Shear Modulus Distribution Research Articles

Spatial distribution of local elastic moduli in nanocrystalline metals

Elastoplastic properties of nanocrystalline metals are nonuniform on the scale of the grain size, and this nonuniformity affects macroscopic quantities as, in these systems, a significant part of the material is at or adjacent to a grain boundary. We use molecular dynamics simulations to study the spatial distributions of local elastic moduli in nanograined pure metals and analyze their dependence on grain size. Calculations are performed for copper and tantalum with grain sizes ranging from 5 to 20 nm. Shear-modulus distributions for grain and grain-boundary atoms were calculated. It is shown that the noncrystalline grain boundary has a wide shear-modulus distribution, which is grain-size independent, while grains have a peaked distribution, which becomes sharper with increasing grain size. Average elastic moduli of the bulk, grains, and grain boundary are calculated as a function of grain size. The atomistic simulations show that the reduction of total elastic moduli with decreasing grain size is mainly due to a resulting larger grain-boundary atoms fraction, and that the total elastic moduli can be approximated by a simple weighted average of larger grain elastic moduli and a lower grain-boundary elastic moduli. Published by the American Physical Society 2024

Static response of vertically loaded pile in inhomogeneous soil

In this study, an analytical model is developed to obtain the static response of the vertically loaded pile in the inhomogeneous soil with arbitrary properties varying along the depth. The soil displacement is expressed by the product of the axial displacement function and the shape function attenuating with the distance from the pile periphery. Through the minimum potential energy principle, the governing equations are established for the axial displacement function and the shape function. The governing equation for the shape function is independent of soil properties and it can be explicitly solved. The governing equation for the axial displacement function contains variable coefficients due to the inhomogeneity of soil and it is solved using the weighted residual method. An iterative procedure is introduced to implement the solution. The proposed model is validated by comparing the present solutions with those obtained by the available methods. The parametric study shows that the pile head stiffness is affected by the distribution of soil shear modulus heavily, in which the average shear moduli are the same for these distributions. Moreover, the relationship between the pile head stiffness and the pile slenderness ratio for a floating pile is different from that for an end-bearing pile.

2D boundary-condition-free nonlinear inversion technique applied to optical shear vibration induced microelastography

Optical microelastography (OME) has emerged as a new technique for quantifying cellular mechanical properties. However, accurately reconstructing viscoelastic properties at the microscale level from noisy 2D displacement fields remains a challenge. This study introduces a 2D boundary-condition-free nonlinear inversion (2D-NoBC-NLI) approach, addressing challenges of interpreting noisy data and deducing full-field 3D displacements from 2D measurements. OME requires vibrating the cell and mapping the shear modulus based on wave-induced displacements within the cell. The shear modulus distribution is recovered via a coupled adjoint field NLI reconstruction to allow 2D-NoBC-NLI. Validation was conducted through numerical simulations at 36 kHz on a homogeneous sphere of 75 μm diameter and an assigned viscoelastic modulus, G*, of 800 + i150 Pa. The same reconstruction approach was also applied to experimental data obtained from polyacrylamide (PAAm) microbeads of the same diameter. Results demonstrated relative differences from true simulated values of 0.7% and 45% for storage and loss moduli, respectively, with a coefficient of variation under 1% for homogeneous regions. When applying this method to PAAm microbeads, viscoelastic reconstructions showed the potential of OME under experimental conditions. These findings highlight the accuracy of 2D-No BC-NLI reconstruction in OME for precise microscale characterization and mapping of the viscoelastic cell structure.

