Shear Modulus Distribution Research Articles

In Situ Profiling of Soil Stiffness Parameters Using High-Resolution Fiber-Optic Distributed Sensing

AbstractThis paper explores the possibility of using high-resolution fiber-optic distributed sensing for in situ geotechnical estimation of soil shear modulus distribution with depth. It is shown that a recursive analysis of an elastic problem together with a measured vertical strain can assist in evaluating the sought stiffness values. It is suggested that high-resolution fiber-optic distributed sensing can provide the necessary strain for the proposed process. The approach was demonstrated in a field trial, entailing a stratified soil profile including a thin sand layer encapsulated between two clay layers. Results of the suggested profiling method are compared against data from a geophysical survey and against correlations with conventional in situ testing. Excellent agreement is exhibited between the different methods, indicating aptitude and viability of the idea for implementation as a geotechnical investigative tool.

Estimating the non-homogeneous elastic modulus distribution from surface deformations

We present a novel approach to solve the inverse problem in finite elasticity for the non-homogeneous shear modulus distribution solely from known surface deformation fields. The inverse problem is posed as a constrained optimization problem under regularization and solved utilizing the adjoint equations. Hypothetical “measured” surface displacement fields are created, by inducing indentations on the exterior of the specimen. These surface displacement fields are used to test the inverse strategy on a problem domain consisting of a stiff circular inclusion in a softer homogeneous background. We observe that the shear modulus reconstruction as well as the shape of the circular inclusion improves with an increasing number of surface displacement fields. Furthermore, with increasing noise level in the surface displacement field, the contrast of the reconstructions decreases.

The mechanical response of the porcine lens to a spinning test

The pig lens has been used as a model for presbyopia as pigs lack accommodative ability. Previous studies using microindentation have indicated that the shear modulus distribution is qualitatively similar to that of the aged human lens and that the lens does not alter its refractive power due to equatorial stretching. A lens spinning test was used to determine whether prior lens stiffness data obtained from a sectioned porcine lens were reliable and whether the testing conditions significantly influence the lens’ mechanical properties. The elastic modulus distribution determined for fresh lenses closely matched that measured previously using a microindentation test. Confocal scanning laser microscopy was used to evaluate changes to the lens’ structure arising from mechanical stress and following storage for up to one week.

A dimensionless analysis of shear modulus and stress distribution for anisotropic materials

ABSTRACTThe effect of various off­axis angles and lamina material properties on the shear modulus and the stress distribution of a circular cut-out in an orthotropic composite plate under shear loadings are presented. Based on the generalized Hooke's law, the generalized plane stress condition and the complex variable method, a dimensionless analysis is used to evaluate the influence of various elastic moduli E1, E2, G12 and ν12 on the laminate shear modulus and the stress distribution along the boundary of the circular cut-out within the laminate under shear loading. The results obtained from this dimensionless analysis provide a set of general design guidelines for structural laminates with high-precision requirements in engineering applications.

Stability analysis in magnetic resonance elastography II

We consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues. In the sequel to [5], we use pseudodifferential methods for the problem of recovering the shear modulus for Stokes systems from internal data. We prove stability estimates in d=2,3 with reduced regularity on the estimates and show that the presence of a finite dimensional kernel can be removed. This implies the convergence of the Landweber numerical iteration scheme. We also show that these hypotheses are natural for experimental use in constructing shear modulus distributions.

Stability analysis for Magnetic Resonance Elastography

We consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues. The measurements are made on the entirety of a smooth domain. Since tissues can be modeled as quasi-incompressible fluids, we examine the Stokes system and consider only the recovery of shear modulus distributions. Our main result is to establish Lipschitz stable estimates on the shear modulus distributions from internal measurements of displacement fields. These estimates imply convergence of a numerical scheme known as the Landweber iteration scheme for reconstructing the shear modulus distributions.

Mathematical Modeling in Full-Field Optical Coherence Elastography

We provide a mathematical analysis of and a numerical framework for full-field optical coherence elastography, which has unique features including micron-scale resolution, real-time processing, and noninvasive imaging. We develop a novel algorithm for transforming volumetric optical images before and after the mechanical solicitation of a sample with subcellular resolution into quantitative shear modulus distributions. This has the potential to improve sensitivities and specificities in the biological and clinical applications of optical coherence tomography.

