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# Orthotropic material

About: Orthotropic material is a(n) research topic. Over the lifetime, 11945 publication(s) have been published within this topic receiving 214691 citation(s).

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TL;DR: Once Deff is estimated from a series of NMR pulsed-gradient, spin-echo experiments, a tissue's three orthotropic axes can be determined and the effective diffusivities along these orthotropic directions are the eigenvalues of Deff.

Abstract: This paper describes a new NMR imaging modality--MR diffusion tensor imaging. It consists of estimating an effective diffusion tensor, Deff, within a voxel, and then displaying useful quantities derived from it. We show how the phenomenon of anisotropic diffusion of water (or metabolites) in anisotropic tissues, measured noninvasively by these NMR methods, is exploited to determine fiber tract orientation and mean particle displacements. Once Deff is estimated from a series of NMR pulsed-gradient, spin-echo experiments, a tissue's three orthotropic axes can be determined. They coincide with the eigenvectors of Deff, while the effective diffusivities along these orthotropic directions are the eigenvalues of Deff. Diffusion ellipsoids, constructed in each voxel from Deff, depict both these orthotropic axes and the mean diffusion distances in these directions. Moreover, the three scalar invariants of Deff, which are independent of the tissue's orientation in the laboratory frame of reference, reveal useful information about molecular mobility reflective of local microstructure and anatomy. Inherently tensors (like Deff) describing transport processes in anisotropic media contain new information within a macroscopic voxel that scalars (such as the apparent diffusivity, proton density, T1, and T2) do not.

5,179 citations

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01 Jan 2004

Abstract: Equations of Anisotropic Elasticity, Virtual Work Principles, and Variational Methods Fiber-Reinforced Composite Materials Mathematical Preliminaries Equations of Anisotropic Entropy Virtual Work Principles Variational Methods Summary Introduction to Composite Materials Basic Concepts and Terminology Constitutive Equations of a Lamina Transformation of Stresses and Strains Plan Stress Constitutive Relations Classical and First-Order Theories of Laminated Composite Plates Introduction An Overview of Laminated Plate Theories The Classical Laminated Plate Theory The First-Order Laminated Plate Theory Laminate Stiffnesses for Selected Laminates One-Dimensional Analysis of Laminated Composite Plates Introduction Analysis of Laminated Beams Using CLPT Analysis of Laminated Beams Using FSDT Cylindrical Bending Using CLPT Cylindrical Bending Using FSDT Vibration Suppression in Beams Closing Remarks Analysis of Specially Orthotropic Laminates Using CLPT Introduction Bending of Simply Supported Rectangular 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 In-Plane Shear Load Vibration of Simply Supported Plates Buckling and Vibration of Plates with Two Parallel Edges Simply Supported Transient Analysis Closure Analytical Solutions of Rectangular Laminated Plates Using CLPT Governing Equations in Terms of Displacements Admissible Boundary Conditions for the Navier Solutions Navier Solutions of Antisymmetric Cross-Ply Laminates Navier Solutions of Antisymmetric Angle-Ply Laminates The Levy Solutions Analysis of Midplane Symmetric Laminates Transient Analysis Summary Analytical Solutions of Rectangular Laminated Plates Using FSDT Introduction Simply Supported Antisymmetric Cross-Ply Laminated Plates Simply Supported Antisymmetric Angle-Ply Laminated Plates Antisymmetric Cross-Ply Laminates with Two Opposite Edges Simply Supported Antisymmetric Angle-Ply Laminates with Two Opposite Edges Simply Supported Transient Solutions Vibration Control of Laminated Plates Summary Theory and Analysis of Laminated Shells Introduction Governing Equations Theory of Doubly-Curved Shell Vibration and Buckling of Cross-Ply Laminated Circular Cylindrical Shells Linear Finite Element Analysis of Composite Plates and Shells Introduction Finite Element Models of the Classical Plate Theory (CLPT) Finite Element Models of Shear Deformation Plate Theory (FSDT) Finite Element Analysis of Shells Summary Nonlinear Analysis of Composite Plates and Shells Introduction Classical Plate Theory First-Order Shear Deformation Plate Theory Time Approximation and the Newton-Raphson Method Numerical Examples of Plates Functionally Graded Plates Finite Element Models of Laminated Shell Theory Continuum Shell Finite Element Postbuckling Response and Progressive Failure of Composite Panels in Compression Closure Third-Order Theory of Laminated Composite Plates and Shells Introduction A Third-Order Plate Theory Higher-Order Laminate Stiffness Characteristics The Navier Solutions Levy Solutions of Cross-Ply Laminates Finite Element Model of Plates Equations of Motion of the Third-Order Theory of Doubly-Curved Shells Layerwise Theory and Variable Kinematic Model Introduction Development of the Theory Finite Element Model Variable Kinematic Formulations Application to Adaptive Structures Layerwise Theory of Cylindrical Shell Closure Subject Index

