Indentation Stiffness Research Articles

The anomalous crack growth behaviour of an elastic-brittle octet-truss architected solid

Strongly rising R-curves are observed for octet-truss architected material specimens comprising ∼1 million unit cells and made from an elastic-brittle parent material. The measurements are supported by finite element (FE) calculations where the octet-truss is modelled discretely and show that the pseudo-ductility is not related to the usual toughening mechanisms such as crack bridging or crack-tip plasticity. Rather, the toughening is attributed to the three-dimensional (3D) structure of the octet-truss specimen: remarkably, no R-curve effect is observed in a quasi two-dimensional (2D) octet-truss specimen. This anomalous behaviour is clarified via FE calculations of the indentation stiffness of the octet-truss which reveal a strong indentation size effect in 3D with the stiffness decreasing with decreasing indenter size relative to the cell size of the octet-truss. In the 3D octet-truss, the size independent continuum stiffness is only attained for indenter sizes greater than about 20 unit cells while the size effect is almost absent in the quasi-2D octet-truss. The strong elastic softening in the 3D octet-truss in the presence of strain gradients gives rise to a large elastic crack-tip process zone. This in turn implies that a specimen geometry independent fracture toughness (i.e., a material property) can only be measured in specimens with large numbers of unit cells. We report a fracture mechanism map that reveals that measurements of specimen geometry independent fracture toughness can only be made in specimens with more than ∼10 million unit cells. This map is also used to rationalise the measured rising R-curves in the 3D specimens with ∼ 1 million unit cells. We end with a discussion on the implication of these findings for the laboratory measurement of the fracture toughness of both architected materials as well as biological cellular materials such as bone.

Compositional MR imaging of cartilage and joint mechanics

MRI-based compositional measurements of cartilage have been developed to characterize the biochemical composition of articular cartilage and the menisci, allowing the assessment of cartilage quality before cartilage loss has occurred and changes are still potentially reversible. They have also been used to investigate the complex relationship between biomechanics and cartilage health. This review focuses on cartilage T1rho and T2 cartilage compositional measurements as these techniques have been most frequently used in studies involving joint biomechanics. Recent clinical and in vitro studies are covered, including those investigating the relationship of cartilage composition with physical activity, gait, muscle strength, and biomechanical loading of specimens. These studies clearly illustrate the impact of physical activity on cartilage, how gait alterations after anterior cruciate ligament reconstruction may lead to early degenerative disease, and the complex interplay between muscle strength, gait, and cartilage composition. In vitro work also demonstrates how indentation stiffness correlates with cartilage composition. The review further consolidates the important role of cartilage compositional imaging in understanding the biomechanics and pathophysiology of degenerative joint disease.

Functional Assessment of Human Articular Cartilage Using Second Harmonic Generation (SHG) Imaging: A Feasibility Study.

Many arthroscopic tools developed for knee joint assessment are contact-based, which is challenging for in vivo application in narrow joint spaces. Second harmonic generation (SHG) laser imaging is a non-invasive and non-contact method, thus presenting an attractive alternative. However, the association between SHG-based measures and cartilage quality has not been established systematically. Here, we investigated the feasibility of using image-based measures derived from SHG microscopy for objective evaluation of cartilage quality as assessed by mechanical testing. Human tibial plateaus harvested from nine patients were used. Cartilage mechanical properties were determined using indentation stiffness (Einst) and streaming potential-based quantitative parameters (QP). The correspondence of the cartilage electromechanical properties (Einst and QP) and the image-based measures derived from SHG imaging, tissue thickness and cell viability were evaluated using correlation and logistic regression analyses. The SHG-related parameters included the newly developed volumetric fraction of organised collagenous network (Φcol) and the coefficient of variation of the SHG intensity (CVSHG). We found that Φcol correlated strongly with Einst and QP (ρ = 0.97 and - 0.89, respectively). CVSHG also correlated, albeit weakly, with QP and Einst, (|ρ| = 0.52-0.58). Einst and Φcol were the most sensitive predictors of cartilage quality whereas CVSHG only showed moderate sensitivity. Cell viability and tissue thickness, often used as measures of cartilage health, predicted the cartilage quality poorly. We present a simple, objective, yet effective image-based approach for assessment of cartilage quality. Φcol correlated strongly with electromechanical properties of cartilage and could fuel the continuous development of SHG-based arthroscopy.