Experimental model updating of slope considering spatially varying soil properties and dynamic loading

AbstractThe widespread threat posed by slope structure failures to human lives and property safety is widely acknowledged. Additionally, natural soil often displays spatial variability due to geological deposition and other factors. Therefore, predicting the seismic response of slopes subjected to ground motions and inversely analyzing the spatial distribution of soils remains an unresolved issue. In the present work, a shaking table experimental test is first designed and carried out, in which a soft‐soil slope dynamic system is established. To capture the seismic response of the soft‐soil slope, specifically the experimental characteristic of acceleration and soil pressure response in both spatial domain and time domain, a series of sensors were pre‐embedded in the slope. Subsequently, a model updating approach is proposed for slope seismic analysis that incorporates spatial variability of soil. In addition, to enhance computational efficiency, the dimensionality reduction of Karhunen–Loève expansion method is introduced to reduce inverse analysis parameters. On the basis of 34 samples collected from experimental data, it is shown that near‐fault pulse‐like ground motions deliver greater concentrated energy, causing more severe damage to slope structures, especially the topsoil layer. Furthermore, using data obtained from a shaking table test subjected to ground motion Recorded Sequence Number 158H1 from the Pacific Earthquake Engineering Research Center NGA‐West2 database as an example, it is also shown that the proposed approach demonstrates high accuracy in predicting the spatial distribution of the maximum shear modulus in soil slope dynamic systems. The present work not only addresses the challenges posed by mainshock–aftershock effects but also highlights the potential of model updating approaches to enhance the understanding of slope behavior under seismic loading conditions.

Local and global measures of the shear moduli of jammed disk packings.

Strain-controlled isotropic compression gives rise to jammed packings of repulsive, frictionless disks with either positive or negative global shear moduli. We carry out computational studies to understand the contributions of the negative shear moduli to the mechanical response of jammed disk packings. We first decompose the ensemble-averaged, global shear modulus as 〈G〉=(1-F_{-})〈G_{+}〉+F_{-}〈G_{-}〉, where F_{-} is the fraction of jammed packings with negative shear moduli and 〈G_{+}〉 and 〈G_{-}〉 are the average values from packings with positive and negative moduli, respectively. We show that 〈G_{+}〉 and 〈|G_{-}|〉 obey different power-law scaling relations above and below pN^{2}∼1. For pN^{2}>1, both 〈G_{+}〉N and 〈|G_{-}|〉N∼(pN^{2})^{β}, where β∼0.5 for repulsive linear spring interactions. Despite this, 〈G〉N∼(pN^{2})^{β^{'}} with β^{'}≳0.5 due to the contributions from packings with negative shear moduli. We show further that the probability distribution of global shear moduli P(G) collapses at fixed pN^{2} and different values of p and N. We calculate analytically that P(G) is a Γ distribution in the pN^{2}≪1 limit. As pN^{2} increases, the skewness of P(G) decreases and P(G) becomes a skew-normal distribution with negative skewness in the pN^{2}≫1 limit. We also partition jammed disk packings into subsystems using Delaunay triangulation of the disk centers to calculate local shear moduli. We show that the local shear moduli defined from groups of adjacent triangles can be negative even when G>0. The spatial correlation function of local shear moduli C(r[over ⃗]) displays weak correlations for pn_{sub}^{2}<10^{-2}, where n_{sub} is the number of particles within each subsystem. However, C(r[over ⃗]) begins to develop long-ranged spatial correlations with fourfold angular symmetry for pn_{sub}^{2}≳10^{-2}.

Reconstructing the Spatial Distribution of the Relative Shear Modulus in Quasi-static Ultrasound Elastography: Plane Stress Analysis.

Quasi-static ultrasound elastography (QSUE) is an imaging technique that mainly provides axial strain maps of tissues when the latter are subjected to compression. In this article, a method for reconstructing the relative shear modulus distribution within a linear elastic and isotropic medium, in QSUE, is introduced. More specifically, the plane stress inverse problem is considered. The proposed method is based on the variational formulation of the equilibrium equations and on the choice of adapted discretization spaces, and only requires displacement fields in the analyzed media to be determined. Results from plane stress and 3-D numerical simulations, as well as from phantom experiments, showed that the method is able to reconstruct the different regions within a medium, with shear modulus contrasts that unambiguously reveal whether inclusions are stiffer or softer than the surrounding material. More specifically, for the plane stress simulations, inclusion-to-background modulus ratios were found to be very accurately estimated, with an error lower than 3%. For the 3-D simulations, for which the plane stress conditions are no longer satisfied, these ratios were, as expected, less accurate, with an error that remained lower than 10% for two of the three cases analyzed but was around 34% for the last case. Concerning the phantom experiments, a comparison with a shear wave elastography technique from a clinical ultrasound scanner was also made. Overall, the inclusion-to-background shear modulus ratios obtained with our approach were found to be closer to those given by the phantom manufacturer than the ratios provided by the clinical system.

