Viscoelastic Constants Research Articles

Efficient and optimized identification of generalized Maxwell viscoelastic relaxation spectra

Viscoelastic relaxation spectra are essential for predicting and interpreting the mechanical responses of materials and structures. For biological tissues, these spectra must usually be estimated from viscoelastic relaxation tests. Interpreting viscoelastic relaxation tests is challenging because the inverse problem is expensive computationally. We present here an efficient algorithm that enables rapid identification of viscoelastic relaxation spectra. The algorithm was tested against trial data to characterize its robustness and identify its limitations and strengths. The algorithm was then applied to identify the viscoelastic response of reconstituted collagen, revealing an extensive distribution of viscoelastic time constants.

The nano-epsilon dot method for strain rate viscoelastic characterisation of soft biomaterials by spherical nano-indentation

Nano-indentation is widely used for probing the micromechanical properties of materials. Based on the indentation of surfaces using probes with a well-defined geometry, the elastic and viscoelastic constants of materials can be determined by relating indenter geometry and measured load and displacement to parameters which represent stress and deformation.Here we describe a method to derive the viscoelastic properties of soft hydrated materials at the micro-scale using constant strain rates and stress-free initial conditions. Using a new self-consistent definition of indentation stress and strain and corresponding unique depth-independent expression for indentation strain rate, the epsilon dot method, which is suitable for bulk compression testing, is transformed to nano-indentation. We demonstrate how two materials can be tested with a displacement controlled commercial nano-indentor using the nano-espilon dot method (nano-ε̇M) to give values of instantaneous and equilibrium elastic moduli and time constants with high precision. As samples are tested in stress-free initial conditions, the nano-ε̇M could be useful for characterising the micro-mechanical behaviour of soft materials such as hydrogels and biological tissues at cell length scales.

Three-parameter shear wave inversion in MR elastography of incompressible transverse isotropic media: Application to in vivo lower leg muscles.

To develop and demonstrate MR elastography (MRE) for the measurement of three independent viscoelastic constants of skeletal muscle according to the theory of linear elasticity of incompressible materials with transverse isotropy (TI). Three-dimensional multifrequency MRE was applied to soleus, gastrocnemius, and tibialis anterior muscles in 10 healthy volunteers. The rotational wave fields were solved for complex-valued viscoelastic parameters μ12, μ13, and E3 corresponding to two shear moduli (within the planes of isotropy and symmetry of TI materials) and Young's modulus (along the principal fiber axis). Anisotropy was represented by the inequality μ12 < μ13 < 1/3E3 considering storage and loss properties of the soleus and gastrocnemius muscles, whereas storage shear moduli of tibialis were indistinguishable. Storage moduli were: 1.06 ± 0.12, 1.33 ± 0.10, 6.92 ± 0.95 kPa (soleus); 0.90 ± 0.11, 1.30 ± 0.15, 8.22 ± 1.37 kPa (gastrocnemius); 1.26 ± 0.16, 1.27 ± 0.11, 9.29 ± 1.42 kPa (tibialis), for μ12, μ13, and E3, respectively. The muscles were different in their μ12 and E3 values, whereas μ13 was less sensitive to the muscle type. Leg differences were observed in the soleus and gastrocnemius muscles. Recovery of the full elasticity tensor in incompressible TI materials is feasible by three-dimensional inversion of the time-harmonic shear wave equation. The method is potentially useful for the clinical evaluation of skeletal muscle anisotropy.

Characterisation of Elastic and Acoustic Properties of an Agar-Based Tissue Mimicking Material.

As a first step towards an acoustic localisation device for coronary stenosis to provide a non-invasive means of diagnosing arterial disease, measurements are reported for an agar-based tissue mimicking material (TMM) of the shear wavepropagation velocity, attenuation and viscoelastic constants, together with one dimensional quasi-static elastic moduli and Poisson's ratio. Phase velocity and attenuation coefficients, determined by generating and detecting shear waves piezo-electrically in the range 300Hz-2kHz, were 3.2-7.5ms(-1) and 320dBm(-1). Quasi-static Young's modulus, shear modulus and Poisson's ratio, obtained by compressive or shear loading of cylindrical specimens were 150-160kPa; 54-56kPa and 0.37-0.44. The dynamic Young's and shear moduli, derived from fitting viscoelastic internal variables by an iterative statistical inverse solver to freely oscillating specimens were 230 and 33kPa and the corresponding relaxation times, 0.046 and 0.036s. The results were self-consistent, repeatable and provide baseline data required for the computational modelling of wave propagation in a phantom.

