Calculation of the slowness vector from the ray vector in anisotropic media

2006 ◽  
Vol 462 (2067) ◽  
pp. 883-896 ◽  
Author(s):  
Václav Vavryčuk
Keyword(s):  
Symmetry Plane ◽  
Closed Form Solution ◽  
Transversely Isotropic ◽  
Form Solution ◽  
Polynomial Equations ◽  
Wave Surface ◽  
Wave Fields ◽  
Slowness Vector ◽  
Arbitrary Strength ◽  
Anisotropic Elastic

The wave quantities needed in constructing wave fields propagating in anisotropic elastic media are usually calculated as a function of the slowness vector, or of its direction called the wave normal. In some applications, however, it is desirable to calculate the wave quantities as a function of the ray direction. In this paper, a method of calculating the slowness vector for a specified ray direction is proposed. The method is applicable to general anisotropy of arbitrary strength with arbitrary complex wave surface. The slowness vector is determined by numerically solving a system of multivariate polynomial equations of the sixth order. By solving the equations, we obtain a complete set of slowness vectors corresponding to all wave types and to all branches of the wave surface including the slowness vectors along the acoustic axes. The wave surface can be folded to any degree. The system of equations is further specified for rays shot in the symmetry plane of an orthorhombic medium and for a transversely isotropic medium. The system is decoupled into two polynomial equations of the fourth order for the P –SV waves, and into equations for the SH wave, which yield an explicit closed-form solution. The presented approach is particularly advantageous in constructing ray fields, ray-theoretical Green functions, wavefronts and wave fields in strong anisotropy.

Use of Resultants and the Bernshtein Formula in the Dimensional Synthesis of Linkage Components

1994 ◽  
Vol 116 (4) ◽  
pp. 1171-1172 ◽  
Author(s):  
Chuen-Sen Lin ◽  
Bao-Ping Jia
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Polynomial Equations ◽  
Dimensional Synthesis ◽  
Fourth Degree ◽  
Finite Precision ◽  
Closed Form Solutions ◽  
System Of Polynomial Equations ◽  
Resultant Theory

The applications of resultants and the Bernshtein formula for the dimensional synthesis of linkage components for finite precision positions are discussed. The closed-form solutions, which are derived from systems of polynomials in multiple unknowns by applying resultant theory, are in forms of polynomial equations of a single unknown. For the case of two compatibility equations, the closed form solution is a sixth degree solution polynomial. For the case of three compatibility equations, the solution is a fifty-fourth degree solution polynomial. For each case, the Bernshtein formula is applied to calculate the number of solutions of the system of polynomial equations. The calculated numbers of solutions match the degrees of the solution polynomials for both cases.

scholarly journals On 3D Anticrack Problem of Thermoelectroelasticity

2018 ◽  
Vol 12 (2) ◽  
pp. 109-114 ◽  
Author(s):  
Andrzej Kaczyński
Keyword(s):  
Arbitrary Shape ◽  
Closed Form Solution ◽  
Transversely Isotropic ◽  
Static Problem ◽  
Form Solution ◽  
Boundary Integral ◽  
The Body ◽  
Circular Rigid Inclusion ◽  
Stress Discontinuity ◽  
Suitable Potential

Abstract A solution is presented for the static problem of thermoelectroelasticity involving a transversely isotropic space with a heat-insulated rigid sheet-like inclusion (anticrack) located in the isotropy plane. It is assumed that far from this defect the body is in a uniform heat flow perpendicular to the inclusion plane. Besides, considered is the case where the electric potential on the anticrack faces is equal to zero. Accurate results are obtained by constructing suitable potential solutions and reducing the thermoelectromechanical problem to its thermomechanical counterpart. The governing boundary integral equation for a planar anticrack of arbitrary shape is obtained in terms of a normal stress discontinuity. As an illustration, a closed-form solution is given and discussed for a circular rigid inclusion.

scholarly journals Centrifugal stresses in arbitrarily laminated, rectangular—anistropic circular discs

1975 ◽  
Vol 10 (2) ◽  
pp. 84-92 ◽  
Author(s):  
C W Bert
Keyword(s):  
Composite Materials ◽  
Closed Form ◽  
Linear Theory ◽  
Closed Form Solution ◽  
Form Solution ◽  
Numerical Results ◽  
Current Interest ◽  
Elastic Plates ◽  
Organic Fibre ◽  
Anisotropic Elastic

The problem is formulated as one in the linear theory of thin, laminated, anisotropic elastic plates. A direct force-and-moment formulation is used, simplifying approximation is introduced and a closed-form solution is obtained. This solution exhibits bending-stretching coupling if the plate is asymmetrically laminated with respect to mass or stiffness or both. Numerical results typical of certain composite materials of current interest are presented. Specific laminates considered as examples include (1) glass—epoxy/steel, (2) cross-ply graphite—epoxy, and (3) various quasi-isotropic layups of organic fibre—epoxy.

