Project Final 17-10-2010

Final Year Systems Engineering Project on the

EFFECTS OF INHOMOGENEITY AND

ANISOTROPY IN NATURAL STRUCTURAL

MATERIALS

Submitted by:

Omogbai Aileme .A

Olowosulu Emmanuel .O

Onuoha Ogechi .B

Under the Supervision of:

Professor O. A. Fakinlede

September, 2010

In partial fulfillment of the requirements for the Degree of Bachelor of Science in Systems Engineering,

University of Lagos, Nigeria

ii

Certification

This is to certify that this project, Effects of Inhomogeneity and Anisotropy in Natural Structural

Materials, was carried out by Olowosulu Emmanuel O., Omogbai Aileme A. and Onuoha Ogechi B. of

the Department of Systems Engineering, University of Lagos.

________________________ _______________

Olowosulu Emmanuel O. Date

________________________ _______________

Omogbai Aileme A. Date

________________________ _______________

Onuoha Ogechi B. Date

________________________ _______________

Prof. O.A. Fakinlede Date

Project Supervisor

iii

Abstract

This project began with a comparative analysis of the stress fringes resulting from a photoelasticity

experiment [1] and a finite element simulation of the experiment we carried out. Then the symbolic

evaluations of some tensor equations were carried out in different coordinate systems using

Wolfram Mathematica. Finally, the effects of anisotropy and inhomogeneity in natural materials

were studied with Bamboo (Phyllostachys pubescens) as a case study based on an earlier work [2].

iv

Acknowledgement

Our appreciation goes to Professor O. A. Fakinlede for his valuable and constructive suggestions

during the planning and development of this work. His willingness to give his time so generously has

been very much appreciated.

Our families, friends and well-wishers are also not forgotten.

v

Table of Contents

CERTIFICATION ................................................................................................................................................ II

ABSTRACT .......................................................................................................................................................III

ACKNOWLEDGEMENT .................................................................................................................................... IV

TABLE OF CONTENTS ....................................................................................................................................... V

LIST OF FIGURES ............................................................................................................................................ VII

LIST OF TABLES .............................................................................................................................................. IX

INTRODUCTION ............................................................................................................................................... 1

STRESS ANALYSIS..................................................................................................................................................... 1

State of Stress ................................................................................................................................................ 1

State of Stress at a Point ................................................................................................................................ 2

Graphical Representation of Stress at a Point ............................................................................................... 2

Graphical Representation of the Stress Field ................................................................................................. 3

Principal Stresses and Directions ................................................................................................................... 4

PLANE STRESS AND PLANE STRAIN .............................................................................................................................. 4

Plane State of Stress....................................................................................................................................... 4

Plane Strain .................................................................................................................................................... 4

BASIC MATERIAL PROPERTIES .................................................................................................................................... 5

Anisotropy and Isotropy ................................................................................................................................. 5

Homogeneity .................................................................................................................................................. 5

FINITE ELEMENT ANALYSIS ........................................................................................................................................ 6

Overview ........................................................................................................................................................ 6

History ............................................................................................................................................................ 7

Applications ................................................................................................................................................... 8

DESCRIPTION OF SOFTWARE PACKAGES ....................................................................................................................... 8

COMSOL Multiphysics .................................................................................................................................... 8

Wolfram Mathematica .................................................................................................................................. 9

BASIC THEORETICAL CONCEPTS ................................................................................................................................ 10

Refraction ..................................................................................................................................................... 10

Double Refraction ........................................................................................................................................ 10

Polarization .................................................................................................................................................. 12

PHOTOELASTICITY METHOD .................................................................................................................................... 15

Overview ...................................................................................................................................................... 15

Photoelasticity as a Method of Stress Analysis ............................................................................................ 15

THE POLARISCOPE ........................................................................................................................................ 17

DESCRIPTION OF THE PHOTOELASTICITY EXPERIMENT ................................................................................................... 18

Verification of Photoelasticity Experiment ................................................................................................... 20

Using COMSOL Multiphysics V3.5a .............................................................................................................. 20

RESULTS .............................................................................................................................................................. 24

DISCUSSION ......................................................................................................................................................... 24

FURTHER WORK ................................................................................................................................................... 24

vi

TENSOR ANALYSIS ......................................................................................................................................... 25

INTRODUCTION ..................................................................................................................................................... 25

HISTORY .............................................................................................................................................................. 25

TENSORS: A DEFINITION ......................................................................................................................................... 26

BACKGROUND MATERIALS ...................................................................................................................................... 26

Summation Convention ................................................................................................................................ 26

Contraction .................................................................................................................................................. 27

SPECIAL TENSORS .................................................................................................................................................. 28

Kronecker Delta ............................................................................................................................................ 28

Metric Tensor ............................................................................................................................................... 28

Levi-Civita Tensor ......................................................................................................................................... 29

Christoffel Symbols ....................................................................................................................................... 29

COVARIANT DERIVATIVE ......................................................................................................................................... 30

Scalars .......................................................................................................................................................... 30

Vectors ......................................................................................................................................................... 30

Higher Order Tensors ................................................................................................................................... 31

MATHEMATICA IMPLEMENTATION ............................................................................................................................ 32

ANALYSIS OF NATURAL STRUCTURAL MATERIALS (BAMBOO) ....................................................................... 42

STRUCTURE AND PROPERTIES OF BAMBOO ................................................................................................................. 42

Inhomogeneity Study ................................................................................................................................... 48

RESULTS ............................................................................................................................................................ 48

DISCUSSION ...................................................................................................................................................... 49

CONCLUSION ................................................................................................................................................. 50

APPENDIX ...................................................................................................................................................... 51

REFERENCES ................................................................................................................................................... 64

vii

List of Figures

Chapter 1.

Figure 1.1 Stress Vector T, Acting on a Plane with Normal Vector n.

Figure 1.2 The Nine Components of the Second Order Stress Tensor 𝜎 (σ )

Figure 1.3 Stress Components at a Plane Passing Through a Point In a Continuum

Under Plane Stress Conditions.

Chapter 2.

Figure 2.1: Stress Fringes in a Protractor

Figure 2.2: Linear polarization

Figure 2.3: Circularly polarized light

Figure 2.4: Elliptically polarized light

Figure 2.5: Digital photoelasticity equipment showing stress distribution in a loaded disk

Figure 2.6: Automatic Polariscope for Stress Measurement in Glass.

Figure 2.7: Restored Photographs of the Models

Figure 2.8: Photographs after Enhancement to Highlight the Stress Fringes

Figure 2.9: Sub-domain Settings and Boundary Conditions in COMSOL Multiphysics

Figure 2.10: Square hook showing stress pattern and deformation

Figure 2.11: Sharp-edged hook showing stress pattern and deformation

Figure 2.12: Round hook design with optimum stress distribution

viii

Chapter 4.

Figure 4.1: Typical load-deflection curves for a loaded bamboo specimen

Figure 4.2: stress distribution of homogenous and isotropic 2D bamboo model

Figure 4.3: stress distribution of homogenous and anisotropic 3D bamboo model

Figure 4.4: Load Deflection curves for anisotropic model with different circumferential young’s modulus

Appendix. Figure A1: Model for 50 -50 volume fraction proportion

Figure A2: Graph for results of 50-50 volume fraction

Figure A3: Model for 60-40 proportion of volume fraction

Figure A4: Graph of results of 60-40 volume fraction







ix

LIST OF TABLES

Chapter 2

Table 2.1: Properties of PSM-8F

Chapter 4

Table 4.1: Table of results for anisotropic bamboo model

Table 4.2: Table of error values for anisotropic bamboo model

Table 4.3: Table showing the properties of the current computer system

Appendix Table T1: Table of results for 50-50 volume fraction

Table T2: Table of error values for 50-50 volume fraction

Table T3: Table of results for 60-40 volume fraction








1

Introduction

Stress Analysis

Stress analysis is an engineering discipline that determines the stress in materials and structures

subjected to static or dynamic forces or loads. A stress analysis is required for the study and design

of structures under prescribed or expected loads. Stress analysis may be applied as a design step to

structures that do not yet exist.

The aim of the analysis is usually to determine whether the element or collection of elements,

usually referred to as a structure, can safely withstand the specified forces. Analysis may be

performed through mathematical modeling or simulation, through experimental testing

procedures, or a combination of techniques.

STATE O F S TRESS

Mathematically, the state of stress at a point in an elastic body is determined by six independent

stress components and is specified by a second-order symmetric Cartesian tensor, also known as

the stress tensor [3].

The values of these stress components change with the orientation of the coordinate system in

which each stress component is defined. The coordinate system may be rotated so as to study some

of the practical issues of the stress state at a point.

For example, certain stress components may be reduced to zero in a particular orientation of the

coordinate system. Such information is useful in calculating the stress concentrations in the stress

analysis of part, assembly or structure.

Figure 1.1: Stress Vector T, Acting on a Plane with Normal Vector n. [4]

2

STATE O F S TRESS AT A PO INT

At a given point on a surface, a stress vector depends on the orientation (unit normal) of the

surface. For a different normal vector in the same stress field, the stress vector on the associated

surface must also be different. The stress vector on three mutually perpendicular planes at a point

specifies the state of stress at that point in a continuum. The components of these stress vectors

form a tensor, the stress tensor. [3]

Mathematically, the stress tensor is a second-order Cartesian tensor with nine stress components.