Wave modes of a pre-stressed thick tube conveying blood on the viscoelastic foundation

Wave propagation of an artery is a fluid–structure interaction problem. It is very complicate. Therefore, the conventional theories for circulation of arteries are emphasized on fluid behavior, some simplified models for experimental utility or the thin-walled tube theory. Based on the geometry of an artery, the thick-walled tube theory is reasonable. In this study, a new mathematical model is proposed to describe the wave propagation through the isotropic elastic thick tube filled with viscous and incompressible fluid. Moreover, the tube is supported by the elastic muscle and simulated as the viscoelastic foundation. The radial, axial and flexural vibrations of a tube wall are introduced simultaneously. These wave modes are generally called as the flexural, Young and Lamb modes. In the literatures according to different assumptions, the Young and Lamb modes were independently derived and independent to the wave frequency. Because these conventional models are over simplified, the corresponding investigations are incomplete and inaccurate. Moreover, the present thick-walled tube theory is compared with the thin-walled tube theory and these conventional limiting theories. The dispersion curves and the energy transmissions of the three modes are investigated. It is illustrate that the energy transmitted through the artery tube is consistent to the experiment. Moreover, it is found that the effects of the viscoelastic foundation constants on the wave speed and the transmission is significant. When the foundation constant is large enough, some corresponding mode will disappear.

High Resolution Imaging of Viscoelastic Properties of Intracranial Tumours by Multi-Frequency Magnetic Resonance Elastography.

In recent years Magnetic Resonance Elastography (MRE) emerged into a clinically applicable imaging technique. It has been shown that MRE is capable of measuring global changes of the viscoelastic properties of cerebral tissue. The purpose of our study was to evaluate a spatially resolved three-dimensional multi-frequent MRE (3DMMRE) for assessment of the viscoelastic properties of intracranial tumours. A total of 27 patients (63 ± 13 years) were included. All examinations were performed on a 3.0 T scanner, using a modified phase-contrast echo planar imaging sequence. We used 7 vibration frequencies in the low acoustic range with a temporal resolution of 8 dynamics per wave cycle. Post-processing included multi-frequency dual elasto-visco (MDEV) inversion to generate high-resolution maps of the magnitude |G*| and the phase angle φ of the complex valued shear modulus. The tumour entities included in this study were: glioblastoma (n = 11), anaplastic astrocytoma (n = 3), meningioma (n = 7), cerebral metastasis (n = 5) and intracerebral abscess formation (n = 1). Primary brain tumours and cerebral metastases were not distinguishable in terms of |G*| and φ. Glioblastoma presented the largest range of |G*| values and a trend was delineable that glioblastoma were slightly softer than WHO grade III tumours. In terms of φ, meningiomas were clearly distinguishable from all other entities. In this pilot study, while analysing the viscoelastic constants of various intracranial tumour entities with an improved spatial resolution, it was possible to characterize intracranial tumours by their mechanical properties. We were able to clearly delineate meningiomas from intraaxial tumours, while for the latter group an overlap remains in viscoelastic terms.

Hydration Dependent Viscoelastic Tensile Behavior of Cornea

The cornea is a protective transparent connective tissue covering the front of the eye. The standard uniaxial tensile experiments are among the most popular techniques for investigating biomechanical properties of the cornea. This experimental method characterizes the stress-strain response of corneal strips immersed in a bathing solution. In the present study, the important roles of corneal hydration on tensile viscoelastic properties were investigated. The thickness was used as a surrogate for hydration and uniaxial tensile experiments were performed on bovine corneal samples with four different average thickness (hydration), i.e., 1100 μm (4.87 mg water/mg dry tissue), 900 μm (4.13 mg water/mg dry tissue), 700 μm (3.20 mg water/mg dry tissue), and 500 μm (1.95 mg water/mg dry tissue). The samples were immersed in mineral oil in order to prevent their swelling during the experiments. A quasilinear viscoelastic (QLV) model was used to analyze the experimental measurements and determine viscoelastic material constants. It was observed that both maximum and equilibrium (relaxed) stresses were exponentially increased with decreasing tissue thickness (hydration). Furthermore, the QLV model successfully captured the corneal viscoelastic response with an average R (2) value greater than 0.99. Additional experiments were conducted in OBSS in order to confirm that these significant changes in viscoelastic properties were because of corneal hydration and not the bathing solution. The findings of this study suggest that extra care must be taken in interpreting the results of earlier uniaxial tensile testings and their correspondence to the corneal biomechanical properties.

