Stiff Inclusions Research Articles

Asymptotic analysis of the stress concentration between two adjacent stiff inclusions in all dimensions

In the region between two closely located stiff inclusions, the stress, which is the gradient of a solution to the Lamé system with partially infinite coefficients, may become arbitrarily large as the distance between interfacial boundaries of inclusions tends to zero. The primary aim of this paper is to give a sharp description in terms of the asymptotic behavior of the stress concentration, as the distance between interfacial boundaries of inclusions goes to zero. For that purpose, we capture all the blow‐up factor matrices, whose elements comprise of some certain integrals of the solutions in the case when two inclusions are touching. Then, we are able to establish the asymptotic formulas of the stress concentration in the presence of two close‐to‐touching ‐convex inclusions in all dimensions. Furthermore, an example of curvilinear squares with rounded‐off angles is also presented for future application in numerical computations and simulations.

Computational Investigation of Vessel Injury Due to Catheter Tracking During Transcatheter Aortic Valve Replacement.

Catheter reaction forces during transcatheter valve replacement (TAVR) may result in injury to the vessel or plaque rupture, triggering distal embolization or thrombosis. In vitro test methods represent the arterial wall using synthetic proxies to determine catheter reaction forces during tracking, but whether they can account for reaction forces within the compliant aortic wall tissue in vivo is unknown. Moreover, the role of plaque inclusions is not well understood. Computational approaches have predicted the impact of TAVR positioning, migration, and leaflet distortion, but have not yet been applied to investigate aortic wall reaction forces and stresses during catheter tracking. In this study, we investigate the role that catheter design and aorta and plaque mechanical properties have on the risk of plaque rupture during TAVR catheter delivery. We report that, for trackability testing, a rigid test model provides a reasonable estimation of the peak reaction forces experienced during catheter tracking within compliant vessels. We investigated the risk of rupture of both the aortic tissue and calcified plaques. We report that there was no risk of diseased aortic tissue rupture based on an accepted aortic tissue stress threshold (4.2MPa). However, we report that both the aortic and plaque tissue exceed a rupture stress threshold (300kPa) with and without the presence of stiff and soft plaque inclusions. We also highlight the potential risks associated with shorter catheter tips during catheter tracking and demonstrate that increasing the contact surface will reduce peak contact pressures experienced in the tissue.

Modeling and Simulation of Robotic Palpation to Detect Subsurface Soft Tissue Anomaly for Presurgical Assessment

Abstract Surgical Haptics is an emergent field of research to integrate and advance the sense of robotic touch in laparoscopic tools in robot-assisted minimally invasive surgery. Haptic feedback from the tooltip and soft tissue surface interaction during robotic palpation can be leveraged to detect the texture and contour of subsurface geometry. However, precise force modulation of the robotic palpating probe is necessary to determine stiff inclusions of the anatomy and maneuver successive manipulation tasks during surgery. This paper focuses on investigating the layered deformations associated with different force profiles involved in manipulating the superficial anatomy of soft tissues during dynamic robotic palpation to determine the underlying anomaly. A realistic three-dimensional (3D) cross-sectional soft tissue phantom with anatomical layers and tumor, as an anomaly, is designed, modeled, and analyzed to examine the effects of oriented palpating forces (0–5 N) of a 7 DOF robot arm equipped with a contoured palpation probe. Finite element static structural analysis of oriented robotic palpation on the developed 3D soft tissue phantoms (with and without anomaly) reveals the soft tissue layer deformations and associated strains needed to identify presence of stiffer inclusions or anomaly during Robotic palpation. The finite element analysis study shows that the difference in deformations of soft tissue layers (e.g., underlying myofascial layers) under stiffer inclusions at different force levels can facilitate haptic feedback to acquire information about subsurface tumors. The deformation variations are further compared to assess better palpation orientations for subsurface anomaly detection.

