A transversely isotropic viscoelastic constitutive equation for brainstem undergoing finite deformation

The objective of this study was to define the constitutive response of brainstem undergoing finite shear deformation. Brainstem was characterized as a transversely isotropic viscoelastic material and the material model was formulated for numerical implementation. Model parameters were fit to shear data obtained in porcine brainstem specimens undergoing finite shear deformation in three directions: parallel, perpendicular, and cross sectional to axonal fiber orientation and determined using a combined approach of finite element analysis (FEA) and a genetic algorithm (GA) optimizing method. The average initial shear modulus of brainstem matrix of 4-week old pigs was 12.7 Pa, and therefore the brainstem offers little resistance to large shear deformations in the parallel or perpendicular directions, due to the dominant contribution of the matrix in these directions. The fiber reinforcement stiffness was 121.2 Pa, indicating that brainstem is anisotropic and that axonal fibers have an important role in the cross-sectional direction. The first two leading relative shear relaxation moduli were 0.8973 and 0.0741, respectively, with corresponding characteristic times of 0.0047 and 1.4538 s, respectively, implying rapid relaxation of shear stresses. The developed material model and parameter estimation technique are likely to find broad applications in neural and orthopaedic tissues.     

Numerical analysis of maximal bat performance in baseball

Metal baseball bats have been experimentally demonstrated to produce higher ball exit velocity (BEV) than wooden bats. In the United States, all bats are subject to BEV tests using hitting machines that rotate the bat in a horizontal plane. In this paper, a model of bat-ball impact was developed based on 3-D translational and rotational kinematics of a swing performed by high-level players. The model was designed to simulate the maximal performance of specific models of a wooden bat and a metal bat when swung by a player, and included material properties and kinematics specific to each bat. Impact dynamics were quantified using the finite element method (ANSYS/LSDYNA, version 6.1). Maximum BEV from both a metal (61.5 m/s) and a wooden (50.9 m/s) bat exceeded the 43.1 m/s threshold by which bats are certified as appropriate for commercial sale. The lower BEV from the wooden bat was attributed to a lower pre-impact bat linear velocity, and a more oblique impact that resulted in a greater proportion of BEV being lost to lateral and vertical motion. The results demonstrate the importance of factoring bat linear velocity and spatial orientation into tests of maximal bat performance, and have implications for the design of metal baseball bats.     

A covalently cross-linked gel derived from the epidermis of the pilot whale Globicephala melas

The rheological properties of the stratum corneum of the pilot whale (Globicephala melas) were investigated with emphasis on their significance to the self-cleaning abilities of the skin surface smoothed by a jelly material enriched with various hydrolytic enzymes. The gel formation of the collected fluid was monitored by applying periodic-harmonic oscillating loads using a stress-controlled rheometer. In the mechanical spectrum of the gel, the plateau region of the storage modulus G' (<1200 Pa) and the loss modulus G" (>120 Pa) were independent of frequency (omega = 43.98 to 0.13 rad x s(-1), tau = 15 Pa, T = 20 degrees C), indicating high elastic performance of a covalently cross-linked viscoelastic solid. In addition, multi-angle laser light scattering experiments (MALLS) were performed to analyse the potential time-dependent changes in the weight-average molar mass of the samples. The observed increase showed that the gel formation is based on the assembly of covalently cross-linked aggregates. The viscoelastic properties and the shear resistance of the gel assure that the enzyme-containing jelly material smoothing the skin surface is not removed from the stratum corneum by shear regimes during dolphin jumping. The even skin surface is considered to be most important for the self-cleaning abilities of the dolphin skin against biofouling.     

Neutrophil transit times through pulmonary capillaries: the effects of capillary geometry and fMLP-stimulation

The deformations of neutrophils as they pass through the pulmonary microcirculation affect their transit time, their tendency to contact and interact with the endothelial surface, and potentially their degree of activation. Here we model the cell as a viscoelastic Maxwell material bounded by constant surface tension and simulate indentation experiments to quantify the effects of (N-formyl-L-methionyl-L-leucyl-L-phenylalanine (fMLP)-stimulation on its mechanical properties (elastic shear modulus and viscosity). We then simulate neutrophil transit through individual pulmonary capillary segments to determine the relative effects of capillary geometry and fMLP-stimulation on transit time. Indentation results indicate that neutrophil viscosity and shear modulus increase by factors of 3.4, for 10(-9) M fMLP, and 7.3, for 10(-6) M fMLP, over nonstimulated cell values, determined to be 30.8 Pa.s and 185 Pa, respectively. Capillary flow results indicate that capillary entrance radius of curvature has a significant effect on cell transit time, in addition to minimum capillary radius and neutrophil stimulation level. The relative effects of capillary geometry and fMLP on neutrophil transit time are presented as a simple dimensionless expression and their physiological significance is discussed.     

