Nonlinear Viscoelasticity Research Articles

Machine learning-based constitutive modelling for material non-linearity: A review

Machine learning (ML) models are widely used across numerous scientific and engineering disciplines due to their exceptional performance, flexibility, prediction quality, and ability to handle highly complex problems if appropriate data are available. One example of such areas which has attracted a lot of attentions in the last couple of years is the integration of data-driven approaches in material modeling. There has been several successful researches in implementing ML-based constitutive models instead of classical phenomenological models for various materials, particularly those with non-linear mechanical behaviors. This review paper aims to systematically investigate the literature on ML-based constitutive models for materials and classify these models based on their suitability for material non-linearity including Non-linear elasticity (hyperelasticity), plasticity, visco-elasticity, and visco-plasticity. Furthermore, we also reviewed and compared the ML-based approaches that have been applied for architectured materials as these groups of materials are designed to represent specific material behaviors that might not exist in classical and conventional material categories. The other goal of this review paper is to provide initial steps in understanding of various ML-based approaches for material modeling, including artificial neural networks (ANN), Gaussian processes, random forests (RF), generated adversarial networks (GANs), support vector machines (SVM), different regression models and physics-informed neural networks (PINN). This paper also outlines different data collection methods, types of data, data processing approaches, the theoretical background of the ML models, advantage and limitations of various models, and potential future research directions. This comprehensive review will provide researchers with the knowledge necessary to develop high-fidelity, robust, adaptable, flexible, and accurate data-driven constitutive models for advanced materials.

Enhancing the viscoelastic properties of bacterial cellulose hydrogels through ultrasonic and enzymatic modification of xyloglucan

Bacterial cellulose (BC) hydrogels exhibit nanofibril porous network with good viscoelasticity for use as food ingredients and medical materials. Xyloglucan (XG), a hemicellulose with branching residues, can hybridize with BC to improve the hydrogel's extensibility. Thus, modifying the molecular structure of XG can fine-tune the viscoelastic properties of BC hydrogels. In this study, tamarind seed XG subjected to ultrasonic and enzymatic treatment was hybridized with BC to form composite materials. The results indicated that incorporating modified XG reduced the modulus and enhanced the viscous behaviour of BC to varying degrees. XG modified via ultrasonic treatment demonstrated a higher binding efficiency (19–22 %) with cellulose compared to enzymatically treated XG (11–13 %). The enzymatically treated XG improved the maximum elongation ratio to 57 %, but reduced the storage modulus to 30 kPa. Although ultrasonic-treated XG had a similar effect on the shear modulus, it had less impact on the extensibility of BC, with an elongation ratio of 38 %. Additionally, the incorporation of modified XG also regulated the nonlinear viscoelasticity of BC. These findings advance our understanding of the application of XG as a regulator of mechanical and rheological properties, broadening its utility in BC hydrogel formulations for the food industry and medical material development.

On the Piezomagnetism of Magnetoactive Elastomeric Cylinders in Uniform Magnetic Fields: Height Modulation in the Vicinity of an Operating Point by Time-Harmonic Fields.

Soft magnetoactive elastomers (MAEs) are currently considered to be promising materials for actuators in soft robotics. Magnetically controlled actuators often operate in the vicinity of a bias point. Their dynamic properties can be characterized by the piezomagnetic strain coefficient, which is a ratio of the time-harmonic strain amplitude to the corresponding magnetic field strength. Herein, the dynamic strain response of a family of MAE cylinders to the time-harmonic (frequency of 0.1-2.5 Hz) magnetic fields of varying amplitude (12.5 kA/m-62.5 kA/m), superimposed on different bias magnetic fields (25-127 kA/m), is systematically investigated for the first time. Strain measurements are based on optical imaging with sub-pixel resolution. It is found that the dynamic strain response of MAEs is considerably different from that in conventional magnetostrictive polymer composites (MPCs), and it cannot be described by the effective piezomagnetic constant from the quasi-static measurements. The obtained maximum values of the piezomagnetic strain coefficient (∼102 nm/A) are one to two orders of magnitude higher than in conventional MPCs, but there is a significant phase lag (35-60°) in the magnetostrictive response with respect to an alternating magnetic field. The experimental dependencies of the characteristics of the alternating strain on the amplitude of the alternating field, bias field, oscillation frequency, and aspect ratio of cylinders are given for several representative examples. It is hypothesized that the main cause of observed peculiarities is the non-linear viscoelasticity of these composite materials.

