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PubMed2006

Changes in the physical properties of human premolar cementum after application of 4 weeks of controlled orthodontic forces.

This study was performed to assess the relationship between the magnitude of orthodontic force and physical properties of individual human cementum, and to identify the sites that may be predisposed to root resorption. The findings may assist in relating physical properties of dental root cementum and its susceptibility to root resorption. Sixteen maxillary first premolar teeth were selected in eight orthodontic patients (three males and five females), mean age 14.8 years (range 11.2-17.5 years), requiring first premolar extractions. In each patient, a light orthodontic force of 25 cN was applied buccally using a sectional archwire on the first premolar on one side, while a heavy force of 225 cN was applied to the contralateral side. The teeth were extracted 4 weeks after initial force application. Hardness and elastic modulus were measured on the buccal and the lingual surfaces of the cementum at the cervical, middle, and apical third of the root. The results showed that the mean hardness and elastic modulus of cementum in the light force group were greater than in the heavy force group at all positions. There were highly significant differences in both hardness and elastic modulus between the heavy and light force groups (P < 0.01). The mean hardness and elastic modulus of cementum gradually decreased from the cervical to the apical regions for buccal as well as lingual surfaces in both groups. There was, however, an insignificant difference between hardness and elastic modulus on the buccal surface compared with the lingual surface (P < 0.05). It was concluded that the hardness and elastic modulus of cementum were affected by the application of orthodontics forces.

Adolescent↗

PubMed1991

Paradoxically decreased aortic wall stiffness in response to vitamin D3-induced calcinosis. A biphasic analysis of segmental elastic properties in conscious dogs.

We studied the aortic elastic behavior in response to vitamin D3-induced accelerated calcinosis in conscious dogs chronically instrumented with a pressure microtransducer and a pair of ultrasonic diameter dimension gauges in the upper descending thoracic aorta. The two functional phases of the elastic segmental properties of the aorta in vivo were discriminated by computation on a beat-by-beat basis from the phasic pressure-diameter (P-D) hysteresis loops in basal conditions and during the transient state of a wide range of pressures obtained mechanically (aortic occlusion) or pharmacologically (angiotensin bolus). The overall P-D curve formed by all P-D hysteresis loops was comprised of two linear relations according to a model that assumes that only elastin is stretched at lower pressures, whereas both elastin and collagen are stretched at higher pressures. The slope of the first linear portion of the P-D curve was considered as the elastin P-D elastic modulus, and the slope of the curve obtained by subtraction between the P-D curve and the extrapolation of the elastin straight line was assumed to be the collagen P-D elastic modulus. After vitamin D3-induced calcinosis, the elastin elastic modulus was unaffected, whereas the collagen elastic modulus decreased significantly during occlusion maneuvers (58.6%, p less than 0.01) and during bolus injections of angiotensin (37.2%, p less than 0.05). The collagen elastic modulus correlated with the serum calcium concentration (r = -0.65, p less than 0.001) and with the aortic pulse pressure (r = 0.51, p less than 0.01), and this relation persisted at constant heart rate. Histopathologic analysis evidenced calcium-depositing elastic lamina, focal disappearance of collagen, and rupture of elastic fibers. The present study shows that accelerated, severe, experimental calcinosis-inducing calcium deposition inside the large artery walls is accompanied by a clear-cut paradoxical reduction in arterial rigidity that is mainly due to functional and structural modification of collagen elasticity.

Animals↗

PubMed1987

Elastic moduli, yield stress, and ultimate stress of cancellous bone in the canine proximal femur.

Elastic moduli, yield stress and ultimate compressive stress were determined for cancellous bone from the femoral head and neck regions of the canine femur. Unconfined compression tests were performed on 5 mm cubic samples which were cut from two femurs. Elastic moduli were measured in three orthogonal directions, and the yield stress and ultimate stress were measured along the proximal-distal axis. The results from this investigation support previous assumptions that the mechanical behavior of canine cancellous bone is qualitatively similar to human cancellous bone. The canine cancellous bone was observed to be anisotropic in elastic modulus. For two thirds of the cubic specimens tested, the elastic modulus was largest in the load-bearing, proximal-distal direction. A linear relationship between yield stress and elastic modulus was observed for canine bone, as is typical of human bone. A similar linear relationship between ultimate stress and elastic modulus was observed. Thus, for canine bone as well as for human bone, failure appears to be governed by a strain level which is position independent. The yield strain of 0.0259 and ultimate strain of 0.0288 for canine bone were both less than the yield strain of 0.0395 reported for human bone.