3,456 citations

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Abstract: In this paper we develop a new constitutive law for the description of the (passive) mechanical response of arterial tissue. The artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia (the solid mechanically relevant layers in healthy tissue). Each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. Fiber orientations obtained from a statistical analysis of histological sections from each arterial layer are used. A specific form of the law, which requires only three material parameters for each layer, is used to study the response of an artery under combined axial extension, inflation and torsion. The characteristic and very important residual stress in an artery in vitro is accounted for by assuming that the natural (unstressed and unstrained) configuration of the material corresponds to an open sector of a tube, which is then closed by an initial bending to form a load-free, but stressed, circular cylindrical configuration prior to application of the extension, inflation and torsion. The effect of residual stress on the stress distribution through the deformed arterial wall in the physiological state is examined. The model is fitted to available data on arteries and its predictions are assessed for the considered combined loadings. It is explained how the new model is designed to avoid certain mechanical, mathematical and computational deficiencies evident in currently available phenomenological models. A critical review of these models is provided by way of background to the development of the new model.

2,617 citations

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01 Jan 1980Abstract: Preface. 1 Introduction. 1.1 Definition. 1.2 Characteristics. 1.3 Classification. 1.4 Particulate Composites. 1.5 Fiber-Reinforced Composites. 1.6 Applications of Fiber Composites. Exercise Problems. References. 2 Fibers, Matrices, and Fabrication of Composites. 2.1 Advanced Fibers. 2.1.1 Glass Fibers. 2.1.2 Carbon and Graphite Fibers. 2.1.3 Aramid Fibers. 2.1.4 Boron Fibers. 2.1.5 Other Fibers. 2.2 Matrix Materials. 2.2.1 Polymers. 2.2.2 Metals. 2.3 Fabrication of Composites. 2.3.1 Fabrication of Thermosetting Resin Matrix Composites. 2.3.2 Fabrication of Thermoplastic-Resin Matrix Composites (Short-Fiber Composites). 2.3.3 Fabrication of Metal Matrix Composites. 2.3.4 Fabrication of Ceramic Matrix Composites. Suggested Reading. 3 Behavior of Unidirectional Composites. 3.1 Introduction. 3.1.1 Nomenclature. 3.1.2 Volume and Weight Fractions. 3.2 Longitudinal Behavior of Unidirectional Composites. 3.2.1 Initial Stiffness. 3.2.2 Load Sharing. 3.2.3 Behavior beyond Initial Deformation. 3.2.4 Failure Mechanism and Strength. 3.2.5 Factors Influencing Longitudinal Strength and Stiffness. 3.3 Transverse Stiffness and Strength. 3.3.1 Constant-Stress Model. 3.3.2 Elasticity Methods of Stiffness Prediction. 3.3.3 Halpin-Tsai Equations for Transverse Modulus. 3.3.4 Transverse Strength. 3.4 Prediction of Shear Modulus. 3.5 Prediction of Poisson's Ratio. 3.6 Failure Modes. 3.6.1 Failure under Longitudinal Tensile Loads. 3.6.2 Failure under Longitudinal Compressive Loads. 3.6.3 Failure under Transverse Tensile Loads. 