Viscoelastic Solid Properties Research Articles

Preparation and Effect of CO2 Response Gel for Plugging Low-Permeability Reservoirs.

In order to solve the problem of gas channeling during CO2 flooding in low-permeability reservoirs, a novel CO2 responsive gel channeling system was prepared by using carrageenan, branched polyethylene imide and ethylenediamine under laboratory conditions. Based on the Box-Behnken response surface design method, the optimal synthesis concentration of the system was 0.5 wt% carrageenan, 2.5 wt% branchized polyethylenimide and 6.5 wt% ethylenediamine. The micromorphology of the system before and after response was characterized by scanning electron microscopy. The rheology and dehydration rate were tested under different conditions. The channeling performance and enhanced oil recovery effect of the gel system were simulated by a core displacement experiment. The experimental results show that the internal structure of the system changes from a disordered, smooth and loosely separated lamellae structure to a more uniform, complete and orderly three-dimensional network structure after exposure to CO2. The viscosity of the system was similar to aqueous solution before contact with CO2 and showed viscoelastic solid properties after contact with CO2. The experiment employing dehydration rates at different temperatures showed that the internal structure of the gel would change at a high temperature, but the gel system had a certain self-healing ability. The results of the displacement experiment show that the plugging rate of the gel system is stable at 85.32% after CO2 contact, and the recovery rate is increased by 17.06%, which provides an important guide for the development of low-permeability reservoirs.

Rational understanding of viscoelastic drop impact dynamics on porous surfaces considering rheological properties

Understanding of viscoelastic drop impact on porous solid surfaces emerges as the essential physics for broad industrial and biological systems. However, there have been few studies concerning the post-drop impact transition considering rheological properties and dynamics. Here, we report the post-impact dynamics of viscoelastic drops on porous mesh surfaces using experimental and theoretical analyses. Five concentrations of highly viscous polymer solutions with shear-thinning and viscoelastic characteristics exhibit different rheological properties (viscosity, storage, loss moduli, etc.). These changes in the rheological features result in four distinguishable impact dynamics on two-dimensional mesh surfaces. Viscoelastic drops that exhibit viscoelastic solid properties (storage modulus > loss modulus) resulted in combined elastic solid-like motions and viscous fluid-like motions depending on the impact velocity and anti-penetration effects (such as capillary pressure and yield stress). Meanwhile, decreases in the viscoelastic fluid properties resulted in monotonous viscosity-dominant behaviors. A pressure-balance equation was proposed to theoretically predict the filament column penetration length of viscoelastic solid drops. The theoretical results correlated with the experimentally measured values until necking occurred in the filament column. We believe that this study on viscoelastic drop impact dynamics can shed light on future applications that involve various mechanical behaviors of non-Newtonian fluid drops, such as 3D printing, coating, bio-fluid treatments, and polymeric solutions.

Modelling of Electro-Viscoelastic Materials through Rate Equations.

Models of dielectric solids subject to large deformations are established by following a thermodynamic approach. The models are quite general in that they account for viscoelastic properties and allow electric and thermal conduction. A preliminary analysis is devoted to the selection of fields for the polarization and the electric field; the appropriate fields are required to comply with the balance of angular momentum and to enjoy the Euclidean invariance. Next, the thermodynamic restrictions for the constitutive equations are investigated using a wide set of variables allowing for the joint properties of viscoelastic solids, electric and heat conductors, dielectrics with memory, and hysteretic ferroelectrics. Particular attention is devoted to models for soft ferroelectrics, such as BTS ceramics. The advantage of this approach is that a few constitutive parameters provide a good fit of material behaviour. A dependence on the gradient of the electric field is also considered. The generality and the accuracy of the models are improved by means of two features. The entropy production is regarded as a constitutive property per se, while the consequences of the thermodynamic inequalities are made explicit by means of representation formulae.

Strain softening of concentrated cohesive particulate suspensions prior to yield

We study the viscoelastic solid properties of cohesive particulate suspensions using creep and constant rate tests in a vane-in-large-cup geometry. A cup-to-vane diameter ratio larger than 4 is used to ensure that wall effects are minimized. In both the creep and constant rate tests, the modulus becomes nonlinear at strains consistent with scaled interparticle bond distances. Yielding and subsequent flow do not occur until strains of order 1, corresponding to a cage melting or particle crowding failure mechanism. In between the bond and cage melting strains, the modulus shows power-law softening with an index of approximately −0.8, corresponding to progressive bond breakage. The observed behavior in creep correlates with constant rate experiments and demonstrates that strain softening of the modulus prior to yielding is an important component of the rheology of particulate gels and suspensions. The assumption of linear behavior up to yielding appears to be inappropriate. We observe that the transient creep behavior initially shows power-law or Andrade creep that transforms to an exponential decay at long times. For creep stresses that show time-dependent yield, the break time decreases exponentially with stress.

