Orthotropic Elastic Constants Research Articles

Orthotropic elastic constants for polyimide film

AbstractThe orthotropic constants of polyimide film have been characterized using the theory of elasticity of an anisotropic material. Experimental techniques coupled with the mechanics of orthotropic materials are used to determine all 9 independent orthotropic elastic constants (3 tensile moduli, 3 shear moduli, and 3 Poisson's ratios) and 3 coefficients of thermal expansion. Vibrational holographic interferom‐etry is used to determine the orthotropic axes of symmetry. For this polyimide film, the two principal axes coincided with the machine and transverse directions. It is also used to evaluate the 2 in‐plane Poisson's ratios by measuring residual stresses in 2‐D and 1‐D square membranes. Using other instruments such as a high pressure gas dilatometry apparatus, a tensile tester, a pressure‐volume‐temperature apparatus, a thermornechanical analyzer, and a torsion pendulum, the 7 other orthotropic constants and the 3 coefficients of thermal expansion are determined.

Finite Element Analysis of Non-Laminated Composite Pin-Loaded Straps for Civil Engineering

Presented is a finite element analysis for carbon fiber reinforced plastic pin-loaded straps of non-laminated form. TheMARCK6.2 code was used. The goal was the determination of tensile strength from a knowledge of the peak stresses in the region where a (multi-layer) strap makes contact with the two steel circular pins. The paper illustrates difficulties other researchers and analysts may encounter if they are to analyze similar composite material problems. Numerical problems were found because of computing resource limitations, the highly orthotropic elastic constants, the geometric non-linearity, the geometric and material discontinuities, and the presence of friction and sliding contact. Their influence on stresses is illustrated by way of a simplified model for a single-layer strap. Peak stresses are shown to change significantly on varying a finite element model parameter that is not physical. Acceptable finite element stresses, away from the region with the peak stresses, are found on comparing them with a classical stress solution and experimental measurement.

The inplane elastic properties of circular cell and elliptical cell honeycombs

The inplane elastic properties of perfectly circular and elliptic cell honeycombs are derived through an analytical method and validated numerically. In the case of perfectly circular cell hexagonally packed honeycomb, the inplane elastic properties are shown to be isotropic. However, a departure from circularity of the cells leading to cell ellipticity results in the inplane properties becoming orthotropic. The orthotropic elastic constants are also derived analytically and validated numerically.

Determination of the orthotropic elastic constants of thin PET film by an ultrasonic micro-spectrometer

The elastic constants of a thin polyethylene terephthalate (PET) film, whose thickness was 12 mum, were determined from the acoustic reflection coefficients. The reflection coefficients were measured by an ultrasonic micro-spectrometer (UMSM), and all of the elastic constants were obtained through the fitting of a theory to the dip points of the magnitude of the reflection constants where leaky Lamb waves were excited. As a result, the film was found to be an orthotropic material, and the observed elastic constants were consistent with those measured by tensile tests. Besides, the in-plane shear modulus was found to be much larger than the out-of-plane shear moduli. Then, the reflection coefficients were calculated using the obtained stiffness matrix. The calculated values reproduced not only the dip points but also the change in magnitude of measured reflection coefficients depending on the frequency and the incident angle. These agreements suggest that the stiffness matrix components were determined accurately.

Parameter uncertainty and design of optimal experiments for the estimation of elastic constants

This paper investigates an inverse technique for the identification of orthotropic elastic constants from measured plate natural frequencies. In general, the accuracy of the identified parameters depends on the method of estimation, modelling errors and measurement errors. The paper addresses the parameter uncertainty due to errors in the measurements. Based on assumptions of the measurement errors, second-order statistics of parameters are approximated by linearization schemes. The main focus is on the possibility of designing the experiment to minimize the uncertainty of the estimated parameters. The uncertainty of each estimate as function of the experimental design variables is investigated. Also the overall optimality of the experimental design defined as the hypervolume of the confidence region is considered. The results show that not all parameters are estimated with a sufficient precision in the general case, but by carefully designing the experiment, the parameter uncertainties can be greatly reduced. Both thin and thick plates are considered with focus on single-layer plates, but the results for laminated plates are also discussed.

Relationships between bone morphology and bone elastic properties can be accurately quantified using high-resolution computer reconstructions.