Effect of twelve weeks of medicine ball training on high school baseball players

This study examined the effect of 12 weeks of medicine ball training on high school baseball players. Forty-nine baseball players (age 15.4 +/- 1.2 years) were randomly assigned using a stratified sampling technique to 1 of 2 groups. Group 1 (n = 24) and group 2 (n = 25) performed the same full-body resistance exercises according to a stepwise periodized model and took 100 bat swings a day, 3 days per week, with their normal game bat for 12 weeks. Group 2 performed additional rotational and full-body medicine ball exercises 3 days per week for 12 weeks. Pre- and post-testing consisted of a 3 repetition maximum (RM) dominant and nondominant torso rotational strength and sequential hip-torso-arm rotational strength (medicine ball hitter's throw). A 3RM parallel squat and bench press were measured at 0 and after 4, 8, and 12 weeks of training. Although both groups made statistically significant increases (p < or = 0.05) in dominant (10.5 vs. 17.1%) and nondominant (10.2 vs. 18.3%) torso rotational strength and the medicine ball hitter's throw (3.0 vs. 10.6%), group 2 showed significantly greater increases in all 3 variables than group 1. Furthermore, both groups made significant increases in predicted 1RM parallel squat and bench press after 4, 8, and 12 weeks of training; however, there were no differences between groups. These data indicate that performing a 12-week medicine ball training program in addition to a stepwise periodized resistance training program with bat swings provided greater sport-specific training improvements in torso rotational and sequential hip-torso-arm rotational strength for high school baseball players.     

A numerical study on hemodynamics in the left coronary bifurcation with normal and hypertension conditions

In this study, a three-dimensional analysis of the non-Newtonian blood flow was carried out in the left coronary bifurcation. The Casson model and hyperelastic and rigid models were used as the constitutive equation for blood flow and vessel wall model, respectively. Physiological conditions were considered first normal and then compliant with hypertension disease with the aim of evaluating hemodynamic parameters and a better understanding of the onset and progression of atherosclerosis plaques in the coronary artery bifurcation. Two-way fluid–structure interaction method applying a fully implicit second-order backward Euler differencing scheme has been used which is performed in the commercial code ANSYS and ANSYS CFX (version 15.0). When artery deformations and blood pressure are associated, arbitrary Lagrangian–Eulerian formulation is employed to calculate the artery domain response using the temporal blood response. As a result of bifurcation, noticeable velocity reduction and backflow formation decrease shear stress and made it oscillatory at the starting point of the LCx branch which caused the shear stress to be less than 1 and 2 Pa in the LCx and the LAD branches, respectively. Oscillatory shear index (OSI) as a hemodynamic parameter represents the increase in residence time and oscillatory wall shear stress. Because of using the ideal 3D geometry and realistic physiological conditions, the values obtained for shear stress are more accurate than the previous studies. Comparing the results of this study with previous clinical investigations shows that the regions with low wall shear stress less than 1.20 Pa and with high OSI value more than 0.3 are in more potential risk to the atherosclerosis plaque development, especially in the posterior after the bifurcation.     