Measurement of temperature dependent shear modulus using frequency dependent SAWs ToF delay in laser-induced temperature gradient

Phase velocity of surface acoustic waves(SAWs) is frequency dependent when propagate on medium with inhomogeneous elasticity over depth. In this work, frequency dependent Time-of-Flight (ToF) variation of SAWs induced by a local and dynamic heating was applied for measurement of temperature dependent shear modulus. Laser-generated broad-band SAWs propagated through the material with elastic properties and density modified by dynamic inhomogeneous temperature field induced by a millisecond laser heating. Sample with spatial dependent material properties introduces phase velocity dispersion in the SAW propagation. As consequence, ToF of SAW becomes frequency dependent. Frequency dependent ToF variation curves at two time instants respected to laser heating were measured by time–frequency analysis together with a differential technique. Free fitted parameters, temperature dependent shear modulus and surface temperature distribution were evaluated by solving the inverse problem by fitting the experimental ToF variation curves into the theoretical ones by means of the differential evolution method. The inversed temperature dependent parameter of shear modulus of Ti–6Al–4V alloy was in good agreement with literature value.

Optical coherence elastography of 3D bilayer soft solids using full-field and partial displacement measurements

Characterizing nonhomogeneous elastic property distribution of soft tissues plays a crucial role in disease diagnosis and treatment. In this paper, we will apply the optical coherence elastography to reconstruct the shear modulus elastic property distribution of a bilayer solid. In the computational aspect, we adopt a well-established inverse technique that solves for every nodal shear modulus in the problem domain (NO method). Additionally, we also propose a novel inverse method that assumes the shear moduli merely vary along the thickness of the bilayer solid (TO method). The inversion tests using simulated data demonstrate that TO method performs better in reconstructing the shear modulus distribution. Further, we utilize the experimental data obtained from the optical coherence tomography to reconstruct the shear modulus distribution of a bilayer phantom. We observe that the quality of the reconstructed shear modulus distribution obtained by the partial displacement measurement is better than that obtained by the full-field displacement measurement. Particularly, merely using the displacement component along the loading direction significantly improves the reconstructed results. This work is of great significance in applying optical coherence elastography (OCE) to characterize the elastic property distribution of layered soft tissues such as skins and corneas.

Analysis of torsionally loaded non-uniform circular piles in multi-layered non-homogeneous elastic soils

A new method for the torsional analysis of non-uniform circular piles partially or fully embedded in multi-layered elastic soils is developed. The mechanical response of non-uniform piles such as those with a linear variation in cross-section (i.e., tapered piles) or discontinuous variations (i.e., stepped piles), and reacting against a non-homogeneous elastic soil can be easily investigated with the proposed formulation. The non-homogeneity of the soil is incorporated by assuming a shear modulus distribution that fits a quadratic equation (i.e., G(z)=Go+sz+tz2). The governing differential equation (GDE) of a single non-uniform circular pile segment is derived and solved using the differential transformation method (DTM); then, the stiffness matrix of the segment is found by applying equilibrium and compatibility conditions at the ends of the element. The analysis of non-uniform piles in multi-layered soils is conducted by dividing the pile into multiple segments (i.e., each segment representing a change in soil properties or pile geometry) and then assembling them using classic matrix methods. The proposed matrix formulation is simple to implement, easy to incorporate into already available structural matrix analysis codes and provides accurate results when compared with more cumbersome analytical and numerical solutions. The simplicity, practicality and accuracy of the method is validated with five illustrative examples.