T1rho MR properties of human patellar cartilage: correlation with indentation stiffness and biochemical contents.

Cartilage degeneration involves structural, compositional, and biomechanical alterations that may be detected non-invasively using quantitative MRI. The goal of this study was to determine if topographical variation in T1rho values correlates with indentation stiffness and biochemical contents of human patellar cartilage. Cadaveric patellae from unilateral knees of 5 donors with moderate degeneration were imaged at 3-Telsa with spiral chopped magnetization preparation T1rho sequence. Indentation testing was performed, followed by biochemical analyses to determine water and sulfated glycosaminoglycan contents. T1rho values were compared to indentation stiffness, using semi-circular regions of interest (ROIs) of varying sizes at each indentation site. ROIs matching the resected tissues were analyzed, and univariate and multivariate regression analyses were performed to compare T1rho values to biochemical contents. Grossly, superficial degenerative change of the cartilage (i.e., roughened texture and erosion) corresponded with regions of high T1rho values. High T1rho values correlated with low indentation stiffness, and the strength of correlation varied slightly with the ROI size. Spatial variations in T1rho values correlated positively with that of the water content (R2 = 0.10, p < 0.05) and negatively with the variations in the GAG content (R2 = 0.13, p < 0.01). Multivariate correlation (R2 = 0.23, p < 0.01) was stronger than either of the univariate correlations. These results demonstrate the sensitivity of T1rho values to spatially varying function and composition of cartilage and that the strength of correlation depends on the method of data analysis and consideration of multiple variables.

Nanoindentation response of small-volume piezoelectric structures and multi-layered composites: modeling the effect of surrounding materials

With piezoelectric small-volume composites gaining importance in smart device applications and nanoindentation being recognized as a versatile method for assessing the properties of layer materials, the present study is focused on the indentation response of the small-volume piezoelectric structures multi-layered composites. In particular, the effects of the nature of the substrate and surrounding materials, on the indentation response of piezoelectric nanocomposites, such as nanoislands, nanowires, and multi-layered composites are investigated. By developing three-dimensional finite element modeling, the complex interaction between the fundamental elastic, piezoelectric and dielectric properties of the piezoelectric materials and the elastic, plastic and electrically conducting or insulating properties of the surrounding materials, on the indentation response of the layered composites is analyzed. It is found that: (i) a substrate material that is elastically stiffer enhances the mechanical indentation stiffness and the electric indentation stiffness while plastic deformation in the substrate causes a reduction in the mechanical and electrical indentation stiffness; (ii) the effective piezoelectric and mechanical indentation stiffnesses of piezoelectric multi-layered composites are bounded by the corresponding characteristics of the bulk material counterparts from which the individual layers are constructed; (iii) electrically conducting surrounding materials produce a softening effect while insulating materials enhance the electrical indentation stiffness resulting in more charges being accumulated during the indentation process.

Self-consistent approximations for the tangential-displacement correction to the incremental indentation stiffness

An axisymmetric problem of unilateral frictionless contact between a paraboloidal or conical indenter and a transversely isotropic elastic half-space is considered in the refined formulation by accounting for the tangential (radial) displacements of the surface points of the elastic body. Using the idea of self-consistent approximation, a system of two coupled relations for the main contact variables (contact force, indenter displacement, and contact radius) is derived. The concept of the true contact radius is discussed, and it has been argued that the incremental stiffness relation should be expressed in its terms. Correction factors for the force–displacement relation and the incremental indentation stiffness (as a function of the true contact radius) are evaluated in explicit form.

Role of multi-layer tissue composition of musculoskeletal extremities for prediction of in vivo surface indentation response and layer deformations.

