Local Elastic Modulus Research Articles

Cavitation and Crazing in Rod-Containing Nanocomposites

Addition of nanoparticles to a polymer can drastically affect the mechanical properties and structure of the nanocomposite. Here we investigate the nucleation and growth of voids that precede craze formation and the early crazing itself by use of coarse-grained Monte Carlo and molecular dynamics simulations. We investigate the role of deformation rate, local density, local rod orientation, nonaffine displacements, and local elastic moduli. We observe that for both pure polymers and nanocomposites, regions of low local elastic modulus are more prone to failure. Additionally, it is found that Voronoi volume can anticipate void formation and that it is also a predictor of failure, particularly in composites. After the onset of crazing, it is found that nanorods incorporated into the crazes rapidly orient themselves to match the direction of the polymer fibrils, but nanorods in bulk regions remain randomly oriented. We further find that attractive rods resist incorporation into the developing crazes and that this effect is stronger with increasing rod length.

Respective roles of organic and mineral components of human cortical bone matrix in micromechanical behavior: An instrumented indentation study

Bone is a multiscale composite material made of both a type I collagen matrix and a poorly crystalline apatite mineral phase. Due to remodeling activity, cortical bone is made of Bone Structural Units (BSUs) called osteons. Since osteon represents a fundamental level of structural hierarchy, it is important to investigate the relationship between mechanical behavior and tissue composition at this scale for a better understanding of the mechanisms of bone fragility. The aim of this study is to analyze the links between ultrastructural properties and the mechanical behavior of bone tissue at the scale of osteon. Iliac bone biopsies were taken from untreated postmenopausal osteoporotic women, embedded, sectioned and microradiographed to assess the degree of mineralization of bone (DMB). On each section, BSUs of known DMB were indented with relatively high load ( ∼ 500 mN) to determine local elastic modulus ( E ), contact hardness ( H c ) and true hardness ( H ) of several bone lamellae. Crystallinity and collagen maturity were measured by Fourier Transform InfraRed Microspectroscopy (FTIRM) on the same BSUs. Inter-relationships between mechanical properties and ultrastructural components were analyzed using multiple regression analysis. This study showed that elastic deformation was only explained by DMB whereas plastic deformation was more correlated with collagen maturity. Contact hardness, reflecting both elastic and plastic behaviors, was correlated with both DMB and collagen maturity. No relationship was found between crystallinity and mechanical properties at the osteon level.

Phonon-deformational generalized ginzburg-landau functionals for describing martensite transitions between FCC and BCC structures in metals

A method is developed for theoretical description of martensite transitions between fcc and bcc phases of metals based on generalized Ginzburg-Landau functionals (GGLFs), in which the transformation parameters are the “phonon” parameter describing relative slips of close-packed atomic planes and fcc lattice deformations in the region of a martensite inclusion. General expressions are proposed for calculating the deformational contributions to the GGLF. Local strains and local elastic moduli in the region of martensite inclusions are estimated numerically. The possibility of constructing phenomenological expressions for the GGLF using interpolations of available experimental data on the structure and elastic moduli in the fcc and bcc phases is discussed.

Nanomechanical and Structural Properties of Native Cellulose Under Compressive Stress

Cellulose is an important biopolymer with applications ranging from its use as an additive in pharmaceutical products to the development of novel smart materials. This wide applicability arises in part from its interesting mechanical properties. Here we report on the use of high pressure X-ray diffraction and Raman spectroscopy in a diamond anvil cell to determine the bulk and local elastic moduli of native cellulose. The modulus values obtained are 20 GPa for the bulk modulus and 200-355 and 15 GPa for the crystalline parts and the overall elastic (Young's) modulus, respectively. These values are consistent with those calculated from tensile measurements. Above 8 GPa, the packing of the cellulose chains within the fibers undergoes significant structural distortion, whereas the chains themselves remain largely unaffected by compression.