Measurements of the dielectric and viscoelastic constants in mixtures of 4,4'-n-octyl-cyanobiphenyl and biphenyl.

We performed measurements of the dielectric constants, splay elastic constant, and rotational viscosity of the nematic phase of mixtures of 4,4'-n-octyl-cyanobiphenyl (8CB) and biphenyl (BP). In contrast with previous results of DasGupta etal. [Phys. Rev. E 63, 041703 (2001); Phys. Lett. A 288, 323 (2001)], we do not find any anomaly of these constants when the smectic-A phase is approached at all concentrations of BP. These results are compatible with recent calorimetric measurements of Denolf etal. [Phys. Rev. Lett. 97, 107801 (2006); Phys. Rev. E 76, 051702 (2007)] and the absence of a tricritical point in the phase diagram. The origin of the anomalies observed by DasGupta etal. at large concentration of BP is also briefly discussed and a likely explanation in terms of biphenyl evaporation is proposed.

Wideband MRE and static mechanical indentation of human liver specimen: Sensitivity of viscoelastic constants to the alteration of tissue structure in hepatic fibrosis

Despite the success of elastography in grading hepatic fibrosis by stiffness related noninvasive markers the relationship between viscoelastic constants in the liver and tissue structure remains unclear. We therefore studied the mechanical properties of 16 human liver specimens with different degrees of fibrosis, inflammation and steatosis by wideband magnetic resonance elastography (MRE) and static indentation experiments providing the specimens׳ static Young׳s modulus (E), dynamic storage modulus (G′) and dynamic loss modulus (G″). A frequency-independent shear modulus μ and a powerlaw exponent α were obtained by fitting G′ and G″ using the two-parameter sprinpot model. The mechanical parameters were compared to the specimens׳ histology derived parameters such as degree of Fibrosis (F), inflammation score and fat score, amount of hydroxyproline (HYP) used for quantification of collagen, blood markers and presurgery in vivo function tests.The frequency averaged parameters G′, G″ and μ were significantly correlated with F (G′: R=0.762, G″: R=0.830; μ: R=0.744; all P<0.01) and HYP (G′: R=0.712; G″: R=0.720; μ: R=0.731; all P<0.01). The powerlaw exponent α displayed an inverse correlation with F (R=–0.590, P=0.034) and a trend of inverse correlation with HYP (R=–0.470, P=0.089). The static Young׳s modulus E was less correlated with F (R=0.587, P=0.022) and not sensitive to HYP. Although inflammation was highly correlated with F (R=0.773, P<0.001), no interaction was discernable between inflammation and mechanical parameters measured in this study. Other histological and blood markers as well as liver function test were correlated with neither F nor the measured mechanical parameters.In conclusion, viscoelastic constants measured by wideband MRE are highly sensitive to histologically proven fibrosis. Our results suggest that, in addition to the amount of connective tissue, subtle structural changes of the viscoelastic matrix determine the sensitivity of mechanical tissue properties to hepatic fibrosis.