Investigations on the finite strain behavior of standard PVB: experiment and modeling

The numerical simulation of the residual load-bearing capacity of laminated glass is one of the most relevant topics of current research in structural glass engineering. The mechanical description of the behavior of the interlayer plays a crucial role. Several approaches and models already exist in automotive engineering and structural protection. Structural glass design assuming quasi-static loads misses a detailed model so far. The aim of this work is the mechanical description of the time-dependent behavior of Polyvinylbutyral (PVB) under large deformations considering quasi-static loading. The results of experiments on single-layer Standard PVB are presented, and a model of finite nonlinear viscoelasticity is derived. In particular, the formulation of phenomenologically motivated viscosity functions is described. Subsequently, the model parameters are calibrated on the experimental results, and the uniaxial stress state is simulated.

Nonlinear viscoelastic effect on the fatigue performance of wood tar-based rejuvenated asphalt

The nonlinear viscoelasticity (NLVE) effects of rejuvenated asphalt are often neglected during fatigue degradation, potentially leading to misjudgment of its true fatigue performance. To more accurately quantify the fatigue life of rejuvenated asphalt under NLVE effects, frequency sweep (FS), incremental stress sweep (ISS), and linear amplitude sweep (LAS) tests were used to study two types of rejuvenated asphalt. FS and ISS tests respectively obtained the dynamic modulus |G*|LVE and |G*|NLVE representing the linear viscoelasticity (LVE) and NLVE of rejuvenated asphalt, and incorporated them into the simplified viscoelastic continuum damage (S-VECD) model to establish the fatigue failure criterion and predict the fatigue life of rejuvenated asphalt under high strain levels in LAS tests. The findings indicate that the NLVE-based-S-VECD model quantifies fatigue damage in rejuvenated asphalt lower than the LVE model, and the fatigue life of rejuvenated asphalt increases significantly after considering the NLVE effect. Besides, the failure difference (∆GR) and failure area difference (∆A) defined during the modeling process explain the impact of NLVE on the fatigue failure criterion average pseudo strain energy release rate (GR) of rejuvenated asphalt, validating the necessity of incorporating NLVE into the fatigue performance analysis of rejuvenated asphalt. Therefore, it's necessary to account for the impact of NLVE effect on the fatigue performance of rejuvenated asphalt in engineering practice, which will contribute to the development and application of rejuvenator.

Universal phase-field mixture representation of thermodynamics and shock-wave mechanics in porous soft biologic continua.

A continuum mixture theory is formulated for large deformations, thermal effects, phase interactions, and degradation of soft biologic tissues suitable at high pressures and low to very high strain rates. Tissues consist of one or more solid and fluid phases and can demonstrate nonlinear anisotropic elastic, viscoelastic, thermoelastic, and poroelastic physics. Under extreme deformations or shock loading, tissues may fracture, tear, or rupture. Existing models do not account for all physics simultaneously, and most poromechanics and soft-tissue models assume incompressibility of some or all constituents, generally inappropriate for modeling shock waves or extreme compressions. Motivated by these prior limitations, a thermodynamically consistent formulation that combines a continuum theory of mixtures, compressible nonlinear anisotropic thermoelasticity, viscoelasticity, and phase-field mechanics of fracture is constructed to resolve the pertinent physics. A metric tensor of generalized Finsler space supplies geometric insight on effects of rearrangements of microstructure, for example degradation, growth, and remodeling. Shocks are modeled as singular surfaces. Hugoniot states and shock decay are analyzed: Solutions account for concurrent viscoelasticity, fracture, and interphase momentum and energy exchange not all contained in previous analyses. Suitability of the framework for representing blood, skeletal muscle, and liver is demonstrated by agreement with experimental data and observations across a range of loading rates and pressures. Insight into previously unresolved physics is obtained, for example importance of rate sensitivity of damage and quantification of effects of dissipation from viscoelasticity and phase interactions on shock decay.