Method for characterizing viscoelasticity of human gluteal tissue

Characterizing compressive transient large deformation properties of biological tissue is becoming increasingly important in impact biomechanics and rehabilitation engineering, which includes devices interfacing with the human body and virtual surgical guidance simulation. Individual mechanical in vivo behaviour, specifically of human gluteal adipose and passive skeletal muscle tissue compressed with finite strain, has, however, been sparsely characterised. Employing a combined experimental and numerical approach, a method is presented to investigate the time-dependent properties of in vivo gluteal adipose and passive skeletal muscle tissue. Specifically, displacement-controlled ramp-and-hold indentation relaxation tests were performed and documented with magnetic resonance imaging. A time domain quasi-linear viscoelasticity (QLV) formulation with Prony series valid for finite strains was used in conjunction with a hyperelastic model formulation for soft tissue constitutive model parameter identification and calibration of the relaxation test data. A finite element model of the indentation region was employed. Strong non-linear elastic but linear viscoelastic tissue material behaviour at finite strains was apparent for both adipose and passive skeletal muscle mechanical properties with orthogonal skin and transversal muscle fibre loading. Using a force-equilibrium assumption, the employed material model was well suited to fit the experimental data and derive viscoelastic model parameters by inverse finite element parameter estimation. An individual characterisation of in vivo gluteal adipose and muscle tissue could thus be established. Initial shear moduli were calculated from the long-term parameters for human gluteal skin/fat: G(∞,S/F)=1850 Pa and for cross-fibre gluteal muscle tissue: G(∞,M)=881 Pa. Instantaneous shear moduli were found at the employed ramp speed: G(0,S/F)=1920 Pa and G(0,M)=1032 Pa.     

Cell traction forces on soft biomaterials. I. Microrheology of type I collagen gels

A laser-trap microrheometry technique was used to determine the local shear moduli of Type I collagen gels. Embedded 2.1 microm polystyrene latex particles were displaced 10-100 nm using a near-infrared laser trap with a trap constant of 0.0001 N/m. The trap was oscillated transversely +/- 200 nm using a refractive glass plate mounted on a galvanometric scanner. The displacement of the microspheres was in phase with the movement of the laser trap at frequencies less than 1 rad/s, indicating that at least locally, the gels behaved as elastic media. The local shear modulus was measured at various positions throughout the gel, and, for gels at 2.3 mg/mL and 37 degrees C, values ranged from G = 3 to 80 Pa. The average shear modulus G = 55 Pa, which compares well with measurements from parallel plate rheometry.     

Viscoelastic properties of a mixed culture biofilm from rheometer creep analysis

The mechanical properties of mixed culture biofilms were determined by creep analysis using an AR1000 rotating disk rheometer. The biofilms were grown directly on the rheometer disks which were rotated in a chemostat for 12 d. The resulting biofilms were heterogeneous and ranged from 35 microns to 50 microns in thickness. The creep curves were all viscoelastic in nature. The close agreement between stress and strain ratio of a sample tested at 0.1 and 0.5 Pa suggested that the biofilms were tested in the linear viscoelastic range and supported the use of linear viscoelastic theory in the development of a constitutive law. The experimental data was fit to a 4-element Burger spring and dashpot model. The shear modulus (G) ranged from 0.2 to 24 Pa and the viscous coefficient (eta) from 10 to 3000 Pa. These values were in the same range as those previously estimated from fluid shear deformation of biofilms in flow cells. A viscoelastic biofilm model will help to predict shear related biofilm phenomena such as elevated pressure drop, detachment, and the flow of biofilms over solid surfaces.     

Direct measurement of shear strain in adherent vascular endothelial cells exposed to fluid shear stress

Functional and morphological responses of endothelial cells (ECs) to fluid shear stress are thought to be mediated by several mechanosensitive molecules. However, how the force due to fluid shear stress applied to the apical surface of ECs is transmitted to the mechanosensors is poorly understood. In the present paper, we performed an analysis of an intracellular mechanical field by observation of the deformation behaviors of living ECs exposed to shear stress with a novel experimental method. Lateral images of human umbilical vein ECs before and after the onset of flow were obtained by confocal microscopy, and image correlation and finite element analysis were performed for quantitative analyses of subcellular strain due to shear stress. The shear strain of the cells changed from 1.06 ± 1.09% (mean ± SD) to 4.67 ± 1.79% as the magnitude of the shear stress increased from 2 to 10 Pa. The nuclei of ECs also exhibited shear deformation, which was similar to that observed in cytoplasm, suggesting that nuclei transmit forces from apical to intracellular components, as well as cytoskeletons. The obtained strain-stress relation resulted in a mean shear modulus of 213 Pa for adherent ECs. These results provide a mechanical perspective on the investigation of flow-sensing mechanisms of ECs.     