Animals↗

PubMed1995

Arterial wall mechanics in conscious dogs. Assessment of viscous, inertial, and elastic moduli to characterize aortic wall behavior.

To evaluate arterial physiopathology, complete arterial wall mechanical characterization is necessary. This study presents a model for determining the elastic response of elastin (sigma E, where sigma is stress), collagen (sigma C), and smooth muscle (sigma SM) fibers and viscous (sigma eta) and inertial (sigma M) aortic wall behaviors. Our work assumes that the total stress developed by the wall to resist stretching is governed by the elastic modulus of elastin fibers (EE), the elastic modulus of collagen (EC) affected by the fraction of collagen fibers (fC) recruited to support wall stress, and the elastic modulus of the maximally contracted vascular smooth muscle (ESM) affected by an activation function (fA). We constructed the constitutive equation of the aortic wall on the basis of three different hookean materials and two nonlinear functions, fA and fC: sigma = sigma E + sigma C + sigma SM + sigma eta + sigma M = EE. (epsilon - epsilon 0E) + EC.fC.epsilon + ESM.fA.epsilon + eta. [equation: see text] + M.[equation: see text] where epsilon is strain and epsilon 0E is strain at zero stress. Stress-strain relations in the control state and during activation of smooth muscle (phenylephrine, 5 micrograms.kg-1.min-1 IV) were obtained by transient occlusions of the descending aorta and the inferior vena cava in 15 conscious dogs by using descending thoracic aortic pressure (microtransducer) and diameter (sonomicrometry) measurements. The fC was not linear with strain, and at the onset of significant collagen participation in the elastic response (break point of the stress-strain relation), 6.02 +/- 2.6% collagen fibers were recruited at 23% of stretching of the unstressed diameter. The fA exhibited a skewed unimodal curve with a maximum level of activation at 28.3 +/- 7.9% of stretching. The aortic wall dynamic behavior was modified by activation increasing viscous (eta) and inertial (M) moduli from the control to active state (viscous, 3.8 +/- 1.3 x 10(4) to 7.8 +/- 1.1 x 10(4) dyne.s.cm-2, P < .0005; inertial, 61 +/- 42 to 91 +/- 23 dyne.s2.cm-2, P < .05). Finally, the purely elastic stress-strain relation was assessed by subtracting the viscous and inertial behaviors.

Animals↗

PubMed2003

On the sensitivity of wall stresses in diseased arteries to variable material properties.

Accurate estimates of stress in an atherosclerotic lesion require knowledge of the material properties of its components (e.g., normal wall, fibrous plaque, calcified regions, lipid pools) that can only be approximated. This leads to considerable uncertainty in these computational predictions. A study was conducted to test the sensitivity of predicted levels of stress and strain to the parameter values of plaque used in finite element analysis. Results show that the stresses within the arterial wall, fibrous plaque, calcified plaque, and lipid pool have low sensitivities for variation in the elastic modulus. Even a +/- 50% variation in elastic modulus leads to less than a 10% change in stress at the site of rupture. Sensitivity to variations in elastic modulus is comparable between isotropic nonlinear, isotropic nonlinear with residual strains, and transversely isotropic linear models. Therefore, stress analysis may be used with confidence that uncertainty in the material properties generates relatively small errors in the prediction of wall stresses. Either isotropic nonlinear or anisotropic linear models provide useful estimates, however the predictions in regions of stress concentration (e.g., the site of rupture) are somewhat more sensitive to the specific model used, increasing by up to 30% from the isotropic nonlinear to orthotropic model in the present example. Changes resulting from the introduction of residual stresses are much smaller.

Anisotropy↗

PubMed1998

Assessment of cancellous bone quality in severe osteoarthrosis: bone mineral density, mechanics, and microdamage.