Numerical Simulation on Crack–Inclusion Interaction for Rib-to-Deck Welded Joints in Orthotropic Steel Deck

Weld defects such as porosity, inclusion, burn-through, and lack of penetration are difficult to detect and control effectively in an orthotropic steel deck (OSD), which will be a fatigue crack initiation site and lead to several fatigue cracking. The crack growth behavior in defective welded joints is different from that of defect-free joints. This study investigates crack–inclusion interaction for rib-to-deck welded joints in OSDs based on numerical simulation and linear elastic fracture mechanics (LEFM). A refined finite element model of a half U-rib with cracks and inclusions was established by using the FRANC3D-ABAQUS interactive technology. The full processes of the crack–inclusion interaction from approaching and penetrating were accurately simulated. Critical parameters, including the stress intensity factor (SIF), the shape factor, the growth rate, and the growth direction were analyzed. The stiff and soft inclusions amplify and shield the SIF of cracks when the crack grows to the local area of inclusions. During the entire process of crack growth, the soft and stiff inclusion accelerate and inhibit the crack growth, respectively. The stiff inclusion will lead to asymmetric growth of the crack shape, where the portion of the crack away from the inclusions has a higher growth rate. The soft and stiff inclusions will attract and repel the direction of crack growth at the proximal point, respectively.

Dynamic remodeling of fiber networks with stiff inclusions under compressive loading.

The ability of tissues to sustain and withstand mechanical stress is critical to tissue development and healthy tissue maintenance. The mechanical properties of tissues are typically considered to be dominated by the fibrous extracellular matrix (ECM) component of tissues. Fiber network mechanics can capture certain mechanical features of tissues, such as shear strain stiffening, but is insufficient in describing the compressive response of certain tissues and blood clots that are rich in extracellular matrix. To understand the mechanical response of tissues, we employ a contemporary mechanical model, a fibrous network of fibrin embedded with inert bead inclusions that preserve the volume-conserving constraints of cells in tissues. Combining bulk mechanical rheology and a custom imaging device, we show that the presence of inclusions alters the local dynamic remodeling of the networks undergoing uniaxial compressive strains and demonstrate non-affine correlated motion within a fiber-bead network, predicted to stretch fibers in the network and lead to the ability of the network to stiffen under compression, a key feature of real tissues. These findings have important implications for understanding how local structural properties of cells and ECM fibers impact the bulk mechanical response of real tissues. STATEMENT OF SIGNIFICANCE: To understand why real tissue stiffens under compression, we study a model tissue system which also stiffens: a fibrin network embedded with stiff beads. We design a device that images compression of both fiber and fiber-bead networks. Distinct from previous imaging studies, this setup can dynamically capture network deformation on scales larger than single fibers. From the videos, we see fluid flow and network remodeling are both critical to compression behavior. The fiber-bead network has faster fluid flow, reduced network recovery, and correlated motion during network relaxation. We hypothesize that the beads hinder network relaxation of stretched fibers, thus retaining the applied stress and exhibiting stiffening. Our findings reveal important details for modeling tissue mechanics towards optimizing healthcare.

On optimization of heterogeneous materials for enhanced resistance to bulk fracture

We propose a novel approach to optimize the design of heterogeneous materials, with the goal of enhancing their effective fracture toughness under mode-I loading. The method employs a Gaussian processes-based Bayesian optimization framework to determine the optimal shapes and locations of stiff elliptical inclusions within a periodic microstructure in two dimensions. To model crack propagation, the phase-field fracture method with an efficient interior-point monolithic solver and adaptive mesh refinement, is used. To account for the high sensitivity of fracture properties to initial crack location with respect to heterogeneities, we consider multiple cases of initial crack and optimize the material for the worst-case scenario. We also impose a minimum clearance constraint between the inclusions to ensure design feasibility. Numerical experiments demonstrate that the method significantly improves the fracture toughness of the material compared to the homogeneous case.

Fat-filled lipid droplets with high curvature act as stiff inclusions that disrupt the lamina, rupture the nucleus and increase DNA damage.

