Modelling Young’s modulus for porous bones with microstructural variation and anisotropy

A structural model with three compositional phases and two levels of hierarchical organization is proposed for predicting Young’s modulus of porous bones with microstructural variations and anisotropy based on their geometric similarity to metal foams. It has been shown that the proposed single model provides predictions of Young’s modulus with high accuracy up to ±30% for cortical and cancellous bones compared with measured data from the literature. In addition, the conversion of the solid bone shape from “Plate-like” to “Rod-like” at a porosity of 70% or higher (BV/TV 30% or lower)—verified by observations—can be predicted using the proposed model.     

Resilient modulus–moisture content relationships for pavement engineering applications

In recent years, there has been an increased interest in determining the influence of moisture changes on the resilient modulus (M_R) of subgrade soils beneath pavement structures. Efforts have also been made to develop mathematical models that predict the change in M_R values with moisture. These models are expected to account for seasonal variations in subgrade moisture content. This study evaluates the variation of resilient modulus with post-compaction moisture content of soils in the State of Oklahoma and the State of Pennsylvania. A series of specimens was compacted at optimum moisture content and then tested for resilient modulus; other series of specimens were prepared at optimum moisture content and then either wetted or dried prior to M_R testing. Employed wetting and drying procedures are time-efficient in developing the M_R–moisture relationships. Results showed that M_R–moisture content relationships varied with soil types and M_R values varied inversely with changes of moisture content. In addition, an M_R–moisture model predicting the variation of resilient modulus with moisture contents is proposed. This model can be used to predict changes in the bearing capacity of pavements due to seasonal variations of moisture content.     

Prediction of Young’s modulus and Poisson ratio for foamed metals using octahedron model

Abstract

The present paper deals with the mathematical–physical expression of Young’s modulus and Poisson ratio of foamed metals. As it is known that, Young’s modulus and Poisson ratio are two basic mechanical parameters of engineering materials. Foamed metal is a class of excellent engineering materials with dual attributes of structural and functional characteristics; therefore, these two parameters are investigated for these materials, and the relevant mathematical–physical expressions are derived from the ‘octahedron model’ of porous materials in the present paper. The results show that the apparent Young’s modulus displays a quite complicated mathematical relationship to porosity of the porous body, and the apparent Poisson ratio is just a characteristic of the material constant almost not relative to porosity of the foamed metal.     

On Young’s modulus of multi-walled carbon nanotubes

Carbon nanotubes (CNTs) were discovered by Iijima in 1991 as the fourth form of carbon. Carbon nanotubes are the ultimate carbon fibres because of their high Young’s modulus of ≈ 1 TPa which is very useful for load transfer in nanocomposites. In the present work, CNT/Al nanocomposites were fabricated by the powder metallurgy technique and after extrusion of the nanocomposites bright field transmission electron microscopic (TEM) studies were carried out. From the TEM images so obtained, a novel method of ascertaining the Young’s modulus of multi-walled carbon nanotubes is worked out in the present paper which turns out to be 0·9 TPa which is consistent with the experimental results.     

Measurement and prediction of Young’s modulus of a Pb-free solder

Accurate and reproducible measurements of the Youngs modulus of solders are complicated by the early onset of yielding, microporosity, and variations in cooling rate. In this study, we report measurements of Youngs modulus of an Sn–3.5 wt % Ag solder by two techniques: (a) loading–unloading measurements in tension, and (b) non-destructive resonant ultrasound spectroscopy. Both techniques yielded similar values of Youngs modulus. The modulus decreased with increasing microporosity, in accordance with predicted values. Cooling rate affected the Ag₃Sn intermetallic morphology, but not Youngs modulus since the distribution of the particles was relatively random. This result was confirmed by microstructure-based finite element modeling.     

Scaling of conductivity and Young’s modulus in replicated microcellular materials

Scaling exponents for the conductivity and stiffness of replicated microcellular materials exceed commonly predicted values of 1 and 2. We show here that this is caused by the fact that, in replicated microcellular materials, the solid architecture varies with the relative density: a simple derivation based on the physics of powder consolidation returns and explains the observed scaling behaviour. The same derivation also gives an explanation for Archie’s law, known to describe the conductivity of wet soils.     

Young’s modulus and Poisson’s ratio of concrete at high temperatures: Experimental investigations

When exposed to fire, Young’s modulus of concrete degrades. Thus, exact knowledge of temperature-dependent reduction is important to determine the fire-resistance of concrete or composite members. Nevertheless, existing material properties for the Young’s and shear modulus of concrete are linked with some incertitudes.In addition, normative regulations lack information on the temperature-dependent Poisson’s ratio. In an attempt to overcome some of the existing uncertainties, experimental work was conducted to investigate elastic material properties of fire-exposed concrete. For this purpose, the Impulse Excitation Technique was used as an innovative testing technique. Based on experimental results, the authors propose new elastic material formulations for fire-exposed concrete.     