Analysis of high performing graphene oxide nanosheets based non-damaging drilling fluids through rheological measurements and CFD studies

This research presents a detailed study on the effect of synthesized graphene oxide nanosheet (GO-NS) on Non-Damaging Drilling Fluid (NDDF) through steady-state rotational and dynamic oscillation rheological measurements along with fluid loss estimation. Synthesis of GO-NS was conducted by following Hummers' method. It was found that GO-NS acts as an excellent thermal stabilizer for NDDF which better preserves viscosity at high temperatures up to 98%. Apart from strengthening viscoelastic solid properties, addition of >0.5 wt% GO-NS also induces superior structural stability showing a recovery of 93.84%. More than 50% reduction in filtrate loss volume was observed with an addition of 0.8 wt% GO-NS in NDDF. Computational Fluid Dynamics (CFD) simulations showed that the overall cutting retention is least with 0.096% for 0.5 wt% GO-NS NDDF with less eddy viscosity, higher velocity and reduced skewness. Pressure drop calculations indicated assured flowability of GO-NS NDDF even though it bears viscoelastic solid nature.

Alternative Form of Standard Linear Solid Model for Characterizing Stress Relaxation and Creep: Including a Novel Parameter for Quantifying the Ratio of Fluids to Solids of a Viscoelastic Solid

The standard linear solid model (SLSM) is a typical and useful model for analyzing stress relaxation and creep behaviors of viscoelastic solids for obtaining the corresponding viscoelastic properties. However, the analysis results cannot be directly compared to the parameters commonly adopted for defining the mechanical properties of viscoelastic solids in the finite element simulation package such as the modulus of elasticity (Ee) and the two parameters in the dimensionless form of the relaxation modulus (g and τ1). The purpose of this paper is to introduce an alternative form of SLSM in terms of Ee, g and τ1 for characterizing stress relaxation and creep behaviors. A series of stress relaxation and creep curves with different Ee, g and τ1 was simulated by finite element simulation. The derived alternative form of SLSM was used to curve fit the simulated stress relaxation and creep curves to obtain the corresponding values of Ee, g and τ1. The results showed that the values of Ee, g and τ1 obtained from the simulation were approximately equal to the theoretical ones (i.e., those set in the simulation), showing that the alternative form of SLSM can accurately evaluate the corresponding Ee, g and τ1. In conclusion, the alternative form is formulated in terms of the parameters used to define the mechanical properties in the finite element simulation package, so that the parameters obtained by curve fitting can be directly compared to those set in the finite element simulation package. It was also found that the physical meaning of g is associated with the ratio of viscous fluids to solids of a viscoelastic solid.

Mechanical and Viscoelastic Properties of Mozzarella Cheese

The aim of this study is to determine the mechanical properties and the viscoelastic characteristics of the mozzarella cheese made from buffalo milk. Mozzarella cheese has a characteristic structure as the curd undergoes a stretching process in hot water that leads to the formation of a fibrous structure. This affects the number, strength and type of bond between the protein molecules that form the basis for the rheological properties exhibited. The mechanical properties and the viscoelastic characteristics of buffalo mozzarella were measured in the present study. Uniaxial compression tests between parallel plates with an extra 5.8 cm compression were carried out in cheese cubes of 2 cm edge in order to estimate Peleg model parameters such as the compressive stress, deformation modulus, degree of elasticity and A and B constants. Cheese cubes were compressed and decompressed at a deformation rate of 20 cm/min up to 40% of their height through a complete compression-decompression cycle. For a compression area of 4.0×10^-4 m² a maximum compressive strength of 28.0±3 N, a compressive stress of 7.0×10⁴ Pa and a deformation modulus of E_D = 175.0 kPa were obtained. The degree of elasticity was 73.0±8.5%, which indicates that mozzarella cheese has a very elastic texture. The values for A and B constants of the Peleg model were as follows: A = 0.83 and B = 0.11. Buffalo mozzarella exhibitselastic recovery levels suitable for technological use, as well as viscoelastic solid properties.

