Plane Solutions for the Displacement and Traction-Displacement Problems for Anisotropic Elastic Wedges

1974 ◽  
Vol 41 (1) ◽  
pp. 197-202 ◽  
Author(s):  
M. C. Kuo ◽  
D. B. Bogy
Keyword(s):  
Mellin Transform ◽  
Symmetry Axis ◽  
Wedge Angle ◽  
Complex Function ◽  
Material Symmetry ◽  
Stress Singularities ◽  
Material Parameters ◽  
Plane Solution ◽  
Anisotropic Elastic ◽  
Plane Displacement

The plane displacement and traction-displacement problems for anisotropic elastic wedges are solved by use of the complex function representation of the plane solution in conjunction with the Mellin transform. The special forms of the solutions pertinent to orthotropic wedges with a material symmetry axis along the wedge bisector is also presented and the dependence of the order of the stress singularities at the apex on the wedge angle and material parameters is shown graphically for this case.

Plane Solutions for Traction Problems on Orthotropic Unsymmetrical Wedges and Symmetrically Twinned Wedges

1974 ◽  
Vol 41 (1) ◽  
pp. 203-208 ◽  
Author(s):  
M. C. Kuo ◽  
D. B. Bogy
Keyword(s):  
Stress Concentration ◽  
Mellin Transform ◽  
Wedge Angle ◽  
Complex Function ◽  
Anisotropic Materials ◽  
Function Representation ◽  
Material Constants ◽  
Plane Solution ◽  
Composite Wedge

The plane traction problem for a composite wedge formed from two identical orthotropic wedges is solved by use of the complex function representation of the plane solution for anisotropic materials in conjunction with the Mellin transform. As a preliminary to this solution the traction problem for a single unsymmetrical orthotropic wedge is also studied. For both problems the stress concentration at the wedge apex is examined and the dependence of the order of the singularity on the wedge angle and material constants is exhibited graphically.

The Plane Solution for Anisotropic Elastic Wedges Under Normal and Shear Loading

1972 ◽  
Vol 39 (4) ◽  
pp. 1103-1109 ◽  
Author(s):  
D. B. Bogy
Keyword(s):  
Half Plane ◽  
Complex Function ◽  
Stress Singularity ◽  
Shear Loading ◽  
Elementary Functions ◽  
Linear Elastostatics ◽  
Full Plane ◽  
Plane Solution ◽  
Logarithmic Singularities ◽  
Anisotropic Elastic

The plane traction problem for an anisotropic wedge is solved within the theory of linear elastostatics. The technique employs the complex function representation of the plane solution in conjunction with the Mellin transform. Special attention is given to the orthotropic wedge; the uniform load solution is given in terms of elementary functions for wedge angles less than π, the logarithmic singularities in the stress field resulting from discontinuous loads on the half plane are studied, and the stress singularity at the apex is investigated for the reentrant wedges. Simplified results for the anisotropic half plane and cracked full plane are also presented.

Representation of the Microstructural Dependence of the Anisotropic Elastic Constants of Textured Materials

1990 ◽  
Vol 207 ◽  
Author(s):  
Stephen C. Cowin
Keyword(s):  
Elastic Constants ◽  
Material Symmetry ◽  
Hooke’S Law ◽  
Mechanical Models ◽  
Hooke's Law ◽  
Micromechanical Models ◽  
Anisotropic Elastic ◽  
Anisotropic Form ◽  
Textured Materials ◽  
Elastic Coefficients

AbstractThis paper addresses the question of representing the dependence of the elastic coefficients in the anisotropic form of Hooke's law upon the microstructure of a material. The concern is with textured material symmetries, that is to say materials such as natural and man-made composites whose material symmetry is determined by microstructural organization. The approach is to relate the anisotropic elastic coefficients to local geometric or stereological measures of the microstructure. The predictions of micromechanical models and continuum mechanical models are compared and are found to be consistent with each other.

scholarly journals GENERAL SCALARIZATION METHOD OF DYNAMIC ELASTIC FIELDS IN TRANSVERSALLY ISOTROPIC MEDIA AND ITS NEW APPLICATIONS