Equivalent Elastic Constants of Truss Core Sandwich Plates

2011 ◽  
Vol 133 (4) ◽  
Author(s):  
Hong-Xia Wang ◽  
Samuel W. Chung
Keyword(s):  
Elastic Constants ◽  
A Priori ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Elastic Stiffness ◽  
Form Solution ◽  
Laminated Plates ◽  
Element Analysis ◽  
Truss Core ◽  
Anisotropic Elastic

A plate structure of a triangular truss core sandwiched by two panels is treated as an equivalent homogeneous laminated plate by obtaining equivalent anisotropic elastic constants. The equivalent elastic constants are obtained by considering generalized Hook’s law of a three dimensional elastic body with no a priori assumption and the equilibrium of a segment deformed by bending moments. To verify the accuracy of the equivalent elastic constants, a linear static analysis of sandwiched aluminum plates subjected to lateral pressure is carried out. The results of the finite element analysis applied to the equivalent laminated plates are compared with those of a NASTRAN analysis of the original structural layouts. The results are also compared with a closed-form solution, which simplifies the sandwiched plate as a homogeneous orthotropic thick plate continuum (Lok and Cheng, 2000, “Elastic Stiffness Properties and Behavior of Truss-Core Sandwich Panel,” J. Struct. Eng., 126(5), pp. 552–559). As the maximum deflections of three analyses agreed closely, one has assurance that the method of the homogeneous plate with equivalent elastic constants is valid and useful.

Exact solutions for the macro-, meso- and micro-scale analysis of composite laminates and sandwich structures

2018 ◽  
Vol 52 (22) ◽  
pp. 3109-3124 ◽  
Author(s):  
Yang Yan ◽  
Alfonso Pagani ◽  
Erasmo Carrera ◽  
Qingwen Ren
Keyword(s):  
Composite Laminates ◽  
Computational Cost ◽  
Three Dimensional ◽  
Sandwich Beam ◽  
Closed Form Solution ◽  
Single Layer ◽  
Transversely Isotropic ◽  
Global Scale ◽  
Form Solution ◽  
Lagrange Polynomials

The present work proposes a closed-form solution based on refined beam theories for the static analysis of fiber-reinforced composite and sandwich beams under simply supported boundary conditions. The higher-order beam models are developed by employing Carrera Unified Formulation, which uses Lagrange-polynomials expansions to approximate the kinematic field over the cross section. The proposed methodology allows to carry out analysis of composite structure analysis through a single formulation in global-local sense, i.e. homogenized laminates at a global scale and fiber-matrix constituents at a local scale, leading to component-wise analysis. Therefore, three-dimensional stress/displacement fields at different scales can be successfully detected by increasing the order of Lagrange polynomials opportunely. The governing equations are derived in a strong-form and solved in a Navier-type sense. Three benchmark numerical assessments are carried out on a single-layer transversely isotropic beam, a cross-ply laminate [Formula: see text] beam and a sandwich beam. The results show that accurate displacement and stress values can be obtained in different parts of the structure with lower computational cost in comparison with traditional, enhanced as well as three-dimensional finite element methods. Besides, this study may serve as benchmarks for future assessments in this field.

scholarly journals Enhancing the electrical and thermal conductivities of polymer composites via curvilinear fibers: An analytical study

2019 ◽  
Vol 24 (10) ◽  
pp. 3231-3253 ◽  
Author(s):  
Marco Salviato ◽  
Sean E Phenisee
Keyword(s):  
Closed Form ◽  
Electric Conductivity ◽  
Electrostatic Potential ◽  
Mechanical Performance ◽  
Transverse Isotropy ◽  
Closed Form Solution ◽  
Transversely Isotropic ◽  
Form Solution ◽  
Closed Form Expression ◽  
Element Analysis

The new generation of manufacturing technologies such as additive manufacturing and automated fiber placement has enabled the development of material systems with desired functional and mechanical properties via particular designs of inhomogeneities and their mesostructural arrangement. Among these systems, particularly interesting are materials exhibiting curvilinear transverse isotropy (CTI), in which the inhomogeneities take the form of continuous fibers following curvilinear paths designed to, for example, optimize the electric and thermal conductivity, and the mechanical performance of the system. In this context, the present work proposes a general framework for the exact, closed-form solution of electrostatic problems in materials featuring CTI. First, the general equations for the fiber paths that optimize the electric conductivity are derived, leveraging a proper conformal coordinate system. Then, the continuity equation for the curvilinear transversely isotropic system is derived in terms of electrostatic potential. A general exact, closed-form expression for the electrostatic potential and electric field is derived and validated by finite element analysis. Finally, potential avenues for the development of materials with superior electric conductivity and damage sensing capabilities are discussed.

Stress Concentration Around a Hyperboloidal Notch Under Tension in a Transversely Isotropic Material

1971 ◽  
Vol 38 (1) ◽  
pp. 185-189 ◽  
Author(s):  
W. T. Chen
Keyword(s):  
Stress Concentration ◽  
Tensile Stress ◽  
Isotropic Material ◽  
Tensile Force ◽  
Closed Form Solution ◽  
Elastic Solid ◽  
Transversely Isotropic ◽  
Form Solution ◽  
Transversely Isotropic Material ◽  
Numerical Examples

An elastic solid is composed of a transversely isotropic material bounded by a single-sheeted hyperboloid of revolution, which is traction free. This solid is subjected to a finite tensile force at infinity. A closed-form solution based upon the potential functions approach is obtained. Numerical examples of the tensile stress at the narrowest section are presented.

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