Properties of starch–polyglutamic acid (PGA) graft copolymer prepared by microwave irradiation − Fourier transform infrared spectroscopy (FTIR) and rheology studies

The rheological properties of waxy starch‐γ‐polygutamic acid (PGA) graft copolymers were investigated. Grafting was confirmed by FTIR spectroscopy. The starch–PGA copolymers absorbed water and formed gels, which exhibited concentration‐dependent viscoelastic solid properties. Higher starch–PGA concentrations resulted in stronger viscoelastic properties. The analysis for moduli of different concentrations indicated that the starch–PGA graft copolymer gels were physical gels, meaning the cross‐links between the molecules were physical junctions. However, stress relaxation measurements showed that the starch–PGA graft copolymer gels had long relaxation times, which result from the presence of the chemical cross‐links that were evident in the FTIR spectra. The non‐linear steady shearing rheological properties studies indicated that starch–PGA gels exhibited shear thinning behavior, which can be well fitted with the power law constitutive equation. The function and behavior of the starch–PGA graft copolymer gels suggest that this kind of starch‐based biomaterial could be a potential candidate for applications in cosmetic gels, skin wound care materials, and agricultural products.

Effect of normal compression on the shear modulus of soft tissue in rheological measurements

While the effect of normal compression on the measured shear material properties of viscoelastic solids has been already acknowledged in rheological studies in the literature, to our knowledge, no systematic study has been conducted to investigate this effect in detail to date. In this study, we perform two sets of experiments to investigate the effect of normal strain and strain rate on the dynamic shear moduli of bovine liver. First, we apply normal compressive strain to the cylindrical bovine samples up to 20% at loading rates of v=0.000625, 0.00625, 0.0625, 0.315, 0.625mm/s. Second, we perform torsional shear loading experiments, in the frequency range of ω=0.1–10Hz, under varying amounts of compressive pre-strain (ε=1%, 2.5%, 5%, 7.5%, 10%, 12.5%, 15%, 17.5% and 20%) applied at the quasi-static loading rate of v=0.000625mm/s. The results of the experiments show that the shear moduli of bovine liver increase with compressive pre-strain. A hyper-viscoelastic constitutive model is developed and fit to the experimental data to estimate the true shear moduli of bovine liver for zero pre-compression. With respect to this reference value, the mean relative error in the measurement of shear moduli of bovine liver varies between 0.2% and 243.1% for the compressive pre-strain varying from ε=1% to 20%. The dynamic shear modulus of bovine liver for compressive pre-strain values higher than ε>2.5% are found to be statistically different than the true shear moduli estimated for zero compressive strain (p<0.05).

Rheological properties of reactive extrusion modified waxy starch and waxy starch-polyacrylamide copolymer gels

The rheological properties of modified waxy starch and waxy starch–polyacrylamide graft copolymers prepared by reactive extrusion were investigated. Both materials can absorb huge amount of water and form gels. The modified waxy starch and waxy starch–polyacrylamide graft copolymer gels all exhibited viscoelastic solid properties. The waxy starch–polyacrylamide graft copolymer gels with the concentration ≤10% showed weaker viscoelastic behaviors than those of the same concentrations of modified waxy starch gels. However, at concentration ≥11.25%, the waxy starch–polyacrylamide graft copolymer gels displayed much stronger viscoelastic properties than those of the same concentrations of modified waxy starch gels. The analysis of modulus and concentration dependence and stress relaxation measurements indicated that both modified waxy starch and waxy starch–polyacrylamide graft copolymer gels were physical gels meaning the cross-linkers between the molecules were physical junctions. The nonlinear steady shearing rheological properties studies indicated that both modified waxy starch and waxy starch–polyacrylamide graft copolymer gels exhibited shear thinning behavior, which can be well fitted with the power law constitutive equation. The function and behavior of the modified waxy starch and waxy starch–polyacrylamide graft copolymer gels suggest that these starch-based biomaterials should be potential candidates for applications in cosmetic gels, wound skin care materials, and agricultural products.

