Evolution of Phase Strains During Tensile Loading of Bovine Cortical Bone

Synchrotron X‐ray scattering is used to measure average strains in the two main nanoscale phases of cortical bone – hydroxyapatite (HAP) platelets and collagen fibrils – under tensile loading at body temperature (37 °C) and under completely hydrated conditions. Dog‐bone shaped specimens from bovine femoral cortical bone were prepared from three anatomical quadrants: anterio‐medial, anterio‐lateral, and posterio‐lateral. The apparent HAP and fibrillar elastic moduli – ratios of tensile stress as applied externally and phase strains as measured by diffraction – exhibit significant correlations with the (i) femur quadrant from which the samples are obtained, (ii) properties obtained at the micro‐scale using micro‐computed tomography, i.e., microstructure, porosity and attenuation coefficient, and (iii) properties at the macro‐scale using thermo‐gravimetry and tensile testing, i.e., volume fraction and Young's modulus. Comparison of these tensile apparent moduli with compressive apparent moduli (previously published for samples from the same animal and tested under the same temperature and irradiation conditions) indicates that collagen deforms plastically to a greater extent in tension. Greater strains in the collagen fibril and concomitant greater load transfer to the HAP result in apparent moduli that are significantly lower in tension than in compression for both phases. However, tensile and compressive Young's moduli measured macroscopically are not significantly different during uniaxial testing.     

Micromechanics and effective elastic moduli of particle-reinforced composites with near-field particle interactions

A micromechanical framework is proposed to predict effective elastic moduli of particle-reinforced composites. First, the interacting eigenstrain is derived by making use of the exterior-point Eshelby tensor and the equivalence principle associated with the pairwise particle interactions. Then, the near-field particle interactions are accounted for in the effective elastic moduli of spherical-particle-reinforced composites. On the foundation of the proposed interacting solution, the consistent versus simplified micromechanical field equations are systematically presented and discussed. Specifically, the focus is upon the effective elastic moduli of two-phase composites containing randomly distributed isotropic spherical particles. To demonstrate the predictive capability of the proposed micromechanical framework, comparisons between the theoretical predictions and the available experimental data on effective elastic moduli are rendered. In contrast to higher-order formulations in the literature, the proposed micromechanical formulation can accommodate the anisotropy of reinforcing particles and can be readily extended to multi-phase composites.     

A generalized method for characterizing elastic anisotropy in solid living tissues

The microstructure of cortical bone may exhibit either transverse isotropic or orthotropic symmetry, thus requiring either five or nine independent elastic stiffness coefficients (or compliances), respectively, to describe its elastic anisotropy. Our previous analysis to describe this anisotropy in terms of two scalar quantities for the transverse isotropic case is extended here to include orthotropic symmetry. The new results for orthotropic symmetry are compared with previous calculations using the transverse isotropic analysis on the same sets of anisotropic elastic constants for bone, determined either by mechanical or by ultrasonic experiments. In addition, the orthotropic calculation has been applied to full sets of orthotropic elastic stiffness coefficients of a large variety of wood species. Although having some resemblance to plexiform bone in microstructural organization, there is a dramatic difference in both the shear and the compressive elastic anisotropy between the two materials: wood is at least one order of magnitude more anisotropic than bone.     

Composite materials whose phases have partially equal elastic moduli

Hill [J. Mech. Phys. Solids 11 (1963) 357, 12 (1964) 199] discovered that, regardless of its microstructure, a linearly elastic composite of two isotropic phases with identical shear moduli is isotropic and has the effective shear modulus equal to the phase ones. The present work generalizes this result to anisotropic phase composites by showing and exploiting the fact that uniform strain and stress fields exist in every composite whose phases have certain common elastic moduli. Precisely, a coordinate-free condition is given to characterize this specific class of elastic composites; an efficient algebraic method is elaborated to find the uniform strain and stress fields of such a composite and to obtain the structure of the effective elastic moduli in terms of the phase ones; sufficient microstructure-independent conditions are deduced for the orthogonal group symmetry of the effective elastic moduli. These results are applied to elastic composites consisting of isotropic, transversely isotropic and orthotropic phases.     