Some closed-form solutions of functionally graded beams undergoing nonuniform torsion

Torsion of linearly elastic isotropic beams, with both cross-sectional and axial inhomogeneities, is analyzed. Twist (rate of torsional rotation along the beam axis) and warping of cross-sections are not uniform if arbitrary axial variations of elastic properties are considered. Composite beams undergoing nonuniform torsion are commonly investigated by finite and boundary element methods. New closed-form solutions are found in the present paper, by detecting axial distributions of longitudinal and shear moduli inducing an axially uniform warping field. The warping is evaluated by Saint-Venant beam theory, while twist and axial distribution of shear moduli are inversely proportional. Coordinate-free expressions of displacement, normal and shear stress fields are given for simply and multiply connected cross-sections. Exact solutions are obtained for elliptic and equilateral triangle beams, by assuming exponentially graded longitudinal and shear moduli. New benchmarks for numerical analyses are thus also provided.

The effect of high yttrium solute concentration on the twinning behaviour of magnesium alloys

The deformation behaviour of two single phase binary alloys, Mg–5Y and Mg–10Y, have been examined. In compression, two twin types were observed, the common {101¯2} twin as well as the less common {112¯1} extension twin. It is shown that the {112¯1} twin is much less sensitive to solute concentration than the {101¯2} twin, and it is suggested that the simple atomic shuffle of the {112¯1} twin reduces the solute strengthening imparted by Y additions. The common {101¯2} twin showed significant hardening as a result of alloying with Y. An analysis of solute behaviour has indicated that of the four chemical parameters investigated, i.e. atomic size, shear modulus, electronegativity and solute distribution, it appears to be the larger atomic radius of Y compared to Mg that increases the stress required to activate the {101¯2} twin. It is suggested that the large atomic radius inhibits the atomic shuffling process which accompanies the twinning shear in this twin type.

Quantitative sparse array vascular elastography: the impact of tissue attenuation and modulus contrast on performance.

Quantitative sparse array vascular elastography visualizes the shear modulus distribution within vascular tissues, information that clinicans could use to reduce the number of strokes each year. However, the low transmit power sparse array (SA) imaging could hamper the clinical usefulness of the resulting elastograms. In this study, we evaluated the performance of modulus elastograms recovered from simulated and physical vessel phantoms with varying attenuation coefficients (0.6, 1.5, and [Formula: see text]) and modulus contrasts ([Formula: see text], [Formula: see text], and [Formula: see text]) using SA imaging relative to those obtained with conventional linear array (CLA) and plane-wave (PW) imaging techniques. Plaques were visible in all modulus elastograms, but those produced using SA and PW contained less artifacts. The modulus contrast-to-noise ratio decreased rapidly with increasing modulus contrast and attenuation coefficient, but more quickly when SA imaging was performed than for CLA or PW. The errors incurred varied from 10.9% to 24% (CLA), 1.8% to 12% (SA), and [Formula: see text] (PW). Modulus elastograms produced with SA and PW imagings were not significantly different ([Formula: see text]). Despite the low transmit power, SA imaging can produce useful modulus elastograms in superficial organs, such as the carotid artery.

Analytical test functions for free vibration analysis of rotating non-homogeneous Timoshenko beams

In this paper, the governing equations for free vibration of a non-homogeneous rotating Timoshenko beam, having uniform cross-section, is studied using an inverse problem approach, for both cantilever and pinned-free boundary conditions. The bending displacement and the rotation due to bending are assumed to be simple polynomials which satisfy all four boundary conditions. It is found that for certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, the assumed polynomials serve as simple closed form solutions to the coupled second order governing differential equations with variable coefficients. It is found that there are an infinite number of analytical polynomial functions possible for material mass density, shear modulus and elastic modulus distributions, which share the same frequency and mode shape for a particular mode. The derived results are intended to serve as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of rotating non-homogeneous Timoshenko beams.

Closed-form solutions for axially functionally graded Timoshenko beams having uniform cross-section and fixed–fixed boundary condition

In this paper, we study the free vibration of axially functionally graded (AFG) Timoshenko beams, with uniform cross-section and having fixed–fixed boundary condition. For certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, there exists a fundamental closed form solution to the coupled second order governing differential equations with variable coefficients. It is found that there are an infinite number of non-homogeneous Timoshenko beams, with various material mass density, elastic modulus and shear modulus distributions having simple polynomial variations, which share the same fundamental frequency. The derived results can be used as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of non-homogeneous Timoshenko beams. They can also be useful for designing fixed–fixed non-homogeneous Timoshenko beams which may be required to vibrate with a particular frequency.