Effect of spatial heterogeneity on level of rejuvenation in Ni80P20 metallic glass

Understanding the relation between spatial heterogeneity and structural rejuvenation is one of the hottest topics in the field of metallic glasses (MGs). In this work, molecular dynamics (MD) simulation is implemented to discover the effects of initial spatial heterogeneity on the level of rejuvenation in the Ni80P20MGs. For this purpose, the samples are prepared with cooling rates of 1010 K/s–1012 K/s to make glassy alloys with different atomic configurations. Firstly, it is found that the increase in the cooling rate leads the Gaussian-type shear modulus distribution to widen, indicating the aggregations in both elastically soft and hard regions. After the primary evaluations, the elastostatic loading is also used to transform structural rejuvenation into the atomic configurations. The results indicate that the sample with intermediate structural heterogeneity prepared with 1011 K/s exhibits the maximum structural rejuvenation which is due to the fact that the atomic configuration in an intermediate structure contains more potential sites for generating the maximum atomic rearrangement and loosely packed regions under an external excitation. The features of atomic rearrangement and structural changes under the rejuvenation process are discussed in detail.

Stiffness matrix method for analysis of torsionally loaded piles in multi-layered soil

A new, simple and practical method to investigate the response of torsionally loaded piles on homogeneous or non-homogeneous multi-layered elastic soil was developed. Soil non-homogeneity is accounted for by assuming, for each layer, a shear modulus distribution that fits a quadratic function. Analysis of piles in multi-layered soil is carried out by subdividing the pile, at the soil–soil layer and soil–air interfaces, into multiple elements and then using conventional matrix methods – such as those commonly implemented in structural analysis – to connect them. The governing differential equation of an individual structural element is solved using the differential transformation method. The stiffness matrix is then derived by applying compatibility conditions at the ends of the element. Piles partially or fully embedded in multiple layers and subjected to torsion can be analysed in a simple manner with the proposed formulation – a tedious endeavour with other available solutions. Explicit expressions for the coefficients of the matrix are provided and four examples are presented to show the simplicity, accuracy and capability of the proposed formulation.

Influence of pia-arachnoid complex on the indentation response of porcine brain at different length scales

Brain tissues are surrounded by two tightly adhering thin membranes known as the pia-arachnoid complex (PAC), which is pivotal in regulating brain mechanical response upon mechanical impact. Despite the crucial role of PAC as a structural damper protecting the brain, its mechanical contribution has received minimal attention. In this work, the mechanical contribution of PAC on brain tissues against mechanical loading is characterized by using a custom-built indentation apparatus. The indentation responses of the isolated and PAC-overlaid brains are quantitatively compared at different length scales and strain rates. Results show that PAC substantially affects the indentation response of brain tissues at micro- and macro-scales and provides better protection against mechanical impact at a relatively small (μm) length scale. The modulus of the PAC-overlaid brain shows a threefold stiffening at the microscale compared with that of the isolated brain (with instantaneous shear modulus distribution means of 0.85 ± 0.14 kPa versus 2.64 ± 0.43 kPa at the strain rate of 0.64 s−1 and 1.40 ± 0.31 kPa versus 4.02 ± 0.51 at 1.27 s−1). These findings indicate that PAC seriously affects the mechanical response of brain tissues, especially at the microscale, and may have important implications for the studies of brain injury.

Site Investigations of the Lacustrine Clay in Taihu Lake, China, Using Self-Boring Pressuremeter Test

