Stiff Inclusions Research Articles

Tunable microstructure transformations and auxetic behavior in 3D-printed multiphase composites: The role of inclusion distribution

We experimentally and numerically investigate instability-induced pattern transformations and switchable auxetic behavior in multiphase composites consisted of circular voids and stiff inclusions periodically distributed in a soft elastomer. We specifically focus on the role of inclusion distribution on the behavior of the soft transformative composites. The inclusions are distributed in either square or triangular periodic configurations, while the voids are distributed in the triangular periodic array – the configurations enabling cooperative buckling induced transformations of the unit cells. Through the survey of microstructure parameter, we show that tailored positioning of the stiff inclusions can be exploited to expand the set of admissible switchable patterns in multiphase composites. Thus, extreme values of negative Poisson's ratio can be attained through applied strains; moreover, the onset of instabilities, and the corresponding switches to extremely soft behavior are shown to be controlled by the inclusion arrangements and volume fractions. Furthermore, the dependence of the microstructure buckling and post-buckling behavior on loading direction is investigated, and the composite anisotropic properties depending on the microstructure parameters are discussed.

Tunable Mechanical Metamaterial with Constrained Negative Stiffness for Improved Quasi‐Static and Dynamic Energy Dissipation

This paper presents the computational design, fabrication, and experimental validation of a mechanical metamaterial in which the damping of the material is significantly increased without decreasing the stiffness by embedding a small volume fraction of negative stiffness (NS) inclusions within it. Unlike other systems that dissipate energy primarily through large‐amplitude deformation of nonlinear structures, this metamaterial dissipates energy by amplifying linear strains in the viscoelastic host material. By macroscopically tuning the pre‐strain of the metamaterial via mechanical loading, the embedded NS inclusions operate about a constrained buckling instability. When further macroscopic vibrational excitation is applied, the inclusions amplify the strains of the surrounding viscoelastic medium. This results in enhanced dissipation of mechanical energy when compared to voided or neat comparison media. Microstereolithography, an emerging high‐resolution additive manufacturing (AM) technology, is employed to fabricate the deeply subwavelength inclusions which ensures broadband damping behavior. The mechanically induced broadband energy dissipation and manufacturing approach further differentiate the metamaterial from other approaches that exploit resonances, large deformations, or non‐mechanical instabilities. The computational design, fabrication, and experimental evaluation reported is the first dynamic demonstration of such a mechanically tunable NS metamaterial, potentially enabling components with integrated structural and damping capabilities.

Domain Formations and Pattern Transitions via Instabilities in Soft Heterogeneous Materials.

Experimental observations of domain formations and pattern transitions in soft particulate composites under large deformations are reported herein. The system of stiff inclusions periodically distributed in a soft elastomeric matrix experiences dramatic microstructure changes upon the development of elastic instabilities. In the experiments, the formation of microstructures with antisymmetric domains and their geometrically tailored evolution into a variety of patterns of cooperative particle rearrangements are observed. Through experimental and numerical analyses, it is shown that these patterns can be tailored by tuning the initial microstructural periodicity and concentration of the inclusions. Thus, these fully determined new patterns can be achieved by fine tuning of the initial microstructure.

Inherent negative refraction on acoustic branch of two dimensional phononic crystals

Guided by theoretical predictions, we have demonstrated experimentally the existence of negative refraction on the two lowest acoustic-branch passbands (shear and longitudinal modes) of a simple two-dimensional phononic crystal consisting of an isotropic stiff (aluminum) matrix and square-patterned isotropic compliant (PMMA) circular inclusions. We experimentally distinguished the shear and longitudinal modes to ensure that there are no couplings between the two modes and that the shear-mode negative refraction is an inherent property of periodic composites with stiff matrix and compliant inclusions. We have also discovered that a composite with compliant (PMMA) matrix and stiff (aluminum) inclusions does not display negative shear-mode refraction on its first branch. At frequencies and wave vectors where the refraction on the acoustic-branch passbands is negative, the effective mass-density and the effective stiffness tensors of the crystal are positive-definite, and this is an inherent property of such phononic crystals.The equi-frequency contours and energy flux vectors as functions of the phase-vector components, reveal a rich body of refractive properties that can be exploited to realize, for example, beam splitting, focusing, and frequency filtration on the lowest passbands of the crystal where the dissipation is minimum. By proper selection of material and geometric parameters these phenomena can be realized at remarkably low frequencies (large wave lengths) using rather small simple two-phase unit cells.