Potential of polymer’s viscosity and viscoelasticity for accessible oil recovery during low salinity polymer flooding in heterogeneous carbonates

Despite the usual improvement in sweep efficiency during polymer flooding, the polymer’s viscoelastic characteristics were reported to contribute to residual oil recovery in sandstone reservoirs at high flux conditions. In carbonate reservoirs, besides high-salinity, high-temperature, and non-water-wet nature, heterogeneity is a key variable that may impair the viscoelastic fluid flow in the porous media and recovery performance. However, no attempts were made to study the polymer’s viscoelastic influence on residual oil recovery in heterogeneous carbonate formations and the objective of this paper is to systematically examine the recovery performance of high molecular weight viscoelastic sulfonated polymer (AN-125-SH) in comparison to viscous xanthan gum polymer with the low salinity injection water of 5957 ppm total dissolved solids (TDS) in heterogeneous outcrop carbonate rocks at fluxes ranging from shear thinning to shear thickening using the combination of bulk shear rheometry, NMR, single-phase in-situ studies and two-phase core flooding experiments.Nuclear magnetic resonance (NMR) imaging ensured that the chosen cores are characterized with similar pore-size distribution. The steady-shear rheological experiments showed that 2500 ppm AN-125-SH polymer exhibits a lower viscosity at the lower shear rate and an early shear thickening (indicating higher non-linear viscoelasticity) than 2000 ppm xanthan gum. Then single-phase flow experiments are conducted using two polymer systems to determine the desired flux rates before the eventual multi-rate two-phase core flood experiments. Core flooding results revealed following an extensive secondary water flooding, at the shear thinning flux of 4 ft/day, 2000 ppm high viscous xanthan gum injection leads to an incremental recovery factor of 8.3 % with the generated pressure gradient of 20.48 psi/ft compared to 3.6 % recovery obtained using 2500 ppm AN 125-SH polymer with a pressure gradient of 16.2 psi/ft. However, at shear thickening fluxes of 20 ft/day and 30.5 ft/day, the higher elastic polymer enhanced the recovery by 3.3 % and 1.2 %, respectively. Conversely, the less elastic xanthan gum showed minimal to no incremental recovery at these fluxes.The novel findings of this work convey that the viscosity of xanthan gum provides a comparative advantage over the viscoelasticity of synthetic sulfonated polymers on oil recovery at relatively low fluxes in the studied heterogeneous formation at low salinity conditions.

On more insightful dimensionless numbers for computational viscoelastic rheology

Abrupt contraction flows involving viscoelastic fluids represent a longstanding computational challenge within the field of non-Newtonian fluid mechanics. Despite the apparent simplicity of the geometry, these flows have given rise to intricate discussions in the study of viscoelastic phenomena. This study aims to re-examine the numerical solutions for flows through abrupt contractions, offering a fresh interpretation through the lens of reformulated dimensionless numbers. These numbers are designed to consider the characteristic shear rate of the problem, providing a more comprehensive understanding of the underlying dynamics.When investigating models with intermediate levels of complexity, such as the Giesekus and Phan-Thien-Tanner constitutive equations, the usual comparison with the corresponding Oldroyd-B model becomes inadequate because it tends to rely on the nominal relaxation time (λ) and the nominal total viscosity (η) instead of their effective counterparts when defining the Reynolds number (Re), the Weissenberg number (Wi) and the ratio of solvent to total viscosities (β) (β plays a role only in rheological models involving a solvent contribution). If these dimensionless numbers are tailored to account for the characteristic shear rate specific to the problem under investigation, the choice of the corresponding Oldroyd-B flow, at the adequate values of Re, Wi, and β allows for significantly better quantification of the correct effects of nonlinear viscoelasticity of the original model.We show the conventional approach tends to overemphasize the role of the nonlinear parameter in nonlinear constitutive equations, like the Giesekus and PTT models, when examining standard abrupt contraction flow outputs such as the Couette correction and vortex size. This overestimation occurs because the conventional method does not allow the Reynolds and Weissenberg numbers (and possibly β) to carry the portion of the nonlinear effect that can potentially be captured by the linear Oldroyd-B model through the use of characteristic shear rate-based values. We believe the present approach provides a better perspective of the role played by the nonlinear parameter and its extension to more general flows is also discussed.

A variational approach to frame-indifferent quasistatic viscoelasticity of rate type.