A consistent approach for characterising asphalt concrete based on generalised Maxwell or Kelvin model

The generalised Maxwell (GM) model and generalised Kelvin (GK) model are widely used in characterising the linear viscoelastic (LVE) properties of asphalt concrete. Traditionally, the GM and GK model coefficients are determined by fitting the analytical LVE functions to pre-smoothed storage component data. The loss component data are usually disregarded primarily due to the fact that a well-accepted loss component master curve of asphalt concrete is not available to pre-smooth the experimental results. This study shows that the results obtained from the traditional approach may not satisfy some basic LVE requirements, namely the Kronig–Kramers relation between the storage and loss components, the convolution integral relation between the relaxation modulus and creep compliance and the applicability of the same time-temperature shift factor to various LVE functions. In addition, the relationships between the viscoelastic constants and model coefficients are usually neglected in the traditional approach which may lead to inconsistent LVE functions. A new procedure was proposed in this study to determine the GM and GK model coefficients using the collocation method. The experimental results are pre-smoothed using a complex modulus master curve represented by the modified Havriliak–Negami (MHN) model and then the model coefficients are solved from either the storage or loss component collocation points. The relation between viscoelastic constants and model coefficients is satisfied by introducing an additional linear equation besides those for the collocation points. It is shown that more densely spaced time constants (typically 0.5 decades apart) are necessary to eliminate any oscillation in the loss component master curves. The results show that the proposed approach effectively correct the deficiencies associated with the traditional approach. The GM model and GK model obtained from the approach are equivalent for the same material. The new approach provides a consistent and simple way for determining the coefficients of GM and GK model for asphalt concrete.

Skin viscoelasticity studied in vitro by microprobe-based techniques

Time-dependent deformation of porcine skin was studied in vitro using specialized microprobe instruments. The deformation behavior of stratum corneum, dermis, and whole skin is examined in the context of results of creep strain, elastic stiffness, and viscoelastic constants obtained in terms of the hold time, loading/unloading rate, and maximum indentation depth (load). Skin time-dependent deformation is significantly influenced by dermis viscoelasticity up to a critical indentation depth (load) beyond which it is controlled by the outermost hard epidermis, particularly stratum corneum. Skin viscoelastic behavior under constant load (creep) and constant displacement (stress relaxation) is interpreted in the light of phenomenological observations and experimental trends.

Thermo-mechanical characterization of glass at high temperature using the cylinder compression test. Part II: No-slip experiments, viscoelastic constants, and sensitivity

The limited availability of viscoelastic material constants for moldable glasses in the transition temperature range makes it difficult to apply computational mechanics to processes such as precision lens molding. In Part I of this two-part study, the cylinder compression test and standard parallel plate viscometer procedures were analyzed, since this test is convenient and relatively easy to conduct. The computational results showed that deformation of the glass cylinder is strongly dependent on both frictional slip at the interface and the viscosity of the glass, which is the key material property for shape change in forming processes. Motivated by these results, in the current study, a specimen was designed that did not allow slip, which results in a very accurate determination of viscosity from the cylinder compression test data and in turn enables a sufficient characterization of the viscoelastic component of the creep curve. The resulting combined experimental and computational procedure was validated with N-BK7 glass using accurate values of viscosity provided by the supplier and then this method was applied to L-BAL35 glass. Given accurate creep data and an accurate viscosity, the shear relaxation functions for the two glasses were determined. In order understand the requirements for a valid test and to best take advantage of the procedure, sensitivity analysis is performed on the height to diameter ratio of the cylinder, the degree of compressibility, imperfections in specimen geometry, and the effects of nonuniform temperature.

Towards an Elastographic Atlas of Brain Anatomy

Cerebral viscoelastic constants can be measured in a noninvasive, image-based way by magnetic resonance elastography (MRE) for the detection of neurological disorders. However, MRE brain maps of viscoelastic constants are still limited by low spatial resolution. Here we introduce three-dimensional multifrequency MRE of the brain combined with a novel reconstruction algorithm based on a model-free multifrequency inversion for calculating spatially resolved viscoelastic parameter maps of the human brain corresponding to the dynamic range of shear oscillations between 30 and 60 Hz. Maps of two viscoelastic parameters, the magnitude and the phase angle of the complex shear modulus, |G*| and φ, were obtained and normalized to group templates of 23 healthy volunteers in the age range of 22 to 72 years. This atlas of the anatomy of brain mechanics reveals a significant contrast in the stiffness parameter |G*| between different anatomical regions such as white matter (WM; 1.252±0.260 kPa), the corpus callosum genu (CCG; 1.104±0.280 kPa), the thalamus (TH; 1.058±0.208 kPa) and the head of the caudate nucleus (HCN; 0.649±0.101 kPa). φ, which is sensitive to the lossy behavior of the tissue, was in the order of CCG (1.011±0.172), TH (1.037±0.173), CN (0.906±0.257) and WM (0.854±0.169). The proposed method provides the first normalized maps of brain viscoelasticity with anatomical details in subcortical regions and provides useful background data for clinical applications of cerebral MRE.

