Rheological Memory in Materials

• Rheological memory is the dependence of a material’s response on its deformation history, expressed through hereditary kernels and evolving internal states.
• It is represented mathematically by models such as jerk-elasticity, fractional viscoelasticity, and stochastic memory kernels that capture logarithmic relaxation and power-law creep.
• Experimental protocols reveal memory effects through measurable features like stress overshoots, residual stresses, and nonlocal structural rearrangements that influence material behavior.

Rheological memory denotes the dependence of a material’s current stress, strain, viscosity, yield response, or fluctuation spectrum on prior deformation, waiting time, thermal path, or internal structural evolution. In linear viscoelasticity this dependence is commonly expressed through hereditary kernels, whereas in aging, trainable, or actively driven systems it may instead be carried by time-varying constitutive coefficients, internal variables, evolving contact networks, or preparation-dependent microstructures. Recent work identifies rheological memory in logarithmic stress relaxation and power-law creep, Kovacs-like non-monotonic stress evolution, cyclic-shear training, stress-selective viscosity states, thermo-rheological history effects during gelation, and hydrodynamic long-time tails in Brownian motion (Pandey, 4 Jun 2025, Ghadai et al., 15 Dec 2025, Makris, 2 Feb 2026).

1. Definitions and conceptual boundaries

In its most classical form, rheological memory is the statement that the present response depends on deformation history rather than only on instantaneous strain or strain rate. A standard linear-viscoelastic representation is

$$\sigma(t)=\int_0^t G(t-\tau)\,\dot{\varepsilon}(\tau)\,d\tau,$$

with $G$ as a memory kernel. In aging systems, however, strict time-translational invariance can break down: the response to a given strain history depends not only on how long ago that history occurred, but also on how old the material was when it occurred. This explicit age dependence is central to the time-varying “jerkity” framework proposed for logarithmic relaxation and universal creep (Pandey, 4 Jun 2025).

The term also covers memory encoded outside conventional linear kernels. In hydrolyzed polyacrylamide, the fractional derivative order $q$ is treated as a memory index whose value changes with salinity, temperature, concentration, and pH, linking rheological memory to a nonlocal constitutive operator rather than to a finite set of relaxation times (Bhattacharyya et al., 2020). In $\kappa$-carrageenan fluid gels, by contrast, the retained history of the cooling shear rate is revealed mainly in the nonlinear viscoelastic regime, while the linear plateau modulus remains comparatively insensitive; the memory is therefore present, but not primarily in small-amplitude linear spectra (Bauland et al., 17 Mar 2026). In skeletal muscle work loops, memory arises from the combination of fixed-stimulus rheology and time-varying activation, so that present force depends on prior length and prior stimulus through internal state evolution rather than through a passive kernel alone (Nguyen et al., 2020).

A recurrent misconception is that rheological memory is identical either to thixotropy or to nonlocal cooperativity. Recent continuum work states the distinction explicitly: nonlocality alone cannot encode history, and memory alone cannot encode spatial cooperativity; their coupling is essential for history-dependent nonlocal flows (Zolfaghari et al., 13 Feb 2026). A second misconception is that memory must be visible in steady viscosity or linear moduli. Several systems instead store memory mainly in peak times, overshoots, residual stresses, nonlinear dissipation, or topological reversibility under strain reversal.

2. Mathematical representations of memory

Several distinct mathematical constructions are used to represent rheological memory. A time-varying constitutive law is provided by jerk-elasticity, where a Hookean spring is placed in parallel with a dissipative “jerk” element obeying

$$\dot{\sigma}_j(t)=E\,\dot{\varepsilon}(t)-\lambda(t)\,\varepsilon(t), \qquad \lambda(t)=\frac{1}{\xi+\theta t}.$$

Here the linearly increasing $1/\lambda(t)=\xi+\theta t$ acts as an internal state encoding aging. In this framework the relaxation modulus is logarithmic and the creep compliance is a Bernstein function, while the specific time variation of $1/\lambda$ and the parallel spring are required for thermodynamic admissibility (Pandey, 4 Jun 2025).