Emergent mechanics of musculoskeletal extremities (surface indentation stiffness and tissue deformation characteristics) depend on the underlying composition and mechanics of each soft tissue layer (i.e. skin, fat, and muscle). Limited experimental studies have been performed to explore the layer specific relationships that contribute to the surface indentation response. The goal of this study was to examine through statistical modeling how the soft tissue architecture contributed to the aggregate mechanical surface response across 8 different sites of the upper and lower extremities. A publicly available dataset was used to examine the relationship of soft tissue thickness (fat and muscle) to bulk tissue surface compliance. Models required only initial tissue layer thicknesses, making them usable in the future with only a static ultrasound image. Two physics inspired models (series of linear springs), which allowed reduced statistical representations (combined locations and location specific), were explored to determine the best predictability of surface compliance and later individual layer deformations. When considering the predictability of the experimental surface compliance, the physics inspired combined locations model showed an improvement over the location specific model (percent difference of 25.4 +/- 27.9% and 29.7 +/- 31.8% for the combined locations and location specific models, respectively). While the statistical models presented in this study show that tissue compliance relies on the individual layer thicknesses, it is clear that there are other variables that need to be accounted for to improve the model. In addition, the individual layer deformations of fat and muscle tissues can be predicted reasonably well with the physics inspired models, however additional parameters may improve the robustness of the model outcomes, specifically in regard to capturing subject specificity.

Indentation stiffness tomography of fibrous inhomogeneities — An asymptotic model

AFM (Atomic Force Microscopy)-based stiffness tomography poses novel indentation problems for a heterogeneous elastic sample containing a single or multiple heterogeneities. A relatively stiff infinite elastic fiber buried in an elastic half-space represents a simple model for fibrous organelles inside a living cell. The leading-order asymptotic model for the frictionless unilateral indentation is constructed using the method of matched asymptotic expansions under the assumption that the diameter of the contact area is small compared to the depth of the fiber below the surface subjected to the indentation imaging. The fiber deformation is described in the framework of Euler’s theory of bending. The resulting system of integro-differential equations is solved by means of the Fourier transform. The approximate relation between the indenter displacement and the contact force is derived in explicit form. The fiber influence factor is introduced to evaluate the incremental indentation stiffness.

Cyclic loading history alters the joint compression tolerance and regional indentation responses in the cartilaginous endplate

This study quantified the effect of subthreshold loading histories that differed by joint posture (neutral, flexed), peak loading variation (10%, 20%, 40%), and loading duration (1000, 3000, 5000 cycles) on the post-loading Ultimate Compressive Tolerance (UCT), yield force, and regional Cartilaginous End Plate (CEP) indentation responses (loading stiffness and creep displacement). One hundred and fourteen porcine spinal units were included. Following conditioning and cyclic compression exposures, spinal units were transected and one endplate from each vertebra underwent subsequent UCT or microindentation testing. UCT testing was conducted by compressing a single vertebra at a rate of 3 kN/s using an indenter fabricated to a representative intervertebral disc size and shape. Force and actuator position were sampled at 100 Hz. Non-destructive uniaxial CEP indentation was performed at five surface locations (central, anterior, posterior, right, left) using a Motoman robot and aluminum indenter (3 mm hemisphere). Force and end-effector position were sampled at 10 Hz. A significant three-way interaction was observed for UCT (p = 0.038). Compared to neutral, the UCT was, on average, 1.9 kN less following each flexed loading duration. No effect of variation was observed in flexion; however, 40% variation caused the UCT to decrease by an average of 2.13 kN and 2.06 kN following 3000 and 5000 cycles, respectively. The indentation stiffness in the central CEP mimicked the UCT response. These results demonstrate a profound effect of posture on post-loading UCT and CEP behaviour. Control of peak compression exposures became particularly relevant only when a neutral posture was maintained and beyond the midpoint of the predicated lifespan.

Mechanical Behavior of Subcutaneous and Visceral Abdominal Adipose Tissue in Patients with Obesity