The nucleus in many cell types is a stiff organelle, and yet fat-filled lipid droplets (FD's) in the cytoplasm can indent and displace the nucleus. FD's are phase-separated liquids with a poorly understood interfacial tension that determines how FD's interact with other organelles. Here, micron-sized FD's remain spherical while deforming both the nuclear lamina and peri-nuclear actomyosin. Nuclear lamins are intermediate filaments with a persistence length in the range of ∼0.2-1.5 μm with Lamin-B1 directly attached to the inner nuclear membrane via farnesylation groups that are lacking in mature lamin-A,C. Such rigidity can compromise the attachment of Lamin-B1 to the membrane in cases of high Gaussian curvature as seen through local dilution of Lamin-B1 at sites of indentation which is independent of Lamin-A,C. A single filament model of curvature-driven membrane detachment fits to Lamin-B1's relative density with the association energy being the key parameter to fit the experimental data. Lamin-B1 depletion triggers rupture as indicated by persistent, local accumulation of cytosolic DNA sensor cGAS at the nuclear boundary. FD-nucleus interactions also initiate rapid mis-localization of the DNA repair factor KU80, confirming nuclear rupture, and associate with heightened DNA damage and perturbed cell cycle. Similar results are evident in FD-laden cells after constricted 3D-migration, which is impeded by FD's. Spherical shapes of small FD's are consistent with a high interfacial tension that we measure for FD's mechanically isolated from fresh adipose tissue as ∼40 mN/m - which is far higher than other liquid condensates, but typical of oils in water and sufficiently rigid to disrupt cell structures.

Elasticity imaging using physics-informed neural networks: Spatial discovery of elastic modulus and Poisson's ratio.

Elasticity imaging is a technique that discovers the spatial distribution of mechanical properties of tissue using deformation and force measurements under various loading conditions. Given the complexity of this discovery, most existing methods approximate only one material parameter while assuming homogeneous distributions for the others. We employ physics-informed neural networks (PINN) in linear elasticity problems to discover the space-dependent distribution of both elastic modulus (E) and Poisson's ratio (ν) simultaneously, using strain data, normal stress boundary conditions, and the governing physics. We validated our model on three examples. First, we experimentally loaded hydrogel samples with embedded stiff inclusions, representing tumorous tissue, and compared the approximations against ground truth determined through tensile tests. Next, using data from finite element simulation of a rectangular domain containing a stiff circular inclusion, the PINN model accurately localized the inclusion and estimated both E and ν. We observed that in a heterogeneous domain, assuming a homogeneous ν distribution increases estimation error for stiffness as well as the area of the stiff inclusion, which could have clinical importance when determining size and stiffness of tumorous tissue. Finally, our model accurately captured spatial distribution of mechanical properties and the tissue interfaces on data from another computational model, simulating uniaxial loading of a rectangular hydrogel sample containing a human brain slice with distinct gray matter and white matter regions and complex geometrical features. This elasticity imaging implementation has the potential to be used in clinical imaging scenarios to reliably discover the spatial distribution of mechanical parameters and identify material interfaces such as tumors. STATEMENT OF SIGNIFICANCE: Our work is the first implementation of physics-informed neural networks to reconstruct both material parameters - Young's modulus and Poisson's ratio - and stress distributions for isotropic linear elastic materials by having deformation and force measurements. We comprehensively validate our model using experimental measurements and synthetic data generated using finite element modeling. Our method can be implemented in clinical elasticity imaging scenarios to improve diagnosis of tumors and for mechanical characterization of biomaterials and biological tissues in a minimally invasive manner.

Intravascular optical coherence elastography.

Optical coherence elastography (OCE), a functional extension of optical coherence tomography (OCT), visualizes tissue strain to deduce the tissue's biomechanical properties. In this study, we demonstrate intravascular OCE using a 1.1 mm motorized catheter and a 1.6 MHz Fourier domain mode-locked OCT system. We induced an intraluminal pressure change by varying the infusion rate from the proximal end of the catheter. We analysed the pixel-matched phase change between two different frames to yield the radial strain. Imaging experiments were carried out in a phantom and in human coronary arteries in vitro. At an imaging speed of 3019 frames/s, we were able to capture the dynamic strain. Stiff inclusions in the phantom and calcification in atherosclerotic plaques are associated with low strain values and can be distinguished from the surrounding soft material, which exhibits elevated strain. For the first time, circumferential intravascular OCE images are provided side by side with conventional OCT images, simultaneously mapping both the tissue structure and stiffness.