Debonding at the fiber/matrix interface in carbon nanotube reinforced composites: Modelling investigation

Debonding at the fiber/matrix interface is an important failure mechanism in composite materials. It controls, among others, the intralaminar cracking process. A large amount of experimental data exists indicating that the addition of carbon nanotubes (CNTs) in a composite can hinder onset and development of these cracks. Here we report results of a modelling study aiming to better understand the effect of carbon nanotubes on the fiber/matrix debonding. The study is performed on an example of a carbon fiber/epoxy unidirectional composite with two practically realizable CNT configurations: agglomerated CNTs in the matrix and CNTs grown on fibers. The composites are subjected to transverse tension. The stress distribution prior to debonding is predicted using a two-scale finite element model based on the embedded element technique. The onset and propagation of the debond are then analyzed using a surface-based cohesive zone model with a linear traction-separation law. Our results show that CNTs have a strong effect on the stress distribution in the matrix and the debonding process, although their effect on the latter is less pronounced. CNT agglomerates are found to act as stiff microscopic inclusions leading to magnification of stress concentrations and earlier initiation of the debonding process. The composites with CNTs grown on fibers, on the other hand, show suppression of stress concentrations at the interface and slight delay in the debonding onset compared to the composite without CNTs.

Bone Ultrastructure as Composite of Aligned Mineralized Collagen Fibrils Embedded Into a Porous Polycrystalline Matrix: Confirmation by Computational Electrodynamics

Micromechanical representation of bone ultrastructure as a composite of aligned mineralized collagen fibrils embedded in a porous polycrystalline matrix has allowed for successfully predicting the (poro/visco-)elastic and strength properties of bone tissues throughout the entire vertebrate animal kingdom, based on the „universal“ mechanical properties of the material’s elementary components: molecular collagen, hydroxyapatite, and water-type fluids. We here check whether the explanatory power of this schematic representation might extend beyond the realm of mechanics; namely, towards electrodynamics and X-ray physics. This requires knowledge about the electron density distribution across the bone ultrastructure, reflecting the organization of collagen molecules, hydroxyapatite (mineral) crystals, and water with noncollageneous organics. The latter follow three principal, mathematically formulated, ”universal” rules, namely (i) a unique bilinear relationship between mineral and collagen concentrations found in bone tissues throughout the vertebrate animal kingdom, (ii) the precipitation of mineral from a ionic solution under closed thermodynamic conditions, governing mass density-dependent lateral distances between the long collagen molecules, and (iii) the identity of the extracollageneous mineral concentration in the fibrillar and extrafibrillar, as well as in the gap and the overlap compartments of bone ultrastructure. The corresponding electron density distributions are then inserted into Fourier transform-type solutions of the Maxwell equations specificied for a Small Angle X-ray Scattering setting. The aforementioned mineral distribution, as well as random fluctuations of fibrils, both within their transverse plane around a hexagonal lattice and in form of axial shifts, turn out to be the key for successfully predicting experimentally observed X-ray diffraction patterns. This marks a new level of quantitative, mathematized understanding of the organization of bone ultrastructure. In particular, earlier interpretations of SAXS data, leading to the idea of bone being a soft organic matrix with stiff mineral inclusions may have been overcome, in favor of a more complex, but also more realistic modeling concept concerning the ultrastructural organization bone.