Fractional constitutive laws provide a second major representation. In the HPAM study, the dashpot law is generalized to

$\sigma(t)=\eta\,\frac{d^{q}\epsilon(t)}{dt^{q}},$

with $0$q$ is treated as a memory index, the complex moduli are fitted with a fractional Maxwell model, and the associated creep compliance takes a power-law form. The same work links larger $G$0 to smaller power-law exponent $G$1 in viscosity–shear-rate scaling, making the memory parameter experimentally inferable from oscillatory and flow data (Bhattacharyya et al., 2020).

A third formulation uses stochastic memory kernels. For Brownian motion in linear viscoelastic media, the power spectral density is proportional to the real part of a complex dynamic fluidity,

$G$2

where $G$3 is the complex modulus of a rheological analogue built from the host material in parallel with an inerter. In dense viscous fluids, hydrodynamic memory appears as a $G$4-order fractional derivative in the generalized Langevin equation; the corresponding rheological analogue is a dashpot, a fractional Scott–Blair element, and an inerter in parallel (Makris, 2021, Makris, 2 Feb 2026).

A fourth representation is internal-variable continuum theory. In memory-augmented nonlocal granular fluidity, a structural parameter $G$5 obeys

$G$6

while the nonlocal fluidity field $G$7 evolves through a diffusion-like equation supplemented by a structure-dependent suppression term. The model is explicitly designed so that $G$8 stores temporal history and $G$9 carries spatial cooperativity (Zolfaghari et al., 13 Feb 2026).

3. Physical mechanisms that encode memory

The proposed carriers of rheological memory vary widely across materials, but recent work converges on a limited set of mechanisms. In aging solids, one mechanism is the slow evolution of frictional contacts. The jerk-elasticity model interprets $q$0 as the effective frictional resistance associated with microstructural stick–slip at rough internal interfaces; logarithmic strengthening of contact area and bond strength then translates into logarithmic stress relaxation and power-law creep (Pandey, 4 Jun 2025).

In dense suspensions, memory is generated by the coexistence of two stress-activated microstructural processes: dynamic covalent bridging at lower stress and frictional contacts at higher stress. The former produces rheopectic viscosity growth, while the latter produces thixotropic weakening. Because different stress windows activate different mechanisms, the same suspension can store multiple, stress-selective memories and can be trained toward targeted viscosity and impact resistance (Kim et al., 12 Mar 2025).

In collagen networks, the mechanism is tied to the nonlinear strain-stiffening regime. A two-step strain protocol produces a Kovacs-like hump only when large strain generates strong negative normal stresses. In-situ imaging shows that relaxation after the first strain step involves boundary-localized compression near both plates, whereas the second step induces expansion localized mainly near the moving plate. The memory therefore correlates with negative normal stress and spatially distinct relaxation modes rather than with a homogeneous linear spectrum alone (Ghadai et al., 15 Dec 2025).

In sheared colloidal gels, memory is often associated with anisotropic or hierarchical restructuring. In boehmite gels, subcritical rejuvenation rates produce stronger gels with glassy-like spectra, larger terminal moduli, and yield strains that depend on the pre-shear rate; the interpretation is directed aging into larger, denser, anisotropic clusters, rationalized by a critical Mason number (Sudreau et al., 2021). In $q$1-carrageenan fluid gels, thermo-rheological memory arises during the sol–gel transition from competition between shear and interparticle adhesion, summarized by an Adhesion number; the cooling shear rate selects a characteristic particle size and thereby the later nonlinear response (Bauland et al., 17 Mar 2026). In carbon black gels, shear history is retained as a double-fractal microstructure whose cluster size, network dimension, and heterogeneity depend on the Mason number, and the post-shear elasticity follows that stored architecture (Bauland et al., 4 Aug 2025).

4. Experimental signatures and readout protocols

Rheological memory is operationally identified through protocols that write a state, allow partial relaxation or reorganization, and then read out a history-dependent response. The observables are diverse: peak times, hysteresis loop areas, crossover moduli, nonlinear overshoots, residual stresses, dissipation, and topology retention.