Revealing the Combined Nanoconfinement Effect by Soft and Stiff Inclusions in PMMA/Silica CO2 Blown Foams

AbstractThis study shows the importance of nanoparticle (NP) spatial arrangement on the nucleation, morphology, and properties of nanocomposite foams. It probes CO2 blown PMMA with three types of nanosilica structures with exactly the same chemical composition: aggregates, chain bound clusters, and individually dispersed NPs. A systematic but nontrivial scaling of morphology, glass transition temperature, static and impact mechanical properties, and thermal conductivity is discussed with special regard to the combined nanoconfinement by soft and stiff inclusions. The presented results suggest a limited potential of simultaneous toughening by nanoreinforcement and shear banding, which can draw significant implications for both the fundamental science and industrial practice dealing with nanoreinforced cellular materials.

Probabilistic failure mechanisms via Monte Carlo simulations of complex microstructures

A probabilistic approach to phase-field brittle and ductile fracture with random material and geometric properties is proposed within this work. In the macroscopic failure mechanics, materials properties and spatial quantities (of different phases in the geometrical domain) are assumed to be homogeneous and deterministic. This is unlike the lower scale with strong fluctuation in the material and geometrical properties. Such a response is approximated through some uncertainty in the model problem. The presented contribution is devoted to providing a mathematical framework for modeling uncertainty through stochastic analysis of a microstructure undergoing brittle/ductile failure. Hereby, the proposed model employs various representative volume elements with random distribution of stiff inclusions and voids within the composite structure. We develop an allocating strategy to allocate the heterogeneities and generate the corresponding meshes in two- and three-dimensional cases. Then the Monte Carlo Finite Element Method (MC-FEM) is employed for solving the stochastic PDE-based model and approximate the expectation and the variance of the solution field of brittle/ductile failure by evaluating a large number of samples. For the prediction of failure mechanisms, we rely on the phase-field approach which is a widely adopted framework for modeling and computing the fracture phenomena in solids. Incremental perturbed minimization principles for a class of gradient-type dissipative materials are used to derive the perturbed governing equations. This analysis enables us to study the highly heterogeneous microstructure and monitor the uncertainty in failure mechanics. Several numerical examples are given to examine the efficiency of the proposed method.

Quantitative homogenization theory for random suspensions in steady Stokes flow

This work develops a quantitative homogenization theory for random suspensions of rigid particles in a steady Stokes flow, and completes recent qualitative results. More precisely, we establish a large-scale regularity theory for this Stokes problem, and we prove moment bounds for the associated correctors and optimal estimates on the homogenization error; the latter further requires a quantitative ergodicity assumption on the random suspension. Compared to the corresponding quantitative homogenization theory for divergence-form linear elliptic equations, substantial difficulties arise from the analysis of the fluid incompressibility and the particle rigidity constraints. Our analysis further applies to the problem of stiff inclusions in (compressible or incompressible) linear elasticity and in electrostatics; it is also new in those cases, even in the periodic setting.

Interactions between multiple rigid lamellae in a ductile metal matrix: Shear band magnification and attenuation in localization patterns

A ductile matrix material containing an arbitrary distribution of parallel and stiff lamellar (’rigid-line’) inclusions is considered, subject to a prestress state provided by a simple shear aligned parallel to the inclusion lines. Because the lamellae have negligible thickness, the simple shear prestress state remains uniform and its amount can be high enough to drive the matrix material on the verge of ellipticity loss. Close to this critical stage, a uniform remote Mode I perturbation realizes shear band formation, growth, interaction, thickening or thinning. This two-dimensional problem is solved through the derivation of specific boundary integral equations, holding for a nonlinear elastic matrix material uniformly prestressed; the related numerical treatment is specifically tailored to capture the stress singularity present at the inclusion tips. Results show how complex localized deformation patterns form, so explaining features related to the failure mechanisms of ductile materials reinforced with stiff and thin inclusions. In particular, the influence of the inclusion distribution on the shear bands pattern is disclosed. Conditions for the magnification (the attenuation) of the localized deformations are revealed, fostering the progress (the setback) of the failure process.

