Sufficient Symmetry Conditions for Isotropy of the Elastic Moduli Tensor

1987 ◽  
Vol 54 (4) ◽  
pp. 772-777 ◽  
Author(s):  
R. M. Christensen
Keyword(s):  
Elastic Moduli ◽  
Fiber Composite ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Axial Deformation ◽  
Space Network ◽  
Plane Normal ◽  
Rank Tensor ◽  
Three Space

Symmetry conditions are found that assure isotropy of the fourth rank tensor of elastic moduli. Crystallography provides the answer to this problem in the two-dimensional context, namely one axis of three-fold symmetry assures the isotropy of properties in the plane normal to the axis. The present work provides the answer in the three-dimensional problem: 6 axes of five-fold symmetry are sufficient to give isotropy of the elastic moduli. An important restriction must accompany the present result. The derivation is given in the special form appropriate to low density materials which have a microstructure that transmits load according to the axial deformation of a space network of material distributed into micro-struts. The corresponding fiber composite idealization is that of a fiber dominated system, it therefore follows that if the fibers take the 6 specific orientations in three-space then isotropy is obtained.

THREE‐DIMENSIONAL GRAVITY INSTABILITY DERIVED FROM TWO‐DIMENSIONAL SOLUTIONS

Geophysics ◽  
1966 ◽  
Vol 31 (1) ◽  
pp. 153-166 ◽  
Author(s):  
M. A. Biot
Keyword(s):  
Three Dimensional ◽  
Time History ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Periodic Patterns ◽  
Numerical Work ◽  
Characteristic Distance ◽  
General Validity ◽  
Dimensional Gravity ◽  
History Of

The theory of three‐dimensional gravity instability of multilayers is developed with particular application to salt structures. It is shown that three‐dimensional solutions are immediately obtained without further numerical work from the solution of the corresponding two‐dimensional problem. Application to a number of typical three‐dimensional structures yields the characteristic distance between peaks and crests and shows that this distance does not differ significantly from the wavelength of the two‐dimensional solution. Various periodic patterns are examined corresponding to rectangular and hexagonal cells. The time history of nonperiodic structures corresponding to initial deviations from perfect horizontality is also derived. The method is applied to the three‐dimensional problem of generation of salt structures when the time‐history of sedimentation is taken into account with variable thickness and compaction of the overburden and establishes the general validity of the geological conclusions derived from the previous two‐dimensional treatment of the same problem (Biot and Odé, 1965). The present method of deriving three‐dimensional solutions, which is developed here in the special context of gravity instability, is valid for a wide variety of problems in theoretical physics.

A numerical scheme for shock propagation in three dimensions

1988 ◽  
Vol 416 (1850) ◽  
pp. 179-198 ◽  
Keyword(s):  
Numerical Scheme ◽  
Three Dimensional ◽  
Experimental Results ◽  
Three Dimensions ◽  
Shock Propagation ◽  
Two Dimensional ◽  
Shock Surface ◽  
Shock Dynamics ◽  
Geometrical Shock Dynamics ◽  
Three Space

A numerical scheme for shock propagation in three space dimensions is presented. The motion of the leading shock surface is calculated by using Whitham’s theory of geometrical shock dynamics. The numerical scheme is used to examine the focusing of initially curved shock surfaces and the diffraction of shocks in a pipe with a 90° bend. Numerical and experimental results for the corresponding two-dimensional or axi-symmetrical cases are used to compare with the new and more complicated three-dimensional results.

A contribution to the application of photoelasticity to the micromechanics of composite materials

1969 ◽  
Vol 4 (2) ◽  
pp. 88-94 ◽  
Author(s):  
D E W Stone
Keyword(s):  
Composite Materials ◽  
Three Dimensional ◽  
Maximum Amount ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Casting Technique ◽  
Intensity Measurements ◽  
Photoelastic Model ◽  
Composite Models

Photoelastic-model methods can prove advantageous for the investigation of microstresses in composite materials. Some two-dimensional investigations of this type are discussed and the extension of this work into three dimensions is considered. It is suggested that more than one approach to the three-dimensional problem may be practicable, and special attention is paid to obtaining the maximum amount of information from a sandwiched polariscope by means of light-intensity measurements. A cold-casting technique for the fabrication of composite models is also described.