Three stress components that are perpendicular to the planes are called normal stresses. Those

components acting tangent to these planes are called shear stresses.

Figure 1.2: The Nine Components of the Second Order Stress Tensor 𝜎 (σ ). [4]

You assume the sign of a stress component is positive when its direction and the normal vector of

the surface, on which the component of the stress tensor is acting, are of the same sign. Otherwise,

assume the sign of the stress component is negative.

GRAPHIC AL REPRESENTA TION O F STRESS AT A PO INT

Mohr's Circle

Mohr's circle, named after Christian Otto Mohr, is the locus of points that represent the state of

stress on individual planes at all their orientations. The abscissa , and ordinate , of each point

on the circle are the normal stress and shear stress components, respectively, acting on a particular

cut plane with a unit vector with components ( ).

3

Lame's Stress Ellipsoid

The surface of the ellipsoid represents the locus of the endpoints of all stress vectors acting on all

planes passing through a given point in the continuum body. In other words, the endpoints of all

stress vectors at a given point in the continuum body lie on the stress ellipsoid surface, i.e., the

radius-vector from the center of the ellipsoid, located at the material point in consideration, to a

point on the surface of the ellipsoid is equal to the stress vector on some plane passing through the

point. In two dimensions, the surface is represented by an ellipse.

Cauchy's Stress Quadric

The Cauchy's stress quadric, also called the stress surface, is a surface of the second order that

traces the variation of the normal stress vector as the orientation of the planes passing through a

given point is changed.

GRAPHIC AL REPRESENTA TION O F THE STRESS F IEL D

The complete state of stress in a body at a particular deformed configuration implies knowing the

six independent components of the stress tensor (𝜎 𝜎 𝜎 𝜎 𝜎 𝜎 ) or the three principal

stresses (𝜎 𝜎 𝜎 ) at each point in the body at that time.

However, numerical analysis and analytical methods allow only for the calculation of the stress

tensor at a certain number of discrete material points. To graphically represent the stress

distribution in two dimensions this partial picture of the stress field different sets of contour

lines can be used:

Isobars are curves along which the principal stress, e.g., 𝜎 is constant.

Isochromatics are curves along which the maximum shear stress is constant. These curves

can be determined by photoelasticity methods.

Isopachs are curves along which the mean normal stress is constant

Isostatics or stress trajectories are a system of curves which are at each material point

tangent to the principal axes of stress.

Isoclinics are curves on which the principal axes make a constant angle with a given fixed

reference direction. These curves can also be obtained directly by photoelasticity methods.

Slip lines are curves on which the shear stress is a maximum.

4

PRINC IPAL S TRESSES A ND DIREC TIO NS

When the normal vector of a surface and the stress vector acting on that surface are collinear, the

direction of the normal vector is called a principal stress direction. The magnitude of the stress

vector on the surface is called the principal stress value.

In the case where the values of all shear stresses are zero, the principal stresses are the normal

stresses. The principal stress directions are the unit vectors of the coordinate system in the case

where no shear stress component is present.

Plane S tress and Plane Strain

PLANE STATE OF S TRES S

A class of common engineering problems involves stresses in a thin plate or on the free surface of a

structural element. These elements include the surfaces of thin-walled pressure vessels under

external or internal pressure, the free surfaces of shafts in torsion and beams under transverse load.

A common characteristic of problems in this class is a principal stress that is much smaller than the

other two. By assuming that this small principal stress is zero, the three-dimensional stress state

can be reduced to two dimensions. Since the remaining two principal stresses lie in a plane, these

simplified 2D problems are called plane stress problems. [5] This is the basis for the photoelasticity

methods.

Figure 1.3: Stress Components at a Plane through a Point in a Continuum under Plane Stress Conditions. [6]

PLANE STRAIN

If one dimension is very large compared to the others, the principal strain in the direction of the

longest dimension is constrained and can be assumed as zero, yielding a plane strain condition. In

this case, though all principal stresses are non-zero, the principal stress in the direction of the

longest dimension can be disregarded for calculations. [6] Thus, allowing a two dimensional

analysis of stresses, e.g. a dam analyzed at a cross section loaded by the reservoir. The structural

analysis of bamboo in a latter chapter is a plane strain problem as a typical bamboo plant has a

length that is much greater than its diameter.

5

Basic Mater ial Propert ies

ANISO TROPY AND ISO TR O PY

In a single crystal, the physical and mechanical properties often differ with orientation. When the

properties of a material vary with different crystallographic orientations, the material is said to be

anisotropic.

Conversely, when the properties of a material are the same in all directions and orientations, the

material is said to be isotropic. In other words, none of the properties depend the orientation. For

many polycrystalline materials the grain orientations are random before any working (deformation)

of the material is done.

For instance, rubber is an isotropic material. A rubber ball will feel the same and bounce the same

way however it is rotated. On the other hand, some materials, such as wood and fiber-

reinforced composites are very anisotropic, being much stronger along the grain/fiber than across.

HO MOG ENEITY

A material is homogeneous if at a given time, all points within it are equivalent. That is, no

difference is seen when any portion of the material when compared with any other portion of the

material. The consequence of homogeneity is that there is nothing special about a location in a

material. Homogeneity does not imply isotropy.

An example of a scenario which is homogeneous but not isotropic is a Universe pervaded by a

uniform electric or magnetic field. Each point is the same as any other, but the field lines will select

out a special direction. Likewise, isotropy does not imply homogeneity; any spherically symmetric

distribution is isotropic as seen from its center. Only if there is isotropy around every point is

homogeneity implied.

6

Finite Element Analys is

OVERVIEW

The finite element method (FEM) is a numerical technique for finding approximate solutions of

partial differential equations (PDEs) as well as of integral equations that describe, or approximately

describe a wide variety of physical problems. Physical problems range from solid, fluid and soil

mechanics, to electromagnetism or dynamics. Its practical application is often known as finite

element analysis (FEA).

The solution approach is based either on eliminating the differential equation completely (steady

state problems), or rendering the PDE into an approximating system of ordinary differential

equations, which are then numerically integrated using standard numerical techniques..

The underlying premise of the method states that a complicated domain can be sub-divided into a

series of smaller regions in which the differential equations are approximately solved. By

assembling the set of equations for each region, the behavior over the entire problem domain is

determined.

Each region is referred to as an element and the process of subdividing a domain into a finite

number of elements is referred to as discretization. Elements are connected at specific points,

called nodes, and the assembly process requires that the solution be continuous along common

boundaries of adjacent elements.

In solving partial differential equations, the primary challenge is to create an equation that

approximates the equation to be studied, but is numerically stable, meaning that errors in the input

and intermediate calculations do not accumulate and cause the resulting output to be meaningless.

There are many ways of doing this, all with advantages and disadvantages.

Finite element method is a very flexible and powerful method for solving differential equations

involving field variables over a continuous but finite domain, which represents most engineering

applications. However, the details of the numerical method are highly complex. A large variety of

software implementations of the method exists, each of which has its own interpretation of the

method and specific formulation of features.

Another method that may be used to solve complex equations is the finite difference method.

Although they use similar ideas, the finite difference method is different from Finite element

method. A few differences are:

7

Finite element method can handle complex geometries and boundary conditions with ease;

Finite difference method is basically restricted to regular shapes.

Finite difference method only tries to minimize error at discrete points, whereas Finite

element method tries to minimize the error over the entire element

The mathematics behind Finite element method is more involved than Finite difference

method.

Finite Element Method is a good choice for solving partial differential equations over complicated

domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with

a moving boundary), when the desired precision varies over the entire domain, or when the solution

lacks smoothness.

HIS TORY

The finite element method originated from the need for solving complex elasticity and structural

analysis problems in civil and aeronautical engineering. Its development can be traced back to the

work by Alexander Hrennikoff (1941) and Richard Courant (1942). While the approaches used by

these pioneers are dramatically different, they share one essential characteristic: mesh

discretization of a continuous domain into a set of discrete sub-domains, usually called elements.

Hrennikoff's work discretizes the domain by using a lattice analogy while Courant's approach

divides the domain into finite triangular subregions for solution of second order elliptic partial

differential equations (PDEs) that arise from the problem of torsion of a cylinder. Courant's

contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by

Rayleigh, Ritz, and Galerkin.

Development of the finite element method began in earnest in the middle to late 1950s for airframe

and structural analysis and gathered momentum at the University of Stuttgart through the work of

John Argyris and at Berkeley through the work of Ray W. Clough in the 1960s for use in civil

engineering. By late 1950s, the key concepts of stiffness matrix and element assembly existed

essentially in the form used today.

The National Aeronautics and Space Agency (NASA) issued a request for proposals for the

development of the finite element software NASTRAN in 1965. The method was again provided

with a rigorous mathematical foundation in 1973 with the publication of Strang and Fix's An Analysis

of The Finite Element Method has since been generalized into a branch of applied mathematics for

8

numerical modeling of physical systems in a wide variety of engineering disciplines, e.g.,

electromagnetism, thanks to Peter P. Silvester and fluid dynamics.

APPLIC ATIO NS

A variety of specializations under the umbrella of the mechanical engineering discipline (such as

aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in design

and development of their products.

FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of

stresses and displacements. FEM allows entire designs to be constructed, refined, and optimized

before the design is manufactured.

The introduction of FEM has substantially decreased the time to take products from concept to the

production line. It is primarily through improved initial prototype designs using FEM that testing

and development have been accelerated.