Elasticity of Developing Cardiac Tissue and Influence on Early Cardiomyocyte Beating

Recent studies have demonstrated the effects of cardiomyocyte mechanosensitivity to ECM elasticity on beating physiology of 10 day chick embryo cardiomyocytes[1] and also of neonatal rat cardiomyocytes[2], developmental stages for which the heart mechanics are established and the Young's modulus is E∼10-20 kPa. Here we study the evolution of embryonic cardiac tissue mechanics in parallel with the effects of microenvironment mechanics on individual cardiomyocyte function throughout early development. To measure the changing mechanical properties of cardiac tissue, we used micropipette aspiration to measure local and average bulk elastic moduli of embryonic avian heart tissue. We observe stiffening of the looping heart tube (days 2-4) from 1 kPa average elasticity to 2 kPa. We measure a greater local elastic modulus in the inner curvature of the looping heart tube than the outer curvature. Treating heart tubes with blebbistatin led to a uniform 30% decrease in elasticity, indicating that the local elasticity is not solely due to actomyosin contractility. We performed a proteomic analysis of various anatomical positions along the 2-4 day heart tubes by mass spectrometry, to measure the relative spaciotemporal changes in protein expression. To explore the effects of the immediate microenvironment mechanics on early cardiomyocyte function, heart cells were isolated from 2-4 day chicken embryos, cultured on collagen-coated polyacrylamide gels of varying stiffness, and observed for effects of substrate elasticity on spontaneous beating and morphology. After 24 hours in culture, cardiomyocyte beating magnitude was larger on softer 1 kPa gels than on stiffer 5-34 kPa gels, with no measurable difference in behavior with developmental stage. [1] A. Engler et al. Journal of Cell Science 121: 3794-3802 (2008). [2] J. Jacot, et al. Biophysical Journal 95(7): 3479-3487 (2008).

Novel hydroxyapatite–forsterite–bioglass nanocomposite coatings with improved mechanical properties

The purpose of this work was fabrication and evaluation of mechanical properties of hydroxyapatite–forsterite–bioactive glass nanocomposite coatings. The novel hydroxyapatite–forsterite–bioactive glass nanocomposite coating on 316L stainless steel (SS) was prepared via a sol–gel process. The X-ray diffraction (XRD), scanning electron microscopy (SEM), atomic force microscopy (AFM) and energy dispersive X-ray analysis (EDX) techniques were used to investigate the microstructure and morphology of the coatings. The local mechanical properties, e.g. nano-hardness, elastic modulus and fracture toughness of four type coatings with different amounts of forsterite, were determined by nano-indentation method using load and displacement data. The nano-indentation tester was equipped with Atomic Force Microscopy to image the indenter mark. The crack-free and homogeneous nanocomposite coatings with no observable defects were prepared. The coatings showed the hardness and elastic modulus values in the range of 2.4–3.4 (GPa) and 46–96 (GPa), respectively. The fracture toughness values of nanocomposite coatings also varied in the range of 0.56–0.972 (MPa m 0.5). The results showed that the elastic modulus, hardness and fracture toughness values of prepared composite coatings increased with increase in forsterite amounts. Results suggested that novel composite coatings might be potentially useful for biomedical applications especially as an implant coating for hard tissue treatment.

Multi-scale characterization of swine femoral cortical bone

Multi-scale experimental work was carried out to characterize cortical bone as a heterogeneous material with hierarchical structure, which spans from nanoscale (mineralized collagen fibril), sub-microscale (single lamella), microscale (lamellar structures), to mesoscale (cortical bone) levels. Sections from femoral cortical bone from 6, 12, and 42 months old swines were studied to quantify the age-related changes in bone structure, chemical composition, and mechanical properties. The structural changes with age from sub-microscale to mesoscale levels were investigated with scanning electron microscopy and micro-computed tomography. The chemical compositions at mesoscale were studied by ash content method and dual energy X-ray absorptiometry, and at microscale by Fourier transform infrared microspectroscopy. The mechanical properties at mesoscale were measured by tensile testing, and elastic modulus and hardness at sub-microscale were obtained using nanoindentation. The experimental results showed age-related changes in the structure and chemical composition of cortical bone. Lamellar bone was a prevalent structure in 6 months and 12 months old animals, resorption sites were most pronounced in 6 months old animals, while secondary osteons were the dominant features in 42 months old animals. Mineral content and mineral-to-organic ratio increased with age. The structural and chemical changes with age corresponded to an increase in local elastic modulus, and overall elastic modulus and ultimate tensile strength as bone matured.