The three-dimensional dynamical model for nonlinear viscoelasticity of strain-rate type is investigated in a quasistatic setting under the assumption of higher-order regularity of the deformation, which in the literature is referred to as the case of non-simple materials. The existence of weak solutions is proven using a time-discretization technique while respecting the condition of dynamical frame indifference. Some observations on frame indifference for strain-rate-type stresses are made, and corrections are proposed for some related work in the literature. Finally, a counterexample is given to show that the assumed higher-order regularity is necessary in order to obtain the required compactness.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Data‐driven variational method for discrepancy modeling: Dynamics with small‐strain nonlinear elasticity and viscoelasticity

AbstractThe effective inclusion of a priori knowledge when embedding known data in physics‐based models of dynamical systems can ensure that the reconstructed model respects physical principles, while simultaneously improving the accuracy of the solution in the previously unseen regions of state space. This paper presents a physics‐constrained data‐driven discrepancy modeling method that variationally embeds known data in the modeling framework. The hierarchical structure of the method yields fine scale variational equations that facilitate the derivation of residuals which are comprised of the first‐principles theory and sensor‐based data from the dynamical system. The embedding of the sensor data via residual terms leads to discrepancy‐informed closure models that yield a method which is driven not only by boundary and initial conditions, but also by measurements that are taken at only a few observation points in the target system. Specifically, the data‐embedding term serves as residual‐based least‐squares loss function, thus retaining variational consistency. Another important relation arises from the interpretation of the stabilization tensor as a kernel function, thereby incorporating a priori knowledge of the problem and adding computational intelligence to the modeling framework. Numerical test cases show that when known data is taken into account, the data driven variational (DDV) method can correctly predict the system response in the presence of several types of discrepancies. Specifically, the damped solution and correct energy time histories are recovered by including known data in the undamped situation. Morlet wavelet analyses reveal that the surrogate problem with embedded data recovers the fundamental frequency band of the target system. The enhanced stability and accuracy of the DDV method is manifested via reconstructed displacement and velocity fields that yield time histories of strain and kinetic energies which match the target systems. The proposed DDV method also serves as a procedure for restoring eigenvalues and eigenvectors of a deficient dynamical system when known data is taken into account, as shown in the numerical test cases presented here.

Yielding behaviour of chemically treated Pseudomonas fluorescens biofilms

The mechanics of biofilms are intrinsically shaped by their physicochemical environment. By understanding the influence of the extracellular matrix composition, pH and elevated levels of cationic species on the biofilm rheology, novel living materials with tuned properties can be formulated. In this study, we examine the role of a chaotropic agent (urea), two divalent cations and distilled deionized water on the nonlinear viscoelasticity of a model biofilm Pseudomonas fluorescens. The structural breakdown of each biofilm is quantified using tools of non-linear rheology. Our findings reveal that urea induced a softening response, and displayed strain overshoots comparable to distilled deionized water, without altering the microstructural packing fraction and macroscale morphology. The absorption of divalent ferrous and calcium cations into the biofilm matrix resulted in stiffening and a reduction in normalized elastic energy dissipation, accompanied by macroscale morphological wrinkling and moderate increases in the packing fraction. Notably, ferrous ions induced a predominance of rate dependent yielding, whereas the calcium ions resulted in equal contribution from both rate and strain dependent yielding and structural breakdown of the biofilms. Together, these results indicate that strain rate increasingly becomes an important factor controlling biofilm fluidity with cation-induced biofilm stiffening. The finding can help inform effective biofilm removal protocols and in development of bio-inks for additive manufacturing of biofilm derived materials.

On the Schapery nonlinear viscoelastic model: A review

Among the various models presented so far to describe the nonlinear viscoelastic behavior of materials, the model proposed by Schapery in the 1960s stands out for its wide acceptance and significant contributions to the field. Through a detailed inspection of the main articles presented by Schapery and other related studies, the present paper reviews the Schapery nonlinear viscoelastic model. One-dimensional and three-dimensional formulations of the model for both isotropic and anisotropic materials, along with model accuracy, are discussed. The present article highlights the challenges of material characterization methods, such as creep-recovery testing, stress relaxation testing, and dynamic mechanical analysis. The article also addresses the issues of characterizing the material's behavior under non-recoverable strain and varying temperatures. Finally, the future research needs in this field are outlined, emphasizing the potential for further advancements in our understanding of nonlinear viscoelasticity.

