Showing papers on "Orthotropic material published in 1996"

C0 reissner–mindlin multilayered plate elements including zig-zag and interlaminar stress continuity

Abstract: SUMMARY Concerning composites plate theories and FEM (Finite Element Method) applications this paper presents some multilayered plate elements which meet computational requirements and include both the zig-zag distribution along the thickness co-ordinate of the in-plane displacements and the interlaminar continuity (equilibrium) for the transverse shear stresses. This is viewed as the extension to multilayered structures of well-known Co Reissner-Mindlin finite plate elements. Two different fields along the plate thickness co-ordinate are assumed for the transverse shear stresses and for the displacements, respectively. In order to eliminate stress unknowns, reference is made to a Reissner mixed variational theorem. Sample tests have shown that the proposed elements, named RMZC, numerically work as the standard Reissner-Mindlin ones. Furthermore, comparisons with other results related to available higher-order shear deformation theories and to three-dimensional solutions have demonstrated the good performance of the RMZC elements. Major portions of aerospace structures, as well as automotive and ship vehicles consist of flat and curved panels that are used as primary load-carrying components. Due to their obvious advantages, such as critical strength/stiffness-to-weight ratios, an increasing number of these panels are made of laminated composite material. This has led to extensive research activities in the mechanical properties, loading behaviour, structural modelling, and failure assessment of multilayered composite structures. Due to the geometry of laminated structural components, two-dimensional approaches have been extensively used to trace their response. The classical Kirchhoff's plate theory (CLT, Classical Lamination Theory) has revealed its limits when applied to thick panels with high orthotropic ratio.' - The shear deformation theories of Reissner-Mindlin-type (FSDT, First Shear Deformation Theories), even though, they are quite acceptable to study global response of high shear deformable thick composite structures, are not adequate for forecasting local stress-strain characteristics. In fact, some representative problems, exact three-dimensional have shown the failure of FSDT both to fulfill the interlaminar transverse shear stresses continuity at each interface and to describe the so-called zig-zag form' of the

Thermoelastic solutions for orthotropic and anisotropic composite laminates

Abstract: Solutions, within the framework of linear uncoupled thermoelasticity, are presented for certain problems of flexure of composite laminates. Benchmark numerical results, useful for the validation or otherwise of approximate laminate models, are tabulated. Finally, these results are used to examine the accuracy of the classical lamination theory based on Kirchhoff's hypothesis.

Buckling of orthotropic plates with free and rotationally restrained unloaded edges

Abstract: Solutions and parametric studies are presented for the buckling of rectangular plates whose axes of material orthotropy coincide with the axes of the plate. The results presented apply to homogeneous orthotropic plates, stiffened orthotropic plates, and laminated composite material plates having flexural orthotropy (i.e. single ply and multiply unidirectional, and symmetric cross-ply composite plates). The plates considered are subjected to uniform uniaxial compression and simply supported on the loaded edges. The boundary conditions are different on the two unloaded edges; one edge being free and the other edge being elastically restrained against rotation. Parametric studies showing the effect of the orthotropic properties of the plate materials, the plate aspect ratio, the rotational restraint of the one unloaded edge and the buckle half-wavelength are discussed. Results in the form of nondimensional buckling curves are given in terms of orthotropy ratios and in terms of properties of common unidirectionally reinforced composite material. The use of the solution in conjunction with experimental data to predict the edge rotational restraint coefficient for thin-walled composite material beams is described.

Finite element analysis of nonlinear orthotropic hyperelastic membranes

Abstract: A finite element method is presented for geometrically and materially nonlinear orthotropic hyperelastic membranes. The constitutive relations are formulated in terms of the invariants of the 2D right Cauchy-Green strain tensor and the resulting system of nonlinear equations solved using a Newton-Raphson approach. Both axisymmetric and nonaxisymmetric versions of the method are developed and validated. Example problems are solved for isotropic and orthotropic membranes, and the effect of various parameters investigated. Finally, convergence studies are performed for various degrees of anisotropy.