Critical factors for determining a first estimate of fatigue limit of bearing steels under rolling contact fatigue

This article investigates the variables that influence the fatigue limit of bearing steels. The experimental evaluation of the fatigue limit of rolling element bearings is a difficult, time consuming and an expensive process. Hence, an analytical method is desirable. Modern bearing steels have a complex microstructure with various non-metallic inclusions, which act as local stress risers and may cause the matrix to accumulate micro-plastic strain. This eventually results in the nucleation of fatigue micro-cracks and ultimately bearing failure. Therefore, one interpretation of the fatigue limit is as a macro-scale stress below which no local yielding occurs. Such a fatigue limit will be dependent upon the mechanical properties of both the matrix and the inclusions as well as the inclusion geometry and distribution. A finite element analysis (FEA) based study was conducted to determine the critical factors that have the largest impact on the fatigue limit of bearing steels during rolling contact fatigue. The effects considered include static vs rolling contact, inclusion geometry (size, shape, orientation, and distribution), the temperature dependence of the mechanical properties of both the matrix and the inclusion (elastic modulus, thermal expansion coefficient, and yield strength), and the possibility of an inclusion becoming debonded from the matrix. The results of this study showed that a static contact model instead of rolling contact model is sufficient to capture the fatigue limit in the case of hard, stiff inclusions. The stress concentration due to debonded inclusions was much more dramatic than any of the other terms considered followed by the change in the elastic modulus due to temperature. The effect of the geometry (size, shape, and orientation) of an individual stiff inclusion on the fatigue limit was found to be relatively minor.

Control of fracture at the interface of dissimilar materials using randomly oriented inclusions and networks

Fracture at or near the interface between two isotropic materials has been a subject of extensive research relevant to the problems encountered in aerospace, naval, electronic packaging, biomechanical engineering and other applications. The problems of edge cracks, semi-infinite interface cracks and substrate cracks under a thermally stressed film considered in the paper are representative of such analyses. We consider a possible improvement in the fracture resistance achieved by embedding randomly distributed stiff inclusions such as fibers, nanotubes, fiber or nanotube networks and ellipsoidal or spherical particles in the more compliant material. This results in a smaller mismatch between the stiffness of the joined materials that may prevent or alleviate fracture. Numerical examples demonstrate both the benefits and the limitations of enhancing the stiffness of the compliant material. In particular, the examples refer to the boundary between singular and non-singular interfacial edge stresses attempting to avoid the stress singularity by embedding random carbon nanotubes or networks. In the semi-infinite interfacial crack problem it is demonstrated that a small reduction in the stiffness mismatch of two materials at the interface causes a significant decrease in the strain energy release rate. The approach to the analysis of substrate cracks under a thermally stressed film is outlined accounting for the history of thermal loading and the effect of temperature on the properties of the film, substrate and embedded inclusions. The solutions presented in the paper rely on effective properties of randomly reinforced materials. The limitations of such approach in fracture problems and an estimate of its validity based on a comparison of scales at the tip of the crack and in the representative volume cell are discussed. It is suggested that fracture involving nanoparticle and nanotube reinforced materials can be characterized using effective properties.

Characterization of the stiffness distribution in two and three dimensions using boundary deformations: a preliminary study

We present for the first time the feasibility to recover the stiffness (here shear modulus) distribution of a three-dimensional heterogeneous sample using measured surface displacements and inverse algorithms without making any assumptions about local homogeneities and the stiffness distribution. We simulate experiments to create measured displacements and augment them with noise, significantly higher than anticipated measurement noise. We also test two-dimensional problems in plane strain with multiple stiff inclusions. Our inverse strategy recovers the shear modulus values in the inclusions and background well, and reveals the shape of the inclusion clearly.

Micromechanical characterization and modeling of cement pastes containing waste marble powder

Abstract A comprehensive study consisting of a microstructure investigation, nanoindentation, and micromechanical modeling was performed to identify phases and assess their impact on macroscopic properties of cement pastes containing waste marble powder. The proposed model, built on the Mori-Tanaka scheme, was used to estimate the effective Young's modulus and compressive strength at a low computational cost. After model validation, the effects of an interfacial transition zone (ITZ) and increased porosity of marble powder rich-pastes were quantified. The study revealed a fundamental role of the ITZ formed around stiff marble powder inclusions, responsible for performance deterioration. These findings indicate that elimination of the ITZ and reduction of porosity would considerably enhance the strength of cement-based materials containing waste marble powder. As a consequence, larger amounts of waste marble could be incorporated as a cement replacement without sacrificing structural performance.