Influences of non-rubber components on the molecular network and viscoelasticity of natural rubber gum

The molecular network of natural rubber (NR) gum is determined by the interparticle diffusion when NR latex coagulates. However, the roles played by the non-rubber components (NRC) species on the heterogeneous network structure and on the viscoelasticity of NR gum have not been throughly investigated. Here, the molecular relaxation was studied by linear rheology. In addition, the inter- and intra-cycle nonlinearities under large amplitude oscillatory shear (LAOS) were studied for gums with extracted and added NRC. It was found that the high molecular weight free protein restricts the interdiffusion and improves the entanglement density, while the added lipid induces homogeneous distribution of NRC. The physical associations coming from free proteins increase the segment friction and retard the terminal relaxation. Onset of the strain softening was found to be induced by entangled network density. Both the elastic and viscous LAOS nonlinearities can be regulated by adjusting the NRC species. This work sheds light on the contributions of major NRC species on the network structure and nonlinear viscoelasticity of NR gum.

Effects of filler content on non-linear viscoelasticity of high fluorine content fluoroelastomer mixture investigated through large amplitude oscillatory shear tests

Effects of filler content on non-linear viscoelasticity of high fluorine content fluoroelastomer mixture investigated through large amplitude oscillatory shear tests

Linking Chemical Structure to the Linear and Nonlinear Properties of Asphalt Binders

The study described in this paper aims to establish the relationship between the chemical composition and rheological properties of asphalt binders in both the linear viscoelastic (LVE) and nonlinear viscoelastic (NLVE) domains. Two asphalt binders with different penetration grades were subjected to four distinct aging treatments of varying severity, resulting in changes in their chemical fingerprints. Chemical characteristics of the asphalt binders were analyzed using thin-layer chromatography (TLC), Fourier Transform infrared spectroscopy (FTIR), and gel permeation chromatography (GPC). Frequency sweep tests were conducted to evaluate the LVE behavior of the asphalt binders, while multiple stress creep recovery (MSCR) tests were employed to assess their NLVE properties. With regard to the LVE properties, rheological index ( R) and zero shear viscosity (ZSV) derived from the Christensen-Anderson-Marasteanu (CAM) model exhibited high correlations with FTIR and molecular weight distribution (MWD) parameters. Additionally, the polydispersity index (PDI) displayed a stronger correlation with LVE-based parameters. Findings from the MSCR tests revealed that the sensitivity of the percentage recovery ( %R) to chemical composition, as opposed to nonrecoverable creep compliance ( Jnr), can be largely attributed to the degree of nonlinearity. Furthermore, it was observed that lower molecular weight molecules exert a greater influence on %R in the nonlinear domain. Finally, it was found that Jnrslope is a more reliable parameter than Jnrdiff for assessing the effects of chemical makeup on stress and temperature sensitivity.

Nonlinear viscoelasticity of Melaleuca alternifolia essential oil Pickering emulsion stabilized with cellulose nanofibrils

Pickering emulsions are suitable technological approaches to stabilize the bioactive compounds of essential oils (EOs) with economic interests for the pharmaceutical, cosmetic and food industries using atoxic and renewable stabilizers, such as cellulose nanostructures. The Melaleuca alternifolia essential oil (MaEO) is recognized for its pronounced antimicrobial performance, which has attracted your interest in developing eco-friendly antiseptic products. In this contribution, we report the nonlinear viscoelasticity behavior of MaEO-in-water Pickering emulsions stabilized by cellulose nanofibrils (CNF) using Large Amplitude Oscillatory Shear (LAOS) tests. The rheological characterizations by Fourier Transform (FT) rheology spectroscopy and Lissajous plots enabled the correlation of the viscous and elastic behaviors with the physical processes involving the deformation of the emulsion. Also, the internal structure of the emulsion was elucidated by fluorescence optical microscopy. Several findings relative to the effects of the shear deformation amplitude over the nonlinear viscoelasticity of the emulsified system are reported here, which are essential to the industrial processes used to develop technological products.

Erratum: "A Numerical Scheme for Anisotropic Reactive Nonlinear Viscoelasticity" [ASME J. Biomech Eng., 2023, 145(1), p. 011004; DOI: 10.1115/1.4054983

In this erratum, we correct a mistake in a subcomponent of the numerical algorithm proposed in our recent study for modeling anisotropic reactive nonlinear viscoelasticity (doi:10.1115/1.4054983), for the special case where multiple weak bond families may be recruited with loading. This correction overcomes a nonphysical response noted under uni-axial cyclical loading.

