Analytical Methods in Anisotropic Elasticity






Preface

Prior to the computer era, analytical methods in elasticity had already been developed and improved

up to impressive levels. Relevant mathematical techniques were extensively exploited,

contributing significantly to the understanding of physical phenomena. In recent decades, numerical

computerized techniques have been refined and modernized, and have reached high

levels of capabilities, standardization and automation. This trend, accompanied by convenient

and high resolution graphical visualization capability, has made analytical methods less attractive,

and the amount of effort devoted to them has become substantially smaller. Yet, with some

tenacity, the tremendous advances in computerized tools have yielded various mature programs

for symbolic manipulation. Such tools have revived many abandoned analytical methodologies

by easing the tedious effort that was previously required, and by providing additional capabilities

to perform complex derivation processes that were once considered impractical.

Generally speaking, it is well recognized that analytical solutions should be applied to relatively

simple problems, while numerical techniques may handle more complex cases. However,

it is also agreed that analytical solutions provide better insight and improved understanding of

the involved physical phenomena, and enable a clear representation of the role taken by each of

the problem parameters. Nowadays, analytical and numerical methods are considered as complementary:

that is, while analytical methods provide the required understanding,numerical

solutions provide accuracy and the capability to deal with cases where the geometry and other

characteristics impose relatively complex solutions.

Nevertheless, from a practical point of view, analytic solutions are still considered as “art”,

while numerical codes (such as codes that are based on the finite-element method) seem to offer

a “straightforward” solution for any type and geometry of a new problem. One of the reasons

for this view emerges from the variety of techniques that are used for analytical solutions. For

example, one has the option to select either the deformation field or the stress field to construct

the initial solution hypothesis, or, one has the option to formulate the governing equations using

differential equilibrium, or by employing more integral energy methodologies for the same

task. Hence, the main obstacle to using analytical approaches seems to be the fact that many

researchers and engineers tend to believe that, as far as analytic solutions are considered, each

problem is associated with a specific solution type and that a different solution methodology

has to be tailored for every new problem.

In light of the above, the objective of this book is twofold. First, it bringstog

ether and

refreshes the fundamentals of anisotropic elasticity and reviews various mathematical tools

and analytical solution trails that are encountered in this area. Then, it presents a collection



of classical and advanced problems in anisotropic elasticity that encompasses varioustwodimensional

problems and different types of three-dimensional beam models. The book includes

models of various mathematical complexity and physical accuracy levels, and provides

the theoretical background for composite material analysis. One of the most advanced formulations

presented is a complete analytical model and solution scheme for an arbitrarily loaded

non-homogeneous beam structure of generic anisotropy.

All classic and modern analytic solutions are derived using symbolic computational techniques.Emphasisisput

on the basic principles of the analytic approach (problem statement,

setting of simplifying assumptions, satisfying the field and boundary conditions, proof of solution,

etc.), and their implementation using symbolic computational tools, so that the reader will

be able to employ the relevant approach to new problems that frequently arise. Discussions are

devoted to the physical interpretation of the presented mathematical solutions.

From a format point of view, the book provides the background and mathematical formulation

for each problem or topic. The main steps of the analytical solution and the graphical

results are discussed as well, while the complete system of symbolic codes (written in Maple)

are available on the enclosed disc.

A unique characteristic of this book is the fact that the entire analytical derivation and

all

solution expressions are symbolically proved by suitable (computerized) codes. Hence, the

chance for (human) error or typographical mistake is eliminated. The symbolic worksheets are

therefore absolute and firm testimony to the exactness of the presented expressions. For that

reason, the specific solutions included in the text should be viewed as illustrative examples

only, while the solution exactness and its generic applicability are proved symbolically in the

most generic manner.

The book is aimed at graduate and senior undergraduate students, professors, engineers, applied

mathematicians, numerical analysis experts, mechanics researchers and composite materials

scientists.

Chapter description:

The first part of the book ( Chapters 1–4) contains the fundamentals of anisotropic elasticity.

The second part (Chapters 5–10) is devoted to various beam analyses and contains recent and

advanced models developed by the authors.