The role of bone microdamage (microscopic cracks or microcracks and ultrastructural collagen matrix and bone mineral damage) in diseases such as osteoarthrosis and osteoporosis is poorly understood. Microdamage accumulation in vivo is influenced by age and cyclic loading, therefore, it would be useful if the burden of microdamage in bone could be assessed by noninvasive measures such as the radiological measurement of bone mineral density (BMD). The aim of this study was to investigate the relationship between BMD, compressive strength and stiffness, and microdamage in the cancellous bone of the proximal femur in patients with severe osteoarthrosis. Trabecular bone core samples, from the intertrochanteric region of the femur, were obtained from 34 patients, with a mean age of 70.3 +/- 11.1 years, undergoing total hip arthroplasty for osteoarthrosis. Cores selected from contact X-ray images were used for BMD measurement, compressive mechanical testing or left untested (uncrushed), en bloc staining for microdamage, and bone histomorphometry. The study shows a strong dependence of both the elastic modulus and ultimate failure stress of the bone samples on BMD and a significant relationship between the elastic modulus and trabecular anisotropy (Tr. An). In multiple linear regression, BMD and Tr. An together account for about 70% of the variance in the elastic modulus. Then including microcrack crack density (Cr.Dn) and damage volume fraction (DxV/BV) variables, Tr. An alone accounts for a relatively small amount of the variation (8.5%) in ultimate failure stress and elastic modulus. The Cr.Dn accounts for more of the variation in the ultimate failure stress than in the elastic modulus (50% vs. 7%). In this experiment, data for Cr.Dn provide a measure of damage associated with the ultimate failure of cancellous bone. In specimens that were not mechanically tested, in vivo microcrack accumulation increases exponentially with age. In conclusion, data from this study suggest that BMD and Cr.Dn are the major determinants of cancellous bone strength, whereas BMD and Tr. An are major determinants of cancellous bone stiffness. In bone specimens subjected to compressive testing there was no relationship between microdamage and BMD, suggesting that BMD cannot be used to monitor changes in the mechanical properties of bone due to microdamage accumulation.

Absorptiometry, Photon↗

PubMed2003

[Effects of bone morphogenetic protein and transforming growth fractor-beta on biomechanical property for fracture healing in rabbit ulna].

OBJECTIVE: To investigate the effects of exogenous bone morphogenetic protein (BMP) and transforming growth factor-beta (TGF-beta) on biomechanical property for ulna of fracture healing. METHODS: Thirty-six adult rabbits were made the model of right ulnar fracture and treated locally with TGF-beta/PLA, BMP/PLA, TGF-beta + BMP/PLA or PLA (as control group). Fracture healing was evaluated by measurement of the mechanical parameters and geometric parameters. RESULTS: As compared with control group, the geometric parameters, the bending broken load, the ultimate bending strength, the bending elastic modulus, the ultimate flexural strength, the flexural elastic modulus, the ultimate compressing strength, the compressing elastic modulus, and the ultimate tensile strength for ulna of fracture healing increased significantly in the treatment groups(P < 0.01). These parameters were higher in TGF-beta + BMP/PLA group than in TGF-beta/PLA group or in BMP/PLA group and in TGF-beta/PLA group than in BMP/PLA group(P < 0.05). There was no significant difference in bone density between the treatment groups and control group. CONCLUSION: Local application of exogenous TGF-beta and BMP can increase the callus formation and enhance biomechanical strength of bone after fracture healing. A combination of TGF-beta and BMP has synergetic effect in enhancing fracture healing.

Animals↗

PubMed1989

Role of physical properties of resistance vessel wall in regulating peripheral blood flow--II. Structural and mechanical behavior.

In order to examine the structural and mechanical properties of the vessel wall resistance when subjected to autoregulatory flow control, a mechanical model for the vascular wall was derived from a mathematical model. The mechanical model was an analogue model which connected in series the Maxwell model (elastic modulus: K3) with the parallel elements of Hill's model (elastic modules: K2) and Hooke's elastic model (elastic modulus: K1); it was also mathematically equivalent to the Spring model (see part I). The structural and mechanical properties of the resistance vessel wall were characterized by the three elastic moduli (K1, alpha K2 and K3) [mmHg]. The parameter alpha was a modification factor of the elastic modulus K2 given by the myogenic mechanism. After a numerical analysis of the experimental data given by the mechanical model, we confirmed that the arterial pressure range for autoregulatory flow controls shifted to the upper region with an increase of the elastic modulus K1 and the flow regulation was reduced.

Animals↗

PubMed1999

Effects of mucokinetic drugs on rheological properties of reconstituted human nasal mucus.