Site investigations of the soils are considered very important for evaluation of the site conditions, as well as the design and construction for the project built in it. Taihu tunnel is thus far the longest tunnel constructed in the lake in China, with an entire length of over 10 km. However, due to the very insufficient site data obtained for the lacustrine clay in the Taihu lake area, a series of self-boring pressuremeter (SBPM) field tests was therefore carried out. Undrained shear strengths were deduced from the SBPM test, with the results showing generally higher than those obtained from the laboratory tests, which may be attributed to the disturbance to the soil mass during the sampling process. Degradation characteristics of the soil shear modulus (Gs) were mainly investigated, via a thorough comparison between different soil layers, and generally, the shear modulus would cease its decreasing trends and become stable when the shear strain reaches over 1%. Meanwhile, it was found that a linear relationship between the plasticity index and the shear modulus, and between the decay rate of the shear modulus and the plasticity index as well, could be developed. Further statistical analysis over the undrained shear strength and shear modulus distribution of the soils shows that the undrained shear strength of the soils follows a normal distribution, while the shear modulus follows a log-normal distribution. More importantly, the spatial correlation length of the shear modulus is found much smaller than that of the undrained strength.

Impact of axisymmetric deformation on MR elastography of a nonlinear tissue-mimicking material and implications in peri-tumour stiffness quantification.

Solid tumour growth is often associated with the accumulation of mechanical stresses acting on the surrounding host tissue. Due to tissue nonlinearity, the shear modulus of the peri-tumoural region inherits a signature from the tumour expansion which depends on multiple factors, including the soft tissue constitutive behaviour and its stress/strain state. Shear waves used in MR-elastography (MRE) sense the apparent change in shear modulus along their propagation direction, thereby probing the anisotropic stiffness field around the tumour. We developed an analytical framework for a heterogeneous shear modulus distribution using a thick-shelled sphere approximation of the tumour and soft tissue ensemble. A hyperelastic material (plastisol) was identified to validate the proposed theory in a phantom setting. A balloon-catheter connected to a pressure sensor was used to replicate the stress generated from tumour pressure and growth while MRE data were acquired. The shear modulus anisotropy retrieved from the reconstructed elastography data confirmed the analytically predicted patterns at various levels of inflation. An alternative measure, combining the generated deformation and the local wave direction and independent of the reconstruction strategy, was also proposed to correlate the analytical findings with the stretch probed by the waves. Overall, this work demonstrates that MRE in combination with non-linear mechanics, is able to identify the apparent shear modulus variation arising from the strain generated by a growth within tissue, such as an idealised model of tumour. Investigation in real tissue represents the next step to further investigate the implications of endogenous forces in tissue characterisation through MRE.

Understanding the scaling of boson peak through insensitivity of elastic heterogeneity to bending rigidity in polymer glasses

Amorphous materials exhibit peculiar mechanical and vibrational properties, including non-affine elastic responses and excess vibrational states, i.e., the so-called boson peak (BP). For polymer glasses, these properties are considered to be affected by the bending rigidity of the constituent polymer chains. In our recent work [Tomoshige, et al 2019, Sci. Rep. 9 19514], we have revealed simple relationships between the variations of vibrational properties and the global elastic properties: the response of the BP scales only with that of the global shear modulus. This observation suggests that the spatial heterogeneity of the local shear modulus distribution is insensitive to changes in the bending rigidity. Here, we demonstrate the insensitivity of elastic heterogeneity by directly measuring the local shear modulus distribution. We also study transverse sound wave propagation, which is also shown to scale only with the global shear modulus. Through these analyses, we conclude that the bending rigidity does not alter the spatial heterogeneity of the local shear modulus distribution, which yields vibrational and acoustic properties that are controlled solely by the global shear modulus of a polymer glass.

Salinity effect on the compaction behaviour, matric suction, stiffness and microstructure of a silty soil

To better understand the salinity effect on the compaction behaviour of soil, standard Proctor compaction test was conducted on soil samples with different salinities. Matric suction and small-strain shear modulus, Gmax, were determined and pore size distribution was also investigated on samples statically compacted at different water contents. Results showed that with the decrease of soil salinity from initial value of 2.1‰ (g of salt/kg of dry soil) to zero, the maximum dry density increased and the optimum water content decreased, whereas there was no significant change with the increase of soil salinity from 2.1‰ to 6.76‰. Interestingly, it was observed that Gmax also decreased when the soil salinity decreased from initial value of 2.1‰ to zero and kept almost constant when the soil salinity increased from 2.1‰ to 6.76‰, for dry samples with similar matric suction and also for samples compacted at optimum state and on wet side whose matric suctions were slightly different due to the difference in remoulded water content. Furthermore, the effect of salinity on compaction behaviour and Gmax decreased for samples compacted from dry side to wet side. The pore size distribution exhibited bi-modal characteristics with two populations of micro- and macro-pores not only for samples compacted on dry side and at optimum state, but also for those compacted on wet side. Further examination showed that the modal size of micro-pores shifted to lower values and that of macro-pores shifted to higher values for saline soil compared to the soil without salt.