It would be advantageous if the mechanical properties of trabecular bone could be directly inferred from stereomorphometric parameters. For that purpose, apparent density and mean intercept length, as measures of bone mass and directionality (fabric), are commonly correlated with the elastic characteristics of bone samples, as determined in compression tests. However, complete and accurate relationships have not yet been established in this way. This may be due not only to the occurrence of artifacts in both the stereomorphometric and the mechanical assessments but also to an inherent inadequacy of mean intercept length in characterizing the full mechanical significance of bone architecture or nonhomogeneities in trabecular tissue properties not accounted for in stereomorphometry. In this study, we introduce a computer modeling approach allowing these biases to be eliminated. With use of high-resolution three-dimensional computer reconstructions of trabecular bone specimens for stereomorphometry and for microstructural finite element models to simulate mechanical tests, unbiased comparisons become feasible. The purpose was to investigate if accurate and complete relationships can be established in this way. Four different fabric measures were considered: mean intercept length and three volume-based ones. Compliance matrices were calculated from fabric tensors, with use of the mathematical relationship proposed by Cowin for 29 vertebral whale-bone specimens. These were correlated with the compliance constants determined directly from the microstructural finite element model simulation. The nine orthotropic elastic constants of all 29 specimens were well predicted from their stereomorphometric fabric and volume fraction values, with correlation coefficients ranging from R2adj = 0.9934 to 0.9963. When individual compliance components were considered (1/Ei, 1/Gij, or -v[ij]/Ei), correlation coefficients ranged from R2adj = 0.924 to 0.982. All four fabric measures performed equally well. It is concluded that volume fraction and fabric measures correlate highly with the apparent elastic properties of bone samples, provided that anisotropy and nonhomogeneity in the elastic properties of the trabecular tissue itself have negligible effects on the apparent properties. Whether this is true for bone in general remains to be seen, as only a subset was analyzed here. These methods, however, can be valuable in similar assessments of other subsets.

Direct mechanics assessment of elastic symmetries and properties of trabecular bone architecture

A method is presented to find orthotropic elastic symmetries and constants directly from the elastic coefficients in the overall stiffness matrix of trabecular bone test specimens. Contrary to earlier developed techniques, this method does not require pure orthotropic behavior or additional fabric measurements. The method uses high-resolution computer reconstructions of trabecular bone specimens as input for large-scale FE-analyses to determine all the 21 elastic coefficients in the overall stiffness matrix of the specimen, using a direct mechanics approach. An optimization procedure is then used to find the coordinate transformation that yields the best orthotropic representation of this matrix. The method is illustrated here relative to two trabecular bone specimens. The techniques developed here can be used to obtain a complete characterization of the mechanical properties of trabecular architecture. With the development of in vivo reconstruction techniques, even in vivo measurements will be possible.

Direct mechanics assessment of elastic symmetries and properties of trabecular bone architecture

A method is presented to find orthotropic elastic symmetries and constants directly from the elastic coefficients in the overall stiffness matrix of trabecular bone test specimens. Contrary to earlier developed techniques, this method does not require pure orthotropic behavior or additional fabric measurements. The method uses high-resolution computer reconstructions of trabecular bone specimens as input for large-scale FE-analyses to determine all the 21 elastic coefficients in the overall stiffness matrix of the specimen, using a direct mechanics approach. An optimization procedure is then used to find the coordinate transformation that yields the best orthotropic representation of this matrix. The method is illustrated here relative to two trabecular bone specimens. The techniques developed here can be used to obtain a complete characterization of the mechanical properties of trabecular architecture. With the development of in vivo reconstruction techniques, even in vivo measurements will be possible.

Analytical and Experimental Results of the Coefficient of Thermal Expansion of High-Modulus Graphite-Epoxy Materials

The coefficient of thermal expansion (CrE) of an arbitrary laminate can be determined by the Classical Laminate Theory. However, the results obtained have shown that the CFE is very sensitive to some orthotropic elastic constants and to the laminate layup. In particular, the non-Hookean behaviour of an unidirectional lamina in the fibres direction should have to be taken into account to exactly predict the CTE. In order to verify the theoretical analysis, a new test facility has been designed to carefully measure the CTE in specimens made of advanced composite materials having a quasi zero value of CTE. The facility is basically composed of two thermostats and contained in a vacuum chamber (10−2 Pa). Four specimens are enclosed in the first thermostat and the operating temperature range is limited to −20°C up to 80°C. Four displacement transducers are enclosed in the second one at a temperature of 20°C and kept rigorously constant within 0.1°C. The specimens are mounted on Ultra Low Expansion (ULE) frames having a CTE of 1 × 10−8/°C. It has been possible to miniize the measurement error in the CTE, by a careful choice of displacement sensors and the high control of their thermal stability. The following CTEs have been obtained along the fibre direction, normal to the fibre direction and for a quasi-isotropic laminate, respectively, for two High-Modulus (HM) graphite/epoxy materials. GY70/M18: α1 = −1.04 × 10−6/°C; α2 = 33.9 × 10−6/°C; α x = -0.095 × 10−6/C M55J/M18: α1 = −1.02 × 10−6/°C; α2 = 35.2 × 10−6/°C; α x = −0.162 × 10−6/°C The analytical results have shown as a variation of ±1° only in the lamina orientation can change the CTE of the quasi-isotropic laminate up to ±50.5% of the theoretical value; as well as, a variation of ±5% in the physical and mechanical properties can change the CTE up to ±48%.