Non-linear viscoelasticity of hagfish slime

We report here the first experimental measurements of non-linear rheological material properties of hagfish slime, a hydrated biopolymer/biofiber network, and develop a microstructural constitutive model to explain the observed non-linear viscoelastic behavior. The linear elastic modulus of the network is observed to be G ′ ≈ 0.02 Pa for timescales 0.1 s ≤ t ≤ 10 s , making it one of the softest elastic biomaterials known. Non-linear rheology is examined via simple shear deformation, and we observe a secant elastic modulus which strain-softens at large input strain while the local tangent elastic modulus strain-stiffens simultaneously. This juxtaposition of simultaneous softening and stiffening suggests a general network structure composed of non-linear elastic strain-stiffening elements, here modeled as finite extensible non-linear elastic (FENE) springs, in which network connections are destroyed as elements are stretched. We simulate the network model in oscillatory shear and creep, including instrument effects from rotational inertia. The network model captures the simultaneous softening of the secant modulus and stiffening of tangent modulus as the model enters the non-linear viscoelastic regime.

The mechanical heterogeneity of the hard callus influences local tissue strains during bone healing: A finite element study based on sheep experiments

During secondary fracture healing, various tissue types including new bone are formed. The local mechanical strains play an important role in tissue proliferation and differentiation. To further our mechanobiological understanding of fracture healing, a precise assessment of local strains is mandatory. Until now, static analyses using Finite Elements (FE) have assumed homogenous material properties. With the recent quantification of both the spatial tissue patterns (Vetter et al., 2010) and the development of elastic modulus of newly formed bone during healing (Manjubala et al., 2009), it is now possible to incorporate this heterogeneity. Therefore, the aim of this study is to investigate the effect of this heterogeneity on the strain patterns at six successive healing stages. The input data of the present work stemmed from a comprehensive cross-sectional study of sheep with a tibial osteotomy (Epari et al., 2006). In our FE model, each element containing bone was described by a bulk elastic modulus, which depended on both the local area fraction and the local elastic modulus of the bone material. The obtained strains were compared with the results of hypothetical FE models assuming homogeneous material properties. The differences in the spatial distributions of the strains between the heterogeneous and homogeneous FE models were interpreted using a current mechanobiological theory ( Isakson et al., 2006). This interpretation showed that considering the heterogeneity of the hard callus is most important at the intermediate stages of healing, when cartilage transforms to bone via endochondral ossification.

Assessment of elastic behavior of nanometric coatings with static force spectroscopy

The local elastic properties of fine coatings are assessed with the method of static force spectroscopy using contact deformation of laminar systems. The values of the local modulus of elasticity of the hydroxyapatite coating 20–25 nm thick are obtained in response to the indenter penetration depth. An assumption is made of the gradient change of the modulus of elasticity of the fine layer through depth.

Icosahedral clustering with medium-range order and local elastic properties of amorphous metals

The paper focuses on the relationship between icosahedral clustering with medium-range order in amorphous metals and local elastic properties. Using a Cu–Zr binary amorphous model constructed by rapid quenching in the framework of molecular dynamics simulations, Voronoi polyhedra analysis shows that the icosahedra are not randomly distributed in space, but form characteristic interpenetrating icosahedral clustering with medium-range order. It is shown that the center atoms of icosahedra within the highly developed interpenetrating icosahedral clusters have higher local elastic moduli than those of independently existing icosahedra and other polyhedra with lower coordination numbers.

Influence of substrate stiffness on the phenotype of heart cells.