Synthesis and rheological properties of hydrophobically modified poly(vinyl alcohol)

A novel series of water-soluble hydrophobically modified poly(vinyl alcohol) (HMPVA) with various hydrophobe contents was prepared by grafting poly(vinyl alcohol) (PVA) using 1-dodecanol and toluene-2,4-diisocyanate as hydrophobic monomer and coupling agent, respectively. The chemical structure of HMPVA was analyzed by Fourier Transform Infrared Spectrometer (FTIR) and 1H NMR. Rheological properties of the aqueous solutions also confirmed the incorporation of hydrophobic groups into PVA. In dilute concentration regime, HMPVAs exhibited lower intrinsic viscosity than PVA, suggesting that HMPVA molecules were more shrunken. While the aqueous solution viscosity was enhanced due to hydrophobic modification at a high concentration, and HMPVAs with higher hydrophobe contents exhibited lager values of apparent viscosities. Over a frequency range of 1 to102 rad/s, the dynamic storage modulus of PVA solution was smaller than the dynamic loss modulus whereas the dynamic storage modulus of HMPVAs solutions was greater than the dynamic loss modulus, indicating the evolution of viscoelastic solid properties in HMPVAs solutions. The yield stress of PVA was nearly zero whereas that of HMPVAs represented positive values, implying that networks were present in HMPVAs solutions.

Normal age-related viscoelastic properties of chondrons and chondrocytes isolated from rabbit knee.

The mechanical microenvironment of the chondrocytes plays an important role in cartilage homeostasis and in the health of the joint. The pericellular matrix, cellular membrane of the chondrocytes, and their cytoskeletal structures are key elements in the mechanical environment. The aims of this study are to measure the viscoelastic properties of isolated chondrons and chondrocytes from rabbit knee cartilage using micropipette aspiration and to determine the effect of aging on these properties. Three age groups of rabbit knees were evaluated: (1) young (2 months, n = 10); (2) adult (8 months, n = 10); and (3) old (31 months, n = 10). Chondrocytes were isolated from the right knee cartilage and chondrons were isolated from left knees using enzymatic methods. Micropipette aspiration combined with a standard linear viscoelastic solid model was used to quantify changes in the viscoelastic properties of chondrons and chondrocytes within 2 hours of isolation. The morphology and structure of isolated chondrons were evaluated by optical microscope using hematoxylin and eosin staining and collagen-6 immunofluorescence staining. In response to an applied constant 0.3 - 0.4 kPa of negative pressure, all chondrocytes exhibited standard linear viscoelastic solid properties. Model predictions of the creep data showed that the average equilibrium modulus (E(∞)), instantaneous modulus (E(0)), and apparent viscosity (m) of old chondrocytes was significantly lower than the young and adult chondrocytes (P < 0.001); however, no difference was found between young and adult chondrocytes (P > 0.05). The adult and old chondrons generally possessed a thicker pericellular matrix (PCM) with more enclosed cells. The young and adult chondrons exhibited the same viscoelastic creep behavior under a greater applied pressure (1.0 - 1.1 kPa) without the deformation seen in the old chondrons. The viscoelastic properties (E(∞), E(0), and m) of young and adult chondrons were significantly greater than that observed in young and adult cells, respectively (P < 0.001). The adult chondrons were stiffer than the young chondrons under micropipette aspiration (P < 0.001). Our findings provide a theoretical model to measure the viscoelastic properties of the chondrons as a whole unit by micropipette aspiration, and further suggest that the properties of the chondrocytes and PCM have an important influence on the biomechanical microenvironment of the knee joint cartilage degeneration that occurs with aging.

Application of Rheological Models to Modified Roofing Asphalts

Abstract The increased understanding of material behavior with rheology has existed for approximately 80 years as a science and has been applied to roofing products for over 50 years. With asphalt binders, several techniques exist for understanding rheology via the use of models. One of the best recognized models for the evaluation of the rheology of unmodified asphalt binders is the Christensen-Anderson model, and various developments of that model have occurred, enabling its use with filled systems such as those used in roofing applications. In addition to this model, the Rowe-Baumgardner-Sharrock model (developed by the authors) is introduced as an alternate method for defining the properties of roofing products that exhibit visco-elastic solid properties. When rheological data are fitted to master curve functional forms, changes in the model parameters can be rapidly visualized. The performance of various products can be assessed using this technique and can be related to the rheological parameters developed from this model (for example, the rheological index, crossover frequency, etc.). An analysis of the data is presented that includes testing using various rheometers over a wide range of temperatures. These data have been combined into a single master curve and plotted in various formats. The data show how the rheological index and the crossover frequency of the materials change over time, particularly with aging. Performance products can then be assessed and evaluated from both the historical data and models that enable the prediction of cracking and deformation. Adiscussion of the key parameters and a trend analysis are shown. Furthermore, the analysis is applied toward understanding the effectiveness of polymer networks in the roofing materials. As materials age, their properties change, and this can be observed, allowing the study of the effects of polymer network degradation with aging. This paper offers some further understanding of how key rheological parameters change.

