Nonlinear Behavior of Greases: Experiment, Identification, Model

1. Introduction

Greases are widely used in tribological systems to reduce the negative effects of friction at the interfaces of metal-to-metal, metal-to-polymer, metal-to-composite, etc. The mechanism of converting external friction between bodies into internal friction in the grease is used [1]. This significantly reduces wear. Approximately 80% of equipment downtime is associated with failure of parts and components due to friction and wear [2]. More than 30% of the energy consumed is used to reduce friction and wear of contact units [3]. Selection of greases for temperature and power operating conditions and functional features of products maybe can significantly increase the working life of structures and reduce energy consumption. The creation of highly efficient greases for difficult operating conditions is also one way to solve the problem [4].

The research trend toward obtaining universal multi-purpose greases that would be effective over a wide range of operating conditions existed in the 1970s and 1980s [5]. This trend has lost its relevance at the present time. This is due to the diversity of the greases requirements set, which is formed in relation to the application scope, operating conditions and functionality of the structures and systems in which they are used. The development of greases or their selection for specific applications or for temperature-force and/or temperature-time operating conditions is relevant now.

The composition of the grease and the technology of its creation have a significant impact on the material behavior under different operating conditions [6]. The technological process of grease production is related. Temperature, force loads and time have a significant impact on the microstructure, physicomechanical and physicochemical properties of the final product. For example, in the production of polyurea greases, static heat treatment results in the formation of nanotubular structures, while heat treatment with stirring creates twisted fibers [6]. The grease quality may also depend on other technological process components [7]: the environment, instability of processing modes, sensor drift, human factor, etc. The introduction of additional operations (stages) can facilitate the development of highly efficient greases, especially with wide temperature adaptation [6, 8]. For example, introducing a mechanical shear into the flow sheet results in the evolution of twisted fibers into long-pitch twisted fibers [6]. Changing the production process and introducing additional operations can be aimed at obtaining functional greases for the required operating conditions or to obtain the required lubrication mechanism: boundary lubrication, lubrication during multiple start-stop cycles and high sliding speeds. Ultrasonication, homogenizing, magnetic force agitation, probe sonication, ball milling, high shear mixing and thermal heating can be used to obtain a homogeneous structure of a multicomponent mixture [6, 9]. The flow sheet can influence the grease quality and rheology. Rheological properties are an important factor in analyzing the grease quality. Grease rheology must be monitored and adjusted at all stages of technological process to obtain a high-quality batch of products [7].

The strong dependence of the grease properties on the base oil and thickener has been noted by many scientific groups [8, 10, 11, 12]. The type of base oil and thickener during the production process can form structures of different geometric configurations, allowing for different processes of lubricant release during deformation or operation of systems and structures [8]. The percentage content of base oil can reach 95% and have a significant impact on the mechanical, thermal, frictional and rheological properties of the grease [10]. Greases exhibit different degradation mechanisms due to weakening or destruction of aliphatic chains of the thickener when operating in aggressive environments [12]. The thickener type also has a significant impact on the grease rheology. For example, the use of calcium stearate allows one to obtain the lowest dynamic viscosity at negative temperatures, that is, it allows one to obtain rheologically stable lubricants [11].

Greases are complex multicomponent fluids that exhibit a transition from a solid to a liquid phase under load [13]. Approximately 75% of the global lubricant market is based on the use of lithium soap as a thickener at present [10]. The market’s remaining part is associated with the production of other soap, inorganic and silicone lubricants, including bioorganic ones. Fully biodegradable lubricants using palm oil as a base and renewable biothickeners also exist [14]. The introduction of different types and nature of the additives, supplements and components to improve the materials tribological and mechanical properties is relevant in connection with the increasing requirements for environmental friendliness and functional characteristics of greases [14, 15].