Adult cardiomyocytes (CM) retain little capacity to regenerate, which motivates efforts to engineer heart tissues that can emulate the functional and mechanical properties of native myocardium. Although the effects of matrix stiffness on individual CM have been explored, less attention was devoted to studies at the monolayer and the tissue level. The purpose of this study was to characterize the influence of substrate mechanical stiffness on the heart cell phenotype and functional properties. Neonatal rat heart cells were seeded onto collagen-coated polyacrylamide (PA) substrates with Young's moduli of 3, 22, 50, and 144 kPa. Collagen-coated glass coverslips without PA represented surfaces with effectively "infinite" stiffness. The local elastic modulus of native neonatal rat heart tissue was measured to range from 4.0 to 11.4 kPa (mean value of 6.8 kPa) and for native adult rat heart tissue from 11.9 to 46.2 kPa (mean value of 25.6 kPa), motivating our choice of the above PA gel stiffness. Overall, by 120 h of cultivation, the lowest stiffness PA substrates (3 kPa) exhibited the lowest excitation threshold (ET; 3.5 +/- 0.3 V/cm), increased troponin I staining (52% positively stained area) but reduced cell density, force of contraction (0.18 +/- 0.1 mN/mm(2)), and cell elongation (aspect ratio = 1.3-1.4). Higher stiffness (144 kPa) PA substrates exhibited reduced troponin I staining (30% positively stained area), increased fibroblast density (70% positively stained area), and poor electrical excitability. Intermediate stiffness PA substrates of stiffness comparable to the native adult rat myocardium (22-50 kPa) were found to be optimal for heart cell morphology and function, with superior elongation (aspect ratio > 4.3), reasonable ET (ranging from 3.95 +/- 0.8 to 4.4 +/- 0.7 V/cm), high contractile force development (ranging from 0.52 +/- 0.2 to 1.60 +/- 0.6 mN/mm(2)), and well-developed striations, all consistent with a differentiated phenotype.

In vivo measurement of proximal pulmonary artery elastic modulus in the neonatal calf model of pulmonary hypertension: development and ex vivo validation

Developing clinical work suggests that vascular stiffening plays a role in the progression of pulmonary hypertension (PH), while recent studies in animal models of hypoxic PH have found significant proximal vascular stiffening in the diseased population. Here, we develop and validate a minimally invasive, clinically realizable method to estimate the local elastic modulus of the proximal pulmonary arteries from pressure-diameter (PD) data. PD measurements were made in the main pulmonary arteries of 16 calves; lumen diameter was assessed using color M-mode tissue Doppler imaging ultrasound, while pressure was measured via catheter. Two methods corresponding to thin-walled pressure vessel theory ("thin") and Lame's equation for a thick-walled cylinder ("thick") were used to approximate the artery elastic modulus from PD measurements. The harvested arteries were tested independently to determine their "true" ex vivo elastic modulus and stiffness. Both approximations displayed excellent correlation with ex vivo elastic modulus of the calf main pulmonary artery (thin r(2) = 0.811; thick r(2) = 0.844; both P < 0.01). Bland-Altman analysis indicated that the thick-walled approximation has better overall agreement with ex vivo modulus. The approximations displayed quantitatively distinct regression slopes that were statistically different (P = 0.02). The elastic modulus of the main pulmonary artery can be reasonably estimated from combined color M-mode tissue Doppler imaging ultrasound and catheter pressure measurements in calves. Such measurements may be a valuable tool in the diagnosis and treatment of human PH.

Ratchet Boundary Determination Using a Noncyclic Method

A simple and systematic procedure is proposed for shakedown analysis using a combination of linear and nonlinear finite element analysis (FEA). The method can identify the boundary between the shakedown and ratcheting domains directly and does not require a cyclic analysis (noncyclic). The proposed method performs an elastic-plastic FEA to determine the reduction in load carrying capacity due to the cyclic secondary loads. An elastic modulus adjustment procedure is then used to generate statically admissible stress distributions and kinematically admissible strain distributions under the applied primary loads. By modifying the local elastic moduli it is possible to obtain inelastic-like stress redistribution. The method is demonstrated with a “two-bar structure” model based on analytical routine. The analysis is then applied to some typical shakedown problems including the “classical Bree problem,” the “bimaterial cylinder,” and the “plate with a hole subjected to radial temperature gradient.”