In summary, benefits of FEM include increased accuracy, enhanced design and better insight into

critical design parameters, virtual prototyping, fewer hardware prototypes, a faster and less

expensive design cycle, increased productivity, and increased revenue.

In many cases, the analytic methods of stress determination in engineering materials become

numerically cumbersome. The advent of superior computer processing power has revolutionized

stress analysis. Finite element modeling (FEM) has become a dominant technique overshadowing

many traditional techniques for stress analysis. The FEM implementation used in this project is

COMSOL Multiphysics.

Descr iption of Software Packages

COMS OL MUL TIPHYS ICS

COMSOL Multiphysics (formerly FEMLAB) is a powerful interactive environment for modeling and

solving all kinds of scientific and engineering problems based on partial differential equations

(PDEs). With COMSOL conventional models for one type of physics can be extended into

Multiphysics models that solve coupled physics phenomena — and do so simultaneously. [8]

Accessing this power does not require an in-depth knowledge of mathematics or numerical

analysis. It has built-in physics modes which make it possible to build models by defining the

relevant physical quantities — such as material properties, loads, constraints, sources, and fluxes —

9

rather than by defining the underlying equations. COMSOL Multiphysics then internally compiles a

set of PDEs representing the entire model.

To solve the PDEs, COMSOL Multiphysics uses the proven finite element method (FEM). The

software runs the finite element analysis together with adaptive meshing and error control using a

variety of numerical solvers.

Models of all types can be built in the COMSOL Multiphysics user interface. For additional flexibility,

COMSOL Multiphysics also provides a seamless interface to MATLAB. This gives you the freedom

to combine PDE-based modeling, simulation, and analysis with other modeling techniques. For

instance, it is possible to create a model in COMSOL Multiphysics and then export it to Simulink as

part of a control-system design. Compared to other methods, COMSOL Multiphysics is:

Relatively easy to use in terms of its graphical user interface

Uses state-of-the-art solvers and optimizers.

Runs well on a suitably – equipped computer

Module based: its interface changes based on what type of physics is being solved for.

WOLF RAM MATHEMATIC A

Mathematica, first released in 1988 by Wolfram Research, Inc., is a system for doing mathematics

on a computer. It combines symbolic manipulation, numerical mathematics, outstanding graphics,

and a sophisticated programming language.

Because of its versatility, Mathematica has established itself as the computer algebra system of

choice for many computer users. Among the over 100,000 users of Mathematica, 28% are

engineers, 21% are computer scientists, 20% are physical scientists, 12% are mathematical

scientists, and 12% are business, social, and life scientists. Two-thirds of the users are in industry

and government with a small (8%) but growing number of student users.

Mathematica's interactive interface also allows immediate testing of components, as well as

making it possible for small programs to be immediately integrated into large enterprise-scale

projects. Mathematica's unparalleled built-in analysis and visualization capabilities provide a new

level of code visualization and management for large software projects.

In addition, Mathematica's ability to handle programs symbolically, together with its unique

pattern-matching capabilities and new built-in theorem proving, allows Mathematica to be used for

a range of important new forms of code analysis and validation.

10

Photoelasticity

Basic Theoretical Concepts

REF RAC TIO N

Refraction is the bending of a wave when it enters a medium where its speed is different. Hence,

refraction of light occurs when a light ray changes mediums. Light traveling from air and going into

water would be an example. The speed of the light ray changes upon changing mediums. In almost

every case the direction of the light ray changes also.

The amount of bending depends on the indices of refraction of the two media. The refractive

index or index of refraction of a substance is a measure of the speed of light in that substance. The

index of refraction characterizes not only the light propagation speed, but also the bending angle

and the amount of radiation transmitted and reflected by a material.

Crystals are classified as being optically isotropic or anisotropic depending upon their optical

behavior. All isotropic crystals have equivalent axes that interact with light in a similar manner,

regardless of the crystal orientation with respect to incident light waves. Light entering an isotropic

crystal is refracted at a constant angle and passes through the crystal at a single velocity without

being polarized.

Anisotropic crystals, on the other hand, interact with light in a manner that is dependent on the

orientation of the crystalline lattice with respect to the incident light. When light enters the optical

axis of anisotropic crystals, it acts in a manner similar to interaction with isotropic crystals and

passes through at a single velocity. However, when light enters a non-equivalent axis, it is refracted

into two rays each polarized with the vibration directions oriented at right angles to one another,

and traveling at different velocities. This phenomenon is termed "double" or "bi" refraction and is

seen to a greater or lesser degree in all anisotropic crystals.

DO UBLE REF RACTIO N

Double refraction or Birefringence is the splitting of a ray of light into two rays when it passes

through certain types of material called birefringent materials. This occurs because the ray of light

experiences two distinct refractive indices. The two rays, called the ordinary ray and

the extraordinary ray, travel at different speeds. The ray with its electric field vibrating perpendicular

to the optical axis is called the ordinary ray, and is characterized by an index of refraction . The ray

11

that vibrates parallel to the optical axis is called the extraordinary ray, with its index of refraction

designated ne.The birefringence magnitude (which can be positive or negative) is then defined by

[9]

The conventional refraction occurs when the ordinary ray and the extraordinary ray are equal.

Transparent anisotropic materials are known to exhibit birefringence naturally because the

material's optical properties are not the same in all directions. The best-known birefringent crystal

is the mineral calcite (CaCO3).

Figure 2.1: Stress Fringes in a Protractor

Isotropic materials do not exhibit birefringence naturally. However, certain isotropic materials

under mechanical stress exhibit birefringence. The stress can be applied externally or is ‘frozen’ in

after a birefringent plastic ware is cooled after it is manufactured using injection molding. When

such a sample is placed between two crossed polarizers, color patterns can be observed due

to stress induced birefringence. The reason is that polarization of a light ray is usually rotated after

passing through a birefringent material and the amount of rotation is dependent on wavelength. [9]

12

POL ARIZATIO N

Polarization is a property of certain types of waves (transverse) that describes the orientation of

their oscillations. Electromagnetic waves, such as light, and gravitational waves exhibit polarization;

acoustic waves (sound waves) in a gas or liquid do not have polarization because the direction of

vibration and direction of propagation are the same.

By convention, the polarization of light is described by specifying the orientation of the wave's

electric field at a point in space over one period of the oscillation. When light travels in free space, in

most cases it propagates as a transverse wave — the polarization is perpendicular to the wave's

direction of travel.

In this case, the electric field may be oriented in a single direction (linear polarization), or it may

rotate as the wave travels (circular or elliptical polarization). For longitudinal waves such as sound

waves in fluids, the direction of oscillation is by definition along the direction of travel, so there is no

polarization.

Linear Polarization

Figure 2.2: Linear polarization

In linear polarization, the two orthogonal (perpendicular) components are in phase. Also, in this

case the ratio of the strengths of the two components is constant, so the direction of the electric

vector (the vector sum of these two components) is constant. The tip of the vector traces out a

single line in the plane and this is a special case. The direction of this line depends on the relative

amplitudes of the two components.

13

Circular Polarization

Figure 2.3: Circularly polarized light [11]

In circular polarization, the two orthogonal components have exactly the same amplitude and are

exactly ninety degrees out of phase. In this case one component is zero when the other component

is at maximum or minimum amplitude. There are two possible phase relationships that satisfy this

requirement: the x component can be ninety degrees ahead of the y component or it can be ninety

degrees behind the y component. [10]

In this special case the electric vector traces out a circle in the plane hence the name: circular

polarization. The direction the field rotates in depends on which of the two phase relationships

exists. These cases are called right-hand circular polarization and left-hand circular polarization,

depending on which way the electric vector rotates and the chosen convention.

To avoid confusion, it is good practice to specify "as seen from the receiver" (or transmitter) when

discussing polarization matters. In physics, astronomy, and optics, polarization is defined as seen

from the receiver. Using this convention, left or right handedness is determined by pointing one’s

left or right thumb toward the source, against the direction of propagation, and matching the

curling of one's fingers to the temporal rotation of the field as the wave passes a given point in

space.

In electrical engineering polarization is defined as seen from the source e.g. a transmitting antenna.

Elliptical Polarization

Elliptical polarization is when the two components are not in phase and either do not have the same

amplitude or are not ninety degrees out of phase, though their phase offset and their amplitude

ratio are constant. This kind of polarization is called elliptical polarization because the electric

vector traces out an ellipse in the plane (the polarization ellipse). This is shown in the above figure

above. [10]

14

Figure 2.4: Elliptically polarized light [10]

The "Cartesian" decomposition of the electric field into x and y components is, of course, arbitrary.

Plane waves of any polarization can be described instead by combining any two orthogonally

polarized waves, for instance waves of opposite circular polarization.

15

Photoelast ici ty Method

OVERVIEW

Photoelasticity is an experimental method to determine stress distribution in a material. The

method is mostly used in cases where mathematical methods become quite cumbersome. Unlike

the analytical methods of stress determination, photoelasticity gives a fairly accurate picture of

stress distribution even around abrupt discontinuities in a material. The method serves as an

important tool for determining the critical stress points in a material and is often used for

determining stress concentration factors in irregular geometries. [12]

Figure 2.5: Digital photoelasticity equipment showing stress distribution in a loaded disk [27]

At any point in a loaded component there is stress acting in every direction. The directions in which

the stresses have maximum and minimum value for the point are known as principal directions. The

corresponding stresses are known as maximum and minimum principal stresses. [13]

Polarised light entering a loaded transparent component is split into two beams. Both beams travel

along the same path, but each vibrates along a principal direction and travels at a speed

proportional to the associated principal stress. Consequently the light emerges as two beams

vibrating out of phase with one another which when combined produce an interference pattern.