The integration of various nano-additives into the consistent greases composition is associated with the problem of its functionality when operating under extreme conditions [16]: high temperatures, high pressures, operating speeds, etc. Modification of greases is also associated with an increase in the service life of both the materials themselves and the structures and systems in which they are used [17]. Modification of greases using special chemical elements (additives), nano-additives or particles is aimed at improving one or more functional properties of materials and their adaptation to specific operating conditions [18, 19]. Nano-additive of Li2O, Fe3O4 and P2O5 to the grease made it possible to obtain samples resistant to deformation under high-speed shear [20]. Modified carbon quantum dots (CQDs) with different alkyl chain lengths can be used to create hybrid grease [16]. Functional materials for specific operating conditions, especially from a tribological point of view, can be created by selecting the length of the alkyl chain. Nanomagnesium hydroxysilicate (nano-MSH) powders are considered as grease components [21]. Nano-MSH is a silicate mineral with a unique crystalline structure. It has the ability to self-repair worn metal surfaces. Layered rare earth sulfate is a new type of functional inorganic compound that has attracted more and more attention recently. The use of rare earth element sulfates as nano-additives in greases is promising [22]. The use of carbon nanocomposite obtained from metal-organic frameworks (MOFs) as a nano-additive is also considered [2]. The expansion of the additives and components range as modifiers is associated not only with an increase in temperature and force operating conditions and requirements for the materials resource, but also with the need to move from non-renewable to renewable resources.

Molybdenum disulfide is a traditional solid lubricant and performance-enhancing component of greases [22]. But reserves of molybdenum disulfide are limited. This leads to the need to search for alternative materials, including renewable ones, both as additives, and as a base and thickener. For example, the use of biochar as a grease additive has potential to improve rheological and tribological performance [18, 23]. The percentage content, pyrolysis temperatures and raw materials for biochar production affect the greases properties. Technologies for obtaining bio-additives influence the final rheological, thermomechanical and operational properties of greases [23].

Ionic liquids are salts that are liquid at temperatures below 100°C. They are a chemically universal component that allows the design of different cations and anions combinations, as well as structural modifications. The use of waste ionic liquids as additives in consistent greases improves the frictional properties of materials and reduces wear [24]. Greases with ionic liquids can find wide application in friction unit designs where retention and replenishment of lubricant is not available.

The experience of modifying greases using additives in the form of graphene, graphene oxide (GO) and reduced graphene oxide (rGO) is reviewed in [25]. It is noted that when introducing graphene additives into the grease composition, problems with dispersion are observed. Mechanical stirring, samples heating or use of chemical agents is used to improve dispersion. The graphene oxide dispersion into grease is carried out using ultrasound and does not require additional technological operations. Research has shown the versatility and effectiveness of graphene oxide as a grease additive. GO allows to improve the materials’ performance properties. Reduced graphene oxide has exceptional frictional advantages. However, current synthesis methods allow rGO to be obtained with a significant defect percentage. It is noted that rGO can become an additive for the creation of energy-efficient greases employed in harsh operating conditions. However, extensive research into optimal rGO mixing methods and bulk synthesis routes is required. Graphene can improve the greases frictional properties, but the concentration and type of component to optimize the material properties remains unclear [25, 26]. The technological production process result largely depends on the graphene type, the method of its introduction into the grease composition and the thickener type. The possibility of introducing two-component graphene-based composites into greases is also being considered. A two-dimensional (2D) layered binary composite (GO/Talc) with a ratio of 1:2 is considered in [27]. The composite is a combination of graphene oxide (GO) with silica talc via a silane coupling agent 3-aminopropyltriethoxysilane (KH-550). Adding 0.1 wt.% GO/Talc to the grease composition allows to significantly reduce the friction coefficient and wear by 50 and 24.6%, respectively.

Analysis of the use of chemical compounds of different nature as additives is one of the research areas. β-Tricalcium phosphate (β-TCP) is a potential grease additive and has excellent biodegradability, biocompatibility and thermal stability. Modification of the grease with ultra-dispersed β-TCP is considered in the study [15]. This is done to improve the tribological characteristics of the material, including at high operating temperatures. A reduction in the friction coefficient by more than 20% and wear by more than 80% were achieved by adding β-TCP to the grease base composition. Zinc dialkyl dithiophosphate (ZDDP) is a long-used additive that allows obtaining a stable protective tribofilm on metal surfaces [28]. The use of the additive is restrained by a number of disadvantages, such as: harmful emissions when in contact with other environments, increased friction losses and micro-pitting corrosion in highly loaded systems. But many studies have been done with oil lubricants. The excellent antiwear properties of ZDDP are noted in the case of greases. Tribofilm does not form under low loads. But this does not prevent the grease from showing good antiwear properties. More severe operating conditions are required for the tribofilm formation than with oil lubricants. Hexagonal boron nitride (h-BN) is a solid lubricant and can be used as a modifier to improve the greases tribological properties. The use of different percentage ratios of h-BN from 1 to 10% as an additive for base lubricants of different types was considered in [1]. It has been established that big and small content of h-BN additive leads to deterioration of greases friction properties. The optimal percentage of h-BN content is 3%. But the improvement level of tribological characteristics depends on the base lubricant type. This is because the base lubricant type affects the dispersion of the additive and its ability to form protective surface layers.