2018 ◽  
Vol 18 (3) ◽  
pp. 258-264
Author(s):  
I. P. Miroshnichenko ◽  
V. P. Sizov
Keyword(s):  
Symmetry Axis ◽  
Layered Structures ◽  
Mutual Arrangement ◽  
Material Symmetry ◽  
Displacement Fields ◽  
General Technique ◽  
Elastic Fields ◽  
Scalarization Method ◽  
Isotropic Media ◽  
Transversally Isotropic

Introduction. An efficient technique of tensor field scalarization  is  successfully  used  while  investigating  tensor  elastic fields of displacements, stresses and deformations in the layered structures of different materials, including transversally isotropic composites. These fields can be expressed through the scalar potentials corresponding to the quasi-longitudinal, quasi-transverse, and transverse-only waves. Such scalarization is possible if the objects under consideration are tensors relating to  the subgroup  of general coordinate conversions, when the local affine basis has one invariant vector that coincides with the material symmetry axis of the material. At this, the known papers consider structures where this vector coincides with the normal to the boundary between layers. However, other cases of the mutual arrangement of the material symmetry axis of the  material  and  the boundaries between layers are of interest on the practical side.Materials and Methods. The work objective is further development of the scalarization method application in the boundary value problems of the dynamic  elasticity theory for the cases of an arbitrary arrangement of the material symmetry axis relative to the boundary between layers. The present research and methodological apparatus are developed through the general technique of scalarization of the dynamic elastic fields of displacements, stresses and strains in the transversally isotropic media.Research Results. New design ratios for the determination of the displacement fields, stresses and deformations in the transversally isotropic media are obtained for the cases of an arbitrary arrangement of the material symmetry axes of the layer materials with respect to the boundaries between layers. Discussion and Conclusions. The present research and methodological apparatus are successfully used in determining the stress-strain  state  in  the  layered  structures  of  transversally isotropic materials, and in analyzing the diagnosis results of the state of the plane-layered and layered cylindrical structures under operation.

Linear Anisotropic Elastic Materials

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
Elastic Constants ◽  
Elastic Material ◽  
Positive Definite ◽  
Material Symmetry ◽  
Two Dimensional ◽  
Elastic Materials ◽  
Rectangular Coordinate System ◽  
Full Symmetry ◽  
Anisotropic Elastic ◽  
Anisotropic Elastic Material

The relations between stresses and strains in an anisotropic elastic material are presented in this chapter. A linear anisotropic elastic material can have as many as 21 elastic constants. This number is reduced when the material possesses a certain material symmetry. The number of elastic constants is also reduced, in most cases, when a two-dimensional deformation is considered. An important condition on elastic constants is that the strain energy must be positive. This condition implies that the 6×6 matrices of elastic constants presented herein must be positive definite. Referring to a fixed rectangular coordinate system x1, x2, x3, let σij and εks be the stress and strain, respectively, in an anisotropic elastic material. The stress-strain law can be written as . . . σij = Cijksεks . . . . . .(2.1-1). . . in which Cijks are the elastic stiffnesses which are components of a fourth rank tensor. They satisfy the full symmetry conditions . . . Cijks = Cjiks, Cijks = Cijsk, Cijks = Cksij. . . . . . .(2.1-2). . .

Behavior of a Sequence of Geometric Transformations for a Truncated Ellipsoid Geometry in Transport Theory

2009 ◽  
Author(s):  
Rube´n Panta Pazos
Keyword(s):  
Boundary Conditions ◽  
Symmetry Axis ◽  
Transport Theory ◽  
Neutron Transport ◽  
Complex Function ◽  
Computational Techniques ◽  
Neutron Transport Equation ◽  
Geometric Transformations ◽  
The Right ◽  
Transport Problems

The neutron transport equation has been studied from different approaches, in order to solve different situations. The number of methods and computational techniques has increased recently. In this work we present the behavior of a sequence of geometric transformations evolving different transport problems in order to obtain solve a transport problem in a truncated ellipsoid geometry and subject to known boundary conditions. This scheme was depicted in 8, but now is solved for the different steps. First, it is considered a rectangle domain that consists of three regions, source, void and shield regions 5. Horseshoe domain: for that it is used the complex function: f:D→C,definedasf(z)=12ez+1ezwhereD=z∈C−0.5≤Re(z)≤0.5,−12π≤Im(z)≤12π(0.1) The geometry obtained is such that the source is at the focus of an ellipse, and the target coincides with the other focus. The boundary conditions are reflective in the left boundary and vacuum in the right boundary. Indeed, if the eccentricity is a number between 0,95 and 0,99, the distance between the source and the target ranges from 20 to 100 length units. The rotation around the symmetry axis of the horseshoe domain generates a truncated ellipsoid, such that a focus coincides with the source. In this work it is analyzed the flux in each step, giving numerical results obtained in a computer algebraic system. Applications: in nuclear medicine and others.

