Three-dimensional Elastic Constants Research Articles

Input data for test cases used in benchmarking triaxial failure theories of composites

This article gives details of the input data provided for use in the Second World-Wide Failure Exercise (WWFE-II) for benchmarking triaxial failure criteria. It includes (a) three-dimensional elastic constants, ultimate strains and strengths and the nonlinear stress–strain curves for five unidirectional laminae and their constituents and (b) a description of 12 challenging test cases of 5 composite laminates, the lay-ups, layer thicknesses, stacking sequences and the loading conditions. The originators of 3D failure theories were requested to use the exact data provided here in their blind predictions of the test cases. The instructions issued to the contributors are also presented at the end of this article.

Elastic Properties of PEEK/IM7 Related to Temperature

An approach using laminate Coefficient of Thermal Expansion (CTE) measurements and two measured elastic constants is applied to predict the elastic properties of PEEK-IM7 prepreg laminae. This study assesses the feasibility of the use of this method to compute elastic properties over a range of temperatures. Three-dimensional elastic constants of PEEK/IM7 are determined for the temperature range of 20°Cto 120°C. The results indicate good agreement with quoted values measured by standard techniques, at room temperature. Moreover, the technique allows the study of the temperature dependence of the elastic properties over a range of temperatures. Thus, the technique demonstrates an ability to generate elastic properties data for a transversely isotropic material, over a wide temperature range, which ultimately can be used for improved performance prediction of structural composites.

Determination of the Elastic Constants of a Transversely Isotropic Lamina Using Laminate Coefficients of Thermal Expansion

Three-dimensional elastic constants for a fiber-reinforced lamina are often difficult to obtain from traditional mechanical tests. The relationships developed in thiswork enable the determination of any three of the five independent elastic constants, of a transversely isotropic lamina, from measurements of the remaining two elastic constants and the coefficients of thermal expansion (CTE’s) of [(+θ/−θ)n]s and unidirectional laminates. The approach is used to predict the elastic propertiesof a carbon fiber reinforced epoxy material and the predicted properties are shown to be in good agreement with quoted values. Further, the developed relationships can be used to determine elastic constants over a range of temperatures and, potentially, to study the effects of manufacturing on elastic properties. Therefore, this technique constitutes an additional method for evaluating the elastic constants that characterize a transversely isotropic, fiber-reinforced lamina.

Prediction of Effective Three-Dimensional Elastic Constants of Translaminar Reinforced Composites

A Trans-Laminar-Reinforced (TLR) composite is defined as composite laminate with up to five percent volume of fibrous reinforcement oriented in a trans-laminar fashion in the through-thickness direction. The objective of this work was to examine the effect of important parameters on elastic response. Detailed finite element models of unit cells were used to study the effects of adding TLR on the elastic constants. Parameters investigated included TLR material, TLR volume fraction, TLR diameter, TLR through-thickness angle, ply stacking sequence, and the microstructural features of pure resin regions and curved in-plane fibers. The work was limited to materials with at least one ply interface. Adding a few percent TLR had a small negative effect on the in-plane extensional and shear moduli, Ex, Ey and Gxy but a large positive effect (up to 60 percent) on the thickness direction extensional modulus, E,. The volume fraction and the axial modulus of the TLR were the controlling parameters affecting E,. The out-of-plane shear moduli, G, and Gy, were significantly affected only with the use of a TLR with a shear modulus an order of magnitude greater than that of the composite lamina. A simple stiffness averaging method for calculating the elastic constants was found to compare closely with the finite element results, with the greatest difference being found in the inter-laminar shear moduli, G., and Gy.

Finite Element Based Degradation Model for Composites with Transverse Matrix Cracks

A degradation model for predicting the effective homogenized threedimensional elastic constants of an interior ply with transverse matrix cracks was developed by analyzing a three-ply representative volume element using the finite element method. The degradation model was first evaluated by comparing the homogenized extensional modulus for a cracked [0/(90)3]s laminate with experimental results and finite element discrete models in which each crack was modeled discretely. Further, using the degradation model, the prediction of load redistribution for a cantilevered laminated plate with matrix cracks subjected to transverse end load was compared with a discrete crack model. Also, the prediction of load redistribution in a cracked plain weave composite subjected to uniaxial stress was compared with a discrete crack model. The degradation model showed good agreement in predicting the homogenized elastic constants and load redistributions.