Correlation between Young’s modulus and impregnation quality of epoxy-impregnated SWCNT buckypaper

Carbon nanotube (CNT) sheets, also known as buckypaper, have high potential for structural applications due to their high volume fraction of CNT, the strongest and stiffest materials known. In this work, two different techniques, one based on positive pressure and another based on vacuum infiltration, are utilized to impregnate single-walled carbon nanotube (SWCNT) buckypaper sheets of 50–70 μm in thickness, resulting in a Young’s modulus of up to 15.4 GPa. Scanning electron microscopy demonstrates that the vacuum-based technique results in more effective impregnation of the buckypaper than the positive pressure technique. Thermogravimetry analysis of vacuum-impregnated specimens indicated a void content ranging from 5% to 32%. An advanced Mori–Tanaka-based micromechanics technique is also utilized to predict the effect of SWCNT volume fraction and void content on Young’s modulus of nanocomposites. These calculations suggest a higher void content of around 40% for the vacuum-impregnated composites.     

Bond calibration method for Young’s modulus determination in the discrete element method framework

A new methodology for the calibration of bond microparameters in rocks represented by a package of joined random spherical particles in the discrete element method (DEM) framework is presented. Typically, calibration is achieved through a trial-and-error procedure using several DEM simulations of uniaxial compressive tests (UCTs). The bond calibration model (BCM) does not need a time-dependent UCT-DEM simulation to establish the relation between the microproperties of the bond and the macroproperties of the rock specimen. The BCM uses matrices to describe the interaction forces exerted by bonds and, by means of an assembly process similar to the finite element method, it can describe the complex network of bonds, enabling the model to capture small variations in particle size and bond distribution as demonstrated in this work. In this work, the BCM is presented and compared with UCT simulations performed using Esys Particle software. Multiple simulations are done with constant bond properties and different particle size ratios (\(D_{MAX}/D_{MIN})\) that cause small variations in the specimen’s Young’s modulus; these variations are well captured by the BCM with an error of <10%.     

Analysis of effect of fiber orientation on Young’s modulus for unidirectional fiber reinforced composites

Young’s modulus of unidirectional glass fiber reinforced polymer (GFRP) composites for wind energy applications were studied using analytical, numerical and experimental methods. In order to explore the effect of fiber orientation angle on the Young’s modulus of composites, from the basic theory of elastic mechanics, a procedure which can be applied to evaluate the elastic stiffness matrix of GFRP composite as an analytical function of fiber orientation angle (from 0° to 90°), was developed. At the same time, different finite element models with inclined glass fiber were developed via the ABAQUS Scripting Interface. Results indicate that Young’s modulus of the composites strongly depends on the fiber orientation angles. A U-shaped dependency of the Young’s modulus of composites on the inclined angle of fiber is found, which agree well with the experimental results. The shear modulus is found to have significant effect on the composites’ Young’s modulus, too. The effect of volume content of glass fiber on the Young’s modulus of composites was investigated. Results indicate the relation between them is nearly linear. The results of the investigation are expected to provide some design guideline for the microstructural optimization of the glass fiber reinforced composites.     

Young’s modulus evolution at high temperature of SiC refractory castables

The evolution of Young’s modulus versus temperature has been evaluated in SiC-based hydraulically bonded refractories used in waste-to-energy (WTE) plants. Two types of low cement castables (LCC) with 60 and 85 wt% of SiC aggregates have been considered. The study was conducted by the way of a high temperature ultrasonic pulse-echo technique which allowed in situ measurement of Young’s modulus during heat treatment starting from the as-cured state up to 1400 °C in air or in neutral atmosphere (Ar) and during thermal cycles at intermediate temperatures (1000 and 1200 °C). For comparison in order to facilitate interpretation, thermal expansion has also been followed by dilatometry performed in the same conditions. Results are discussed in correlation with phase transformations occurring in the oxide matrix (dehydration at low temperature, crystallization of phases in the CaO–Al₂O₃–SiO₂ system) above 800 °C and damage occurring when cooling. The influence of oxidation of SiC aggregates on elastic properties is also discussed.     

Predictions of Young’s modulus of short inorganic fiber reinforced polymer composites

An expression of Young’s modulus of short inorganic fiber reinforced polymer composites was derived based on the tensile strength equation proposed in the previous paper, and the factor affecting the Young’s modulus was analyzed. This equation was applied to estimate the Young’s modulus of short inorganic fiber reinforced polymer composites. The results showed that the relative Young’s modulus increased nonlinearly with increasing fiber volume fraction, while increased linearly with an increase of fiber length-diameter ratio. Finally, the equation was verified preliminarily by using the measured Young’s modulus of the short glass fiber (SGF) reinforced polycarbonate/acrylnitrile–butadiene–styrene copolymer composites and the polypropylene reinforced respectively with SGF and short carbon fiber reported from literature, good agreement was found between the predictions and the experimental data.     