Remetallizing powders of low-modulus metals, their oxides, hydroxides or salts are used as additives in greases [29]. The introduction of metal powders into the grease composition has one or two mechanisms of influence on friction [30]: One is dissolution in the friction process with a transfer selective mode of particles into thin metal films with a high degree of adhesion and a low friction coefficient; second is distribution of particles over the surface with microrelief smoothing. Metal oxide nanoparticles, such as CeO2 and CuO, exhibit unique interfacial interactions and effects when incorporated into the grease structure compared to traditional additives [31]. The introduction of 3 and 2% zinc and cadmium powder, respectively, into the grease composition made it possible to reduce the friction force by 5 times [29]. The use of dispersed serpentinite powder based on magnesium hydrosilicate as a functional additive to grease is discussed in [30]. It was found that at a concentration of 0.5%, the friction force was reduced by 69%, but when the concentration was increased to more than 0.6% the friction force was not reduced as effectively. The use of metal powders as additives in greases can have other effects as well. For example, the introduction of Fe3O4/WS2 and CuO/WS2 into the grease composition leads to a reduction in the probability of spark discharge in the electrical machines bearings [32].

Polymer additives are often used to improve the properties of greases [33]. Polymer additives can change the grease viscosity, reduce the resistance on the friction units mating surfaces and the friction coefficient, and also increase the service life and materials storage [34]. Ultra-dispersive polytetrafluoroethylene (PTFE) or nano-PTFE has been widely used as a nano-additive [17]. Nano-PTFE additives are used as a lubricant thickener for long service lives, including under difficult conditions [35]. Particle morphology, size and distribution of PTFE particle parameters influence dispersion with the material and the grease properties. The introduction of 27–31% PTFE additive allows obtaining stable greases with low liquid separation and acceptable viscosity [35]. The effect of nano-PTFE additive on grease is still not fully understood. The influence of particle size and geometry, as well as additive concentration on the physical, mechanical, rheological, frictional and operational properties of the grease, is being investigated [36, 37].

Greases are multicomponent complex media that include elements and materials of different nature. Complex nonlinear (rheological) behavior of greases is observed in laboratory studies and during operation in real structures [12, 38]. The description of the greases behavior and their numerical modeling as an elastic or slightly compressible body is not effective and introduces significant error into digital models of research objects [39, 40]. The study of rheology, both of the greases themselves and of the components that make up their composition, is an important factor for the development of the industry and increasing the service life and efficiency of materials [5, 7, 12, 41]. More effective methods for predicting the impact of grease properties on the operation of research objects are required [12]. Numerical models (digital analogs) of greases can become an effective tool for behavior predictive analysis of both the materials themselves and the systems and structures in which they are used [42]. The study of rheological effects on the grease behavior associated with changes in viscosity depending on thermal force loads and non-Newtonian effects is necessary [41]. Empirical studies aimed at determining viscosity, storage modulus, loss modulus and loss tangent are required to effectively describe the grease rheology [20]. The study of grease rheology will allow the formation of materials service life data [42, 43, 44]. Empirical studies and numerical modeling make it possible to determine the greases resource potential under conditions close to the real operating conditions of highly loaded tribosystems [45]. The diffusive wave spectroscopy (DWS) method is positioned as a means for materials rheology effective study [46]. Spectroscopy allows for non-contact investigation of the sample, which eliminates the possibility of sample destruction. This type of technology allows for the evaluation of rheological materials properties over a wider range of temperatures and frequencies compared to analogs. But rotational rheometers have become the most widespread in Russia and the world at the present time [17, 38, 47, 48]. The effectiveness of using a rotational rheometer in analyzing lubricant rheology has been confirmed by a number of studies [17, 20, 42]. Methods and approaches to the study of lubricants using widely available experimental equipment should be replicated. The results of empirical studies form the basis for materials behavior’s mathematical modeling.