Stochastic analysis of seismic ground response for site classification methods verification

A database of numerical ground response analyses has been generated to assess the effectiveness of site classification schemes adopted in seismic building codes. 1-D Ground models have been generated with a stochastic approach based on a Monte Carlo procedure and the corresponding seismic response has been evaluated by the equivalent linear elastic approach for a set of natural strong motion records. The study is intended to contribute to the discussion of optimal subsoil classification schemes in terms of inter-class dispersion and reliability of the predicted amplification factors. Moreover some specific analyses have been performed to show the relevance of stiff layer inclusions and of nonlinear soil behavior on the performances of classification schemes. The variability observed for each class in the different systems indicates that simple classification schemes should be preferred according to Occam's razor statement.

A comparative study of two constitutive models within an inverse approach to determine the spatial stiffness distribution in soft materials

A comparative study is presented to solve the inverse problem in elasticity for the shear modulus (stiffness) distribution utilizing two constitutive equations: (1) linear elasticity assuming small strain theory, and (2) finite elasticity with a hyperelastic neo-Hookean material model. Assuming that a material undergoes large deformations and material nonlinearity is assumed negligible, the inverse solution using (2) is anticipated to yield better results than (1). Given the fact that solving a linear elastic model is significantly faster than a nonlinear model and more robust numerically, we posed the following question: How accurately could we map the shear modulus distribution with a linear elastic model using small strain theory for a specimen undergoing large deformations? To this end, experimental displacement data of a silicone composite sample containing two stiff inclusions of different sizes under uniaxial displacement controlled extension were acquired using a digital image correlation system. The silicone based composite was modeled both as a linear elastic solid under infinitesimal strains and as a neo-Hookean hyperelastic solid that takes into account geometrically nonlinear finite deformations. We observed that the mapped shear modulus contrast, determined by solving an inverse problem, between inclusion and background was higher for the linear elastic model as compared to that of the hyperelastic one. A similar trend was observed for simulated experiments, where synthetically computed displacement data were produced and the inverse problem solved using both, the linear elastic model and the neo-Hookean material model. In addition, it was observed that the inverse problem solution was inclusion size-sensitive. Consequently, an 1-D model was introduced to broaden our understanding of this issue. This 1-D analysis revealed that by using a linear elastic approach, the overestimation of the shear modulus contrast between inclusion and background increases with the increase of external loads and target shear modulus contrast. Finally, this investigation provides valuable information on the validity of the assumption for utilizing linear elasticity in solving inverse problems for the spatial distribution of shear modulus associated with soft solids undergoing large deformations. Thus, this work could be of importance to characterize mechanical property variations of polymer based materials such as rubbers or in elasticity imaging of tissues for pathology.

N-Phase micromechanical framework for the conductivity and elastic modulus of particulate composites: Design to microencapsulated phase change materials (MPCMs)-cementitious composites

The smart design of microencapsulated phase change materials (MPCMs) in cementitious composites requires an explicit understanding of effects of soft microcapsule particles, stiff aggregates and their surrounding weak interfaces on the physico-mechanical properties of particulate composites. This paper devises a n-phase micromechanical framework to predict the effective thermal conductivity and elastic modulus of multicomponent particulate composites that consist in stiff and soft anisotropic-shaped inclusions, their surrounding weak interfaces and matrix. In this micromechanical model, the volume fraction of weak interfaces treated as the interphase model is quantified and incorporated into the n-phase differential effective medium model. It is found that the structural configuration of interfaces has a significant effect on the effective physico-mechanical properties of particulate composites. The micromechanical model leads to predictions of the effective conductivity and elastic modulus of multicomponent particulate composites to a good accuracy by comparing with available experimental data for regular concrete, quartz mortar and MPCMs-cementitious composites. By utilizing the micromechanical model, the authors further develop a theoretical design rule for the robust overall performance of MPCMs-cementitious composites with the better thermal resistance and elastic modulus. These results can also be used to design other multiphase particulate composites and porous media with the cherry-pit structure.