Biomechanical imaging of cell stiffness and prestress with subcellular resolution

Knowledge of cell mechanical properties, such as elastic modulus, is essential to understanding the mechanisms by which cells carry out many integrated functions in health and disease. Cellular stiffness is regulated by the composition, structural organization, and indigenous mechanical stress (or prestress) borne by the cytoskeleton. Current methods for measuring stiffness and cytoskeletal prestress of living cells necessitate either limited spatial resolution but with high speed, or spatial maps of the entire cell at the expense of long imaging times. We have developed a novel technique, called biomechanical imaging, for generating maps of both cellular stiffness and prestress that requires less than 30 s of interrogation time, but which provides subcellular spatial resolution. The technique is based on the ability to measure tractions applied to the cell while simultaneously observing cell deformation, combined with capability to solve an elastic inverse problem to find cell stiffness and prestress distributions. We demonstrated the application of this technique by carrying out detailed mapping of the shear modulus and cytoskeletal prestress distributions of 3T3 fibroblasts, making no assumptions regarding those distributions or the correlation between them. We also showed that on the whole cell level, the average shear modulus is closely associated with the average prestress, which is consistent with the data from the literature. Data collection is a straightforward procedure that lends itself to other biochemical/biomechanical interventions. Biomechanical imaging thus offers a new tool that can be used in studies of cell biomechanics and mechanobiology where fast imaging of cell properties and prestress is desired at subcellular resolution.

Study of deformation and shape recovery of NiTi nanowires under torsion

The nanomechanical properties, deformation, and shape recovery mechanism of NiTi nanowires (NWs) under torsion are studied using molecular dynamics simulations. The effects of loading rate, aspect ratio of NWs, and NW shape are evaluated in terms of atomic trajectories, potential energy, torque required for deformation, stress, shear modulus, centro-symmetry parameter, and radial distribution function. Simulation results show that dislocation nucleation starts from the surface and then extends to the interior along the {110} close-packed plane. For a high loading rate, the occurrence of torsional buckling of a NW is faster, and the buckling gradually develops near the location of the applied external loading. The critical torsional angle and critical buckling angle increase with aspect ratio of the NWs. Square NWs have better mechanical strength than that of circular NWs due to the effect of shape. Shape recovery naturally occurs before buckling.

Solution of the time‐harmonic viscoelastic inverse problem with interior data in two dimensions

SUMMARYWe consider the problem of determining the distribution of the complex‐valued shear modulus for an incompressible linear viscoelastic material undergoing infinitesimal time‐harmonic deformation, given the knowledge of the displacement field in its interior. In particular, we focus on the two‐dimensional problems of anti‐plane shear and plane stress. These problems are motivated by applications in biomechanical imaging, where the material modulus distributions are used to detect and/or diagnose cancerous tumors. We analyze the well‐posedness of the strong form of these problems and conclude that for the solution to exist, the measured displacement field is required to satisfy rather restrictive compatibility conditions. We propose a weak, or a variational formulation, and prove the existence and uniqueness of solutions under milder conditions on measured data. This formulation is derived by weighting the original PDE for the shear modulus by the adjoint operator acting on the complex‐conjugate of the weighting functions. For this reason, we refer to it as the complex adjoint weighted equation (CAWE). We consider a straightforward finite element discretization of these equations with total variation regularization, and test its performance with synthetically generated and experimentally measured data. We find that the CAWE method is, in general, less diffusive than a corresponding least squares solution, and that the total variation regularization significantly improves its performance in the presence of noise. Copyright © 2012 John Wiley & Sons, Ltd.

Effects of temperature, loading rate and nanowire length on torsional deformation and mechanical properties of aluminium nanowires investigated using molecular dynamics simulation

Single-crystal aluminium nanowires under torsion are studied using molecular dynamics simulations based on the many-body tight-binding potential. The effects of temperature, loading rate and nanowire length are evaluated in terms of atomic trajectories, potential energy, von Mises stress, a centrosymmetry parameter, torque, shear modulus and radial distribution function. Simulation results clearly show that torsional deformation begins at the surface, extends close to the two ends and finally diffuses to the middle part. The critical torsional angle which represents the beginning of plastic deformation varies with different conditions. Before the critical torsional angle is reached, the potential energy and the torque required for the deformation of a nanowire significantly increase with the torsional angle. The critical torsional angle increases with increasing nanowire length and loading rate and decreasing temperature. The torque required for the deformation decreases and the shear modulus increases with increasing nanowire length. For higher temperatures and higher loading rates, torsional buckling more easily occurs at the two ends of a nanowire, whereas it occurs towards the middle part at or below room temperature with lower loading rates. Geometry instability occurs before material instability (buckling) for a long nanowire.