Antiplasticization and the elastic properties of glass-forming polymer liquids

We investigate the effect of antiplasticizer additives on long-wavelength thermodynamic properties relating to the efficiency of molecular packing (density ρ and isothermal compressibility κT) and material stiffness (bulk modulus K, shear modulus G, and Poisson ratio ν) of model glass-forming polymer melts. Variations in the stiffness (G and molecular force constant km from the Debye–Waller factor 〈u2〉) and density ρ at a molecular scale are also considered. We find that antiplasticizer additives cause significant changes in the long-wavelength properties that are associated with an enhanced packing efficiency in the glass state, such as a decrease in κT and an increase in K, G, and ν. Moreover, values of the local elastic moduli in the glass state follow a nearly universal (approximately log-normal) distribution and a Fourier transform of the elastic constant fluctuations in space reveals a general −2 wave vector q scaling, as in order parameter fluctuations in critical fluids and magnetic systems near their critical point for phase separation and ordering, respectively. No discernible large-scale fluctuations in the density were observed, however, indicating that the local elastic constant fluctuations do not reflect density (free volume) fluctuations, as often assumed. These observations provide essential insights into how varying the fragility of glass formation, through the addition of antiplasticizer additives, alters the local mechanical properties of polymer melts and other glass-forming liquids.

Analysis of micro fracture in human Haversian cortical bone under transverse tension using extended physical imaging

AbstractWe propose a procedure to investigate local stress intensity factors at the scale of the osteons in human Haversian cortical bone. The method combines a specific experimental setting for a three‐point bending millimetric specimen and a numerical method using the eXtended Finite Element Method (X‐FEM). The interface between the experimental setting and the numerical method is ensured through an imaging technique that analyses the light microscopy observations to import the geometrical heterogeneity of the Haversian microstructures, the boundary conditions and appearing crack discontinuities into the numerical model. The local mechanical elastic Young's moduli are measured by nano‐indentation, and the Poisson ratios are determined by an imaging technique of the stress–strain fields. The model is able to access three scales of measurement: the macro scale of the material level (mm), the micro scale inside the Haversian material for stress–strain fields (10–100µm), and the sub‐micro scale for the crack opening profiles (1–10µm ) and fracture parameters (stress intensity factors). The model is applied to several patients at different aging stages. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Local elasticity map and plasticity in a model Lennard-Jones glass

In this work we calculate the local elastic moduli in a weakly polydispersed two-dimensional Lennard-Jones glass undergoing a quasistatic shear deformation at zero temperature. The numerical method uses coarse-grained microscopic expressions for the strain, displacement, and stress fields. This method allows us to calculate the local elasticity tensor and to quantify the deviation from linear elasticity (local Hooke's law) at different coarse-graining scales. From the results a clear picture emerges of an amorphous material with strongly spatially heterogeneous elastic moduli that simultaneously satisfies Hooke's law at scales larger than a characteristic length scale of the order of five interatomic distances. At this scale, the glass appears as a composite material composed of a rigid scaffolding and of soft zones. Only recently calculated in nonhomogeneous materials, the local elastic structure plays a crucial role in the elastoplastic response of the amorphous material. For a small macroscopic shear strain, the structures associated with the nonaffine displacement field appear directly related to the spatial structure of the elastic moduli. Moreover, for a larger macroscopic shear strain we show that zones of low shear modulus concentrate most of the strain in the form of plastic rearrangements. The spatiotemporal evolution of this local elasticity map and its connection with long term dynamical heterogeneity as well as with the plasticity in the material is quantified. The possibility to use this local parameter as a predictor of subsequent local plastic activity is also discussed.