Separation of crack extension modes in orthotropic delamination models

Abstract: In modeling a crack along a distinct interface between dissimilar elastic materials, the ratio of mode I to mode II stress intensity factors or energy release rates is typically not unique, due to oscillatory behavior of near-tip stresses and displacements. Although methods have been developed for comparing mode mixes for isotropic interfacial fracture problems, this behavior currently limits the applicability of interfacial fracture mechanics in predicting delamination in layered materials without isotropic symmetry. The virtual crack closure technique (VCCT) is a method used to extract mode I and mode II energy release rate components from numerical fracture solutions. Energy release rate components extracted from an oscillatory solution using the VCCT are not unique due to their dependence on the virtual crack extension length, Δ. In this work, a method is presented for using the VCCT to extract Δ-independent energy release rate quantities for the case of an interface crack between two in-plane orthotropic materials. The method does not involve altering the analysis to eliminate its oscillatory behavior and it is similar to existing methods for extracting a mode mix from isotropic interfacial fracture models. Knowledge of near-tip fields is used to determine the explicit Δ dependence of energy release rate parameters. Energy release rates are then defined that are separated from the oscillatory dependence on Δ. A modified VCCT using these energy release rate definitions is applied to results from finite element analyses, showing that Δ-independent energy release rate quantities result. The modified technique has potential as a consistent method for extracting a mode mix from numerical solutions. The Δ-independent energy release rate quantities extracted using this technique can also aid numerical modelers, serving as guides for testing the convergence of finite element models. Direct applications of this work include the analysis of planar composite delamination problems, where plies or debonded laminates are modeled as in-plane orthotropic materials.

Failure Criteria for Masonry Panels under In-Plane Loading

Abstract: Failure of masonry panels under in-plane loading can be attributed to three simple modes: slipping of mortar joints, cracking of clay bricks and splitting of mortar joints, and middle plane spalling. In this paper a suitable strength criterion is connected to each collapse mode. In more detail, a frictional law is associated with the slipping, which accounts for the shear strength depending nonlinearly on normal stress (modified Mohr-Coulomb criterion of intrinsic curve). Splitting can be expected by the maximum tensile strain criterion (Saint Venant), orthotropic nonsymmetric elasticity being assumed for the material. Eventually panels exhibit spalling when the maximum compressive stress (Navier criterion) is attained under biaxial loading. Strength parameters are then identified on the basis of experimental results and a comparison with the reliable criteria found in the literature is carried out. The validity of the proposed failure criteria to predict the experimental failure modes in a nondimensional...

Bounding elastic constants of an orthotropic polycrystal using measurements of the microstructure

Abstract: We show how information about the elastic stiffness and compliance of an orthotropic polycrystal may be obtained from measurements of the statistical properties of the microstructure We begin by discussing the statistical properties of the governing equations and the hierarchy that results when the equations are averaged Perturbation solutions are obtained in terms of low-order statistical information Using methods previously developed, we derive bounds for the elastic stiffnesses and compliances A hierarchy of bounds is derived using the statistical information obtained from measurement of the microstructure We discuss the methods used to obtain the correlation functions of the microstructure and provide a specific example, using data obtained from a copper sample

Rayleigh waves in an orthotropic thermoelastic medium under gravity field and initial stress

Abstract: The aim of the present paper is to investigate the influence both of gravity field and initial stress on the propagation of Rayleigh waves in an orthotropic thermoelastic medium subject to certain boundary conditions. We suppose that the body is under initial stress alonqx 1-direction and incremental thermal stresses. The wave velocity equation has been obtained. Many special cases and comparison with the previous results have been studied.

Finite element analyses of anisotropic poroelasticity : A generalized Mandel's problem and an inclined borehole problem

Abstract: The finite element equations for non-linear, anisotropic poroelasticity are cast in the form of measurable engineering constants. Two problems of importance to the rock and petroleum industry are analysed by the FEM. First, the classical Mandel's problem with an extension to transversely isotropic case is investigated. Second, the problem of an inclined borehole is explored. In particular, the effect of material anisotropy on stress concentration near the wall with implication to borehole instability is examined in detail.

Semicontinuous mathematical model for bending of multilayered wire strands

Abstract: Semicontinuous wire strand modeling is an approach by which each layer of a strand is mathematically represented by an orthotropic cylinder whose mechanical properties are chosen to match the behavior of its corresponding layer of wires. Such a semicontinuous model is herein proposed for the analysis of multilayered wire strands under bending, tensile, and torsion loads. It is based on continuum mechanics and elasticity of orthotropic materials. The model premits the evaluation of strand stiffness, contact stress, interlayer shear stress, and interlayer slip. Results are obtained and given for a seven-wire strand and for selected steel reinforced aluminum conductors (ACSRs). Comparisons are made with results from two existing models that use a more classical approach, from another model that uses a semicontinuous approach, and from published experimental works. Under tensile and torsion loads, it is shown that the present semicontinuous model gives very accurate results. Under bending load, validation is more difficult to establish, but this new model is very promising.

