Showing papers on "Orthotropic material published in 1997"

Mechanics of laminated composite plates : theory and analysis

Abstract: Introduction and Mathematical Preliminaries Fiber-Reinforced Composite Materials. Vectors and Tensors. Matrices. Transformation of Vector and Tensor Components. Integral Relations. Equations of Anisotropic Elasticity Classification of Equations. Kinematics. Kinetics. Constitutive Equations. Equations of Thermoelasticity and Electroelasticity. Summary. Virtual Work Principles and Variational Methods Virtual Work. The Variational Operator and Functionals. Extrema of Functionals. Virtual Work Principles. Variational Methods. Summary. Introduction to Composite Materials Basic Concepts and Terminology. Constitutive Equations of a Lamina. Transformation of Stresses and Strains. Plane Stress Constitutive Relations. Classical and First-Order Theories of Laminated Composite Plates Introduction. An Overview of ESL Laminate Theories. The Classical Laminated Plate Theory. The First-Order Laminated Plate Theory. Stiffness Characteristics for Selected Laminates. One-Dimensional Analysis of Laminated Plates Introduction. Analysis of Laminated Beams Using CLPT. Analysis of Laminated Beams Using FSDT. Cylindrical Bending Using CLPT. Cylindrical Bending Using FSDT. Closing Remarks. Analysis of Specially Orthotropic Plates Using CLPT Introduction. Bending of Simply Supported Plates. Bending of Plates with Two Opposite Edges Simply Supported. Bending of Rectangular Plates with Various Boundary Conditions. Buckling of Simply Supported Plates Under Compressive Loads. Buckling of Rectangular Plates Under Inplane Shear Load. Vibration of Simply Supported Plates. Buckling and Vibration of Plates with Two Parallel Edges Simply Supported. Closure. Analytical Solutions of Rectangular Laminates Using CLPT Governing Equations in Terms of Displacements. Admissible Boundary Conditions for the Navier Solutions. Navier Solutions of Antisymmetric Cross-Ply Laminates. The Navier Solutions of Antisymmetric Angle-Ply Laminates. The LTvy Solutions. Analysis of Midplane Symmetric Laminates. Transient Analysis. Summary. Analytical Solutions of Rectangular Laminates Using FSDT Introduction. Simply Supported Antisymmetric Cross-Ply Laminates. Simply Supported Antisymmetric Angle-Ply Laminates. Antisymmetric Cross-Ply Laminates with Two Opposite Edges Simply Supported. Antisymmetric Angle-Ply Laminates with Two Opposite Edges Simply Supported. Transient Solutions. Summary. Finite Element Analysis of Composite Laminates Introduction. Laminated Beams and Plate Strips by CLPT. Timoshenko Beam/Plate Theory. Numerical Results for Beams and Plate Strips. Finite Element Models of Laminated Plates (CLPT). Finite Element Models of Laminated Plates (FSDT). Summary. Refined Theories of Laminated Composite Plates Introduction. A Third-Order Plate Theory. Higher-Order Laminate Stiffness Characteristics. The Navier Solutions. LTvy Solutions of Cross-Ply Laminates. Displacement Finite Element Model. Layerwise Theories and Variable Kinematic Models In troduction. Development of the Theory. Finite Element Model. Variable Kinematic Formulations. Nonlinear Analysis of Composite Laminates Introduction. Nonlinear Stiffness Coefficients. Solution Methods for Nonlinear Algebraic Equations. Computational Aspects and Numerical Examples. Closure. Index Most chapters include Exercises and References for Additional Reading

Anisotropic parameters and P-wave velocity for orthorhombic media

Abstract: Although orthorhombic (or orthotropic) symmetry is believed to be common for fractured reservoirs, the difficulties in dealing with nine independent elastic constants have precluded this model from being used in seismology. A notation introduced in this work is designed to help make seismic inversion and processing for orthorhombic media more practical by simplifying the description of a wide range of seismic signatures. Taking advantage of the fact that the Christoffel equation has the same form in the symmetry planes of orthorhombic and transversely isotropic (TI) media, we can replace the stiffness coefficients by two vertical (P and S) velocities and seven dimensionless parameters that represent an extension of Thomsen's anisotropy coefficients to orthorhombic models. By design, this notation provides a uniform description of anisotropic media with both orthorhombic and TI symmetry. The dimensionless anisotropic parameters introduced here preserve all attractive features of Thomsen notation in treatin...