3D printed two-dimensional periodic structures with tailored in-plane dynamic responses and fracture behaviors

High-performance biological structural materials such as bone and spider silk have evolved soft and stiff components to optimize the load transfer path. This inspired many researchers to achieve engineering composites with significantly improved properties. In this paper, we designed a class of 2D elastomer filled composites with periodic units. The designs of these composites were realized by 3D printing of a soft elastomer and a stiff plastic. The plastic constructed a honeycomb-like mesh, in which the soft elastomer formed isolated inclusions. Variation of the mesh geometries, the elastomer content, and the constituent material properties could tune the in-plane dynamic responses and fracture behaviors. Dynamic mechanical analysis measurements showed two tanδ peaks, corresponding to the elastomer and the plastic. By changing the geometry, a very wide range of storage modulus values (about three orders of magnitudes) could be achieved at room temperature. The fracture strain of the composites was found to increase with the elastomer content, and an obvious brittle-ductile transition was observed. Tough samples showed several stress plateaus and a trumpet-shaped crack profile. Increasing the vertex angles of the rhombus filler geometry was found to result in a brittle-ductile transition. In both experiments and simulations, a forward-backward crack propagation mode was observed, indicating that soft elastomer inclusions stabilize the crack propagation. Compared to staggered composites with stiff inclusions, the current design with soft inclusions showed higher modulus, tensile strength and toughness.

Auxetic multiphase soft composite material design through instabilities with application for acoustic metamaterials.

We investigate the instability-induced pattern transformations in 3D-printed soft composites consisting of stiff inclusions and voids periodically distributed in a soft matrix. These soft auxetic composites are prone to elastic instabilities giving rise to negative Poisson's ratio (NPR) behavior. Upon reaching the instability point, the composite microstructure rearranges into a new morphology attaining an NPR regime. Remarkably, identical composites can morph into distinct patterns depending on the loading direction. These fully determined instability-induced distinct patterns are characterized by significantly different NPR behaviors, thus, giving rise to enhanced tunability of the composite properties. Finally, we illustrate a potential application of these reversible pattern transformations as tunable acoustic-elastic metamaterials capable of selectively filtering low frequency ranges controlled by deformation.

Discrete-element modeling of nacre-like materials: Effects of random microstructures on strain localization and mechanical performance

Natural materials such as nacre, collagen, and spider silk are composed of staggered stiff and strong inclusions in a softer matrix. This type of hybrid microstructure results in remarkable combinations of stiffness, strength, and toughness and it now inspires novel classes of high-performance composites. However, the analytical and numerical approaches used to predict and optimize the mechanics of staggered composites often neglect statistical variations and inhomogeneities, which may have significant impacts on modulus, strength, and toughness. Here we present an analysis of localization using small representative volume elements (RVEs) and large scale statistical volume elements (SVEs) based on the discrete element method (DEM). DEM is an efficient numerical method which enabled the evaluation of more than 10,000 microstructures in this study, each including about 5,000 inclusions. The models explore the combined effects of statistics, inclusion arrangement, and interface properties. We find that statistical variations have a negative effect on all properties, in particular on the ductility and energy absorption because randomness precipitates the localization of deformations. However, the results also show that the negative effects of random microstructures can be offset by interfaces with large strain at failure accompanied by strain hardening. More specifically, this quantitative study reveals an optimal range of interface properties where the interfaces are the most effective at delaying localization. These findings show how carefully designed interfaces in bioinspired staggered composites can offset the negative effects of microstructural randomness, which is inherent to most current fabrication methods.

Prostate Cancer Detection with a Tactile Resonance Sensor-Measurement Considerations and Clinical Setup.

Tumors in the human prostate are usually stiffer compared to surrounding non-malignant glandular tissue, and tactile resonance sensors measuring stiffness can be used to detect prostate cancer. To explore this further, we used a tactile resonance sensor system combined with a rotatable sample holder where whole surgically removed prostates could be attached to detect tumors on, and beneath, the surface ex vivo. Model studies on tissue phantoms made of silicone and porcine tissue were performed. Finally, two resected human prostate glands were studied. Embedded stiff silicone inclusions placed 4 mm under the surface could be detected in both the silicone and biological tissue models, with a sensor indentation of 0.6 mm. Areas with different amounts of prostate cancer (PCa) could be distinguished from normal tissue (p < 0.05), when the tumor was located in the anterior part, whereas small tumors located in the dorsal aspect were undetected. The study indicates that PCa may be detected in a whole resected prostate with an uneven surface and through its capsule. This is promising for the development of a clinically useful instrument to detect prostate cancer during surgery.