On Lamb and Rayleigh wave convergence in viscoelastic tissues

Characterization of the viscoelastic material properties of soft tissue has become an important area of research over the last two decades. Our group has been investigating the feasibility of using a shear wave dispersion ultrasound vibrometry (SDUV) method to excite Lamb waves in organs with plate-like geometry to estimate the viscoelasticity of the medium of interest. The use of Lamb wave dispersion ultrasound vibrometry to quantify the mechanical properties of viscoelastic solids has previously been reported. Two organs, the heart wall and the spleen, can be readily modeled using plate-like geometries. The elasticity of these two organs is important because they change in pathological conditions. Diastolic dysfunction is the inability of the left ventricle (LV) of the heart to supply sufficient stroke volumes into the systemic circulation and is accompanied by the loss of compliance and stiffening of the LV myocardium. It has been shown that there is a correlation between high splenic stiffness in patients with chronic liver disease and strong correlation between spleen and liver stiffness. Here, we investigate the use of the SDUV method to quantify the viscoelasticity of the LV free-wall myocardium and spleen by exciting Rayleigh waves on the organ's surface and measuring the wave dispersion (change of wave velocity as a function of frequency) in the frequency range 40–500 Hz. An equation for Rayleigh wave dispersion due to cylindrical excitation was derived by modeling the excised myocardium and spleen with a homogenous Voigt material plate immersed in a nonviscous fluid. Boundary conditions and wave potential functions were solved for the surface wave velocity. Analytical and experimental convergence between the Lamb and Rayleigh waves is reported in a finite element model of a plate in a fluid of similar density, gelatin plate and excised porcine spleen and left-ventricular free-wall myocardium.

Viscoelastic properties of chondrons enzymatically isolated from rabbit knee articular cartilage in virto

Objective To characterize the biomechanical behavior and properties of the chondrons enzymatically isolated from rabbit knee articular cartilage in virto. Methods Eight months old New Zealand white rabbits were randomly divided into chondroctye and chondron groups (4 rabbits in each group). In chondrocyte groups, the full articular cartilages from both knees were enzymatically isolated to chondrocytes by 0.4% pronase and 0.025% collagenase type-Ⅱ in turn. In chondron groups, chondrons were obtained from articular cartilage using the mixture of 0.3% dispase (a neutral protease) and 0.2% collagenase type-Ⅱin at 37C for 3 h. The micropipette aspiration was used to quantify changes in biomechanical properties of chondrons and chondrocytes and the viscoelastic parameters, including K1, K2, E∞ (equilibrium modulus), E0(instantaneous modulus), and μ (apparent viscosity), were calculated coupled with standard linear half-space viscoelastic solid model. Results In response to a constant negative pressure of 0.2-0.4 kPa, the chondrocytes exhibited standard linear viscoelastic solid properties. Namely, the cells showed an initial elastic response followed by a viscoelastic creep response. then cells continued to enter into the micropipette with a monotonically decreasing rate of deformation, until reaching equilibrium within about (110±18) s. Comparing with chondrocytes, the chondrons exhibited significant viscoelasticity under a greater negative pressure of 1.0-1.2 kPa. But the instantaneous length deformed into the micropipette significantly reduced, and the equilibrium time reduced to (36.5±4.5) s. The equilibrium modulus (E∞), the instantaneous modulus (E0) and the apparent viscosity (μ) of chondrons were significantly higher than the those of chondrocytes. Conclusion Comparing with chondrocytes, the chondrons exhibited significant viscoelastic properties, and viscoelastic properties of chondrons have increased in vitro. Key words: Knee joint; Cartilage; Biomechanics

Lamb wave dispersion ultrasound vibrometry (LDUV) method for quantifying mechanical properties of viscoelastic solids