3.6.4 Failure under Transverse Compressive Loads. 3.6.5 Failure under In-Plane Shear Loads. 3.7 Expansion Coefficients and Transport Properties. 3.7.1 Thermal Expansion Coefficients. 3.7.2 Moisture Expansion Coefficients. 3.7.3 Transport Properties. 3.7.4 Mass Diffusion. 3.8 Typical Unidirectional Fiber Composite Properties. Exercise Problems. References. 4 Short-Fiber Composites. 4.1 Introduction. 4.2 Theories of Stress Transfer. 4.2.1 Approximate Analysis of Stress Transfer. 4.2.2 Stress Distributions from Finite-Element Analysis. 4.2.3 Average Fiber Stress. 4.3 Modulus and Strength of Short-Fiber Composites. 4.3.1 Prediction of Modulus. 4.3.2 Prediction of Strength. 4.3.3 Effect of Matrix Ductility. 4.4 Ribbon-Reinforced Composites. Exercise Problems. References. 5 Analysis of an Orthotropic Lamina. 5.1 Introduction. 5.1.1 Orthotropic Materials. 5.2 Stress-Strain Relations and Engineering Constants. 5.2.1 Stress-Strain Relations for Specially Orthotropic Lamina. 5.2.2 Stress-Strain Relations for Generally Orthotropic Lamina. 5.2.3 Transformation of Engineering Constants. 5.3 Hooke's Law and Stiffness and Compliance Matrices. 5.3.1 General Anisotropic Material. 5.3.2 Specially Orthotropic Material. 5.3.3 Transversely Isotropic Material. 5.3.4 Isotropic Material. 5.3.5 Specially Orthotropic Material under Plane Stress. 5.3.6 Compliance Tensor and Compliance Matrix. 5.3.7 Relations between Engineering Constants and Elements of Stiffness and Compliance Matrices. 5.3.8 Restrictions on Elastic Constants. 5.3.9 Transformation of Stiffness and Compliance Matrices. 5.3.10 Invariant Forms of Stiffness and Compliance Matrices. 5.4 Strengths of an Orthotropic Lamina. 5.4.1 Maximum-Stress Theory. 5.4.2 Maximum-Strain Theory. 5.4.3 Maximum-Work Theory. 5.4.4 Importance of Sign of Shear Stress on Strength of Composites. Exercise Problems. References. 6 Analysis of Laminated Composites. 6.1 Introduction. 6.2 Laminate Strains. 6.3 Variation of Stresses in a Laminate. 6.4 Resultant Forces and Moments: Synthesis of Stiffness Matrix. 6.5 Laminate Description System. 6.6 Construction and Properties of Special Laminates. 6.6.1 Symmetric Laminates. 6.6.2 Unidirectional, Cross-Ply, and Angle-Ply Laminates. 6.6.3 Quasi-isotropic Laminates. 6.7 Determination of Laminae Stresses and Strains. 6.8 Analysis of Laminates after Initial Failure. 6.9 Hygrothermal Stresses in Laminates. 6.9.1 Concepts of Thermal Stresses. 6.9.2 Hygrothermal Stress Calculations. 6.10 Laminate Analysis Through Computers. Exercise Problems. References. 7 Analysis of Laminated Plates and Beams. 7.1 Introduction. 7.2 Governing Equations for Plates. 7.2.1 Equilibrium Equations. 7.2.2 Equilibrium Equations in Terms of Displacements. 7.3 Application of Plate Theory. 7.3.1 Bending. 7.3.2 Buckling. 7.3.3 Free Vibrations. 7.4 Deformations Due to Transverse Shear. 7.4.1 First-Order Shear Deformation Theory. 7.4.2 Higher-Order Shear Deformation Theory. 7.5 Analysis of Laminated Beams. 7.5.1 Governing Equations for Laminated Beams. 7.5.2 Application of Beam Theory. Exercise Problems. References. 8 Advanced Topics in Fiber Composites. 8.1 Interlaminar Stresses and Free-Edge Effects. 8.1.1 Concepts of Interlaminar Stresses. 