A simple computer program for calculating stress and strain rate in 2D viscous inclusion-matrix systems

Computer-based numerical solutions of geomechanical problems are important to understand the processes forming rock structures as well as to quantify the associated pressure, stresses and strain rates. However, the development of such computer programs and the underlying mathematical methods are commonly not taught in a standard structural geology curriculum. Here, we present a simple computer program to calculate the stress, pressure, velocity and strain rate fields for two-dimensional (2D) viscous inclusion-matrix systems under pure shear and simple shear. The main aim of our contribution is to explain this computer program in a simple and transparent way, so that it can be used for introductory courses on geomechanical numerical modelling in structural geology. We present the governing equations of 2D viscous deformation and program the equations in the same order and style, so that the equations are still recognizable in the computer program. The computer program can treat stiff and weak inclusions of any shape and considers both linear and power-law viscous flow laws. We present numerical calculations for various inclusion-matrix scenarios. The program is written with the software MATLAB, is provided as supplementary material, and can also be run with the freely available software GNU Octave.

Self-consistent assessments for the effective properties of two-phase composites within strain gradient elasticity

Analytical method for the second-order homogenization of two-phase composites within Mindlin–Toupin strain gradient elasticity theory is proposed. Direct approach and self-consistent approximation are used to reduce the homogenization problem to the problem of determination of averaged Cauchy stresses, double stresses and static moments of Cauchy stresses inside the inclusions under prescribed quadratic boundary conditions. The ellipsoidal shape of inclusions and orthotropic properties of phases are assumed. Extended equivalent inclusion method with linear eigenstrain is proposed to derive the explicit relations between the Eshelby-like tensors and corresponding concentration tensors that are used to define the averaged field variables inside the inclusions. Obtained analytical solutions allow to evaluate the effective classical and gradient elastic moduli of composite materials accounting for the phases properties, volume fraction, shape and size of inclusions. Presented solution for the effective gradient moduli covers the full range of volume fraction and correctly predicts the absence of gradient effects for the homogeneous classical Cauchy medium. Examples of calculations for the composites with spherical inclusions are given. Micro-scale definition of the well-known simplified strain gradient elasticity theory is provided. Namely, it is shown that this theory is the phenomenological continuum model for the composites with isotropic matrix and with the small volume fraction of stiff isotropic spherical inclusions, which have the same Poisson’s ratio to those one of matrix phase.

Interface History on Strain Field Evolution in Epoxy Resins

To determine if full-strain measurements are sufficient in measuring interfacial phenomena in additively manufactured thermosets, we devised a testing method to examine the effect of interface history (i.e., time between interface formation and original material deposition) on localized mechanical properties. Interfaces were formed in diglycidyl ether of bisphenol A (DGEBA)/diethylenetriamine (DETA) epoxy resins with various DETA concentrations. Times to the gel point and full cure were assessed by rheology and Fourier transform infrared spectroscopy (FTIR) for each composition. From here, dogbone samples were fabricated with an interface formed at either the gel point or at full cure for each composition. Key findings show that tensile strength and Young’s modulus deteriorate globally with the presence of an interface regardless of the history or initiator (DETA) concentration. Samples with an interface demonstrate high strain regions near the interface prior to fracture. Micro X-ray computed tomography revealed high density regions at the interface that increased in number with both cure times and initiator concentrations. FTIR revealed that the interface demonstrated a higher cure completion than the sample interior, resulting in a stiffer epoxy localized at the interface versus the bulk. These findings were confirmed by atomic force microscopy modulus mapping at the interface. Finally, computational modeling of epoxy in uniaxial tension with an increasing number of stiff inclusions demonstrated that inclusion contents correlated with increased, localized stress concentrations. These findings will aid in the understanding of fracture phenomenon in additively manufactured thermosets and point to digital image correlation as a useful tool in epoxy interface detection.

Curvature-Mediated Forces on Elastic Inclusions in Fluid Interfaces.