Characterization of anisotropie elastic constants of silicon-carbide participate reinforced aluminum metal matrix composites: Part I. Experiment

The anisotropic elastic properties of silicon-carbide particulate (SiCp) reinforced Al metal matrix composites were characterized using ultrasonic techniques and microstructural analysis. The composite materials, fabricated by a powder metallurgy extrusion process, included 2124, 6061, and 7091 Al alloys reinforced by 10 to 30 pct ofα-SiCp by volume. Results were presented for the assumed orthotropic elastic constants obtained from ultrasonic velocities and for the microstructural data on particulate shape, aspect ratio, and orientation distribution. All of the composite samples exhibited a systematic anisotropy: the stiffness in the extrusion direction was the highest, and the stiffness in the out-of-plane direction was the lowest. Microstructural analysis suggested that the observed anisotropy could be attributed to the preferred orientation of SiCp. The ultrasonic velocity was found to be sensitive to internal defects such as porosity and intermetallic compounds. It has been observed that ultrasonics may be a useful, nondestructive technique for detecting small directional differences in the overall elastic constants of the composites since a good correlation has been noted between the velocity and microstructure and the mechanical test. By incorporating the observed microstructural characteristics, a theoretical model for predicting the anisotropic stiffnesses of the composites has been developed and is presented in a companion article (Part II).

Identification of Anisotropic Elastic Parameter with the Aid of Extended Kalman Filter.

The paper deals with the inverse problem of identifying the elastic constants of anisotropic elastic materials. To solve the problem, the extended Kalman filter algorithm is combined with the finite element method. The effectiveness of the method is demonstrated by two simulations, I and II. The isotropic elastic constants are identified in simulation I. The orthotropic elastic constants are identified in simulation II. At the beginning of the identification processes, the materials are assumed to be orthotropic in simulation I and isotropic in simulation II. It was confirmed through the two simulations that the material constants were identified correctly.

Toward laser-ultrasonic characterization of an orthotropic–isotropic interface

Unidirectional boron–epoxy composite overlays are used to reinforce highly stressed or damaged regions in aircraft structures. This paper describes work ultimately directed toward an assessment of the adhesive bond strength between the composite overlay and the aircraft substrate. The potential use of leaky interface waves for the adhesive bond assessment is addressed, by considering the existence of such waves along the boundary between an orthotropic composite overlay and an isotropic metallic substrate. Numerical procedures are described to calculate complex leaky interface wave velocities, for propagation parallel or perpendicular to the fibers. It is shown that seven of the nine orthotropic elastic constants are required to calculate these velocities. Next, a new scheme is proposed to determine the seven required orthotropic elastic constants, based on laser-ultrasonic line sources and interferometric detection. Simplifications to the scheme, including the use of piezoelectric detection, are possible for a plate where there is access to both sides. This simplified scheme is demonstrated experimentally for a boron–epoxy plate. For the case where this boron–epoxy plate is well bonded to aluminum, titanium, or steel substrates, leaky interface waves are predicted for propagation perpendicular to the fibers. By contrast, for propagation parallel to the fibers, an interface wave is predicted only for the steel substrate; no wave is predicted for the aluminum substrate, while the titanium substrate is a borderline case.

A note on the anisotropy and fabric of highly porous materials

The dependence of the orthotropic elastic constants of a highly porous material upon the stereological parameters characterizing the anisotropy of the porous microstructure has been considered in two recent papers in this journal. In the first paper [1] dimensional arguments were employed to develop to a relationship between ratios of the orthotropic elastic constants and ratios of the mean intercept lengths for a class of cell wall bending models of highly porous materials. In the second paper [2] the general tensorial form of the relationship between the orthotropic elastic constants and the mean intercept length was described without reference to a specific form or type of porous microstructure. The purpose of this note is to observe that the particular relationships obtained from the class of cell wall bending models used in the first paper are proper special cases of the general relationships given in the second paper.