An automated ultrasonic immersion technique for the determination of three-dimensional elastic constants of polymer composites

Abstract An automated, ultrasonic immersion system based upon an original National Physics Laboratory design has been built and evaluated. Ultrasonic transducers operating at a frequency of 2.25 MHz have been used to investigate both tensile and shear velocities in fibre reinforced composite samples. The automation of the system allows fast and efficient large area scanning to be performed. Hence both large-scale, spatial variations in the tensile velocities can be investigated and also, all of the three-dimensional elastic moduli of fibre-reinforced polymer composites can be deduced by monitoring the variation of ultrasonic velocity with angle of incidence on orthogonal planes. Preliminary results indicate that the system will yield most of the three-dimensional elastic moduli to absolute accuracies of better than ± 1 % and in the fast scanning mode, will detect spatial variations of the time-of-flight of ultrasound with a fractional timing error of ± 0.001%. Work is in progress to correlate the ultrasonic propagation to three-dimensional mesostructural features. An automated, two-dimensional image analyser has been developed in-house and a Noran, three-dimensional confocal laser scanning microscope system is being automated at Leeds to determine the three-dimensional fibre orientation states and voidage within composite samples. Therefore it will soon be possible to evaluate existing models linking three-dimensional mesostructure to ultrasonic propagation and the stiffness constants of anisotropic materials.

Modelling the three-dimensional elastic constants of parallel-fibred and lamellar bone

The complex hierarchical structure of lamellar bone makes understanding structure–mechanical function relations, very difficult. We approach the problem by first using the relatively simple structure of parallel-fibred bone to construct a mathematical model for calculating Young's moduli in three-dimensions. Parallel-fibred bone is composed essentially of arrays of mineralized collagen fibrils, which are also the basic structural motif of the individual lamellae of lamellar bone. Parallel-fibred bone structure has orthotropic symmetry. As the sizes and shapes of crystals in bone are not well known, the model is also used to compare the cases of platelet-, ribbon- and sheet-reinforced composites. The far more complicated rotated plywood structure of lamellar bone results in the loss of the orthotropic symmetry of individual lamellae. The mathematical model used circumvents this problem by sub-dividing the lamellar unit into a thin lamella, thick lamella, transition zone between them, and the recently observed “back-flip” lamella. Each of these is regarded as having orthotropic symmetry. After the calculation of their Young's moduli they are rotated in space in accordance with the rotated plywood model, and then the segments are combined to present the overall modulus values in three-dimensions. The calculated trends compare well with the trends in microhardness values measured for circumferential lamellar bone. Microhardness values are, as yet, the only measurements available for direct comparison. Although the model is not directly applicable to osteonal bone, which is composed of many hollow cylinders of lamellar bone, the range of calculated modulus values and the trends observed for off-axis calculations, compare well with measured values. © 1998 Chapman & Hall

Elastic Constants of Thick Orthotropic Composite Plates

This paper gives a method of obtaining the three-dimensional elastic constants of a completely free orthotropic plate from experimental plate vibration data, using an optimized three-mode Rayleigh formulation that incorporates through-the-thickness shear and rotatory inertia. The author believes this is the first reported attempt employing this formulation for obtaining the frequencies of thick plates with the incorporation of these two phenomena. The primary purpose of this paper is to develop the basic theory and method for this class of plates. The author had earlier utilized the classical lamination theory (which does not include these two effects) with both three and six optimized modes. The forward problem of computing plate vibration frequencies from given elastic constants by the Rayleigh method is solved for three thickness ratios (.075, .1, .125) for an isotropic (aluminum) and an orthotropic (graphite/epoxy) material. The inverse method for extracting the elastic constants from experimental vibration data is then applied to two thick plates fabricated from these materials. The results suggest that the proposed method is potentially useful for rapid, inexpensive extraction of in situ elastic constants of composite plates. The major original contribution of this paper is the novel approach for formulating and solving both the forward and inverse problems for thick plates.