Fatigue creep damage at the cement–bone interface: An experimental and a micro-mechanical finite element study

The goal of this study was to quantify the micromechanics of the cement–bone interface under tensile fatigue loading using finite element analysis (FEA) and to understand the underlying mechanisms that play a role in the fatigue behavior of this interface. Laboratory cement–bone specimens were subjected to a tensile fatigue load, while local displacements and crack growth on the specimen's surface were monitored. FEA models were created from these specimens based upon micro-computed tomography data. To accurately model interfacial gaps at the interface between the bone and cement, a custom-written erosion algorithm was applied to the bone model. A fatigue load was simulated in the FEA models while monitoring the local displacements and crack propagation. The results showed the FEA models were able to capture the general experimental creep damage behavior and creep stages of the interface. Consistent with the experiments, the majority of the deformation took place at the contact interface. Additionally, the FEA models predicted fatigue crack patterns similar to experimental findings. Experimental surface cracks correlated moderately with FEA surface cracks (r²=0.43), but did not correlate with the simulated crack volume fraction (r²=0.06). Although there was no relationship between experimental surface cracks and experimental creep damage displacement (r²=0.07), there was a strong relationship between the FEA crack volume fraction and the FEA creep damage displacement (r²=0.76). This study shows the additional value of FEA of the cement–bone interface relative to experimental studies and can therefore be used to optimize its mechanical properties.     

A method for analysis of pulmonary arterial and venous occlusion data.

Recently, we presented a compartmental model of the pulmonary vascular resistance (R) and compliance (C) distribution with the configuration C1R1C2R2C3 (J. Appl. Physiol. 70: 2126-2136, 1991). This model was used to interpret the pressure vs. time data obtained after the sudden occlusion of the arterial inflow (AO), venous outflow (VO), or both inflow and outflow (DO) from an isolated dog lung lobe. In the present study, we present a new approach to the data analysis in terms of this model that is relatively simple to carry out and more robust. The data used to estimate the R's and C's are the steady-state arterial [Pa(0)] and venous [Pv(0)] pressures, the flow rate (Q), the area (A2) encompassed by Pa(t) after AO and the equilibrium pressure (Pd) after DO, and the average slope (m) of the Pa(t) and Pv(t) curves after VO. The following formulas can then be used to calculate the 2 R's and 3 C's: [Pa(0) - Pv(0)]/Q = R1 + R2 = RT, R1C1 congruent to to A2/[Pa(0) - Pd], R1 congruent to [Pa(0) - Pd]/Q, Q/m = C1 + C2 + C3 = CT, and C2 = CT - (RTC1/R2).     

Effect of torso rotational strength on angular hip, angular shoulder, and linear bat velocities of high school baseball players

This investigation examined the effect of torso rotational strength on angular hip (AHV), angular shoulder (ASV), linear bat-end (BEV), and hand velocities (HV) and 3 repetition maximum (RM) torso rotational and sequential hip-torso-arm rotational strength (medicine ball hitter's throw) in high school baseball players (age 15.4 +/- 1.2 y). Participants were randomly assigned to 1 of 2 training groups. Group 1 (n = 24) and group 2 (n = 25) both performed a stepwise periodized resistance exercise program and took 100 swings a day, 3 days a week, for 12 weeks with their normal game bat. Group 2 performed additional rotational and full-body medicine ball exercises 3 days a week for 12 weeks. A 3RM parallel squat and bench press were measured at 0 and after 4, 8, and 12 weeks. Participants were pre- and posttested for 3RM dominant and nondominant torso rotational strength and medicine ball hitter's throw. Angular hip velocities, ASV, BEV, and HV were recorded pre- and posttraining by a motion capture system that identified and digitally processed reflective markers attached to each participant's bat and body. Groups 1 and 2 increased (p < or = 0.05) BEV (3.6 and 6.4%), HV (2.6 and 3.6%), 3RM dominant (10.5 and 17.1%) and nondominant (10.2 and 18.3%) torso rotational strength, and medicine ball hitter's throw (3.0 and 10.6%) after 12 weeks. Group 2 increased AHV (6.8%) and ASV (8.8%). Group 2 showed greater improvements in BEV, AHV, ASV, 3RM dominant and nondominant torso rotational strength, and medicine ball hitter's throw than group 1. Groups 1 and 2 increased predicted 1RM parallel squat (29.7 and 26.7%) and bench press (17.2 and 16.7%) strength after 12 weeks. These data indicate that performing additional rotational medicine ball exercises 2 days a week for 12 weeks statistically improves baseball performance variables.     