Experimental and theoretical studies of the elastic behavior of knitted-fabric composites

This paper presents experimental and theoretical studies of the elastic behavior of knitted-fabric composites. In the experimental studies, two types of weft-knit preforms based upon plain-stitch and rib-stitch fabrics were first fabricated and fabric composites were consolidated by using a hand lay-up process. Tensile and rail shear tests were performed, and Young's moduli along the warp and weft directions and shear modulus determined. In order to correlate the preform microstructure with composite elastic properties, geometric models for plain-stitch and rib-stitch fabric composites were developed. Modeling of the elastic behavior was conducted by using an averaging method. The predicted elastic constants are in reasonably good agreement with experimental values. Finally, the limitation and potential of knitted-fabric composites are discussed.     

The effect of particle shape on the mechanical properties of filled polymers

The existing models for predicting the elastic moduli of polymers dispersed with particles of shape other than spheres and continuous fibres are reviewed. The applicability and limitation of these equations are discussed. The emphasis of the review is to seek a unified understanding and approach to the effect of particle shape at finite concentration on the elastic moduli, thermal expansion coefficient, stress concentration factor, viscoelastic relaxation modulus and creep compliance of filled polymers. The effects of anisotropic particle shape on mechanical properties of polymeric composites are clearly illustrated. Attention is also drawn to the relationship between elastic moduli, thermal expansion, creep elongation and stress relaxation moduli.     

Elastic anisotropy and magnetic phase transition of yttrium garnet ferrite

Temperature dependences of elastic moduli describing elastic anisotropy in the rage from room temperature to 600 °C are shown. These temperature dependences are in agreement with available values of elastic moduli and temperature coefficients of the elastic moduli at room temperature. They are in agreement with modern quasi-harmonic theory of crystal lattices. Near 285 °C critical changes of effective elastic moduli due to magnetic phase transition are observed. A correlation was made between experimental and theoretical critical changes of longitudinal and transverse elastic wave velocities for different crystallographic directions. It is proved that the critical change of ultrasound wave velocities of this crystal is connected with the single-ion magnetostriction.     

Effective Elastic and Plastic Properties of Interpenetrating Multiphase Composites

In interpenetrating phase composites, there are at least two phases that are each interconnected in three dimensions, constructing a topologically continuous network throughout the microstructure. The dependence relation between the macroscopically effective properties and the microstructures of interpenetrating phase composites is investigated in this paper. The effective elastic moduli of such kind of composites cannot be calculated from conventional micromechanics methods based on Eshelby's tensor because an interpenetrating phase cannot be extracted as dispersed inclusions. Using the concept of connectivity, a micromechanical cell model is first presented to characterize the complex microstructure and stress transfer features and to estimate the effective elastic moduli of composites reinforced with either dispersed inclusions or interpenetrating networks. The Mori–Tanaka method and the iso-stress and iso-strain assumptions are adopted in an appropriate manner of combination by decomposing the unit cell into parallel and series sub-cells, rendering the calculation of effective moduli quite easy and accurate. This model is also used to determine the elastoplastic constitutive relation of interpenetrating phase composites. Several typical examples are given to illustrate the application of this method. The obtained analytical solutions for both effective elastic moduli and elastoplastic constitutive relations agree well with the finite element results and experimental data.     

Elastic moduli of colony-based lamellar solids

A means of calculating the effective elastic moduli of a material with a colony-based lamellar microstructure is presented. A two-step averaging process is employed in which the moduli of an individual colony are calculated using three-dimensional lamination theory and are then incorporated into a self-consistent scheme. Attention is focused on the effect of the colony orientation distribution and the moduli of the individual lamellae. Numerical results are presented for a range of material parameters, and analytic expressions are given for some special cases.     