The multiparameter Carreau-Yasuda model is often used to describe the lubricants’ nonlinear behavior [20, 49, 50]. The model allows one to determine the critical shear rate and the shear stress magnitude when a Newtonian fluid transforms into a non-Newtonian fluid [49]. The Carreau-Yasuda model correlates the complex viscosity with zero complex viscosity, relaxation time, power-law index and the width of the transition region between Newtonian and power-law behavior [50]. The model establishes the dependence of the material complex viscosity on the shear frequency. The relaxation time can be described as the ratio of the zero complex viscosity to the shear stress at the transition between the Newtonian and non-Newtonian fluid states. Lubricant modeling as a viscoelastic fluid based on the Herschel-Bulkley model, which is used in hydrodynamics to describe non-Newtonian fluids, is also considered [51]. The model is related to the materials yield stress. The model includes the flow phenomenon, where a fluid becomes thicker or fluider under load. A new rheological model using an algebraic equation with four unknown parameters based on structure parameters is formulated in [52]. The model is sufficiently functional to describe the shear loads’ influence on the greases behavior, including creep. Selection of the dynamic equilibrium structure parameter allows the model to be reduced to the Bingham or Herschel-Bulkley model with a yield strength dependent on the shear rate, or to the Newton model under steady-state conditions. Thixotropic models based on time and stress/strain rate are also considered. A modification of the model [52] based on deformation is proposed in the article [13]. A comparative analysis between the Herschel-Bulkley fluid model and the Bingham model was carried out in [53], and the limitations of the models were established. Bingham’s rheology model lacks physical meaning, which may lead to unphysical results in viscous material modeling [52]. Modeling of the grease viscoelastic behavior based on a Reynolds equation modification that allows describing the materials behavior using non-Maxwellian equations, including taking into account the polymer additives viscosity, is considered [41]. Models of materials viscoelastic behavior are used to describe the greases nonlinear behavior [54]: Maxwell, Kelvin-Voigt and Maxwell-Wiechert. Maxwell’s model can adequately demonstrate the stress relaxation phenomenon of linear viscoelastic materials, but does not take into account creep and recovery. The Maxwell or Kelvin-Voigt models allow only a qualitative description of materials due to their simplicity. The Maxwell-Wiechert model is more complex and allows one to describe the real viscoelastic behavior of materials with more than one relaxation time. A semi-analytical model based on the numerical equivalents method of inclusions and fast Fourier transforms to grease described as a Maxwell body is also considered [55].

Rheological effects are characterized by the dependence of the shear rate on the shear stress or apparent viscosity [56]. The rheological governing equation must describe the regions below and above the yield strength. Complex models with a large number of correction parameters can provide a fairly accurate description of experimental data. But this makes it difficult to find experimental certainty about these parameters. This fact indicates the need for a correct selection of models with satisfactory correspondence to experimental data, including for describing the materials behavior in engineering problems. Rheological models are constructed for stress rather than shear rate and often include a Maxwell-type viscoelastic element. The Maxwell model is suitable for describing highly viscous materials operating under long periods of loading [57]. The model is well suited to changes in material viscosity over time or under different temperature and force operating conditions. Many modern materials have a viscoelastic pattern. The determination of materials viscoelastic properties and their mathematical description play an important role in the mechanical systems optimal design. Classical viscoelastic models, constructed using linear springs and linear shock absorbers, are the most widely used at present [58]: Maxwell, Kelvin, Burger, generalized Maxwell and Kelvin models, and so on. The precise correspondence of material constants to physical phenomena is an important factor for the materials description. The Maxwell model is more efficient: all its parameters have significant physical values related to the viscoelastic solid structure. A viscoelastic model with parameters that have significant physical values is recommended for describing materials behavior.