Antifouling performance of cross-linked hydrogels: refinement of an attachment model

Poly(ethylene glycol) dimethacrylate (PEGDMA), PEGDMA-co-glycidyl methacrylate (PEGDMA-co-GMA), and PEGDMA-co-hydroxyethyl methacrylate (PEGDMA-co-HEMA) hydrogels were polymerized using ammonium persulfate and ascorbic acid as radical initiators. Surface energies of the hydrogels and a standard, poly(dimethylsiloxane) elastomer (PDMSe), were characterized using captive bubble and sessile drop measurements, respectively (γ = 52 mN/m, γ(0) = 19 mN/m). The chemical composition of the hydrogels was characterized by attenuated total reflectance Fourier transform infrared (ATR-FTIR) spectroscopy. All three hydrogel compositions reduced significantly (p = 0.05) initial attachment of zoospores of the green alga Ulva linza (up to 97%), cells of the diatom Navicula incerta (up to 58%) and the bacterium Cobetia marina (up to 62%), compared to a smooth PDMSe standard. A shear stress (45 Pa), generated in a water channel, eliminated up to 95% of the initially attached cells of Navicula from the smooth hydrogel surfaces relative to smooth PDMSe surfaces. Compared to the PDMSe standard, 79% of the cells of C. marina were removed from all smooth hydrogel compositions when exposed to a 50 Pa wall shear stress. Attachment of spores of the green alga Ulva to microtopographies replicated in PEGDMA-co-HEMA was also evaluated. The Sharklet AF microtopography patterned, PEGDMA-co-HEMA surfaces reduced attachment of spores of Ulva by 97% compared to a smooth PDMSe standard. The attachment densities of spores to engineered microtopographies in PDMSe and PEGDMA-co-HEMA were shown to correlate with a modified attachment model through the inclusion of a surface energy term. Attachment densities of spores of Ulva to engineered topographies replicated in a material other than PDMSe are now correlated with the attachment model (R(2) = 0.80).     

Including surrounding tissue improves ultrasound-based 3D mechanical characterization of abdominal aortic aneurysms

ObjectivesIn this study the influence of surrounding tissues including the presence of the spine on wall stress analysis and mechanical characterization of abdominal aortic aneurysms using ultrasound imaging has been investigated.MethodsGeometries of 7 AAA patients and 11 healthy volunteers were acquired using 3-D ultrasound and converted to finite element based models. Model complexity of externally unsupported (aorta-only) models was complemented with inclusion of both soft tissue around the aorta and a spine support dorsal to the aorta. Computed 3-D motion of the aortic wall was verified by means of ultrasound speckle tracking. Resulting stress, strain, and estimated shear moduli were analyzed to quantify the effect of adding surrounding material supports.ResultsAn improved agreement was shown between the ultrasound measurements and the finite element tissue and spine models compared to the aorta-only models. Peak and 99-percentile Von Mises stress showed an overall decrease of 23–30%, while estimated shear modulus decreased with 12–20% after addition of the soft tissue. Shear strains in the aortic wall were higher in areas close to the spine compared to the anterior region.ConclusionsImproving model complexity with surrounding tissue and spine showed a homogenization of wall stresses, reduction in homogeneity of shear strain at the posterior side of the AAA, and a decrease in estimated aortic wall shear modulus. Future research will focus on the importance of a patient-specific spine geometry and location.     

Alteration of red cell membrane viscoelasticity by heat treatment: effect on cell deformability and suspension viscosity

The membrane shear elastic modulus (mu) and the time constant for extensional shape recovery (tc) were measured for normal, control human red blood cells (RBC) and for RBC heat treated (HT) at 48 degrees C. Three separate methods for the measurement of mu were compared (two used a micropipette and one employed a flow channel), and the membrane viscosity (n) was calculated from the relation n = mu. tc. The deformability of HT and control cells was evaluated using micropipette techniques, and the bulk viscosity of RBC suspensions at 40% hematocrit was measured. The shear elastic modulus, or "membrane rigidity", was more than doubled by heat treatment, although both the absolute value for mu and the estimate of the increase induced by heat treatment varied depending on the method of measurement. Heat treatment caused smaller increases in membrane viscosity and in membrane bending resistance, and only minimal changes in cell geometry. The deformability of HT cells was reduced: 1) the pressure required for cell entry (Pe) into 3 micrometers pipettes was increased, on average, by 170%; 2) at an aspiration pressure (Pa) exceeding Pe, longer times were required for cell entry into the same pipettes. However, when Pa was scaled relative to the mean entry pressure for a given sample (i.e, Pa/Pe), entry times were similar for control and HT cells. Bulk viscosity of HT RBC suspensions was elevated by approximately 12% on average (shear rates 75 to 1500 inverse seconds). These findings suggest that alteration of RBC membrane mechanical properties, similar to those induced by heat treatment, would most affect the in vivo circulation in regions where vessel dimensions are smaller than cellular diameters.     

