Orthotropic Material Properties Research Articles

The vibration and stability of non-homogeneous orthotropic conical shells with clamped edges subjected to uniform external pressures

In this paper an analytical procedure is given to study the free vibration and stability characteristics of homogeneous and non-homogeneous orthotropic truncated and complete conical shells with clamped edges under uniform external pressures. The non-homogeneous orthotropic material properties of conical shells vary continuously in the thickness direction. The governing equations according to the Donnell’s theory are solved by Galerkin’s method and critical hydrostatic and lateral pressures and fundamental natural frequencies have been found analytically. The appropriate formulas for homogeneous orthotropic and isotropic conical shells and for cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Several examples are presented to show the accuracy and efficiency of the formulation. The closed-form solutions are verified by accurate different solutions. Finally, the influences of the non-homogeneity, orthotropy and the variations of conical shells characteristics on the critical lateral and hydrostatic pressures and natural frequencies are investigated, when Young’s moduli and density vary together and separately. The results obtained for homogeneous cases are compared with their counterparts in the literature.

Non-stationary vibrations of a thin viscoelastic orthotropic beam

Non-stationary vibrations of a thin viscoelastic beam with orthotropic material properties are solved in this work. Concretely, analytical and numerical solutions of the problem of a finite simple supported beam transversely excited are presented. The analytical solution of the problem solved is derived based on the approximate Timoshenko’s beam theory and the discrete model of the generalized standard viscoelastic solid is used for the description of beam viscoelastic properties. Spatio-temporal functions describing distribution of beam deflection and slope of the beam represent the main results of this work. Moreover, the problem is solved numerically using FEM and the corresponding results are presented, as well. The comparison of results obtained shows excellent correspondence between analytical and numerical solutions and the ability of Timoshenko’s beam theory to describe non-stationary wave phenomena also for short times and in the immediate vicinity of external excitation.

Resonant-Based Identification of the Poisson’s Ratio of Orthotropic Materials

The resonant-based identification of the in-plane elastic properties of orthotropic materials implies the estimation of four principal elastic parameters: E 1 , E 2 , G 12 , and ν 12 . The two elastic moduli and the shear modulus can easily be derived from the resonant frequencies of the flexural and torsional vibration modes, respectively. The identification of the Poisson’s ratio, however, is much more challenging, since most frequencies are not sufficiently sensitive to it. The present work addresses this problem by determining the test specimen specifications that create the optimal conditions for the identification of the Poisson’s ratio. Two methods are suggested for the determination of the Poisson’s ratio of orthotropic materials: the first employs the resonant frequencies of a plate-shaped specimen, while the second uses the resonant frequencies of a set of beam-shaped specimens. Both methods are experimentally validated using a stainless steel sheet.

J1201-2-1 電動機固定子の振動解析モデル化に関する検討(解析・設計の高度化・最適化II)

In this study, a vibration analysis model of a motor stator with divided cores was created with the finite element method to examine the core material properties and a method of connecting the cores. The cores were composed of bonded laminated steel sheets of an orthotropic material having a different elastic modulus in the laminated direction and the in-plane direction. A method was devised for calculating the orthotropic material properties by using the material properties of the steel sheets and insulating layer and their lamination factor, instead of identifying the properties on the basis of experimental results. The vibration of the divided cores alone was accurately estimated with this method. Based that result, the motor stator vibration was accurately estimated by connecting the cores directly, instead of using springs to connect them.

Mathematical modelling and the study of the influence of initial stresses on the SIF and ERR at the crack tips in a plate-strip of orthotropic material

A plate-strip fabricated from the orthotropic material and containing a crack whose edges are parallel to the face planes of the plate is considered. It is assumed that the strip is stretched (or compressed) initially along the crack edges by uniformly distributed external normal forces acting on the simply supported ends of the plate-strip. After this initial stretching (or compression) the crack edges are loaded by additional uniformly distributed normal (opening) forces. As a result of the action of these additional forces the stress concentration characterized by the stress intensity factor (SIF) of mode I or by the energy release rate (ERR) of mixed mode arises at the crack tips. In this paper, the influence of the initial stresses on the SIF or ERR is modelled mathematically by the use of the three-dimensional linearized theory of elasticity. The aim of the present investigations is to study the effect of the mechanical–orthotropic properties of the plate-strip material on this influence by the use of the finite element method (FEM) modelling of the corresponding boundary-value problem.