A deep learning approach to the inverse problem of modulus identification in elasticity

The inverse elasticity problem of identifying elastic modulus distribution based on measured displacement/strain fields plays a key role in various non-destructive evaluation (NDE) techniques used in geological exploration, quality control, and medical diagnosis (e.g., elastography). Conventional methods in this field are often computationally costly and cannot meet the increasing demand for real-time and high-throughput solutions for advanced manufacturing and clinical practices. Here, we propose a deep learning (DL) approach to address this challenge. By constructing representative sampling spaces of shear modulus distribution and adopting a conditional generative adversarial net, we demonstrate that the DL model can learn high-dimensional mapping between strain and modulus via training over a limited portion of the sampling space. The proposed DL approach bypasses the costly iterative solver in conventional methods and can be rapidly deployed with high accuracy, making it particularly suitable for applications such as real-time elastography and highthroughput NDE techniques.

Linear one-dimensional site response analysis in the presence of stiffness-less free surface for certain power-law heterogeneities

We revisit previous results in small-strain One-dimensional Site Response Analysis of heterogeneous soil deposits. Specifically, we focus on sites whose shear modulus distribution is described by means of a power law that yields zero stiffness at the free surface. First, we show that in some cases (which we characterize in detail) considerations of energy finitude should prevail over considerations of vanishing tractions at the free-surface, as these may pose acuter constrains. We re-evaluate previous contributions in light of this result. Second, we analyze the previously-reported occurrence of “energy accumulation in upper layers”, providing a physical explanation for it. In passing, we supply estimates of the natural frequencies, and compare these with our previous results.

Moving Morphable Inclusion Approach: An Explicit Framework to Solve Inverse Problem in Elasticity

Abstract In this work, we present a novel inverse approach to characterize the nonhomogeneous mechanical behavior of linear elastic solids. In this approach, we optimize the geometric parameters and shear modulus values of the predefined moving morphable inclusions (MMIs) to solve the inverse problem. Thereby, the total number of the optimization parameters is remarkably reduced compared with the conventional iterative inverse algorithms to identify the nonhomogeneous shear modulus distribution of solids. The proposed inverse approach is tested by multiple numerical examples, and we observe that this approach is capable of preserving the shape and the shear moduli of the inclusions well. In particular, this inverse approach performs well even without any regularization when the noise level is not very high. Overall, the proposed approach provides a new paradigm to solve the inverse problem in elasticity and has potential of addressing the issue of computational inefficacy existing in the conventional inverse approaches.

Theoretical solution for the axial vibration of functionally graded double-lap adhesive joints

Adhesively bonded joints have widely been used in a number of engineering applications, owing to their improved mechanical performance as compared with other mechanical joining techniques, such as rivets or bolts. In this study, a theoretical solution of double-lap joints is established, with functionally graded adhesive and isotopic adherends, under harmonic loads. Assuming a parabolic distribution of the adhesive shear modulus along the overlap length, the analytical harmonic solution is expressed in terms of the solution of the nonlinear Heun differential equation. Furthermore, a two-dimensional finite-element model was developed to validate the analytical solution. We show that using softer adhesive at the edges of the bond line leads to less stress concentration. We also conclude that the adhesive’s shear modulus gradient along the bond line should be greater as the ratio of the adhesive’s shear stiffness to the adherends’ shear stiffness increases. An equation has also been established to determine the optimal shear modulus gradient in terms of the stiffness ratio.