OBJECTIVE: To investigate the effects of mucokinetic drugs on the rheological properties of human nasal mucus in patients with chronic sinusitis. DESIGN: We reconstituted human nasal mucus obtained from 74 patients with chronic sinusitis and determined the effects of 4 mucokinetic drugs, including acetylcysteine, deoxynuclease I, 2% sodium bicarbonate, and a combination product containing tyloxapol (Alevaire), on rheological properties of reconstituted human nasal mucus (RHNM). We used 5% RHNM dissolved in phosphate-buffered solution as the optimal buffer and concentration of RHNM for the study because it showed a viscoelastic response similar to that of freshly collected nasal mucus from patients with chronic sinusitis. METHODS: Four experiments were performed to determine the influence of each drug on dynamic viscosity and elasticity of 5% RHNM. Distilled water was used as a control. RESULTS: Acetylcysteine and deoxynuclease I significantly decreased both dynamic viscosity and elastic modulus, while distilled water had no effect on rheological properties of 5% RHNM in vitro. Alevaire significantly reduced both dynamic viscosity and elastic modulus. Sodium bicarbonate significantly reduced elastic modulus but not dynamic viscosity. Reduction of elastic modulus by Alevaire was significantly greater than that by sodium bicarbonate, while there was no difference in reduction of dynamic viscosity between them. CONCLUSION: Our results indicate that RHNM may be useful for studying the topical effects of various drugs on nasal mucus from patients with chronic sinusitis.

Acetylcysteine↗

PubMed1991

Design considerations for cushion form bearings in artificial hip joints.

Lubrication mechanisms and contact mechanics have been analysed in a new generation of 'cushion form' bearings for artificial hip joints, which comprise low elastic modulus layers on the articulating surfaces. Comparisons have been made with 'hard' bearings used in existing prostheses and also with the natural hip joint. Lubricating film thicknesses are enhanced by larger contact areas and lower contact pressures. For a fixed contact area, simultaneous changes in layer thickness and radial clearance have been shown to have a small effect on elastohydrodynamic film thickness. Hard bearings designed with the same contact area as the cushion bearings produced a similar film thickness, but lubricant film thickness is not optimized in current designs. The main advantage of using a cushion bearing with low elastic modulus layers was found to be associated with microelastohydrodynamic lubrication. Careful selection of the elastic modulus is important in order to ensure that this lubrication regime was effective. Low elastic modulus layers may also produce local deformations, which enhance squeeze film action. The elastic modulus of the material should not be lower than necessary to produce effective microelastohydrodynamic lubrication, as a further reduction in modulus only increases the strain distribution in the material. A lubricant film thickness of 0.3 microns has been predicted for a cushion hip prosthesis with a femoral head diameter of 32 mm and radius of contact zone of 16 mm, using a 2 mm thick layer with an elastic modulus of 20 MPa.

Biomechanical Phenomena↗

PubMed1994

Bovine bladder compliance increases with normal fetal development.

In this study we characterized the elastic properties of the normal bovine bladder throughout fetal life, the newborn period and into adulthood. The elasticity of the bladder was measured with a novel circularly clamped bladder testing system. Pressurization of a circularly clamped bladder tissue sample caused the tissue to deflect upward repetitively in the shape of a spherical cap. If the centerline deflection is much larger than the tissue thickness, the elastic modulus, considered the inverse of compliance, can be determined using linear regression techniques. The results of our analyses showed that the elastic modulus decreases in direct proportion with increasing gestational age. These data suggest that during normal development of the bovine fetal bladder there is a progressive change from a rather stiff noncompliant bladder characterized by a high elastic modulus to a compliant bladder with a lower elastic modulus. Moreover, the increase in compliance appears in the developmental period when urine production first occurs. These observations suggest that volume work may be a significant event in the normal development process of the bovine bladder and results in an increase in bladder compliance. Conversely, the poorly compliant fetal bladder may explain some of the transient dilatations of the upper urinary tract which have been documented in utero. Finally, from the newborn period to the mature adult bovine we documented a relatively modest increase in the elastic modulus or decrease in bladder compliance which may reflect the normal aging process.

Animals↗

PubMedDate not supplied

Effect of pressurization on mechanical properties of mesenteric small arteries from spontaneously hypertensive rats.