A plane stress softening plasticity model for orthotropic materials

Abstract: A plane stress model has been developed for quasi-brittle orthotropic materials. The theory of plasticity, which is adopted to describe the inelastic behaviour, utilizes modern algorithmic concepts, including an implicit Euler backward return mapping scheme, a local Newton-Raphson method and a consistent tangential stiffness matrix. The model is capable of predicting independent responses along the material axes. It features a tensile fracture energy and a compressive fracture energy, which are different for each material axis. A comparison between calculated and experimental results in masonry shear walls shows that a successful implementation has been achieved. © 1997 John Wiley & Sons, Ltd.

A three-dimensional finite element formulation for thermoviscoelastic orthotropic media

Abstract: SUMMARY This paper is concerned with the development of a numerical algorithm for the solution of the uncoupled, quasistatic initial/boundary value problem involving orthotropic linear viscoelastic media undergoing thermal and/or mechanical deformation. The constitutive equations, expressed in integral form involving the relaxation moduli, are transformed into an incremental algebraic form prior to development of the nite element formulation. This incrementalization is accomplished in closed form and results in a recursive relationship which leads to the need of solving a simple set of linear algebraic equations only for the extraction of the nite element solution. Use is made of a Dirichlet{Prony series representation of the relaxation moduli in order to derive the recursive relationship and thereby eliminate the storage problem that arises when dealing with materials possessing memory. Three illustrative example problems are included to demonstrate the method. ? 1997 by John Wiley & Sons, Ltd.

Singular Stress and Electric Fields of a Piezoelectric Ceramic Strip with a Finite Crack under Longitudinal Shear

Abstract: Following the theory of linear piezoelectricity, we consider the problem of determining the singular stress and electric fields in an orthotropic piezoelectric ceramic strip containing a Griffith crack under longitudinal shear. The crack is situated symmetrically and oriented in a direction parallel to the edges of the strip. Fourier transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate for piezoelectric ceramics are obtained, and the results are graphed to display the influence of the electric field.

Experimental procedure and results for the identification of elastic constants of thick orthotropic plates

Abstract: Ability to determine six independent material parameters of thick orthotropic plates by a non-destructive numerical/experimental method is demonstrated. The parameters, which are the four in-plane elastic constants and the two transverse shear moduli, are deduced simultaneously from natural frequencies of the completely free plate by a nonlinear least squares approach. The main new contribution of the present paper is the consideration of higher mode natural frequencies of thick plates linked with an accurate numerical model based on a higher-order shear deformation theory for the theoretical predictions. This has enabled both transverse shear moduli to be estimated along with the in-plane parameters. Focus is here on the experimental technique and the application of the method in real tests. High quality frequency measurements are performed by the impulse technique employing a non-contacting microphone combined with advanced curve fitting of the frequency response function. The first 12 to 15 natural fre...

Evolution of anisotropy under plane stress

Abstract: Yield loci have been measured for cold rolled steel sheets prestrained by two stage loading. During the first loading, the sheets have been stretched by 3 and 6% tensile strain in the rolling direction. The second loading was at angles to the rolling direction with varying amounts of tensile strain. Then a set of tensile test specimens has been prepared from each of the prestrained sheets. From tensile tests, effects of the two-stage prestrains on the subsequent yielding has been investigated. Experiments show that initial orthotropic symmetry is maintained and that the orientations of orthotropy axes change continuously throughout the prestraining process. A simple phenomenological rule for the rotation of orthotropy axes is suggested.

New finite difference formulations for general inhomogeneous anisotropic bioelectric problems

Abstract: Due to its low computational complexity, finite difference modeling offers a viable tool for studying bioelectric problems, allowing the field behaviour to be observed easily as different system parameters are varied. Previous finite difference formulations, however, have been limited mainly to systems in which the conductivity is orthotropic, i.e., a strictly diagonal conductivity tensor. This in turn has limited the effectiveness of the finite difference technique in modeling complex anatomies with arbitrarily anisotropic conductivities, e.g., detailed fiber structures of muscles where the fiber can lie in any arbitrary direction. Here, the authors present both two-dimensional and three dimensional finite difference formulations that are valid for structures with an inhomogeneous and nondiagonal conductivity tensor. A data parallel computer, the connection machine CM-5, is used in the finite difference implementation to provide the computational power and memory for solving large problems. The finite difference grid is mapped effectively to the CM-5 by associating a group of nodes with one processor. Details on the new approach and its data parallel implementation are presented together with validation and computational performance results. In addition, an application of the new formulation in providing the potential distribution inside a canine torso during electrical defibrillation is demonstrated.