The mechanical characterization of adipose tissues is important for various medical purposes, including plastic surgery and biomechanical applications, such as computational human body models for the simulation of surgical procedures or injury prediction, for example, in the evaluation of vehicle crashworthiness. In this context, the measurement of human subcutaneous adipose tissue (SAT) and visceral adipose tissue (VAT) mechanical properties in relation to subject characteristics may be really relevant. The aim of this work was to properly characterize the mechanical response of adipose tissues in patients with obesity. Then, the data were exploited to develop a reliable finite element model of the adipose tissues characterized by a constitutive material model that accounted for nonlinear elasticity and time dependence. Mechanical tests have been performed on both SAT and VAT specimens, which have been harvested from patients with severe obesity during standard laparoscopic sleeve gastrectomy intervention. The experimental campaign included indentation tests, which permitted us to obtain the initial/final indentation stiffnesses for each specimen. Statistical results revealed a higher statistical stiffness in SAT than in VAT, with an initial/final indentation stiffness of 1.65 (SD ± 0.29) N/30.30 (SD ± 20) N compared to 1.29 (SD ± 0.30) N/21.00 (SD ± 16) N. Moreover, the results showed that gender, BMI, and age did not significantly affect the stiffness. The experimental results were used in the identification of the constitutive parameters to be inserted in the constitutive material model. Such constitutive characterization of VAT and SAT mechanics can be the starting point for the future development of more accurate computational models of the human adipose tissue and, in general, of the human body for the optimization of numerous medical and biomechanical procedures and applications.

Recovery of information on the depth-dependent profile of elastic FGMs from indentation experiments

A perturbation technique is applied to construct the first-order perturbation model for the incremental indentation stiffness in normal frictionless indentation of an isotropic functionally graded medium (FGM) with a constant Poisson’s ratio and an in-depth variable elastic shear modulus. The FGM identification problem of determining the shear modulus function from the effective indentation shear modulus, which is measured as a result of indentation testing, is formulated and solved. It is shown that both the effective shear modulus and the relative adhesive (pull-off) force are sensitive to the type of material grading.

Investigating the effects of activation state and location on lower limb tissue stiffness

Lower limb tissue stiffness is contingent on various factors, including location, tissue composition, loading rates, and the geometry of the indenting object. Previous studies demonstrated that tissue stiffness varies greatly between individuals and between locations on an individual. Additionally, some studies have shown that activation of underlying muscle tissue increases bulk soft tissue stiffness. Yet, few studies have simultaneously considered both location and activation; this could be particularly important for measuring and predicting the function of devices such as prostheses and exoskeletons that interact with limbs at various locations during dynamic movement. In the present study, a custom handheld indentation device was used to explore changes in bulk leg tissue stiffness at rest and during isometric contractions. The indentation force-displacement curves were modelled using a Hertz model. At each level of activation (active/inactive), the shank had dramatically (∼150%) greater tissue stiffness than the thigh (p < 0.001). However, results suggested location independence for stiffness ratio (active/inactive, p = 0.42); for either location, stiffness was approximately 2x greater for active vs inactive muscle. These results should be considered during the development of biomechanical models to simulate human tissue indentation stiffness across a range of activation states and locations.

Mechanical behavior of infrapatellar fat pad of patients affected by osteoarthritis

The infrapatellar fat pad (IFP) is an adipose tissue present in the knee that lies between the patella, femur, meniscus and tibia, filling the space between these structures. IFP facilitates the distribution of the synovial fluid and may act to absorb impulsive actions generated through the joint. IFP in osteoarthritis (OA) pathology undergoes structural changes characterized by inflammation, hypertrophy and fibrosis. The aim of the present study is to analyze the mechanical behavior of the IFP in patients affected by end-stage OA. A specific test fixture was designed and indentation tests were performed on IFP specimens harvested from OA patients who underwent total knee arthroplasty. Experiments allowed to assess the typical features of mechanical response, such as non-linear stress–strain behavior and time-dependent effects. Results from mechanical experimentations were implemented within the framework of a visco-hyperelastic constitutive theory, with the aim to provide data for computational modelling of OA IFP role in knee mechanics.Initial and final indentation stiffness were calculated for all subjects and statistical results reveled that OA IFP mechanics was not significantly influenced by gender, BMI and sample preparation. OA IFP mechanical behavior was also compared to that of other adipose tissues. OA IFP appeared to be a stiffer adipose tissue compared to subcutaneous, visceral adipose tissues and heel fat pads. It is reasonable that fibrosis induces a modification of the tissue destabilizing the normal distribution of forces in the joint during movement, causing a worsening of the disease.

