Showing papers on "Orthotropic material published in 1994"

Principles of Composite Material Mechanics

Abstract: Introduction Basic Concepts Constituent Materials for Composites Structural Applications of Composites Multifunctional Applications of Composites Fabrication Processes Elements of Mechanical Behavior of Composites Review of Basic Mechanics of Materials Equations Lamina Stress-Strain Relationships Introduction Effective Moduli in Stress-Strain Relationships Symmetry in Stress-Strain Relationships Orthotropic and Isotropic Engineering Constants The Specially Orthotropic Lamina The Generally Orthotropic Lamina Effective Moduli of a Continuous Fiber-Reinforced Lamina Introduction Elementary Mechanics of Materials Models Improved Mechanics of Materials Models Elasticity Models Semiempirical Models Strength of a Continuous Fiber-Reinforced Lamina Introduction Multiaxial Strength Criteria Micromechanics Models for Lamina Strength Analysis of Lamina Hygrothermal Behavior Introduction Hygrothermal Degradation of Properties Lamina Stress-Strain Relationships Including Hygrothermal Effects Micromechanics Models for Hygrothermal Properties Analysis of a Discontinuously Reinforced Lamina Introduction Aligned Discontinuous Fibers Off-Axis-Aligned Discontinuous Fibers Randomly Oriented Discontinuous Fibers Nanofibers and Nanotubes Particulates Hybrid Multiscale Reinforcements Analysis of Laminates Introduction Theory of Laminated Beams Theory of Laminated Plates with Coupling Stiffness Characteristics of Selected Laminate Configurations Derivation and Use of Laminate Compliances Hygrothermal Effects in Laminates Interlaminar Stresses Laminate Strength Analysis Deflection and Buckling of Laminates Selection of Laminate Designs Application of Laminate Analysis to Composite Structures Analysis of Viscoelastic and Dynamic Behavior Introduction Linear Viscoelastic Behavior of Composites Dynamic Behavior of Composites Nanoenhancement of Viscoelastic and Dynamic Properties Analysis of Fracture Introduction Fracture Mechanics Analysis of Through-Thickness Cracks Stress Fracture Criteria for Through-Thickness Notches Interlaminar Fracture Nanoenhancement of Fracture Toughness Mechanical Testing of Composites and Their Constituents Introduction Measurement of Constituent Material Properties Measurement of Basic Composite Properties Measurement of Viscoelastic and Dynamic Properties Measurement of Hygrothermal Properties Appendix A: Matrix Concepts and Operations Appendix B: Stress Equilibrium Equations Appendix C: Strain-Displacement Equations Index Problems and References appear at the end of each chapter.

Elastic and viscoelastic properties of trabecular bone by a compression testing approach.

Abstract: In this review the advantages and disadvantages of different variants of compression testing of trabecular bone are discussed. Factors affecting the precision and the accuracy of mechanical properties of trabecular bone derived from such tests are analysed. Below are listed some of the important conclusions which can be drawn. Conclusions based on the author's previous studies (I-IX) are shown in italic. 1) Trabecular bone is a viscoelastic solid. 2) Stiffness, strength, ultimate strain, and failure energy are derived from a standard compression test to failure. Viscoelastic properties such as energy dissipation and the relative energy loss (loss tangent) can be obtained from non-destructive cyclic tests. 3) A non-destructive test conducted between a lower load level (zero strain) and an upper strain limit of about 0.8% specimen strain has been developed. The reproducibility of such a test technique has been assessed at different conditions. The reproducibility was best after a number of conditioning cycles in order to achieve a viscoelastic steady state. Orthotropic properties can be determined by non-destructive testing in different directions of cubic specimens. The reproducibility of such testing has been established. 4) The stiffness derived from non-destructive tests will be lower than that obtained from a destructive test because of the non-linearity of the load-deformation curve, but the stiffnesses will be strongly correlated. 5) Stiffnesses derived from destructive and non-destructive tests have an elastic and a viscoelastic contribution. Since the viscoelastic contribution is time dependent, the results will be dependent on strain rate and loading frequency in cyclic tests. 6) Standard testing of small trabecular bone specimens is associated with systematic errors. The most significant of these errors are believed to be related to trabecular disintegrity at the surface of the specimen and to friction at the specimen-platen interface. Structural disintegrity causes an axial strain inhomogeneity resulting in a overestimation of axial strain and a corresponding underestimation of specimen stiffness. Friction at the interface causes an uneven stress and strain distribution in the layer nearest to the test platen resulting in a overestimation of stiffness. The net result of these systematic errors is a 20-40 per cent underestimation of stiffness. 7) The specimen geometry has a highly significant influence on mechanical properties such as stiffness, ultimate strain and energy absorption. A cube with a side length of 6.5 mm and a cylindrical specimen with a length of 6.5 mm and a diameter of 7.5 mm are suggested as standard geometries providing comparable results.