3D structure–2D plate interaction model

2019 ◽  
Vol 24 (10) ◽  
pp. 3354-3377 ◽  
Author(s):  
Matko Ljulj ◽  
Josip Tambača
Keyword(s):  
Elastic Layer ◽  
3D Structure ◽  
Three Dimensional ◽  
Interaction Model ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Dimensional Model ◽  
Limit Model ◽  
Two Dimensional Model ◽  
Thin Elastic Layer

In this paper, we derive models for the interaction of a linearized three-dimensional elastic structure with a thin elastic layer of possibly different material attached to it. Rigorous derivation is performed by considering a thin three-dimensional layer and the asymptotics of the solution of the full remaining three-dimensional problem when the thickness [Formula: see text] of the thin layer tends to zero. Furthermore, the attached thin material is assumed to have the elasticity coefficients which are of order [Formula: see text], for [Formula: see text] with respect to the coefficients of the three-dimensional body. In the limit, five different models are obtained with respect to different choices of p, namely [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. Furthermore a three-dimensional–two-dimensional model is proposed that has the same asymptotics as the original three-dimensional problem. This is convenient for applications because one does not have to decide in advance which limit model to use.

On The Theory Of Bending Of A Thick Orthotropic Plate Without Consideration Of Simplifying Hypothesis

2011 ◽  
Vol 3 ◽  
Author(s):  
Makhamatali Koraboyevich Usarov
Keyword(s):  
Analytical Solution ◽  
Orthotropic Plate ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Orthotropic Plates

The problem of bending of a thick orthotropic plate is considered as a three-dimensional problem of the theory ofelasticity. On the basic of the method of expansion of thesolution into the series, a three-dimensional problem isreduced to two independent two –dimensional problems.The theory of thick orthotropic plates free from simplifiedhypothesis is developed: An analytical solution of equationis given. Maximum values of displacements and stressesfor upper, middle and lower surfaces of the plate are calculated.

ITPE Technique Applications to Time-Varying Three-Dimensional Ground-Coupling Problems

1990 ◽  
Vol 112 (4) ◽  
pp. 849-856 ◽  
Author(s):  
M. Krarti ◽  
D. E. Claridge ◽  
J. F. Kreider
Keyword(s):  
Heat Loss ◽  
Parametric Analysis ◽  
Three Dimensional ◽  
Time Dependent ◽  
Time Varying ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Dimensional Model ◽  
Approximate Analytical Solutions ◽  
Profile Estimation

Approximate analytical solutions for the three-dimensional heat transfer between slab-on-grade floors and rectangular basements under steady-periodic conditions are developed using the Interzone Temperature Profile Estimation (ITPE) method. The slab-on-grade solution is the first analytical slab-on-grade solution that treats the presence of insulation on/under the floor, while the basement solution is the first analytical solution of the time-dependent three-dimensional problem for basements. Solutions are given for the temperature field and expressions are derived for the annual heat loss. Parametric analysis is used to emphasize the effect of geometric dimensions on the magnitude and phase of heat loss relative to ambient temperature. The results obtained are compared with those from the two-dimensional model, and the three-dimensional characteristics of heat flow from slabs and basements are examined.

scholarly journals Stresses in Coating with Gradient Interlayer caused by Contact Loading

2014 ◽  
Vol 8 (1) ◽  
pp. 33-37
Author(s):  
Roman Kulchytsky-Zhyhailo ◽  
Adam Stanisław Bajkowski
Keyword(s):  
Half Space ◽  
Solution Method ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Properties Of Coatings ◽  
Contact Loading ◽  
Integral Fourier Transform ◽  
Dimensional Integral ◽  
Layer Coating

Abstract The three-dimensional problem of elasticity concerning inhomogeneous half-space under normal and tangential loading applied in circular region was considered. The half-space is composed of the homogeneous body and double-layer coating which includes a homogeneous top coat and a gradient interlayer. The solution method is based on the two-dimensional integral Fourier transform. The influence of mechanical properties of coatings component and coefficient of friction on the first principal stress distribution was considered.

Envelope Theory Applied to Globoidal Cam Surface Geometry

1990 ◽  
Vol 204 (6) ◽  
pp. 409-416 ◽  
Author(s):  
C J Backhouse ◽  
J Rees Jones
Keyword(s):  
Cutting Tool ◽  
Three Dimensional ◽  
Illustrative Case ◽  
Two Dimensional ◽  
Surface Form ◽  
Dimensional Problem ◽  
Surface Geometry ◽  
Geometric Properties ◽  
Envelope Theory

This paper describes a technique based on envelope theory for deriving geometric properties of a globoidal type cam. The cam surface is taken to be the spatial envelope formed by a cylindrical cutting tool as it moves relative to the cam. The method employs a variation on classical envelope theory in that the three-dimensional problem of defining the cam surface is reduced to a set of two-dimensional ones. Results are presented to show that the cam surface form can be defined including values of pressure angles and cam surface curvatures for an illustrative case.

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