System	Writing protocol / memory carrier	Readout
Collagen network (Ghadai et al., 15 Dec 2025)	Two-step shear $q$2 in the nonlinear strain-stiffening regime	Non-monotonic $q$3, $q$4, correlated normal-stress dip
Granular packing (Murphy et al., 2019)	Step strain with waiting times and decompression/recompression	Turnaround times in nonmonotonic stress relaxation; multiple memories
Nanocolloidal soft glass (Chen et al., 2023)	Cyclic shear at $q$5	Minima in MSD, dissipation, and residual-stress change near $q$6
Dense suspension (Kim et al., 12 Mar 2025)	Stress-selected bridging or aggregate destruction	Rheopexic or thixotropic $q$7; trainable $q$8 and $q$9
$\kappa$0-carrageenan fluid gel (Bauland et al., 17 Mar 2026)	Cooling under shear $\kappa$1	Yield stress and $\kappa$2 overshoot encode $\kappa$3
Boehmite gel (Sudreau et al., 2021)	Shear rejuvenation at $\kappa$4 before cessation	Gel-like versus glassy-like spectra; $\kappa$5 and $\kappa$6 depend on $\kappa$7
Carbon black gel (Bauland et al., 4 Aug 2025)	Pre-shear at selected Mason number and time	Retained USAXS structure; $\kappa$8 set by double-fractal microstructure

These protocols show that memory can be read out from either transient or quasi-steady quantities. The collagen and granular studies use non-monotonic relaxation: a waiting time is encoded in the peak or turnaround time during later relaxation. The nanocolloidal glass uses a trained amplitude that is retrieved through minima in irreversible displacement and dissipation when the readout amplitude matches the training amplitude. The suspension study uses repeated impacts or shear sweeps to write viscosity states that later persist for tens to hundreds of seconds. The fluid-gel, boehmite, and carbon-black studies instead read out history from the rheology of a reformed or arrested material, showing that memory may be embedded in a prepared structure rather than only in a transient relaxation curve.

A plausible implication is that “memory” is best treated operationally rather than by a single constitutive signature. In some systems it is the location of a peak, in others the sign of a slope, the magnitude of an overshoot, or the persistence of a structural state after cessation of flow.

5. Structural, topological, and spatially nonlocal memory

Recent work increasingly locates rheological memory in spatial organization rather than only in scalar constitutive parameters. In carbon black gels, USAXS shows that the scattering curve after flow cessation is essentially the same as during flow at the final imposed shear rate, so the shear-formed structure is retained in the solid state. The gel is described by a double-fractal architecture with cluster scale $\kappa$9, network scale $$\dot{\sigma}_j(t)=E\,\dot{\varepsilon}(t)-\lambda(t)\,\varepsilon(t), \qquad \lambda(t)=\frac{1}{\xi+\theta t}.$$0, and fractal dimensions $$\dot{\sigma}_j(t)=E\,\dot{\varepsilon}(t)-\lambda(t)\,\varepsilon(t), \qquad \lambda(t)=\frac{1}{\xi+\theta t}.$$1 and $$\dot{\sigma}_j(t)=E\,\dot{\varepsilon}(t)-\lambda(t)\,\varepsilon(t), \qquad \lambda(t)=\frac{1}{\xi+\theta t}.$$2; within the reported regime, the modulus is determined by this microstructure through recently developed scaling laws (Bauland et al., 4 Aug 2025).

In nanocolloidal soft glasses under cyclic shear, memory is carried by a trained limit-cycle-like state. XPCS reveals that irreversible displacements decrease cycle by cycle toward enhanced reversibility, while the residual irreversibility behaves as a random walk. When the amplitude is changed after training, the microscopic mean-squared displacement and macroscopic dissipation become non-monotonic functions of the readout amplitude, with minima near the training amplitude. The memory is therefore distributed across both microstructural trajectories and macroscopic rheology (Chen et al., 2023).