Analysis of filament-wound cylindrical shells under combined centrifugal, pressure and axial loading

Abstract: An analytical procedure is developed to assess the stresses and deformations of filament-wound structures under loading conditions particular to centrifuge rotors and to assess the effects of wind angle variation through the centrifuge wall. This procedure is based on classical laminated plate theory and models both plane stress and plane strain states of a cylindrical shell comprising a number of cylindrical sublayers, each of which is cylindrically orthotropic. Available loading conditions are: radial body force due to rotation about the cylinder axis, internal and external pressures and axial force. The analysis is applied to three examples: a pressure vessel, a centrifuge rotor and a flywheel. It is shown that the benefit of wind angle variation is more significant for applications in which there is no axial loading to the cylindrical shell. It is also shown that, where axial loading is present, the benefits of wind angle variation are more significant under the last ply failure criterion than under the first ply failure criterion.

The equilibrium shape of an elastically inhomogeneous inclusion

Abstract: A method is proposed for determining the equilibrium shape of an elastically inhomogeneous coherent precipitate in an anisotropic medium. Arbitrary eigenstrains and an isotropic interfacial energy density are taken into account. Use is made of Eshelby's notion of a “force on an interface” and boundary integral techniques are employed. Results for the plane strain case of one single inclusion and a regular array of inclusions with dilatational eigenstrains in an orthotropic matrix are presented. We find that the three factors: difference of elastic moduli in matrix and inclusion, interfacial energy and particle-particle interaction can affect the equilibrium shape in much the same way. The elastic inhomogeneity can cause strongly non-convex particle shapes and considerably affects the stability of fourfold symmetric equilibrium shapes.

Derivation of Plate Theory Accounting Asymptotically Correct Shear Deformation

Abstract: The focus of this paper is the development of linear, asymptotically correct theories for inhomogeneous orthotropic plates, for example, laminated plates with orthotropic laminae. It is noted that the method used can be easily extended to develop nonlinear theories for plates with generally anisotropic inhomogeneity. The development, based on variational-asymptotic method, begins with three-dimensional elasticity and mathematically splits the analysis into two separate problems: a one-dimensional through-the-thickness analysis and a two-dimensional “plate” analysis. The through-the-thickness analysis provides elastic constants for use in the plate theory and approximate closed-form recovering relations for all truly three-dimensional displacements, stresses, and strains expressed in terms of plate variables. In general, the specific type of plate theory that results from variational-asymptotic method is determined by the method itself. However, the procedure does not determine the plate theory uniquely, and one may use the freedom appeared to simplify the plate theory as much as possible. The simplest and the most suitable for engineering purposes plate theory would be a “Reissner-like” plate theory, also called first-order shear deformation theory. However, it is shown that construction of an asymptotically correct Reissner-like theory for laminated plates is not possible in general. A new point of view on the variational-asymptotic method is presented, leading to an optimization procedure that permits a derived theory to be as close to asymptotical correctness as possible while it is a Reissner-like. This uniquely determines the plate theory. Numerical results from such an optimum Reissner-like theory are presented. These results include comparisons of plate displacement as well as of three-dimensional field variables and are the best of all extant Reissner-like theories. Indeed, they even surpass results from theories that carry many more generalized displacement variables. Although the derivation presented herein is inspired by, and completely equivalent to, the well-known variational-asymptotic method, the new procedure looks different. In fact, one does not have to be familiar with the variational-asymptotic method in order to follow the present derivation.

Finite elements based on a first-order shear deformation moderate rotation shell theory with applications to the analysis of composite structures

Abstract: The first-order shear deformation moderate rotation shell theory of Schmidt and Reddy [R. Schmidt and J. N. Reddy, J. Appl. Mech. 55, 611–617 (1988)] is used as a basis for the development of finite element models for the analysis of the static, geometrically non-linear response of anisotropic and laminated structures. The incremental, total Lagrangian formulation of the theory is developed, and numerical solutions are obtained by using the isoparametric Lagrangian 9-node and Serendipity 8-node shell finite elements. Various integration schemes (full, selective reduced, and uniformly reduced integration) are applied in order to detect and to overcome the effects of shear and membrane locking on the predicted structural response. A number of sample problems of isotropic, orthotropic, and multi-layered structures are presented to show the accuracy of the present theory. The von Karman-type first-order shear deformation shell theory and continuum 2D theory are used for comparative analyses.

Identification of the damping properties of orthotropic composite materials using a mixed numerical experimental method

Abstract: A mixed numerical experimental approach is the basis of a new method for the identification of the material damping properties of fibre reinforced polymers, which provides an answer to many problems encountered in experimental damping characterization. Experimental modal parameters, measured on a plate specimen, are compared with corresponding results from a numerical calculation, thus allowing to determine the stiffness and damping properties of the material. The relation between the modal parameters (structural parameters) and the material parameters, is obtained by using a numerical model of the specimen in combination with the modal strain energy method.