Exact solutions for variable-thickness inhomogeneous elastic plates under various boundary conditions

In this paper, an exact solution to the governing equations of the bending of a variable-thickness inhomogeneous rectangular plate is presented. The procedure is applicable to variable-thickness inhomogeneous rectangular plates with two opposite edges simply supported. The remaining ones subjected to a combination of clamped, simply supported, and free boundary conditions and between these two edges the plate may have varying thickness. The procedure is valuable in view of the fact that tables of deflections and stresses cannot be presented for variable-thickness inhomogeneous orthotropic plates as for uniform-thickness homogeneous isotropic plates even for commonly encountered loads because the results depend on the inhomogeneity coefficient and the orthotropic material properties instead of a single flexural rigidity. Numerical results, useful for the validation or otherwise of approximate solutions, are tabulated. The influences of the degree of the inhomogeneity, aspect ratio, thickness parameter and degree of non-uniformity on the deflections and stresses are investigated.

Axial buckling of an orthotropic circular cylinder: Application to orthogrid concept

A novel re-defining of the orthotropic material properties in terms of a so-called associated geometric mean isotropic (GMI) material is used to develop a thorough buckling analysis of an axially-loaded orthotropic circular cylinder. A membrane prebuckling condition is assumed and an expression for the buckling stress is derived in terms of cylinder geometry, orthotropic material properties, and the number of waves in the buckling deformation pattern in the axial and circumferential directions. By assuming the number of waves in each direction are real-valued variables, as opposed to integers, conditions which result in stationary values of the buckling stress are sought, and once found, examined for their character as regards representing minima, maxima, or saddle points. Three quite different buckling characteristics are predicted, the particulars depending on the shear modulus of the orthotropic material relative to that of the associated GMI material. It is shown that if the shear modulus of the orthotropic material is greater than the shear modulus of the associated GMI material, the cylinder buckles into a unique axisymmetric deformation pattern. If the shear modulus of the orthotropic material is less than the shear modulus of the associated GMI material, the cylinder buckles into a unique nonaxisymmetric deformation pattern. If the shear modulus of the orthotropic material is exactly equal to the shear modulus of the associated GMI material (this is the situation for an isotropic cylinder), the cylinder can buckle into either axisymmetric or nonaxisymmetric deformation patterns. Moreover it is shown that, in this case, there exists a number of deformation patterns, all at essentially the same stress level. Closed-form lower-bound expressions for the buckling stress are developed using the adopted notation, the value of the shear modulus relative to the shear modulus of the GMI material determining which expression is applicable. The results of this analysis are applied to a circular cylinder constructed of a lattice structure consisting of helical and circumferential ribs, a so-called orthogrid lattice cylinder, where it is assumed that the ribs of the lattice structure are dense enough to be able to represent the elastic properties of the lattice with an equivalent homogenized orthotropic material. An isogrid cylinder, where the helical rib angle is 30° relative to the axial direction, is a special case. The orthotropic cylinder analysis is reformulated in terms of the material properties of the ribs and the angle of the helical ribs. For this situation the isogrid case is the GMI material, and the rib angle determines whether the shear modulus of the equivalent orthotropic material is greater than or less than the GMI material. This translates into the character of the buckling deformations depending directly on the rib angle.

Representative volume element of anisotropic unidirectional carbon–epoxy composite with high-fibre volume fraction

The knowledge of thermal properties of orthotropic layered composite materials is very important in the aeronautics industry. They can be measured with a dedicated apparatus but numerical computations develop strongly. It is well known that the microstructure has a strong impact on the calculation of the effective properties of these heterogeneous materials. The representative volume element (RVE) is then a crucial input to predict reliable results. In composites, fibre distribution often exhibits fibre-rich and fibre-poor regions. In this contribution, we first investigated a quantitative characterization of the microstructure using optical images and spatial descriptors. They give rise to microstructural features but also anisotropy. In a second time, we defined the RVEs of this microstructure for statistical parameters and effective thermal conductivities.