PHO TO ELASTICITY A S A METHO D OF STRES S ANAL YS IS

If the part to be analyzed is made of a transparent birefringent material, stress analysis is simple. All

transparent plastics, if birefringent, lend themselves to photoelastic stress analysis. The transparent

part is placed between two polarizing mediums and viewed from the opposite side of the light

source. The fringe patterns are observed without applying external stress. This allows the observer

to study the stresses in the part.

16

High fringe order indicates the area of high stress level whereas low fringe order represents an

unstressed area. Also, close spacing of fringes represents a high stress gradient. A uniform color

indicates uniform stress in the part. Next, the part should be stressed by applying external force and

simulating actual-use conditions. The areas of high stress concentration can be easily pinpointed by

observing changes in fringe patterns brought forth by external stress.

Another technique known as the photoelastic coating technique can be used stress-analyze opaque

plastic parts using photoelasticity. The part to be analyzed is coated with a photoelastic coating,

service loads are applied to the part, and coating is illuminated by polarized light from the reflection

polariscope. Molded-in or residual stresses cannot be observed with this technique.

Alternatively, photoelasticity may depend on fabricating models of the problem (prototype)

required to be solved using a transparent material. Structural materials are usually metals or

ceramics which are opaque. As transparent materials are the only ones suitable for photoelastic

analysis, it is questionable if the stresses yielded in the transparent model can be related to those in

a dimensionally similar model. This question is answered by considering the equations of elasticity

as they apply to two dimensional stress systems.

In transiting from Model to Prototype, it is important to know that for many practical two-

dimensional elastic problems with forces applied to external boundaries, stresses depend upon

geometry and external forces only, and not upon the physical properties of the material (except, of

course, that the material must be elastic, homogeneous and isotropic).

A model must then be geometrically similar to the prototype, but not necessarily of the same size;

the loads must be similarly distributed, but they may differ in magnitude by a factor of

proportionality.

17

Figure 2.6: Automatic Polariscope for Stress Measurement in Glass.

THE POL ARISCO PE

The Polariscope is a laboratory tool that can be used for direct visual observation and analysis of

stress patterns in transparent materials. Ordinary monochromatic light vibrates in an infinite

number of planes all passing through the direction of the propagation of the ray of light.

A polariscope is composed chiefly of a light source and two crossed polarized lenses. The

polariscope light source is mounted beneath one lens. The material to be examined is placed

between the two polariscope lenses and viewed through the lens opposite the light source lens.

To utilize the polariscope light source, the mixture of light waves must be linearly polarized in one

particular direction. When light is incident on a polarizer only the component parallel to the

transmission direction of the polarizer will be transmitted. When two polarizers are mounted in the

crossed position, i.e., with their transmission directions at right angles to each other, no light is

transmitted through the combination.

If a material that refracts light is inserted between the crossed polarizers, the linearized light

passing through the first polarizer will have its direction altered by the refracting material and some

light will pass through the second polarizer to the viewer. The areas of the doubly refracting

material that refracting light from the first polarizer will be seen as patches of color when viewed

through the second polarizer.

18

In many cases, the analytic methods of stress determination in engineering materials become

numerically cumbersome. The advent of superior computer processing power has revolutionized

stress analysis. Finite element modeling (FEM) has become a dominant technique overshadowing

many traditional techniques for stress analysis. When using FEM, it is crucial to access the accuracy

of the numerical model and ultimately this can only be achieved by experimental verification.

Photoelasticity therefore remains a major tool in modern stress analysis. [14]

Thus, the first part of this work is aimed at verifying the results obtained in a photoelastic

experiment [1] using the Finite Element Method with COMSOL Multiphysics.

Descr iption of the Photoelastic i ty Experiment

The experiment was aimed at obtaining a pictorial stress pattern of three models, cut from a

photoelastic material (PSM-8F) and subjected to tensile loading. The models were three hooks with

different geometries as shown in the figure below. [1]

Figure 2.7: Restored Photographs of the Models [1]

The photographs in figure 2.1 were enhanced with Corel® PaintShop Photo Pro X3 as they were

originally taken 33 years ago in 1977. The photographs in figure 2.2 were further processed with an

edge detection algorithm to give a clearer visualization of the stress fringe patterns.

19

Figure 2.8: Photographs after Enhancement to Highlight the Stress Fringes

The models when viewed under a polariscope were seen to have stress distribution patterns as

shown in figure2.1. The material properties of PSM-8F are as shown in table 2.1.

Temperature(oF) C (psi/fring/in) E (psi) U.T.S. (psi)

α (per oF) µ

Room temperature 80 400 x 103 7500 39 x 10-6 0.36

Stress freezing temperature (160 – 180)

1.8 1.8 x 103 150 90 x 10-6 0.50

Table 2.1: Properties of PSM-8F [1]

Where

The three hooks were loaded and examined under the polariscope to note the regions of high

stresses and those of lower stress using the concentration of the fringes. It is known that where the

fringes are closely packed, there are higher values of the stress. Thus the photoelastic method does

not give a numerical value of the stress but knowledge of the points of higher and lower stress

within a material

20

VERIFIC ATIO N O F PH OTO EL ASTICITY EXPERI MENT

USING CO MS OL MUL TIPHYSICS V 3.5A

3Dimensional solid models of the hooks were detailed in Siemens Solid Edge to the specifications

given in the original work. The hooks were originally modeled as Solid Edge parts in their

proprietary file format, .par, but had to be exported in their Parasolid exchange format for

compatibility with the CAD Import Module of COMSOL Multiphysics.

Parasolid is a geometric modeling kernel originally developed by ShapeData, now owned by

Siemens PLM Software (formerly UGS Corp.), that can be licensed by other companies for use in

their 3D computer graphics software products.

When exported from the parent software package, a Parasolid commonly has the file extension

.x_t. Another format is .x_b, which is in binary format so it is more machine independent and not

subject to binary-to-text conversion errors. [15]

The CAD Import Module allows engineers and designers alike to create complex geometrical forms

in CAD software of their choice and then import the finished part into COMSOL for analysis.

Geometric models do not always pass flawlessly between different file formats due to the different

representations they use. This implies that the quality of a translation from a CAD model to

COMSOL Multiphysics geometry depends heavily on the file format: this is why the native

SolidEdge parts were converted to the Parasolid format.

The models were analysed in the Plane Stress section of the Solid Mechanics Module and boundary

conditions were specified as part of the subdomain settings: some edges and faces were fixed and

the geometries were loaded on the relevant faces as in the experiment described above. The

material constants were supplied and the mesh was generated.

In verifying the results obtained in the Experiment described above, 3Dimensional solid models of

the hooks were detailed in Siemens Solid Edge to the specifications given in the original work. The

models originally in Siemens Solid Edge’s proprietary .par format were exported in their Parasolid

exchange format for compatibility with the CAD Import Module of COMSOL Multiphysics.

The models were analysed in the Plane Stress mode of the Structural Mechanics Module of the FEM

software. Within the COMSOL environment, the material properties of the hooks were specified as

those shown in table 2.1.

21

The values for the young’s modulus of the material and its thermal coefficient of were converted to

the SI system of units compatible with COMSOL Multiphysics. The new values and the conversions

used are shown below.

( ) =

( )

Figure 2.9: Sub-domain Settings and Boundary Conditions in COMSOL Multiphysics

In the boundary settings window, the imported hooks were each loaded in the Fz direction on the

relevant faces which corresponds to a vertical load acting downwards on those faces. In an attempt

to properly model the actual experiment done, as part of the boundary settings, the upper parts of

the hooks were specified to be fixed as no displacement is expected at those areas.

The problem as described was solved by the FEM software. After solving the problem, the stress

distribution plot was obtained from the post-processing menu of COMSOL Multiphysics to give the

results as shown in figure 2.4-6 below.

22

Figure 2.10: Square hook showing stress pattern and deformation

Figure 2.11: Sharp-edged hook showing stress pattern and deformation

23

Figure 2.12: Round hook design with optimum stress distribution

24

Results

The results show the stress patterns or distributions within the hooks (figures 2.4-6) and how the

hooks deform under the particular load applied. These patterns will change with the magnitude

and/or direction of the load applied. The result also shows the points of maximum and minimum

stress.

This is an interesting advantage of using the FEM software over the actual performance of the

experiment. In performing the actual experiment, measurements and mathematical

computations/analysis have to be carried out in order to determine the stresses at any point within

the loaded model. These cumbersome calculations are eliminated and desired values can be

obtained by a click of the mouse.

Discuss ion

From figure 2.4, it can be seen that the stress is highest at the inner corners of the square hook and

also at the sharp points. This is indicated by the red coloring of the fringes at those points. The

maximum stress also appears at the corner which is closer to the point of application of the load.

It should also be noted that the areas of little or no stress (whose fringes are deep blue in color)

contribute little or nothing in bearing the load applied. This suggests a better design of the hook can

be obtained by removing those unnecessary parts thus eliminating wastage and optimizing the

material used in designing the hooks.