Spherical Indentation of a Micropolar Solid: A Numerical Investigation Using the Local Point Interpolation–Boundary Element Method

The load-penetration curve in elastic nanoindentation of an elastic micropolar flat by a diamond spherical punch is analyzed. The presented results are obtained by a specifically developed numerical tool based on a judicious combination of the conventional boundary element method and strong form local point interpolation method. The results show that the usual linear relationship between the material depression and the square of the radius of the contact area is also valid in this case of micropolar elastic material. It is also shown that the relation between the indentation stress (applied load over the contact surface) and the indentation strain (ratio of contact radius by the punch radius) is linear. The proportionality coefficient which is none other than the indentation stiffness varies with the coupling factor of the micropolar elastic medium. A relation between the indentation stiffness of a micropolar solid and that of a conventional solid with the same Young modulus and Poisson ratio is derived.

Indentation responses of pressurized ellipsoidal and cylindrical elastic shells: Insights from shallow-shell theory.

Pressurized elastic shells are ubiquitous in nature and technology, from the outer walls of yeast and bacterial cells to artificial pressure vessels. Indentation measurements simultaneously probe the internal pressure and elastic properties of thin shells and serve as a useful tool for strength testing and for inferring internal biological functions of living cells. We study the effects of geometry and pressure-induced stress on the indentation stiffness of ellipsoidal and cylindrical elastic shells using shallow-shell theory. We show that the linear indentation response reduces to a single integral with two dimensionless parameters that encode the asphericity and internal pressure. This integral can be numerically evaluated in all regimes and is used to generate compact analytical expressions for the indentation stiffness in various regimes of technological and biological importance. Our results provide theoretical support for previous scaling and numerical results describing the stiffness of ellipsoids, reveal a new pressure scale that dictates the large-pressure response, and give new insights to the linear indentation response of pressurized cylinders.

Cervical endplate bone density distribution measured by CT osteoabsorptiometry and direct comparison with mechanical properties of the endplate.

Intervertebral device subsidence is one of the complications of anterior cervical discectomy and fusion. The biomechanical properties of vertebral bony endplate may be related to device subsidence. The aim of this study is to measure the cervical endplate bone density distribution using a novel 3D measurement method. Eight human cadaver cervical spines were obtained and levels C3-C7 were dissected and CT scanned. Three-dimensional (3D) CT model was created with the same 3D coordinates of the original DICOM dataset. The regional strength and stiffness of the endplate were determined by indentation testing. The indentation points were recorded by a photograph and the location of the indentation points was projected to the 3D CT model. Three-dimensional coordinates of the indentation point was obtained in the 3D space determined by the DICOM dataset. The area underneath the indentation point was calculated by a trilinear interpolation method directly. Data in HU and correlations with the indentation strength and stiffness were analysed. A positive correlation was found between HU and strength (r = 0.52) and between HU and stiffness (r = 0.41). Overall, mechanical strength and stiffness and HU in the superior endplate of the caudal vertebra were lower than those in the inferior endplate of the cranial vertebra in the same intervertebral disc. The mechanical properties and the HU were found to be significantly correlated, which employed a novel 3D HU measurement method, thus demonstrating potential to predict cervical endplate failure risk in a clinical setting.

Effect of laser power on the microstructure and mechanical properties of laser deposited titanium aluminide composite

ABSTRACT In this work, titanium aluminide alloy have been fabricated via the laser deposition technique. The effect of some selected deposition parameters on the microstructure and mechanical properties of produced deposits were studied. The relationship between the laser power, and the microhardness of deposited samples on laser preheated substrate showed an incremental change in laser power from 200 to 600 W. This led to an overall decrease in microhardness of deposited samples from 426 to 373 HV. Sample deposited at 500 W gave the lowest Icorr of 1.8 x 10-8 and the highest Ecorr of -0.138 V. It is evident from the nanoindentation results that indentation modulus and stiffness of sample deposited at 600 W laser power had a lower value compared with 400 W laser power. However, the modulus of both samples fell within titanium alloy modulus range between 105-120 GPa. The microstructures of the deposits are mainly characterized with γ-TiAl and α2-Ti3Al phases and an improved hardness property almost two times higher than that of commercially pure titanium were achieved. It was concluded that changes in the laser power directly causes changes in the microstructure, hardness, stiffness, modulus of elasticity and corrosion resistance of the deposits.