Vibrational Spectra and Theoretical Three-Dimensional Elastic Constants of Isotactic Polypropylene Crystal: An Important Role of Anharmonic Vibrations

Calculation of normal mode frequencies and three-dimensional elastic constants is made for the first time for isotactic polypropylene crystal on the basis of lattice dynamical theory. The vibrational frequencies are found in relatively good agreement with the observed values for both the internal and external modes. The calculated Young’s modulus along the chain axis is 40.1 GPa which agrees well with the X-ray observed crystallite modulus ca. 40 GPa. The anisotropic curves of Young’s modulus and linear compressibility in the plane perpendicular to the chain axis are also calculated but agreement with the values observed at room temperature is not good. This discrepancy is erased successfully by considering a large anharmonic effect of methyl torsional modes.

Theoretical evaluation of three-dimensional elastic constants of native and regenerated celluloses: role of hydrogen bonds

Three-dimensional elastic constants were calculated for cellulose crystalline forms I and II (native and regenerated celluloses, respectively). The calculated Young's modulus E l along the chain axis is 167.5 GPa for form I and 162.1 GPa for form II. The E l of form II has only a slightly lower value than that for form I. This is consistent with the X-ray data ( c. 120–140 GPa for form I and c. 110 GPa for form II) although the absolute values are large. The E l was found not to be affected by intermolecular interactions but by intramolecular hydrogen bonds along the chain axis, especially the bond between the hydroxyl side group and the ether oxygen atom of the glucose ring (type a). A calculation neglecting this hydrogen bond gives the largely reduced E l of c. 70 GPa. Anisotropy of the Young's modulus and linear compressibility in the planes perpendicular to the molecular chain axis were also calculated. In the form I crystal, where the hydrogen-bonded sheet planes are stacked together by non-bonded van der Waals interactions, the modulus is large within the sheet plane and small in the direction perpendicular to the sheet: the anisotropy is similar to that reported for the nylon 6 α and γ forms. In form II the modulus is large but the anisotropy is not so remarkable, which is similar to atactic poly(vinyl alcohol) and poly( m-phenylene isophthalamide). The same tendency is seen also for the linear compressibility. In parallel with the calculation of the elastic constants, the lattice vibrational frequencies were calculated for forms I and II and compared with the observed infra-red and Raman spectral data so as to confirm whether the force fields used were reasonable.

Calculation of three-dimensional elastic constants of polymer crystals. 3. α and γ Forms of nylon 6

Calculation of three-dimensional elastic constants of polymer crystals. 3. α and γ Forms of nylon 6

Calculation of Three-Dimensional Elastic Constants of Polymer Crystals. 1. Method of Calculation

Calculation of Three-Dimensional Elastic Constants of Polymer Crystals. 1. Method of Calculation

Calculation of Three-Dimensional Elastic Constants of Polymer Crystals. 2. Application to Orthorhombic Polyethylene and Poly(vinyl alcohol)

Calculation of Three-Dimensional Elastic Constants of Polymer Crystals. 2. Application to Orthorhombic Polyethylene and Poly(vinyl alcohol)

Analysis of a thick-walled, layered anisotropic elastic cylinder subjected to a radial pressure.

U, V, W w x, y, z A method is developed for analyzing the stresses and strains in an anisotropic multilayered thick-walled circular cylinder subjected to a radial pressure. The basic assumption is made that each individual layer behaves in a homogeneous and anisotropic manner and that the plane strain assumption accurately describes the behavior of each layer. However, the analysis is simplified by also assuming that the basic element in equilibrium can be oriented so that it has, at all times, at least one plane of symmetry normal to the radius for the coordinate transformations required in the application of the analysis. The analysis is applied to a filament-reinforced structure by the derivation of a procedure for determination of the equivalent three-dimensional elastic constants in terms of the matrix and filament properties. This procedure is based on the assumption that opposite sides of the composite elemental cube remain straight and parallel after deformation. The analytic procedures have been programed for the IBM 7094 computer; the output yields the pressures between layers and the composite and individual stresses and strains in each layer for a cylinder-oriented and a filament-oriented coordinate system.