Nonlinear dynamics and forced vibrations of simply-supported fractional viscoelastic microbeams using a fractional differential quadrature method

In this paper, free and forced nonlinear vibrations of fractional viscoelastic microbeams are modeled based on Euler-Bernoulli theory, the Von Karman’s nonlinear strain relations, the modified couple stress theory (MCST), and the fractional Kelvin-Voigt viscoelastic model. In the present work, the nonlinear-fractional order governing equations are discretized in the space domain by Galerkin’s method. Two different approaches are introduced to solve the resulting nonlinear fractional-order Duffing equation in the time domain. In the first approach, a time marching fractional finite difference method is presented to compute the transient time response starting from the initial conditions. This approach is time consuming if the frequency-amplitude curves of steady state response are required since it provides just one point on the frequency-amplitude curve. More importantly, this approach usually converges only for the stable branch with smallest amplitude of frequency-amplitude curve. In the second approach, we introduce a novel fractional differential quadrature method (FDQM) to discretize the fractional Duffing equation and apply a pseudo-arc length algorithm to directly construct the frequency-amplitude curves. Effects of the fractional-order, linear and nonlinear viscoelasticity coefficients, viscous damping parameter, microstructure parameters, and the micro-beam thickness on the nonlinear dynamics of the viscoelastic micro-beam are analyzed numerically. Numerical results show that each of these parameters can change natural frequency and/or the damping behavior of the structure. The present model can be used for designing and analyzing the nonlinear microstructure viscoelastic beam under dynamic loads.

The relationship between nonlinear viscoelasticity and baking performance in low-saturated puff pastry margarine

The oil phase obtained by blending and oleogel methods has potential for the production of non‑hydrogenated and low-saturated puff pastry margarine, thereby reducing intakes of both types of dietary fat. The crystal form, microstructure, rheology, and baking applications of puff pastry margarines prepared with anhydrous milk fat (AMF)/palm stearin (POs), POs/palm oil (PO), beef tallow (BT)/PO, or AMF/POs/diacetyl tartaric acid ester of mono(di)glycerides (DATEM) oleogels were investigated using X-ray scattering, polarized light microscope, and rheometer, respectively. All margarines exhibited β’-form crystal and strongly viscoelastic at low strain. With the addition of DATEM oleogel, their crystal microstructure became more uniform and finer, and the croissants were less hard (1690) and chewiness (160). The chewiness of croissants produced using the margarines was significantly improved with POs content. The theoretical basis for preparation and application in non‑hydrogenated and low-saturated puff pastry margarine was provided in the present study.

Acid-induced gels from mixtures of micellar casein and pea protein: Effect of protein ratio and preheating route

Enhancing the sustainability of dairy products through the partial substitution of dairy proteins with plant proteins requires exploring formulation and processing strategies. This study investigates the gluconic-δ-lactone (GDL)-induced hybrid gels from commercial micellar casein isolate (M) and pea protein isolate (P) dispersions (5% w/w protein content). Variations in the M/P ratios (3:1, 2:2, and 1:3) and preheating routes (Route 1: preheating M and P dispersions together; Route 2: preheating them separately) impacted the physical/conformational properties of protein dispersions as well as the rheological/structural features of resulting gels. Small and large amplitude oscillatory shear (SAOS and LAOS) tests revealed that lower M/P ratio led to earlier gelling points and increased the stiffness and elasticity of hybrid gels in the linear viscoelastic (LVE) region. All gels transitioned from elastic to plastic behavior in the non-linear viscoelastic (NLVE) region, with lower M/P ratio showing reduced stretchability and faster structural breakdown. Regardless of routes, the preheating step (95 °C, 30 min) disintegrated inherent aggregates/agglomerates in these commercial protein ingredients, leading to smaller particle size, but higher protein solubility, surface hydrophobicity, and |ζ-potential|. Pea proteins formed soluble aggregates during preheating, but the presence of micellar caseins (Route 1) hindered this process. Consequently, mixtures with lower M/P ratio and from preheating route 2, possessed higher quantities of pea protein soluble aggregates, forming a compact gel network with low water mobility, as observed by CLSM, STED, and LF-NMR. These findings show the potential for using pea proteins in acid-induced gel foods like yogurt and paneer-type cheeses.