Increased tissue-level storage modulus and hardness with age in male cortical bone and its association with decreased fracture toughness.

The incidence of bone fracture increases with age, due to both declining bone quantity and quality. Toward the goal of an improved understanding of the causes of the age-related decline in the fracture toughness of male cortical bone, nanoindentation experiments were performed on femoral diaphysis specimens from men aged 21-98years. Because aged bone has less matrix-bound water and dry bone is less viscoelastic, we used a nanoindentation method that is sensitive to changes in viscoelasticity. Given the anisotropy of bone stiffness, longitudinal (n=26) and transverse (n=25) specimens relative to the long axis of the femur diaphysis were tested both dry in air and immersed in phosphate buffered saline solution. Indentation stiffness (storage modulus) and hardness increased with age, while viscoelasticity (loss modulus) was independent of donor age. The increases in indentation stiffness and hardness with age were best explained by increased mineralization with age. Indentation stiffness and hardness were negatively correlated with previously acquired fracture toughness parameters, which is consistent with a tradeoff between material strength and toughness. In keeping with the complex structure of bone, a combination of tissue-level storage modulus or hardness, bound water, and osteonal area in regression models best explained the variance in the fracture toughness of male human cortical bone. On the other hand, viscoelasticity was unchanged with age and was not associated with fracture toughness. In conclusion, the age-related increase in stiffness and hardness of male cortical bone may be one of the multiple tissue-level characteristics that contributes to decreased fracture toughness.

Indentation Stiffness Measurement by an Optical Coherence Tomography-Based Air-Jet Indentation System Can Reflect Type I Collagen Abundance and Organisation in Diabetic Wounds

There is a lack of quantitative and non-invasive clinical biomechanical assessment tools for diabetic foot ulcers. Our previous study reported that the indentation stiffness measured by an optical coherence tomography-based air-jet indentation system in a non-contact and non-invasive manner may reflect the tensile properties of diabetic wounds. As the tensile properties are known to be contributed by type I collagen, this study was aimed to establish the correlations between the indentation stiffness, and type I collagen abundance and organisation, in order to further justify and characterise the in vivo indentation stiffness measurement in diabetic wounds. In a male streptozotocin-induced diabetic rat model, indentation stiffness, and type I collagen abundance and organisation of excisional wounds were quantified and examined using the optical coherence tomography-based air-jet indentation system and picrosirius red polarised light microscopy, respectively, on post-wounding days 3, 5, 7, 10, 14, and 21. The results showed significant negative correlations between indentation stiffness at the wound centre, and the collagen abundance and organisation. The correlations between the indentation stiffness, as well as collagen abundance and organisation of diabetic wounds suggest that the optical coherence tomography-based air-jet indentation system can potentially be used to quantitatively and non-invasively monitor diabetic wound healing in clinical settings, clinical research or preclinical research.

Pressure-Induced Formation and Mechanical Properties of 2D Diamond Boron Nitride.

Understanding phase transformations in 2D materials can unlock unprecedented developments in nanotechnology, since their unique properties can be dramatically modified by external fields that control the phase change. Here, experiments and simulations are used to investigate the mechanical properties of a 2D diamond boron nitride (BN) phase induced by applying local pressure on atomically thin h‐BN on a SiO2 substrate, at room temperature, and without chemical functionalization. Molecular dynamics (MD) simulations show a metastable local rearrangement of the h‐BN atoms into diamond crystal clusters when increasing the indentation pressure. Raman spectroscopy experiments confirm the presence of a pressure‐induced cubic BN phase, and its metastability upon release of pressure. Å‐indentation experiments and simulations show that at pressures of 2–4 GPa, the indentation stiffness of monolayer h‐BN on SiO2 is the same of bare SiO2, whereas for two‐ and three‐layer‐thick h‐BN on SiO2 the stiffness increases of up to 50% compared to bare SiO2, and then it decreases when increasing the number of layers. Up to 4 GPa, the reduced strain in the layers closer to the substrate decreases the probability of the sp2‐to‐sp3 phase transition, explaining the lower stiffness observed in thicker h‐BN.