The Maxwell viscoelastic model is widely used to describe stress relaxation and creep [59, 60, 61, 62]. Prony series and particular case Anand model are used to describe grease as a Maxwell body [63]. Temperature dependences of the storage modulus and loss modulus are the main indicators of material rheology [20]. The storage modulus and loss modulus dependences can reflect structural changes in the grease [64]. The storage modulus reflects the elastic response of the material upon phase entry, i.e. it characterizes the solid material performance [65]. The loss modulus characterizes the viscous component, which lags 90 degrees behind the input phase. The linear viscous region is observed at small oscillation amplitudes. The intersection of the storage modulus and the loss modulus is the flow point (phase angle 45°). This is the transition point between the solid and the fluid.

However, the research team previously found the following limitations [38]: using Maxwell’s equations, the model works only at high strain rates; the simplified Anand model is only suitable for frequencies greater than 1 Hz. Significant error is observed at small frequencies (0 to 1 Hz). Greases can exhibit more complex behavior that cannot be described by simple viscoelastic models [56]. Phenomenological constitutive models must take into account temperature-time factors. Existing models have certain limitations, which lead to the need for their modification. Modifications of the standard linear Maxwell model allow for expansion of functionality: for the case of finite loading rate [59]; for integrating the hysteresis effect [66]; for taking into account temperature dependences of material constants [38], etc. Assessing the potential for practical application of mathematical models as a predictive analysis tool is an important area of research.

The range of lubricants is expanding every year. Attempts to modify the greases composition to obtain unique functional materials exist, both successful and unsuccessful. Experimental studies for comparative analysis of grease behavior are needed. Methods and approaches to describing greases behavior are important for predicting their performance over long time periods and a wide range of temperatures and loading conditions. Prediction of lubricant performance, both in laboratory conditions and in real structures, is required. A mathematical description of the materials behavior taking into account the temperature dependences of key parameters is required to form a more complete picture of the greases nonlinear deformation. Smart selection of greases is required to ensure the functionality of components while taking into account the specifics of their operation [67]. An experimental and theoretical description of the materials behavior taking into account their nonlinear nature can become a stable basis for the formation of lubricant selection technologies. Approaches to conducting experimental studies of greases using a rotational rheometer, mathematical description of material behavior models from simple to more functional, as well as their testing on a greases set are discussed in this chapter.

2. Materials and methods

2.1 Materials

Four greases developed by two Russian research and production companies were considered in the study (Figure 1): CIATIM-221 and CIATIM-221F are from the Central Institute of Aviation Fuels and Oils (Moscow, Russian Federation); TOMFLON SK 170 and TOMFLON HSK 240 are from Fluoropolymer Technologies LLC (Tomsk, Russian Federation, https://tomflon.ru/).

CIATIM-221 (mat. 1) is the heat-resistant multi-purpose plastic grease based on silicone oil. The base is organosilicon liquid 132-24 (a liquid based on a mixture of polyethylsiloxanes). Calcium stearate acetate is thickener. Diphenylamine is antioxidant additive.

CIATIM-221F (mat. 2) is the modification of CIATIM-221 by adding ultra-dispersive polytetrafluoroethylene (PTFE) or nano-PTFE.

TOMFLON CK 170 (mat. 3) is multi-purpose synthetic PTFE grease. The base is a mixture of polyalphaolefin oil and organosilicon liquid. Nano-PTFE is thickener.

TOMFLON HCK 240 (mat. 4) is chemical storage grease. The base is organosilicon liquid with thickener nano-PTFE.

Greases may also contain additives, such as antioxidants, antiwear, extreme pressure and water-resistant ones. The nano-PTFE used in the grease production may also vary. For example, Fluoropolymer Technologies LLC has developed TOMFLON powder, which can be used to modify greases [68, 69]. TOMFLON powder was obtained by combining irradiation and subsequent mechanical processing of PTFE [69]. The percentage content of base oil and thickener, as well as the concentration of various additives, is a commercial secret and is not disclosed in open sources. For example, molybdenum disulfide imparts a black color to TOMFLON CK 170 and TOMFLON HCK 240 greases. But this grease component is not listed on the developer’s website.