Measurement of the dynamic Young’s modulus of porous titanium and Ti6Al4V

The dynamic Young’s modulus of porous titanium and Ti6Al4V with various porosities was measured using the electromagnetic acoustic resonance method. The dependence of Young’s modulus (E) on the porosity (P) has been analysed in detail based on Phani–Niyogi relation and Pabst–Gregorová relation . We find that both Phani–Niyogi relation and Pabst–Gregorová relation with fixed material constant n = 2 or a = 1 but varying P _C can correctly account for the dependence of Young’s modulus on the porosity for porous titanium and Ti6Al4V.     

Definition of diagonal Poisson’s ratio and elastic modulus for infill masonry walls

The prediction of the response of infilled frames through the simplified approach of substituting the infill with an equivalent pin-jointed strut is treated. In this framework the results of an experimental study for the mechanical characterization of different types of masonry infills having the aim of estimating strength, Young modulus and Poisson’s ratio are presented. Four types of masonry were investigated and subjected to ordinary compressive tests orthogonally to the mortar beds and along the directions of the mortar beds. The experimental campaign confirmed the possibility of using an orthotropic plate model for prediction of the Poisson’s ratio and Young modulus along the diagonal direction of infills (these parameters are requested by a model already known in the literature for the identification of struts equivalent to masonry infills). The experimental campaign made it possible to recognise a correlation between the Poisson’s ratios and the strengths of masonries investigated along the orthotropic axes and to obtain the diagonal Poisson’s ratio without specific experimental tests. Finally, the experimental responses of some infilled frames were used to test the reliability of the model proposed here.     

Simplified fractal FEA model for the estimation of the Young’s modulus of Ti foams obtained by powder metallurgy

This work reports the introduction of a new finite elements model, for the estimation of the Young’s modulus (E) of metallic foams obtained by means of powder metallurgy using a Space Holder Phase (SHP). The model is based on the reduction of the volume, and the replication of the expected porosity distribution as fractal using two different pore sizes, which depend on the SHP. Ti foams with porosities ranging from 30% to 70% were experimentally obtained and characterized to validate the model. The effect of the porosity on the Young’s modulus for the Ti foams was estimated using the proposed model, and compared to the experimental values. Estimations obtained were in excellent agreement with the experimental results, with relative errors ranging from 0.8% to 11.2%. Both, experimental and estimated values for E showed important decreases as the porosity increases: e.g. E estimations were 37.1 and 9.3 GPa for porosities of 30% and 70%, respectively. The results demonstrated the fractal behavior of the porosity for the experimental metallic foams, as well as the efficacy of the proposed model for predicting the mechanical properties of these materials, being an important tool for their design and manufacture.     

Effect of a heating–cooling cycle on elastic strain and Young’s modulus of high performance and ordinary concrete

In this paper, the variations of elastic strain and Young modulus of high performance concrete and ordinary concrete during a heating?Ccooling cycle is presented. For the HPC, two heating rates are applied: 1.5 and 0.1?°C/min corresponding respectively to accidental and service conditions. For ordinary concrete, the results of service conditions are given. The temperatures of 400 and 220?°C are the heating??s final temperature phase of the accidental and service conditions respectively. The present work analyses the differences between the value of the elastic strain and the Young??s modulus at the beginning of the test (at ambient temperature), the end of the heating part and the end of the cooling part of each variation. Indeed, during the heating phase, the corresponding heating rates are applied until successive constant temperature levels are achieved: 150, 200, 300 and 400?°C for the high-performance concrete under accidental conditions and 140, 190 and 220?°C for both high-performance and ordinary concrete under service conditions. Those applied temperatures are maintained for several hours to ensure the stabilisation of internal temperature and physico-chemical thermo dependent processes. Moreover, the influence of the difference in mix concretes between the two types of concretes and the heating rate influence on those variations is also presented.     

The generalized Hamilton’s principle for a non-material volume

Fundamental principles of mechanics were primarily conceived for constant mass systems. Since the pioneering works of Meshcherskii, efforts have been made in order to elaborate an adequate mathematical formalism for variable mass systems. This is a current research field in theoretical mechanics. In this paper, attention is focused on the derivation of the generalized Hamilton’s principle for a non-material volume. First studies on the subject go back at least four decades with the article of McIver (J Eng Math 7(3):249–261, 1973). However, it is curious to note that the extended form of Hamilton’s principle that is derived by McIver does not recover the Lagrange’s equation for a non-material volume which is demonstrated by Irschik and Holl (Acta Mech 153(3–4):231–248, 2002). This does suggest additional theoretical investigations. In the upcoming discussion, Reynolds’ transport theorem is consistently considered regarding the original form of the principle of virtual work, and so the generalized Hamilton’s principle for a non-material volume is properly derived. It is finally shown that the generalization of Hamilton’s principle that is here proposed is in harmony with the Lagrange’s equation which is demonstrated by Irschik and Holl.