A Generalized Interaction Integral Method for the Evaluation of the T-Stress in Orthotropic Functionally Graded Materials Under Thermal Loading

The interaction integral method that is equipped with the nonequilibrium formulation is generalized to evaluate the nonsingular T-stress as well as mixed-mode stress intensity factors in orthotropic functionally graded materials under thermomechanical loads. This paper addresses both Mode-I and mixed-mode fracture problems and considers various types of orthotropic material gradation. The orthotropic thermomechanical material properties are graded spatially and integrated into the element stiffness matrix using the direct Gaussian formulation. The types of orthotropic material gradation considered include exponential, power-law, and hyperbolic-tangent functions, and the numerical formulation is generalized for any type of smooth material gradation. The T-stress and mixed-mode stress intensity factors are evaluated by means of the interaction integral method developed in conjunction with the finite element method. The accuracy of numerical results is assessed by means of thermomechanically equivalent problems.

Optimization of laminated composite structures using delamination criteria and hierarchical models

In this paper, the problem of maximizing the safety against failure of a fully three dimensional laminated composite structure is studied. The geometrically linear formulation of an eight node equivalent single layer solid shell finite element which utilizes fully three dimensional linear elastic orthotropic material properties is presented. Sensitivity analysis with respect to localized failure criteria functions is performed through use of semi-analytically calculated design sensitivities derived by the direct differentiation approach. Models are hierarchically refined in a two stage procedure, where a coarse mesh model is refined through the laminate thickness to obtain fully three dimensional descriptions of the detailed stress-strain state in localized zones of interest, hereby making it possible to take into account in-plane and transverse delamination failure effects. Examples illustrate the methodology.

Active fiber composites for the generation of Lamb waves

Active fiber composites (AFC) are thin and conformable transducer elements with orthotropic material properties, since they are made of one layer of piezoelectric ceramic fibers. They are suitable for applications in structural health monitoring systems (SHM) with acoustic non-destructive testing methods (NDT). In the presented work the transfer behavior of an AFC as an emitter of transient elastic waves in plate-like structures is investigated. The wave field emitted by an AFC surface bonded on an isotropic plate was simulated with the finite-difference method. The model includes the piezoelectric element and the plate and allows the simulation of the elastic wave propagation. For comparison with the model experiments using a laser interferometer for non-contact measurements of particle velocities at different points around the AFC on the surface of the plate were performed. Transfer functions defined as the ratio of the electric voltage excitation signal and the resulting surface velocity at a specific point are separately determined for the two fundamental Lamb wave modes. In order to take the orthotropic behavior of the AFC into account the transfer functions are determined for several points around the AFC. Results show that the AFC is capable to excite the fundamental symmetric and antisymmetric Lamb wave mode. The antisymmetric mode is mainly radiated in the direction of the piezoelectric fibers, while the symmetric mode is spread over a larger angle. The amplitudes of the emitted waves depend on the frequency of the excitation as well as on the geometric dimensions of the transducer.

FEA-Aided Design for a Working Model of a Split Florida-Helix

The NHMFL is developing a high-field split resistive magnet for use in far-infrared photon scattering experiments. The magnet will include four large scattering ports of elliptical shape at the midplane. Such a magnet configuration results in unique design challenges being especially severe for the windings in the midplane region of the innermost coils. Consequently, the NHMFL developed a new technology called split Florida-helix. To test and demonstrate those new concepts, the NHMFL recently completed the design and fabrication of a working model of a split Florida- helix. Advanced 3-D finite element analysis (FEA) served as the essential tool guiding the design optimization. This paper provides a systematic discussion of the critical features and techniques utilized in the complex model-based analysis including orthotropic material properties, actual non-symmetric current and load distributions as well as turn-to-turn frictional contact and sliding. Furthermore, the authors present a variety of detailed FEA results that prove to be fundamental for a successful application and integration of the split Florida-helix. Critical design decisions are discussed.

Caustics method in dynamic fracture problem of orthotropic materials

Caustics method is a powerful optical technique in fracture mechanics because of its high sensitivity to stress gradients. In this paper, it is applied to resolve dynamic fracture problems in orthotropic composites. Considering most orthotropic materials are opaque, reflective caustics method is derived here by combining the fundamental principle of caustics method with the mechanical properties of orthotropic materials. Meanwhile, corresponding experiments are carried out for typical glass fiber-reinforced composites, where mode I and mixed-mode fracture states are taken into account. By recording and analyzing shadow spot patterns during the crack propagation process carefully, crack onset time, dynamic fracture toughness and crack growth velocity of orthotropic composite are determined. These results will be useful to evaluate the dynamic fracture properties of composites and further to optimize their designs.