Local measurements of viscoelastic parameters of adherent cell surfaces by magnetic bead microrheometry.

A magnetic bead microrheometer has been designed which allows the generation of forces up to 10(4) pN on 4.5 micron paramagnetic beads. It is applied to measure local viscoelastic properties of the surface of adhering fibroblasts. Creep response and relaxation curves evoked by tangential force pulses of 500-2500 pN (and approximately 1 s duration) on the magnetic beads fixed to the integrin receptors of the cell membrane are recorded by particle tracking. Linear three-phasic creep responses consisting of an elastic deflection, a stress relaxation, and a viscous flow are established. The viscoelastic response curves are analyzed in terms of a series arrangement of a dashpot and a Voigt body, which allows characterization of the viscoelastic behavior of the adhering cell surface in terms of three parameters: an effective elastic constant, a viscosity, and a relaxation time. The displacement field generated by the local tangential forces on the cell surface is visualized by observing the induced motion of assemblies of nonmagnetic colloidal probes fixed to the membrane. It is found that the displacement field decays rapidly with the distance from the magnetic bead. A cutoff radius of Rc approximately 7 micron of the screened elastic field is established. Partial penetration of the shear field into the cytoplasm is established by observing the induced deflection of intracellular compartments. The cell membrane was modeled as a thin elastic plate of shear modulus mu * coupled to a viscoelastic layer, which is fixed to a solid support on the opposite side; the former accounts for the membrane/actin cortex, and the latter for the contribution of the cytoskeleton to the deformation of the cell envelope. It is characterized by the coupling constant chi characterizing the elasticity of the cytoskeleton. The coupling constant chi and the surface shear modulus mu * are obtained from the measured displacements of the magnetic and nonmagnetic beads. By analyzing the experimental data in terms of this model a surface shear modulus of mu * approximately 2 . 10(-3) Pa m to 4 . 10(-3) Pa m is found. By assuming an approximate plate thickness of 0.1 micron one estimates an average bulk shear modulus of mu approximately (2 / 4) . 10(-4) Pa, which is in reasonable agreement with data obtained by atomic force microscopy. The viscosity of the dashpot is related to the apparent viscosity of the cytoplasm, which is obtained by assuming that the top membrane is coupled to the bottom (fixed) membrane by a viscous medium. By application of the theory of diffusion of membrane proteins in supported membranes we find a coefficient of friction of bc approximately 2 . 10(9) Pa s/m corresponding to a cytoplasmic viscosity of 2 . 10(3) Pa s.     

Microscopic investigation of erythrocyte deformation dynamics

The understanding of erythrocyte deformation under conditions of high shear stress and short exposure time is central to the study of hemorheology and hemolysis within prosthetic blood contacting devices. A combined computational and experimental microscopic study was conducted to investigate the erythrocyte deformation and its relation to transient stress fields. A microfluidic channel system with small channels fabricated using polydimethylsiloxane on the order of 100 mum was designed to generate transient stress fields through which the erythrocytes were forced to flow. The shear stress fields were analyzed by three-dimensional computational fluid dynamics. Microscopic images of deforming erythrocytes were experimentally recorded to obtain the changes in cell morphology over a wide range of fluid dynamic stresses. The erythrocyte elongation index (EI) increased from 0 to 0.54 with increasing shear stress up to 123 Pa. In this shear stress range, erythrocytes behaved like fluid droplets, and deformed and flowed following the surrounding fluid. Cells exposed to shear stress beyond 123 Pa (up to 5170 Pa) did not exhibit additional elongation beyond EI=0.54. Two-stage deformation of erythrocytes in response to shear stress was observed: an initial linear elongation with increasing shear stress and a plateau beyond a critical shear stress.