A better design of the hook is shown in figure 2.5. The maximum stress is reduced and the points of

highest stress are reduced to one point since the numbers of sharp corners have been reduced. The

best design for the hook is as shown in figure 2.6 where the corners have been totally eliminated.

Further Work

The method of photoelasticity using FEM software such as the one used in this work can be further

applied to the calibration of the photoelastic experiment. The method may be made quantitative if

the photoelastic coefficient of the material is known.

25

Tensor Analysis

Introduction

The field equations of mechanics (including photoelasticity and stress analysis) are best expressed

in tensor form. When that is done, the resulting expressions are simple and independent of

coordinates. General tensor theory is elegant and provides useful insight and economy of space.

The main drawback is in its steep learning curve. It is our belief that the investment will more than

pay for itself in the time saved deriving the same equations over again as the coordinate systems of

reference change. [16]

In this chapter, most of the important results of general tensor theory are presented. Proofs of

theorems are given only when they are simple enough and do not distract from our chief objective -

to present a clear picture of what tensors are and how they are used.

History

The concepts of later tensor analysis arose from the work of C. F. Gauss in differential geometry,

and the formulation was much influenced by the theory of algebraic forms and invariants developed

in the middle of the nineteenth century. The word "tensor" itself was introduced in 1846 by William

Rowan Hamilton to describe something different from what is now meant by a tensor. The

contemporary usage was brought in by W. Voigt in 1898. [17]

Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro (also called just Ricci)

under the title absolute differential calculus, and originally presented by Ricci in 1892. It was made

accessible to many mathematicians in French [19] and translations followed.

In the 20th century, the subject came to be known as tensor analysis, and achieved broader

acceptance with the introduction of Einstein's theory of general relativity, around 1915. General

relativity is formulated completely in the language of tensors. Einstein had learned about them,

with great difficulty, from the geometer Marcel Grossmann. [19] Levi-Civita then initiated a

correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis.

The correspondence lasted 1915–17, and was characterized by mutual respect, with Einstein at one

point writing: [20]

"I admire the elegance of your method of computation; it must be nice to ride through these fields

upon the horse of true mathematics while the like of us have to make our way laboriously on foot."

26

Tensors: A Definit ion

Tensors are geometric entities introduced into mathematics and physics to extend the notion of

scalars, geometric vectors, and matrices to higher orders. In this chapter we will limit ourselves to

the discussion of rank 2 tensors, unless stated otherwise. The precise definition of the rank of a

tensor will become clear later. [19]

A scalar can be described by a single number and a vector can be described by a list of numbers;

tensors in general can be considered as a multidimensional array of numbers, which are known as

its "scalar components" or simply "components." The entries of such an array are symbolically

denoted by the name of the tensor with indices giving the position in the array.

The total number of indices is equal to the dimension of the array and is called the order or the rank

of the tensor. For example, the entries (also called components) of an order 2 tensor would be

denoted , where and are indices running from 1 to the dimension of the related vector space.

Many physical quantities are naturally regarded not as vectors themselves, but as correspondences

between one set of vectors and another. For example, the stress tensor takes a direction as

input and produces the stress ( ) on the surface normal to this vector as output and so expresses a

relationship between these two vectors. Because they express a relationship between vectors,

tensors themselves are independent of a particular choice of coordinate system.

Background Materials

SUMMATION C ONVENTIO N

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or

Einstein summation convention is a notational convention useful when dealing with coordinate

formulae. It was introduced by Albert Einstein in 1916. [21][22]

According to this convention, when an index variable appears twice in a single term, once in an

upper (superscript) and once in a lower (subscript) position, it implies that we are summing over all

of its possible values. In typical applications, the index values are 1, 2, 3 (representing the three

dimensions of physical Euclidean space), or 0, 1, 2, 3 or 1, 2, 3, 4 (representing the four dimensions

of space-time, or Minkowski space), but they can have any range, even (in some applications) an

infinite set. Thus in three dimensions:

,

27

actually means

∑

.

The upper indices are not exponents, but instead different axes. Thus, for example, should be

read as "x-two", not "x squared", and corresponds to the traditional y-axis. Abstract index notation

is a way of presenting the summation convention so that it is made clear that it is independent of

coordinates.

In general relativity, the Greek alphabet and the Roman alphabet are used to distinguish whether

summing over 1,2,3 or 0,1,2,3 (usually Roman, , , ... for 1,2,3 and Greek, , , ... for 0,1,2,3). As in

sign conventions, the convention used in practice varies: Roman and Greek may be reversed.

When there is a fixed basis, one can work with only subscripts, but in general one must distinguish

between superscripts and subscripts. In some fields, Einstein notation is referred to simply as index

notation, or indicial notation. The use of the implied summation of repeated indices is also referred

to as the Einstein Sum Convention.

CONTRAC TION

Tensor contraction is an operation that reduces the total order of a tensor by two. More precisely, it

reduces a type (n, m) tensor (having n contravariant indices and m covariant indices) to a type (n-1,

m-1) tensor. In terms of components, the operation is achieved by summing over one contravariant

and one covariant index of tensor.

For example, a second order tensor can be contracted to a scalar through

, where the

summation is implied. The operation is carried out as

and the order of the

tensor is reduced from 2 to 1.

28

Special Tensors

KRO NEC KER DEL TA

The simplest interpretation of the Kronecker delta is as the discrete version of the delta function

defined by

{

It has the contour integral representation

∮

Where is a contour corresponding to the unit circle and and are integers.

METRIC TENSO R

Consider the -dimensional space covered by the coordinate system, . It is not

necessarily essential that there be any concept of distance or length of an arc in this space. The

space may for example define the thermodynamic states of a system. The coordinates are

properties of the system. [16]

New values of any, some or all properties take the system to a new state of equilibrium. The space

may be the configuration space of a dynamical system. In this case, each of the 's are the

generalised coordinates of the system. There would then be as many coordinates as the number of

degrees of freedom in the system. In the former, the concept of the length of an arc or length of a

curve embedded in the space is completely meaningless.

Consider that in the space under examination, we somehow desire to have a consistent way to

measure length. We look at an arc of a curve defined by the parametric equations,

( ) ( )

We define the length of this arc as the integral,

∫ [

]

⁄

Where are some prescribed functions of the coordinate variables for every given value of and .

The space is said to be metrized by the formula above. Notice at once that if

, we

recover a generalisation of the well-known Pythagorean formula. In general,

. The quantity

29

is a covariant tensor of the second order. The contravariant form of the metric tensor is the

inverse of .

This assertion follows easily from the fact that ( ) ; this is known as the

fundamental quadratic form. The tensors and are called fundamental or metric tensors and

they give the concept of distance to the space . The metric tensor is symmetric.

LEVI -CIVITA TENSO R

The Levi-Civita tensor (also called the isotropic tensor) is anti-symmetric under the exchange of two

indices. It is a tensor of rank three where [

{

The Levi-Civita tensor obeys the following identities

where Einstein summation is implicitly assumed. The vector cross product may be defined in terms

of the Levi-Civita tensor as .

CHRISTOFF EL SYMBOL S

These are the three-index Christoffel symbols: The Christoffel symbol of the first kind is defined as

the set of derivative sums: [

[ ]

{

}.

While the sum,

{ } [ ].

are the Christoffel symbols of the second kind. The symmetry of both Christoffel symbols in the

indices i and j follow from the symmetry of the fundamental tensors themselves: for,

[ ]

{

}

{

}

{

} [ ],

after invoking the symmetry of the covariant fundamental tensor. Symmetry of the Christoffel

symbols of the second kind in these same indices follows immediately. This symmetry reduces the

number of the Christoffel symbols from to

( ).

So we have:

{ }

[ ] [ ] [ ].

30

Covariant Derivat ive

SCAL ARS

The covariant derivative of a scalar reduces to its partial derivative as it is an invariant (an absolute

tensor of order zero) and we obtain a covariant tensor of order one. This is a special case. In general,

partial derivatives of tensors do not yield other tensors.

The covariant derivative of a scalar is the same as taking the gradient of the scalar.

The Laplacian

Find an expression for the Laplacian operator in Orthogonal coordinates. For a given scalar , the

Laplacian is defined as,

√

(√

).

If the coordinate system is orthogonal then , for . Expanding the computation

formula, we can write,

[

(

)

(

)

(

)]

[

(

)

(

)

(

)]

VECTORS

Given a a covariant vector , the quantities,

{

} ,

form the covariant derivative of the above vector with while for a contravariant vector, , the

covariant derivative is given by,

{

} .

The summation convention has been used in the last terms. The quantities represented by the

formulas above transform covariantly in accordance with tensor laws.

Gradient

The gradient of a vector is defined as its covariant derivative. It is an operation that produces a

second order tensor having nine components.

.

31

Divergence

The divergence of the contravariant vector is the contraction of the covariant derivative,

.This is equivalent to taking the trace of its gradient and the operation results in a scalar.

Curl

The curl of a vector is gotten by contracting the Levi-Civita tensor with the second order tensor

obtained by taking the gradient of the vector. Mathematically, this is expressed as:

.

HIG HER ORDER TENS ORS

The formula for the covariant differentiation of higher order tensors follows the same kind of logic

as the above definitions. Each covariant index will produce an additional term similar to the second

term of the formula for a covariant vector.