Mode II fracture testing method for highly orthotropic materials like wood

Abstract: In this paper a mode II fracture testing method has been developed for wood from analytical, experimental and numerical investigations. Analytical results obtained by other researchers showed that the specimen geometry and loading type used for the proposed mode II testing method results in only mode II stress intensity and no mode I stress intensity at the crack tip. Experiments have been carried out to determine mode II fracture toughness K IIC and fracture energy G IIF from the test data collected from both spruce (pice abies) and poplar (populus nigra) specimens. It was found that there existed a very good relation between fracture toughness KIIC and fracture energy G IIF when the influence of orthotropic stiffness E II * in mode II was taken into account. It verified that for this mode II testing method the formula of LEFM can be employed for calculating mode II fracture toughness even for highly orthotropic materials like wood. In the numerical studies for the tested spruce specimen, the crack propagation process, stress and strain fields in front of crack tips and the stress distributions along the ligament have been investigated in detail. It can be seen that the simulated crack propagating process along the ligament is a typical shear cracking pattern and the development of cracks along the ligament is due to shear stress concentrations at the crack tips of the specimen. It has been shown that this mode II fracture testing method is suitable for measuring mode II fracture toughness K IIC for highly orthotropic materials like wood.

Isothermal flow of an anisotropic ice sheet in the vicinity of an ice divide

Abstract: Simulations of glacier flow are commonly based on the assumption that ice has an isotropic viscosity. Here we examine the plane flow of ice in the special region of an ice divide using a constitutive relation for an anisotropic, incompressible viscous body that is orthotropic and transversally isotropic. Ice is assumed to be isotropic at the ice sheet surface, with the continuous development of a vertical single maximum c axis fabric with increasing depth. We consider the theoretical case of an isothermal ice sheet over a horizontal bedrock, with no slip at the ice-bedrock interface. The ice sheet surface elevation is imposed, and the flow corresponding to the steady state is calculated, using a two-dimensional finite difference model based on the resolution of a pressure-Poisson equation. In this model, all components of the stress and strain rate tensor are calculated. The main conclusion is that for a fixed surface elevation, the general flow pattern accelerates when the anisotropic behavior of the ice is taken into account due to the greater fluidity with respect to shear stress. The downward motion of the ice is faster, despite a higher resistance to vertical deformation. As a result, the dominance of shear strain rate in the flow of polar ice is stronger in the anisotropic case than in the isotropic case. The shear stresses are slightly relaxed, while the longitudinal stresses are significantly increased in the anisotropic case.

In-Plane Fracture of Paper

Abstract: The in-plane failure of paper is studied in this work by means of a cohesive crack model from experimental as well as theoretical perspectives. Localized damage at in-plane tension of short paper strips is studied for low strain rates. It is observed that under uniaxial in-plane tensile tests, the evolution of the failure is stable and the damage of the paper strip is localized into a narrow zone. The damage in the paper strip develops only after the tensile strength has been reached. The uniaxial fracture properties of paper are defined and characterized by a descending stress-crack widening curve. From this curve the fracture energy can be obtained. A characteristic material parameter of a length dimension is introduced and depends on the fracture energy, the elastic modulus and the fracture strength. The material parameters are found to vary with the material orientation of the paper. A method to measure the fracture material parameters is proposed, where only the load and the elongation of the test specimen need to be recorded. Newsprint, kraft paper and paperboard are examined in this investigation. The cohesive crack model is used as a basis in the formulation of an orthotropic smeared crack constitutive relation to be used in finite element codes. The advantage of this approach is that it provides a theoretical tool in the study of the initiation and stable growth of a localized damage zone or crack in an arbitrary structure subjected to an arbitrary in-plane loading. The model proposed includes a failure criterion and a failure potential. The failure criterion changes its size and shape during the course of fracture softening. The failure potential determines the orientation of the fracture zone and the subsequent crack. The cohesive crack constitutive model is calibrated against one newsprint and one board paper. Simulation results from a single central notch specimen loaded in mode I are compared with experimental results. It was found that the fracture process region is of significant size and that the deviation from an autonomous fracture performance is considerable. The constitutive model developed is used in the investigation of the behavior of a fiber-based package material, with a punched opening, in the converting process at constant web tension. Finite element simulations of the converting process are made in order to understand how punched paperboard behaves in converting processes. The simulations are compared with experimental results and a reasonable agreement is obtained.