All lubricants show good thermostable, mechanical stability and nonhygroscopic properties. Operating temperatures range from – 60°C to +150°C (CIATIM-221), +160°C (CIATIM-221F), + 170°C (TOMFLON CK 170) and + 240°C (TOMFLON HCK 240). Greases are compatible with plastics and polymers and work well by mating metal-to-metal and metal-to-rubber. Greases have found wide application in bearing manufacturing, gear transmissions and experimental installations for various purposes [70, 71, 72, 73].

CIATIM-221F, TOMFLON CK 170 and TOMFLON HCK 240 are alternatives to CIATIM-221 with improved properties, including the addition of nano-PTFE. A tribofilm forms on the mating surfaces when CIATIM-221F, TOMFLON CK 170 and TOMFLON HCK 240 greases are used. The material decomposition was not observed (long-term observation of the grease consistency). This indicates the compositional stability.

2.2 Experimental study

Experimental studies are conducted at the premises of the Perm National Research Polytechnic University (PNRPU) plastics laboratory. Thermophysical, rheological and thermomechanical characteristics of materials and products are tested in the laboratory [74, 75]. A Discovery Hybrid Rheometer (TA Instruments - Waters LLC, New Castle, Delaware, USA) (Figure 2a) was used for experimental investigation of the thermophysical and thermomechanical properties of greases. The experimental study stages are shown in Figure 2b–e. An electrically heated plane is used for the active heating and cooling of the study samples. This design allows specimens to be heated to 400°C and cooled to −70°C using liquid nitrogen (Figure 2d). The temperature change range in our case was T∈−40;+80 °C. This corresponds to the operating temperatures of the environment for spherical bridge bearings. The design allows for uniform temperature distribution throughout the sample. As part of the operation, the temperature tolerance limit is ±1°C. Dynamic modal analysis (DMA) allows conducting research within the framework of the linear viscoelasticity theory without destroying samples.

The experimental research methodology consists of three main stages (Figure 3): analysis of the grease aggregate state (strain sweep experiment); dependence of material properties on temperature; and dependence of the material stresses on the shear rate. An experiment to select a measurement system must be carried out in advance: cone-plane or plane-plane.

When analyzing the material aggregate state, it was found that the greases have a constant aggregate state (elastic or gel) over the temperature entire range from −40 to +80°C. The heating rate during the experiments was 2°C/min.

The influence of plane-to-plane and cone-to-plane mating surfaces was investigated to ensure the required accuracy and quality of test results. A constant angle of cone inclination to the plate allows maintaining a constant shear rate. This ensures the experimental data quality. The shear rate in a plane-to-plane system depends on the surface radial position. This leads to differences between experimental data and the actual material behavior. The cone-plane measurement system is chosen in our case to ensure uniform distribution of the strain rate in the sample.

Two types of experimental studies were conducted:

1. The analysis of the temperature change effect T∈−40;+80 °C (T.cooling/heating=2 °C/min) at a fixed sample deformation rate of 1 Hz;

2. The analysis of the influence of sample deformation rate changing γ.∈0.01100 Hz at a fixed temperature (6 constant temperatures −40, −20, 0, +20, +50 and + 80° C).

Each experiment was repeated 4 times.

The experiment first type allows obtaining data on the dependence of viscosity ηT and shear stress τT on temperature.

The second experiment type is aimed at assessing the elastic, viscous and plastic behavior of materials at constant temperature. The Ostwald-de Waele flow law τ=K˜TγgT is used to describe the shear stress on the strain rate dependence τγ., where K˜T is the empirical constant is related to the material consistency and gT is the degree of material deviation flow in the highly elastic state.

Experimental studies also made it possible to describe temperature dependences on complex shear modulus G∗T, storage modulus G′T and loss modulus G″T.

3. Results and discussion

3.1 Results of experimental studies

The material parameters’ dependences on temperature and a number of material constants were established. The temperature dependences of storage modulus G′T and loss modulus G″T, shear modulus GT, viscosity ηT and yield strength σ0T for all the materials considered are presented as an example in Figure 4.