Residual-stress measurement in orthotropic materials using the hole-drilling method

Abstract: The hole-drilling method is used here to measure residual stresses in an orthotropic material. An existing stress-calculation method adapted from the isotropic case is shown not to be valid for orthotropic materials. A new stress-calculation method is described, based on the analytical solution for the displacement field around a hole in a stressed orthotropic plate. The validity of this method is assessed through a series of experimental measurements. A table of elastic compliances is provided for practical residual-stress measurements in a wide range of orthotropic materials.

Energy criterion for crack deflection at an interface between two orthotropic media

Abstract: To achieve toughness in many brittle composites, crack deflection at interfaces between fibers and matrix, or between fibers and their protective coatings is essential. In this paper, an energy criterion for crack deflection at an interface between two aligned, orthotropic media is obtained by using the method of singular integral equations. Only the case of a perpendicularly impinging interface crack is considered. Since the main crack can either be deflected on one (singly deflected) or both (doubly deflected) sides of the interface, the energy release rate ratios (ratio of energy released for crack deflecting along the interface to that penetrating the interface) are calculated for both cases and the maximum of the two is assumed to govern the interface design. We provide results for the G d ⧸ G p ratio for both singly and doubly deflected cracks as a function of the orthotropic parameters /gr and /gl of each material and the two bi-material constants α and β. Unlike the strength criterion, the energy release rate criterion is very sensitive to the values of λ, ρ, α and β. Thus, it is not possible to provide generalized delamination charts as a function of the bi-material constant α alone. Instead, conditions for interface crack deflection are provided for several fiber-matrix pairs, chosen from different metal, ceramic and intermetallic matrix composites. We recover the previously published results for the singly deflected crack case when the orthotropic media approach the corresponding isotropic material idealization. In addition, we present the correct solutions for the doubly deflected case and show that for positive values of the Dundurs parameter α, the G d ⧸ G p ratio is higher than that obtained for the singly deflected case, albeit by only 3–4%. Finally, the results of the above analysis are demonstrated by growing wedge-induced cracks normal to the grain boundary of a freshwater ice bi-crystal with c -axes of the individual crystals oriented normal to each other.

Analytical solution for bending of coaxial orthotropic cylinders

Abstract: A general analytical solution is obtained for the stresses and displacements of an elastic body consisting of an assembly of coaxial hollow circular cylinders made of orthotropic material, with or without a core, and subjected to bending, tensile and torsion loads. Two types of conditions are considered at the interfaces between cylinders: no slip and no friction. A numerical application is used to illustrate the theoretical results. Results show that there is no coupling between bending and tension‐torsion and that there is no deviation in the bending curvature. It was finally found that some warping of the cross section develops under bending, meaning that the Bernoulli‐Euler hypothesis would not strictly apply in the case of orthotropic cylinders.

On determination of orthotropic material moduli from ultrasonic velocity data in nonsymmetry planes

Abstract: This paper reports analysis of the reconstruction of elastic constants from ultrasonic velocity measurements in nonsymmetry planes of unidirectional composite materials. It is shown that the nonlinear least‐square optimization method is stable to initial guess selection and data scatter and can be used routinely to measure the full set of nine elastic constants for orthotropic materials. Simple analytical expressions are derived for phase velocities in arbitrary directions in an orthotropic material with low transverse‐to‐fiber anisotropy. Using these, coefficients of sensitivity of phase velocities to elastic constants are obtained in closed form and used to find the optimal wave propagation directions for elastic constant measurement. The changes in velocity due to elastic constant variation calculated by the exact theory agree well with the predictions from the sensitivity coefficients. When all nine elastic constants are reconstructed from velocity data in nonsymmetry planes, the inversion is highly d...