Non-invasive characterization of intracranial tumors by magnetic resonance elastography

Presurgical, non-invasive methods of differentiating brain tumors have remained unsatisfactory even for specialized academic hospitals. Despite major advances in clinical and neuroradiological diagnostic techniques, the majority of neurooncology patients still need to undergo a brain biopsy for diagnosis. Recent single cell experiments suggested that biomechanical cell properties might be very sensitive in detecting cellular malignancy. Accordingly, we investigated magnetic resonance elastography (MRE) as an investigative tool for the clinical routine diagnostic work-up of intracranial neoplasm. In order to obtain sufficient spatial resolution for the biomechanical characterization of intracranial tumors, we modified a recently introduced least-squares solution of the stationary wave equation, facilitating stable solutions of the magnitude |G*| and the phase angle φ of the complex shear modulus G*. MRE was added to a routine diagnostic or presurgical neuroradiological magnetic resonance imaging work-up in 16 prospective patients and it was well tolerated in all cases. Our preliminary tumor MRE data revealed alterations in viscoelastic constants, e.g. a loss of stiffness in malignancies compared to healthy reference tissue, or benign variants. Based on larger studies on selected tumor entities to establish threshold and reference values for future diagnostic purposes, MRE may thus provide a predictive marker for tumor malignancy and thereby contribute to an early non-invasive clinical assessment of suspicious cerebral lesions.

Dynamic characteristics of damped viscoelastic nonlocal Euler–Bernoulli beams

The dynamic characteristics of damped viscoelastic nonlocal beams are studied in this paper. The Kelvin–Voigt and three-parameter standard viscoelastic models, velocity-dependent external damping and nonlocal Euler–Bernoulli beam theory are employed to establish the governing equations of motion for the bending vibration of nanobeams. A transfer function method (TFM) is developed to obtain closed-form and uniform solution for the vibration analysis of Euler–Bernoulli beams with different boundary conditions. New analytical expressions for critical viscoelastic parameters, damping parameters and limiting frequencies are obtained. Considering a carbon nanotube as a numerical example, the effects of the nonlocal and viscoelastic constants on the natural frequencies and damping factors are discussed. The results demonstrate the efficiency of the proposed modeling and analysis methods for free vibration analysis of viscoelastic damped nonlocal Euler–Bernoulli beams.

Vibration of nonlocal Kelvin–Voigt viscoelastic damped Timoshenko beams

This paper investigates the dynamic behavior of nonlocal viscoelastic damped nanobeams. The Kelvin–Voigt viscoelastic model, velocity-dependent external damping and Timoshenko beam theory are employed to establish the governing equations and boundary conditions for the bending vibration of nanotubes. Using transfer function methods (TFM), the natural frequencies and frequency response functions (FRF) are computed for beams with different boundary conditions. Unlike local structures, taking into account rotary inertia and shear deformation, the nonlocal beam has maximum frequencies, called the escape frequencies or asymptotic frequencies, which are obtained for undamped and damped nonlocal Timoshenko beams. Damped nonlocal beams are also shown to possess an asymptotic critical damping factor. Taking a carbon nanotube as a numerical example, the effects of the nonlocal parameter, viscoelastic material constants, the external damping ratio, and the beam length-to-diameter ratio on the natural frequencies and the FRF are investigated. The results demonstrate the efficiency of the proposed modeling and analysis methods for the free vibration and frequency response analysis of nonlocal viscoelastic damped Timoshenko beams.