In the same way, each contravariant index produces an additional term similar to the second term of

the formula for a contravariant vector. The covariant derivative of the mixed tensor, is

the most general case for the covariant derivative of an absolute tensor: [16]

{

}

{ }

{ }

{ }

{ }

{ }

32

Mathematica Implementation

Defining a tensor

Dummy Variables

The Kronecker Delta

33

Calculating the Laplacian in the Spherical Polar and Polar Cylindrical Coordinates

34

The Christoffel Symbols of the Second Kind in the Spherical Polar Coordinate System

The Christoffel Symbols of the Second Kind in the Polar Cylindrical Coordinate System

35

Calculating the Jacobian and Metric Tensor in the Elliptical Coordinate System

36

Calculating the Holonomic Bases for the Elliptical Coordinate System

Defining a Scalar and Obtaining its Gradient in the Spherical Coordinate System

37

Defining a Contravariant Tensor and Obtaining its Gradient in the Spherical Coordinate System

38

Defining a Doubly Contravariant Tensor and Obtaining its Gradient in the Spherical Coordinate System

39

Defining a Mixed Tensor and Obtaining its Gradient in the Spherical Coordinate System

40

Calculating the Contraction of the Gradient of a Second Order Mixed Tensor in the Spherical

Coordinate System

41

Defining a Doubly Covariant Tensor and Obtaining its Gradient in the Spherical Coordinate System

42

Analysis of Natural Structural Materials (Bamboo)

A natural material is any product or physical matter that comes from plants, animals, or the ground.

Minerals and the metals that can be extracted from them (without further modification) are also

considered to belong into this category.

Biotic materials

Wood (bamboo, tree bark, etc.)

Natural fibers (wool, cotton, hemp, jute, etc.)

Inorganic material

Stone (flint, granite, sandstone, sand, gems, glass, etc.)

Metal (copper, bronze, iron, gold, silver, steel, etc.)

Composites (clay, porcelain, plasticine, etc.)

Other natural materials

Soil

Structure and Propert ies of Bamboo

Bamboo, the fastest-growing woody plant on Earth, is known to be a naturally strong material

suitable for various purposes. [25] Bamboo is a hollow cylinder reinforced by strong fibers in the

longitudinal direction, dividing the bamboo into sections.

It is common practice to make use of bamboo when constructing scaffolds. It has also been used in

constructing bridges, floors and houses. Other areas of application of bamboo include textiles,

medicine, music and food. The use of bamboo in the area of construction has prompted the

investigation of its material properties by engineers and researchers.

Based on the definitions of homogeneity and isotropy in the opening chapter, bamboo can be said

to be a heterogeneous and anisotropic material. A simple example will explain the heterogeneity of

the bamboo. It is known that the outer surface region of the bamboo is harder compared to the

inner. This is because the fibers on the outer surface region are densely distributed and those inside

are sparsely distributed. Thus a sample taken from a region near the outer surface will be different

from another taken near the inner surface.

43

The anisotropy of bamboo is readily observed as it is more resistant to axial loading than diametric

loading. Bamboo is known to be stronger when loaded in the longitudinal direction than in the

transverse direction.

Currently, the only known elastic constant of the Bamboo is its longitudinal young’s modulus. As to

the knowledge of the authors, no experiments have been conducted to determine the

circumferential or radial Young’s modulus of bamboo. One of the reasons may be the difficulty to

prepare specimens in the circumferential or radial directions given the relatively small thickness of

the wall. [2]

In a work by Torres et al, an attempt is made to determine the circumferential Young’s modulus by

assuming transverse isotropy for two bamboo specimens – Guadua angustifolia and Phyllostachys

pubescens. Diametric compression tests were carried out on the specimens and were validated

using finite element methods.

The proposed test, designed to suit bamboo geometry, was the first reported procedure for the

determination of the circumferential young’s modulus. Finite element models were used to

simulate the tests and to evaluate the influence of geometric irregularities in the results. [2] A graph

of the load applied against the resulting deformation shows a reduction in slope after some point as

the load increases.

Figure 4.1: Typical load-deflection curves for a loaded bamboo specimen

The results obtained by Torres et al are 485 ± 172 MPa and 1685 ± 132 MPa for Guadua angustifolia

and Phyllostachys pubescens, respectively. An inspection of the change in the value of the

circumferential Young’s modulus along the length of the bamboo was also done using the same

method of diametric compression.

44

For Guadua angustifolia the circumferential modulus showed a significant increase with height,

registering values of 651 ± 153 MPa for the top, 465 ± 126 MPa for the middle and 344 ± 98 MPa for

the bottom parts of the culms. For Phyllostachys pubescens the circumferential modulus was

independent of the length of the specimen.

Figure 4.2: stress distribution of homogenous and isotropic 2D bamboo model

The transversely isotropic model does not fully capture the anisotropic features of bamboo. Also, a

closer look at the structure of bamboo shows that the outer surface layer is harder than its inner

surface. This shows an inherent inhomogeneity in the structure of the bamboo which has so far

been neglected.

Thus it is proposed here that the values for the circumferential Young’s modulus as obtained by

Torres et al is only an average of the circumferential young’s modulus of the individual concentric

rings (representing the different layers off fiber) making up the bamboo with the outermost ring

having the largest value for the circumferential young’s modulus and the innermost layer the least.

The objective of this section is to determine the values of the different Young’s moduli

(circumferential and radial) that will reproduce the results of the compression tests performed using

a fully anisotropic model. COMSOL Multiphysics gives us the ability to experiment with a wider

range of values than would have been possible with a physical experiment

45

For the simulation, one of the groups of specimens was taken as the model. The bottom specimen

of Guadua angustifolia, with internal and external diameters of 86 and 110 mm respectively was

chosen. The length used in the experiment was one-fourth the external diameter. The average

value of the circumferential Young’s modulus gotten for this specimen is 344 ± 98 MPa.

These values were used to create the 3D finite element model. For the subdomain settings of the

model, an orthotropic model was used to characterize the bamboo, using the well-known

longitudinal Young’s modulus and the average value gotten by Torres et al for the moduli in the

transverse direction.

Figure 4.3: stress distribution of homogenous and anisotropic 3D bamboo model

The experiment was performed by varying the value of the Force from N to N as in the physical

tests for this specimen and noting the deflection in the model.

46

The results of the first test conducted using the average value of MPa for when compared

with the experimental values, showed slight variance. Therefore, other values for were used to

determine the value which would best approximate the experimental results. As initial estimates,

the values 700 MPa (twice the original value) and 175 MPa (half the original value) were used for .

It was noticed that an increase in from the original value, pushed the results closer to the

experimental values. Therefore, other values between 350 and 700 MPA were tested. The figure

below shows the Load – Deflection plots for the different values of used.

Figure 4.4: Load Deflection curves for anisotropic model with different circumferential young’s modulus

From the correlation functions from the plot of the experimental values, the discrete values

corresponding to the values of the Load used in the simulation were estimated. The table 4.1 below

shows the results for all the tests conducted. Table 4.2 shows the error in each result (compared to

the experimental values). From the plot, it is observed that with ( ) , the

plot of the simulation and experimental values almost perfectly coincide for load values ranging

from 25 to 150 N; that is until the observed softening behavior. The table (Number) above also

highlights this in that the error values for ( ) are very minimal in that load

range.

The line representing one fourth the value of shows very little variation from the deflection

values obtained by Torres et al for loads below 200N after which a large variation is observed.

47

It is proposed that this change is as a result of the assumption of homogeneity in the finite element

model used. Thus a better model will be one which incorporates the inhomogeneity of the bamboo.

Force Exp. Values E = 350MPa E = 700MPa

(2 * 350)

E = 175MPa

(0.5 * 350)

E = 385MPa

(1.1 * 350)

E = 490MPa

(1.4 * 350)

25 0.259 -0.353 -0.164 -0.508 0.301 -0.241

50 0.504 -0.706 -0.328 -1.016 -0.602 -0.482

75 0.749 -1.059 -0.492 1.524 -0.903 -0.723

100 0.993 -1.412 -0.656 -2.032 -1.204 -0.964

125 1.238 -1.764 -0.820 -2.540 -1.505 -1.205

150 1.482 -2.117 -0.984 -3.048 -1.805 -1.446

175 1.907 -2.470 -1.148 -3.557 -2.106 -1.687

200 2.321 -2.823 -1.312 -4.065 -2.407 -1.928

225 2.735 -3.176 -1.476 -4.573 -2.708 -2.169

250 3.148 -3.529 -1.640 -5.081 -3.009 -2.411

275 3.562 -3.882 -1.804 -5.589 -3.310 -2.652

Table 4.1: Table of results for anisotropic bamboo model

Force ( )

E = 350 MPa

( )

E = (2 * 350)

MPa

( )

E = (0.5 * 350)

MPa

( )

E = (1.1 * 350)

MPa

( )

E = (1.4 * 350)

MPa

25 0.009 0.009 0.062 0.002 0.000

50 0.041 0.031 0.262 0.010 0.000

75 0.096 0.066 0.602 0.024 0.001

100 0.175 0.114 1.080 0.044 0.001

125 0.277 0.174 1.697 0.071 0.001

150 0.403 0.248 2.453 0.104 0.001

175 0.317 0.576 2.721 0.040 0.048

200 0.252 1.018 3.041 0.007 0.154

225 0.195 1.584 3.378 0.001 0.320

250 0.145 2.276 3.734 0.019 0.544

275 0.102 3.092 4.107 0.064 0.829

Error Sum 2.012 9.187 23.135 0.386 1.900

Table 4.2: Table of error values for anisotropic bamboo model

48

INHO MO GENEITY S TUDY

Bamboo is inhomogeneous in that the strength differs along the radial component. The exterior

portion of it is visibly stronger than the interior portion and this affects the overall strength of the

bamboo. Thus an assumption of homogeneity as in the preceding simulations is erroneous and is

done only for purpose of simplification.