Chapter 1 addresses fundamentalissues of anisotropic elasticity and a

nalytical methodologies.

It provides a review of deformation measures and strain in generic orthogonal curvilinear

coordinates, and reaches the complete nonlinear compatibility equations in such systems. It

then introduces fundamental stress measures and the associated equilibriumequations. Later

on, energy theorems and variational analyses are derived, followed by a general discussion of

analytical methodologies and typical solution trails.

Chapter 2 reviews the mathematical representation of general anisotropic materials, including

the special cases of Monoclinic, Orthotropic, Tetragonal, Transversely Isotropic, Cubic and

Isotropic materials. Later, transformations between coordinate systems of the compliance and

stiffness matrices (or tensors) are presented. The chapter also addresses issues such as planes

of elastic symmetry, principal directions of anisotropy and non-Cartesian anisotropy.

Chapter 3 defines two-dimensional homogeneous and non-homogeneous domain topologies,

and presents various plane deformation problems and analyses, includin

g detailed formulation

of plane-strain/stress and plane-shear states. The derivation yields formal definitions of

generalized Neumann/Dirichlet and biharmonic boundary value problems (BVPs). The chapter



also addresses Coupled-Plane BVP for materials of general anisotropy. Along the same lines,

the classical anisotropic laminated plate theory is then presented.

Chapter 4 presents various solution methodologies for the BVPs derived in Chapter 3, and

establishes solution schemes that facilitate applications presented later on. Explicit analytic

expressions for low-order exact/conditional polynomial solutions, and approximate high-order

polynomial solutions in a homogeneous simply connected domain are derived and illustrated.

Aformulation based on complex potentials is also thoroughly derived and demonstrated by

Fourier series solutions.

Chapter 5 reviews some basic aspects and general definitions of anisotropic beam analysis,

approximate analysis techniques and relevant literature. It discusses the associated coupling

characteristics at both the material and structural levels.

Chapter 6 presents an analysis of general anisotropic beams that may be viewed as a levelbased

extension of the classical Lekhnitskii formulation, and is capable of handling beams

of

general anisotropy and cross-section geometry that undergo generic distribution of surface,

body-force and tip loading. The derivation is founded on the BVPsdeduced in Chapter 3, and

despite its complexity, it provides a clear insight into the associated structural behavior and

coupling mechanisms.

Chapter 7 contains a closed-form formulation for uncoupled monoclinic homogeneous

beams. It first presents solutions for tip loads, and then a generic formulation for axially nonuniform

distribution of surface and body loads. Later on, analysis and examples of beams of

cylindrical anisotropy are presented.

The entire reasoning of the approach in this chapter is founded on St. Venant’s semi-inverse

method of solution and may be considered as dual (thoughlessgeneric) to the method presented

in Chapter 6.

Chapter 8 is focused on problems in various non-homogeneousdoma

ins. It first reviews

generic formulations of plane BVPs, and then extends the analysis of Chapter 7 to the case

of monoclinic non-homogeneousbeams under tip loading,whichisfounded on extending the

classical definition of the auxiliary problems of plane deformation to the anisotropic case. The

discussion encompasses the determination of the principal axis of extension, principal planes

of bending and shear center. The chapter also presents a generalization of the derivation in

Chapter 7 to the case of uncoupled non-homogeneous beams that undergo generic distributed

loading.

Chapter 9 discusses coupled solid monoclinic beams. The analysis presents an approximate

model that provides insight into and fundamental understanding of the coupling mechanisms

within anisotropic beams at the structural level. The model also supplies a simplified but relatively

accurate tool for qu

antitative estimation of coupled beam behavior.

In addition, the chapter presents an exact, level-based solution scheme for coupled beams.

The derivation employs a series of properly interconnected solution levels and reaches the

exact solution in an iterative manner.

Chapter 10 handles coupled thin-wall monoclinic beams in a similar (approximate) manner

to Chapter 9. The analysis encompasses beams having either multiply connected domain

(“closed”) or simply connected domain (“open”) cross-sections.

Chapter 11 presents instructions for the symbolic and illustrative programs included in this

book (implemented in Maple).



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