Greases are in a state of all-round compression in the spherical bearing for a long time period. Therefore, the solid mechanics relationships are chosen to describe models of nonlinear material behavior. The experimental results confirm the assumption that the materials can be described in terms of viscoelasticity or viscoelastoplasticity [63]: the storage modulus is greater than the loss modulus over the entire temperature range (Figure 4a); the viscosity of the materials corresponds to that of amorphous bodies (Figure 4c). The maximum viscosity at negative temperatures is observed in material 4. The viscosity of other materials over the entire temperature range does not exceed 0.05 MPa•c. Material 1 (CIATIM-221) has the minimum resistance to the movement of some particles relative to others by the entire temperature range.

The temperature dependence of the shear modulus at strain rate values equal to 1000 Hz was found based on the Ostwald-de Waele law (Figure 4b). The shear modulus of materials 1 and 4 is significantly higher, approximately 3–4 times, than that of materials 2 and 3 at T∈−40;−20 °C. This means that CIATIM-221 (mat. 1) and TOMFLON HSK 240 (mat. 4) materials are stiffer and more resistant to shear loads at temperatures below −20°C. CIATIM-221F (mat. 2) and TOMFLON CK 170 (mat. 3), on the contrary, better withstand shear loads at temperatures above −20°C.

The higher the material viscosity, the lower the yield strength (Figure 4d). An active increase in the yield strength of materials is observed at temperatures below 20°C, similar to viscosity. CIATIM-221 (mat. 1) and CIATIM-221F (mat. 2) are more resistant to plastic deformation. This reflects the dependence of the material yield strength on temperature.

The glass transition temperatures were also found: 207.96, 219.55, 211.06 and 221.99°K for materials 1–4, respectively.

3.2 Digital models of material behavior

A material behavior comparison described within the framework of three models (Prony series, particular case Anand and modified Anand) is presented in Figure 5 using the example of the temperature dependence of shear stresses. The dependences were obtained by simulating on pure shear experiment. The results were obtained at a constant sample deformation rate of 1 Hz. The models comparison is shown using the example of material 1 (CIATIM-221).

All models describe the grease behavior at a constant shear rate quite accurately. The error is no more than 2–3%.

The dependence of shear stresses on the rate of sample deformation is considered (Figure 6).

Simplified models of the Maxwell body have a strong deviation from the experimental data when taking into account the rate of sample deformation: 40% is Prony series; 80% is special case Anand model. The shear stress values of the Prony series are higher than the experimental data. The values are several orders of magnitude lower than the experimental data for the particular case Anand model. The error of the modified Anand model from the experimental data in all temperature ranges does not exceed 2.64%. The average error was 1.69%.

All greases considered were described using three models. However, the use of simplified viscoelastic models of material behavior to describe greases is not effective. The operation of the modified Anand model for all materials considered is shown in Figure 7.

The arithmetic mean error of the dependence shear stresses on the sample strain rate: 1.69, 0.90, 0.56 and 0.26% for materials 1–4, respectively.

4. Conclusion

Experimental data on the greases behavior confirm the significant influence of composition and additives on the materials properties. The introduction of nano-PTFE into the composition makes it possible to obtain greases with improved physical and mechanical properties over a wide range of temperatures and deformation rates. CIATIM-221 and CIATIM-221F are more resistant to plastic deformation over the entire temperature range from −40 to +80°C. CIATIM-221 and TOMFLON HSK 240 are more resistant to shear loads at temperatures below −20°C. CIATIM-221F and TOMFLON CK 170 are more resistant to shear loads at temperatures above −20°C. The experimental results confirm the need for greases rational selection with the temperature and force conditions and the climatic zones of their use.

The greases description of this type using classical simple viscoelastic models leads to significant errors with experimental data. Greases description as Maxwell bodies is possible at deformation rates from 0 to 1 Hz. The construction of a numerical analog with a grease description as a Maxwell body can lead to result unreliability by large range of deformation rates.

The use of more complex models for greases behavior describing is required to effectively predict the structures behavior while minimizing error. Models must take into account the temperature dependence of material properties. The modified Anand model allows one to describe greases behavior with an error not exceeding 3% for a wide range of temperatures and deformation rates.

Acknowledgments

The study was supported by the Russian Science Foundation grant No. 25-29-00470, https://rscf.ru/project/25-29-00470/.

Conflict of interest

The authors declare no conflict of interest.