Determination of a Constitutive Relation for Passive Myocardium: I. A New Functional Form

1990 ◽  
Vol 112 (3) ◽  
pp. 333-339 ◽  
Author(s):  
J. D. Humphrey ◽  
R. K. Strumpf ◽  
F. C. P. Yin
Keyword(s):  
Constitutive Relation ◽  
Invariant Measures ◽  
Functional Form ◽  
Transversely Isotropic ◽  
The Other ◽  
Material Symmetry ◽  
Material Parameters ◽  
Experimental Protocols ◽  
Theory And Experiment

The specific aim of this study is to determine a constitutive relation for non-contracting myocardium in terms of a pseudostrain-energy function W whose form is guided by both theory and experiment. We assume that the material symmetry of myocardium is initially and locally transversely-isotropic, and seek a W which depends upon only two coordinate invariant measures of the finite deformation. The specific functional form of such a W is inferred directly from experimental protocols in which one invariant is held constant while the other is varied, and vice versa. On the basis of data from families of these “constant invariant” tests on thin slabs of myocardium taken from the mid-walls of six canine left ventricles, we propose a new polynomial form of W containing only five material parameters.

Fourier integral representation of the Green function for an anisotropic elastic half-space

1993 ◽  
Vol 443 (1918) ◽  
pp. 367-389 ◽  
Keyword(s):  
Green Function ◽  
Integral Representation ◽  
Half Space ◽  
Isotropic Material ◽  
Fourier Integral ◽  
Elastic Half Space ◽  
Material Symmetry ◽  
Vertical Force ◽  
Anisotropic Elastic ◽  
The Green Function

This paper describes the development of a Fourier integral representation of the Green function for an anisotropic elastic half-space. The representation for an isotropic material is integrated in closed form and shown to reduce to Mindlin’s solution. An application of the anisotropic representation is made to deduce the exact displacement caused by a two-dimensional periodic vertical force distribu­tion applied to the interior of a half-space with cubic material symmetry.

The Analysis of Plane Flat Lap Interface End of Orthotropic Bi-Material

2011 ◽  
Vol 233-235 ◽  
pp. 1950-1953
Author(s):  
Cai Xia Ren ◽  
Jun Lin Li
Keyword(s):  
Stress Intensity Factor ◽  
Function Method ◽  
Complex Function ◽  
Stress Fields ◽  
Stress Singularities ◽  
Displacement Fields ◽  
Multiple Stress ◽  
Stress Functions ◽  
Material Fracture ◽  
Material Plane

The orthotropic bi-material plane interface end of a flat lap is studied by constructing new stress functions and using the composite complex function method of material fracture. When the characteristic equations’ discriminates and, the theoretical formulas of stress fields, displacement fields and the stress intensity factor around the flat lap interface end are derived, indicating that there is no oscillatory singularity. There are multiple stress singularities of the orthotropic bi-material plane flat lap interface end.

The Stress Field of Plane Flat Lap Interface End of Orthotropic Bi-Material

2010 ◽  
Vol 44-47 ◽  
pp. 2827-2831
Author(s):  
Jun Lin Li ◽  
Cai Xia Ren ◽  
Wen Ting Zhao ◽  
Jing Zhao
Keyword(s):  
Stress Intensity Factor ◽  
Function Method ◽  
Complex Function ◽  
Stress Fields ◽  
Stress Singularities ◽  
Displacement Fields ◽  
Multiple Stress ◽  
Stress Functions ◽  
Material Fracture ◽  
Material Plane

The orthotropic bi-material plane interface end of a flat lap is studied by constructing new stress functions and using the composite complex function method of material fracture. When the characteristic equations’ discriminates and , the theoretical formulas of stress fields, displacement fields and the stress intensity factor around the flat lap interface end are derived, indicating that there is no oscillatory singularity. There are multiple stress singularities of the orthotropic bi-material plane flat lap interface end.

Sign in / Sign up

Export Citation Format

Share Document