Structure-sensitive elastography: on the viscoelastic powerlaw behavior of in vivo human tissue in health and disease

Elastography combines medical imaging with soft tissue mechanics and is used for the diagnosis of diseases associated with an altered stiffness of affected tissue. Beyond stiffness, dynamic elastography can measure viscoelastic constants sensitive to the network structure of polymers or biological materials. In this article current applications of in vivo multifrequency magnetic resonance elastography to healthy or diseased tissue are revisited in order to develop a unified framework for the interpretation of disease-related structural changes using viscoelastic powerlaw constants. The generalized view on different organs and processes such as liver fibrosis, neuronal tissue degradation, and muscle contraction reveals systematic signatures of the underlying microstructural changes to viscoelastic powerlaw constants. It is shown that in vivo powerlaw constants measured by elastography scale the mechanical properties of cellular networks into the macroscopic images obtained by magnetic resonance imaging (MRI) or ultrasound. This sensitivity to scales far below image resolution makes dynamic elastography an ideal diagnostic tool for the assessment of subtle alterations in living tissue occult to other medical imaging methods.

Use of wavelet energy for spinal cord vibration analysis during spinal surgery

An online non-contact measurement system using a laser displacement sensor was developed for obtaining the vibration amplitude of spinal cord and hard tissue. The discrete wavelet transform was used to extract the distinctive features of tissue vibration signals. The spinal cord and spinal cancellous bone can be discriminated by the comparison of wavelet energy over a characteristic scale. We also derived the integro-differential equation of motion to describe the spinal cord vibration excited by the motion of bone. Experimental results show that the method works well in identifying spinal cord and bone. However, available viscoelastic constants cannot describe the high-frequency features of spinal cord. The examined issue of tissue vibration due to the operation power device is a significant problem. The proposed method can be used by a surgery robot, and then spinal surgery may greatly benefit from the enhanced safety of robotics.

Numerical simulation of a relaxation test designed to fit a quasi-linear viscoelastic model for temporomandibular joint discs

The main objectives of this work are: (a) to introduce an algorithm for adjusting the quasi-linear viscoelastic model to fit a material using a stress relaxation test and (b) to validate a protocol for performing such tests in temporomandibular joint discs. This algorithm is intended for fitting the Prony series coefficients and the hyperelastic constants of the quasi-linear viscoelastic model by considering that the relaxation test is performed with an initial ramp loading at a certain rate. This algorithm was validated before being applied to achieve the second objective. Generally, the complete three-dimensional formulation of the quasi-linear viscoelastic model is very complex. Therefore, it is necessary to design an experimental test to ensure a simple stress state, such as uniaxial compression to facilitate obtaining the viscoelastic properties. This work provides some recommendations about the experimental setup, which are important to follow, as an inadequate setup could produce a stress state far from uniaxial, thus, distorting the material constants determined from the experiment. The test considered is a stress relaxation test using unconfined compression performed in cylindrical specimens extracted from temporomandibular joint discs. To validate the experimental protocol, the test was numerically simulated using finite-element modelling. The disc was arbitrarily assigned a set of quasi-linear viscoelastic constants (c1) in the finite-element model. Another set of constants (c2) was obtained by fitting the results of the simulated test with the proposed algorithm. The deviation of constants c2 from constants c1 measures how far the stresses are from the uniaxial state. The effects of the following features of the experimental setup on this deviation have been analysed: (a) the friction coefficient between the compression plates and the specimen (which should be as low as possible); (b) the portion of the specimen glued to the compression plates (smaller areas glued are better); and (c) the variation in the thickness of the specimen. The specimen's faces should be parallel to ensure a uniaxial stress state. However, this is not possible in real specimens, and a criterion must be defined to accept the specimen in terms of the specimen's thickness variation and the deviation of the fitted constants arising from such a variation.

파킨슨성 경직의 점탄성에 대한 수술중의 뇌심부 자극의 효과

Deep brain stimulation (DBS) of the subthalamic nucleus (STN) has been found to be effective treatment of Parkinson’s disease (PD). This study aims to evaluate the effect of DBS for rigidity during DBS surgery. Six Parkinsonian patients who received STN-DBS surgery participated in this study. The examiner imposed flexion and extension of a patient’s wrist randomly. Resistance to passive movement was quantified by viscoelastic properties (two damping constants for each of flexion and extension phase and one spring constant throughout both phases). All Viscoelastic constants decreased by DBS (p<0.01). Specifically, reduction in damping constant during flexion (Bf) was greater than those of damping constant during extension (Be) and of spring constant (p<0.05). Bf would be appropriate for evaluation of effect of DBS for rigidity during DBS surgery.