A more striking separation of memory carriers appears in deformable-particle tilings. At large strains these systems enter a weakened regime with near-zero shear stress, elevated mechanical energy, and short-ranged tetratic order. Under strain reversal, the contact network evolves reversibly and recovers the initial topology, while particle-scale dynamics and energy remain irreversible. This identifies a specifically topological memory: the network of neighbors remembers the initial state even when the detailed microstate does not (Pasupalak et al., 2021).

Spatial cooperativity introduces another layer. The minimal nonlocal theory of thixotropic flow argues that a local memory variable is insufficient to reproduce nonlocal flow profiles, while a nonlocal fluidity field without memory cannot reproduce hysteresis or delayed yielding. Their coupling yields both history dependence and spatially cooperative flow. This suggests that in confined or heterogeneous geometries, the memory carrier is not a purely local state but a coupled field structure distributed across the material (Zolfaghari et al., 13 Feb 2026).

6. Relations to viscosity, yielding, and active matter

Rheological memory does not sit outside classical rheology; in several cases it reorganizes familiar limits. In jerk-elasticity, ordinary viscosity appears as a special case reached when $$\dot{\sigma}_j(t)=E\,\dot{\varepsilon}(t)-\lambda(t)\,\varepsilon(t), \qquad \lambda(t)=\frac{1}{\xi+\theta t}.$$3, giving the Maxwell creep law $$\dot{\sigma}_j(t)=E\,\dot{\varepsilon}(t)-\lambda(t)\,\varepsilon(t), \qquad \lambda(t)=\frac{1}{\xi+\theta t}.$$4; pure elasticity is recovered in another limit, while fractional viscoelasticity and logarithmic relaxation occupy the intermediate regime (Pandey, 4 Jun 2025). This places Newtonian viscosity as a short-memory limit of a broader aging rheology rather than as a fully separate constitutive class.

Conversely, some materials show strong memory without thixotropy. A solid-based model for simple yield-stress fluids combines a Zener-type viscoelastic solid, a nonlinear plastic flow rule, an evolution equation for back stress, and the Kröner–Lee decomposition. The internal variables $$\dot{\sigma}_j(t)=E\,\dot{\varepsilon}(t)-\lambda(t)\,\varepsilon(t), \qquad \lambda(t)=\frac{1}{\xi+\theta t}.$$5, $$\dot{\sigma}_j(t)=E\,\dot{\varepsilon}(t)-\lambda(t)\,\varepsilon(t), \qquad \lambda(t)=\frac{1}{\xi+\theta t}.$$6, and $$\dot{\sigma}_j(t)=E\,\dot{\varepsilon}(t)-\lambda(t)\,\varepsilon(t), \qquad \lambda(t)=\frac{1}{\xi+\theta t}.$$7 store deformation history, allowing the model to reproduce stress overshoot during start-up, a creep transition near the Herschel–Bulkley yield stress, and nonzero residual stress after cessation, all without invoking thixotropic structural rebuilding (Choi et al., 3 Apr 2026). This is an explicit counterexample to the idea that memory in yield-stress materials must be thixotropic.

The same broad principle extends to active matter. In skeletal muscle, work loops under simultaneous length change and time-varying stimulus are reconstructed by splicing together force–length loops measured at fixed stimuli and interpolating with an activation variable $$\dot{\sigma}_j(t)=E\,\dot{\varepsilon}(t)-\lambda(t)\,\varepsilon(t), \qquad \lambda(t)=\frac{1}{\xi+\theta t}.$$8. Here the material memory is dual: viscoelastic memory at fixed stimulus and activation memory associated with evolving internal contractile state. The framework explains when fixed-stimulus rheology suffices and when additional muscle-specific phenomena must be invoked (Nguyen et al., 2020).

Current theory therefore points toward a layered view. Rheological memory may reside in hereditary kernels, fractional operators, time-varying coefficients, kinematic hardening variables, contact topology, cyclic attractors, or retained gel microstructures. It may appear primarily in linear response, primarily in nonlinear response, or mainly after cessation of flow. The common feature is not a unique formula but an internal state—scalar, tensorial, topological, or spatially distributed—that preserves part of the loading or preparation history and re-enters later rheological response.