Effects of curvilinear anisotropy on radially symmetric stresses in anisotropic linearly elastic solids

Abstract: It has been known for some time that certain radial anisotropies in some linear elasticity problems can give rise to stress singularities which are absent in the corresponding isotropic problems. Recently related issues were examined by other authors in the context of plane strain axisymmetric deformations of a hollow circular cylindrically anisotropic linearly elastic cylinder under uniform external pressure, an anisotropic analog of the classic isotropic Lame problem. In the isotropic case, as the external radius increases, the stresses rapidly approach those for a traction-free cavity in an infinite medium under remotely applied uniform compression. However, it has been shown that this does not occur when the cylinder is even slightly anisotropic. In this paper, we provide further elaboration on these issues. For the externally pressurized hollow cylinder (or disk), it is shown that for radially orthotropic materials, the maximum hoop stress occurs always on the inner boundary (as in the isotropic case) but that the stress concentration factor is infinite. For circumferentially orthotropic materials, if the tube is sufficiently thin, the maximum hoop stress always occurs on the inner boundary whereas for sufficiently thick tubes, the maximum hoop stress occurs at the outer boundary. For the case of an internally pressurized tube, the anisotropic problem does not give rise to such radical differences in stress behavior from the isotropic problem. Such differences do, however, arise in the problem of an anisotropic disk, in plane stress, rotating at a constant angular velocity about its center, as well as in the three-dimensional problem governing radially symmetric deformations of anisotropic externally pressurized hollow spheres. The anisotropies of concern here do arise in technological applications such as the processing of fiber composites as well as the casting of metals.

A complete characterization of the possible bulk and shear moduli of planar polycrystals

Abstract: This paper investigates the range of possible elastic moduli of two-dimensional isotropic polycrystals. The polycrystals are comprised of grains obtained from a single orthotropic material. The overall elastic properties are described by an effective bulk modulus K 0 and an effective shear modulus μ 0 The pair ( K 0 , μ 0 ) is shown to be confined to a rectangle in the ( K , μ ) plane. Microstructures are identified which correspond to every point within the rectangle, and in particular to the corner points. Optimal bounds on the effective Poisson's ratio and Young's modulus follow immediately. Under a certain constraint on the crystal moduli, the rectangle degenerates to a line segment: the effective shear modulus of such a polycrystal is microstructure independent. This extends earlier work that established microstructure independence for polycrystals constructed from square symmetric crystals.

Theory of laser-generated transient Lamb waves in orthotropic plates

Abstract: This paper presents a theoretical study for modelling the thermoelastic excitation of transient Lamb waves propagating along the principal directions in an orthotropic plate. The normal mode expansion method is employed to express the transient displacement field by a summation of the antisymmetric and symmetric Rayleigh - Lamb wave modes in the surface stress-free orthotropic plate. This method is particularly suitable for waveform analyses of transient Lamb waves in thin sheet materials because one needs only to calculate contributions of the lowest few antisymmetric and symmetric modes. The dispersion characteristics and the transient Lamb waveforms excited by a pulsed laser in machine-made paper are analysed numerically and discussed in detail and attention is focused on the influence of the elastic stiffness constants. This work provides a quantitative analysis for noncontact and nondestructive detection of the elastic stiffness properties of the machine-made paper by the laser-generated Lamb wave technique.

Free bending vibrations of adhesively bonded orthotropic plates with a single lap joint

Abstract: This study is concerned with the free bending vibrations of two rectangular, orthotropic plates connected by an adhesively bonded lap joint. The influence of shear deformation and rotatory inertia in plates are taken into account in the equations according to the Mindlin plate theory. The effects of both thickness and shear deformations in the thin adhesive layer are included in the formulation. Plates are assumed to have simply supported boundary conditions at two opposite edges. However, any boundary conditions can be prescribed at the other two edges. First, equations of motion at the overlap region are derived. Then, a Levy-type solution for displacements and stress resultants are used to formulate the problem in terms of a system of first order ordinary differential equations. A revised version of the Transfer Matrix Method together with the boundary and continuity conditions are used to obtain the frequency equation of the system. The natural frequencies and corresponding mode shapes are obtained for identical and dissimilar adherends with different boundary conditions. The effects of some parameters on the natural frequencies are studied and plotted.