In this work, the inhomogeneity in bamboo will be modeled as a set of concentric cylinders each

representing a portion of the bamboo which is assumed to be isotropic and each of the cylinders will

have a different value for its material property constants.

In order to obtain a close approximation of the load-deflection curve obtained by Torres et al, we

carry out iterations of changes in values of the material properties of the models, discarding values

that produce results with higher deviation from the desired plot.

In varying the values of the material properties of the model, we experience an increase in the

number of degrees of freedom with increase in the number of concentric cylinders. Hence to

simplify our work we restrict ourselves to a model using two concentric cylinders each with a

thickness of 6mm (that is half the thickness of the original cylinder).

RESULTS

In fitting the material properties of the inhomogeneous model, it is important to note that by

inspection, a smaller volume fraction [26] of the bamboo holds more of the reinforcement fibre

making that volume stronger than the other. Thus when assigning material properties to the

cylinders, one must be careful to assign a larger value of the Young’s modulus to the cylinder with

the smaller thickness, making sure that the average of the values assigned corresponds to the initial

young’s modulus.

The graphs shown in the appendix are results obtained from varying the volume fraction of the

model. Each line in each graph is a representation of the results gotten from varying the proportion

of the young’s modulus assigned to each of the concentric cylinders of the model.

Also shown are tables of all the results generated by varying the volume fraction and the

distribution of the Young’s modulus on the two concentric cylinders.

Error plots are given which show the error in the results when compared to the values gotten from

Torres’ experiment. Figures A11 through A15 show the error plots for each of the volume fractions

considered as the Young’s modulus is distributed along the cylinders. Figures A15 through A20

49

show the error plots for the results generated when the Young’s modulus is kept constant as the

volume fraction is varied.

DISCUSSION

From inspection of the results obtained, one can see that none of the patterns is able to match the

desired plot. Hence a better model of the bamboo is recommended. An attempt to increase the

number of concentric cylinders led to a sharp increase in the amount of computer resources

required to obtain a solution.

Thus a more powerful computer system than we have access to at present is required to analyse a

more accurate model of the material properties of Bamboo. The properties of the current system

are presented below:

OS Name: Microsoft Windows 7 Professional Version

System Type: x64-based PC

Processor: Intel(R) Core(TM)2 Duo CPU

Installed Physical Memory (RAM) 4.00 GB

Total Physical Memory 4.00 GB

Total Virtual Memory 8.00 GB

Graphics Card: NVIDIA GeForce 8400M GS

Table 4.3: Table showing the properties of the current computer system

50

CONCLUSION

This project has proven that anisotropy and inhomogeneity are responsible for the response of

bamboo to diametric loading. The aim of the project was to determine the effects of anisotropy and

inhomogeneity natural structural materials. From the error plots, is reasonable to deduce that in a

simple anisotropic model comprising two inhomogeneous cylinders, the ratio of the value of

circumferential Young’s modulus in the outer cylinder to the inner one is between 60:40 and 70:30

and this ratio is independent of the size ratios of the cylinders.

This project was limited in its analyses due to the prohibitive computing power required to analyse

models having more concentric cylinders: a closer approximation of the inhomogeneity of bamboo.

Better approximations of the physical experiment would have been obtained had these models

been analysed.

Further work on this project should involve an attempt to calibrate the photoelastic experiment if

physical photoelastic equipment is available as qualitative as well as quantitative readings can be

made with such calibration. Also, for the bamboo experiment, a more accurate bamboo model

should be analysed with more concentric cylinders giving a higher degree of freedom for changing

parameters and a more powerful workstation should be used for the FEA.