Composite Structures for Civil and Architectural Engineering

Abstract: Introduction. Historical necessity. Basic concept of composites. Matrix. Reinforcements. Filamentary type composites. Composite manufacture. Application, present and future. Further reading. Properties of composites. Introduction. Reinforcements. Matrices. Particulate composites. Other matrix based composites. Mechanical properties of fibrous composite. 'Comingle' and 'FIT' - new concepts for thermoplastic composites. References. Further reading. Classical theory of elasticity and mechanics. Assumptions for classical theory of elasticity. Stress and strain. Hooke's Law. Two-dimensional problems (plane) problems. Stress at a point. Mohr's circle. Strain at a point. Pure shear. Equilibrium equations and boundary conditions for two-dimensional problem. Compatibility equations. Stress function. Application of stress function in rectangular coordinate system. Displacements in two-dimensional problem. Application of stress function using Fourier Series. Equilibrium equations in terms of displacements. Two-dimensional problems in polar coordinates. Solution of two-dimensional problems in polar coordinates. Problems of concentrated loads. Strain energy and principle of virtual work. Examples of use of the energy method. Castigliano's theorem. Equilibrium equations in three dimensions. Boundary conditions in three-dimensional problems. Compatibility equations in three-dimensional problems. Saint Venant', solution of the problem of torsion of a prismatic bar. Membrane analogy for torsional problems. Asymmetric bending of prismatic beams. Asymmetrical bending of a beam with longitudinal stringers. Torsion of thin-walled structures with longitudinal stringers. Determination of rate of twist of a cell. Shear centre of beams with longerons. Shear flow distribution in multicell beams with longerons. Out-of-plane bending of curved girders. Alternative formulation for out-of-plane bending of curved beams with constant radius. Bending in the plane of the ring. Elementary theory of thin shell. Space frame. Space trusses. Cable. Guyed tower. Review of beam theory. Plates with irregular shapes and arbitrary boundary conditions. References. Further reading. Eigenvalue problems of beams and frames of isotropic materials. Stability of beams and frames. Structural vibration of beams and frames. References. Further reading. Anisotropic elasticity. Introduction. Stress-strain relations of anitotropic materials. Engineering constants for orthotropic materials. Stress-strain relations of plane stress and plane strain problems for unidirectionally reinforced lamina. Transformation equations. Invariants. Laminates. Micromechanics - mechanical properties of composites. Numerical examples. References. Further reading. One-dimensional structural elements of composite materials. Equations for beams and rods. Beams with hollow cross-sections. Eigenvalue problems of beams and frames of anisotropic materials. References

Modeling of laser thermoelastic generation of ultrasound in an orthotropic medium