Diastolic dysfunction is the inability of the left ventricle to supply sufficient stroke volumes under normal physiological conditions and is often accompanied by stiffening of the left-ventricular myocardium. A noninvasive technique capable of quantifying viscoelasticity of the myocardium would be beneficial in clinical settings. Our group has been investigating the use of shear wave dispersion ultrasound vibrometry (SDUV), a noninvasive ultrasound-based method for quantifying viscoelasticity of soft tissues. The primary motive of this study is the design and testing of viscoelastic materials suitable for validation of the Lamb wave dispersion ultrasound vibrometry (LDUV), an SDUV-based technique for measuring viscoelasticity of tissues with plate-like geometry. We report the results of quantifying viscoelasticity of urethane rubber and gelatin samples using LDUV and an embedded sphere method. The LDUV method was used to excite antisymmetric Lamb waves and measure the dispersion in urethane rubber and gelatin plates. An antisymmetric Lamb wave model was fitted to the wave speed dispersion data to estimate elasticity and viscosity of the materials. A finite element model of a viscoelastic plate submerged in water was used to study the appropriateness of the Lamb wave dispersion equations. An embedded sphere method was used as an independent measurement of the viscoelasticity of the urethane rubber and gelatin. The FEM dispersion data were in excellent agreement with the theoretical predictions. Viscoelasticity of the urethane rubber and gelatin obtained using the LDUV and embedded sphere methods agreed within one standard deviation. LDUV studies on excised porcine myocardium sample were performed to investigate the feasibility of the approach in preparation for open-chest in vivo studies. The results suggest that the LDUV technique can be used to quantify the mechanical properties of soft tissues with a plate-like geometry.

A large deformation framework for compressible viscoelastic materials: Constitutive equations and finite element implementation

For modeling the constitutive properties of viscoelastic solids in the context of small deformations, the so-called three-parameter solid is often used. The differential equation governing the model response may be derived in a thermodynamically consistent way considering linear spring-dashpot elements. The main problem in generalizing constitutive models from small to finite deformations is to extend the theory in a thermodynamically consistent way, so that the second law of thermodynamics remains satisfied in every admissible process. This paper concerns with the formulation and constitutive equations of finite strain viscoelastic material using multiplicative decomposition in a thermodynamically consistent manner. Based on the proposed constitutive equations, a finite element (FE) procedure is developed and implemented in an FE code. Subsequently, the code is used to predict the response of elastomer bushings. The finite element analysis predicts displacements and rotations at the relaxed state reasonably well. The response to coupled radial and torsional deformations is also simulated.

Nanoindentation and the dynamic characterization of viscoelastic solids

Using a high-damping thermoplastic as a standard reference material, the purpose of this work is to compare measured values of the complex modulus as determined by dynamic nanoindentation and dynamic mechanical analysis (DMA). Experiments were performed at approximately 22 °C and seven frequencies over the range 1–50 Hz. The indentation measurements were performed using a 103 µm diameter flat punch and a newly developed test method that optimizes the accuracy and precision of the measured stiffness and damping. As determined by dynamic nanoindentation, values of the storage modulus and loss factor (tangent delta) ranged from 4.2 to 10.2 MPa, and 0.28 to 1.05, respectively. Over the range 1–25 Hz, DMA confirmed the nanoindentation results to within 15% or better. Collectively, these data and the testing methods used to generate them should help future investigators make more accurate and precise measurements of the dynamic properties of viscoelastic solids using nanoindentation.

Nonlinear Analysis of Oscillatory Indentation in Elastic and Viscoelastic Solids

Determining the mechanical properties at micro- and nanometer length scales using nanoindentation or atomic force microscopy is important to many areas of science and engineering. Here we establish equations for obtaining storage and loss modulus from oscillatory indentations by performing a nonlinear analysis of conical and spherical indentation in elastic and viscoelastic solids. We show that, when the conical indenter is driven by a sinusoidal force, the square of displacement is a sinusoidal function of time, not the displacement itself, which is commonly assumed. Similar conclusions hold for spherical indentations. Well-known difficulties associated with measuring contact area and correcting thermal drift may be circumvented using the newly derived equations. These results may help improve methods of using oscillatory indentation for determining elastic and viscoelastic properties of solids.

General relationship between contact stiffness, contact depth, and mechanical properties for indentation in linear viscoelastic solids using axisymmetric indenters of arbitrary profiles

We derive a relationship between the initial unloading slope, contact depth, and the instantaneous relaxation modulus for indentation in linear viscoelastic solids by a rigid indenter with an arbitrary axisymmetric smooth profile. Although the same expression is well known for indentation in elastic and in elastic-plastic solids, we show that it is also true for indentation in linear viscoelastic solids, provided that the unloading rate is sufficiently fast. Furthermore, the same expression holds true for both fast loading and unloading. These results should provide a sound basis for using the relationship for determining properties of viscoelastic solids using indentation techniques.