., 2010)
Plasmonics	surface plasmon polariton	(Ichiji et al., 2019, Baburin et al., 2018)
Accelerator physics	SANAEM or SNRTC Project Prometheus	(Yasatekin et al., 2014, Turemen et al., 2014, Turemen et al., 2014)
Machine learning	Sequential Neural Processes	(Singh et al., 2019)

In astrophysics, the phrase refers to the sequence $G_t'$4 sedimentation in the liquid white-dwarf core, followed by crystallization, followed by $G_t'$5 phase separation during crystallization; these processes are used to explain the cooling delay and age discrepancy of NGC 6791 (García-Berro et al., 2010). In plasmonics, SPP denotes femtosecond or propagating surface plasmon polaritons interacting with metal–insulator–metal nanocavities or ultra-smooth silver films (Ichiji et al., 2019, Baburin et al., 2018). In accelerator physics, SPP denotes the SANAEM Project Prometheus proof-of-principle proton accelerator and its RFQ beamline studies (Yasatekin et al., 2014, Turemen et al., 2014, Turemen et al., 2014). In machine learning, SPP refers to Sequential Neural Processes, a temporal extension of Neural Processes for a sequence of stochastic processes (Singh et al., 2019).

This terminological overlap does not imply conceptual continuity. In the rheological literature, SPP is specifically a geometric framework for extracting transient moduli and, in its recent extension, for distinguishing genuine rheological nonlinearity from large-deformation geometric effects (Bouthier et al., 2 Mar 2025).