A micromechanical model for interpenetrating multiphase composites

The dependence relation between the macroscopic effective property and the microstructure of interpenetrating multiphase composites is investigated in this paper. The effective elastic moduli of such composites cannot be calculated from conventional micromechanics methods based on Eshelby’s tensor because an interpenetrating phase cannot be extracted as dispersed inclusions. Employing the concept of connectivity, a micromechanical cell model is presented for estimating the effective elastic moduli of composites reinforced with either dispersed inclusions or interpenetrating networks. The model includes the main features of stress transfer of interpenetrating microstructures. The Mori–Tanaka method and the iso-stress and iso-strain assumptions are adopted in an appropriate manner of combination, rendering the calculation of effective moduli quite easy and accurate.     

An extension of the Secant Method for the homogenization of the nonlinear behavior of composite materials

We discuss the consequences of a different application of the principle the Modified Secant Method is [C. R. Acad. Sci. Paris Sér. IIb 320 (1995) 563] based on. In fact, we directly compute the second-order averages of the local fields available from the linear elastic homogenization procedure exploited in order to evaluate the effective elastic moduli. This method, which can be seen either as a simplification of the Modified Secant Method or as an extension of the Secant Method [J. Mech. Phys. Solids 26 (1979) 325], may be useful for any composite whose overall elastic constants need to be estimated by modeling the microstructure through Morphologically Representative Patterns [J. Mech. Phys. Solids 44 (1996) 307], which is for instance the case of syntactic foams [Int. J. Solids and Structures 38/40-41 (2001) 7235]. In order to show the accuracy of the proposed method, we apply it to several examples and compare its results with those obtainable by means of other analytical methods available in the literature, with numerical results of Finite Element simulations, and with experimental results. Closed-form solutions are derived for the effective yield stress of porous metals and incompressible composites reinforced with rigid spheres.     

Determination of third-order elastic moduli via parameters of bulk strain solitons

A method is proposed aimed for determination of the third-order elastic moduli (Murnaghan moduli) based on the estimation of measured parameters of bulk strain solitons in the three main waveguide configurations, a rod, a plate, and a shell. Formulas connecting the third-order moduli of the waveguide material and the parameters of a solitary strain wave (amplitude, velocity, full width at half-maximum) are derived. If the soliton parameters measured in three waveguide types manufactured from the same material are available, determination of the third-order elastic moduli is reduced to the solution of a system of three algebraic equations with a nondegenerate matrix.     

Numerical modelling of oxidized microstructure and degraded properties of 2D C/SiC composites in air oxidizing environments below 800 °C

In this paper, a numerical model which incorporates the modelling of oxidized microstructure and computing of degraded elastic moduli is presented for simulation of the oxidation behaviors of 2D C/SiC composites exposed to air oxidizing environments below 800 °C. Regarding the multi-scale characteristics of 2D C/SiC composite, the microstructure modelling is carried out on microscopic and macroscopic scale, respectively to compute the degraded elastic properties in terms of time duration, temperature and pressure, whose influences upon the oxidation microstructure morphology and degraded properties of 2D C/SiC composites are also investigated. It is shown that the simulation models well the microstructure morphology after oxidation and numerical results are found to be in good agreement with experimental data.     

Modeling the elastic properties of concrete composites: Experiment,differential effective medium theory,and numerical simulation

Concrete is a mixture of cement, water and aggregates. In terms of microstructure, besides the cement paste matrix and aggregate inclusions, there is a third phase, which is called the interfacial transition zone (ITZ), which forms due to the wall effect and can be thought of as a thin shell that randomly forms around each aggregate. Thus, concrete can be viewed as a bulk paste matrix containing composite inclusions. To compute the elastic properties of a concrete composite, a differential effective medium theory (D-EMT) is used in this study by assigning elastic moduli to corresponding bulk paste matrix, ITZ and aggregate. In this special D-EMT, each aggregate particle, surrounded by a shell of ITZ of uniform thickness and properties, is mapped onto an effective particle with uniform elastic moduli. The resulting simpler composite, with a bulk paste matrix, is then treated by the usual D-EMT. This study shows that to assure the accuracy of the D-EMT calculation, it is important to consider the increase in the water:cement mass ratio (w/c) of the ITZ and the corresponding decrease in w/c ratio of the bulk matrix. Because of this difference in w/c ratio, the contrast of elastic moduli between the ITZ and the bulk paste matrix needs to be considered as a function of hydration age. The Virtual Cement and Concrete Testing Laboratory (VCCTL) cement hydration module is used to simulate the microstructure of cement paste both inside and outside the ITZ. The redistribution of calcium hydroxide between ITZ and bulk paste regions can further affect the elastic contrast between ITZ and bulk paste. The elastic properties of these two regions are computed with a finite element technique and used as input into the D-EMT calculation. The D-EMT predictions of the elastic properties of concrete composites are compared with the results measured directly with a resonant frequency method on corresponding composites. This comparison shows that the D-EMT predictions agree well with experimental measurements of the elastic properties of a variety of concrete mixtures.     