51

Appendix

Figure A1: Model for 50 -50 proportion of volume fraction


0.000

50.000

100.000

150.000

200.000

250.000

300.000

0.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000

Torres

50 / 50

60 / 40

70 / 30

80 / 20

90 / 10

52

Force Exp. Values 50-50 60-40 70-30 80-20 90-10

25 0.259 0.243 0.311 0.259 0.346 0.521

50 0.504 0.485 0.621 0.518 0.693 1.041

75 0.749 0.728 0.932 0.777 1.039 1.562

100 0.993 0.970 1.243 1.036 1.385 2.083

125 1.238 1.213 1.554 1.295 1.732 2.603

150 1.493 1.456 1.864 1.554 2.078 3.124

175 1.907 1.698 2.175 1.813 2.424 3.644

200 2.321 1.941 2.486 2.071 2.770 4.165

225 2.735 2.184 2.797 2.330 3.117 4.686

250 3.148 2.426 3.107 2.589 3.463 5.206

275 3.562 2.669 3.418 2.848 3.809 5.727


Force 50-50 60-40 70-30 80-20 90-10

25 0.000 0.003 0.000 0.008 0.068

50 0.000 0.014 0.000 0.036 0.289

75 0.000 0.034 0.001 0.084 0.662

100 0.001 0.062 0.002 0.154 1.187

125 0.001 0.100 0.003 0.244 1.864

150 0.001 0.138 0.004 0.342 2.659

175 0.044 0.072 0.009 0.268 3.019

200 0.144 0.027 0.062 0.202 3.401

225 0.304 0.004 0.163 0.146 3.807

250 0.522 0.002 0.313 0.099 4.235

275 0.798 0.021 0.510 0.061 4.686

Error Sum 1.815 0.476 1.067 1.643 25.877


53


Figure A4: Graph of results of 60-40 volume fraction

0

50

100

150

200

250

300

0.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000

Torres

50 / 50

60 / 40

70 / 30

80 / 20

90 / 10

54

Force Exp. Values 50-50 60-40 70-30 80-20 90-10

25 0.259 0.242 0.277 0.297 0.384 0.549

50 0.504 0.484 0.554 0.595 0.768 1.098

75 0.749 0.725 0.832 0.892 1.152 1.647

100 0.993 0.967 1.109 1.189 1.536 2.195

125 1.238 1.209 1.386 1.487 1.919 2.744

150 1.493 1.451 1.663 1.784 2.303 3.293

175 1.907 1.693 1.940 2.081 2.687 3.842

200 2.321 1.935 2.218 2.379 3.071 4.391

225 2.735 2.176 2.495 2.676 3.455 4.940

250 3.148 2.418 2.772 2.973 3.839 5.489

275 3.562 2.660 3.049 3.271 4.223 6.038


Force 50-50 60-40 70-30 80-20 90-10

25 0.000 0.000 0.001 0.016 0.084

50 0.000 0.003 0.008 0.070 0.353

75 0.001 0.007 0.021 0.163 0.807

100 0.001 0.013 0.038 0.294 1.446

125 0.001 0.022 0.062 0.465 2.270

150 0.002 0.029 0.085 0.656 3.240

175 0.046 0.001 0.030 0.609 3.745

200 0.149 0.011 0.003 0.563 4.286

225 0.312 0.057 0.003 0.519 4.863

250 0.533 0.142 0.031 0.477 5.477

275 0.814 0.263 0.085 0.436 6.127

Error Sum 1.858 0.548 0.368 4.266 32.696


55



0

50

100

150

200

250

300

0.000 2.000 4.000 6.000 8.000

Torres

50 / 50

60 / 40

70 / 30

80 / 20

90 / 10

56

Force Exp. Values 50-50 60-40 70-30 80-20 90-10

25 0.259 0.258 0.255 0.299 0.059 0.643

50 0.504 0.516 0.510 0.598 0.119 1.286

75 0.749 0.775 0.764 0.896 0.178 1.929

100 0.993 1.033 1.019 1.195 0.237 2.571

125 1.238 1.291 1.274 1.494 0.296 3.214

150 1.493 1.549 1.529 1.793 0.356 3.857

175 1.907 1.807 1.784 2.092 0.415 4.500

200 2.321 2.066 2.038 2.390 0.474 5.143

225 2.735 2.324 2.293 2.689 0.533 5.786

250 3.148 2.582 2.548 2.988 0.593 6.428

275 3.562 2.840 2.803 3.287 0.652 7.071


Force 50-50 60-40 70-30 80-20 90-10

25 0.000 0.000 0.002 0.040 0.147

50 0.000 0.000 0.009 0.149 0.611

75 0.001 0.000 0.022 0.326 1.392

100 0.002 0.001 0.041 0.572 2.491

125 0.003 0.001 0.066 0.886 3.906

150 0.003 0.001 0.090 1.294 5.588

175 0.010 0.015 0.034 2.227 6.723

200 0.065 0.080 0.005 3.410 7.963

225 0.169 0.195 0.002 4.846 9.308

250 0.321 0.361 0.026 6.533 10.758

275 0.521 0.577 0.076 8.471 12.313

Error Sum 1.095 1.231 0.371 28.753 61.201


57



0.000

50.000

100.000

150.000

200.000

250.000

300.000

0.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000

Torres

50 / 50

60 / 40

70 / 30

80 / 20

90 / 10

58

Force Exp. Values 50-50 60-40 70-30 80-20 90-10

25 0.259 0.228 0.266 0.313 0.367 0.658

50 0.504 0.455 0.533 0.627 0.734 1.315

75 0.749 0.683 0.799 0.940 1.101 1.973

100 0.993 0.910 1.066 1.253 1.467 2.631

125 1.238 1.138 1.332 1.566 1.834 3.289

150 1.493 1.365 1.598 1.879 2.201 3.946

175 1.907 1.593 1.865 2.193 2.568 4.604

200 2.321 1.820 2.131 2.506 2.935 5.262

225 2.735 2.048 2.398 2.819 3.302 5.919

250 3.148 2.275 2.664 3.132 3.669 6.577

275 3.562 2.503 2.930 3.445 4.036 7.235


Force 50-50 60-40 70-30 80-20 90-10

25 0.001 0.000 0.003 0.012 0.159

50 0.002 0.001 0.015 0.053 0.659

75 0.004 0.003 0.037 0.124 1.500

100 0.007 0.005 0.068 0.225 2.682

125 0.010 0.009 0.108 0.356 4.206

150 0.016 0.011 0.149 0.501 6.018

175 0.099 0.002 0.082 0.437 7.274

200 0.251 0.036 0.034 0.377 8.648

225 0.472 0.114 0.007 0.322 10.142

250 0.763 0.235 0.000 0.271 11.755

275 1.123 0.399 0.014 0.224 13.486

Error Sum 2.749 0.814 0.516 2.901 66.527


59



0.000

50.000

100.000

150.000

200.000

250.000

300.000

0.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000

Torres

50 / 50

60 / 40

70 / 30

80 / 20

90 / 10

60

Force Exp. Values 50-50 60-40 70-30 80-20 90-10

25 0.259 0.242 0.281 0.332 0.418 0.661

50 0.504 0.485 0.741 0.665 0.836 1.322

75 0.749 0.727 1.022 0.997 1.253 1.982

100 0.993 0.969 1.304 1.329 1.671 2.643

125 1.238 1.212 1.585 1.661 2.089 3.304

150 1.493 1.454 1.866 1.994 2.507 3.965

175 1.907 1.696 2.147 2.326 2.925 4.626

200 2.321 1.938 2.429 2.658 3.343 5.286

225 2.735 2.181 2.710 2.991 3.760 5.947

250 3.148 2.423 2.991 3.323 4.178 6.608

275 3.562 2.665 3.272 3.655 4.596 7.269


Force 50-50 60-40 70-30 80-20 90-10

25 0.000 0.000 0.005 0.025 0.161

50 0.000 0.056 0.026 0.110 0.669

75 0.000 0.075 0.062 0.255 1.522

100 0.001 0.096 0.113 0.460 2.723

125 0.001 0.120 0.180 0.725 4.269

150 0.002 0.139 0.251 1.028 6.109

175 0.044 0.058 0.176 1.036 7.391

200 0.146 0.012 0.114 1.044 8.794

225 0.307 0.001 0.066 1.052 10.320

250 0.526 0.025 0.030 1.060 11.968

275 0.804 0.084 0.009 1.069 13.737

Error Sum 1.832 0.667 1.030 7.864 67.663


61

Figure A11: Plot of Error against Young Modulus for 50-50 volume fraction




0.000

5.000

10.000

15.000

20.000

25.000

30.000

0 1 2 3 4 5 6

"50 / 50"

"60 / 40"

"70 / 30"

"80 / 20"

"90 / 10"

0.000

5.000

10.000

15.000

20.000

25.000

30.000

35.000

0 2 4 6

"50 / 50"

"60 / 40"

"70 / 30"

"80 / 20"

"90 / 10"

0.000

10.000

20.000

30.000

40.000

50.000

60.000

70.000

0 1 2 3 4 5 6

"50 / 50"

"60 / 40"

"70 / 30"

"80 / 20"

"90 / 10"

0.000

10.000

20.000

30.000

40.000

50.000

60.000

70.000

0 1 2 3 4 5 6

"50 / 50"

"60 / 40"

"70 / 30"

"80 / 20"

"90 / 10"

62


Figure A16: Plot of Error against Volume fraction for 50-50 Young’s modulus



0.000

10.000

20.000

30.000

40.000

50.000

60.000

70.000

80.000

0 1 2 3 4 5 6

"50 / 50"

"60 / 40"

"70 / 30"

"80 / 20"

"90 / 10"

0.000

0.500

1.000

1.500

2.000

2.500

3.000

0 1 2 3 4 5 6

"50 / 50"

"60 / 40"

"70 / 30"

"80 / 20"

"90 / 10"

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

0 1 2 3 4 5 6

"50 / 50"

"60 / 40"

"70 / 30"

"80 / 20"

"90 / 10"

0.000

0.200

0.400

0.600

0.800

1.000

1.200

0 1 2 3 4 5 6

"50 / 50"

"60 / 40"

"70 / 30"

"80 / 20"

"90 / 10"

63



0.000

5.000

10.000

15.000

20.000

25.000

30.000

35.000

0 1 2 3 4 5 6

"50 / 50"

"60 / 40"

"70 / 30"

"80 / 20"

"90 / 10"

0.000

20.000

40.000

60.000

80.000

0 1 2 3 4 5 6

"50 / 50"

"60 / 40"

"70 / 30"

"80 / 20"

"90 / 10"

64

REFERENCES

[1] O. A. Fakinlede. "Design and Construction of a Set for Photoelasticity". Department of

Mechanical Engineering, University of Lagos. (1977). Unpublished

[2] L. A. Torres, K. Ghavami and J. J. García. "A Transversely Isotropic Law for the Determination of

the Circumferential Young’s Modulus of Bamboo with Diametric Compression Tests". Latin

American Applied Research. 37:255-260 (2007)

[3] Wolfram Mathematica Documentation Center. "Chapter 6. Analysis of Stress". Structural

Mechanics. URL:http://reference.wolfram.com/applications/structural/AnalysisofStress.html

[4] Wikipedia, the free encyclopedia. (2010). "Stress (mechanics)". Retrieved [2010] from

URL:http://www.wikipedia.com/Stress_(mechanics).html

[5] eFunda Inc. "Plane Stress and Coordinate Transformations". Retrieved [2010] from

URL:http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/plane_stress.cfm (2010).

[6] Wikipedia, the free encyclopedia. (2009). "Stress analysis". Retrieved [2010] from

URL:http://en.wikipedia.org/wiki/Stress_analysis.html

[7] NEi Software. "New to FEA". Retrieved [2010] from

URL:http://www.nenastran.com/newnoran/newToFEA.html (2005).

[8] COMSOL AB. "Introduction to COMSOL Multiphysics". Included with the COMSOL Multiphysics

3.5a installation. comsol_introduction.pdf

[9] Wikipedia, the free encyclopedia. "Birefringence". Retrieved [2010] from

URL:http://en.wikipedia.org/wiki/Birefringence.html

[10] Wikipedia, the free encyclopedia. "Polarization". Retrieved [2010] from

URL:http://en.wikipedia.org/wiki/Polarization_(waves).html

[11] Wikipedia, the free encyclopedia. "Circular Polarization". Retrieved [2010] from

URL:http://en.wikipedia.org/wiki/Circular_polarization.html

[12] Wikipedia, the free encyclopedia. "Photoelasticity". Retrieved [2010] from

URL:http://en.wikipedia.org/wiki/Photoelasticity.html

65

[13] Experimental Stress Analysis Group. "Photoelasticity". Retrieved [2010] from

URL:http://www.experimentalstress.com/topic/photos.html

[14] University of Cambridge, www.doITpoms.ac.uk/tlplib/photoelasticity/history.html

[15] COMSOL AB. "COMSOL CAD Import Module User’s Guide". Included with the COMSOL

Multiphysics 4.0a installation. cad_ug.pdf (2010).

[16] O. A. Fakinlede. "Chapter 2: Tensor Analysis". Unpublished

[17] W. Voigt, "Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer

Darstellung". Leipzig. (1898)

[18] Ricci and Tullio Levi-Civita. "Méthodes de calcul différentiel absolu et leurs applications",

(Methods of absolute differential calculus and their applications). (1900).

[19] Wikipedia, the free encyclopedia. "Tensor". Retrieved [2010] from

URL:http://en.wikipedia.org/wiki/Tensor.html

[20] J. R. Goodstein. "The Italian Mathematicians of Relativity". (2007).

[21] Wikipedia, the free encyclopedia. "Einstein notation". Retrieved [2010] from

URL:http://en.wikipedia.org/wiki/Einstein_notation.html

[22] A. Einstein. "The Foundation of the General Theory of Relativity". (1916).

[23] Wolfram MathWorld. "Kronecker Delta". Retrieved [2010] from

URL:http://mathworld.wolfram.com/KroneckerDelta.html

[24] Wolfram MathWorld. "Permutation Symbol". Retrieved [2010] from

URL:http://mathworld.wolfram.com/PermutationSymbol.html

[25] Bamboo World. http://www.bambooworld.com

[26] S. Amada and S. Untao. “Fracture properties of bamboo”. Elsevier Science Ltd. (2001).

[27] Automatic Polariscope for Stress Measurement in Glass.

URL:http://www.ioc.ee/tech2000/epolar.html

Documents

Project Final 17-10-2010