8.1.2 Determination of Interlaminar Stresses. 8.1.3 Effect of Stacking Sequence on Interlaminar Stresses. 8.1.4 Approximate Solutions for Interlaminar Stresses. 8.1.5 Summary. 8.2 Fracture Mechanics of Fiber Composites. 8.2.1 Introduction. 8.2.2 Fracture Mechanics Concepts and Measures of Fracture Toughness. 8.2.3 Fracture Toughness of Composite Laminates. 8.2.4 Whitney-Nuismer Failure Criteria for Notched Composites. 8.3 Joints for Composite Structures. 8.3.1 Adhesively Bonded Joints. 8.3.2 Mechanically Fastened Joints. 8.3.3 Bonded-Fastened Joints. Exercise Problems. References. 9 Performance of Fiber Composites: Fatigue, Impact, and Environmental Effects. 9.1 Fatigue. 9.1.1 Introduction. 9.1.2 Fatigue Damage. 9.1.3 Factors Influencing Fatigue Behavior of Composites. 9.1.4 Empirical Relations for Fatigue Damage and Fatigue Life. 9.1.5 Fatigue of High-Modulus Fiber-Reinforced Composites. 9.1.6 Fatigue of Short-Fiber Composites. 9.2 Impact. 9.2.1 Introduction and Fracture Process. 9.2.2 Energy-Absorbing Mechanisms and Failure Models. 9.2.3 Effect of Materials and Testing Variables on Impact Properties. 9.2.4 Hybrid Composites and Their Impact Strength. 9.2.5 Damage Due to Low-Velocity Impact. 9.3 Environmental-Interaction Effects. 9.3.1 Fiber Strength. 9.3.2 Matrix Effects. Exercise Problems. References. 10 Experimental Characterization of Composites. 10.1 Introduction. 10.2 Measurement of Physical Properties. 10.2.1 Density. 10.2.2 Constituent Weight and Volume Fractions. 10.2.3 Void Volume Fraction. 10.2.4 Thermal Expansion Coefficients. 10.2.5 Moisture Absorption and Diffusivity. 10.2.6 Moisture Expansion Coefficients. 10.3 Measurement of Mechanical Properties. 10.3.1 Properties in Tension. 10.3.2 Properties in Compression. 10.3.3 In-Place Shear Properties. 10.3.4 Flexural Properties. 10.3.5 Measures of In-Plane Fracture Toughness. 10.3.6 Interlaminar Shear Strength and Fracture Toughness. 10.3.7 Impact Properties. 10.4 Damage Identification Using Nondestructive Evaluation Techniques. 10.4.1 Ultrasonics. 10.4.2 Acoustic Emission. 10.4.3 x-Radiography. 10.4.4 Thermography. 10.4.5 Laser Shearography. 10.5 General Remarks on Characterization. Exercise Problems. References. 11 Emerging Composite Materials. 11.1 Nanocomposites. 11.2 Carbon-Carbon Composites. 11.3 Biocomposites. 11.3.1 Biofibers. 11.3.2 Wood-Plastic Composites (WPCs). 11.3.3 Biopolymers. 11.4 Composites in "Smart" Structures. Suggested Reading. Appendix 1: Matrices and Tensors. Appendix 2: Equations of Theory of Elasticity. Appendix 3: Laminate Orientation Code. Appendix 4: Properties of Fiber Composites. Appendix 5: Computer Programs for Laminate Analysis. Index.

1,854 citations

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Abstract: In a continuing study, three-dimensional elasticity solutions are constructed for rectangular laminates with pinned edges. The lamination geometry treated consists of arbitrary numbers of layers which can be isotropic or orthotropic with material symmetry axes parallel to the plate axes. Several specific example problems are solved, including a sandwich plate, and compared to the analogous results in classical laminated plate theory.

1,654 citations