Ultrasonic measurements of microelectronic resins

The elastic moduli, measured with the ultrasonic technique, of commercial silica filled epoxy resins used in the electronic circuits are reported. Measurements of velocity propagation and attenuation were carried out in large temperature and frequency ranges. Predictions of the theoretical models were compared with the experimental values. Explicit expressions of the elastic moduli were derived as functions of filler content and the properties of the matrix and the fillers. The influences of frequency and temperature on the elastic moduli and attenuation are discussed.     

New higher-order bounds on effective transverse elastic moduli of three-phase fiber-reinforced composites with randomly located and interacting aligned circular fibers

A higher-order structure for three-phase composites containing randomly located yet unidirectionally aligned circular fibers is proposed to predict effective transverse elastic moduli based on the probabilistic spatial distribution of circular fibers, the pairwise fiber interactions, and the ensemble-area homogenization method. Specifically, the two inhomogeneity phases feature distinct elastic properties and sizes. In the special event, two-phase composites with same elastic properties and sizes of fibers are studied. Two non-equivalent formulations are considered in detail to derive effective transverse elastic moduli of two-phase composites leading to new higher-order bounds. Furthermore, the effective transverse elastic moduli for an incompressible matrix containing randomly located and identical circular rigid fibers and voids are derived. It is demonstrated that significant improvements in the singular problems and accuracy are achieved by the proposed methodology. Numerical examples and comparisons among our theoretical predictions, available experimental data, and other analytical predictions are rendered to illustrate the potential of the present method.     

Dependence of the elastic moduli of porous silica gel prepared by the sol-gel method on heat-treatment

The elastic moduli of silica gel monolith prepared by the sol-gel method from a tetramethoxysilane solution have been measured before and after heat treatment. The elastic moduli showed a drastic change with heat treatment; for example, Young's modulus changed from 0.95 GPa for the gel before heating to 72.5 GPa for the densified product after heating to 1050 ° C. The change in the Young's modulus of the gel with heating temperature is discussed on the basis of changes of porosity and strength of the silica skeleton.     

微观结构对复合材料弹性有效性能的影响

在渐近均匀化方法的基础上,用ANSYS参数设计语言建立了周期性边界条件,用ANSYS有限元程序对单胞进行求解,得到了复合材料的有效性能。分析了不同微观结构对材料有效性能的影响,并与实验和其它理论结果进行比较。得到了不同方向的方形纤维对于材料的有效模量和有效泊松比的影响。     

Evaluation of thermo-viscoelastic property of CFRP laminate based on a homogenization theory

The thermo-viscoelastic constitutive equation of unidirectional carbon fiber reinforced plastic (CFRP) is evaluated using a numerical approach based on the finite element method (FEM) and homogenization theory. The constitutive equation of the CFRP is considered in the Laplace-transformed domain, and it is discussed on the basis of the correspondence principle, which is satisfied by each of the Laplace-transformed elastic moduli. Homogenization theory is employed to estimate the ‘homogenized elastic moduli’ of the composite composed of matrix resin and carbon fibers. Using the approximation of a generalized Maxwell model, the relaxation moduli of CFRP are obtained by numerical computation using the FEM. From the relaxation modulus of epoxy resin and elastic moduli of carbon fiber, thermo-viscoelastic properties of CFRP laminates at several temperatures can be estimated using the FEM with homogenization theory. The effectiveness of the present study is verified by comparing the experimental results and numerical calculations for the relaxation moduli of the CFRP laminates.