Static and Dynamic Analysis of Shallow Shells with Functionally Graded and Orthotropic Material Properties

The first-order shear deformation theory is used for description of shear deformable shallow shells with orthotropic material properties and continuously varying material properties through the shell thickness. Static and dynamic loads are considered here. For transient elastodynamic cases the Laplace-transform is used to eliminate the time dependence of the field variables. The first-order shear deformation theory reduces the original three-dimensional (3-D) problem into a two-dimensional (2-D) one. A meshless local Petrov-Galerkin (MLPG) formulation is applied to solve the 2-D problem. A weak formulation with a unit test function transforms the set of governing equations into local integral equations on local subdomains in the basic plane of the shell. Nodal points are randomly spread in that domain and each node is surrounded by a circular subdomain to which local integral equations are applied. The meshless approximation based on the Moving Least-Squares (MLS) method is employed for the implementation. The unknown Laplace-transformed quantities are computed from the local boundary-domain integral equations. The time-dependent values are obtained by the Stehfest's inversion technique.

Full-Scale Flexural Testing on Fiber-Reinforced Polymer (FRP) Poles

An extensive research project is currently carried out at the University of Sherbrooke to develop and evaluate the flexural behavior of lightweight fiber reinforced polymer (FRP) poles. In this project, a total of 23 full-scale prototypes of FRP poles with length ranging from 5 to 12 m were submitted to static flexural testing. The load carrying capacity, the failure modes and the deflection of these FRP poles, having hollow circular cross section and variable wall thickness, are being investigated experimentally and theoretically. The FRP poles were produced with the filament winding process, using epoxy resin reinforced with E-glass fibers. Each type of the poles tested in this study is constituted by three zones where the geometrical and the mechanical properties are different in each zone. The difference of these properties is due to the number of layers used in each zone and the fiber orientation of each layer. A new test setup designed and built according to ASTM-D4923–01 and ANSI-C136.20 standards recommendations was used to conduct full-scale flexural testing. Test parameters include the geometrical properties of FRP poles, the type of fibers, presence and positioning (compression side compared to tension side) of the hole are also investigated. Experimental results show that the use of low linear density glass-fibers could provide an increase of the ultimate load carrying capacity up to 38 % for some FRP poles. Also, the positioning of the hole in the compression side compared to the tension side leads to an increase of the ultimate load carrying capacity up to 22 % for the 5.4m (18 feet) FRP poles and no significant effect (3,5%) for the 12m (40 feet) FRP poles. This is mainly due to the stacking sequence and the stress states generated around the hole. Theoretical predictions of the deflection at the loading position are also presented using the theory of linear elasticity and the orthotropic material properties of the composite materials. Good agreement is found between experimental and theoretical results.

Finite element analysis of thin-walled composite two-span wood-based loadbearing stressed skin roof panels and experimental validation

The aim of this work is to study the structural behavior of prefabricated thin-walled wood-based loadbearing stressed skin panels for use in roofs. These stressed skin panels are composed of double thin skins, made of waterproof agglomerate and oriented strandboard, with a rigid insulating core made of extruded polystyrene without a vapor control layer or breather membrane. Current design criteria for these panels are often governed by calculating shear strength rather than bending strength. However, bending is often observed as the controlling factor in real structures and experimental studies. This work describes a design procedure based on predicting whether bending or shear will control their structural behavior. In order to get this purpose, we have studied both experimentally and theoretically the different elements that constitute this construction system to determine accurately its response in the presence of the external loads. The numerical analysis of the different variables using the finite element method (FEM) was validated by means of real tests on prototypes. The biggest difficulties in the simulation were found in the supports, due to the contacts between the different elements that compose them, and in the orthotropic material properties. Finally, the conclusions and suggested simplified procedures of calculation to be applied in similar structures are given. Comparisons with experimental data and with predictions using the quasi-analytic formulas are provided in order to support the validity of the proposed models.