Correction formulae for the misalignment of axes in the measurement of the orthotropic elastic constants

In the experimental determination of the orthotropic elastic constants, one often encounters the situation in which the symmetry axes of the material are not coincident with specimen axes along which the material testing is accomplished. The problem of calculating the compliance coefficients in the symmetry coordinate system from measurements of the compliance coefficients made in an arbitrary, specimen fixed, coordinate system is considered here.

直交異方性材料積層板の材料定数の光学的測定法

This parer presents an optical method for the measurement of the bending stiffness ratio of orthotropic laminates and elastic constants of orthotropic laminae. Cornu's method for the measurement of Poisson's ratio of isotropic plates is elaborated with the aid of laser holographic interferometry to measure the pure bending deformation of orthotropic laminates. A four-point bending device was made, which enables us to adjust the bending load distribution so as to eliminate distortion in the pure bending deformation. The thermoplastic plate was used in place of the photographic plate for the realtime holographic interferometry. Quadratic expressions for the fringe curves are determined by the method of least mean square. Elastic constants are determined from the coefficients of the quadratic expressions. A method is described for the confirmation of the state of the pure bending deformation. The results indicate a need for further study on the conditions for the establishment of the pure bending deformations in experiment.

The fabric dependence of the orthotropic elastic constants of cancellous bone

It has been proposed that the orthotropic elastic constants of cancellous bone depend upon a tensorial measure of anisotropy called fabric as well as the tissue's structural density. Cowin (1985, Mechanics Mater. 4, 137–147; 1986, J. biomech. Engng 108, 83–88) developed explicit relationships for the elastic constant, structural density and fabric relationship. In this study the orthotropic elastic moduli, structural density, and fabric components were measured for 11 cancellous bone specimens from five bovine femora and for 75 specimens from three human proximal tibiae and fitted to these relationships using a least squares analysis. The relationships explained between 72 and 94% of the variance in the elastic constants. The relationships between the elastic constants and squared or cubed power functions of structural density had better predictive value over the entire distribution of the data than did expressions with linear functions of structural density.

Errors induced by off-axis measurement of the elastic properties of bone.

Misalignment between the axes of measurement and the material symmetry axes of bone causes error in anisotropic elastic property measurements. Measurements of Poisson's ratio were strongly affected by misalignment errors. The mean errors in the measured Young's moduli were 9.5 and 1.3 percent for cancellous and cortical bone, respectively, at a misalignment angle of 10 degrees. Mean errors of 1.1 and 5.0 percent in the measured shear moduli for cancellous and cortical bone, respectively, were found at a misalignment angle of 10 degrees. Although, cancellous bone tissue was assumed to have orthotropic elastic symmetry, the possibility of the greater symmetry of transverse isotropy was investigated. When the nine orthotropic elastic constants were forced to approximate the five transverse isotropic elastic constants, errors of over 60 percent were introduced. Therefore, it was concluded that cancellous bone is truly orthotropic and not transversely isotropic. A similar but less strong result for cortical bone tissue was obtained.

The elastic properties of a human mandible.

This paper is a report on a study of the elastic properties of the human mandible. Using a continuous-wave ultrasound technique, we determined the elastic constants of ten small specimens taken from various locations within a single mandible. Seven of the nine independent orthotropic elastic constants were determined for each specimen. The data indicate that human mandibular bone is elastically homogeneous but anisotropic. In the elastic properties it exhibits, it is similar to, if slightly less stiff than, human and canine femora. Indeed, elastically it seems like a long bone bent into the shape of a horseshoe.

Dependence of elastic constants of an anisotropic porous material upon porosity and fabric

The elastic properties of an anisotropic porous material can be represented as functions of the material's solid volume fraction (or porosity) and the principal diameters of the material's fabric ellipsoid. The fabric ellipsoid is a measure of the anisotropy of the microstructure of a material. The definitions and measurement techniques for fabric ellipsoids in granular materials, foams, cancellous bone, and rocks are discussed. The principal results presented in this work are algebraic expressions for the dependence of the orthotropic elastic constants upon both solid volume fraction and the fabric ellipsoid.

The equivalent orthotropic elastic properties of plywood

Plywood, in the form of a system of orthogonally oriented plies, is analyzed using strain energy considerations in order to define its elastic properties in terms of a homogeneous ‘equivalent material’. Because of the alignment of the plies these properties are orthotropic. The results obtained are either more general or more soundly based than those previously available. For a range of timbers, complete sets of nine orthotropic elastic constants are tabulated for both a three ply system and a system consisting of an even number of plies. The Poisson's ratios applicable within the plane of the plywood are significantly lower than those given by one previously available method.