Abstract: We present a numerical model that calculates the surface displacements generated by the absorption of a laser pulse by an orthotropic medium. This model solves the heat and acoustic wave equations using temporal Laplace and spatial two-dimensional Fourier transformations. This model allows us to calculate the normal and in-plane displacements on the front or back surface of an orthotropic plate over a complete area and for virtually any time and beam profiles of the laser excitation. Numerical simulations are compared to experimental results obtained on an aluminum sample and on a graphite-eppxy plate. The experimental and numerical results are in good agreement. Laser generation of ultrasound, especially when coupled to optical detection, has been recognized to be a powerful tool for nondestructive materials evaluation.'-3 In the thermoelastic generation regime, a localized temperature elevation in the sample induced by the absorption of a laser pulse results in a localized thermal expansion which, in turn, generates ultrasonic waves. Several approaches were proposed for modeling this thermoelastic process, essentially using Green functions4-7 or Laplace and Hankel transformations.8'9 However, the use of Green functions or Hankel transformations for modeling this process cannot be easily extended to the case of anisotropic materials such as materials with an orthotropic symmetry. Other numerical techniques, like the finite element technique, " are not suitable for the resolution of the laser generation of ultrasound. In this letter, we present a model based on temporal Laplace and spatial two-dimensional (2D) Fourier transformations that allows us to describe the thermoelastic generation of ultrasound in the case of an orthotropic sample. In our model, we consider an infinite plate of finite thickness L made of an orthotropic material. The coordinate =es XI, x2, and x3 correspond to the principal axes of the medium, with x3 being the optical axis of the incident laser radiation. The stiffness tensor [C] of the orthotropic medium has then only nine different nonzero components when the abbreviated notation is used.'r To describe the laser generation of ultrasound in the ther-moelastic regime, one must solve simultaneously the heat equation and the acoustic wave equation. In our model, we neglect the mechanical heat sources in the heat equation. This assumption is valid for the time scales of our simulations.7~gP12 We use the hyperbolic heat equation' even if in the cases studied, the classical one would be adequate. The temperature elevation field 6 is then given by d6 Bs v([k]w)=pc, at +7x-%dxlkk-l " 3f(t>, i 1 (1)

A fourth-order plastic potential for anisotropic metals and its analytical calculation from the texture function

Abstract: A fourth-order potential is proposed for anisotropic metals, assuming orthotropic symmetry. It may be used as a yield criterion, but here it is used as a dual potential depending on the strain-rate. The concept of dual potential is recalled and the implementation in simulation codes is discussed, along with the extension to viscoplasticity. The coefficients depend linearly on the texture coefficients, up to a scale factor determinable by one mechanical test. The adjustment of the dual potential from texture data is numerically tested for steels, and compared with mechanical tests. A significant improvement of the flow rule (strain ratio R) is obtained as compared with both the Taylor model and the texture-adjusted Hill criterion. The predicted stress ratios are close to those predicted with the latter criterion, which were previously found to be in a good experimental agreement.

Properties of carbon fibers

Abstract: The mechanical properties for carbon fibers are determined in terms of the elastic moduli of single crystal graphite. This is done for the two basic fiber microstructures, radial and onion skin. The final results are found through a transformation from cylindrically orthotropic to transversely isotropic properties. One of the five independent properties is found to be strongly influenced by the microstructure type.

New approach for simple prediction of impact force history on composite laminates

Abstract: A new method for simple prediction of the impact force history on composite laminates subjected to low-velocity impact is proposed. First, the impact force history is calculated from the impact response analysis using the finite element method incorporated with the modified Hertzian contact law. Frequency characteristics of the numerical impact force history are investigated from modal analysis and compared with the natural frequencies of the system in which the mass of an impactor is lumped with the plate. The present method can be efficiently applied to isotropic or orthotropic plates with unknown contact laws as well as composite laminates with no restraint on the material properties and the geometrical shape

Anisotropic plastic potentials for polycrystals and application to the design of optimum blank shapes in sheet forming

Abstract: Recently, some potentials were proposed to analytically describe the plastic behavior of orthotropic metals. These potentials, when expressed in six-dimensional stress space, were called yield functions or, when expressed in six-dimensional strain-rate space, were called strain-rate potentials. It was shown that these phenomenological potentials provide good approximations of the plastic potentials calculated with polycrystal models. They can be used for any type of loading condition, and they can account for orthotropic anisotropy. In a parallel effort, called ideal forming theory, a forming design theory that optimizes processes and initial blank shapes in sheet forming was developed. This ideal forming theory was implemented in a finite element modeling code in order to design the blank shape directly from the final part shape. The main input to this model includes the final part geometry and the constitutive behavior of the material. In the present article, the constitutive equations describing the plastic behavior of metals as well as the main features of the ideal forming theory are briefly summarized. Then, application of the strain-rate potential to the design of a blank shape for a circular cup drawn from an anisotropic Al-Li sheet is presented. It is shown that the design code efficiently predicts the shape of the blank needed to obtain a cup with minimal earing from a highly anisotropic material.