Forward and inverse viscoelastic wave scattering by irregular inclusions for shear wave elastography.

Inversion methods in shear wave elastography use simplifying assumptions to recover the mechanical properties of soft tissues. Consequently, these methods suffer from artifacts when applied to media containing strong stiffness contrasts, and do not provide a map of the viscosity. In this work, the shear wave field recorded inside and around an inclusion was used to estimate the viscoelastic properties of the inclusion and surrounding medium, based on an inverse problem approach assuming local homogeneity of both media. An efficient semi-analytical method was developed to model the scattering of an elastic wave by an irregular inclusion, based on a decomposition of the field by Bessel functions and on a decomposition of the boundaries as Fourier series. This model was validated against finite element modeling. Shear waves were experimentally induced by acoustic radiation force in soft tissue phantoms containing stiff and soft inclusions, and the displacement field was imaged at a high frame rate using plane wave imaging. A nonlinear least-squares algorithm compared the model to the experimental data and adjusted the geometrical and mechanical parameters. The estimated shear storage and loss moduli were in good agreement with reference measurements, as well as the estimated inclusion shape. This approach provides an accurate estimation of geometry and viscoelastic properties for a single inclusion in a homogeneous background in the context of radiation force elastography.

A numerical study on improving the specific properties of staggered composites by incorporating voids

Despite being porous, natural materials such as bone, nacre and tooth enamel possess interesting combinations of stiffness, strength and toughness, an outstanding performance which is hard to achieve in engineering materials. Recent studies show that this performance is rooted in their architectures in which stiff and strong inclusions are arranged in a staggered structure and placed within a softer matrix. These studies however focus on void-free staggered composites. Here, we use finite element analysis and exhaustive exploration of the design space to show that the specific modulus and yield strength of staggered composites can be improved by removing the non-load bearing regions of the matrix material and to show the morphologies emerging from this process. This improvement increases with aspect ratio of the inclusions and with mismatch of properties between the phases and decreases with inclusion content. Particularly, 20–30% increase in the specific modulus and 10–20% increase in specific yield strength can be achieved considering the actual properties of carbon nanotube (or ceramic/mineral platelet) reinforced polymers. The morphologies emerging from our exploration can be used to develop lightweight composites of interest in aerospace, defence, and packaging industries.

Mechanics Based Tomography: A Preliminary Feasibility Study.

We present a non-destructive approach to sense inclusion objects embedded in a solid medium remotely from force sensors applied to the medium and boundary displacements that could be measured via a digital image correlation system using a set of cameras. We provide a rationale and strategy to uniquely identify the heterogeneous sample composition based on stiffness (here, shear modulus) maps. The feasibility of this inversion scheme is tested with simulated experiments that could have clinical relevance in diagnostic imaging (e.g., tumor detection) or could be applied to engineering materials. No assumptions are made on the shape or stiffness quantity of the inclusions. We observe that the novel inversion method using solely boundary displacements and force measurements performs well in recovering the heterogeneous material/tissue composition that consists of one and two stiff inclusions embedded in a softer background material. Furthermore, the target shear modulus value for the stiffer inclusion region is underestimated and the inclusion size is overestimated when incomplete boundary displacements on some part of the boundary are utilized. For displacements measured on the entire boundary, the shear modulus reconstruction improves significantly. Additionally, we observe that with increasing number of displacement data sets utilized in solving the inverse problem, the quality of the mapped shear moduli improves. We also analyze the sensitivity of the shear modulus maps on the noise level varied between 0.1% and 5% white Gaussian noise in the boundary displacements, force and corresponding displacement indentation. Finally, a sensitivity analysis of the recovered shear moduli to the depth, stiffness and the shape of the stiff inclusion is performed. We conclude that this approach has potential as a novel imaging modality and refer to it as Mechanics Based Tomography (MBT).