Finite Element Method for Nonlinear Free Vibrations of Composite Plates

Abstract: A multimode time-domain modal formulation based on the finite element method for large-amplitude free vibration of thin composite plates is presented. Accurate frequency-maximum deflection relations can be predicted for the fundamental and the higher nonlinear modes. A modal participation is defined, and accurate and convergent frequencies can be determined with minimum number of linear modes. A procedure for the selection of initial conditions for periodic plate response is presented. Convergence of frequency with gridwork refinement and number of linear modes is studied. The classical single-mode elliptic function frequency solutions for simply supported beams and square plates are assessed. Examples of orthotropic and composite plates are given, and the characteristics of nonlinear response are studied.

Elasto-plastic interface model for 3d-frictional orthotropic contact problems

Abstract: The present study deals with the solution of the fully three-dimensional contact/friction problem taking into account microstructural characteristics of the surfaces. An incremental non-associated hardening friction law model analogous to the classical theory of plasticity is used. Two different non-linear friction functions in the orthogonal directions are used to account for the orthotropic properties of the contacting bodies. A frontal solver processing unsymmetric matrices is adopted. Two numerical examples have been selected to show applicability of the method proposed. © 1997 by John Wiley & Sons, Ltd.

Seismic traveltime analysis for azimuthally anisotropic media: Theory and experiment

Abstract: Natural fractures in reservoirs, and in the caprock overlying the reservoir, play an important role in determining fluid flow during production. The density and orientation of sets of fractures is therefore of great interest. Rocks possessing an anisotropic fabric and a preferred orientation of fractures display both polar and azimuthal anisotropy. Sedimentary rocks containing several sets of vertical fractures may be approximated as having monoclinic symmetry with symmetry plane parallel to the layers if, in the absence of fractures, the rock is transversely isotropic with symmetry axis perpendicular to the bedding plane. A nonhyperbolic traveltime equation, which can be used in the presence of azimuthally anisotropic layered media, can be obtained from an expansion of the inverse-squared ray velocity in spherical harmonics. For a single set of aligned fractures, application of this equation to traveltime data acquired at a sufficient number of azimuths allows the strike of the fractures to be estimated. Analysis of the traveltimes measured in a physical model simulation of a reverse vertical seismic profile in an azimuthally anisotropic medium shows the medium to be orthorhombic with principal axes in agreement with those given by an independent shear-wave experiment. In contrast to previous work, no knowledge of the orientation of the symmetry planes is required. The method is therefore applicable to P-wave data collected at multiple azimuths using multiple offset vertical seismic profiling (VSP) techniques.

A Unified Damage Approach for Predicting Forming Limit Diagrams

Abstract: Plastic deformation in sheet metal consists of four distinct phases, namely, uniform deformation, diffuse necking, localized necking, and final rupture. The last three phases are commonly known as nonuniform deformation. A proper forming limit diagram (FLD) should include all three phases of the nonuniform deformation. This paper presents the development of a unified approach to the prediction of FLD to include all three phases of nonuniform deformation. The conventional method for predicting FLD is based on localized necking and adopts two fundamentally different approaches. Under biaxial loading, the Hill's plasticity method is often chosen when α (= ∈ 2 /∈ 1 ) 0 or when the biaxial stretching of sheet metal is significant. The M-K method, however, suffers from the arbitrary selection of the imperfection size, thus resulting in inconsistent predictions. The unified approach takes into account the effects of micro-cracks/voids on the FLD. All real-life materials contain varying sizes and degrees of micro-cracks/voids which can be characterized by the theory of damage mechanics. The theory is extended to include orthotropic damage, which is often observed in extensive plastic deformation during sheet metal forming. The orthotropic FLD model is based on an anisotropic damage model proposed recently by Chow and Wang (1993). Coupling the incremental theory of plasticity with damage, the new model can be used to predict not only the forming limit diagram but also the fracture limit diagram under proportional or nonpropor-tional loading. In view of the two distinct physical phenomena governing the cases when α (=∈ 2 /∈ 1 ) 0, a set of instability criteria is proposed to characterize all three phases of nonuniform deformation. The orthotropic damage model has been employed to predict the FLD of VDIF steel (Chow et at., 1996) and excellent agreement between the predicted and measured results has been achieved as shown in Fig. 1. The damage model is extended in this paper to examine its applicability and validity for another important engineering material, namely aluminum alloy 6111-T4.