P-wave propagation in weakly anisotropic media

Abstract: SUMMARY The variation of P-wave speed with direction in a weakly anisotropic homogeneous elastic medium of arbitrary symmetry is expanded as a series of spherical harmonics. The results given may be used to assess the information content of measurements of the P-wave speed acquired over a restricted angular range, and to design experiments with the objective of imaging a limited number of anisotropy parameters. Simplifications which occur for various symmetries typical of sedimentary rocks with and without fractures are discussed. Application is made to the determination of the azimuthal anisotropy of sedimentary rocks. For propagation direction n with polar angle χ and azimuthal angle χ defined with respect to some convenient choice of reference axes Ox1x2x3, the variation of P-wave speed νp(χ, η) with η for fixed χ is described by 12 parameters in the absence of symmetry. If the material contains a plane of mirror symmetry, and the axes are chosen with OX3 perpendicular to the symmetry plane, the variation of νp(χ, η) with η for fixed χ depends on six anisotropy parameters. For orthotropic symmetry the number of required parameters is reduced to three. For small polar angles χ, νp(χ, η) then varies with azimuth as cos 2η, with amplitude determined by a single anisotropy parameter (c13–c23–2c44+ 2c55).

Optimization of the strain energy density in linear anisotropic elasticity

Abstract: The problem considered here is that of extremizing the strain energy density of a linear anisotropic material by varying the relative orientation between a fixed stress state and a fixed material symmetry. It is shown that the principal axes of stress must coincide with the principal axes of strain in order to minimize or maximize the strain energy density in this situation. Specific conditions for maxima and minima are obtained. These conditions involve the stress state and the elastic constants. It is shown that the symmetry coordinate system of cubic symmetry is the only situation in linear anisotropic elasticity for which a strain energy density extremum can exist for all stress states. The conditions for the extrema of the strain energy density for transversely isotropic and orthotropic materials with respect to uniaxial normal stress states are obtained and illustrated with data on the elastic constants of some composite materials. Not surprisingly, the results show that a uniaxial normal stress in the grain direction in wood minimizes the strain energy in the set of all uniaxial stress states. These extrema are of interest in structural and material optimization.

A Numerical and Experimental Study of Delaminated Layered Composites

Abstract: This paper describes the development of a finite element-based computer model for the determination of stress fields and energy release rates around a delamination in a layered composite. The model is fully three-dimensional. Each layer of a composite material is modelled as homogeneous and orthotropic. The basic finite elements are isoparametric 20-noded bricks with collapsed, prismatic quarter-point elements around the delamination front. Contact theory is used to prevent interpenetration between the faces of a delamination. The modified crack closure method is used to calculate the energy release rates around a delamination.Preliminary experiments are performed to study the growth of a delamination between the layers of a two-layer laminate-one epoxy and the other graphite/epoxy. Initial results from the experiments and numerical simulations are shown to be consistent.

Characterization of anisotropie elastic constants of silicon-carbide participate reinforced aluminum metal matrix composites: Part I. Experiment

Abstract: The anisotropic elastic properties of silicon-carbide particulate (SiCp) reinforced Al metal matrix composites were characterized using ultrasonic techniques and microstructural analysis. The composite materials, fabricated by a powder metallurgy extrusion process, included 2124, 6061, and 7091 Al alloys reinforced by 10 to 30 pct ofα-SiCp by volume. Results were presented for the assumed orthotropic elastic constants obtained from ultrasonic velocities and for the microstructural data on particulate shape, aspect ratio, and orientation distribution. All of the composite samples exhibited a systematic anisotropy: the stiffness in the extrusion direction was the highest, and the stiffness in the out-of-plane direction was the lowest. Microstructural analysis suggested that the observed anisotropy could be attributed to the preferred orientation of SiCp. The ultrasonic velocity was found to be sensitive to internal defects such as porosity and intermetallic compounds. It has been observed that ultrasonics may be a useful, nondestructive technique for detecting small directional differences in the overall elastic constants of the composites since a good correlation has been noted between the velocity and microstructure and the mechanical test. By incorporating the observed microstructural characteristics, a theoretical model for predicting the anisotropic stiffnesses of the composites has been developed and is presented in a companion article (Part II).