The mechanical properties of the wall of isolated perfused arterial segments of mesenteric small arteries from 17-week-old spontaneously hypertensive rats (SHR) and age-matched Wistar Kyoto rats (WKY) were investigated. Third-order branches of mesenteric arteries were mounted in a pressure myograph chamber and pressurized from 1 to 140 mm Hg. Under isobaric conditions, the outer diameter and the lumen of small arteries studied were smaller in SHR than in WKY, whereas media width, media cross-sectional area and media-lumen ratio were greater in SHR. Under passive conditions, the total change in internal and external diameter in response to increasing intravascular pressure was smaller in arteries from SHR. Incremental distensibility was significantly lower in arteries of SHR at intravascular pressures between 1 and 40 mm Hg, but was significantly greater between pressures of 40-100 mm Hg. Wall stress generated by intravascular pressure was significantly smaller in arteries from SHR. As a function of wall strain (under isometric conditions), stress and incremental elastic modulus were shifted to the left in SHR vessels. Under isobaric conditions or in relation to wall stress, the slope of elastic modulus was smaller in SHR. This decrease in elastic modulus may confer additional elasticity to the vascular wall of resistance arteries from SHR. The presence of a greater distensibility at physiological levels of intravascular pressure and decreased incremental elastic modulus indicates that the changes in the structure of small mesenteric arteries in SHR can be defined as the result of a combination of eutrophic and hypertrophic remodeling.

Animals↗

PubMed2004

Application of dynamic mechanical analysis (DMA) to determine the mechanical properties of pellets.

Pellets of a wide range of mechanical properties were produced by the process of extrusion and spheronisation using various formulation factors. A range of mechanical properties from a simple fracture load to detailed load/displacement curves obtained when pellets were subjected to diametral compression test and a bed of pellets was compacted, were used to provide measure of tensile strength, deformability, linear strain, elastic modulus, yield and shear strength. Such conventional techniques resulted in irreversible damage to the structure of the pellets and were unable to establish the viscoelastic properties of the pellets. The application of the dynamic mechanical analysis (DMA), however, allowed the determination of (1) an accurate Young's modulus of elasticity, which was found to be between 8.4 and 24-fold higher than that determined from the diametral compression test, (2) the presence of a reversible elastic deformation even after the yield point in terms of storage modulus and (3) a change in the values of the phase angle, which illustrates the increase in viscoelasticity of the pellets formed with ethanol, glyceryl monostearate (GMS) or glycerol, while a decrease in viscoelasticity with the incorporation of lactose into the microcrystalline cellulose (MCC) pellets. This work further demonstrated that the only feasible technique for determining the elastic and plastic deformability of the pellets is the one which subjects the specimen to stress/relaxation cycles and can determine the dissipated energy in terms of loss modulus or phase angle, and that is DMA.

Analysis of Variance↗

PubMed2001

Endothelial, cardiac muscle and skeletal muscle exhibit different viscous and elastic properties as determined by atomic force microscopy.

This study evaluated the hypothesis that, due to functional and structural differences, the apparent elastic modulus and viscous behavior of cardiac and skeletal muscle and vascular endothelium would differ. To accurately determine the elastic modulus, the contribution of probe velocity, indentation depth, and the assumed shape of the probe were examined. Hysteresis was observed at high indentation velocities arising from viscous effects. Irreversible deformation was not observed for endothelial cells and hysteresis was negligible below 1 microm/s. For skeletal muscle and cardiac muscle cells, hysteresis was negligible below 0.25 microm/s. Viscous dissipation for endothelial and cardiac muscle cells was higher than for skeletal muscle cells. The calculated elastic modulus was most sensitive to the assumed probe geometry for the first 60 nm of indentation for the three cell types. Modeling the probe as a blunt cone-spherical cap resulted in variation in elastic modulus with indentation depth that was less than that calculated by treating the probe as a conical tip. Substrate contributions were negligible since the elastic modulus reached a steady value for indentations above 60 nm and the probe never indented more than 10% of the cell thickness. Cardiac cells were the stiffest (100.3+/-10.7 kPa), the skeletal muscle cells were intermediate (24.7+/-3.5 kPa), and the endothelial cells were the softest with a range of elastic moduli (1.4+/-0.1 to 6.8+/-0.4 kPa) depending on the location of the cell surface tested. Cardiac and skeletal muscle exhibited nonlinear elastic behavior. These passive mechanical properties are generally consistent with the function of these different cell types.