Heterogeneous fluid interfaces often include two-dimensional solid domains that mechanically respond to changes in interfacial curvature. While this response is well-characterized for rigid inclusions, the influence of solid-like elasticity remains essentially unexplored. Here, we show that an initially flat, elastic inclusion embedded in a curved, fluid interface will exhibit qualitatively distinct behavior depending on its size and stiffness. Small, stiff inclusions are limited by bending and experience forces directed up gradients of Gaussian curvature, in keeping with prior findings for rigid discoids. By contrast, larger and softer inclusions are driven down gradients of squared Gaussian curvature in order to minimize the elastic penalty for stretching. Our calculations of the force on a solid inclusion are shown to collapse onto a universal curve spanning the bending- and stretching-limited regimes. From these results, we make predictions for the curvature-directed motion of deformable solids embedded within a model interface of variable Gaussian curvature.

A study for effect of steel fibre on strength of conventional concrete

Discrete discontinuous fibre reinforced concrete (FRC) is a composite material made up of Portland cement, aggregate, and discrete discontinuous fibres used to construct bridges and other structural components. Why would we want to put such fibres in concrete in the first place, anyway? Plain, unreinforced concrete is discouraged because it is a fragile, low-tensile-strength, low-strain-capacity material that should be avoided if possible. A random distribution of discontinuous fibres was intended to bridge over fractures as they expanded, restoring the material's “ductility” after the cracks had developed. Fibre reinforced concrete (FRC) can endure considerable pressures with a reasonably high strain capacity at the post-cracking stage due to the strength and connection of the fibres to the concrete, provided the fibres are sufficiently strong and well-bonded to the concrete. There are, however, other (and more cost-effective) ways to increase the strength of concrete than the ones listed above. The fibres' most significant contribution is to enhance the toughness of the concrete, regardless of the kind of force applied to it. As a consequence, although the fibres have a tendency to increase strain at peak load, they also absorb a significant amount of energy throughout operation in the post-peak section of the load vs. deflection curve. Fiber reinforcement in the form of short, discrete fibres functions as stiff inclusions in the concrete matrix and is a very efficient means of reinforcing concrete. In terms of particle size and number, they are comparable to aggregate inclusions. It is believed that conventionally reinforced structural elements will be more resistant to cracking, deflection, and other issues that impact serviceability.

Homogenization of stiff inclusions through network approximation

<p style='text-indent:20px;'>We investigate the homogenization of inclusions of infinite conductivity, randomly stationary distributed inside a homogeneous conducting medium. A now classical result by Zhikov shows that, under a logarithmic moment bound on the minimal distance between the inclusions, an effective model with finite homogeneous conductivity exists. Relying on ideas from network approximation, we provide a relaxed criterion ensuring homogenization. Several examples not covered by the previous theory are discussed.</p>

Shear Wave Elasticity Imaging Using Nondiffractive Bessel Apodized Acoustic Radiation Force.

The acoustic radiation force impulse (ARFI) has been widely used in transient shear wave elasticity imaging (SWEI). For SWEI based on focused ARFI, the highest image quality exists inside the focal zone due to the limitation of the depth of focus and diffraction. Consequently, the areas outside the focal zone and in the near field present poor image quality. To address the limitations of the focused beam, we introduce Bessel apodized ARFI that enhances image quality and improves the depth of focus. The objective of this study is to evaluate the feasibility of SWEI based on Bessel ARF in simulation and experiment. We report measurements of elastogram image quality and depth of field in tissue-mimicking phantoms and ex vivo liver tissue. Our results demonstrate improved depth of field, image quality, and shear wave speed (SWS) estimation accuracy using Bessel push beams. As a result, Bessel ARF enlarges the field of view of elastograms. The signal-to-noise ratio (SNR) of Bessel SWEI is improved 26% compared with focused SWEI in homogeneous phantom. The estimated SWS by Bessel SWEI is closer to the measured SWS from a clinical scanner with an error of 0.3% compared to 2.4% with a focused beam. In heterogeneous phantoms, the contrast-to-noise ratios (CNRs) of shallow and deep inclusions are improved by 8.79 and 3.33 dB, respectively, under Bessel ARF. We also compare the results between Bessel SWEI and supersonic shear imaging (SSI), and the SNR of Bessel SWEI is improved by 8.1%. Compared with SSI, Bessel SWEI shows more accurate SWS estimates in high stiffness inclusions. Finally, Bessel SWEI can generate higher quality elastograms with less energy than conventional SSI.