The Elastic Constants of a Solid containing Spherical Holes

MacKenzie, J.K.

doi:10.1088/0370-1301/63/1/302

Cited by 712 publications

(225 citation statements)

References 6 publications

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“…17 Cartilage treated with collagenase for 12 h exhibited a significantly lower measured modulus (1.5 -0.2 MPa) than native cartilage. Histology further confirmed that the digestion of cartilage with collagenase leads to depletion of type-2 collagen and the loss of proteoglycan content as shown in Figure 6B, D. The highest modulus biological tissue tested, cancellous bone, had a measured average modulus of 123 MPa, with a standard deviation of 27 MPa, which is within the range previously published by Cowin et al 18 Modulus for cancellous bone, which is a porous material, was calculated using MacKenzie's Equation, 19 a well-established correlation of measured apparent modulus and porosity (Eq. 5), with an assumed porosity of 20% or a void space of 80%, which is a conservative, average estimate for cancellous bone beneath the articular cartilage of the distal femur head.…”

Section: Young's Modulus Of Biological Tissues Cartilage and Cancelsupporting

confidence: 83%

Design and Validation of a Compressive Tissue Stimulator with High-Throughput Capacity and Real-Time Modulus Measurement Capability

Salvetti

Pino²,

Manuel³

et al. 2012

Tissue Engineering Part C: Methods

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Mechanical stimulation has been shown to impact the properties of engineered hyaline cartilage constructs and is relevant for engineering of cartilage and osteochondral tissues. Most mechanical stimulators developed to date emphasize precision over adaptability to standard tissue culture equipment and protocols. The realization of mechanical characteristics in engineered constructs approaching native cartilage requires the optimization of complex variables (type of stimulus, regimen, and bimolecular signals). We have proposed and validated a stimulator design that focuses on high construct capacity, compatibility with tissue culture plastic ware, and regimen adaptability to maximize throughput. This design utilizes thin force sensors in lieu of a load cell and a linear encoder to verify position. The implementation of an individual force sensor for each sample enables the measurement of Young's modulus while stimulating the sample. Removable and interchangeable Teflon plungers mounted using neodymium magnets contact each sample. Variations in plunger height and design can vary the strain and force type on individual samples. This allows for the evaluation of a myriad of culture conditions and regimens simultaneously. The system was validated using contact accuracy, and Young's modulus measurements range as key parameters. Contact accuracy for the system was excellent within 1.16% error of the construct height in comparison to measurements made with a micrometer. Biomaterials ranging from bioceramics (cancellous bone, 123 MPa) to soft gels (1% agarose, 20 KPa) can be measured without any modification to the device. The accuracy of measurements in conjunction with the wide range of moduli tested demonstrate the unique characteristics of the device and the feasibility of using this device in mapping real-time changes to Young's modulus of tissue constructs (cartilage, bone) through the developmental phases in ex vivo culture conditions.

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Section: Young's Modulus Of Biological Tissues Cartilage and Cancelsupporting

confidence: 83%

Design and Validation of a Compressive Tissue Stimulator with High-Throughput Capacity and Real-Time Modulus Measurement Capability

Salvetti

Pino²,

Manuel³

et al. 2012

Tissue Engineering Part C: Methods

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“…which comes from the viscous analogy of shear modulus in a porous medium (McKenzie, 1950) one obtains the dependence of apparent viscosity measured at low porosity and volume fraction of voids in form:…”

Section: Discussionmentioning

confidence: 99%

“…Theoretical analysis and experimental testing of different relationships between the viscosity and porosity of glasses (Sura and Panda, 1990 ) illustrate the decrease of glass viscosity with porosity increase. McKenzie's (1950) analysis of dilute pores results in a viscosity dependence (r/s) vs. porosity (f) of:…”

Section: Theoretical Backgroundmentioning

confidence: 99%

A rheological investigation of vesicular rhyolite

Bagdassarov

Dingwell

1992

Journal of Volcanology and Geothermal Research

137

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Res., 50: 307-322.The rheology of vesiculating rhyolitic systems exerts a strong control on the transport of silicic magmas in the subvolcanic to volcanic environments. We present here an investigation of vesiculating and vesiculated rhyolites using dilatometric methods. This study examines the effect of vesicle content on the viscosity of a natural supercooled rhyolitic liquid with 0-70% vesicles.The experimental samples of rhyolitic glass are derived from fusion of a natural obsidian from Little Glass Butte, Oregon. Crystal-free rhyolite glasses of varying porosity were prepared by fusing obsidian powder in a Pt crucible. Differing porosities were obtained by varying the temperature (1300-1650°C) and duration (0.5-6 h) of the fusions. Cylindrical samples of the resulting vesiculated rhyolites were cored from the crucible using diamond tools and their ends were ground flat and parallel for dilatometry. The porosity of each sample was determined from Archimedean buoyancy density determinations and comparison with bubble-free rhyolite (2.331 g/cm 3, porosity= 1 -P/Po). The density of foamed samples was determined using their mass, volume and regular geometry.Viscosities were determined in the parallel plate mode at stresses of 5 × 103 to 105 Pa. The viscosimeter was calibrated using NBS 711 glass. The bubble contents were microscopically investigated using a video-reflected light system and image analysis software. Distribution functions of the size, orientation, aspect ratio and surface porosity were obtained.The viscosity of rhyolite decreases with increasing bubble content. A general relationship of the form: r/(f) = q(0)/ ( 1 + CJ), describes the effect of porosity, f (in volume fraction) on the viscosity, q, where C is a dimensionless constant (=22.4+_2.9) and logmr/(0) = 10.94+0.04 Pa s at 850°C.

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“…The first of these was by Einstein (1906Einstein ( , 1911 in which the viscosity of a suspersion was determined, assuming that it may be described by rigid spheres suspended in a viscous fluid and that the volume concentration is so small that particles do not interact. The case of dilute concentration, assuming that the particles are spherical, has been solved for a variety of materiaLs: Liquid droplets in another liquid, Taylor (1932); Elastic particles in '.scous fluid; Froehlich and Sack (1946); Empty holes in elastic solid, Mackenzie (1950); Rigid particles in elastic solid, Hashin (1955); Elastic p.rti4les in another elastic materi, Eshelby (1957), Hashin (1958); Viscous liquid' inclusions in elastic solid, Oldroyd (1956). Methods .of analysis for small concentration have been recently applied br Buediansky, Hashin and Sanders (1960) In the case of small voli_,rns concentration the fundamental assumption is that particles do not interact.…”

Section: Introductionmentioning

confidence: 99%