Animals↗

PubMed1980

Material properties of femoral cancellous bone in axial loading. Part I: Time independent properties.

The time independent material behavior of cylindrical specimens obtained from the cancelous bone of 20 cadaveric human femora were determined. In this part of the publication, the nominal values for compressive strength, limits of elasticity (yield point), strain, elastic modulus and apparent density are being reported for the cancellous bone of the femoral head and condyle. The correlations between the various parameters are analysed. A positive linear correlation between the four parameters compressive stength, limit of elasticity, modulus of elasticity and apparent density could not be excluded. The material properties vary considerably both within one single bone and between individuals. Compressive strength, modulus of elasticity and apparent density found for cancellous bone of the femoral head are greater than those found in the condyles. Within the condyles, compressive strength, elastic modulus and apparent density increase from the proximal parts to the parts closer to the joint. The medial femoral condyle showed higher compressive strength than the lateral one. Relating each of the three other parameters to the apparent density of the individual specimen did not result in equalizing the data for the material properties. This indicates that the mechanical properties of cancellous bone are strongly related to the direction of loading.

Aged↗

PubMed1977

The modulus of elasticity and flexural strength of some acrylic bone cements.

The modulus of elasticity and flexural strength were measured for three bone cements, CMW Bone Cement, Surgical Simplex, and Palacos R. with various additions and prepared as in actual use. A distinct afterpolymerization was demonstrated in all three brands. The addition of blood lowered the E-modulus wheras radio-opaque media seemed to render them somewhat stiffer. The rupture values demonstrated the inhomogeneity of the final product in all three brands. The possible clinical significance of the findings is discussed.

Bone Cements↗

PubMed2004

The use of acoustic microscopy to study the mechanical properties of glass-ionomer cement.

OBJECTIVES: The aim of the study was to investigate the relationship between microstructure, acoustic and mechanical properties of hardened dental cement samples, prepared with different powder/water ratios. METHODS: Glass-ionomer dental cement samples, prepared with a standard amount of cement powder and different amounts of water have been examined after being hardened. Surface microstructure and ultrasound, longitudinal and shear velocities were obtained with a scanning acoustic microscope. Conditional effective elastic modulus and Poisson's ratio have been calculated using longitudinal and shear sound velocity values. Then on the same samples elastic modulus and microhardness have been determined by standard tests. Additional samples have been used to determine compressive strength. RESULTS: Density; conditionally instantaneous elastic modulus; high-elasticity modulus and compression strength of the samples decrease when large amounts of water were used for their preparation. At the same time porosity and microhardness of the cement matrix increase. Acoustic parameters and parameters of elasticity, calculated on the basis of sound velocity, demonstrated changes, similar to those obtained in standard mechanical tests. SIGNIFICANCE: The established relation between microstructure, acoustic and mechanical parameters demonstrated a high capacity of acoustic microscopy application for non-destructive characterization of dental materials. A particular advantage of the acoustic microscopy is the opportunity to evaluate microstructure and mechanical properties on the same sample.

Compressive Strength↗

PubMed2003

Elastohydrodynamic lubrication analysis of a functionally graded layered bearing surface, with particular reference to 'cushion form bearings' for artificial knee joints.

Elastohydrodynamic lubrication of a functionally graded layered (FGL) bearing surface, whose elastic modulus increases with depth from the bearing surface, was investigated in this study. The finite difference method was employed to solve the Reynolds equation, simultaneously with the elasticity equation of the bearing surface, under circular point contacts. The finite element method was adopted to solve the elasticity equation for the FGL bearing surface. The displacement coefficients thus obtained were used to calculate the elastic deformation of the bearing surface, required for the elastohydrodynamic lubrication analysis. Good agreement of the predicted film thickness and pressure distribution was obtained, between the present method and a previous study for a single layered bearing surface with a uniform elastic modulus. The general numerical methodology was then applied to an FGL bearing surface with both linear and exponential variations in elastic modulus, with particular reference to the 'cushion form bearing' for artificial knee joints. The predicted film thickness and pressure distribution were shown to be quite close to those obtained for a single layer under typical operating conditions representative of artificial knee joints, provided that the elastic modulus of the single layer was chosen to be the average elastic modulus of the graded layer.

Coated Materials, Biocompatible↗