The elastic modulus of Matrigel™ as determined by atomic force microscopy

Recent studies indicate that the biophysical properties of the cellular microenvironment strongly influence a variety of fundamental cell behaviors. The extracellular matrix's (ECM) response to mechanical force, described mathematically as the elastic modulus, is believed to play a particularly critical role in regulatory and pathological cell behaviors. The basement membrane (BM) is a specialization of the ECM that serves as the immediate interface for many cell types (e.g. all epithelial cells) and through which cells are connected to the underlying stroma. Matrigel is a commercially available BM-like complex and serves as an easily accessible experimental simulant of native BMs. However, the local elastic modulus of Matrigel has not been defined under physiological conditions. Here we present the procedures and results of indentation tests performed on Matrigel with atomic force microscopy (AFM) in an aqueous, temperature controlled environment. The average modulus value was found to be approximately 450 Pa. However, this result is considerably higher than macroscopic shear storage moduli reported in the scientific literature. The reason for this discrepancy is believed to result from differences in test methods and the tendency of Matrigel to soften at temperatures below 37 degrees C.

Propagation and localization of acoustic and elastic waves in heterogeneous materials: renormalization group analysis and numerical simulations

We describe and discuss the recent progress in the study of propagation and localization of acoustic and elastic waves in heterogeneous media. The heterogeneity is represented by a spatial distribution of the local elastic moduli. Both randomly distributed elastic moduli as well as those with long-range correlations with a nondecaying power-law correlation function, are considered. The motivation for the study is twofold. One is that recent analysis of experimental data for the spatial distribution of the elastic moduli of rock indicated that the distribution is characterized by the type of long-range correlations that we consider in this study. The second motivation for the problem is to understand whether localization of electrons (which, in quantum mechanics, are described by wave functions) has any analogy in the propagation of classical waves in disordered media. The problem is studied by two approaches. One of them is based on developing a dynamic renormalization group (RG) approach to analytical analysis of the governing equations for wave propagation. The RG analysis indicates that, depending on the type of the disorder (correlated vs. uncorrelated), one may have a transition between localized and extended regimes in any spatial dimension. The second approach utilizes numerical simulations of the governing equations in two- and three-dimensional media. The results obtained by the two approaches are in agreement with each other. Using numerical simulations, we also describe how the characteristics of a propagating wave may be used for probing the differences between heterogeneous media with short- and long-range correlations. To do so, we study the evolution of several distinct characteristics of the waves, such as the amplitude of the coherent wave front, its width, the spectral densities, the scalogram (wavelet transformation of the waves’ amplitudes at different scales and times), and the dispersion relation. It is demonstrated that such properties have completely different characteristics in uncorrelated and correlated media. Finally, it is shown how wave propagation may be used for establishing a link between the static and dynamical properties of heterogeneous media.

Contact-resonance atomic force microscopy for nanoscale elastic property measurements: Spectroscopy and imaging

Quantitative measurements of the elastic modulus of nanosize systems and nanostructured materials are provided with great accuracy and precision by contact-resonance atomic force microscopy (CR-AFM). As an example of measuring the elastic modulus of nanosize entities, we used the CR-AFM technique to measure the out-of-plane indentation modulus of tellurium nanowires. A size-dependence of the indentation modulus was observed for the investigated tellurium nanowires with diameters in the range 20–150 nm. Over this diameter range, the elastic modulus of the outer layers of the tellurium nanowires experienced significant enhancement due to a pronounced surface stiffening effect. Quantitative estimations for the elastic moduli of the outer and inner parts of tellurium nanowires of reduced diameter are made with a core–shell structure model. Besides localized elastic modulus measurements, we have also developed a unique CR-AFM imaging capability to map the elastic modulus over a micrometer-scale area. We used this CR-AFM capability to construct indentation modulus maps at the junction between two adjacent facets of a tellurium microcrystal. The clear contrast observed in the elastic moduli of the two facets indicates the different surface crystallography of these facets.