Modeling shear modulus distribution in magnetic resonance elastography with piecewise constant level sets

Magnetic resonance elastography (MRE) is designed for imaging the mechanical properties of soft tissues. However, the interpretation of shear modulus distribution is often confusing and cumbersome. For reliable evaluation, a common practice is to specify the regions of interest and consider regional elasticity. Such an experience-dependent protocol is susceptible to intrapersonal and interpersonal variability. In this study we propose to remodel shear modulus distribution with piecewise constant level sets by referring to the corresponding magnitude image. Optimal segmentation and registration are achieved by a new hybrid level set model comprised of alternating global and local region competitions. Experimental results on the simulated MRE data sets show that the mean error of elasticity reconstruction is 11.33% for local frequency estimation and 18.87% for algebraic inversion of differential equation. Piecewise constant level set modeling is effective to improve the quality of shear modulus distribution, and facilitates MRE analysis and interpretation.

Estimating uncertainty in inverse elasticity with application to quantitative elastography

We consider an inverse elasticity problem motivated by medical ultrasound imaging: Given a displacement field measured in a 2D domain, determine the modulus distribution in that domain. An iterative approach to solve the inverse problem can be formulated by repeated solutions of the forward problem. That is, the shear modulus distribution sought is that which predicts a displacement field most consistent with the measured displacement field and any assumed a priori knowledge of the modulus distribution. All such inverse problem solutions are subject to uncertainties in the data, however, which results in uncertainties in the predictions. For diagnostic purposes, it is desirable to know the confidence intervals within which the stiffness at a point might reside. The focus of this presentation is the computation of said confidence intervals. We discuss the formulation of the problem within a Bayesian context. We derive a formal solution for the a posteriori probability distribution of the modulus. We prove bounds on uncertainty in terms of the data at the continuous level and discuss the computational solution of the problem at the discrete level.

Rheological Properties of Growth-Arrested Fibroblast Cells under Serum Starvation Measured by Atomic Force Microscopy

The rheological properties of growth-arrested and quiescent (G0 phase) mouse fibroblast cells under serum starvation were investigated by atomic force microscopy (AFM) with a microarray technique. The number distribution of complex shear modulus, G*, of quiescent cells at the serum concentration, CS=0.1%, followed a log-normal distribution, and the frequency dependence of G* exhibited a power law behavior, which were similar to those under a control condition at CS=10%. On the other hand, we found that the Newtonian viscosity coefficient of the quiescent cells significantly increased, and the distribution broadened, as compared with the control cells, whereas the power-law exponent was unchanged. The result indicated that the rheological properties of quiescent fibroblast cells were not identical to those in the G1 phase during cell cycle. This finding suggests that the Newtonian viscosity of cells is one of the useful indicators for evaluating growth-arrested cells under serum starvation.

Change in the Number Distribution of Complex Shear Modulus of Single Cells by Actin Cytoskeleton Modifications Measured by Atomic Force Microscopy

The rheological properties of living cells strongly depend on their cytoskeletal structures, which are composed of polymer networks and responsible for fundamental cellular functions. In particular, the actin network plays a major role in determining the rheological properties of living cells. In order to elucidate how the rheological properties of individual cells are affected by actin filamentous structures, we measured the number distribution of complex shear modulus of single cells, which were treated by actin modification drugs and cultured on microarray substrates, by atomic force microscopy. A force modulation mode experiment was employed to measure the complex shear modulus of single cells in a frequency range of 2-200Hz. When the cells were treated with actin-stabilizing drug, jasplakinolide, and actin-disrupting drug, cytochalasin D (CD), the storage modulus G’ and loss modulus G” increased and decreased, respectively. The changes in G” were smaller comparing to those in G’. The moduli exhibited a weak power-law dependence on frequency [1], whereas the increasing and decreasing of G’ and G” were accompanied by a decreasing and increasing power-law exponent respectively. Furthermore, their corresponding logarithmic standard deviation σ showed a slight change in the case of jasplakinolide treatment whereas it became small and attained a constant value at higher frequencies in CD treatment [2]. The results indicated that individual differences of cell rheology enhanced as actin cytoskeletal structures were stabilized. Furthermore, it was implied that the observed frequency dependence of σ was attributed to a frequency susceptibility of actin filaments.[1] B. Fabry et al., Phys. Rev. E., vol. 68, pp. 041914-041917, 2003.[2] S. Hiratsuka et al., Ultramicroscopy, vol. 109, pp. 937-941, 2009.