Determination of Calibration Factors of the Hole Drilling Method for Orthotropic Composites using an Exact Solution

The hole drilling method has been used for measurement of the residual stresses in various materials. The released strains measured by the hole drilling method are converted to residual stresses using calibration factors. There is a standard for the hole drilling method for isotropic materials. Also for orthotropic materials, there are various methods for converting the released strains to residual stresses. In these methods the numbers of calibration factors vary from two to nine (a three by three matrix). In this article, calibration factors of the orthotropic plates are determined based on an exact solution method. For this purpose, assuming a plane stress condition, stresses and strains around the area under a rosette strain gage mounted to an orthotropic plate are calculated. A hole in the center of the rosette strain gage changes the distribution of the stresses and strains. Using the closed form solution proposed in this research the released strains and the calibration factors are calculated, and the residual stresses are found. Also, calibration factors for any orthotropic material with any level of orthotropy are calculated. These factors are independent of the loading and depend on the material properties of orthotropic materials and the ratio of the rosette diameter to the hole diameter. The material properties and the ratio of the rosette diameter to the hole diameter are considered in the equations, therefore there is no need for presenting the results in tabular form or any interpolations. The results for an orthotropic unidirectional ply made of carbon/epoxy are presented. Also, the calibration factors for three different unidirectional composites are calculated. The results obtained from the proposed closed form solution are compared with the simulated hole drilling method. A good agreement between the results shows the reliability of the closed form solution presented in this study.

Elastic characterization of orthotropic plates of any shape via static testing

The paper presents an inverse procedure for identifying elastic properties of isotropic or orthotropic materials from the full-field measurement of the surface displacements of plates under flexural loading configurations. The procedure is based on a numerical–experimental optimisation process which minimizes an error function defined by subtracting the experimental data from the outputs of the numerical analysis. In each iteration the optimisation process updates the values of the elastic constants in a finite element model of the specimen used in the experimental tests. The unknown parameters are simultaneously identified by a single test and without damaging the structural integrity of the specimen. The possibility of using the methodology for characterizing any-shaped plates was investigated. The applicability and the robustness of the procedure were carried out on aluminum and unidirectional Graphite/PEEK laminate specimens. Phase-shifting speckle interferometry was employed to detect the out-of-plane displacement field of a portion of the observed surface of the specimen.

LINEAR BENDING ANALYSIS OF INHOMOGENEOUS VARIABLE-THICKNESS ORTHOTROPIC PLATES UNDER VARIOUS BOUNDARY CONDITIONS

An exact solution to the bending of variable-thickness orthotropic plates is developed for a variety of boundary conditions. The procedure, based on a Lévy-type solution considered in conjunction with the state-space concept, is applicable to inhomogeneous variable-thickness rectangular plates with two opposite edges simply supported. The remaining ones are subjected to a combination of clamped, simply supported, and free boundary conditions, and between these two edges the plate may have varying thickness. The procedure is valuable in view of the fact that tables of deflections and stresses cannot be presented for inhomogeneous variable-thickness plates as for isotropic homogeneous plates even for commonly encountered loads because the results depend on the inhomogeneity coefficient and the orthotropic material properties instead of a single flexural rigidity. Benchmark numerical results, useful for the validation or otherwise of approximate solutions, are tabulated. The influences of the degree of inhomogeneity, aspect ratio, thickness parameter, and the degree of nonuniformity on the deflections and stresses are investigated.

Three-dimensional analysis of multi-layered functionally graded anisotropic cylindrical panel under thermomechanical loading

A general method is presented for the analysis of a thermoelastic multi-layered cylindrical panel made of an oblique pile of functionally graded layers having orthotropic material properties. The two edges of the panel are under simply-supported and zero temperature conditions. Steady-state thermal and mechanical boundary conditions can be applied on the innermost and outermost surfaces of the cylindrical panel. In addition, the functionally graded material is assumed to be of a power law type in the radial direction and the homogeneous solution in each layer is obtained in terms of the pseudo-Stroh formalism. The solution for the multi-layered cylindrical panel is derived based on the transfer matrix method. Novel to this work, is an alternative but equivalent formulation for the transfer matrix method which is more concise in expression. The derived solution is subsequently applied to investigate a five-layered cylindrical panel. The influence of the gradation of the material and ply angle on the distribution of temperature, displacements and stresses are investigated. The results clearly show that material gradation and ply angle significantly influence the elastic fields and temperature.